"The goal is to maximize the total profit from investing in the real-estate market"

"The goal is to maximize the total profit from producing regular and special sandwiches"

"The goal is to maximize the exposure for the new product by determining the optimal number of radio and social media advertisements within the budget and given constraints."

"The goal is to maximize the profit from producing long and short cables"

"The goal is to determine the number of advertisements on each type of media to maximize the total audience while adhering to these constraints."

The goal is to maximize the total profit from producing chairs and dressers.

"The goal is to minimize the cost of the mixture of Feed A and Feed B while meeting the required minimum units of protein and fat."

"The goal is to maximize the total profit from producing terracotta jars"

"The goal is to maximize the total profit from preparing banana-haters packages and combo packages"

"The goal is to maximize the total profit from producing bread and cookies"

"The goal is to maximize the total profit from producing regular and tempered glass panes."

"The goal is to determine the optimal investment in apartments and townhouses to maximize profit."

"The goal is to determine the quantity of Zodiac and Sunny pills that should be taken to meet the medicine requirements at the lowest cost"

"The goal is to maximize the total profit from producing model trains and planes"

The goal is to maximize the profit by optimizing the inventory of phones and laptops.

"The goal is to determine how many kilograms of each type of product should be created to maximize the total profit given the available candy resources."

"The goal is to maximize total net revenue from the allocation of space between phone and laptop production"

"The goal is to maximize the total profit from producing desktops, both premium and regular."

"The goal is to maximize the total profit from producing regular and premium car models"

"The goal is to maximize the seating availability in the office space"

"The goal is to minimize the cost of producing the plant growth compound"

"The goal is to maximize the total profit from producing different types of car oils"

The goal is to determine the number of mangos and guavas to be sold to maximize the total profit.

"The goal is to minimize the amount of Vitamin K in the vitamin mix"

The goal is to maximize the total profit from producing chocolate and vanilla ice cream.

"The goal is to maximize the total profit from growing potatoes and cucumbers"

"The goal is to determine the number of square feet of each type of flooring to produce each week to maximize the company’s profit."

"The goal is to minimize the total operation cost while meeting the DeliveryRequirement for each type of berry"

The goal is to maximize the monthly profit from producing scooters and bikes.

"The goal is to minimize the total cost of consuming servings of vegetables and fruits while meeting the minimum required amounts of vitamins and minerals"

"The goal is to maximize the profit using the available capacity"

"The goal is to minimize the cost of production while fulfilling the demand"

"The goal is to maximize the total revenue from growing turnips and pumpkins."

"The goal is to minimize the total number of teachers hired (both staff and substitute teachers) while satisfying teaching hour requirements and staying within the budget."

"The goal is to minimize the total number of stores the company opens"

"The goal is to maximize the total number of coins plated by running Process A and Process B"

"The goal is to determine how many of each type of saw should be purchased to minimize the total number of saws needed"

"The goal is to maximize the amount of hay processed on the farmer's land"

"The goal is to minimize the total production time for producing Wraps and Platters"

"The goal is to minimize the total number of ticket machines needed to meet the constraints."

"The goal is to maximize the total number of jars filled with honey"

"The goal is to minimize the total number of furnaces purchased"

"The goal is to minimize the total number of machines needed to meet the ice cream production and heat output constraints"

"The goal is to maximize the total oil production from the land"

"The goal is to minimize the total production time of croissants"

The goal is to determine the number of each type of item to make in order to minimize the total amount of fiber needed.

"The goal is to minimize the total number of healthcare workers scheduled"

The goal is to minimize the total number of vans needed to meet the delivery and pollution constraints.

"The goal is to maximize the total production of carbon dioxide"

"The goal is to maximize the amount of tea leaves that can be picked."

"The goal is to minimize the total number of fire fighters hired while meeting the required hours of fire fighter time within the given budget."

"The goal is to minimize the total number of hand sanitizer machines"

"The goal is to minimize the total number of shops that the company builds"

The goal is to maximize the amount of fish the fisherman can catch.

"The goal is to minimize the total production time for making mochas and regular coffees"

The goal is to determine the number of each slicer to minimize the total number of slicers in the shop.

The objective is to minimize the total number of machines used in the factory while meeting the production requirements and constraints on electricity, machine type proportions, and minimum machines of type A.

"The goal is to maximize the total amount of the rare compound produced from chemical reactions A and B"

"The goal is to maximize the daily production of rare earth oxide"

"The goal is to minimize the total production time of the chocolate bars."

"The goal is to minimize the total number of printers in the office"

"The goal is to minimize the total amount of discharge caused by the pills taken by the patient."

"The goal is to find the number of bottles of each brand he should drink to minimize his sugar intake"

"The goal is to maximize the total vitamin D intake for the scientist"

"The goal is to minimize the total time it takes for a house to be cleaned"

"The goal is to determine the number of hours each factory should operate to minimize the total operating time while satisfying all constraints"

Maximize the number of patients whose blood pressure can be taken

"The goal is to maximize the total amount of minty foam produced"

"The goal is to maximize the baby's fat intake from consuming apple and carrot flavored baby food servings"

"The goal is to minimize the total time needed to meet the minimum production requirements for each product"

The goal is to maximize the number of people that can be supplied.

"The goal is to maximize the total number of patients whose temperature is taken"

"The goal is to minimize the total amount of waste produced by using the two machines."

The goal is to determine the number of servings of almonds and cashews she should eat to minimize her fat intake.

"The goal is to maximize the total number of hands that can be cleaned."

"The goal is to maximize the amount of medicine delivered to the back"

"The goal is to minimize the total amount of sleep medicine required"

"The goal is to minimize the sugar intake from consuming oranges and grapefruits"

"The goal is to minimize the total radiation received by the benign area of the pancreas."

"The goal is to minimize the total fat intake from consuming turkey dinners and tuna salad sandwiches."

"The goal is to maximize the total amount of green gas produced."

The goal is to determine how many hours each of Process1 and Process2 should be run to meet the production requirements while minimizing the total hours of operation.

"The goal is to minimize the total sugar intake from consuming packs of blueberries and strawberries."

"The goal is to minimize the total number of trips required to transport the boxes"

"The goal is to minimize the total amount of fuel consumed for transporting the goods"

"The objective is to minimize the total amount of time needed to transport a certain number of ducks using two transportation methods: boats and canoes."

"The goal is to minimize the total amount of pollution produced by using old and new vans to deliver at least the minimum required number of soda bottles"

"The goal is to maximize the total number of tricks that can be performed"

"The goal is to minimize the total number of containers and trucks used"

"The goal is to minimize the total number of trips required to ship the patties"

"The goal is to determine the number of each bouquet type he should transport to maximize the total number of flowers that reach the stores"

"The goal is to determine how many trips each company should make to minimize the total amount of diesel used."

"The goal is to minimize the total time required to transport the monkeys"

"The goal is to minimize the total amount of pollution produced while transporting NumCows."

"The goal is to maximize the number of meals that can be delivered"

"The goal is to minimize the total number of planes used while delivering at least TotalCars cars and ensuring the number of large planes is less than the number of small planes."

The goal is to minimize the total number of wagons needed to transport the specified amount of ore.

"The goal is to minimize the total number of scooters used to transport the visitors"

"The goal is to minimize the total number of animals used for delivering the packages"

"The goal is to minimize the total amount of gas consumed by the mail delivery service"

"The goal is to minimize the total transportation time for transporting the chickens"

"The goal is to minimize the total pollution produced by the vans and minibuses used to transport the children."

"The goal is to minimize the total number of tractors and cars needed to send at least the minimum required weight of corn to the city."

"The goal is to minimize the total number of bikes needed to transport at least MinPeople people."

"The goal is to maximize the total amount of glacial water that can be transported"

"The goal is to minimize the total gas usage while meeting the constraints."

"The goal is to minimize the total time needed to transport at least MinFishRequired fish"

The goal is to minimize the total pollution produced by the transportation methods.

The goal is to maximize the amount of rice that can be transported by the carts.

The goal is to maximize the number of newspapers that can be delivered.

"The goal is to minimize the number of animals, cows and elephants, used to transport at least MinBricks."

"The goal is to minimize the total number of trips needed to transport at least the minimum required boxes of corn."

"The goal is to minimize the total number of trips (NumHPTT + NumLHT)"

"The goal is to maximize the total number of grapes transported"

The goal is to minimize the total time to transport the patients.

"The goal is to minimize the total pollution produced from transporting the employees"

The goal is to determine the number of each type of truck to use in order to maximize the total amount of snow transported

"The goal is to maximize the total number of letters that can be sent"

"The goal is to minimize the total number of canoes and diesel boats needed to transport at least MinFish fish to shore"

"The goal is to maximize the total profit from producing different products"

"The goal is to minimize the daily cost of obtaining the required amounts of calcium and iron"

"The goal is to minimize the cost of Bob's diet while ensuring the meals provide at least the specified minimum calories and protein per day."

"The goal is to minimize the total cost of the supplements while meeting the daily iron and calcium requirements"

"The goal is to determine the amount of money to put on each sport bet to maximize her expected payout"

"The goal is to maximize the total earnings from repairing washing machines and freezers."

The goal is to maximize the total profit from producing crepe cakes, sponge cakes, and birthday cakes.

"The goal is to maximize the total profit from producing the two snack mix products."

"The goal is to maximize the total profit from producing personal and commercial software licenses."

"The goal is to minimize the total effectiveness time sum of all pills taken"

"The goal is to maximize the number of catered guests at the science fair"

"The goal is to minimize the total number of pipes required to meet the minimum daily demand with the given constraints."

The goal is to maximize the tips received from shifts on bikes and scooters.

"The goal is to determine the number of each type of concert to minimize the total number of concerts"

"The goal is to determine the number of each meal to be promoted by the waiting staff to minimize the total cooking time while complying with waste constraints"

"The goal is to minimize the total number of metal-working equipment needed"

"The goal is to maximize the total number of units (sets of small packets and jugs) of jam that can be sold to customers"

"The goal is to minimize the number of retail stores and factory outlets that need to be open"

"The goal is to maximize the total number of units (cans and bottles) produced."

"The goal is to minimize the total number of trips required for transporting the candles."

"The goal is to minimize the total number of parking spots required by this production"

"The goal is to minimize the total number of boxes needed to distribute the required number of masks"

"The goal is to maximize the amount of cars processed every hour"

"The goal is to minimize the total number of transportation units (trains and trams)."

"The goal is to maximize the total amount of conversion to carbon dioxide"

"The goal is to maximize the total metal extracted from the ores"

"The goal is to minimize the total units of space taken by escalators and elevators"

"The goal is to minimize the total time needed to produce at least the minimum required units of fabric and plastic, subject to the availability of the special element."

"The goal is to minimize the total amount of time needed to transport all the locals across the lake."

"The goal is to minimize the total amount of water used in producing the smoothies"

"The goal is to maximize the total number of sports equipment produced"

"The goal is to minimize the total radiation received by the researcher from performing in-vivo and ex-vivo experiments"

"The goal is to maximize the total amount of mail that can be delivered"

"The goal is to maximize the woman's protein intake by choosing the appropriate quantities of smoothies and protein bars"

"The goal is to minimize the total operational time while satisfying the plastic usage constraint and the production requirements for both anti-itch injections and topical cream"

"The goal is to maximize the number of people treated by producing batches of medication patches and anti-biotic creams"

"The goal is to minimize the total number of hours required to deliver the TotalGoods units of goods"

"The goal is to minimize the total time it takes for the mixed bread to be ready"

"The goal is to maximize the total number of keyboards produced"

"The goal is to minimize the maximum deviation of all the observed values of y from the value predicted by the linear relationship y = bx + a."

The objective is to minimize the sum of absolute deviations of each observed value of y from the value predicted by the quadratic relationship y = c * x^2 + b * x + a.

"The goal is to minimize the total cost of running the power stations over all periods of the day, including running costs, extra costs for operating above the minimum level, and start-up costs for starting the generators."

"The goal is to maximize the total profit from producing the products while satisfying all production constraints."

"The goal is to minimize the total prices of the foods to purchase"

The goal is to maximize the total profit from buying and selling the stock of a certain commodity over multiple periods.

"The goal is to minimize the total cost of purchasing the various alloys to produce the target alloy with the specified metal ratios."

The goal is to maximize the total profit from producing spare automobile parts.

"The goal is to minimize the total cost of production, including regular production cost, overtime production cost, and storage cost, while satisfying the monthly demand."

"The goal is to minimize the total transportation cost of sending empty containers from depots to ports"

"The goal is to maximize the profit from producing parts, which is calculated as the total revenue from selling the parts minus the total machine usage costs."

The goal is to minimize the cost of purchasing clicks for different ad types.

"The goal is to maximize the profit over the season from producing gift baskets."

The goal is to maximize the profit from producing different parts.

"The goal is to minimize the total cost of hiring translators while ensuring that all required languages are covered."

"The goal is to minimize the total cost of assigning consultants to projects, which includes both the fixed cost of hiring consultants and the additional cost for assigning projects to consultants."

"The goal is to minimize the total cost of interventions to halt the intruder's progress within the network while adhering to the time constraints on central and distributed processing."

$\max \left( \text{ProfitPerDollarCondos} \cdot \text{investmentCondos} + \text{ProfitPerDollarDetachedHouses} \cdot \text{investmentDetachedHouses} \right)$

$\max (\text{ProfitRegular} \times x + \text{ProfitSpecial} \times y)$

$\max \text{ExposureRadioAd} \times \text{numRadioAds} + \text{ExposureSocialMediaAd} \times \text{numSocialMediaAds}$

$\max (\text{ProfitPerLongCable} \times \text{longCables} + \text{ProfitPerShortCable} \times \text{shortCables})$

$\max \text{numAdsZTube} \cdot \text{ViewersZTube} + \text{numAdsSoorchle} \cdot \text{ViewersSoorchle} + \text{numAdsWassa} \cdot \text{ViewersWassa}$

$\max (\text{ProfitChair} \times \text{numChairs} + \text{ProfitDresser} \times \text{numDressers})$

$\min \left( \text{CostFeedA} \times x_A + \text{CostFeedB} \times x_B \right)$

$\max ProfitThinJar \cdot numThinJars + ProfitStubbyJar \cdot numStubbyJars$

$\max \big( \text{ProfitBananaHatersPackage} \times \text{quantityBananaHatersPackages} + \text{ProfitComboPackage} \times \text{quantityComboPackages} \big)$

$\max (\text{ProfitBread} \times \text{breadProduced}) + (\text{ProfitCookies} \times \text{cookiesProduced})$

$\max (\text{ProfitRegular} \times \text{numRegularPanes} + \text{ProfitTempered} \times \text{numTemperedPanes})$

$\max (\text{ProfitPerElephant} \cdot \text{numElephants} + \text{ProfitPerTiger} \cdot \text{numTigers})$

$\max \left( \text{investmentApt} \times \text{ReturnRateApt} + \text{investmentTown} \times \text{ReturnRateTown} \right)$

\[
\min \, \text{CostZodiac} \times x + \text{CostSunny} \times y
\]

$\max \text{ProfitPerTrain} \cdot \text{trains} + \text{ProfitPerPlane} \cdot \text{planes}$

$\max \, (\text{ProfitPhone} \cdot \text{numPhones} + \text{ProfitLaptop} \cdot \text{numLaptops})$

$\max \left( \text{ProfitRegularMix} \times \text{regularMix} + \text{ProfitSourMix} \times \text{sourMix} \right)$

$\max \left(\text{PhoneRevenue} \times \text{phoneSpaceAllocation} + \text{LaptopRevenue} \times \text{laptopSpaceAllocation}\right)$

$\max \ \text{ProfitPremium} \times \text{numPremium} + \text{ProfitRegular} \times \text{numRegular}$

$\max (\text{ProfitRegular} \cdot x_1) + (\text{ProfitPremium} \cdot x_2)$

$\max \left( \text{SeatingLongDesk} \times \text{numLongDesks} + \text{SeatingShortDesk} \times \text{numShortDesks} \right)$

$\min \left( \text{CostPerKgFertilizerC} \times \text{amountFertilizerC} + \text{CostPerKgFertilizerY} \times \text{amountFertilizerY} \right)$

$\max \sum_{j=1}^{\text{NumOilTypes}} \text{Profit}_j \cdot \text{production}_j$

$\max \ \text{MangoProfit} \times \text{mangosSold} + \text{GuavaProfit} \times \text{guavasSold}$

\[
\max \, \left( \text{ProfitPerGallonChocolate} \times \text{chocolateGallons} + \text{ProfitPerGallonVanilla} \times \text{vanillaGallons} \right)
\]

$\max (\text{ProfitPerAcrePotato} \times \text{acresPotatoes} + \text{ProfitPerAcreCucumber} \times \text{acresCucumbers})$

$\max \left( ProfitHardwood \times hardwoodProduction + ProfitVinyl \times vinylProduction \right)$

$\min (\text{OldFarmCost} \times \text{daysOldFarm} + \text{NewFarmCost} \times \text{daysNewFarm})$

$\min \left( \text{CostRice} \times \text{servingsRice} + \text{CostKebab} \times \text{servingsKebab} \right)$

$\max \left( \text{ProfitPerScooter} \times \text{scooters} + \text{ProfitPerBike} \times \text{bikes} \right)$

$\min \left( \text{CostPerServingVegetable} \cdot \text{numServingsVegetable} + \text{CostPerServingFruit} \cdot \text{numServingsFruit} \right)$

$\max \left( \text{ProfitBagels} \times \text{bagelsBatches} + \text{ProfitCroissants} \times \text{croissantsBatches} \right)$

$\min \left( \text{CostFactory1} \cdot \text{hoursFactory1} + \text{CostFactory2} \cdot \text{hoursFactory2} \right)$

$\max \left( \text{RevenuePerAcreTurnips} \times \text{acresTurnips} + \text{RevenuePerAcrePumpkins} \times \text{acresPumpkins} \right)$

$\min \left( \text{highIntensityDrills} + \text{lowIntensityDrills} \right)$

$\max (\text{CoinsPlatedA} \times \text{processesRunA} + \text{CoinsPlatedB} \times \text{processesRunB})$

$\max \left( \text{WindrowerHayPerAcre} \cdot \text{acresWindrower} + \text{HarvesterHayPerAcre} \cdot \text{acresHarvester} \right)$

$\min (\text{TimePerWrap} \times \text{numberOfWraps}) + (\text{TimePerPlatter} \times \text{numberOfPlatters})$

$\min \left( \text{numCounterTopMachines} + \text{numFridgeMachines} \right)$

$\max \sum_{i=1}^{\text{NumContainers}} \text{PasteProduced}[i] \cdot \text{numContainersUsed}[i]$

$\max \left( \text{OilProductionSmall} \cdot \text{acresSmall} + \text{OilProductionLarge} \cdot \text{acresLarge} \right)$

$\min \left( \text{TimeAlmond} \times \text{numAlmond} + \text{TimePistachio} \times \text{numPistachio} \right)$

$\min (\text{FiberPerBrownie} \times \text{numBrownies} + \text{FiberPerLemonSquare} \times \text{numLemonSquares})$

$\max (\text{CO2WithCatalyst} \times \text{numCatalystProcessUses} + \text{CO2WithoutCatalyst} \times \text{numNonCatalystProcessUses})$

$\max \left( \text{TealeavesTraditional} \cdot \text{acresTraditional} + \text{TealeavesModern} \cdot \text{acresModern} \right)$

$\min \left( \text{CrepeMixForChocolateCrepe} \times \text{numChocolateCrepes} + \text{CrepeMixForPeanutButterCrepe} \times \text{numPeanutButterCrepes} \right)$

$\max FishCaughtNet \times acresNet + FishCaughtLine \times acresLine$

$\min \text{TimePerMocha} \times \text{mochas} + \text{TimePerRegular} \times \text{regulars}$

let's think through the problem step by step.

The objective is to minimize the total usage of the curry base.

We have two types of curry bowls:
1. Goat curry bowls, with each bowl using `CurryBasePerGoatCurry` units of curry base.
2. Chicken curry bowls, with each bowl using `CurryBasePerChickenCurry` units of curry base.

The variables are:
- `goatCurryBowls`: the number of goat curry bowls made.
- `chickenCurryBowls`: the number of chicken curry bowls made.

To minimize the total usage of curry base, we need to minimize the sum of the curry base used for both goat curry bowls and chicken curry bowls.

The total curry base used for goat curry bowls is:
\[ \text{CurryBasePerGoatCurry} \times \text{goatCurryBowls} \]

The total curry base used for chicken curry bowls is:
\[ \text{CurryBasePerChickenCurry} \times \text{chickenCurryBowls} \]

Therefore, the objective function to minimize is:
\[ \text{CurryBasePerGoatCurry} \times \text{goatCurryBowls} + \text{CurryBasePerChickenCurry} \times \text{chickenCurryBowls} \]

Now, let's put this in LaTeX format.

-----
$\min (\text{CurryBasePerGoatCurry} \times \text{goatCurryBowls} + \text{CurryBasePerChickenCurry} \times \text{chickenCurryBowls})$
----

$\min ( \text{numFullTimeWorkers} + \text{numPartTimeWorkers} )$

\[
\min \left( \text{numberMachinesA} + \text{numberMachinesB} \right)
\]

$\max \  \text{CompoundA} \cdot x_A + \text{CompoundB} \cdot x_B$

$\max \text{HeapLeachingProduction} \times \text{heapLeachingLand} + \text{VatLeachingProduction} \times \text{vatLeachingLand}$

$\min (TimePerMilkChocolate \cdot milkChocolateBars + TimePerDarkChocolate \cdot darkChocolateBars)$

\[
\min (\text{DischargePill1} \cdot x_1 + \text{DischargePill2} \cdot x_2)
\]

$\min (\text{SugarAlpha} \times \text{numAlphaBottles)} + (\text{SugarOmega} \times \text{numOmegaBottles)}$

$\max \sum_{i=1}^{\text{NumBeakers}} \text{SlimeProduced}[i] \times \text{beakers}[i]$

$\min (\text{MeatSmall} \times \text{smallBones}) + (\text{MeatLarge} \times \text{largeBones})$

$\max \left( \text{VitDOrangeJuice} \times \text{orangeJuiceBoxes} + \text{VitDAppleJuice} \times \text{appleJuiceBoxes} \right)$

$\min (\text{cleansingChemical} \times \text{CleansingTimePerUnit} + \text{odorRemovingChemical} \times \text{OdorRemovingTimePerUnit})$

$\min \sum_{i=1}^{\text{NumSyrups}} \text{numServings}[i] \times \text{SugarContent}[i]$

$\max \sum_{i=1}^{\text{NumDemos}} \text{MintyFoamProduced}[i] \cdot x_i$

$\max \left( \text{FatPerAppleServing} \times \text{appleServings} + \text{FatPerCarrotServing} \times \text{carrotServings} \right)$

$\min \left( \text{CholesterolPerBurger} \times \text{burgers} + \text{CholesterolPerPizza} \times \text{pizzaSlices} \right)$

$\max ( \text{SupplyShots} \cdot \text{numBatchesShots} + \text{SupplyPills} \cdot \text{numBatchesPills} )$

$\min \left( \text{WasteProductionRateMachine1} \times \text{timeMachine1} + \text{WasteProductionRateMachine2} \times \text{timeMachine2} \right)$

$\min \left( \text{RamenSodium} \cdot \text{numRamenMeals} + \text{FriesSodium} \cdot \text{numFriesMeals} \right)$

$\max \left( \text{FoamCheapBox} \cdot \text{numberCheapBoxes} + \text{FoamExpensiveBox} \cdot \text{numberExpensiveBoxes} \right)$

ht, let's carefully think through the problem formulation:

1. **Objective:** We are asked to minimize the woman's fat intake.
2. **Decision Variables:**
   - \( \text{numberOfAlmondServings} \): Number of servings of almonds.
   - \( \text{numberOfCashewServings} \): Number of servings of cashews.
3. **Given Parameters:**
   - \( \text{FatAlmonds} \): Fat per serving of almonds.
   - \( \text{FatCashews} \): Fat per serving of cashews.

To minimize her fat intake, we need to consider the total fat coming from both almonds and cashews. The total fat intake is calculated as:
\[ \text{numberOfAlmondServings} \times \text{FatAlmonds} + \text{numberOfCashewServings} \times \text{FatCashews} \]

So, the objective function to be minimized can be expressed mathematically in LaTeX as:

-----
$\min \left( \text{numberOfAlmondServings} \times \text{FatAlmonds} + \text{numberOfCashewServings} \times \text{FatCashews} \right)$
-----

That's the correct formulation for minimizing the total fat intake based on the given parameters and decision variables

$\max \text{(liquidSanitizer} \times \text{HandsCleanedByLiquid)} + \text{(foamSanitizer} \times \text{HandsCleanedByFoam)}$

$\max \text{BackMedicinePainKiller1} \times \text{dosesPainKiller1} + \text{BackMedicinePainKiller2} \times \text{dosesPainKiller2}$

$\min \left( \text{daytimePills} \times \text{SleepMedicineInDaytimePill} + \text{nighttimePills} \times \text{SleepMedicineInNighttimePill} \right)$

$\max \text{SugarCheesecake} \cdot \text{slicesCheesecake} + \text{SugarCaramelCake} \cdot \text{slicesCaramelCake}$

$\min (numOranges \times SugarOranges + numGrapefruits \times SugarGrapefruits)$

$\min \left( \text{DoseBeam1Pancreas} \cdot \text{TimeBeam1} + \text{DoseBeam2Pancreas} \cdot \text{TimeBeam2} \right)$

$\min \left( \text{FatTurkeyDinner} \times \text{turkeyDinnerCount} + \text{FatTunaSandwich} \times \text{tunaSaladSandwichCount} \right)$

$\max \left( \text{GreenGasExp1} \cdot \text{exp1} + \text{GreenGasExp2} \cdot \text{exp2} \right)$

$\min \left( \text{SugarBlueberries} \times \text{packsBlueberries} + \text{SugarStrawberries} \times \text{packsStrawberries} \right)$

$\max \left( \texttt{smallBottlesUsed} \times \texttt{SmallBottleCapacity} + \texttt{largeBottlesUsed} \times \texttt{LargeBottleCapacity} \right)$

$\min (\text{ShipFuel} \times \text{shipTrips}) + (\text{PlaneFuel} \times \text{planeTrips})$

$\min (BoatTripTime \cdot boatTrips + CanoeTripTime \cdot canoeTrips)$

$\min \left( \text{PollutionOldVan} \cdot \text{oldVans} + \text{PollutionNewVan} \cdot \text{newVans} \right)$

$\max (\text{numSmallTeams} \times \text{AreaSmallTeam} + \text{numLargeTeams} \times \text{AreaLargeTeam})$

$\max \left(\text{otters} \times \text{OtterTricks} + \text{dolphins} \times \text{DolphinTricks}\right)$

$\max (SmallBouquetFlowers \cdot smallBouquetsTransported + LargeBouquetFlowers \cdot largeBouquetsTransported)$

$\min \left( \text{NewCompanyDieselUsage} \times \text{newCompanyTrips} + \text{OldCompanyDieselUsage} \times \text{oldCompanyTrips} \right)$

$\min (\text{TimePerBusTrip} \times \text{busTrips} + \text{TimePerCarTrip} \times \text{carTrips})$

$\min (\text{HelicopterPollution} \times \text{helicopterTrips} + \text{TruckPollution} \times \text{truckTrips})$

$\max \left( \text{bikes} \times \text{MealsPerBike} + \text{scooters} \times \text{MealsPerScooter} \right)$

$\min (\text{GasUsageRegular} \times \text{regularTrips}) + (\text{GasUsageSpeed} \times \text{speedTrips})$

$\min (busTrips \times BusTimePerTrip + carTrips \times CarTimePerTrip)$

$\min ( \text{VanPollution} \times \text{numVans} ) + ( \text{MinibusPollution} \times \text{numMinibuses} )$

$\max \left( \text{SmallKegCapacity} \times \text{smallKegs} + \text{LargeKegCapacity} \times \text{largeKegs} \right)$

$\min \left( SubmarineGasUsage \cdot submarineTrips + BoatGasUsage \cdot boatTrips \right)$

$\min (\text{HelicopterTripTime} \times \text{helicopterTrips} + \text{CarTripTime} \times \text{carTrips})$

$\min (\text{BalloonPollution} \times \text{numBalloons}) + (\text{GondolaPollution} \times \text{numGondolas})$

$\max \text{numMediumCarts} \times \text{CapacityMediumCart} + \text{numLargeCarts} \times \text{CapacityLargeCart}$

$\max (\text{numLabradors} \times \text{NewspapersPerLabrador} + \text{numGoldenRetrievers} \times \text{NewspapersPerGoldenRetriever})$

rmulate the objective mathematically for this MILP problem, we need to minimize the total number of trips. Given the variables defined (`ferryTrips` and `lightRailTrips`), the mathematical formulation for this objective is straightforward.

The total number of trips is the sum of the ferry trips and the light rail trips. Therefore, the objective function aims to minimize the sum of these two variables:

-----
$\min (ferryTrips + lightRailTrips)$
----

$\max \left( \text{SmallCrateCapacity} \times \text{numSmallCrates} + \text{LargeCrateCapacity} \times \text{numLargeCrates} \right)$

$\min \left( \text{HelicopterTripTime} \times \text{helicopterTrips} + \text{BusTripTime} \times \text{busTrips} \right)$

$\min \left( \text{numCars} \times \text{CarPollution} + \text{numBuses} \times \text{BusPollution} \right)$

$\max \left( \text{NumSmallTrucks} \times \text{CapacitySmallTruck} + \text{NumLargeTrucks} \times \text{CapacityLargeTruck} \right)$

$\max \left( \text{numCarrierPigeons} \times \text{CarrierPigeonCapacity} + \text{numOwls} \times \text{OwlCapacity} \right)$

$\min \left(HamburgerCost \cdot numHamburgers + ChickenWrapCost \cdot numChickenWraps\right)$

$\max \sum_{i=1}^{\text{NumProducts}} \text{Profit}[i] \times \text{unitsProduced}[i]$

$\min \left( \text{CostMilk} \times \text{glassesMilk} + \text{CostVegetables} \times \text{platesVegetables} \right)$

$\min \left( \text{CostNoodles} \cdot x + \text{CostProteinBar} \cdot y \right)$

$\min (\text{PriceSupplementA} \times \text{pillsSupplementA} + \text{PriceSupplementB} \times \text{pillsSupplementB})$

$\max \left( \text{betBasketball} \times \text{PayoutBasketball} \times (1 - \text{ProbLoseBasketball}) + \text{betHorseRace} \times \text{PayoutHorseRace} \times (1 - \text{ProbLoseHorseRace}) + \text{betSoccer} \times \text{PayoutSoccer} \times (1 - \text{ProbLoseSoccer}) \right)$

$\max \left( \text{EarningWashingMachine} \times \text{numWashingMachines} + \text{EarningFreezer} \times \text{numFreezers} \right)$

$\max \left( \text{ProfitCrepeCake} \cdot \text{crepeCakes} + \text{ProfitSpongeCake} \cdot \text{spongeCakes} + \text{ProfitBirthdayCake} \cdot \text{birthdayCakes} \right)$

$\max \left( \text{amountFirstMix} \times \text{ProfitFirstMix} + \text{amountSecondMix} \times \text{ProfitSecondMix} \right)$

$\max (ProfitPersonalLicense \cdot personalLicenses + ProfitCommercialLicense \cdot commercialLicenses)$

$\min (CalciumTime \cdot calciumPills + VitaminDTime \cdot vitaminDPills)$

$\max \left( \text{ThrowingGameCustomers} \times \text{throwingGames} + \text{ClimbingGameCustomers} \times \text{climbingGames} \right)$

$\max \left( \text{GuestsCircular} \cdot \text{numCircularTables} + \text{GuestsRectangular} \cdot \text{numRectangularTables} \right)$

$\max \left( \text{bikeShifts} \times \text{TipsPerBikeShift} + \text{ scooterShifts} \times \text{TipsPerScooterShift} \right)$

$\min (\text{TimeLargePizza} \times \text{largePizzas} + \text{TimeMediumPizza} \times \text{mediumPizzas})$

$\min (\text{CartRefills} \times \text{serversCart} + \text{HandRefills} \times \text{serversHand})$

\[
\min \left( \text{OriginalCookingTime} \times \text{numberOfOriginalMeals} + \text{ExperimentalCookingTime} \times \text{numberOfExperimentalMeals} \right)
\]

$\max \text{PotassiumPerSalad} \cdot s + \text{PotassiumPerFruitBowl} \cdot f$

$\min ( \text{numLargeUnits} \times \text{LargeUnitParkingSpots} + \text{numSmallUnits} \times \text{SmallUnitParkingSpots} )$

$\max (\text{numElectricJacks} \times \text{CarsPerHourElectric} + \text{numGasJacks} \times \text{CarsPerHourGas})$

$\max \sum_{i=1}^{\text{NumCatalysts}} \text{ConversionRate}[i] \cdot \text{numCatalystsUsed}[i]$

$\max \left( MetalExtractedJ \cdot numProcessJ + MetalExtractedP \cdot numProcessP \right)$

$\min (n_e \times \text{EscalatorSpace} + n_v \times \text{ElevatorSpace})$

$\min (\text{kayakTrips} \times \text{KayakTime}) + (\text{motorboatTrips} \times \text{MotorboatTime})$

$\min (\text{timeSpecialized} \times \text{SpecializedCost}) + (\text{timeCommon} \times \text{CommonCost})$

$\min \left( \text{WaterPerAcaiSmoothie} \times \text{acaiSmoothies} + \text{WaterPerBananaSmoothie} \times \text{bananaSmoothies} \right)$

$\min \left( \text{numInVivo} \times \text{RadiationInVivo} + \text{numExVivo} \times \text{RadiationExVivo} \right)$

$\max (\text{runnerDeliveries} \times \text{RunnerCapacity} + \text{canoerDeliveries} \times \text{CanoerCapacity})$

$\max \big(\text{PeoplePerMedPatch} \cdot \text{medPatchBatches} + \text{PeoplePerCream} \cdot \text{creamBatches}\big)$

$\min \left( \text{numberOfCamelCaravans} \times \text{HoursCamelCaravan} + \text{numberOfDesertTrucks} \times \text{HoursDesertTruck} \right)$

$\min \left( \frac{\text{TimeA} \cdot \text{unitsA} + \text{TimeB} \cdot \text{unitsB}}{\text{unitsA} + \text{unitsB}} \right)$

$\max \: (\text{CaloriesInSpinach} \times \text{cupsOfSpinach}) + (\text{CaloriesInSoybeans} \times \text{cupsOfSoybeans})$

$\max \left( \text{mechanicalKeyboards} + \text{standardKeyboards} \right)$

$\min \left( \text{FlavouringMatcha} \times \text{matchaOrders} + \text{FlavouringOrange} \times \text{orangeOrders} \right)$

$\min \left( \sum_{k=1}^{K} \sum_{t=1}^{T} runcost_k \cdot isRunning_{k,t} + \sum_{k=1}^{K} \sum_{t=1}^{T} extracost_k \cdot (powerGenerated_{k,t} - minlevel_k \cdot isRunning_{k,t}) + \sum_{k=1}^{K} \sum_{t=1}^{T} startcost_k \cdot \left( isRunning_{k,t} - isRunning_{k,t-1} \right) \right)$

$\max \sum_{k=1}^{\text{NumItems}} \text{Value}_k \cdot \text{packItem}_k$

\[
\max \sum_{n=1}^{\text{NumPeriods}} \left( \text{Price}_n \cdot \text{sellAmount}_n - \text{Cost}_n \cdot \text{buyAmount}_n - \text{HoldingCost} \cdot \text{stockLevel}_n \right)
\]

$\min \sum_{k=1}^{\text{NumAlloys}} \text{Price}[k] \cdot \text{amountOfAlloys}[k]$

$\max \sum_{k=1}^{K} \text{Profit}_{k} \times \text{quantityK}_{k}$

$ \min \sum_{t=1}^{NumMonths} \left( CostRegular \cdot regularProduction_t + CostOvertime \cdot overtimeProduction_t + StoreCost \cdot inventory_t \right) $

\[ \min \sum_{i=1}^{I} \sum_{j=1}^{J} \text{numBarges}_{i,j} \times \text{Distance}_{i,j} \times \text{Price} \]

\[
\max \left( \sum_{p=1}^{P} \text{Prices}_p \times \text{batchCount}_p - \sum_{m=1}^{M} \text{MachineCosts}_m \times \sum_{p=1}^{P} \text{TimeRequired}_{m,p} \times \text{batchCount}_p \right)
\]

$\min \sum_{a=1}^{\text{NumAdTypes}} \text{Costs}[a] \cdot \frac{\text{clicks}[a]}{1000}$

\begin{equation}
\max \sum_{w=1}^{W} \left( \text{SellingPrice} \times \text{Demand}_w - \text{MaterialCost} \times (\text{regularBaskets}_w + \text{overtimeBaskets}_w) - \text{RegularCost} \times \left(\frac{\text{regularBaskets}_w \times \text{AssemblyTime}}{\text{RegularLabor}_w}\right) - \text{OvertimeCost} \times \left(\frac{\text{overtimeBaskets}_w \times \text{AssemblyTime}}{\text{OvertimeLabor}_w}\right) - \text{HoldingCost} \times \text{inventory}_w \right) + \text{SalvageValue} \times \text{surplusEndSeason}
\end{equation}

$\max \left( \sum_{p=1}^{P} \text{Prices}_p \cdot \text{Batches}_p - \sum_{m=1}^{M} \sum_{p=1}^{P} \text{MachineCosts}_m \cdot \text{TimeRequired}_{m,p} \cdot \text{Batches}_p \right)$

$\max \left( \sum_{p=1}^{P} \text{Prices}[p] \times \text{batches}[p] - \sum_{m=1}^{M} \sum_{p=1}^{P} \text{TimeRequired}[m,p] \times \text{batches}[p] \times \text{MachineCosts}[m] - \sum_{m=1}^{M} \text{extraTime}[m] \times \text{ExtraCosts}[m] \right)$

$\min \left( \sum_{j=1}^{\texttt{NumConsultants}} f_j \cdot \texttt{hireConsultant}_j + \sum_{i=1}^{\texttt{NumProjects}} \sum_{j=1}^{\texttt{NumConsultants}} c_{i,j} \cdot \texttt{assignProjToConsultant}_{i,j} \right)$

$\min \sum_{n=1}^{N} \left( \text{IsolateCentral}[n] \times \text{CentralCost} \times \text{IsolateCentral}[n] + \text{ScanCentral}[n] \times \text{CentralCost} \times \text{ScanCentral}[n] + \text{IsolateDistributed}[n] \times \text{DistributedCost} \times \text{IsolateDistributed}[n] + \text{ScanDistributed}[n] \times \text{DistributedCost} \times \text{ScanDistributed}[n] \right)$

The total investment in condos and detached houses should not exceed the TotalBudget

The investment in condos must be at least MinPercentageCondos percent of the total investment

The investment in detached houses must be at least MinInvestmentDetachedHouses

The total number of eggs used by regular and special sandwiches should not exceed the total number of eggs available

The total number of bacon slices used by regular and special sandwiches should not exceed the total number of bacon slices available

Total cost of radio ads and social media ads should not exceed the TotalBudget

The number of radio advertisements should be at least MinRadioAds

The number of radio advertisements should not exceed MaxRadioAds

The number of social media advertisements should be at least MinSocialMediaAds

The total gold used for making long and short cables should not exceed TotalGold

The number of short cables should be at least ShortCableMultiplier times the number of long cables

Total cost of advertisements on all media types (ZTube, Soorchle, Wassa) should not exceed the weekly advertising budget

Number of advertisements on SoorchleEngine should not exceed MaxSzoorchleAds

Proportion of total number of advertisements on Wassa should not exceed MaxWassaProportion

Proportion of total number of advertisements on ZTube should be at least MinZTubeProportion

The total amount of stain used for chairs plus the total amount of stain used for dressers should not exceed the total gallons of stain available per week

The total lengths of oak wood used for chairs plus the total lengths of oak wood used for dressers should not exceed the total lengths of oak wood available per week

Total shaping time for all thin jars and stubby jars should not exceed TotalShapingTime minutes

Total baking time for all thin jars and stubby jars should not exceed TotalBakingTime minutes

The total number of apples used in banana-haters packages and combo packages should not exceed the stock of apples

The total number of bananas used in combo packages should not exceed the stock of bananas

The total number of grapes used in banana-haters packages and combo packages should not exceed the stock of grapes

The total time the mixer is used to produce bread and cookies must not exceed MaxMixerHours

The total time the oven is used to bake bread and cookies must not exceed MaxOvenHours

The total time used on the heating machine should not exceed MaxHeatingTime minutes per day

The total time used on the cooling machine should not exceed MaxCoolingTime minutes per day

Total amount of wood used for making Elephants and Tigers should not exceed the AvailableWood

Total amount of plastic used for making Elephants and Tigers should not exceed the AvailablePlastic

Investment in apartments must be at least MinRatioInvestAptToTown times the investment in townhouses

Total investment in apartments and townhouses should not exceed TotalInvest

The total grams of Z1 from Zodiac and Sunny pills should be at least MinZ1.

The total grams of D3 from Zodiac and Sunny pills should be at least MinD3.

The total wood used by model trains and model planes should not exceed TotalWood

The total paint used by model trains and model planes should not exceed TotalPaint

Total floor space used by phones and laptops should not exceed the total available floor space

The total spending to purchase phones and laptops should not exceed the maximum allowed spending

The percentage of laptops in the inventory should be at least the minimum required percentage

The total amount of regular candy used in the regular mix cannot exceed the available kilograms of regular candy

The total amount of sour candy used in the regular mix cannot exceed the available kilograms of sour candy

The total amount of regular candy used in the sour surprise mix cannot exceed the available kilograms of regular candy

The total amount of sour candy used in the sour surprise mix cannot exceed the available kilograms of sour candy

The total amount of regular candy used in both mixes combined cannot exceed the available kilograms of regular candy

The total amount of sour candy used in both mixes combined cannot exceed the available kilograms of sour candy

Total space allocated to phone and laptop production should not exceed TotalSpace

Total cost of phone and laptop production should not exceed TotalBudget

Total labor hours used for phone and laptop production should not exceed TotalLabor

The total number of premium and regular desktops created should not exceed MaxDesktops

The total cost to produce premium and regular desktops should not exceed MaxBudget

The number of regular models made per day (x1) should be less than or equal to MaxDemandRegular

The number of premium models made per day (x2) should be less than or equal to MaxDemandPremium

The total number of cars made per day (x1 + x2) should be less than or equal to MaxTotalProduction

The number of regular models made per day (x1) should be greater than or equal to 0

The number of premium models made per day (x2) should be greater than or equal to 0

The total cost of the long desks and short desks should not exceed the Budget

The total space occupied by the long desks and short desks should not exceed the MaxSpace

The number of long desks and short desks should be non-negative integers

The mixture must contain at least MinRequiredNitrousOxide units of nitrous oxide

The mixture must contain at least MinRequiredVitaminMix units of vitamin mix

The total amount of each substance used by all oil types should not exceed the total amount of that substance available

The number of containers of each oil type should be non-negative

Total number of guavas sold should be at most MaxGuavasPerMango times the number of mangos sold

The total units of Vitamin A from Drink A and Drink B must be at least MinVitaminA units

The total units of Vitamin D from Drink A and Drink B must be at least MinVitaminD units

The total units of Vitamin E from Drink A and Drink B must be at most MaxVitaminE units

Number of gallons of chocolate ice cream produced per week should be at least MinChocolateGallons

Number of gallons of chocolate ice cream produced per week should be at most MaxChocolateGallons

Number of gallons of vanilla ice cream produced per week should be at least MinVanillaGallons

Number of gallons of vanilla ice cream produced per week should be at most MaxVanillaGallons

Total production time for chocolate ice cream and vanilla ice cream should not exceed TotalAvailableHours

Number of workers assigned to chocolate ice cream should be at least WorkersPerGallonChocolate times the number of gallons of chocolate ice cream produced

Number of workers assigned to vanilla ice cream should be at least WorkersPerGallonVanilla times the number of gallons of vanilla ice cream produced

The total acres of potatoes and cucumbers combined must not exceed TotalLand acres

The acres of cucumbers must not exceed MaxCucumberToPotatoRatio times the acres of potatoes

The demand for hardwood flooring each week is at least MinDemandHardwood square feet

The demand for vinyl planks each week is at least MinDemandVinyl square feet

The total amount of flooring shipped each week must be at least MinTotalShipment square feet

The company can produce no more than MaxProductionHardwood square feet of hardwood flooring each week

The company can produce no more than MaxProductionVinyl square feet of vinyl planks each week

The total quantity of each type of berry harvested from both farms must meet or exceed the DeliveryRequirement for that type of berry

The total operational cost is the sum of the costs of operating both farms, which must be minimized

The total daily calorie intake from Rice and Kebab should be at least MinCalories calories

The total daily protein intake from Rice and Kebab should be at least MinProtein grams

Total design hours required by scooters and bikes should not exceed the available total design hours

Total engineering hours required by scooters and bikes should not exceed the available total engineering hours

The total units of vitamins from the servings of vegetables and fruits must be at least the minimum required units of vitamins

The total units of minerals from the servings of vegetables and fruits must be at least the minimum required units of minerals

The number of servings of vegetables and fruits should be non-negative

The total oven time used for baking bagels and croissants should not exceed the maximum available oven time per day

The total pastry chef time used for baking bagels and croissants should not exceed the maximum available pastry chef time per day

The total number of black teddy bears produced by both factories must be at least DemandBlack

The total number of white teddy bears produced by both factories must be at least DemandWhite

The total number of brown teddy bears produced by the first factory must be at least DemandBrown

The sum of acres allocated to turnips and pumpkins should not exceed TotalAcres

The total watering time for the acres of turnips and pumpkins should not exceed TotalWateringTime

The total pesticide cost for the acres of turnips and pumpkins should not exceed TotalPesticideBudget

Total hours provided by staff and substitute teachers must meet the RequiredHours

Total wage paid to staff and substitute teachers must not exceed the TotalBudget

Number of staff and substitute teachers hired must be non-negative integers

The total number of gems processed by HighIntensityDrills and LowIntensityDrills must meet or exceed the DailyGemProcessingRequirement

The total amount of water required by HighIntensityDrills and LowIntensityDrills must not exceed the TotalAvailableWater

The fraction of HighIntensityDrills must not exceed MaxFractionHighIntensityDrills times the total number of drills

The number of LowIntensityDrills must meet or exceed MinLowIntensityDrills

The number of HighIntensityDrills and LowIntensityDrills must be non-negative integers

The total number of sandwiches produced by dine-in places and food-trucks must be at least the minimum required sandwich production per day

The total number of employees used by all dine-in places and food-trucks must not exceed the total number of available employees

The total units of gold used by both processes combined should not exceed the TotalGoldAvailable

The total number of wires used by both processes combined should not exceed the TotalWiresAvailable

The total cutting capacity of the saws must be at least MinPlanks planks per day

The total sawdust produced by the saws must be at most MaxSawdust units per day

The sum of acres processed by the windrower and the hay harvester should not exceed TotalAcres

The total fuel used by both the windrower and the hay harvester should not exceed TotalFuel

The total methane gas produced by both the windrower and the hay harvester should not exceed MaxMethane

The total units of meat used must be at least MinMeat units. This translates to the constraint: MeatPerWrap * number_of_wraps + MeatPerPlatter * number_of_platters >= MinMeat

The total units of rice used must be at least MinRice units. This translates to the constraint: RicePerWrap * number_of_wraps + RicePerPlatter * number_of_platters >= MinRice

The number of Wraps produced must be at least WrapsToPlattersRatio times the number of Platters produced. This translates to the constraint: number_of_wraps >= WrapsToPlattersRatio * number_of_platters

The amusement park needs to process at least MinTotalPeopleRequirement people per hour

The amusement park can use at most MaxTotalPaperLimit rolls of paper per hour

The number of card-only machines must not exceed the number of cash-based machines

At least MinPlasticPerGlass times as many plastic jars must be filled as glass jars

The total amount of honey used in all jars should not exceed TotalHoney

The total heating capacity of the selected furnaces must be at least 'TotalApartments'

The total daily electricity consumed by the selected furnaces must not exceed 'TotalElectricity'

The total number of managers used by rural and urban factories should not exceed TotalManagers

The total number of phones produced by rural and urban factories should be at least MinPhonesPerDay

The total heat output from all machines should not exceed MaxDailyHeatOutput units of heat per day

The total production of ice cream cones from all machines should be at least MinDailyProduction cones per day

The total amount of water used cannot exceed the total available water

The total amount of powdered pill used cannot exceed the total available powdered pill

The total acres allocated to small wells and large wells should not exceed TotalAcres

The total drill bits used for both small and large wells should not exceed TotalDrillBits

The total pollution produced by both small and large wells should not exceed MaxPollution

The total amount of butter used for almond croissants and pistachio croissants should not exceed the total available units of butter

The total amount of flour used for almond croissants and pistachio croissants should not exceed the total available units of flour

The number of almond croissants produced should be at least MinRatioAlmondToPistachio times the number of pistachio croissants

Number of electric grills must be less than the number of gas grills

Number of patties cooked per minute by gas grills and electric grills combined must be at least MinPattiesPerMinute

Total units of cooking oil used per minute by gas grills and electric grills combined must be at most MaxOilPerMinute

The number of adult size bottles must be at least MinRatioAdultToKids times the number of kids size bottles

The total volume of cough syrup used in kids size bottles and adult size bottles combined must not exceed TotalCoughSyrup

Number of lemon squares made must be larger than the number of brownies made

Total units of chocolate mix required for brownies must not exceed TotalChocolateMix

Total units of lemon mix required for lemon squares must not exceed TotalLemonMix

Total labor hours provided by nurses and pharmacists should be at least equal to TotalLaborHoursRequired

Total payment for nurses and pharmacists should not exceed TotalBudget

The laundromat must be able to wash at least MinWashingCapacityRequired items per day

The total energy consumption per day must not exceed MaxEnergyAvailable kWh per day

At most MaxPercentageTopLoading of the total machines can be top-loading

The total pollution produced by the vans should be at most MaxPollution units per day

The total number of packages delivered by the vans should be at least MinPackages packages per day

The total amount of wood used by both processes should not exceed the total available units of wood: WoodWithCatalyst * number_of_times_process_with_catalyst_is_used + WoodWithoutCatalyst * number_of_times_process_without_catalyst_is_used <= TotalWood

The total amount of oxygen used by both processes should not exceed the total available units of oxygen: OxygenWithCatalyst * number_of_times_process_with_catalyst_is_used + OxygenWithoutCatalyst * number_of_times_process_without_catalyst_is_used <= TotalOxygen

The total number of acres used by the traditional and modern machines should not exceed the TotalLand

The total fuel used by both traditional and modern machines should not exceed the TotalFuel

The total waste generated by both traditional and modern machines should not exceed the MaxWaste

The number of the first model of machines must be less than the number of the second model

The sum of StampingRates of the machines must be at least MinLetters

The sum of GlueConsumption of the machines must be at most MaxGlue

The number of large crates must be at least MinLargeToSmallRatio times the number of small crates

The total number of bananas used by small and large crates should not exceed the TotalBananas available

The number of peanut butter crepes made (NumPeanutButterCrepes) must exceed the number of chocolate crepes made (NumChocolateCrepes)

NumChocolateCrepes should be at least MinimumChocolateCrepeProportion times the total number of crepes made (NumChocolateCrepes + NumPeanutButterCrepes)

The total amount of chocolate spread used by all chocolate crepes must not exceed TotalChocolateSpread

The total amount of peanut butter spread used by all peanut butter crepes must not exceed TotalPeanutButterSpread

The total hours worked by regular and emergency fire fighters must be at least MinHoursRequired

The total earnings of regular and emergency fire fighters must not exceed Budget

The total number of machines must achieve a minimum delivery rate of 'MinDeliverRate'

The total energy consumption of all machines must not exceed 'MaxEnergyConsumption'

The total number of hot dogs produced per day by all small and large shops must be at least MinHotDogsPerDay

The total number of workers needed for all small and large shops should not exceed TotalWorkers

The total luggage capacity from 4-wheeler and 3-wheeler vehicles combined should be at least MinLuggage per day

The total pollution produced by 4-wheeler and 3-wheeler vehicles combined should be at most MaxPollution units per day

The total number of acres fished using net and fishing line should not exceed the total area of the lake

The total bait used for fishing with both methods should not exceed the total available units of bait

The total pain caused by fishing with both methods should not exceed the maximum tolerable units of pain

The number of units of coffee powder used to make mochas and regular coffees should not exceed the total units of coffee powder available

The number of units of milk used to make mochas and regular coffees should not exceed the total units of milk available

The number of mochas produced should be at least a certain multiple of the number of regular coffees produced

The number of manual slicers should be less than the number of automatic slicers

The total slices per minute produced by all slicers should be greater than or equal to the minimum slices required per minute (MinSlices)

The total grease usage per minute by all slicers should be less than or equal to the maximum grease usage allowed per minute (MaxGrease)

The number of plastic bottles must be at least MinPlasticToGlassRatio times the number of glass bottles

The total volume of water in glass bottles plus the total volume of water in plastic bottles must not exceed TotalWater

Total usage of chicken meat should not exceed TotalChickenMeat units

At least MinChickenCurryPercentage percentage of the total bowls made must be chicken curry

The number of goat curry bowls must be larger than the number of chicken curry bowls

The total hours worked by full time and part time workers must meet or exceed the TotalLaborHours required for the project

The total wages paid to full time and part time workers must not exceed the TotalBudget available

The number of full time workers and part time workers should be minimized

The total number of items produced per day by both types of machines must be at least MinItemsPerDay

The total electricity consumption per day by both types of machines must not exceed MaxElectricityPerDay

The number of type B machines used must not exceed MaxPercentageB of the total number of machines

The number of type A machines used must be at least MinMachinesA

CountA * HydrogenProductionA + CountB * HydrogenProductionB >= MinHydrogenRequirement

CountA * PollutantProductionA + CountB * PollutantProductionB <= MaxPollutantAllowance

The total rare inert gas used by reactions A and B should not exceed the available rare inert gas

The total treated water used by reactions A and B should not exceed the available treated water

The total land used for heap leaching and vat leaching must not exceed TotalLand

The total number of extraction machines used must not exceed TotalMachines

The total daily polluted wastewater produced must not exceed MaxWastewater

The total amount of cocoa used for all milk and dark chocolate bars should not exceed the total available units of cocoa

The total amount of milk used for all milk and dark chocolate bars should not exceed the total available units of milk

The number of milk chocolate bars produced should be at least MilkToDarkRatio times the number of dark chocolate bars produced

The total number of premium printers and regular printers should be minimized

The number of regular printers must be less than the number of premium printers

The combined printing speed of all printers must be at least MinPagesPerMinute pages per minute

The combined ink usage of all printers must be at most MaxInkPerMinute units per minute

The total volume used by vintage and regular bottles must not exceed the TotalVolume

The number of regular bottles must be at least MinRatioRegularToVintage times the number of vintage bottles

The number of vintage bottles produced must be at least MinVintageBottles

Total units of pain medication from both Pill 1 and Pill 2 should not exceed MaxPainMed

Total units of anxiety medication from both Pill 1 and Pill 2 should be at least MinAnxietyMed

The total grams of protein consumed from both brands should be at least MinProtein grams

The total calories consumed from both brands should be at least MinCalories calories

The proportion of omega brand drink should be at most MaxOmegaProportion of the total drink intake

The total units of flour used by all beakers should not exceed the total available units of flour

The total units of special liquid used by all beakers should not exceed the total available units of special liquid

The total units of waste produced by all beakers should not exceed the maximum allowable units of waste

The total amount of tooth medication used for small bones and large bones combined should not exceed TotalToothMedication

At least MinPercentageSmallBones percent of the total number of bones produced should be small bones

The number of large bones produced should be at least MinLargeBones

The number of apple juice boxes must be at least MinAppleToOrangeRatio times the number of orange juice boxes

The number of orange juice boxes must be at least MinOrangeJuice

The total amount of vitamin C consumed must not exceed MaxVitaminC units

The amount of cleansing chemical used must be at least MinCleansingChemical

The total amount of chemicals used (cleansing chemical + odor-removing chemical) must be at least MinTotalChemicals

The amount of cleansing chemical used must not exceed MaxCleansingToOdorRatio times the amount of odor-removing chemical

The total operating time of factory 1 times the allergy pills production rate in factory 1 plus the total operating time of factory 2 times the allergy pills production rate in factory 2 should be at least the minimum number of allergy pills needed

The total operating time of factory 1 times the fever reducing pills production rate in factory 1 plus the total operating time of factory 2 times the fever reducing pills production rate in factory 2 should be at least the minimum number of fever reducing pills needed

The total operating time of factory 1 times the rare compound usage rate in factory 1 plus the total operating time of factory 2 times the rare compound usage rate in factory 2 should not exceed the total available units of rare compound

At least ManualToAutomaticRatio times the number of patients processed by the automatic machine must have their blood pressure taken by the manual machine

At least MinPatientsAutomatic patients must be processed by the automatic machine

Total time taken by both the automatic and manual machines must not exceed TotalTime

The total amount of medicine delivered to the throat by all servings must not exceed MaxMedicineThroat

The total amount of medicine delivered to the lungs by all servings must be at least MinMedicineLungs

The amount of mint used for all demonstrations should not exceed AvailableMint

The amount of active ingredient used for all demonstrations should not exceed AvailableActive

The amount of black tar produced by all demonstrations should not exceed MaxBlackTar

The baby must consume AppleCarrotConsumptionRatio times as many apple flavored servings as carrot flavored servings

The baby must eat at least MinCarrotServings servings of carrot flavored baby food

The total folate consumed from both apple and carrot flavored servings must not exceed MaxFolate units

Total hours each factory runs multiplied by its base gel requirement should not exceed the available base gel

Total production of each product across all factories should meet or exceed the minimum production requirement for that product

The total units of fat from burgers and pizza must be at least the RequiredFat

The total calories from burgers and pizza must be at least the RequiredCalories

The number of slices of pizza must be at least twice the number of burgers

Minimize the total cholesterol intake from the burgers and pizza slices consumed

UnitCShots * NumBatchesShots + UnitCPills * NumBatchesPills <= AvailableC

UnitDShots * NumBatchesShots + UnitDPills * NumBatchesPills <= AvailableD

The total time taken by all patients must not exceed TotalOpenTime

At least MinElectronicRegularRatio times as many patients should have their temperature checked by the electronic thermometer than by the regular thermometer

At least MinRegularPatients patients should have their temperature checked by the regular thermometer

The total units of medicine delivered to the heart by Machine 1 and Machine 2 should not exceed MaxMedicineHeart

The total units of medicine delivered to the brain by Machine 1 and Machine 2 should be at least MinMedicineBrain

The total count of ramen meals cannot exceed MaxRamenPercentage times NumMeals

The total amount of metal used for producing cheap and expensive boxes should not exceed the total amount of metal available

The total amount of acid used for producing cheap and expensive boxes should not exceed the total amount of acid available

The total heat given off by the cheap and expensive boxes should not exceed the maximum allowed heat

The total calories consumed per week from almonds and cashews must be at least MinimumCalories

The total protein consumed per week from almonds and cashews must be at least MinimumProtein

The number of servings of almonds consumed per week must be at least MinimumAlmondCashewRatio times the number of servings of cashews consumed per week

Water used for LiquidSanitizer plus water used for FoamSanitizer should not exceed TotalWater

Alcohol used for LiquidSanitizer plus alcohol used for FoamSanitizer should not exceed TotalAlcohol

The number of FoamSanitizer made must be greater than the number of LiquidSanitizer

At most MaxSleepMedicine units of sleeping medicine should be delivered

At least MinLegsMedicine units of medicine should be delivered to the legs

The total units of painkiller medicine used in both daytime and nighttime pills should not exceed TotalPainkillerMedicine

At least MinPercentageDaytimePills percent of the total pills must be daytime pills

The number of daytime and nighttime pills must be non-negative integers

The total calories from cheesecake and caramel cake slices should not exceed MaxCalories

The number of cheesecake slices should be at least MinCheesecakeRatio times the number of caramel cake slices

The number of caramel cake slices should be at least MinCaramelSlices

The total units of vitamin C from oranges and grapefruits must be at least MinVitC

The total units of vitamin A from oranges and grapefruits must be at least MinVitA

The number of oranges consumed must be at least 2 times the number of grapefruits consumed

The dose received by the skin from Beam 1 and Beam 2 combined should not exceed MaxDoseSkin units: DoseBeam1Skin * TimeBeam1 + DoseBeam2Skin * TimeBeam2 ≤ MaxDoseSkin

The dose received by the tumor from Beam 1 and Beam 2 combined should be at least MinDoseTumor units: DoseBeam1Tumor * TimeBeam1 + DoseBeam2Tumor * TimeBeam2 ≥ MinDoseTumor

The time for which Beam 1 is used should be non-negative: TimeBeam1 ≥ 0

The time for which Beam 2 is used should be non-negative: TimeBeam2 ≥ 0

The total grams of protein from the meals must be at least MinProtein

The total grams of carbs from the meals must be at least MinCarbs

The proportion of turkey dinners to the total number of meals must not exceed MaxPercentTurkeyDinner

The total units of red liquid used in both experiments should not exceed the total available units of red liquid

The total units of blue liquid used in both experiments should not exceed the total available units of blue liquid

The total units of smelly gas produced by both experiments should not exceed the maximum allowable units of smelly gas

The total hours of Process1 multiplied by Material1 plus the total hours of Process2 multiplied by Material2 must not exceed AvailableMaterial

The total units of pain killers produced by Process1 plus the total units of pain killers produced by Process2 must be at least MinPainKillers

The total units of sleeping pills produced by Process1 plus the total units of sleeping pills produced by Process2 must be at least MinSleepingPills

The total units of antioxidants from blueberries and strawberries must be at least MinAntioxidants units

The total units of minerals from blueberries and strawberries must be at least MinMinerals units

The number of packs of strawberries must be at least MinRatioStrawberriesBlueberries times the number of packs of blueberries

The total number of boxes transported by vans and trucks must be at least MinBoxes

The total cost of van trips and truck trips must not exceed MaxBudget

The number of van trips must be greater than the number of truck trips

The number of small bottles used must be at most MaxSmallBottles

The number of large bottles used must be at most MaxLargeBottles

The number of small bottles used must be at least twice the number of large bottles used

The total number of bottles used must not exceed MaxTotalBottles

The number of large bottles used must be at least MinLargeBottles

The total number of containers transported by ships and planes must be at least MinContainers

The number of ship trips made should be at least MinShipPercentage fraction of the total number of trips

The total number of ducks taken to shore by boats and canoes should be at least MinDucksToShore

At least MinCanoeTripPercentage percent of the total trips should be by canoe

The total number of soda bottles delivered by old and new vans should be at least MinBottles

The total number of employees used by small teams and large teams should not exceed the total number of employees available

The number of small teams should be at least SmallTeamsFactor times the number of large teams

The total number of employees used by small teams is EmployeesPerSmallTeam multiplied by the number of small teams

The total number of employees used by large teams is EmployeesPerLargeTeam multiplied by the number of large teams

Total number of treats used by otters and dolphins should not exceed TotalTreats

The number of trucks used should be at most MaxTruckContainerRatio times the number of containers used

At least MinOilRequired units of oil need to be sent to the port

The total number of students transported by all hired buses should be at least MinStudents

The fraction of large buses hired should not exceed MaxLargeBusPercentage of the total number of hired buses

The total number of patties shipped by trucks and vans must be at least MinPatties

The total cost of shipping patties using trucks and vans must not exceed MaxBudget

The number of trucks used should not exceed the number of vans used

The number of small bouquets transported should not exceed MaxSmallBouquets

The number of large bouquets transported should not exceed MaxLargeBouquets

The total number of bouquets transported (small + large) should not exceed MaxTotalBouquets

The number of large bouquets transported should be at least MinLargeBouquets

The number of small bouquets transported should be at least SmallLargeRatio times the number of large bouquets transported

The total number of gifts delivered must be at least MinGiftsRequired

The number of trips made by the new company must not exceed MaxNewCompanyTrips

At least MinOldCompanyTripPercentage of all trips must be made by the old company

The number of gifts delivered by the new company per trip times the number of trips by the new company plus the number of gifts delivered by the old company per trip times the number of trips by the old company must be at least MinGiftsRequired

The percentage of trips made by the old company must be at least MinOldCompanyTripPercentage times the total number of trips

The total number of monkeys transported by bus and car should equal TotalMonkeys

At least MinCarTripPercentage percent of the trips should be by car

The number of monkeys transported by bus per trip multiplied by the number of bus trips plus the number of monkeys transported by car per trip multiplied by the number of car trips should equal TotalMonkeys

The total number of cows transported by helicopters and trucks must equal NumCows

The charge required by the total number of bikes and scooters used should not exceed the TotalCharge available

The number of bikes used should not exceed MaxPercentageBikes percentage of the total number of vehicles used

The number of large planes must be less than the number of small planes

The total number of cars delivered by large and small planes must be at least TotalCars

Number of small wagons must be at least MinRatioSmallToLarge times the number of large wagons

Total ore transported by small and large wagons combined must be at least TotalOre units

The total number of visitors transported by scooters and rickshaws must be at least MinVisitors

The proportion of rickshaws used compared to the total number of vehicles must not exceed MaxRickshawProportion

The number of packages delivered by camels and horses together must be at least MinPackages

The total food required by the camels and horses together must not exceed TotalFood

The number of horses used must not exceed the number of camels used

Total number of trips by regular and speed boats must be sufficient to deliver TotalMail pieces of mail

Total number of regular boat trips must not exceed MaxRegularTrips

At least MinSpeedBoatTripPercentage percent of the total trips must be made by speed boats

Pieces of mail delivered by regular boats plus pieces of mail delivered by speed boats must equal TotalMail

The total number of chicken transported by buses and cars must equal TotalChicken

The number of trips by bus times BusCapacity plus the number of trips by car times CarCapacity must equal TotalChicken

The number of trips by car must be at least MinCarTripPercentage of the total number of trips

The total transportation time is the sum of (number of trips by bus times BusTimePerTrip) and (number of trips by car times CarTimePerTrip) and must be minimized

The total number of children transported by vans and minibuses must be at least NumChildren

The total weight of corn carried by tractors and cars must be at least MinCornWeight

The number of cars used must be at least CarToTractorRatio times the number of tractors used

The number of cars used must not exceed MaxPercentageCars of the total number of vehicles used

The total number of people transported by bikes and cars must be at least MinPeople

The number of cars used multiplied by PeoplePerCar plus the number of bikes used multiplied by PeoplePerBike should be at least MinPeople

Number of small kegs used must be less than or equal to MaxSmallKegs

Number of large kegs used must be less than or equal to MaxLargeKegs

Total number of kegs used must be less than or equal to MaxTotalKegs

Number of large kegs used must be greater than or equal to MinLargeKegs

Ratio of small kegs to large kegs must be at least MinSmallToLargeRatio

The number of submarine trips must be less than or equal to MaxSubmarineTrips

At least MinBoatTripPercentage percent of the trips must be made by boat

The number of submarine trips times SubmarineCapacity plus the number of boat trips times BoatCapacity must be at least TotalMail

The total number of trips (submarine trips plus boat trips) times MinBoatTripPercentage (as a percentage) must be less than or equal to the number of boat trips

The number of helicopter trips must not exceed MaxHelicopterTrips

At least MinCarTripPercentage percent of the total trips must be by car

The total number of fish transported by helicopter and car combined must be at least MinFishRequired

The total number of visitors transported by hot air balloons and gondola lifts should be at least MinVisitors

The number of hot air balloon rides should not exceed MaxBalloons

The number of medium sized carts must be MediumToLargeCartRatio times the number of large sized carts

The total number of horses used by medium and large sized carts must not exceed NumHorses

The number of medium sized carts must be at least MinMediumCarts

The total number of small bone treats used by labradors and golden retrievers combined should not exceed the total number of small bone treats available.

The number of golden retrievers used should be at least the minimum number of golden retrievers required.

The number of labradors used should not exceed the maximum proportion of labradors out of the total number of dogs.

The number of cows cannot exceed MaxCowsToElephantsRatio times the number of elephants

The total number of bricks carried by cows and elephants must be at least MinBricks

The number of light rail trips must be at least LightRailToFerryRatio times the number of ferry trips

The total number of boxes of corn sent must be at least MinBoxes

The total number of boxes of corn sent by ferry trips plus the total number of boxes of corn sent by light rail trips must be at least MinBoxes

The total transported volume (VolumeHPTT * NumHPTT + VolumeLHT * NumLHT) must be at least MinVolume

The total cost (CostHPTT * NumHPTT + CostLHT * NumLHT) must not exceed Budget

The number of high pressure tube trailer trips (NumHPTT) must be less than the number of liquefied hydrogen tanker trips (NumLHT)

Number of small crates must be at least MinSmallCratesToLargeCratesRatio times the number of large crates

Total number of crates (small + large) must not exceed MaxTotalCrates

The total number of patients transported by helicopters and buses should be at least MinPatientsToTransport

At least MinHelicopterTripPercentage percent of the total trips should be made by helicopters

The total number of employees transported by cars and buses should be at least NumEmployees

The number of buses used should be less than or equal to MaxBuses

The number of employees transported by cars should be a non-negative integer

The number of employees transported by buses should be a non-negative integer

PeopleForSmallTruck * NumSmallTrucks + PeopleForLargeTruck * NumLargeTrucks <= TotalPeople

The total number of treats used by carrier pigeons and owls should not exceed the number of treats available (NumTreats)

The number of treats used by carrier pigeons is equal to the number of carrier pigeons multiplied by CarrierPigeonTreats

The number of treats used by owls is equal to the number of owls multiplied by OwlTreats

The number of small canoes used has to be at least CanoeToDieselRatio times the number of diesel boats used

The total number of fish transported by small canoes and diesel boats must be at least MinFish

The total calories from hamburgers and chicken wraps must be at least MinCalories

The total protein from hamburgers and chicken wraps must be at least MinProtein

The total carbs from hamburgers and chicken wraps must be at least MinCarbs

The total assembly time required for all products should not exceed the available assembly time

The total sanding time required for all products should not exceed the available sanding time

The total calcium intake from glasses of milk and plates of vegetables should be at least MinCalcium units

The total iron intake from glasses of milk and plates of vegetables should be at least MinIron units

The total units of iron from the pills of supplement A and supplement B should be at least MinIronRequirement

The total units of calcium from the pills of supplement A and supplement B should be at least MinCalciumRequirement

The total amount of money bet on all sports should not exceed TotalMoney

The average probability of losing money weighted by the amount bet on each sport should not exceed MaxAvgProbLose

The total inspection time for all washing machines and freezers should not exceed the available inspection time

The total fixing time for all washing machines and freezers should not exceed the available fixing time

The total amount of batter used for crepe cakes, sponge cakes, and birthday cakes should not exceed TotalBatter

The total amount of milk used for crepe cakes, sponge cakes, and birthday cakes should not exceed TotalMilk

The number of sponge cakes made should be a non-negative integer

The number of birthday cakes made should be a non-negative integer

The amount of cat paw snacks used in the first mix plus the amount of cat paw snacks used in the second mix should not exceed AvailableCatPawSnacks

The amount of gold shark snacks used in the first mix plus the amount of gold shark snacks used in the second mix should not exceed AvailableGoldSharkSnacks

The percentage of cat paw snacks in the first mix plus the percentage of gold shark snacks in the first mix should equal 1

The percentage of cat paw snacks in the second mix plus the percentage of gold shark snacks in the second mix should equal 1

Number of personal licenses plus number of commercial licenses should not exceed MaxLicenses

Total cost of generating personal licenses and commercial licenses should not exceed BudgetLimit

The number of wide pipes can be at most WidePipeFraction times the number of narrow pipes

At least MinWaterTransport units of water need to be transported every minute

The number of vitamin D pills taken must be at least MinVitaminDPills

The number of calcium pills taken must be greater than the number of vitamin D pills

The total effectiveness time sum of all pills taken should be minimized

There must be at least ThrowingToClimbingRatio times as many throwing games as climbing games

The total cost in prizes per hour for both types of games must not exceed MaxPrizeCost

The total number of participants accommodated by circular and rectangular tables must be at least MinParticipants

The total number of poster boards supported by circular and rectangular tables must be at least MinPosterBoards

The total space occupied by circular and rectangular tables must not exceed TotalAvailableSpace

The total daily throughput must be at least MinimumDailyDemand gallons

The total number of technicians required must not exceed TotalTechnicians

The number of high-volume pipes must not exceed MaxHighVolumePipePercentage times the total number of pipes

There must be TechnicianToResearcherRatio times as many ultrasound technician shifts as graduate researcher shifts

The total hours worked by all technicians and researchers combined must be at least TotalRequiredHours

The total payment to all technicians and researchers combined must not exceed TotalBudget

The total number of shifts (bike shifts + scooter shifts) must equal TotalShifts

The total energy used (EnergyPerBikeShift * bike shifts + EnergyPerScooterShift * scooter shifts) must not exceed TotalEnergy

The total number of orders delivered (OrdersPerBikeShift * bike shifts + OrdersPerScooterShift * scooter shifts) must be at least MinOrders

The number of shifts on scooters must be at least MinScooterShifts

The total audience members brought in by pop and R&B concerts must be at least MinAudienceRequired

The total practice days required by pop and R&B concerts must not exceed MaxPracticeDays

The number of R&B concerts must not exceed MaxRnBConcertRate times the total number of concerts

The dough used for large plus the dough used for medium pizzas must be at least MinDough units

The toppings used for large plus the toppings used for medium pizzas must be at least MinToppings units

The number of large pizzas must be at least 2 times the number of medium pizzas

At least MinCartPercentage of delivery shifts must be by the first method

There must be at least MinHandServers servers delivering by the second method

Total number of customer interactions per hour must be equal to TotalInteractions

The total food waste generated by both meals should not exceed MaxFoodWaste

The total wrapping waste generated by both meals should not exceed MaxWrappingWaste

Total metal cut by chop saws and steel cutters should be at least equal to the total metal cutting requirement per day

Total waste generated by chop saws and steel cutters should not exceed the maximum allowed waste per day

At least MinRatioJugToPackets times the number of jugs must be used as sets of small packets

At least MinSetsSmallPackets sets of small packets should be filled

The total volume of jam used by sets of small packets and jugs should not exceed TotalVolumeJam

Number of fruit bowls should not exceed MaxFruitBowlPercentage of the total meals

The total number of customers brought in by the retail stores and factory outlets should be at least MinCustomersPerDay

The total number of employees required to run the retail stores and factory outlets should not exceed MaxEmployees

The total volume of soda bottled every day should be at least MinTotalVolume ml

The number of cans must be at least MinRatioCansToBottles times the number of glass bottles

Total cost of trips should not exceed the budget: FreightCost * number of freight trips + AirCost * number of air trips <= Budget

Total tons transported must be at least TotalTons: TonsPerFreight * number of freight trips + TonsPerAir * number of air trips >= TotalTons

At least MinAirPercentage of the total tons must be transported by air: TonsPerAir * number of air trips >= MinAirPercentage * TotalTons

There must be at least MinFreightTrips trips through freight: number of freight trips >= MinFreightTrips

The total number of people transported by large and small mobile production units must be at least TotalPeople.

The total number of small mobile production units must be at least MinSmallUnits.

The number of large mobile production units must be at least MinPercentageLargeUnits percent of the total number of mobile production units.

The total number of parking spots required for large and small mobile production units must be minimized.

The number of small boxes should be at least MinSmallToLargeRatio times the number of large boxes

The total number of masks distributed should be at least MinMasksRequired

The number of automatic electric jacks must be less than MaxElectricJacks

The total electricity used by all automatic electric jacks must be less than or equal to MaxElectricity

The total gas used by all gas-powered jacks must be less than or equal to MaxGas

The number of trams must be at least MinTramToTrainRatio times the number of trains

The total number of people transported per hour by trains and trams must be at least MinTotalPeople

Minimize the total number of transportation units (trains + trams)

The total units of platinum used by all catalysts should not exceed the total available units of platinum

The total units of palladium used by all catalysts should not exceed the total available units of palladium

Total water used by both processes should not exceed the maximum available units of water

Total pollution produced by both processes should not exceed the maximum allowable units of pollution

The total transport capacity of all installed escalators and elevators must be at least RequiredCapacity

The number of escalators must be at least EscalatorElevatorRatio times the number of elevators

Total units of fabric produced by method A and method B must be at least MinFabric

Total units of plastic produced by method A and method B must be at least MinPlastic

Total units of special element consumed by method A and method B must not exceed TotalSpecialElement

Total number of motorboat trips must be at most MaxMotorboatTrips

At least MinKayakTripProportion of the total trips should be by kayak

The total number of images annotated by both companies must be at least equal to TotalImages

The number of images annotated by the specialized company must be at least one-third of TotalImages

The total time spent by the specialized company is determined by the number of images they annotate divided by SpecializedRate

The total time spent by the common company is determined by the number of images they annotate divided by CommonRate

The cost incurred from the specialized company is the total time spent by the specialized company multiplied by SpecializedCost

The cost incurred from the common company is the total time spent by the common company multiplied by CommonCost

The sum of images annotated by the specialized company and the common company must be exactly equal to TotalImages

The total number of guests transported should be at least MinGuests

The total number of golf carts used should not exceed MaxGolfCartPercentage of the total number of carts used

The number of banana chocolate smoothies made must be more than the number of acai berry smoothies made

At least MinAcaiFraction of the total smoothies made must be acai berry smoothies

The total units of acai berries used for making acai berry smoothies must not exceed TotalAcaiBerries

The total units of banana chocolate used for making banana chocolate smoothies must not exceed TotalBananaChocolate

The total materials used for basketballs and footballs should not exceed TotalMaterial units

The total hours worked on basketballs and footballs should not exceed TotalHours hours

The number of basketballs produced should be at least BasketballToFootballRatio times the number of footballs produced

The number of footballs produced should be at least MinFootballs

The total preparation time for scheduled in-vivo and ex-vivo experiments should not exceed MaxPrepTime minutes

The total execution time for scheduled in-vivo and ex-vivo experiments should not exceed MaxExecTime minutes

The number of protein bars must be equal to RatioBarsToSmoothies times the number of smoothies

The total calories consumed (from both smoothies and protein bars) should not exceed MaxCalories

Total hours run by all factories should satisfy the minimum required production of anti-itch injections

Total hours run by all factories should satisfy the minimum required production of topical cream

Total plastic units used by all factories running for the determined hours should not exceed the total available plastic units

The total time required for preparing batches should not exceed the TotalAvailableMinutes

The total materials required for preparing batches should not exceed the TotalAvailableMaterials

The total number of batches of medication patches and anti-biotic creams should not exceed MaxTotalBatches

The number of anti-biotic creams should be at least MinCreamPatchRatio times the number of medication patches

The total number of units delivered by camel caravans and desert trucks should be at least equal to the total goods required, i.e., GoodsCamelCaravan * number of camel caravans + GoodsDesertTruck * number of desert trucks >= TotalGoods

The number of camel caravans should be greater than or equal to the number of desert trucks

The amount of chemical A in the mixer must be at least MinUnitsA units

The total amount of chemicals in the mixer (A and B combined) must be at least MinTotalChemicals units

The proportion of chemical A to chemical B in the mixer must be at most MaxProportionAtoB

The number of cups of spinach must exceed the number of cups of soybeans

The total units of fibre from spinach and soybeans must be at least the TotalFibreRequired units of fibre

The total mg of iron from spinach and soybeans must be at least the TotalIronRequired mg of iron

The number of mechanical keyboards produced should be MechanicalToStandardRatio times the number of standard keyboards produced

The number of standard keyboards produced should be at least MinStandardKeyboards

The total units of ice cream used for matcha ice cream should not exceed the available units of ice cream

The total units of water used for orange sorbet should not exceed the available units of water

At least MinMatchaPercentage percent of the total dessert orders must be matcha ice cream

The number of orders of orange sorbet must exceed the number of orders of matcha ice cream

The ball with center y and radius r must be entirely contained within the set P, meaning for each inequality (a_i^T * x <= b_i), the farthest point from y in the direction of a_i should also satisfy the inequality, i.e., a_i^T * y + r * ||a_i||_2 <= b_i for all i from 1 to M

The objective is to minimize the maximum deviation of all observed y values from the values predicted by the line y = bx + a

For each k from 1 to NumObs, let d_k be the absolute deviation y_k - (bx_k + a), which must be minimized in terms of maximum deviation

a (intercept) and b (slope) are the decision variables for the linear relationship

d_k (deviation) must be non-negative for each k from 1 to NumObs

The quadratic curve parameters a, b, and c should minimize the sum of absolute deviations of the observed y values from the values predicted by the quadratic relationship y = c * x^2 + b * x + a

Total power generated in any period t should meet or exceed the demand Demand[t]

A generator of type k should not generate less than MinLevel[k] if it is running

A generator of type k should not generate more than MaxLevel[k] if it is running

Only NumGenerator[k] generators of type k are available to operate in any period

Non-negativity constraint for the power levels generated by each generator

Binary constraint that indicates whether each generator is running in each period

The total production time for each product on each machine must not exceed the available time of that machine

The total size of the packed items should not exceed the capacity of the knapsack

Each individual must receive at least demand_m units of nutrient m per day

The total weight of the alloy produced should equal AlloyQuantity.

For each metal m, the combined quantity of that metal in the final alloy should equal Target[m].

The ratios of metals in each alloy k, ratio[k, m], should range between 0 and 1.0.

The total time taken for all spare parts k in shop s should not exceed the capacity of shop s

The production using regular conditions in any month should not exceed MaxRegularAmount

The total production in any month (regular production + overtime production) must meet or exceed the demand for that month

Each month's inventory carried over to the next month should account for the production in that month, the demand for that month, and the storage cost per unit

Each employee works n_working_days consecutive days followed by n_resting_days consecutive days off

The sum of all employees' scheduled work days on any given day must meet or exceed the required num_n for that day

The number of containers sent from depot i to all ports should not exceed the number of containers available at depot i

The number of containers received at port j should meet the container requirement at port j

Containers must be transported in pairs since each barge can carry exactly two containers

The total time spent on any machine, for all parts produced, should not exceed the machine's availability per month: for each machine m, the sum of time required to produce the decided number of batches of each part on that machine must be less than or equal to available_m

The number of batches of each part produced must be at least the minimum number of batches required to fulfill the contract: for each part p, the number of batches must be greater than or equal to min_batches_p

Sum of clicks from ad types for young individuals should be at least GoalYoung thousand clicks

Sum of clicks from ad types for old individuals should be at least GoalOld thousand clicks

Sum of unique clicks from ad types for young individuals should be at least GoalUniqueYoung thousand unique clicks

Sum of unique clicks from ad types for old individuals should be at least GoalUniqueOld thousand unique clicks

Clicks purchased for each ad type should not exceed the maximum allowable clicks for that ad type

The total number of baskets produced each week using both regular and overtime labor must meet the weekly demand.

The total hours of labor used in each week for regular labor should not exceed the available regular labor hours for that week.

The total hours of labor used in each week for overtime labor should not exceed the available overtime labor hours for that week.

The number of baskets assembled each week should be equal to or greater than the total demand from the start of the season up to that week minus the number of baskets carried over from previous weeks.

The number of baskets carried over from one week to the next is equal to the number of baskets left unsold after meeting the weekly demand.

No baskets are available in the initial week, so the initial inventory is zero.

At the end of the season, any unsold baskets have a salvage value and do not incur holding costs.

The time used on each machine m for all parts p must not exceed the available time available_{m} hours per month, except Machine M and Machine M-1 can share total combined availability of available_{M} + available_{M-1} hours per month

The number of batches of each part p produced must be at least min_batches_{p} to fulfill the contract

The total time used by machine m should not exceed available_m hours plus the max_extra_m hours that can be purchased

The number of batches of part p produced each month should be at least min_batches_p

Each required language must be covered by at least one selected translator

The total number of projects assigned to each consultant must not exceed the limit of MaxProjectsPerConsultant

If a consultant is assigned at least one project, the fixed cost for hiring that consultant must be incurred

Each cluster must be treated with one type of intervention consistently, either isolation or scanning

Each intervention can be processed either through the central system or through distributed nodes

The total central processing time for all interventions must not exceed the maximum allotted central hours (CentralMaxHours)

The total distributed processing time for all interventions must not exceed the maximum allotted distributed hours (DistributedMaxHours)

$investmentCondos \\geq MinPercentageCondos \\times (investmentCondos + investmentDetachedHouses)$

$x \times \text{BaconRegular} + y \times \text{BaconSpecial} \leq \text{TotalBacon}$

$CostRadioAd \\times numRadioAds + CostSocialMediaAd \\times numSocialMediaAds \\leq TotalBudget$

$GoldPerLongCable \cdot longCables + GoldPerShortCable \cdot shortCables \leq TotalGold$

$\\text{shortCables} \\geq \\text{ShortCableMultiplier} \\cdot \\text{longCables}$

$CostZTube \times numAdsZTube + CostSoorchle \times numAdsSoorchle + CostWassa \times numAdsWassa \leq Budget$

$\text{numAdsWassa} \leq \text{MaxWassaProportion} \times (\text{numAdsWassa} + \text{numAdsZTube} + \text{numAdsSoorchle})$

$\\text{numAdsZTube} \\geq \\text{MinFractionZTube} \\times (\\text{numAdsZTube} + \\text{numAdsSoorchle} + \\text{numAdsWassa})$

$numChairs \times StainPerChair + numDressers \times StainPerDresser \leq TotalStain$

$\\text{numChairs} \\times \\text{OakWoodPerChair} + \\text{numDressers} \\times \\text{OakWoodPerDresser} \\leq \\text{TotalOakWood}$

$ProteinFeedA \times xA + ProteinFeedB \times xB \geq MinProtein$

$ShapingTimeThinJar \\times numThinJars + ShapingTimeStubbyJar \\times numStubbyJars \\leq TotalShapingTime$

$numThinJars \times BakingTimeThinJar + numStubbyJars \times BakingTimeStubbyJar \leq TotalBakingTime$

$ApplesInBananaHatersPackage \times quantityBananaHatersPackages + ApplesInComboPackage \times quantityComboPackages \leq StockApples$

$BananasInComboPackage \\times quantityComboPackages \\leq StockBananas$

$quantityBananaHatersPackages \cdot GrapesInBananaHatersPackage + quantityComboPackages \cdot GrapesInComboPackage \leq StockGrapes$

$breadProduced \cdot MixerTimeBread + cookiesProduced \cdot MixerTimeCookies \leq MaxMixerHours$

$breadProduced \\cdot OvenTimeBread + cookiesProduced \\cdot OvenTimeCookies \\leq MaxOvenHours$

$HeatingTimeRegular \cdot numRegularPanes + HeatingTimeTempered \cdot numTemperedPanes \leq MaxHeatingTime$

$\text{numRegularPanes} \times \text{CoolingTimeRegular} + \text{numTemperedPanes} \times \text{CoolingTimeTempered} \leq \text{MaxCoolingTime}$

$\\text{WoodRequirement}[0] \\times \\text{numElephants} + \\text{WoodRequirement}[1] \\times \\text{numTigers} \\leq \\text{AvailableWood}$

$PlasticRequirement[0] \\times numElephants + PlasticRequirement[1] \\times numTigers \\leq AvailablePlastic$

$investmentApt \geq MinRatioInvestAptToTown \times investmentTown$

$\\text{Z1InZodiac} \\cdot x + \\text{Z1InSunny} \\cdot y \\geq \\text{MinZ1}$

$WoodPerTrain \cdot trains + WoodPerPlane \cdot planes \leq TotalWood$

$PaintPerTrain \cdot trains + PaintPerPlane \cdot planes \leq TotalPaint$

$numPhones \cdot SpacePhone + numLaptops \cdot SpaceLaptop \leq TotalSpace$

$\\text{numPhones} \\times \\text{CostPhone} + \\text{numLaptops} \\times \\text{CostLaptop} \\leq \\text{MaxSpending}$

$\\text{numLaptops} \\geq \\frac{\\text{MinLaptopPercentage} \\times \\text{numPhones}}{1 - \\text{MinLaptopPercentage}}$

$RegularCandyInRegularMix \\times regularMix + RegularCandyInSourMix \\times sourMix \\leq AvailableRegularCandy$

$SourCandyInRegularMix \\times regularMix \\leq AvailableSourCandy$

$RegularCandyInSourMix \times sourMix \leq AvailableRegularCandy$

$\\text{regularMix} \\times \\text{RegularCandyInRegularMix} + \\text{sourMix} \\times \\text{RegularCandyInSourMix} \\leq \\text{AvailableRegularCandy}$

$\\text{regularMix} \\times \\text{SourCandyInRegularMix} + \\text{sourMix} \\times \\text{SourCandyInSourMix} \\leq \\text{AvailableSourCandy}$

$PhoneCost \cdot phoneSpaceAllocation + LaptopCost \cdot laptopSpaceAllocation \leq TotalBudget$

$\\frac{\\text{phoneSpaceAllocation}}{\\text{PhoneSpace}} \\times \\text{PhoneLabor} + \\frac{\\text{laptopSpaceAllocation}}{\\text{LaptopSpace}} \\times \\text{LaptopLabor} \\leq \\text{TotalLabor}$

$\\text{numPremium} + \\text{numRegular} \\leq \\text{MaxDesktops}$

$CostPremium \\times numPremium + CostRegular \\times numRegular \\leq MaxBudget$

$CostLongDesk \\times numLongDesks + CostShortDesk \\times numShortDesks \\leq Budget$

$numLongDesks \\times SpaceLongDesk + numShortDesks \\times SpaceShortDesk \\leq MaxSpace$

$\\text{NitrousOxidePerKgC} \\times \\text{amountFertilizerC} + \\text{NitrousOxidePerKgY} \\times \\text{amountFertilizerY} \\geq \\text{MinRequiredNitrousOxide}$

$\\text{VitaminMixPerKgC} \\cdot \\text{amountFertilizerC} + \\text{VitaminMixPerKgY} \\cdot \\text{amountFertilizerY} \\geq \\text{MinRequiredVitaminMix}$

$\\forall i \\in [0, \\text{NumSubstances}), \\sum_{j=0}^{\\text{NumOilTypes}-1} \\text{Composition}[i, j] \\times \\text{production}[j] \\leq \\text{TotalSubstances}[i]$

$\\forall j \\in \\{1, \\ldots, \\text{NumOilTypes}\\}, \\text{production}[j] \\geq 0$

$MangoCost \cdot mangosPurchased + GuavaCost \cdot guavasPurchased \leq TotalBudget$

$VitaminAInA \\times xA + VitaminAInB \\times xB \\geq MinVitaminA$

$xA \\times VitaminDInA + xB \\times VitaminDInB \\geq MinVitaminD$

$chocolateGallons \cdot TimePerGallonChocolate + vanillaGallons \cdot TimePerGallonVanilla \leq TotalAvailableHours$

$WorkersPerGallonChocolate \cdot chocolateGallons + WorkersPerGallonVanilla \cdot vanillaGallons \geq MinWorkers$

$workersChocolate \geq WorkersPerGallonChocolate \times chocolateGallons$

$workersVanilla \geq WorkersPerGallonVanilla \times vanillaGallons$

$acresCucumbers \leq MaxCucumberToPotatoRatio \times acresPotatoes$

$\\forall i \\in \\{1, \\ldots, \\text{NumBerryTypes}\\}, \\text{OldFarmOutput}[i] \\cdot \\text{daysOldFarm} + \\text{NewFarmOutput}[i] \\cdot \\text{daysNewFarm} \\geq \\text{DeliveryRequirement}[i]$

$\\text{Minimize } OldFarmCost \\times daysOldFarm + NewFarmCost \\times daysNewFarm$

$CaloriesRice \times servingsRice + CaloriesKebab \times servingsKebab \geq MinCalories$

$ProteinRice \times servingsRice + ProteinKebab \times servingsKebab \geq MinProtein$

$DesignTimePerScooter \times scooters + DesignTimePerBike \times bikes \leq TotalDesignHours$

$scooters \\times EngineeringTimePerScooter + bikes \\times EngineeringTimePerBike \\leq TotalEngineeringHours$

$VitaminsPerServingVegetable \times numServingsVegetable + VitaminsPerServingFruit \times numServingsFruit \geq MinVitamins$

$\\text{MineralsPerServingVegetable} \\times \\text{numServingsVegetable} + \\text{MineralsPerServingFruit} \\times \\text{numServingsFruit} \\geq \\text{MinMinerals}$

$bagelsBatches \times OvenTimeBagels + croissantsBatches \times OvenTimeCroissants \leq MaxOvenTime$

$ChefTimeBagels \times bagelsBatches + ChefTimeCroissants \times croissantsBatches \leq MaxChefTime$

$ProductionBlackFactory1 \times hoursFactory1 + ProductionBlackFactory2 \times hoursFactory2 \geq DemandBlack$

$ProductionWhiteFactory1 \cdot hoursFactory1 + ProductionWhiteFactory2 \cdot hoursFactory2 \geq DemandWhite$

$\text{acresTurnips} \times \text{WateringPerAcreTurnips} + \text{acresPumpkins} \times \text{WateringPerAcrePumpkins} \leq \text{TotalWateringTime}$

$PesticidePerAcreTurnips \times acresTurnips + PesticidePerAcrePumpkins \times acresPumpkins \leq TotalPesticideBudget$

$numStaff \\times StaffHoursPerShift + numSubstitute \\times SubstituteHoursPerShift \\geq RequiredHours$

$numStaff \\times StaffWagePerShift + numSubstitute \\times SubstituteWagePerShift \\leq TotalBudget$

$\\text{numStaff} \\geq 0 \\quad \\text{and} \\quad \\text{numSubstitute} \\geq 0$

$highIntensityDrills \\times HighIntensityDrillCapacity + lowIntensityDrills \\times LowIntensityDrillCapacity \\geq DailyGemProcessingRequirement$

$highIntensityDrills \cdot HighIntensityDrillWaterRequirement + lowIntensityDrills \cdot LowIntensityDrillWaterRequirement \leq TotalAvailableWater$

$highIntensityDrills \leq MaxFractionHighIntensityDrills \cdot (highIntensityDrills + lowIntensityDrills)$

$\\text{highIntensityDrills} \\geq 0 \\\\, \\text{lowIntensityDrills} \\geq 0$

$ProductionDineIn \times dineInPlaces + ProductionFoodTruck \times foodTrucks \geq MinProduction$

$EmployeesDineIn \\times dineInPlaces + EmployeesFoodTruck \\times foodTrucks \\leq TotalEmployees$

$GoldRequiredA \times processesRunA + GoldRequiredB \times processesRunB \leq TotalGoldAvailable$

$processesRunA \\times WiresRequiredA + processesRunB \\times WiresRequiredB \\leq TotalWiresAvailable$

$\\text{numMiterSaws} \\times \\text{MiterCutCapacity} + \\text{numCircularSaws} \\times \\text{CircularCutCapacity} \\geq \\text{MinPlanks}$

$\\text{numMiterSaws} \\times \\text{MiterSawdust} + \\text{numCircularSaws} \\times \\text{CircularSawdust} \\leq \\text{MaxSawdust}$

$WindrowerFuelPerAcre \times acresWindrower + HarvesterFuelPerAcre \times acresHarvester \leq TotalFuel$

$(WindrowerMethanePerAcre \times acresWindrower) + (HarvesterMethanePerAcre \times acresHarvester) \leq MaxMethane$

$\\text{MeatPerWrap} \\times \\text{numberOfWraps} + \\text{MeatPerPlatter} \\times \\text{numberOfPlatters} \\geq \\text{MinMeat}$

$RicePerWrap \cdot numberOfWraps + RicePerPlatter \cdot numberOfPlatters \geq MinRice$

$\\text{numberOfWraps} \\geq \\text{WrapsToPlattersRatio} \\times \\text{numberOfPlatters}$

$numCashMachines \times CashMachineCapacity + numCardMachines \times CardMachineCapacity \geq MinTotalPeopleRequirement$

$CashPaperConsumption \\times numCashMachines + CardPaperConsumption \\times numCardMachines \\leq MaxTotalPaperLimit$

$glassJars \\times HoneyPerGlassJar + plasticJars \\times HoneyPerPlasticJar \\leq TotalHoney$

$\\text{oldFurnaces} \\leq \\text{MaxProportionOld} \\cdot (\\text{newFurnaces} + \\text{oldFurnaces})$

$newFurnaces \cdot HeatingCapacityNew + oldFurnaces \cdot HeatingCapacityOld \geq TotalApartments$

$ElectricityConsumptionNew \\times newFurnaces + ElectricityConsumptionOld \\times oldFurnaces \\leq TotalElectricity$

$ManagersRural \cdot numRuralFactories + ManagersUrban \cdot numUrbanFactories \leq TotalManagers$

$\\text{numRuralFactories} \\times \\text{PhonesPerDayRural} + \\text{numUrbanFactories} \\times \\text{PhonesPerDayUrban} \\geq \\text{MinPhonesPerDay}$

$numCounterTopMachines \times CounterTopHeatOutput + numFridgeMachines \times FridgeHeatOutput \leq MaxDailyHeatOutput$

$numCounterTopMachines \\times CounterTopCapacity + numFridgeMachines \\times FridgeCapacity \\geq MinDailyProduction$

$\\sum_{i=1}^{NumContainers} \\text{WaterRequired}[i] \\cdot \\text{numContainersUsed}[i] \\leq \\text{TotalWater}$

$\\sum_{i=1}^{\\text{NumContainers}} \\text{numContainersUsed}[i] \\times \\text{PillRequired}[i] \\leq \\text{TotalPill}$

$DrillBitsSmall \cdot acresSmall + DrillBitsLarge \cdot acresLarge \leq TotalDrillBits$

$acresSmall \times PollutionSmall + acresLarge \times PollutionLarge \leq MaxPollution$

$numAlmond \times ButterAlmond + numPistachio \times ButterPistachio \leq TotalButter$

$numAlmond \\times FlourAlmond + numPistachio \\times FlourPistachio \\leq TotalFlour$

$PattiesPerGasGrill \times gasGrills + PattiesPerElectricGrill \times electricGrills \geq MinPattiesPerMinute$

$gasGrills \times OilPerGasGrill + electricGrills \times OilPerElectricGrill \leq MaxOilPerMinute$

$VolumeKids \times xKids + VolumeAdult \times xAdult \leq TotalCoughSyrup$

$numBrownies \cdot (1 - MinBrowniesRatio) \geq MinBrowniesRatio \cdot numLemonSquares$

$ChocolateMixPerBrownie \times numBrownies \leq TotalChocolateMix$

$numLemonSquares \times LemonMixPerLemonSquare \leq TotalLemonMix$

$nurses \times NurseHoursPerShift + pharmacists \times PharmacistHoursPerShift \geq TotalLaborHoursRequired$

$(nurses \times NursePaymentPerShift) + (pharmacists \times PharmacistPaymentPerShift) \leq TotalBudget$

$WashingCapacityTopLoading \\times numTopLoading + WashingCapacityFrontLoading \\times numFrontLoading \\geq MinWashingCapacityRequired$

$numTopLoading \times EnergyConsumptionTopLoading + numFrontLoading \times EnergyConsumptionFrontLoading \leq MaxEnergyAvailable$

$\\text{numTopLoading} \\leq \\text{MaxPercentageTopLoading} \\times (\\text{numTopLoading} + \\text{numFrontLoading})$

$regularVans \times RegularVanPollution + hybridVans \times HybridVanPollution \leq MaxPollution$

$\\text{regularVans} \\times \\text{RegularVanCapacity} + \\text{hybridVans} \\times \\text{HybridVanCapacity} \\geq \\text{MinPackages}$

$\\text{WoodWithCatalyst} \\times \\text{numCatalystProcessUses} + \\text{WoodWithoutCatalyst} \\times \\text{numNonCatalystProcessUses} \\leq \\text{TotalWood}$

$OxygenWithCatalyst \\times numCatalystProcessUses + OxygenWithoutCatalyst \\times numNonCatalystProcessUses \\leq TotalOxygen$

$FuelTraditional \times acresTraditional + FuelModern \times acresModern \leq TotalFuel$

$WasteTraditional \cdot acresTraditional + WasteModern \cdot acresModern \leq MaxWaste$

$StampingRate[0] \cdot numModel1 + StampingRate[1] \cdot numModel2 \geq MinLetters$

$\\text{GlueConsumption}[0] \\times \\text{numModel1} + \\text{GlueConsumption}[1] \\times \\text{numModel2} \\leq \\text{MaxGlue}$

$BananasPerSmallCrate \cdot smallCrates + BananasPerLargeCrate \cdot largeCrates \leq TotalBananas$

$\text{NumPeanutButterCrepes} > \text{NumChocolateCrepes}$
```

Let's put it all together correctly:

```json
{
    "FORMULATION": "$\text{NumPeanutButterCrepes} > \text{NumChocolateCrepes}$

$ChocolateSpreadForChocolateCrepe \times numChocolateCrepes \leq TotalChocolateSpread$

$\text{numPeanutButterCrepes} \times \text{PeanutButterSpreadForPeanutButterCrepe} \leq \text{TotalPeanutButterSpread}$

$regularShifts \\times HoursRegularShift + emergencyShifts \\times HoursEmergencyShift \\geq MinHoursRequired$

$EarningsRegular \times regularShifts + EarningsEmergency \times emergencyShifts \leq Budget$

$numManual \leq MaxManualPercentage \times (numManual + numMotionActivated)$

$DeliverRate[0] \\cdot numManual + DeliverRate[1] \\cdot numMotionActivated \\geq MinDeliverRate$

$EnergyConsumption[0] \\times numManual + EnergyConsumption[1] \\times numMotionActivated \\leq MaxEnergyConsumption$

$numSmallShops \\times HotDogsPerDaySmallShop + numLargeShops \\times HotDogsPerDayLargeShop \\geq MinHotDogsPerDay$

$WorkersPerSmallShop \\times numSmallShops + WorkersPerLargeShop \\times numLargeShops \\leq TotalWorkers$

$x4 \times LuggageCapacity4Wheeler + x3 \times LuggageCapacity3Wheeler \geq MinLuggage$

$(\\text{Pollution4Wheeler} \\times x4) + (\\text{Pollution3Wheeler} \\times x3) \\leq \\text{MaxPollution}$

$BaitRequiredNet \times acresNet + BaitRequiredLine \times acresLine \leq TotalBait$

$PainNet \cdot acresNet + PainLine \cdot acresLine \leq MaxPain$

$mochas \times CoffeePowderPerMocha + regulars \times CoffeePowderPerRegular \leq TotalCoffeePowder$

$MilkPerMocha \cdot mochas + MilkPerRegular \cdot regulars \leq TotalMilk$

$\\text{mochas} \\geq \\text{MochaToRegularRatio} \\times \\text{regulars}$

$ManualCutRate \\times numManualSlicers + AutomaticCutRate \\times numAutomaticSlicers \\geq MinSlices$

$ManualGreaseRate \times numManualSlicers + AutomaticGreaseRate \times numAutomaticSlicers \leq MaxGrease$

$numberOfPlasticBottles \geq MinPlasticToGlassRatio \times numberOfGlassBottles$

$\\text{numberOfGlassBottles} \\times \\text{GlassBottleCapacity} + \\text{numberOfPlasticBottles} \\times \\text{PlasticBottleCapacity} \\leq \\text{TotalWater}$

$\text{chickenCurryBowls} \times \text{ChickenMeatPerChickenCurry} \leq \text{TotalChickenMeat}$

$100 \cdot chickenCurryBowls \geq MinChickenCurryPercentage \cdot (goatCurryBowls + chickenCurryBowls)$

$numFullTimeWorkers \\times HoursPerShiftFullTime + numPartTimeWorkers \\times HoursPerShiftPartTime \\geq TotalLaborHours$

$numFullTimeWorkers \\cdot WageFullTime + numPartTimeWorkers \\cdot WagePartTime \\leq TotalBudget$

$ItemsMachineA \\times numberMachinesA + ItemsMachineB \\times numberMachinesB \\geq MinItemsPerDay$

$\\text{numberMachinesA} \\times \\text{ElectricityMachineA} + \\text{numberMachinesB} \\times \\text{ElectricityMachineB} \\leq \\text{MaxElectricityPerDay}$

$\text{CountA} \times \text{HydrogenProductionA} + \text{CountB} \times \text{HydrogenProductionB} \geq \text{MinHydrogenRequirement}$

$\\text{countA} \\cdot \\text{PollutantProductionA} + \\text{countB} \\cdot \\text{PollutantProductionB} \\leq \\text{MaxPollutantAllowance}$

$\\text{TreatedWaterA} \\times xA + \\text{TreatedWaterB} \\times xB \\leq \\text{TotalTreatedWater}$

$\text{heapLeachingLand} \times \text{HeapLeachingMachines} + \text{vatLeachingLand} \times \text{VatLeachingMachines} \leq \text{TotalMachines}$

$HeapLeachingWastewater \times heapLeachingLand + VatLeachingWastewater \times vatLeachingLand \leq MaxWastewater$

$(CocoaPerMilkChocolate \times milkChocolateBars) + (CocoaPerDarkChocolate \times darkChocolateBars) \leq TotalCocoa$

$milkChocolateBars \times MilkPerMilkChocolate + darkChocolateBars \times MilkPerDarkChocolate \leq TotalMilk$

$milkChocolateBars \geq MilkToDarkRatio \times darkChocolateBars$

$\\text{minimize} \\quad totalPrinters = premiumPrinters + regularPrinters$

$\\text{premiumPrinters} \\times \\text{PagesPerMinutePremium} + \\text{regularPrinters} \\times \\text{PagesPerMinuteRegular} \\geq \\text{MinPagesPerMinute}$

$premiumPrinters \\times InkPerMinutePremium + regularPrinters \\times InkPerMinuteRegular \\leq MaxInkPerMinute$

$VolumeVintage \times vintageBottles + VolumeRegular \times regularBottles \leq TotalVolume$

$\\text{regularBottles} \\geq \\text{MinRatioRegularToVintage} \\times \\text{vintageBottles}$

$PainMedPill1 \times x1 + PainMedPill2 \times x2 \leq MaxPainMed$

$AnxietyMedPill1 \\cdot x1 + AnxietyMedPill2 \\cdot x2 \\geq MinAnxietyMed$

$ProteinAlpha \times numAlphaBottles + ProteinOmega \times numOmegaBottles \geq MinProtein$

$CaloriesAlpha \\cdot numAlphaBottles + CaloriesOmega \\cdot numOmegaBottles \\geq MinCalories$

$(1 - \text{MaxOmegaProportion}) \cdot \text{numOmegaBottles} \leq \text{MaxOmegaProportion} \cdot \text{numAlphaBottles}$

$\\sum_{i=1}^{\\text{NumBeakers}} \\text{FlourRequired}[i] \\times \\text{beakers}[i] \\leq \\text{TotalFlour}$

$\\sum_{i=1}^{\\text{NumBeakers}} \\text{LiquidRequired}[i] \\cdot \\text{beakers}[i] \\leq \\text{TotalLiquid}$

$\\sum_{i=1}^{NumBeakers} WasteProduced[i] \\cdot beakers[i] \\leq MaxWaste$

$\\text{ToothMedicationSmall} \\cdot \\text{smallBones} + \\text{ToothMedicationLarge} \\cdot \\text{largeBones} \\leq \\text{TotalToothMedication}$

$\\text{smallBones} \\geq \\frac{\\text{MinPercentageSmallBones}}{100 - \\text{MinPercentageSmallBones}} \\times \\text{largeBones}$

$appleJuiceBoxes \geq MinAppleToOrangeRatio \times orangeJuiceBoxes$

$VitCOrangeJuice \times orangeJuiceBoxes + VitCAppleJuice \times appleJuiceBoxes \leq MaxVitaminC$

$cleansingChemical + odorRemovingChemical \\geq MinTotalChemicals$

$cleansingChemical \leq MaxCleansingToOdorRatio \times odorRemovingChemical$

$AllergyPillsPerHourFactory1 \times t1 + AllergyPillsPerHourFactory2 \times t2 \geq MinAllergyPills$

$\\text{FeverPillsPerHourFactory1} \\times t1 + \\text{FeverPillsPerHourFactory2} \\times t2 \\geq \\text{MinFeverPills}$

$CompoundPerHourFactory1 \cdot t1 + CompoundPerHourFactory2 \cdot t2 \leq TotalCompoundAvailable$

$patientsManual \geq ManualToAutomaticRatio \times patientsAutomatic$

$patientsAutomatic \times TimeAutomatic + patientsManual \times TimeManual \leq TotalTime$

$\\sum_{i=1}^{\\text{NumSyrups}} \\text{numServings}[i] \\times \\text{MedicineToThroat}[i] \\leq \\text{MaxMedicineThroat}$

$\\sum_{i=1}^{\\text{NumSyrups}} \\text{numServings}[i] \\cdot \\text{MedicineToLungs}[i] \\geq \\text{MinMedicineLungs}$

$\\sum_{i=1}^{NumDemos} \\text{MintRequirement}[i] \\cdot x[i] \\leq \\text{AvailableMint}$

$\\sum_{i=1}^{NumDemos} ActiveRequirement[i] \\times x[i] \\leq AvailableActive$

$\\sum_{i=1}^{NumDemos} BlackTarProduced[i] \\cdot x[i] \\leq MaxBlackTar$

$appleServings = AppleCarrotConsumptionRatio \times carrotServings$

$FolatePerAppleServing \\times appleServings + FolatePerCarrotServing \\times carrotServings \\leq MaxFolate$

$\\sum_{f=1}^{\\text{NumFactories}} (\\text{BaseGelRequirement}[f] \\cdot \\text{hoursRun}[f]) \\leq \\text{AvailableBaseGel}$

$\\forall p \\in \\{1, \\ldots, \\text{NumProducts}\\}, \\sum_{f=1}^{\\text{NumFactories}} \\text{ProductionRate}[f][p] \\cdot \\text{hoursRun}[f] \\geq \\text{MinProduction}[p]$

$\\text{hoursRun}[f] \\geq 0 \\quad \\forall f \\in \\{1, 2, \\ldots, \\text{NumFactories}\\}$

$FatPerBurger \\cdot burgers + FatPerPizza \\cdot pizzaSlices \\geq RequiredFat$

$CaloriesPerBurger \cdot burgers + CaloriesPerPizza \cdot pizzaSlices \geq RequiredCalories$

$\\text{UnitCShots} \\times \\text{numBatchesShots} + \\text{UnitCPills} \\times \\text{numBatchesPills} \\leq \\text{AvailableC}$

$UnitDShots \times numBatchesShots + UnitDPills \times numBatchesPills \leq AvailableD$

$numElectronicPatients \times ElectronicThermometerTime + numRegularPatients \times RegularThermometerTime \leq TotalOpenTime$

$\\text{numElectronicPatients} \\geq \\text{MinElectronicRegularRatio} \\times \\text{numRegularPatients}$

$MedicineDeliveryRateMachine1Heart \times timeMachine1 + MedicineDeliveryRateMachine2Heart \times timeMachine2 \leq MaxMedicineHeart$

$timeMachine1 \\times MedicineDeliveryRateMachine1Brain + timeMachine2 \\times MedicineDeliveryRateMachine2Brain \\geq MinMedicineBrain$

$\\text{numRamenMeals} + \\text{numFriesMeals} = \\text{NumMeals}$

$\\text{numRamenMeals} \\leq \\text{MaxRamenPercentage} \\times \\text{NumMeals}$

$RamenCalories \cdot numRamenMeals + FriesCalories \cdot numFriesMeals \geq MinCalories$

$numRamenMeals \\times RamenProtein + numFriesMeals \\times FriesProtein \\geq MinProtein$

$\\text{Minimize } \\text{numRamenMeals} \\times \\text{RamenSodium} + \\text{numFriesMeals} \\times \\text{FriesSodium}$

$MetalCheapBox \\cdot numberCheapBoxes + MetalExpensiveBox \\cdot numberExpensiveBoxes \\leq TotalMetal$

$AcidCheapBox \times numberCheapBoxes + AcidExpensiveBox \times numberExpensiveBoxes \leq TotalAcid$

$numberCheapBoxes \times HeatCheapBox + numberExpensiveBoxes \times HeatExpensiveBox \leq MaxHeat$

$CaloriesAlmonds \\cdot numberOfAlmondServings + CaloriesCashews \\cdot numberOfCashewServings \\geq MinimumCalories$

$\\text{ProteinAlmonds} \\times \\text{numberOfAlmondServings} + \\text{ProteinCashews} \\times \\text{numberOfCashewServings} \\geq \\text{MinimumProtein}$

$numberOfAlmondServings \geq MinimumAlmondCashewRatio \times numberOfCashewServings$

$WaterLiquidSanitizer \cdot liquidSanitizer + WaterFoamSanitizer \cdot foamSanitizer \leq TotalWater$

$AlcoholLiquidSanitizer \times liquidSanitizer + AlcoholFoamSanitizer \times foamSanitizer \leq TotalAlcohol$

$SleepMedicinePainKiller1 \cdot dosesPainKiller1 + SleepMedicinePainKiller2 \cdot dosesPainKiller2 \leq MaxSleepMedicine$

$LegsMedicinePainKiller1 \cdot dosesPainKiller1 + LegsMedicinePainKiller2 \cdot dosesPainKiller2 \geq MinLegsMedicine$

$daytimePills \cdot PainkillerInDaytimePill + nighttimePills \cdot PainkillerInNighttimePill \leq TotalPainkillerMedicine$

$\\left( 1 - \\frac{MinPercentageDaytimePills}{100} \\right) \\times daytimePills \\geq \\left( \\frac{MinPercentageDaytimePills}{100} \\right) \\times nighttimePills$

$\text{daytimePills} \geq 0 \quad \text{and} \quad \text{nighttimePills} \geq 0$

$CaloriesCheesecake \\times slicesCheesecake + CaloriesCaramelCake \\times slicesCaramelCake \\leq MaxCalories$

$slicesCheesecake \geq MinCheesecakeRatio \times slicesCaramelCake$

$\text{VitCOranges} \times \text{numOranges} + \text{VitCGrapefruit} \times \text{numGrapefruits} \geq \text{MinVitC}$

$VitAOranges \times numOranges + VitAGrapefruit \times numGrapefruits \geq MinVitA$

$DoseBeam1Skin * TimeBeam1 + DoseBeam2Skin * TimeBeam2 \leq MaxDoseSkin$

$DoseBeam1Tumor \cdot TimeBeam1 + DoseBeam2Tumor \cdot TimeBeam2 \geq MinDoseTumor$

$ProteinTurkeyDinner \cdot turkeyDinnerCount + ProteinTunaSandwich \cdot tunaSaladSandwichCount \geq MinProtein$

$CarbsTurkeyDinner \cdot turkeyDinnerCount + CarbsTunaSandwich \cdot tunaSaladSandwichCount \geq MinCarbs$

$turkeyDinnerCount \\leq MaxPercentTurkeyDinner \\times (turkeyDinnerCount + tunaSaladSandwichCount)$

$RedLiquidExp1 \cdot exp1 + RedLiquidExp2 \cdot exp2 \leq TotalRedLiquid$

$\\text{BlueLiquidExp1} \\times \\text{exp1} + \\text{BlueLiquidExp2} \\times \\text{exp2} \\leq \\text{TotalBlueLiquid}$

$exp1 \times SmellyGasExp1 + exp2 \times SmellyGasExp2 \leq MaxSmellyGas$

$hoursProcess1 \times Material1 + hoursProcess2 \times Material2 \leq AvailableMaterial$

$PainKillers1 \times hoursProcess1 + PainKillers2 \times hoursProcess2 \geq MinPainKillers$

$SleepingPills1 \cdot hoursProcess1 + SleepingPills2 \cdot hoursProcess2 \geq MinSleepingPills$

$\text{AntioxidantsBlueberries} \times \text{packsBlueberries} + \text{AntioxidantsStrawberries} \times \text{packsStrawberries} \geq \text{MinAntioxidants}$

$\text{MineralsBlueberries} \times \text{packsBlueberries} + \text{MineralsStrawberries} \times \text{packsStrawberries} \geq \text{MinMinerals}$

$packsStrawberries \geq MinRatioStrawberriesBlueberries \cdot packsBlueberries$

$\\text{vanTrips} \\times \\text{BoxesPerVan} + \\text{truckTrips} \\times \\text{BoxesPerTruck} \\geq \\text{MinBoxes}$

$CostPerVan \\times vanTrips + CostPerTruck \\times truckTrips \\le MaxBudget$

$shipTrips \\times ShipCapacity + planeTrips \\times PlaneCapacity \\geq MinContainers$

$shipTrips \geq MinShipPercentage \cdot (shipTrips + planeTrips)$

$boatTrips \times BoatDucksPerTrip + canoeTrips \times CanoeDucksPerTrip \geq MinDucksToShore$

$\\text{canoeTrips} \\geq \\frac{\\frac{\\text{MinCanoeTripPercentage}}{100} \\times \\text{boatTrips}}{1 - \\frac{\\text{MinCanoeTripPercentage}}{100}}$

$oldVans \times CapacityOldVan + newVans \times CapacityNewVan \geq MinBottles$

$\\text{EmployeesPerSmallTeam} \\times \\text{numSmallTeams} + \\text{EmployeesPerLargeTeam} \\times \\text{numLargeTeams} \\leq \\text{TotalEmployees}$

$\\text{numSmallTeams} \\geq \\text{SmallTeamsFactor} \\cdot \\text{numLargeTeams}$

$\\text{totalEmployeesUsedByLargeTeams} = \\text{EmployeesPerLargeTeam} \\times \\text{numLargeTeams}$

$\\text{otters} \\leq \\text{MaxOtterPercentage} \\times (\\text{otters} + \\text{dolphins})$

$\\text{otters} \\times \\text{OtterTreats} + \\text{dolphins} \\times \\text{DolphinTreats} \\leq \\text{TotalTreats}$

$numContainers \times ContainerCapacity + numTrucks \times TruckCapacity \geq MinOilRequired$

$\text{SmallBusCapacity} \cdot \text{numSmallBuses} + \text{LargeBusCapacity} \cdot \text{numLargeBuses} \geq \text{MinStudents}$

$\\text{numLargeBuses} \\leq \\frac{\\text{MaxLargeBusPercentage} \\times \\text{numSmallBuses}}{1 - \\text{MaxLargeBusPercentage}}$

$\\text{TruckCapacity} \\times x + \\text{VanCapacity} \\times y \\geq \\text{MinPatties}$

$smallBouquetsTransported + largeBouquetsTransported \leq MaxTotalBouquets$

$smallBouquetsTransported \geq SmallLargeRatio \times largeBouquetsTransported$

$NewCompanyGiftCapacity \times newCompanyTrips + OldCompanyGiftCapacity \times oldCompanyTrips \geq MinGiftsRequired$

$oldCompanyTrips \\geq \\frac{MinOldCompanyTripPercentage \\times newCompanyTrips}{1 - MinOldCompanyTripPercentage}$

$\\text{NewCompanyGiftCapacity} \\times \\text{newCompanyTrips} + \\text{OldCompanyGiftCapacity} \\times \\text{oldCompanyTrips} \\geq \\text{MinGiftsRequired}$

$\text{oldCompanyTrips} \geq \frac{\text{MinOldCompanyTripPercentage} \times \text{newCompanyTrips}}{1 - \text{MinOldCompanyTripPercentage}}$

$busTrips \times MonkeysPerBusTrip + carTrips \times MonkeysPerCarTrip = TotalMonkeys$

$\text{carTrips} \geq \frac{\text{MinCarTripPercentage}}{1 - \text{MinCarTripPercentage}} \times \text{busTrips}$

$\\text{MonkeysPerBusTrip} \\times \\text{busTrips} + \\text{MonkeysPerCarTrip} \\times \\text{carTrips} = \\text{TotalMonkeys}$

$helicopterTrips \cdot HelicopterCapacity + truckTrips \cdot TruckCapacity = NumCows$

$ChargePerBike \cdot bikes + ChargePerScooter \cdot scooters \leq TotalCharge$

$bikes \\leq \\frac{MaxPercentageBikes}{100} \\times (bikes + scooters)$

$CarsPerLargePlane \times largePlanes + CarsPerSmallPlane \times smallPlanes \geq TotalCars$

$SmallWagonCapacity \\cdot smallWagons + LargeWagonCapacity \\cdot largeWagons \\geq TotalOre$

$scooters \\times ScooterCapacity + rickshaws \\times RickshawCapacity \\geq MinVisitors$

$rickshaws \leq \\frac{MaxRickshawProportion}{1 - MaxRickshawProportion} \\times scooters$

$PackagesPerCamel \cdot camels + PackagesPerHorse \cdot horses \geq MinPackages$

$FoodPerCamel \times camels + FoodPerHorse \times horses \leq TotalFood$

$MailCapacityRegular \\times regularTrips + MailCapacitySpeed \\times speedTrips \\geq TotalMail$

$speedTrips \geq \\frac{MinSpeedBoatTripPercentage}{100} \times (regularTrips + speedTrips)$

$MailCapacityRegular \\times regularTrips + MailCapacitySpeed \\times speedTrips = TotalMail$

$busTrips \\cdot BusCapacity + carTrips \\cdot CarCapacity = TotalChicken$

$busTrips \\times BusCapacity + carTrips \\times CarCapacity = TotalChicken$

$carTrips \\geq \\frac{MinCarTripPercentage \\times busTrips}{1 - MinCarTripPercentage}$

$\\text{Minimize}: \\quad \\text{busTrips} \\times \\text{BusTimePerTrip} + \\text{carTrips} \\times \\text{CarTimePerTrip}$

$\\text{numVans} \\times \\text{VanCapacity} + \\text{numMinibuses} \\times \\text{MinibusCapacity} \\geq \\text{NumChildren}$

$\text{numMinibuses} \times \text{MinibusCapacity} \geq \text{NumChildren} - \text{numVans} \times \text{VanCapacity}$

$\\text{TractorCapacity} \\cdot \\text{numTractors} + \\text{CarCapacity} \\cdot \\text{numCars} \\geq \\text{MinCornWeight}$

$\\text{numCars} \\geq \\text{CarToTractorRatio} \\times \\text{numTractors}$

$\\text{bikes} \\times \\text{PeoplePerBike} + \\text{cars} \\times \\text{PeoplePerCar} \\geq \\text{MinPeople}$

$cars \times PeoplePerCar + bikes \times PeoplePerBike \geq MinPeople$

$submarineTrips \cdot SubmarineCapacity + boatTrips \cdot BoatCapacity \geq TotalMail$

$(100 - MinBoatTripPercentage) \times boatTrips \geq MinBoatTripPercentage \times submarineTrips$

$submarineTrips \times SubmarineCapacity + boatTrips \times BoatCapacity \geq TotalMail$

$(submarineTrips + boatTrips) \times MinBoatTripPercentage \leq boatTrips$

$(100 - \text{MinCarTripPercentage}) \times \text{carTrips} \geq \text{MinCarTripPercentage} \times \text{helicopterTrips}$

$helicopterTrips \times HelicopterFishCapacity + carTrips \times CarFishCapacity \geq MinFishRequired$

$(numBalloons \times BalloonCapacity) + (numGondolas \times GondolaCapacity) \geq MinVisitors$

$\text{HorsesPerMediumCart} \times \text{numMediumCarts} + \text{HorsesPerLargeCart} \times \text{numLargeCarts} \leq \text{NumHorses}$

$\\text{TreatsPerLabrador} \\times \\text{numLabradors} + \\text{TreatsPerGoldenRetriever} \\times \\text{numGoldenRetrievers} \\leq \\text{TotalTreats}$

$numLabradors \\leq MaxLabradorPercentage \\times (numLabradors + numGoldenRetrievers)$

$numCows \\times BricksPerCow + numElephants \\times BricksPerElephant \\geq MinBricks$

$\text{FerryCapacity} \times \text{ferryTrips} + \text{LightRailCapacity} \times \text{lightRailTrips} \geq \text{MinBoxes}$

$FerryCapacity \times ferryTrips + LightRailCapacity \times lightRailTrips \geq MinBoxes$

$\\text{VolumeHPTT} \\cdot \\text{NumHPTT} + \\text{VolumeLHT} \\cdot \\text{NumLHT} \\geq \\text{MinVolume}$

$numSmallCrates \\geq MinSmallCratesToLargeCratesRatio \\times numLargeCrates$

$helicopterTrips \times HelicopterCapacity + busTrips \times BusCapacity \geq MinPatientsToTransport$

$100 \times helicopterTrips \geq MinHelicopterTripPercentage \times (helicopterTrips + busTrips)$

$CarCapacity \\times numCars + BusCapacity \\times numBuses \\geq NumEmployees$

$PeopleForSmallTruck \cdot NumSmallTrucks + PeopleForLargeTruck \cdot NumLargeTrucks \\leq TotalPeople$

$CarrierPigeonTreats \\cdot numCarrierPigeons + OwlTreats \\cdot numOwls \\leq NumTreats$

$\text{numOwls} \leq \text{MaxOwlPercentage} * (\text{numCarrierPigeons} + \text{numOwls})$

$\\text{numTreatsUsedByOwls} = \\text{numOwls} \\times \\text{OwlTreats}$

$\\text{numCanoes} \\times \\text{CapacityCanoe} + \\text{numDieselBoats} \\times \\text{CapacityDieselBoat} \\geq \\text{MinFish}$

$HamburgerCalories \times numHamburgers + ChickenWrapCalories \times numChickenWraps \geq MinCalories$

$numHamburgers \times HamburgerProtein + numChickenWraps \times ChickenWrapProtein \geq MinProtein$

$numHamburgers \times HamburgerCarbs + numChickenWraps \times ChickenWrapCarbs \geq MinCarbs$

$\\sum_{i=1}^{\\text{NumProducts}} \\text{AssemblyTime}[i] \\times \\text{unitsProduced}[i] \\leq \\text{AvailableAssemblyTime}$

$\\sum_{i=1}^{\\text{NumProducts}} \\text{unitsProduced}[i] \\times \\text{SandingTime}[i] \\leq \\text{AvailableSandingTime}$

$CalciumMilk \cdot glassesMilk + CalciumVegetables \cdot platesVegetables \geq MinCalcium$

$glassesMilk \\times IronMilk + platesVegetables \\times IronVegetables \\geq MinIron$

$\\text{minimize } CostNoodles \\cdot x + CostProteinBar \\cdot y$

$x \cdot CaloriesNoodles + y \cdot CaloriesProteinBar \geq MinCalories$

$\\text{ProteinNoodles} \\cdot x + \\text{ProteinProteinBar} \\cdot y \\geq \\text{MinProtein}$

$IronInSupplementA \cdot pillsSupplementA + IronInSupplementB \cdot pillsSupplementB \geq MinIronRequirement$

$CalciumInSupplementA \\cdot pillsSupplementA + CalciumInSupplementB \\cdot pillsSupplementB \\geq MinCalciumRequirement$

$\\frac{\\text{betBasketball} \\times \\text{ProbLoseBasketball} + \\text{betHorseRace} \\times \\text{ProbLoseHorseRace} + \\text{betSoccer} \\times \\text{ProbLoseSoccer}}{\\text{betBasketball} + \\text{betHorseRace} + \\text{betSoccer}} \\leq \\text{MaxAvgProbLose}$

$InspectionTimeWashingMachine \times numWashingMachines + InspectionTimeFreezer \times numFreezers \leq AvailableInspectionTime$

$\text{numWashingMachines} \times \text{FixingTimeWashingMachine} + \text{numFreezers} \times \text{FixingTimeFreezer} \leq \text{AvailableFixingTime}$

$\text{BatterCrepeCake} \times \text{crepeCakes} + \text{BatterSpongeCake} \times \text{spongeCakes} + \text{BatterBirthdayCake} \times \text{birthdayCakes} \leq \text{TotalBatter}$

$MilkCrepeCake \times crepeCakes + MilkSpongeCake \times spongeCakes + MilkBirthdayCake \times birthdayCakes \leq TotalMilk$

$PercentCatPawFirstMix \\times amountFirstMix + PercentCatPawSecondMix \\times amountSecondMix \\leq AvailableCatPawSnacks$

$PercentGoldSharkFirstMix \cdot amountFirstMix + PercentGoldSharkSecondMix \cdot amountSecondMix \leq AvailableGoldSharkSnacks$

$CostPersonalLicense \\times personalLicenses + CostCommercialLicense \\times commercialLicenses \\leq BudgetLimit$

$w \\times \text{WidePipeFlowRate} + n \\times \text{NarrowPipeFlowRate} \\geq \text{MinWaterTransport}$

$\\min \\, ( \\text{calciumPills} \\times \\text{CalciumTime} + \\text{vitaminDPills} \\times \\text{VitaminDTime})$

$throwingGames \\geq ThrowingToClimbingRatio \\times climbingGames$

$\\text{ThrowingGameCost} \\times \\text{throwingGames} + \\text{ClimbingGameCost} \\times \\text{climbingGames} \\leq \\text{MaxPrizeCost}$

$ParticipantsCircular \cdot numCircularTables + ParticipantsRectangular \cdot numRectangularTables \geq MinParticipants$

$\text{PosterBoardsCircular} \times \text{numCircularTables} + \text{PosterBoardsRectangular} \times \text{numRectangularTables} \geq \text{MinPosterBoards}$

$\text{numCircularTables} \cdot \text{SpaceCircularTable} + \text{numRectangularTables} \cdot \text{SpaceRectangularTable} \leq \text{TotalAvailableSpace}$

$x \\times \\text{HighVolumePipeThroughput} + y \\times \\text{LowVolumePipeThroughput} \\geq \\text{MinimumDailyDemand}$

$x \times TechniciansPerHighVolumePipe + y \times TechniciansPerLowVolumePipe \leq TotalTechnicians$

$technicianShifts \geq TechnicianToResearcherRatio \times researcherShifts$

$technicianShifts \\times TechnicianHoursPerShift + researcherShifts \\times ResearcherHoursPerShift \\geq TotalRequiredHours$

$\text{technicianShifts} \times \text{TechnicianPaymentPerShift} + \text{researcherShifts} \times \text{ResearcherPaymentPerShift} \leq \text{TotalBudget}$

$EnergyPerBikeShift \\times bikeShifts + EnergyPerScooterShift \\times scooterShifts \\leq TotalEnergy$

$OrdersPerBikeShift \\times bikeShifts + OrdersPerScooterShift \\times scooterShifts \\geq MinOrders$

$PopAudience \\times popConcerts + RnBAudience \\times rnbConcerts \\geq MinAudienceRequired$

$popConcerts \times PopPracticeDays + rnbConcerts \times RnBPracticeDays \leq MaxPracticeDays$

$(1 - \\text{MaxRnBConcertRate}) \\cdot \\text{rnbConcerts} \\leq \\text{MaxRnBConcertRate} \\cdot \\text{popConcerts}$

$DoughLargePizza \\times largePizzas + DoughMediumPizza \\times mediumPizzas \\geq MinDough$

$ToppingsLargePizza \\cdot largePizzas + ToppingsMediumPizza \\cdot mediumPizzas \\geq MinToppings$

$serversCart \geq MinCartPercentage \times (serversCart + serversHand)$

$serversCart \\times CartInteractions + serversHand \\times HandInteractions = TotalInteractions$

$numberOfOriginalMeals \times OriginalFoodWaste + numberOfExperimentalMeals \times ExperimentalFoodWaste \leq MaxFoodWaste$

$\\text{OriginalWrappingWaste} \\times \\text{numberOfOriginalMeals} + \\text{ExperimentalWrappingWaste} \\times \\text{numberOfExperimentalMeals} \\leq \\text{MaxWrappingWaste}$

$numChopSaws \times ChopSawCapacity + numSteelCutters \times SteelCutterCapacity \geq TotalMetalCuttingRequirement$

$numChopSaws \\times ChopSawWaste + numSteelCutters \\times SteelCutterWaste \\leq MaxAllowedWaste$

$\\text{numSetsSmallPackets} \\times \\text{VolumeSmallPackets} + \\text{numJugs} \\times \\text{VolumeJug} \\leq \\text{TotalVolumeJam}$

$s \cdot VitaminPerSalad + f \cdot VitaminPerFruitBowl \geq MinVitamin$

$s \times FibrePerSalad + f \times FibrePerFruitBowl \geq MinFibre$

$RetailStoreCustomers \cdot x + FactoryOutletCustomers \cdot y \geq MinCustomersPerDay$

$\\text{RetailStoreEmployees} \\times x + \\text{FactoryOutletEmployees} \\times y \\leq \\text{MaxEmployees}$

$numCans \times VolumeCan + numBottles \times VolumeBottle \geq MinTotalVolume$

$\\text{numCans} \\geq \\text{MinRatioCansToBottles} \\times \\text{numBottles}$

$FreightCost \\times freightTrips + AirCost \\times airTrips \\leq Budget$

$TonsPerFreight \\times freightTrips + TonsPerAir \\times airTrips \\geq TotalTons$

$TonsPerAir \times airTrips \geq MinAirPercentage \times TotalTons$

$\\text{numLargeUnits} \\cdot \\text{LargeUnitCapacity} + \\text{numSmallUnits} \\cdot \\text{SmallUnitCapacity} \\geq \\text{TotalPeople}$

$\\text{numLargeUnits} \\geq \\frac{\\text{MinPercentageLargeUnits}}{(1 - \\text{MinPercentageLargeUnits})} \\times \\text{numSmallUnits}$

$Z = \text{numLargeUnits} \times \text{LargeUnitParkingSpots} + \text{numSmallUnits} \times \text{SmallUnitParkingSpots}$

$smallBoxes \times MasksInSmallBox + largeBoxes \times MasksInLargeBox \geq MinMasksRequired$

$\text{numElectricJacks} \times \text{ElectricityPerHourElectric} \leq \text{MaxElectricity}$

$numberOfTrams \\geq MinTramToTrainRatio \\times numberOfTrains$

$numberOfTrains \times PeoplePerTrain + numberOfTrams \times PeoplePerTram \geq MinTotalPeople$

$PeoplePerTrain \\cdot numberOfTrains + PeoplePerTram \\cdot numberOfTrams \\geq MinTotalPeople$

$\sum_{i=1}^{NumCatalysts} PlatinumRequired[i] \cdot numCatalystsUsed[i] \leq TotalPlatinumAvailable$

$\\sum_{i=1}^{\\text{NumCatalysts}} \\text{numCatalystsUsed}[i] \\times \\text{PalladiumRequired}[i] \\leq \\text{TotalPalladiumAvailable}$

$numProcessJ \\times WaterUsedJ + numProcessP \\times WaterUsedP \\leq MaxWater$

$PollutionProducedJ \cdot numProcessJ + PollutionProducedP \cdot numProcessP \leq MaxPollution$

$EscalatorCapacity \\times n_e + ElevatorCapacity \\times n_v \\geq RequiredCapacity$

$FabricUnitsA \\cdot timeA + FabricUnitsB \\cdot timeB \\geq MinFabric$

$PlasticUnitsA \\cdot timeA + PlasticUnitsB \\cdot timeB \\geq MinPlastic$

$SpecialElementA \\cdot timeA + SpecialElementB \\cdot timeB \\leq TotalSpecialElement$

$FabricUnitsA \times timeA + FabricUnitsB \times timeB \geq MinFabric$

$PlasticUnitsA \\times timeA + PlasticUnitsB \\times timeB \\geq MinPlastic$

$SpecialElementA \cdot timeA + SpecialElementB \cdot timeB \leq TotalSpecialElement$

$KayakCapacity \\times kayakTrips + MotorboatCapacity \\times motorboatTrips \\geq MinLocals$

$\\text{kayakTrips} \\geq \\text{MinKayakTripProportion} \\cdot (\\text{kayakTrips} + \\text{motorboatTrips})$

$\\text{timeSpecialized} = \\frac{imagesSpecialized}{SpecializedRate}$

$\\text{numGolfCarts} \\times \\text{GuestsPerGolfCart} + \\text{numPullCarts} \\times \\text{GuestsPerPullCart} \\geq \\text{MinGuests}$

$\\text{numGolfCarts} \\leq \\text{MaxGolfCartPercentage} \\times (\\text{numGolfCarts} + \\text{numPullCarts})$

$(1 - \text{MinAcaiFraction}) \times \text{acaiSmoothies} \geq \text{MinAcaiFraction} \times \text{bananaSmoothies}$

$acaiSmoothies \times AcaiBerriesPerAcaiSmoothie \leq TotalAcaiBerries$

$bananaSmoothies \\times BananaChocolatePerBananaSmoothie \\leq TotalBananaChocolate$

$BasketballMaterial \times numBasketballs + FootballMaterial \times numFootballs \leq TotalMaterial$

$BasketballHours \\times numBasketballs + FootballHours \\times numFootballs \\leq TotalHours$

$numBasketballs \\geq BasketballToFootballRatio \\times numFootballs$

$PrepTimeInVivo \\times numInVivo + PrepTimeExVivo \\times numExVivo \\leq MaxPrepTime$

$\\text{numInVivo} \\times \\text{ExecTimeInVivo} + \\text{numExVivo} \\times \\text{ExecTimeExVivo} \\leq \\text{MaxExecTime}$

$canoerDeliveries \leq \\left( \\frac{CanoeProportion}{1 - CanoeProportion} \\right) \\times runnerDeliveries$

$runnerDeliveries \times RunnerTime + canoerDeliveries \times CanoerTime \leq TotalHours$

$CaloriesPerSmoothie \times numSmoothies + CaloriesPerBar \times numBars \leq MaxCalories$

$\\sum_{i=1}^{\\text{NumFactories}} \\text{ProductionInjections}[i] \\cdot \\text{hoursRun}[i] \\geq \\text{MinInjections}$

$\\sum_{i=1}^{\\text{NumFactories}} \\text{ProductionCream}[i] \\times \\text{hoursRun}[i] \\geq \\text{MinCream}$

$\\sum_{i=1}^{\\text{NumFactories}} \\text{PlasticRequired}[i] \\cdot \\text{hoursRun}[i] \\leq \\text{TotalPlastic}$

$\\text{TimePerMedPatch} \\times \\text{medPatchBatches} + \\text{TimePerCream} \\times \\text{creamBatches} \\leq \\text{TotalAvailableMinutes}$

$MaterialsPerMedPatch \\cdot medPatchBatches + MaterialsPerCream \\cdot creamBatches \\leq TotalAvailableMaterials$

$\text{creamBatches} \geq \text{MinCreamPatchRatio} \times \text{medPatchBatches}$

$GoodsCamelCaravan \cdot numberOfCamelCaravans + GoodsDesertTruck \cdot numberOfDesertTrucks \geq TotalGoods$

$\\text{numberOfCamelCaravans} \\geq \\text{numberOfDesertTrucks}$

$\\text{unitsA} \\leq \\text{MaxProportionAtoB} \\times \\text{unitsB}$

$FibreInSpinach \cdot cupsOfSpinach + FibreInSoybeans \cdot cupsOfSoybeans \geq TotalFibreRequired$

$cupsOfSpinach \times IronInSpinach + cupsOfSoybeans \times IronInSoybeans \geq TotalIronRequired$

$mechanicalKeyboards = MechanicalToStandardRatio \times standardKeyboards$

$PlasticPerMechanicalKeyboard \\times mechanicalKeyboards + PlasticPerStandardKeyboard \\times standardKeyboards \\leq TotalPlastic$

$SolderPerMechanicalKeyboard \\times mechanicalKeyboards + SolderPerStandardKeyboard \\times standardKeyboards \\leq TotalSolder$

$(1 - \text{MinMatchaPercentage}) \times \text{matchaOrders} \geq \text{MinMatchaPercentage} \times \text{orangeOrders}$

$a_i^T y + r \|a_i\|_2 \leq b_i, \forall i \in \{1, \ldots, M\}$

$\\forall k \\in \\{1, \\ldots, \\text{NumObs}\\}, y_k - (b \\cdot x_k + a) \\leq \\text{maxDeviation}$

$\\forall k \\in \\{1, \\ldots, \\text{NumObs}\\}, (b \\cdot x_k + a) - y_k \\leq \\text{maxDeviation}$

$\\forall k \in \{1, 2, \ldots, \\text{NumObs}\}, \\text{maxDeviation} \\ge |y_k - (bx_k + a)|$

$\\forall k \\in \{1, 2, \\ldots, \\text{NumObs}\}, d[k] = y[k] - (b \\cdot X[k] + a)$

$\\forall k \\in \{1, 2, \\ldots, \\text{NumObs}\}, d[k] = -y[k] + (b \\cdot X[k] + a)$

$\\forall k \\in \{1, 2, \\ldots, \\text{NumObs}\}, \\text{maxDeviation} \\ge |d[k]|$

$a, b \\text{ are the decision variables for the linear relationship}$

$\\forall t, \\sum_{k=1}^{K} \\text{powerGenerated}[k, t] \\geq \\text{Demand}[t]$

$\\forall k, \\forall t, \\text{powerGenerated}[k, t] \\leq \\text{MaxLevel}[k] \\cdot \\text{isRunning}[k, t]$

$\\forall k, \\forall t, \\text{powerGenerated}[k, t] \\geq \\text{MinLevel}[k] \\cdot \\text{isRunning}[k, t]$

$\\text{powerGenerated}[k, t] \\geq \\text{MinLevel}[k] \\cdot \\text{isRunning}[k, t]$

$\\text{powerGenerated}[k, t] \\leq \\text{MaxLevel}[k] \\times \\text{isRunning}[k, t]$

$\\sum_{k \in K} \\text{isRunning}[k,t] \\leq \\text{NumGenerator}[k], \quad \\forall t$

$\forall k \in \{1, \ldots, K\}, \forall t \in \{1, \ldots, T\}, powerGenerated[k, t] \geq 0$

$\\forall k \\in \\{1, \\ldots, K\\}, \\forall t \\in \\{1, \\ldots, T\\}, ~ minLevel_k \\cdot isRunning_{k,t} \\leq powerGenerated_{k,t} \\leq maxLevel_k \\cdot isRunning_{k,t}$

$\\forall m, \\sum_{k=1}^{\\text{NumProducts}} \\text{ProduceTime}_{k, m} \\cdot x_k \\leq \\text{AvailableTime}_m$

$x_k \\geq 0 \\quad \\forall k \\in \\{1, 2, \\ldots, \\text{NumProducts}\\}$

$\\sum_{k=1}^{\\text{NumItems}} \\text{Size}[k] \\cdot \\text{packItem}[k] \\leq C$

$\\forall m \\in \\{1, 2, \\ldots, M\\}, \\sum_{k=1}^{K} \\text{quantity}[k] \\cdot \\text{Nutrition}[k, m] \\geq \\text{Demand}[m]$

$\\forall n \\geq 1, \\text{stockLevel}[n] = \\text{stockLevel}[n-1] + \\text{buyAmount}[n] - \\text{sellAmount}[n]$

$\\text{stockLevel}[n] \\geq 0, \\forall n \\in \\{1, \\ldots, \\text{NumPeriods}\\}$

$\\text{buyAmount}[n] \\geq 0 \\quad \\forall n \\in \\{1, 2, \\ldots, \\text{NumPeriods}\\}$

$\forall n \in [0, \\text{NumPeriods} - 1], sellAmount[n] \ge 0$

$\\sum_{k=1}^{NumAlloys} \\text{amountOfAlloys}[k] = \\text{AlloyQuantity}$

$\\forall m \\in \\{1, \\ldots, \\text{NumMetals}\\}, \\quad \\sum_{k=1}^{\\text{NumAlloys}} \\text{amountOfAlloys}[k] \\times \\text{Ratio}[k, m] = \\text{Target}[m]$

$\forall k \in \{1, \ldots, \text{NumAlloys}\}, \text{amountOfAlloys}[k] \geq 0$

$0 \\leq \\text{Ratio}[k, m] \\leq 1 \\quad \\forall k \\in \\{1, \\ldots, \\text{NumAlloys}\\}, \\forall m \\in \\{1, \\ldots, \\text{NumMetals}\\}$

$\\forall s \\in S, \\sum_{k \\in K} \\text{Time}[k][s] \\cdot \\text{quantityK}[k] \\leq \\text{Capacity}[s]$

$\\forall t \\in \\{1, \\ldots, \\text{NumMonths}\\}, \\ \\text{regularProduction}[t] \\leq \\text{MaxRegularAmount}$

$\\forall t, regularProduction[t] + overtimeProduction[t] \\geq Demand[t]$

$\\forall t \\in \\{1, \ldots, NumMonths\\}, \\, inventory[t] = inventory[t-1] + regularProduction[t] + overtimeProduction[t] - demand[t] $

$\\forall t \\in \\{1, \ldots, NumMonths\\}, \\, inventory[0] = inventory0$

$\\forall e, \\forall d \\in [1, Days - WorkingDays - RestingDays + 1], \\quad isWorking[d, e] + isWorking[d + 1, e] + \\ldots + isWorking[d + WorkingDays - 1, e] = WorkingDays$

$\\forall e, \\forall d \\in [1, Days - WorkingDays - RestingDays + 1], \\quad isWorking[d + WorkingDays, e] + isWorking[d + WorkingDays + 1, e] + \\ldots + isWorking[d + WorkingDays + RestingDays - 1, e] = 0$

$\\forall d, totalEmployeesWorking[d] = \\sum_{e} isWorking[d, e]$

$\\forall d \in \\{1, \\ldots, \\text{Days}\\}, \\sum_e \\text{isWorking}[d, e] \\geq \\text{Num}[d]$

$\\forall d \in \\{1, \\ldots, \\text{Days}\\}, \\text{totalEmployeesWorking}[d] = \\sum_e \\text{isWorking}[d, e]$

$\\sum_{j=1}^{J} \\mathrm{containersTransported}[i, j] \\leq \\mathrm{NumDepot}[i] \\quad \\forall i \\in \\{1, 2, \\ldots, I\\}$

$\\forall j \\in \\{1, \\ldots, J\\}, \\sum_{i=1}^I containersTransported[i,j] = NumPort[j]$

$\\forall i \\in \\{1, \\ldots, I\\}, \\forall j \\in \\{1, \\ldots, J\\}, \\text{containersTransported}[i, j] = 2 \\times \\text{numBarges}[i, j]$

$\\sum_{p=1}^{P} TimeRequired[m][p] \\cdot batchCount[p] \\leq Availability[m] \\quad \\forall m \\in \\{1, \\ldots, M\\}$

$\\sum_{a=1}^{\\text{NumAdTypes}} \\text{clicks}[a] \\cdot \\text{YoungClicks}[a] \\geq \\text{GoalYoung}$

$\\sum_{a \\in \\text{NumAdTypes}} \\text{UniqueClicks}[a] \\cdot \\text{clicks}[a] \\ge \\text{GoalUniqueYoung}$

$\\sum_{a=1}^{\\text{NumAdTypes}} \\text{clicks}[a] \\times \\text{UniqueClicks}[a] \\geq \\text{GoalUniqueOld}$

$\\forall a \\in [1, \\text{NumAdTypes}], \\text{clicks}[a] \\leq \\text{MaxClicks}[a]$

$\forall a \in \{1, 2, ..., \text{NumAdTypes}\}, \; \text{clicks}[a] \geq 0$

$\\forall w, regularBaskets[w] + overtimeBaskets[w] \\geq Demand[w]$

$\\forall w, regularBaskets[w] \\leq \\frac{RegularLabor[w]}{AssemblyTime}$

$\\forall w, overtimeBaskets[w] \\leq \\frac{OvertimeLabor[w]}{AssemblyTime}$

$\\forall w \in \\{1, 2, \\ldots, \\text{Weeks}\\}, \\text{regularBaskets}[w] \\times \\text{AssemblyTime} \\leq \\text{RegularLabor}[w]$

$\\forall w \\in \\{1, \\dots, \\text{Weeks}\\}, \\text{overtimeBaskets[w]} \\times \\text{AssemblyTime} \\leq \\text{OvertimeLabor[w]}$

$\\forall w \\in \\{1, \\ldots, W\\}, \\quad \\sum_{i=1}^w (regularBaskets[i] + overtimeBaskets[i]) \\geq \\sum_{i=1}^w Demand[i] - inventory[w-1]$

$\\forall w \\in \\{1, \\ldots, W-1\\}, inventory[w] = regularBaskets[w] + overtimeBaskets[w] - Demand[w] + inventory[w-1]$

$\\forall w \\in \\{1, \\dots, W-1\\}, \\quad inventory[w+1] = regularBaskets[w] + overtimeBaskets[w] - demand[w] + inventory[w]$

$\\forall m \\in \\{1, \\dots, M-2\\}, \\sum_{p=1}^{P} TimeRequired[m][p] \\cdot Batches[p] \\leq Availability[m] \\quad \\text{and} \\quad \\sum_{m=M-1}^{M} \\sum_{p=1}^{P} TimeRequired[m][p] \\cdot Batches[p] \\leq Availability[M-1] + Availability[M]$

$\\forall m \\in \\{1, 2, ... , M\\}, \\sum_{p=1}^{P} \\text{TimeRequired}[m,p] \\times \\text{batches}[p] \\leq \\text{Availability}[m] + \\text{extraTime}[m]$

$\\forall m \\in \\{1, 2, ... , M\\}, \\text{extraTime}[m] \\leq \\text{MaxExtra}[m]$

$\\forall j \\in \\text{RequiredLanguages}, \\sum_{i \\in N | j \\in \\text{Languages}[i]} x[i] \\geq 1$

$\\sum_{i=1}^{NumProjects} assignProjToConsultant[i, j] \\leq MaxProjectsPerConsultant \\quad \\forall j \\in \\{1, 2, \\ldots, NumConsultants\\}$

$\sum_{i=1}^{NumProjects} assignProjToConsultant[i][j] \leq NumProjects \cdot hireConsultant[j], \forall j$

$\forall n \in \{1, 2, \ldots, N\}, isolateCentral[n] + scanCentral[n] + isolateDistributed[n] + scanDistributed[n] = 1$

$\forall n \in \{1, \ldots, N\}, \text{isolateCentral}[n] + \text{scanCentral}[n] + \text{isolateDistributed}[n] + \text{scanDistributed}[n] = 1$

$\\sum_{i=1}^{N} \\left( \\text{IsolateCentral}[i] \\cdot \\text{isolateCentral}[i] + \\text{ScanCentral}[i] \\cdot \\text{scanCentral}[i] \\right) \\leq \\text{CentralMaxHours}$

$\\sum_{n=1}^{N} \\left( \\text{IsolateDistributed}[n] \\cdot \\text{isolateDistributed}[n] + \\text{ScanDistributed}[n] \\cdot \\text{scanDistributed}[n] \\right) \\leq \\text{DistributedMaxHours}$

Mrs. Watson wants to invest in the real-estate market and has a total budget of at most TotalBudget. She has two choices which include condos and detached houses. Each dollar invested in condos yields a ProfitPerDollarCondos profit and each dollar invested in detached houses yields a ProfitPerDollarHouses profit. A minimum of MinPercentageCondos of the total investment must be in condos, and at least MinInvestmentDetached must be invested in detached houses.

A breakfast joint makes two different sandwiches: a regular and a special. Both need eggs and bacon. Each regular sandwich requires EggsPerRegular eggs and BaconPerRegular slices of bacon. Each special sandwich requires EggsPerSpecial eggs and BaconPerSpecial slices of bacon. The joint has a total of TotalEggs eggs and TotalBacon slices of bacon. It makes a profit of ProfitRegular per regular sandwich and a profit of ProfitSpecial per special sandwich. How many of each sandwich should be made to maximize profit?

A cleaning company aims to maximize exposure for promoting their new product without exceeding their TotalBudget. They will advertise using radio ads and social media ads. Each radio ad costs CostRadioAd and provides ExposureRadioAd exposure. Each social media ad costs CostSocialMediaAd and delivers ExposureSocialMediaAd exposure. The company aims to run between MinRadioAds and MaxRadioAds radio ads and at least MinSocialMediaAds social media ads while staying within the TotalBudget.

There is TotalGold available that is needed to make long and short cables. Long cables require GoldPerLongCable while short cables require GoldPerShortCable. Because of their compact size, the number of short cables should be at least ShortCableMultiplier times the number of long cables. In addition, there needs to be at least MinLongCables made. If each long cable sold results in a ProfitPerLongCable and each short cable sold results in a ProfitPerShortCable, how many of each type of cable should be made to maximize profit?

A company is deciding where to promote their product across NumMedia types of media. Each medium has a cost and a number of viewers it attracts per advertisement, defined by CostMedia and ViewersMedia respectively. SzoorchleEngine has a maximum number of advertisements constraint, MaxSzoorchleAds. To balance advertising, at most MaxWassaProportion of the total number of ads should be on Wassa, and at least MinZTubeProportion of the total number of ads should be on ZTube. The total weekly advertising budget is defined by Budget. The goal is to determine the number of advertisements on each type of media to maximize the total audience while adhering to these constraints.

A furniture company produces two types of products, chairs and dressers. The profit from each chair is ChairProfit, while the profit from each dresser is DresserProfit. Every week, the company has TotalStain gallons of stain and TotalOakWood lengths of oak wood available. Each chair requires StainPerChair gallons of stain and WoodPerChair lengths of oak wood, while each dresser requires StainPerDresser gallons of stain and WoodPerDresser lengths of oak wood. The goal is to determine the number of chairs and dressers to produce in order to maximize the total profit, given the constraints on the availability of stain and oak wood.

A farmer wants to mix his animal feeds, Feed A and Feed B, in such a way that the mixture will contain a minimum of MinProtein units of protein and MinFat units of fat. Feed A costs CostFeedA per kilogram and contains ProteinFeedA units of protein and FatFeedA units of fat. Feed B costs CostFeedB per kilogram and contains ProteinFeedB units of protein and FatFeedB units of fat. Determine the minimum cost of the mixture.

An artisan makes two types of terracotta jars: a thin jar and a stubby jar. Each thin jar requires ShapingTimeThinJar minutes of shaping time and BakingTimeThinJar minutes of baking time. Each stubby jar requires ShapingTimeStubbyJar minutes of shaping time and BakingTimeStubbyJar minutes of baking time. Per week, there are TotalShapingTime minutes available for shaping and TotalBakingTime minutes available for baking. The profit per thin jar is ProfitThinJar and the profit per stubby jar is ProfitStubbyJar. How many jars of each type should the artisan make to maximize profit?

A grocery store wants to liquidate its stock of TotalApples apples, TotalBananas bananas, and TotalGrapes grapes. Given past experience, the store knows that they can propose a banana-haters package with ApplesInBananaHaters apples and GrapesInBananaHaters grapes, and that this package will bring a profit of ProfitBananaHaters euros. Similarly, they can prepare a combo package with ApplesInCombo apples, BananasInCombo bananas, and GrapesInCombo grapes, yielding a profit of ProfitCombo euros. They know they can sell any quantity of these two packages within the availability of its stock. What quantity of each package, banana-haters packages and combo packages, should the store prepare to maximize net profit?

A bakery uses a stand-mixer and a slow bake oven to make products. Each machine can run for at most MaxHours per year. To produce one unit of bread takes BreadMixerTime time in the stand mixer and BreadOvenTime time in the oven. A unit of cookies requires CookieMixerTime time in the mixer and CookieOvenTime time in the oven. The profit per unit of bread is ProfitBread and the profit per unit of cookies is ProfitCookie. How should the bakery operate to maximize total profit?

A factory makes two types of glass panes: a regular glass pane and a tempered glass pane. Both require time on a heating and cooling machine. Both machines are available for a maximum of MaxHeatingTime minutes per day for the heating machine and MaxCoolingTime minutes per day for the cooling machine. It takes HeatingTimeRegular minutes in the heating machine and CoolingTimeRegular minutes in the cooling machine to make one regular glass pane. It takes HeatingTimeTempered minutes in the heating machine and CoolingTimeTempered minutes in the cooling machine to make one tempered glass pane. The profit per pane of regular glass is ProfitRegular and the profit per pane of tempered glass is ProfitTempered. How many panes of each glass type should the factory make to maximize profit? What is the maximum profit?

A shop makes two types of souvenirs, Elephants and Tigers. Each Elephant requires WoodPerElephant amount of wood and PlasticPerElephant amount of plastic. Each Tiger requires WoodPerTiger amount of wood and PlasticPerTiger amount of plastic. The shop has TotalWoodAvailable amount of wood and TotalPlasticAvailable amount of plastic available each week. The profit from selling one Elephant is ProfitPerElephant, and the profit from selling one Tiger is ProfitPerTiger. Determine the number of Elephants and Tigers to make to maximize the total profit.

The family has TotalInvest amount of money to invest, with some in apartments and the rest in townhouses. The amount of money invested in apartments must not exceed MaxInvestApt. The investment in apartments must be at least a MinRatioInvestAptToTown times the investment in townhouses. The return rate for investment in apartments is ReturnRateApt, and the return rate for investment in townhouses is ReturnRateTown. The goal is to determine the optimal investment in apartments and townhouses to maximize profit.

An individual needs to take at least MinZ1 grams of Z1 and MinD3 grams of D3 daily. These medicines are available in two types of pills: Zodiac and Sunny. One pill of Zodiac contains Z1InZodiac grams of Z1 and D3InZodiac grams of D3. One pill of Sunny contains Z1InSunny grams of Z1 and D3InSunny grams of D3. The cost per pill of Zodiac is CostZodiac, and the cost per pill of Sunny is CostSunny. The objective is to determine the quantity of Zodiac and Sunny pills that should be taken to meet the medicine requirements at the lowest cost.

A hobbyist makes model trains and planes using TotalWood and TotalPaint. A model train requires WoodPerTrain units of wood and PaintPerTrain units of paint. A model plane requires WoodPerPlane units of wood and PaintPerPlane units of paint. The hobbyist has available TotalWood units of wood and TotalPaint units of paint. If the profit per model train is ProfitPerTrain and the profit per model plane is ProfitPerPlane, how many of each should be made to maximize his profit?

A store wants to maximize its profit by optimizing the inventory of phones and laptops. The profit earned per phone is ProfitPhone, and the profit earned per laptop is ProfitLaptop. Each phone requires SpacePhone floor space, and each laptop requires SpaceLaptop floor space, with a total available floor space given by TotalSpace. Corporate policy mandates that at least MinLaptopPercentage of all items in the inventory be laptops. The cost for the store to purchase each phone is CostPhone, and each laptop costs CostLaptop. The store aims to keep the total spending within MaxSpending.

A candy store mixes different types of candy to prepare two types of products: regular mix and sour surprise mix. Each kilogram of the regular mix contains RegularCandyInRegularMix kilograms of regular candy and SourCandyInRegularMix kilograms of sour candy. The profit per kilogram of the regular mix is ProfitRegularMix. Each kilogram of the sour surprise mix contains RegularCandyInSourMix kilograms of regular candy and SourCandyInSourMix kilograms of sour candy. The profit per kilogram of the sour surprise mix is ProfitSourMix. The candy store has AvailableRegularCandy kilograms of regular candy and AvailableSourCandy kilograms of sour candy available. The goal is to determine how many kilograms of each type of product should be created to maximize the total profit given the available candy.

A factory has TotalSpace of space to allocate between the production of phones and laptops, where each type of product occupies a certain amount of space per unit of production. Phones require PhoneLabor hours of labor and cost PhoneCost per square foot of space allocated for phone production. Laptops require LaptopLabor hours of labor and cost LaptopCost per square foot of space allocated for laptop production. Phones produce a net revenue of PhoneRevenue per square foot while laptops produce a net revenue of LaptopRevenue per square foot. The factory aims to spend at most TotalBudget and use at most TotalLabor hours of labor. The goal is to determine the optimal allocation of TotalSpace between phones and laptops to maximize total net revenue.

An electronics store sells premium desktops with more power as well as regular desktops for everyday use. Each premium desktop costs the store CostPremium to make and yields a profit of ProfitPremium. Each regular desktop costs the store CostRegular to make and yields a profit of ProfitRegular. The store sells at most MaxDesktops desktops each month and wants to spend at most MaxBudget on making the desktops. How many of each should be made and sold to maximize profit?

A car manufacturer makes two versions of the same car, a regular model and a premium model. They make x1 regular models per day and x2 premium models per day. The profit per regular model is ProfitRegular and the profit per premium model is ProfitPremium (x1 and x2 are unknown values both greater than or equal to 0). The daily demand for these cars is limited to at most MaxDemandRegular regular models and MaxDemandPremium premium models. In addition, the manufacturer can make a maximum of MaxTotalProduction cars of either type per day. How many cars of each model should the manufacturer make in order to maximize profit?

You are designing an office space with two types of desks: long desks and short desks. You can spend at most Budget. Long desks cost CostLongDesk, take up SpaceLongDesk of space, and seat SeatingLongDesk employees. Short desks cost CostShortDesk, take up SpaceShortDesk of space, and seat SeatingShortDesk employees. The office can have at most MaxSpace of desks. How many of each desk should you buy in order to maximize the seating availability?

Ayse produces a plant growth compound by mixing two types of fertilizer: C and Y. This growth compound must contain at least MinRequiredNitrousOxide units of nitrous oxide and MinRequiredVitaminMix units of vitamin mix. Fertilizer C and Y cost CostPerKgFertilizerC and CostPerKgFertilizerY per kg respectively. Fertilizer C contains NitrousOxidePerKgC units of nitrous oxide per kg and VitaminMixPerKgC units of vitamin mix per kg. Fertilizer Y contains NitrousOxidePerKgY units of nitrous oxide per kg and VitaminMixPerKgY units of vitamin mix per kg. Determine the minimum cost of Ayse's compound.

A car manufacturer makes NumOilTypes types of car oils. A container of each oil type contains different amounts of NumSubstances substances. The car manufacturer has a certain total amount of each substance available. Additionally, the profit per container of each oil type is given. The manufacturer wants to determine how many containers of each oil type should be made to maximize profit.

A food truck owner can spend at most TotalBudget on purchasing mangos and guavas. A mango costs MangoCost and a guava costs GuavaCost. Each mango is sold for a profit of MangoProfit while each guava is sold for a profit of GuavaProfit. The owner estimates that at least MinMangosSold but at most MaxMangosSold mangos are sold each month. He also estimates that the number of guavas sold is at most MaxGuavasPerMango of the mangos sold. The objective is to determine how many mangos and guavas should be sold to maximize the profit.

A nutritionist is making a special vitamin mix using two drinks: A and B. The vitamin mix must contain at least MinVitaminA units of Vitamin A, at least MinVitaminD units of Vitamin D, and at most MaxVitaminE units of Vitamin E. A cup of drink A contains VitaminAInA units of Vitamin A, VitaminDInA units of Vitamin D, VitaminEInA units of Vitamin E, and VitaminKInA units of Vitamin K. A cup of drink B contains VitaminAInB units of Vitamin A, VitaminDInB units of Vitamin D, VitaminEInB units of Vitamin E, and VitaminKInB units of Vitamin K. How many cups of each drink should be used to make the vitamin mix in order to minimize the amount of Vitamin K?

An ice cream store makes two types of ice cream, chocolate and vanilla. In a week, they must produce at least MinChocolateGallons and at most MaxChocolateGallons of chocolate ice cream, and at least MinVanillaGallons and at most MaxVanillaGallons of vanilla ice cream. The time required to produce one gallon of chocolate ice cream is TimePerGallonChocolate hours, and the time required to produce one gallon of vanilla ice cream is TimePerGallonVanilla hours. The total available time for production in a week is TotalAvailableHours hours. Additionally, at least MinWorkers workers are needed at any time, with at least WorkersPerGallonChocolate workers working on chocolate ice cream and at least WorkersPerGallonVanilla workers working on vanilla ice cream. The profit per gallon of chocolate ice cream is ProfitPerGallonChocolate, and the profit per gallon of vanilla ice cream is ProfitPerGallonVanilla. The objective is to determine the number of gallons of each type of ice cream to produce in order to maximize profit.

Mark has TotalLand acres of land available to grow potatoes and cucumbers that he sells at a farmers' market. He must grow at least MinPotatoes acres of potatoes and MinCucumbers acres of cucumbers to meet his contract. Mark prefers to grow more cucumbers than potatoes, but he only has enough resources to grow at most MaxCucumberToPotatoRatio times the amount of cucumbers as potatoes. If the profit per acre of potatoes is ProfitPerAcrePotato and the profit per acre of cucumbers is ProfitPerAcreCucumber, how many acres of each should he grow to maximize his profit? What is that profit?

A flooring company produces hardwood and vinyl planks. The demand for hardwood flooring each week is at least MinDemandHardwood square feet, and the demand for vinyl planks each week is at least MinDemandVinyl square feet. The total amount of flooring shipped each week must be at least MinTotalShipment square feet. Due to a labor shortage, the company can produce no more than MaxProductionHardwood square feet of hardwood flooring and no more than MaxProductionVinyl square feet of vinyl planks each week. The profit from producing a square foot of hardwood flooring is ProfitHardwood, while the profit from producing a square foot of vinyl planks is ProfitVinyl. The goal is to determine the number of square feet of each type of flooring to produce each week to maximize the company’s profit.

A farmer has two farms, where he grows NumBerryTypes different types of berries. He has a contract to provide specific quantities of each type of berry to a local store (DeliveryRequirement). It costs OldFarmCost to operate the old farm per day, where OldFarmOutput of each type of berry can be harvested and delivered in a day. It costs NewFarmCost to operate the new farm per day, where NewFarmOutput of each type of berry is harvested and delivered in a day. The goal is to meet the DeliveryRequirement for each type of berry while minimizing the total operation cost.

Jordan wants to design a diet consisting of items Rice and Kebab. The cost per serving of Rice is represented by CostRice, and the cost per serving of Kebab is represented by CostKebab. Each serving of Rice provides CaloriesRice calories and ProteinRice grams of protein, while each serving of Kebab provides CaloriesKebab calories and ProteinKebab grams of protein. The goal is to minimize the total cost of the diet while ensuring that the total daily calorie intake is at least MinCalories calories and the total daily protein intake is at least MinProtein grams.

A company sells custom scooters and bikes. The ProfitPerScooter and ProfitPerBike are given as abstract symbols. Each scooter requires DesignTimePerScooter with the design team and EngineeringTimePerScooter with the engineering team. Each bike requires DesignTimePerBike with the design team and EngineeringTimePerBike with the engineering team. Per month, there are TotalDesignHours available on the design team and TotalEngineeringHours available on the engineering team. Determine the monthly production quantities of scooters and bikes to maximize profit.

A person only eats two types of food: vegetables and fruits. Each serving of vegetables contains a certain amount of vitamins and minerals, and each serving of fruits contains a different amount of vitamins and minerals. The person wants to consume a minimum required amount of vitamins and minerals. The cost per serving of vegetables and fruits is specified. The goal is to determine the number of servings of vegetables and fruits to consume in order to meet the minimum vitamin and mineral requirements while minimizing the total cost.

A bakery bakes two types of products: bagels and croissants. A batch of bagels requires OvenTimeBagels of oven time and ChefTimeBagels of pastry chef time. A batch of croissants requires OvenTimeCroissants of oven time and ChefTimeCroissants of pastry chef time. The bakery has a maximum of MaxOvenTime hours available for the oven and MaxChefTime hours available for the pastry chef each day. The profit per batch is ProfitBagels for bagels and ProfitCroissants for croissants. The objective is to maximize profit using the available capacity.

A teddy bear company produces three different colored bears using two factories. Running the first factory for one hour costs CostFactory1 and produces ProductionBlackFactory1 black teddy bears, ProductionWhiteFactory1 white teddy bears, and ProductionBrownFactory1 brown teddy bears. Running the second factory for one hour costs CostFactory2 and produces ProductionBlackFactory2 black teddy bears and ProductionWhiteFactory2 white teddy bears but no brown teddy bears. The company needs to meet a daily demand of at least DemandBlack black teddy bears, DemandWhite white teddy bears, and DemandBrown brown teddy bears. The goal is to minimize the cost of production while fulfilling the demand.

A farmer has TotalAcres acres of land to grow turnips and pumpkins. Turnips require WateringPerAcreTurnips minutes of watering and PesticidePerAcreTurnips worth of pesticide per acre. Pumpkins require WateringPerAcrePumpkins minutes of watering and PesticidePerAcrePumpkins worth of pesticide per acre. The farmer has TotalWateringTime minutes available for watering and TotalPesticideBudget available to spend on pesticide. If the revenue per acre of turnips is RevenuePerAcreTurnips and the revenue per acre of pumpkins is RevenuePerAcrePumpkins, how many acres of each should he grow to maximize his revenue.

A school employs staff and substitute teachers. Per shift, staff teachers work StaffHoursPerShift hours and get paid StaffWagePerShift. Per shift, substitute teachers work SubstituteHoursPerShift hours and get paid SubstituteWagePerShift. The school requires RequiredHours of teaching availability. If the school has a budget of TotalBudget, how many of each teacher should be hired to minimize the total number of teachers?

A factory uses HighIntensityDrills and LowIntensityDrills to process DailyGemProcessingRequirement gems each day. Each HighIntensityDrill has a capacity of HighIntensityDrillCapacity gems and requires HighIntensityDrillWaterRequirement units of water per day. Each LowIntensityDrill has a capacity of LowIntensityDrillCapacity gems and requires LowIntensityDrillWaterRequirement units of water per day. The total available water per day is TotalAvailableWater. At most MaxFractionHighIntensityDrills of the drills can be HighIntensityDrills, and at least MinLowIntensityDrills of the drills must be LowIntensityDrills. The goal is to determine the number of HighIntensityDrills and LowIntensityDrills required to minimize the total number of drills needed.

A sandwich company can open two types of stores: dine-in places, each requiring EmployeesDineIn employees and producing ProductionDineIn sandwiches per day, and food-trucks, each requiring EmployeesFoodTruck employees and producing ProductionFoodTruck sandwiches per day. The company must produce at least MinProduction sandwiches per day and has TotalEmployees employees available. The objective is to determine how many of each type of store to open to minimize the total number of stores.

There are two processes, process A and process B, to plate coins with gold. Process A requires GoldRequiredA units of gold, WiresRequiredA number of wires, and can plate CoinsPlatedA number of coins. Process B requires GoldRequiredB units of gold, WiresRequiredB number of wires, and can plate CoinsPlatedB number of coins. There are TotalGoldAvailable units of gold and TotalWiresAvailable number of wires available. How many processes of each type should be run to maximize the total number of coins that can be plated?

A woodshop can purchase two types of saws, a miter saw and a circular saw. A miter saw has a cutting capacity of MiterCutCapacity and produces MiterSawdust units of sawdust per day. A circular saw has a cutting capacity of CircularCutCapacity and produces CircularSawdust units of sawdust per day. The woodshop must cut at least MinPlanks planks of wood per day. However, to avoid too much pollution, they can produce at most MaxSawdust units of sawdust per day. The goal is to determine how many of each type of saw should be purchased to minimize the total number of saws needed.

A farmer has TotalAcres of land on which he must process hay using either a windrower or hay harvester. For each acre of land, the windrower can process WindrowerHayPerAcre of hay while the hay harvester can process HarvesterHayPerAcre of hay. Per acre, the windrower produces WindrowerMethanePerAcre of methane gas and requires WindrowerFuelPerAcre of fuel. On the other hand, the hay harvester produces HarvesterMethanePerAcre of methane gas per acre and requires HarvesterFuelPerAcre of fuel. There are TotalFuel of fuel available and the farmer can produce at most MaxMethane of methane gas. For how many acres should each machine be used to maximize the amount of hay processed?

A fast food place makes two types of food items. Each 'Wrap' requires 'MeatPerWrap' units of meat and 'RicePerWrap' units of rice, and each 'Platter' requires 'MeatPerPlatter' units of meat and 'RicePerPlatter' units of rice. The time required to make one Wrap is 'TimePerWrap' and the time required to make one Platter is 'TimePerPlatter'. The fast food place must use at least 'MinMeat' units of meat and 'MinRice' units of rice. Additionally, the number of Wraps produced must be at least 'WrapsToPlattersRatio' times the number of Platters produced. The objective is to determine the number of each type of food item to make in order to minimize the total production time.

An amusement park is installing ticket machines, both cash-based and card-only machines. A cash-based machine can process CashMachineCapacity people per hour while a card-only machine can process CardMachineCapacity people per hour. The cash-based machine needs CashPaperConsumption rolls of paper per hour while the card-only machine requires CardPaperConsumption rolls of paper per hour. The amusement park needs to be able to process at least MinTotalPeopleRequirement people per hour but can use at most MaxTotalPaperLimit rolls of paper per hour. Also, the number of card-only machines must not exceed the number of cash-based machines.

A honey farmer sells his honey in glass and plastic jars. A glass jar can hold HoneyPerGlassJar of honey while a plastic jar can hold HoneyPerPlasticJar of honey. Since glass jars are more expensive, at least MinPlasticPerGlass times as many plastic jars must be filled as glass jars. However, at least MinGlassJars glass jars should be filled. If the farmer has TotalHoney of honey, how many jars of each should be filled to maximize the total number of bottles filled?

A high rise building is buying two types of furnaces, a new model and an old model. A new model furnace has a HeatingCapacityNew and consumes ElectricityConsumptionNew per day. An old model furnace has a HeatingCapacityOld and consumes ElectricityConsumptionOld per day. At most MaxProportionOld of the furnaces can be the old model and at least MinNumNew new model furnaces should be used. The building needs to heat at least TotalApartments and has TotalElectricity available. Determine the number of new and old model furnaces to purchase in order to minimize the total number of furnaces.

A company needs to determine the number of rural and urban factories to build in order to minimize the total number of factories. A rural factory produces PhonesPerDayRural phones per day and requires ManagersRural managers. An urban factory produces PhonesPerDayUrban phones per day and requires ManagersUrban managers. The company has TotalManagers managers available and must produce at least MinPhonesPerDay phones per day.

An ice cream store can buy two types of machines, a counter-top one and a fridge-sized one, to make ice cream. The counter-top machine can produce CounterTopCapacity cones worth of ice cream every day while producing CounterTopHeatOutput units of heat. The fridge-sized machine can produce FridgeCapacity cones worth of ice cream every day while producing FridgeHeatOutput units of heat. The ice cream store can output at most MaxDailyHeatOutput units of heat per day and must produce at least MinDailyProduction cones worth of ice cream. How many of each machine should they buy to minimize the total number of machines needed?

There are NumContainers specialized containers that are used to make a pharmaceutical paste. Each type of container requires WaterRequired of water and PillRequired of powdered pill to produce PasteProduced of paste. The pharmacy has a TotalWater amount of water and a TotalPill amount of powdered pill available. How many of each type of container should be used to maximize the amount of paste that can be made?

A company operates on TotalAcres of land where they use oil wells of small and large size. Per acre of land, a small oil well can produce OilProductionSmall units of oil. Per acre of land, a large oil well can produce OilProductionLarge units of oil. However, a small oil well produces PollutionSmall units of pollution and requires DrillBitsSmall drill bits per acre, and a large oil well produces PollutionLarge units of pollution and requires DrillBitsLarge drill bits per acre. The company has TotalDrillBits drill bits available and can produce at most MaxPollution units of pollution due to strict environmental laws. How many acres should be allocated to small and large oil wells to maximize the total oil production?

A bakery makes two types of croissants. The first type of croissant requires ButterAlmond units of butter and FlourAlmond units of flour. The second type of croissant requires ButterPistachio units of butter and FlourPistachio units of flour. The bakery has a total of TotalButter units of butter and TotalFlour units of flour available. The number of the first type of croissants produced should be at least MinRatioAlmondToPistachio times the number of the second type of croissants. The time required to produce one unit of the first type of croissant is TimeAlmond, and the time required to produce one unit of the second type of croissant is TimePistachio. The objective is to determine the number of each type of croissant to produce in order to minimize the total production time.

A burger chain is buying both gas and electric grills to make burger patties. A gas grill can cook PattiesPerGasGrill patties per minute while an electric grill can cook PattiesPerElectricGrill patties per minute. The gas grill requires OilPerGasGrill units of cooking oil per minute while the electric grill requires OilPerElectricGrill units of cooking oil per minute. The number of electric grills must be less than the number of gas grills. The burger chain wants to be able to make at least MinPattiesPerMinute patties per minute and use at most MaxOilPerMinute units of cooking oil per minute. How many of each grill should be bought to minimize the total number of grills?

A company sells cough syrup in kids and adult size bottles. A kids size bottle can hold VolumeKids of cough syrup while an adult size bottle can hold VolumeAdult of cough syrup. Since there is no difference in concentration of medical ingredients between the two bottles, the number of adult size bottles must be at least MinRatioAdultToKids times the number of kids size bottles. However, at least MinKidsBottles kids size bottles must be made. If the company has TotalCoughSyrup of cough syrup available, how many bottles of each should be made to maximize the total number of bottles?

A bakery makes brownies and lemon squares. Each brownie requires ChocolateMixPerBrownie units of chocolate mix and FiberPerBrownie units of fiber. Each lemon square requires LemonMixPerLemonSquare units of lemon mix and FiberPerLemonSquare units of fiber. Lemon squares sell faster, so the number of lemon squares made must be larger than the number of brownies made. At least MinBrowniesRatio of the items made must be brownies. The bakery has TotalChocolateMix units of chocolate mix and TotalLemonMix units of lemon mix. The goal is to determine the number of each type of item to make in order to minimize the total amount of fiber needed.

A clinic employs nurses and pharmacists to deliver shots to patients. A nurse works NurseHoursPerShift per shift while a pharmacist works PharmacistHoursPerShift per shift. Nurses are paid NursePaymentPerShift per shift while pharmacists are paid PharmacistPaymentPerShift per shift. The clinic needs TotalLaborHoursRequired of healthcare labor to meet needs. If the firm has a budget of TotalBudget, how many of each healthcare worker should be scheduled to minimize the total number of workers?

A laundromat can buy two types of washing machines, a top-loading model and a front-loading model. The top-loading model can wash WashingCapacityTopLoading items per day while the front-loading model can wash WashingCapacityFrontLoading items per day. The top-loading model consumes EnergyConsumptionTopLoading kWh per day while the front-loading model consumes EnergyConsumptionFrontLoading kWh per day. The laundromat must be able to wash at least MinWashingCapacityRequired items per day and has available MaxEnergyAvailable kWh per day. Since the top-loading machines are harder to use, at most MaxPercentageTopLoading of the machines can be top-loading. Further, at least MinFrontLoadingMachines should be front-loading. How many of each machine should the laundromat buy to minimize the total number of washing machines?

A shipping company can purchase RegularVans and HybridVans to make deliveries. A RegularVan can deliver RegularVanCapacity packages per day and produces RegularVanPollution units of pollutants. A HybridVan can deliver HybridVanCapacity packages per day and produces HybridVanPollution units of pollutants. Due to a new environmental law, they can produce at most MaxPollution units of pollutants per day. However, the company needs to be able to deliver at least MinPackages packages per day. How many of each type of van should they buy to minimize the total number of vans needed?

The objective is to maximize the amount of carbon dioxide produced using two different processes: one with a catalyst and one without a catalyst. The process with a catalyst requires 'WoodWithCatalyst' units of wood and 'OxygenWithCatalyst' units of oxygen to produce 'CO2WithCatalyst' units of carbon dioxide. The process without a catalyst requires 'WoodWithoutCatalyst' units of wood and 'OxygenWithoutCatalyst' units of oxygen to produce 'CO2WithoutCatalyst' units of carbon dioxide. The total available resources are 'TotalWood' units of wood and 'TotalOxygen' units of oxygen. The decision variables are the number of times the process with a catalyst is used and the number of times the process without a catalyst is used. The aim is to find the optimal number of each process to maximize the total production of carbon dioxide, subject to the constraints on the availability of wood and oxygen.

A tea estate has TotalLand acres of land available and they need to pick the tea leaves either using a traditional machine or modern machine. For each acre of land, the traditional machine can pick TealeavesTraditional kg of tea leaves, creates WasteTraditional kg of waste, and requires FuelTraditional liters of fuel. For each acre of land, the modern machine can pick TealeavesModern kg of tea leaves, creates WasteModern kg of waste, and requires FuelModern liters of fuel. The estate has available TotalFuel liters of fuel and can handle at most MaxWaste kg of waste. For how many acres should each machine be used to maximize the amount of tea leaves that can be picked?

An organization is purchasing 2 different models of stamping machines. Each model has a specific StampingRate and GlueConsumption per minute. The number of the first model of machines must be less than the number of the second model. Additionally, the organization aims to ensure they can process at least MinLetters letters per minute and use at most MaxGlue units of glue per minute. The objective is to determine the number of each model of stamping machines to minimize the total number of stamping machines.

A company sells their bananas in small and large crates. A small crate can hold BananasPerSmallCrate bananas while a large crate can hold BananasPerLargeCrate bananas. The number of large crates must be at least MinLargeToSmallRatio times the number of small crates. However, at least MinSmallCrates small crates should be used. If the company has TotalBananas available bananas, how many of each crate should the company use to maximize the total number of crates used?

A store sells two types of crepes: chocolate crepes and peanut butter crepes. A chocolate crepe requires ChocolateSpreadForChocolateCrepe units of chocolate spread and CrepeMixForChocolateCrepe units of crepe mix. A peanut butter crepe requires PeanutButterSpreadForPeanutButterCrepe units of peanut butter spread and CrepeMixForPeanutButterCrepe units of crepe mix. The number of peanut butter crepes made (NumPeanutButterCrepes) must exceed the number of chocolate crepes made (NumChocolateCrepes). Additionally, at least MinimumChocolateCrepeProportion of the total crepes made should be chocolate crepes, i.e., NumChocolateCrepes should be at least MinimumChocolateCrepeProportion * (NumChocolateCrepes + NumPeanutButterCrepes). The store has TotalChocolateSpread units of chocolate spread and TotalPeanutButterSpread units of peanut butter spread available. The total amount of chocolate spread used by all chocolate crepes must not exceed TotalChocolateSpread, and the total amount of peanut butter spread used by all peanut butter crepes must not exceed TotalPeanutButterSpread. The objective is to determine NumChocolateCrepes and NumPeanutButterCrepes to minimize the total amount of crepe mix needed while satisfying these constraints.

A fire department employs regular and emergency fire fighters. A regular fire fighter works HoursRegularShift per shift and earns EarningsRegular. An emergency fire fighter works HoursEmergencyShift per shift and earns EarningsEmergency. Due to wildfires in the region, the fire department needs at least MinHoursRequired hours of fire fighter time. If the fire department has a budget of Budget, how many of each should the fire department hire to minimize the total number of fire fighters?

A mall buys two types of hand sanitizer machines, which have delivery rates defined by 'DeliverRate' and energy consumptions defined by 'EnergyConsumption'. Due to hygiene concerns, at most 'MaxManualPercentage' of the machines can be manual, and at least 'MinMotionActivated' must be motion activated. The mall must achieve a minimum total delivery rate of 'MinDeliverRate' and adhere to a maximum total energy consumption of 'MaxEnergyConsumption'. The goal is to determine the number of each type of machine to minimize the total number of machines.

A company can build two types of shops: a small shop and a large shop. A small shop can produce HotDogsPerDaySmallShop hot dogs per day and requires WorkersPerSmallShop workers. A large shop can produce HotDogsPerDayLargeShop hot dogs per day and requires WorkersPerLargeShop workers. The company must produce at least MinHotDogsPerDay hot dogs per day but only has TotalWorkers workers available. How many of each type of shop should the company build to minimize the total number of shops?

An airport buys two types of vehicles, a 4-wheeler and 3-wheeler, to help move luggage. A 4-wheeler vehicle can move LuggageCapacity4Wheeler luggage per day and produces Pollution4Wheeler units of pollutant per day. A 3-wheeler vehicle can move LuggageCapacity3Wheeler luggage per day and produces Pollution3Wheeler units of pollutant per day. The airport needs to be able to move at least MinLuggage luggage per day. To avoid over-polluting the airport, they can produce at most MaxPollution units of pollutant per day. How many of each vehicle should the airport buy to minimize the total number of vehicles needed.

A man fishes in a TotalArea lake and can catch fish either using a net or fishing line. For each acre of the lake, using a net will catch FishCaughtNet fish and requires BaitRequiredNet units of bait but also causes PainNet units of pain for the fisherman. For each acre of the lake, using a fishing line will catch FishCaughtLine fish and requires BaitRequiredLine units of bait but also causes PainLine units of pain for the fisherman. The fisherman has available TotalBait units of bait and can tolerate at most MaxPain units of pain. For how many acres each should he use each fishing method to maximize the amount of fish he can catch?

A coffee shop sells two types of coffee: mochas and regular coffees. Each type requires specific units of coffee powder (CoffeePowderPerMocha, CoffeePowderPerRegular) and milk (MilkPerMocha, MilkPerRegular), which are limited in total availability (TotalCoffeePowder, TotalMilk). Producing each type also requires a specific amount of time (TimePerMocha, TimePerRegular). Additionally, there is a constraint specifying that the number of mochas made must be at least a certain multiple (MochaToRegularRatio) of the number of regular coffees made. The goal is to determine the number of each type of coffee to produce in order to minimize the total production time, given these constraints.

A butcher shop is buying slicers for their shop, a manual slicer and an automatic slicer. The manual slicer has a Cutting Rate (ManualCutRate) and uses grease at a rate (ManualGreaseRate). The automatic slicer has a Cutting Rate (AutomaticCutRate) and uses grease at a rate (AutomaticGreaseRate). The number of manual slicers should be less than the number of automatic slicers. The butcher shop must meet a minimum requirement of slices per minute (MinSlices) and adhere to a maximum grease usage per minute (MaxGrease). The goal is to determine the number of each slicer to minimize the total number of slicers in the shop.

A water company sells water in glass and plastic bottles. A glass bottle can hold GlassBottleCapacity of water while a plastic bottle can hold PlasticBottleCapacity of water. Because most customers prefer plastic bottles, the number of plastic bottles must be at least MinPlasticToGlassRatio times the number of glass bottles. However, there must be at least MinGlassBottles glass bottles. If the company has available TotalWater of water, how many of each bottle should be made to maximize the total number of bottles?

A restaurant makes two types of curry: goat curry and chicken curry. Each bowl of goat curry requires GoatMeatPerGoatCurry units of goat meat and CurryBasePerGoatCurry units of curry base. Each bowl of chicken curry requires ChickenMeatPerChickenCurry units of chicken meat and CurryBasePerChickenCurry units of curry base. The restaurant has TotalGoatMeat units of goat meat and TotalChickenMeat units of chicken meat available. At least MinChickenCurryPercentage percentage of the total bowls made must be chicken curry, and the number of goat curry bowls must be larger than the number of chicken curry bowls. The objective is to determine the number of each type of curry bowl to make so as to minimize the total usage of curry base.

An accounting firm employs part time workers and full time workers. Full time workers work HoursPerShiftFullTime per shift while part time workers work HoursPerShiftPartTime per shift. In addition, full time workers are paid WageFullTime per shift while part time workers are paid WagePartTime per shift. Currently, the accounting firm has a project requiring TotalLaborHours of labor. If the firm has a TotalBudget for wages, determine the number of each type of worker to schedule in order to minimize the total number of workers, subject to the constraints of meeting the required labor hours and the budget.

A factory has two types of machines. Each machine of the first type can produce ItemsMachineA items and consumes ElectricityMachineA kWh per day. Each machine of the second type can produce ItemsMachineB items and consumes ElectricityMachineB kWh per day. The factory needs to produce at least MinItemsPerDay items per day and has MaxElectricityPerDay kWh of electricity available per day. At most MaxPercentageB of the machines can be of the second type. Additionally, at least MinMachinesA of the machines must be of the first type. The objective is to find the number of each type of machine to use to minimize the total number of machines.

A chemical plant can buy two types of generators, generator A and generator B, which use two different processes to generate hydrogen. Generator A uses an electrolysis process and produces HydrogenProductionA of hydrogen as well as PollutantProductionA of pollutants every day. Generator B uses a fermentation process to produce HydrogenProductionB of hydrogen per day while outputting PollutantProductionB of pollutants. The chemical plant must make a minimum of MinHydrogenRequirement of hydrogen per day but due to environmental restrictions they can produce at most MaxPollutantAllowance of pollutant. The goal is to determine the quantities, CountA and CountB, of each type of generator to minimize the total number of generators needed.

There are two chemical reactions, chemical reaction A and chemical reaction B. Chemical reaction A requires InertGasA units of rare inert gas and TreatedWaterA units of treated water to produce CompoundA units of a rare compound. Chemical reaction B requires InertGasB units of rare inert gas and TreatedWaterB units of treated water to produce CompoundB units of a rare compound. There are TotalInertGas units of rare inert gas and TotalTreatedWater units of treated water available in the lab. The objective is to determine the number of reactions of type A (XA) and type B (XB) to maximize the total amount of the rare compound produced.

A mining company has available TotalLand square miles of mining sites and is considering the use of two mining techniques: heap leaching and vat leaching. For each square mile of land, heap leaching technique can have a daily production of HeapLeachingProduction of rare earth oxide per square mile but it also creates HeapLeachingWastewater of polluted wastewater and requires HeapLeachingMachines of extraction machines. On the other hand, vat leaching technique produces VatLeachingProduction of rare earth oxide per square mile per day while creating VatLeachingWastewater of polluted wastewater and requiring VatLeachingMachines of extraction machines. There are TotalMachines available and due to environmental regulations, the amount of polluted wastewater must be at most MaxWastewater daily. Find the proportion of lands that use each mining technique in order to maximize the daily production of rare earth oxide.

A small chocolate shop makes two types of chocolate bars: milk and dark. Milk chocolate bars require CocoaPerMilkChocolate units of cocoa and MilkPerMilkChocolate units of milk. Dark chocolate bars require CocoaPerDarkChocolate units of cocoa and MilkPerDarkChocolate units of milk. The shop has TotalCocoa units of cocoa and TotalMilk units of milk available. Additionally, at least MilkToDarkRatio times as many milk chocolate bars need to be produced as dark chocolate bars. To minimize the total production time, where producing one milk chocolate bar takes TimePerMilkChocolate minutes and one dark chocolate bar takes TimePerDarkChocolate minutes, how many of each type of chocolate bar should be produced?

An office is buying two types of printers: a premium model and a regular model. The premium model can print PagesPerMinutePremium pages per minute and the regular model can print PagesPerMinuteRegular pages per minute. Additionally, the premium model requires InkPerMinutePremium units of ink per minute, while the regular model requires InkPerMinuteRegular units of ink per minute. The office wants to ensure at least MinPagesPerMinute pages can be printed per minute, and at most MaxInkPerMinute units of ink can be used per minute. The number of regular printers must be less than the number of premium printers to minimize the total number of printers in the office.

A company sells two types of bottles: a vintage bottle with volume VolumeVintage and a regular bottle with volume VolumeRegular. The company has a total available volume of TotalVolume. The number of regular bottles must be at least MinRatioRegularToVintage times the number of vintage bottles. Additionally, there is a requirement to produce at least MinVintageBottles vintage bottles. The goal is to determine how many of each type of bottle should be produced in order to maximize the total number of bottles.

A patient in the hospital can take two pills, Pill 1 and Pill 2. Per pill, Pill 1 provides PainMedPill1 units of pain medication and AnxietyMedPill1 units of anxiety medication. Per pill, Pill 2 provides PainMedPill2 units of pain medication and AnxietyMedPill2 units of anxiety medication. In addition, Pill 1 causes DischargePill1 units of discharge while Pill 2 causes DischargePill2 units of discharge. At most MaxPainMed units of pain medication can be provided and at least MinAnxietyMed units of anxiety medication must be provided. How many pills of each should the patient be given to minimize the total amount of discharge?

A man on a strict diet consumes drinks from two brands, alpha and omega. Each alpha brand drink contains ProteinAlpha grams of protein, SugarAlpha grams of sugar, and CaloriesAlpha calories per bottle. Each omega brand drink contains ProteinOmega grams of protein, SugarOmega grams of sugar, and CaloriesOmega calories per bottle. The man wants to consume at least MinProtein grams of protein and MinCalories calories. Additionally, the omega brand drink should constitute at most MaxOmegaProportion of his total drink intake. The goal is to find the number of bottles of each brand he should drink to minimize his sugar intake.

A summer camp conducts science experiments using NumBeakers different types of beakers. Each beaker i requires FlourRequired[i] units of flour and LiquidRequired[i] units of special liquid to produce SlimeProduced[i] units of slime, and generates WasteProduced[i] units of waste. The summer camp has a total of TotalFlour units of flour and TotalLiquid units of special liquid available. The maximum allowable waste that can be produced is MaxWaste units. The goal is to determine the number of each beaker type to use in order to maximize the total amount of slime produced.

A dog hospital has TotalToothMedication units of tooth medication to make both small and large bones. A small bone requires ToothMedicationSmall units of tooth medication and MeatSmall units of meat. A large bone requires ToothMedicationLarge units of tooth medication and MeatLarge units of meat. Since most dogs prefer small bones, at least MinPercentageSmallBones percent of the bones made must be small. In addition, the hospital must make at least MinLargeBones large bones. How many of each bone should be made to minimize the amount of meat needed?

A scientist in the arctic needs to consume enough vitamin D and vitamin C by drinking two types of juice: orange juice and apple juice. Each box of orange juice contains VitDOrangeJuice units of vitamin D and VitCOrangeJuice units of vitamin C. Each box of apple juice contains VitDAppleJuice units of vitamin D and VitCAppleJuice units of vitamin C. Since the scientist prefers apple juice, he must drink at least MinAppleToOrangeRatio times the number of apple juice boxes as the number of orange juice boxes. However, he must also drink at least MinOrangeJuice boxes of orange juice. To avoid a vitamin C overdose, the scientist can consume at most MaxVitaminC units of vitamin C. The objective is to determine the number of each type of juice box the scientist should drink to maximize his total vitamin D intake.

A cleaning company uses a cleansing chemical and odor-removing chemical to clean a house. Each unit of the cleansing chemical takes CleansingTimePerUnit to be effective while each unit of the odor-removing chemical takes OdorRemovingTimePerUnit to be effective. The company must use at least MinCleansingChemical of the cleansing chemical. In total, at least MinTotalChemicals of chemicals must be used per house. Further, because the cleansing chemical is strong, there can be at most MaxCleansingToOdorRatio times the amount of odor-removing chemical as cleansing chemical. The objective is to determine the number of units of each chemical to minimize the total time it takes for a house to be cleaned.

A drug company is producing two types of pills: allergy pills and fever reducing pills, using two factories: factory 1 and factory 2. Factory 1 produces AllergyPillsPerHourFactory1 allergy pills and FeverPillsPerHourFactory1 fever reducing pills per hour, requiring CompoundPerHourFactory1 units of a rare compound per hour. Factory 2 produces AllergyPillsPerHourFactory2 allergy pills and FeverPillsPerHourFactory2 fever reducing pills per hour, requiring CompoundPerHourFactory2 units of the rare compound per hour. The company has TotalCompoundAvailable units of the rare compound available. The company must produce at least MinAllergyPills allergy pills and MinFeverPills fever reducing pills. The objective is to determine the number of hours each factory should operate to minimize the total operating time while satisfying all constraints.

A clinic takes patient blood pressure either using an automatic machine or a manual machine. The automatic machine takes TimeAutomatic per patient while the manual machine takes TimeManual per patient. Since the automatic machine frequently breaks, at least ManualToAutomaticRatio times the number of patients processed by the automatic machine must have their blood pressure taken by the manual machine. However, at least MinPatientsAutomatic patients can be processed by the automatic machine. If the clinic is open for TotalTime, maximize the number of patients whose blood pressure can be taken.

A patient with a sore throat can drink NumSyrups different syrups for treatment. Each serving of syrup i delivers MedicineToThroat[i] units of medicine to the throat and MedicineToLungs[i] units of medicine to the lungs. Furthermore, each serving of syrup i contains SugarContent[i] units of sugar. The patient can get at most MaxMedicineThroat units of medicine for the throat and at least MinMedicineLungs units of medicine to the lungs. How many servings of each syrup should the patient take to minimize their sugar intake?

A science show performs NumDemos different demonstrations. Demonstration i requires MintRequirement[i] units of mint and ActiveRequirement[i] units of active ingredient to produce MintyFoamProduced[i] units of minty foam. Additionally, demonstration i produces BlackTarProduced[i] units of black tar. The show has AvailableMint units of mint and AvailableActive units of active ingredient available. The maximum allowable production of black tar is MaxBlackTar units. The task is to determine the optimal number of each demonstration to maximize the total amount of minty foam produced, without exceeding the available resources and the maximum allowable black tar.

A parent feeds their baby two flavors of baby food, apple and carrot, in order to meet the baby's fat and folate requirements. Each serving of apple flavored baby food contains FatPerAppleServing units of fat and FolatePerAppleServing units of folate. Each serving of carrot flavored baby food contains FatPerCarrotServing units of fat and FolatePerCarrotServing units of folate. The baby does not like the carrot flavor, and therefore he must eat AppleCarrotConsumptionRatio times as many apple flavored baby food servings as carrot flavored baby food servings. However, he must eat at least MinCarrotServings servings of carrot flavored baby food. If the baby can consume at most MaxFolate units of folate, how many servings of each should he eat to maximize his fat intake?

A pharmacy has NumFactories factories, where they make NumProducts different products. Each factory produces ProductionRate of each product per hour. Each factory also requires BaseGelRequirement units of base gel per hour. The pharmacy has AvailableBaseGel units of base gel available. Further, they must make at least MinProduction of each product. The goal is to determine how many hours each factory should be run to meet the minimum production requirements and minimize the total time needed.

A man needs to consume at least a minimum required number of units of fat (RequiredFat) and a minimum required number of calories (RequiredCalories) by eating burgers and slices of pizza. Each burger contains given units of fat (FatPerBurger), calories (CaloriesPerBurger), and cholesterol (CholesterolPerBurger). Each slice of pizza contains given units of fat (FatPerPizza), calories (CaloriesPerPizza), and cholesterol (CholesterolPerPizza). He needs to consume at least twice as many slices of pizza as burgers (PizzaToBurgerRatio) and wants to minimize his cholesterol intake.

A clinic makes two types of vitamin batches: shots and pills. Each batch of vitamin shots requires UnitCShots units of vitamin C and UnitDShots units of vitamin D. Each batch of vitamin pills requires UnitCPills units of vitamin C and UnitDPills units of vitamin D. The number of batches of vitamin pills (NumBatchesPills) must be larger than the number of batches of vitamin shots (NumBatchesShots). The clinic can make at most UpperLimitBatchesShots batches of vitamin shots. The clinic has AvailableC units of vitamin C and AvailableD units of vitamin D. Each batch of vitamin shots supplies SupplyShots people and each batch of vitamin pills supplies SupplyPills people. The goal is to determine the number of batches of each type (NumBatchesShots and NumBatchesPills) to maximize the number of people that can be supplied.

A doctor's office takes the temperature of patients one by one either by using an electronic or regular thermometer. The electronic thermometer takes ElectronicThermometerTime to make a reading while the regular thermometer takes RegularThermometerTime to make a reading. Since the electronic thermometer is more accurate, at least MinElectronicRegularRatio times as many patients should have their temperature checked by the electronic thermometer than the regular thermometer. Since the electronic thermometer has a cooldown time, at least MinRegularPatients patients should have their temperature checked by a regular thermometer. If the office is open for TotalOpenTime, maximize the total number of patients whose temperature is taken.

A patient can be hooked up to two machines to have medicine delivered, machine 1 and machine 2. Machine 1 delivers MedicineDeliveryRateMachine1Heart units of medicine to the heart per minute and MedicineDeliveryRateMachine1Brain units of medicine per minute to the brain. Machine 2 delivers MedicineDeliveryRateMachine2Heart units of medicine per minute to the heart and MedicineDeliveryRateMachine2Brain units of medicine per minute to the brain. In addition however, machine 1 creates WasteProductionRateMachine1 units of waste per minute while machine 2 creates WasteProductionRateMachine2 units of waste per minute. If at most MaxMedicineHeart units of medicine can be received by the heart and at least MinMedicineBrain units of medicine should be received by the brain, how many minutes should each machine be used to minimize the total amount of waste produced?

A travelling salesman consumes packs of ramen and fries. Each pack of ramen provides RamenCalories, RamenProtein, and RamenSodium. Each pack of fries provides FriesCalories, FriesProtein, and FriesSodium. The salesman has NumMeals meals, and at most MaxRamenPercentage of these meals can be ramen. The salesman needs to consume at least MinCalories and MinProtein in total across all meals and seeks to minimize his sodium intake.

A science store produces two types of boxes, a cheap box and an expensive box. Each cheap box requires MetalCheapBox units of metal and AcidCheapBox units of acid to produce FoamCheapBox units of foam, while giving off HeatCheapBox units of heat. Each expensive box requires MetalExpensiveBox units of metal and AcidExpensiveBox units of acid to produce FoamExpensiveBox units of foam, while giving off HeatExpensiveBox units of heat. The store has TotalMetal units of metal and TotalAcid units of acid available. The total heat given off by the boxes cannot exceed MaxHeat units. The objective is to determine the number of each type of box to produce in order to maximize the total foam produced.

A woman eats servings of almonds and servings of cashews to meet her weekly intake of MinimumCalories calories and MinimumProtein grams of protein. A serving of almonds provides CaloriesAlmonds calories, ProteinAlmonds grams of protein, and FatAlmonds grams of fat. A serving of cashews provides CaloriesCashews calories, ProteinCashews grams of protein, and FatCashews grams of fat. The woman must consume at least MinimumAlmondCashewRatio times as many servings of almonds as cashews. The objective is to determine the number of servings of almonds and cashews she should eat to minimize her fat intake.

A company makes LiquidSanitizer and FoamSanitizer. Each LiquidSanitizer requires WaterLiquidSanitizer units of water and AlcoholLiquidSanitizer units of alcohol. Each FoamSanitizer requires WaterFoamSanitizer units of water and AlcoholFoamSanitizer units of alcohol. The company has TotalWater units of water and TotalAlcohol units of alcohol available. The number of FoamSanitizer made must be greater than the number of LiquidSanitizer. Additionally, at most MaxLiquidSanitizer units of LiquidSanitizer can be made. Each LiquidSanitizer can clean HandsCleanedByLiquid hands and each FoamSanitizer can clean HandsCleanedByFoam hands. The objective is to determine the number of LiquidSanitizer and FoamSanitizer to produce to maximize the total number of hands that can be cleaned.

A patient in the hospital can take two different pain killers, pain killer 1 and pain killer 2. Per dose, pain killer 1 delivers LegsMedicinePainKiller1 units of medicine to the legs and BackMedicinePainKiller1 units of medicine to the back. Per dose, pain killer 2 delivers LegsMedicinePainKiller2 units of medicine to the legs and BackMedicinePainKiller2 units of medicine to the back. In addition, pain killer 1 delivers SleepMedicinePainKiller1 units of sleeping medicine and pain killer 2 delivers SleepMedicinePainKiller2 units of sleeping medicine. At most MaxSleepMedicine units of sleeping medicine should be delivered, and at least MinLegsMedicine units of medicine should be delivered to the legs. The goal is to determine the number of doses of each pain killer that should be taken to maximize the amount of medicine delivered to the back.

A pharmaceutical company has TotalPainkillerMedicine units of painkiller medicine and makes daytime and nighttime painkiller pills. A daytime pill has PainkillerInDaytimePill units of painkiller medicine and SleepMedicineInDaytimePill units of sleep medicine. A nighttime pill has PainkillerInNighttimePill units of painkiller medicine and SleepMedicineInNighttimePill units of sleep medicine. Since some people don’t want the sleeping effect, at least MinPercentageDaytimePills percent of the pills must be daytime pills. However at least MinNighttimePills nighttime pills must be made. The objective is to determine the number of each pill that should be made to minimize the total amount of sleep medicine required.

A competitive eater challenges himself to eat slices of cheesecake and caramel cake. Each slice of cheesecake contains CaloriesCheesecake and SugarCheesecake. Each slice of caramel cake contains CaloriesCaramelCake and SugarCaramelCake. He prefers cheesecake and decides to eat at least MinCheesecakeRatio times as many slices of cheesecake as there are slices of caramel cake. However, he must also eat at least MinCaramelSlices slices of caramel cake. If he can consume at most MaxCalories in one day, how many slices of each cake should he eat to maximize the total amount of sugar he consumes?

A patient needs to determine the optimal number of oranges and grapefruits to eat in order to minimize their sugar intake. One orange contains VitCOranges units of vitamin C, VitAOranges units of vitamin A, and SugarOranges grams of sugar. One grapefruit contains VitCGrapefruit units of vitamin C, VitAGrapefruit units of vitamin A, and SugarGrapefruit grams of sugar. The patient must get at least MinVitC units of vitamin C and MinVitA units of vitamin A. Additionally, the patient must consume at least 2 times as many oranges as grapefruits.

A patient is undergoing radiation treatment involving two beams, Beam1 and Beam2. Beam1 delivers DoseBeam1Pancreas units of medicine per minute to the benign area of the pancreas and DoseBeam1Skin units of medicine per minute to the benign area of the skin. Beam2 delivers DoseBeam2Pancreas units of medicine per minute to the benign area of the pancreas and DoseBeam2Skin units of medicine per minute to the benign area of the skin. In addition, Beam1 delivers DoseBeam1Tumor units of medicine per minute to the tumor and Beam2 delivers DoseBeam2Tumor units of medicine per minute to the tumor. At most MaxDoseSkin units of medicine should be received by the skin and at least MinDoseTumor units of medicine should be delivered to the tumor. The objective is to determine how many minutes of each beam, denoted as TimeBeam1 and TimeBeam2, should be used to minimize the total radiation received by the pancreas. This requires solving an optimization problem with the constraints TimeBeam1 ≥ 0 and TimeBeam2 ≥ 0.

A body builder buys pre prepared meals, a turkey dinner and a tuna salad sandwich. The turkey dinner contains ProteinTurkeyDinner grams of protein, CarbsTurkeyDinner grams of carbs, and FatTurkeyDinner grams of fat. The tuna salad sandwich contains ProteinTunaSandwich grams of protein, CarbsTunaSandwich grams of carbs, and FatTunaSandwich grams of fat. The bodybuilder wants to get at least MinProtein grams of protein and MinCarbs grams of carbs. In addition, because the turkey dinner is expensive, at most MaxPercentTurkeyDinner percent of the meals should be turkey dinner. How many of each meal should he eat if he wants to minimize his fat intake?

A chemistry teacher teaches her students two experiments, experiment 1 and experiment 2. In experiment 1, RedLiquidExp1 units of the red liquid and BlueLiquidExp1 units of the blue liquid mix to create GreenGasExp1 units of green gas. In experiment 2, RedLiquidExp2 units of the red liquid and BlueLiquidExp2 units of the blue liquid mix to create GreenGasExp2 units of green gas. In addition, experiment 1 produces SmellyGasExp1 units of smelly gas while experiment 2 produces SmellyGasExp2 units of smelly gas. The lab has available TotalRedLiquid units of red liquid and TotalBlueLiquid units of blue liquid. If at most MaxSmellyGas units of smelly gas can be produced, how many experiments of each should be done to maximize the total amount of green gas produced?

A drug company is making pain killers and sleeping pills using two processes, Process1 and Process2. Process1 produces PainKillers1 units of pain killers and SleepingPills1 units of sleeping pills per hour. Process2 produces PainKillers2 units of pain killers and SleepingPills2 units of sleeping pills per hour. Process1 requires Material1 units of preliminary material while Process2 requires Material2 units of preliminary material. The drug company has AvailableMaterial units of preliminary material and must produce at least MinPainKillers units of pain killers and MinSleepingPills units of sleeping pills. The goal is to determine how many hours each of Process1 and Process2 should be run to meet the production requirements while minimizing the total hours of operation.

A dietician recommends her client eat blueberries and strawberries to meet her anti-oxidant and mineral requirement. A pack of blueberries contains AntioxidantsBlueberries units of anti-oxidants and MineralsBlueberries units of minerals. A pack of strawberries contains AntioxidantsStrawberries units of anti-oxidants and MineralsStrawberries units of minerals. The client must get at least MinAntioxidants units of anti-oxidants and MinMinerals units of minerals. In addition, because blueberries are not in season, the dietician recommends she eats at least MinRatioStrawberriesBlueberries times as many packs of strawberries as blueberries. If a pack of blueberries contains SugarBlueberries grams of sugar and a pack of strawberries contains SugarStrawberries grams of sugar, how many packs of each should she consume to minimize her sugar intake?

A company can transport their boxes using their own vans or by renting trucks. Their vans can transport BoxesPerVan per trip while a truck can transport BoxesPerTruck per trip. Since they own their vans, the cost per van trip is CostPerVan while the cost per truck trip is CostPerTruck. The company needs to transport at least MinBoxes and they have a budget of MaxBudget. Since the vans also provide advertising, the number of trips by van must be larger than the number of trips by trucks. How many trips by each should be done to minimize the total number of trips?

A farmer transports his honey in two types of bottles, small and large. A small bottle has a capacity of SmallBottleCapacity units of honey while a large bottle has a capacity of LargeBottleCapacity units of honey. There are at most MaxSmallBottles small bottles and MaxLargeBottles large bottles available. The farmer must use at least twice as many small bottles as large bottles. Additionally, the total number of bottles used must not exceed MaxTotalBottles, and at least MinLargeBottles bottles must be large. The objective is to determine the number of each type of bottle to use to maximize the total amount of honey transported.

An international goods exporter uses ships and planes to transport goods. A ship can take ShipCapacity containers worth of goods and uses ShipFuel liters of fuel per trip. A plane can take PlaneCapacity containers worth of goods and uses PlaneFuel liters of fuel per trip. The company needs to transport at least MinContainers containers worth of goods. In addition, there can be at most MaxPlaneTrips plane trips made and a minimum of MinShipPercentage fraction of the trips must be made by ship. How many of each trip should be made to minimize the total amount of fuel consumed?

The objective is to minimize the total amount of time needed to transport a certain number of ducks using two transportation methods: boats and canoes. Each boat can take BoatDucksPerTrip ducks per trip and takes BoatTripTime minutes per trip. Each canoe can take CanoeDucksPerTrip ducks per trip and takes CanoeTripTime minutes per trip. There is a maximum limit of MaxBoatTrips boat trips allowed, and at least MinCanoeTripPercentage percent of the total trips should be by canoe. Additionally, at least MinDucksToShore ducks need to be taken to shore.

A soda company sends bottles of their soda to stores around the city in old and new vans. An old van can take CapacityOldVan soda bottles while a new van can take CapacityNewVan soda bottles. An old van produces PollutionOldVan units of pollution while a new van only produces PollutionNewVan units of pollution. The company needs to send at least MinBottles bottles. In addition, at most MaxNewVans new vans can be used. How many of each van should be used to minimize the total amount of pollution produced?

A lawn mowing service provides neighborhood services using small teams and large teams. A small team requires EmployeesPerSmallTeam employees and can mow AreaSmallTeam sq ft of lawn. A large team requires EmployeesPerLargeTeam employees and can mow AreaLargeTeam sq ft of lawn. The company has TotalEmployees employees available. The number of small teams must be at least SmallTeamsFactor times the number of large teams. There must be at least MinLargeTeams large teams and at least MinSmallTeams small teams. How many of each team type should be used to maximize the amount of lawn that can be mowed?

An aquarium conducts shows using otters and dolphins. Each otter performs OtterTricks tricks and requires OtterTreats treats to do so. Each dolphin performs DolphinTricks trick and requires DolphinTreats treats to do so. Considering the popularity, at least MinDolphins dolphins must be used, and no more than MaxOtterPercentage of the performers can be otters. With a total of TotalTreats treats available, the task is to maximize the total number of tricks that can be performed.

An oil and gas company is sending their oil to the port using containers and trucks. A container can hold ContainerCapacity units of oil while a truck can hold TruckCapacity units of oil. Due to government restrictions, the number of trucks used has to be at most MaxTruckContainerRatio times the number of containers used. If at least MinOilRequired units of oil need to be sent to the port and at least MinContainers containers need to be used, the objective is to minimize the total number of containers and trucks used.

A school is organizing a field trip and plans to hire buses of two types. One type of bus has a capacity of SmallBusCapacity students, and the other type has a capacity of LargeBusCapacity students. The school needs to provide transportation for at least MinStudents students. In addition, no more than a fraction MaxLargeBusPercentage of the total number of hired buses can be large buses. The objective is to determine the number of each type of bus to hire to minimize the total number of buses.

A meat shop ships their burger patties using refrigerated trucks and vans. Each truck can take TruckCapacity patties at a cost of TruckCost per trip. Each van can take VanCapacity patties at a cost of VanCost per trip. The number of trucks must not exceed the number of vans. The meat shop has to ship at least MinPatties patties and they have a maximum budget of MaxBudget. How should they plan their shipment to minimize the total number of trips?

A florist transports his flowers to stores in small bouquets and large bouquets. Each small bouquet has SmallBouquetFlowers flowers while each large bouquet has LargeBouquetFlowers flowers. The florist can transport at most MaxSmallBouquets small bouquets and MaxLargeBouquets large bouquets. In total, he can transport at most MaxTotalBouquets bouquets, and he must transport at least MinLargeBouquets large bouquets. Additionally, he must transport at least SmallLargeRatio times the number of large bouquets as small bouquets. The goal is to determine the number of each bouquet type he should transport to maximize the total number of flowers that reach the stores.

A toy store decides to deliver gifts using two shipping companies: a new company and an old company. The new company can deliver NewCompanyGiftCapacity gifts per trip, while the old company can deliver OldCompanyGiftCapacity gifts per trip. The new company uses NewCompanyDieselUsage liters of diesel per trip, while the old company uses OldCompanyDieselUsage liters of diesel per trip. The toy store needs to deliver at least MinGiftsRequired gifts. The new company can make at most MaxNewCompanyTrips trips. To ensure the old company does not go out of business, at least MinOldCompanyTripPercentage of all trips must be made by the old company. The goal is to determine how many trips each company should make to minimize the total amount of diesel used.

A zoo needs to transport TotalMonkeys to the vet either by bus or by car. A bus can transport MonkeysPerBusTrip per trip and takes TimePerBusTrip. A car can transport MonkeysPerCarTrip per trip and takes TimePerCarTrip. There can be at most MaxBusTrips. In addition, since the monkeys get aggressive when there are too many in one place, at least MinCarTripPercentage of the trips should be by car. If the zoo needs to transport TotalMonkeys, how many trips of each should be done to minimize the total time required to transport the monkeys?

A farmer decides to move NumCows to a nearby farm using helicopters and trucks. A helicopter can take HelicopterCapacity cows per trip and creates HelicopterPollution units of pollution. A truck can take TruckCapacity cows per trip and creates TruckPollution units of pollution. The farmer only has enough money for at most MaxTruckTrips truck trips. The objective is to minimize the total amount of pollution produced while transporting NumCows.

A meal service company delivers meals to customers either on electric bikes or scooters. A bike can hold MealsPerBike meals and requires ChargePerBike units of charge. A scooter can hold MealsPerScooter meals and requires ChargePerScooter units of charge. Since the city is more friendly towards scooters, at most MaxPercentageBikes percentage of the electric vehicles can be bikes and at least MinScooters scooters must be used. If the company only has TotalCharge units of charge available, how many of each vehicle should be used to maximize the number of meals that can be delivered?

A car manufacturer ships their cars on large and small planes. A large plane can carry CarsPerLargePlane cars while a small plane can carry CarsPerSmallPlane cars. To reduce pollution, the number of large planes must be less than the number of small planes. The manufacturer wants to deliver at least TotalCars cars while minimizing the total number of planes used.

A mine sends ore to the surface in small and large wagons. A small wagon holds SmallWagonCapacity units of ore while a large wagon holds LargeWagonCapacity units of ore. To avoid taking up too much space on the ramps, the number of small wagons must be at least MinRatioSmallToLarge times the number of large wagons. In addition, at least MinLargeWagons large wagons must be used. If TotalOre units of ore need to be taken to the surface, minimize the total number of wagons needed.

A theme park transports its visitors around the park either by scooter or rickshaw. A scooter can carry ScooterCapacity people while a rickshaw can carry RickshawCapacity people. To avoid excessive pollution, at most MaxRickshawProportion of the vehicles used can be rickshaws. If the park needs to transport at least MinVisitors visitors, minimize the total number of scooters used.

A company delivers their packages using two types of animals. One type of animal can carry PackagesPerCamel packages and requires FoodPerCamel units of food. The other type of animal can carry PackagesPerHorse packages and requires FoodPerHorse units of food. The company needs to deliver at least MinPackages packages and has TotalFood units of food available. Additionally, the number of the second type of animal cannot exceed the number of the first type of animal. The goal is to determine how many of each animal to use in order to minimize the total number of animals.

A mail delivery service in an island village delivers mail by regular and speed boats. A regular boat can carry MailCapacityRegular pieces of mail per trip and uses GasUsageRegular liters of gas. A speed boat can carry MailCapacitySpeed pieces of mail per trip and uses GasUsageSpeed liters of gas. There can be at most MaxRegularTrips regular boat trips. Since customers want their mail as fast as possible, at least MinSpeedBoatTripPercentage percent of the trips must be made by speed boats. If the service needs to deliver TotalMail pieces of mail, how many trips of each should be made to minimize the total amount of gas consumed?

A chicken farmer needs to transport TotalChicken using either buses or cars. A bus can carry BusCapacity and takes BusTimePerTrip hours per trip. A car can carry CarCapacity and takes CarTimePerTrip hours per trip. There can be at most MaxBusTrips and at least MinCarTripPercentage of the trips must be by car. The farmer needs to determine the number of trips by car and bus to minimize the total transportation time.

Children can go to school either by van or by minibus. A van can take VanCapacity kids and produces VanPollution units of pollution. A minibus can take MinibusCapacity kids and produces MinibusPollution units of pollution. There are at least NumChildren kids that need to go to school and at most MaxMinibuses minibuses can be used. In addition, the number of vans used must exceed the number of minibuses.

A farmer needs to send at least MinCornWeight of corn to the city using either tractors or cars. A tractor can carry TractorCapacity of corn, and a car can carry CarCapacity of corn. The number of cars used has to be at least CarToTractorRatio times the number of tractors used. The objective is to minimize the total number of tractors and cars needed.

A village provides bike and car transportation for everyone. A bike can take PeoplePerBike people while a car can take PeoplePerCar people. Since cars are more expensive, at most MaxPercentageCars of the vehicles can be cars. If the village needs to transport at least MinPeople people, how many of each vehicle should be used to minimize the total number of bikes needed?

A water salesman collects water from a glacier and transports it in either small or large kegs. A small keg can hold SmallKegCapacity liters of water while a large keg can hold LargeKegCapacity liters of water. The salesman has available at most MaxSmallKegs small kegs and MaxLargeKegs large kegs. Since small kegs are easier to carry, at least MinSmallToLargeRatio small kegs must be used for every large keg. If he can transport at most MaxTotalKegs kegs total and at least MinLargeKegs kegs must be large, how many of each should he use to maximize the total amount of glacial water he can transport?

A city needs to transport at least TotalMail pieces of mail using either submarines or boats. A submarine can carry SubmarineCapacity pieces of mail per trip and uses SubmarineGasUsage liters of gas per trip. A boat can carry BoatCapacity pieces of mail per trip and uses BoatGasUsage liters of gas per trip. The maximum allowable trips by submarine is MaxSubmarineTrips. At least MinBoatTripPercentage of the trips must be made by boat. The objective is to determine the number of submarine trips and boat trips that minimize the total gas usage while meeting these constraints.

The weather is freezing and the fish in the pond need to be transported either by helicopter or car. A helicopter can take HelicopterFishCapacity fish per trip and takes HelicopterTripTime minutes. A car can take CarFishCapacity fish per trip and takes CarTripTime minutes. Since helicopter trips are expensive, there can be at most MaxHelicopterTrips helicopter trips. In addition, at least MinCarTripPercentage of the trips should be by car. If at least MinFishRequired fish need to be transported, how many of each trip should be taken to minimize the total time needed?

A tourist spot in the mountains allows visitors to travel to the top either by hot-air balloon or gondola lift. A hot air balloon can carry BalloonCapacity visitors while a gondola lift can carry GondolaCapacity visitors. Each hot air balloon produces BalloonPollution units of pollution while each gondola lift produces GondolaPollution units of pollution. There can be at most MaxBalloons hot-air balloon rides and at least MinVisitors visitors need to be transported. How many of each type of transport method should be taken to minimize the total pollution produced?

A factory transports rice to the city in horse-drawn carts that are either medium or large size. A medium sized cart requires HorsesPerMediumCart horses and can carry CapacityMediumCart kg of rice. A large sized cart requires HorsesPerLargeCart horses and can carry CapacityLargeCart kg of rice. The factory has NumHorses horses available. The number of medium sized carts must be MediumToLargeCartRatio times the number of large sized carts. There must be at least MinMediumCarts medium sized carts and at least MinLargeCarts large sized carts. How many of each cart size should be used to maximize the amount of rice that can be transported?

A dog school trains labradors and golden retrievers to deliver newspapers. A labrador can deliver NewspapersPerLabrador newspapers at a time and requires TreatsPerLabrador small bone treats for service. A golden retriever can deliver NewspapersPerGoldenRetriever newspapers at a time and requires TreatsPerGoldenRetriever small bone treats per service. The school only has TotalTreats small bone treats available. In addition, at least MinGoldenRetrievers golden retrievers must be used and at most MaxLabradorPercentage of the dogs can be labradors. How many of each dog should be used to maximize the number of newspapers that can be delivered?

A construction company in the tropics uses cows and elephants to carry bricks. A cow can carry BricksPerCow on its back while an elephant can carry BricksPerElephant on its back. To avoid having elephants create too much traffic, the number of elephants cannot exceed the number of cows. In addition, there can be at most MaxCowsToElephantsRatio times the number of cows as elephants (i.e., the maximum ratio of cows to elephants is MaxCowsToElephantsRatio). If the company needs to transport at least MinBricks, find the minimum number of animals, cows and elephants, that can be used.

A farmer on an island sends corn to the main land either by ferry or light rail. Each ferry trip can take FerryCapacity boxes of corn while each light rail trip can take LightRailCapacity boxes of corn. Since ferry trips are slow, the number of light rail trip has to be at least LightRailToFerryRatio times the number of ferry trips. If the farmer wants to send at least MinBoxes boxes of corn, minimize the total number of trips of either type needed.

A chemical company is transporting their hydrogen using two different methods, high pressure tube trailers and liquefied hydrogen tankers. Constants: VolumeHPTT is the volume transported per high pressure tube trailer trip. CostHPTT is the cost per high pressure tube trailer trip. VolumeLHT is the volume transported per liquefied hydrogen tanker trip. CostLHT is the cost per liquefied hydrogen tanker trip. MinVolume is the minimum volume of hydrogen to transport. Budget is the budget available for transportation. Decision Variables: NumHPTT is the number of trips done using the high-pressure tube trailers, and NumLHT is the number of trips done using the liquefied hydrogen tankers. Constraints: The total transported volume (VolumeHPTT * NumHPTT + VolumeLHT * NumLHT) must be at least MinVolume. The total cost (CostHPTT * NumHPTT + CostLHT * NumLHT) must not exceed Budget. The number of high pressure tube trailer trips (NumHPTT) must be less than the number of liquefied hydrogen tanker trips (NumLHT). Objective: Minimize the total number of trips (NumHPTT + NumLHT).

A farmer transports his grapes in either small crates or large crates. Each small crate can take SmallCrateCapacity number of grapes while each large crate can take LargeCrateCapacity number of grapes. The customers prefer smaller crates, so the farmer must use at least MinSmallCratesToLargeCratesRatio times as many small crates as large crates. The farmer has at most MaxSmallCrates small crates and at most MaxLargeCrates large crates available. His truck can transport at most MaxTotalCrates crates in total, using at least MinLargeCrates large crates. The objective is to determine the number of small and large crates to maximize the total number of grapes transported.

Patients need to be transported to the hospital by either a helicopter or a bus. A helicopter can transport HelicopterCapacity patients per trip and takes HelicopterTripTime hours. A bus can transport BusCapacity patients per trip and takes BusTripTime hours. At least MinPatientsToTransport patients need to be transported and at least MinHelicopterTripPercentage percent of the trips should be by helicopter. Additionally, there can be at most MaxBusTrips bus trips. The objective is to minimize the total time to transport the patients.

Employees have the option of car-pooling to work or taking the company bus. A car can take CarCapacity employees and produces CarPollution units of pollution, while a bus can take BusCapacity employees and produces BusPollution units of pollution. At least NumEmployees employees need to be transported and a maximum of MaxBuses buses can be used. How many of each type of transport should be taken to minimize the total pollution produced.

A company removes snow using two types of trucks: small trucks and large trucks. Let NumSmallTrucks represent the number of small trucks and NumLargeTrucks represent the number of large trucks. A small truck requires PeopleForSmallTruck people to shovel the snow and can carry CapacitySmallTruck units of snow. A large truck requires PeopleForLargeTruck people to shovel the snow and can carry CapacityLargeTruck units of snow. The company has a total of TotalPeople available. At least MinSmallTrucks small trucks are required. At least MinLargeTrucks large trucks are required. Additionally, the number of small trucks (NumSmallTrucks) must be twice the number of large trucks (NumLargeTrucks). The goal is to determine the number of each type of truck to use in order to maximize the total amount of snow transported.

A magic school sends letters to students either by carrier pigeons or owls. A carrier pigeon can carry CarrierPigeonCapacity letters at a time and requires CarrierPigeonTreats treats for service. An owl can carry OwlCapacity letters at a time and requires OwlTreats treats for service. At most MaxOwlPercentage of the birds can be owls. In addition, the school only has NumTreats treats available and at least MinCarrierPigeons carrier pigeons must be used. How many of each bird can be used to maximize the total number of letters that can be sent.

A large fishing boat sends fish back to shore either by small canoes or smaller diesel boats. A small canoe can carry back CapacityCanoe fish while a small diesel boat can carry back CapacityDieselBoat fish. In order to follow environmental rules, the number of small canoes used has to be at least CanoeToDieselRatio times the number of diesel boats used. If at least MinFish fish need to be transported to shore, minimize the total number of canoes and diesel boats needed.

A company needs to provide a diet that meets at least MinCalories calories, MinProtein grams of protein, and MinCarbs grams of carbs for its workers. One type of food, referred to as a hamburger, costs HamburgerCost and contains HamburgerCalories calories, HamburgerProtein grams of protein, and HamburgerCarbs grams of carbs. Another type of food, referred to as a chicken wrap, costs ChickenWrapCost and contains ChickenWrapCalories calories, ChickenWrapProtein grams of protein, and ChickenWrapCarbs grams of carbs. The objective is to find the minimum cost combination of these foods that meets the dietary requirements.

A company makes NumProducts different products. Each product i requires AssemblyTime[i] of assembly and SandingTime[i] of sanding. The company has available AvailableAssemblyTime for assembly and AvailableSandingTime for sanding. The profit per unit of product i is Profit[i]. How many of each product should the company make to maximize profit?

A young boy wants to get a minimum of MinCalcium units of calcium and MinIron units of iron per day. A glass of milk costs CostMilk and contains CalciumMilk units of calcium and IronMilk units of iron. A plate of vegetables costs CostVegetables and contains CalciumVegetables units of calcium and IronVegetables units of iron. The goal is to determine the number of glasses of milk and plates of vegetables he should consume to minimize his cost.

Bob wants to design a diet consisting of two types of food items. Assume that the first food item costs CostNoodles per serving and contains CaloriesNoodles calories and ProteinNoodles grams of protein. Assume that the second food item costs CostProteinBar per serving and contains CaloriesProteinBar calories and ProteinProteinBar grams of protein. Bob is interested in minimizing the cost of his diet while ensuring that his meals provide at least MinCalories calories and at least MinProtein grams of protein per day.

An individual takes two different supplements to satisfy daily iron and calcium requirements. Each pill of the first supplement contains IronInSupplementA units of iron and CalciumInSupplementA units of calcium. Each pill of the second supplement contains IronInSupplementB units of iron and CalciumInSupplementB units of calcium. The individual needs at least MinIronRequirement units of iron and at least MinCalciumRequirement units of calcium per day. The cost per pill of the first supplement is PriceSupplementA, and the cost per pill of the second supplement is PriceSupplementB. The objective is to determine the number of pills of each supplement the individual should take to minimize the total cost while meeting the daily iron and calcium requirements.

A woman has TotalMoney to gamble on three sports bets: a basketball tournament, a horse race, and a soccer game. Based on simple analysis, the woman determines her chance of losing her money would be ProbLoseBasketball for the basketball tournament, ProbLoseHorseRace for the horse race, and ProbLoseSoccer for the soccer game. The payout for each dollar put on each bet will be PayoutBasketball for the basketball tournament, PayoutHorseRace for the horse race, and PayoutSoccer for the soccer game. She limits her average chance of losing her money to at most MaxAvgProbLose. The goal is to determine the amount of money to put on each sport bet to maximize her expected payout.

A repairman fixes two types of appliances. Each appliance of the first type (washing machine) takes InspectionTimeWashingMachine for inspection and FixingTimeWashingMachine for fixing. Each appliance of the second type (freezer) takes InspectionTimeFreezer for inspection and FixingTimeFreezer for fixing. The repairman has AvailableInspectionTime for inspection and AvailableFixingTime in his schedule. If each appliance of the first type repaired earns him EarningWashingMachine and each appliance of the second type repaired earns him EarningFreezer, how many of each appliance should he fix to maximize his earnings?

A bakery has TotalBatter of batter and TotalMilk of milk to make their crepe cakes, sponge cakes, and birthday cakes. A crepe cake needs BatterCrepeCake of batter and MilkCrepeCake of milk. A sponge cake requires BatterSpongeCake of batter and MilkSpongeCake of milk. A birthday cake requires BatterBirthdayCake of batter and MilkBirthdayCake of milk. If the profit per crepe cake is ProfitCrepeCake, the profit per sponge cake is ProfitSpongeCake, and the profit per birthday cake is ProfitBirthdayCake, how many of each should the bakery make to maximize their profit?

Super Shop sells two types of snacks in bulk: cat paw snacks and gold shark snacks. It plans to create two mix products. The first mix contains PercentCatPawFirstMix of cat paw snacks and PercentGoldSharkFirstMix of gold shark snacks. The second mix contains PercentCatPawSecondMix of cat paw snacks and PercentGoldSharkSecondMix of gold shark snacks. The store has AvailableCatPawSnacks of cat paw snacks and AvailableGoldSharkSnacks of gold shark snacks on hand. The profit per kilogram of the first mix is ProfitFirstMix and the profit per kilogram of the second mix is ProfitSecondMix. The objective is to determine how many kilograms of each mix should be prepared to maximize profit.

A company sells two types of subscription software packages: a personal license and a commercial license which will cost CostPersonalLicense and CostCommercialLicense to generate respectively. The marketing department estimates that they can sell at most MaxLicenses licenses for both versions combined a month. The profit per personal license is ProfitPersonalLicense and the profit per commercial license is ProfitCommercialLicense. If the company does not want to spend more than BudgetLimit, how many of each software package should they produce to maximize the profits.

The government is reworking pipes to transport water to houses. Water can be transported through wide pipes or narrow pipes. Wide pipes transport WidePipeFlowRate units of water per minute, and narrow pipes transport NarrowPipeFlowRate units of water per minute. The number of wide pipes can be at most WidePipeFraction the number of narrow pipes. There needs to be at least MinWaterTransport units of water transported every minute, and at least MinWidePipes wide pipes must be used. Minimize the total number of pipes required.

A student takes pills, where each pill type has a different effectiveness time. The student must take at least TotalPills pills over a month, with at least MinVitaminDPills of them being vitamin D pills. The effectiveness times for calcium and vitamin D pills are CalciumTime and VitaminDTime respectively. The student must also take more calcium pills than vitamin D pills. The goal is to determine the number of each type of pill to take in order to minimize the total effectiveness time sum of all pills taken.

An amusement park has two types of games: throwing and climbing games. Throwing games attract ThrowingGameCustomers customers every hour and climbing games attract ClimbingGameCustomers customers every hour. Throwing games cost the amusement park ThrowingGameCost in prizes per hour whereas climbing games cost ClimbingGameCost in prizes per hour. Since throwing games yield the most profit, there must be at least ThrowingToClimbingRatio times as many throwing games as climbing games. However, at least MinClimbingGames games must be climbing games. If the amusement park can have at most MaxPrizeCost in prizes every hour, maximize the total number of customers attracted every hour.

Determine the quantity of circular and rectangular tables to set up at a science fair in order to maximize the number of catered guests. Each circular table fits PosterBoardsCircular poster boards and ParticipantsCircular participants, catering to GuestsCircular guests and taking up SpaceCircularTable units of space. Each rectangular table fits PosterBoardsRectangular poster boards and ParticipantsRectangular participants, catering to GuestsRectangular guests and taking up SpaceRectangularTable units of space. The fair must accommodate at least MinParticipants participants and MinPosterBoards poster boards, without exceeding the TotalAvailableSpace units of space.

An oil and gas company has two types of pipes, a high-volume and a low-volume one. Every day, the high-volume pipe allows a throughput of HighVolumePipeThroughput and it is recommended that TechniciansPerHighVolumePipe technicians closely monitor the pipes to ensure that it is functioning properly. Each day, the low-volume pipe allows a throughput of LowVolumePipeThroughput and TechniciansPerLowVolumePipe technicians should closely monitor for safety reasons. Every day, the oil and gas company needs to meet the demands of at least MinimumDailyDemand of gas and they have TotalTechnicians technicians on their staff. Since the high-volume pipe has a higher risk of environmental damage, at most MaxHighVolumePipePercentage of the total number of pipes can be high-volume ones. Additionally, there must be a minimum of MinLowVolumePipes low-volume pipes. How many of each pipe types should be used to reduce the total number of pipes required?

A hospital hires workers to provide ultrasound services. An ultrasound technician works TechnicianHoursPerShift hours per shift and is paid TechnicianPaymentPerShift per shift, while a graduate researcher works ResearcherHoursPerShift hours per shift and is paid ResearcherPaymentPerShift per shift. Due to regulations, there must be TechnicianToResearcherRatio times as many ultrasound technician shifts as graduate researcher shifts. The hospital requires TotalRequiredHours of ultrasound services and has a TotalBudget available. The objective is to schedule the minimum total number of workers.

A delivery person can have shifts on either bikes or scooters. A shift on a bike delivers OrdersPerBikeShift orders, uses EnergyPerBikeShift units of energy, and brings in TipsPerBikeShift in tips because it is faster. A shift on a scooter delivers OrdersPerScooterShift orders, uses EnergyPerScooterShift units of energy, and brings in TipsPerScooterShift in tips. The delivery person has a total of TotalShifts available per month and TotalEnergy units of energy and must deliver at least MinOrders orders. Additionally, they must have at least MinScooterShifts shifts on scooters because bikes are harder to get. The goal is to determine the number of shifts on each type of transportation to maximize the tips received.

A singer can hold two types of concerts: pop and R&B. Each pop concert brings in PopAudience audience members and requires PopPracticeDays practice days. Each R&B concert brings in RnBAudience audience members and requires RnBPracticeDays practice days. The singer must bring in at least MinAudienceRequired total audience members and has a maximum of MaxPracticeDays practice days available. Additionally, the singer can perform at most MaxRnBConcertRate of his total concerts as R&B concerts. The objective is to determine the number of each type of concert to minimize the total number of concerts.

A neighbourhood pizza restaurant sells two sizes of pizzas: large and medium. Each large pizza requires DoughLargePizza units of dough and ToppingsLargePizza units of toppings, and takes TimeLargePizza minutes to bake. Each medium pizza requires DoughMediumPizza units of dough and ToppingsMediumPizza units of toppings, and takes TimeMediumPizza minutes to bake. The restaurant must use at least MinDough units of dough and at least MinToppings units of toppings. Due to a promotion, at least MinMediumPizza medium pizzas need to be made. To cater to regulars, the number of large pizzas should be at least LargeToMediumRatio times the number of medium pizzas, where LargeToMediumRatio represents the value 2. The objective is to minimize the total baking time.

A restaurant can deliver their food by two methods. Servers can deliver by one method and will have CartInteractions customer interactions and will need to refill food CartRefills times per hour. They can also deliver food by another method with increased mobility, achieving HandInteractions customer interactions while needing HandRefills refills per hour. At least MinCartPercentage of delivery shifts must be by the first method. Additionally, there must be at least MinHandServers servers delivering by the second method. The restaurant aims to have TotalInteractions customer interactions per hour. Determine the number of servers for each delivery method to minimize the total number of refills per hour.

A restaurant has two combos available to their customers: an original meal and an experimental meal. The original meal generates OriginalFoodWaste units of food waste, OriginalWrappingWaste units of wrapping waste, and takes OriginalCookingTime minutes to cook. The experimental meal generates ExperimentalFoodWaste units of food waste, ExperimentalWrappingWaste units of wrapping waste, and takes ExperimentalCookingTime minutes to cook. The restaurant must adhere to a maximum of MaxWrappingWaste units of wrapping waste and MaxFoodWaste units of food waste as regulated by the government. The goal is to determine the number of each meal to be promoted by the waiting staff to minimize the total cooking time while complying with waste constraints.

A metal-working shop has access to two types of metal-working equipment, a chop saw and a steel cutter. A chop saw can work ChopSawCapacity of metal and generates ChopSawWaste units of waste every day. A steel cutter can work SteelCutterCapacity of metal and generates SteelCutterWaste units of waste every day. The metal-working shop must cut TotalMetalCuttingRequirement of metal every day. However, there is a waste limitation and they may at most produce MaxAllowedWaste units of waste every day. The objective is to minimize the total number of metal-working equipment needed.

A market owner wants to start a jam business. He can either sell one unit of jam to customers in a set of small packets or jugs. A set of small packets can hold VolumeSmallPackets ml of jam while a jug can hold VolumeJug ml of jam. Since a set of small packets are more expensive to make, at least MinRatioJugToPackets times the number of jugs must be used as sets of small packets. Additionally, at least MinSetsSmallPackets sets of small packets should be filled. If the market owner has TotalVolumeJam ml of jam, how many sets of small packets or jugs should be selected to maximize the total number of units that he can sell to his customers?

The goal is to determine the quantities of two types of meals, one denoted as 'salad' and the other as 'fruit bowl.' Each type of meal contributes different amounts of vitamins, fibres, and potassium. The objective is to maximize the total potassium intake. There are constraints to ensure a minimum required intake of vitamins and fibres: MinVitamin and MinFibre. Each salad provides VitaminPerSalad units of vitamins and FibrePerSalad units of fibre. Each fruit bowl provides VitaminPerFruitBowl units of vitamins and FibrePerFruitBowl units of fibre. Additionally, there is a constraint that at most MaxFruitBowlPercentage of the meals can be fruit bowls. Each salad contains PotassiumPerSalad units of potassium and each fruit bowl contains PotassiumPerFruitBowl units of potassium.

A company wants to determine the number of retail stores and factory outlets to open to satisfy MinCustomersPerDay, while using at most MaxEmployees. Each retail store brings in RetailStoreCustomers and requires RetailStoreEmployees to operate. Each factory outlet brings in FactoryOutletCustomers and requires FactoryOutletEmployees to run. The goal is to minimize the number of retail stores and factory outlets that need to be open.

A soda company needs to bottle at least MinTotalVolume of soda every day. A can holds VolumeCan of soda whereas a bottle holds VolumeBottle of soda. The number of cans must be at least MinRatioCansToBottles times the number of glass bottles, and at least MinBottles glass bottles must be produced due to contracts. How many cans and bottles should be made to maximize the total number of units produced?

A candle-making company can move their inventory of candles using freight and air. Freight can transport TonsPerFreight tons per trip while using air can transport TonsPerAir tons per trip. Since freight takes longer, the cost for each freight trip is FreightCost while the cost over air for each trip is AirCost. The company needs to transport at least TotalTons tons of candles and they have a budget of Budget. Additionally, due to some urgent orders, at least MinAirPercentage of total tons must be transported through air. There must also be at least MinFreightTrips trips through freight. How many of trip by each should be scheduled to minimize the total number of trips?

A production company needs to transport TotalPeople using large mobile production units or small mobile production units. Large mobile production units can hold LargeUnitCapacity people and take up LargeUnitParkingSpots parking spots whereas small mobile production units can hold SmallUnitCapacity people and take up SmallUnitParkingSpots parking spots. Movie stars prefer to have their small mobile production unit to share with a close staff, therefore at least MinSmallUnits units must be small mobile units. However, due to logistic limitations, large mobile production units are more space efficient per spot, they must make up at least MinPercentageLargeUnits of all vehicles. Minimize the total number of parking spots required by this production.

A company ships items to stores using two different types of boxes. A small box holds MasksInSmallBox masks whereas a large box holds MasksInLargeBox masks. Since small boxes are easier to stack and will be used first to stock stores, there must be at least MinSmallToLargeRatio times as many small boxes as large boxes. Additionally, at least MinLargeBoxes large boxes must be used. If at least MinMasksRequired masks are required to be distributed, how many of each size of box should be used to minimize the total number of boxes needed?

An autobody shop needs to purchase two types of car jacks, an automatic electric one, or a gas-powered one. The automatic electric one can process CarsPerHourElectric cars every hour and uses ElectricityPerHourElectric units of electricity whereas the gas-powered one can process CarsPerHourGas cars each hour using GasPerHourGas units of gas. Since there is a limit to how many automatic electric ones there can be due to the limited number of power outlets, the shop must use less than MaxElectricJacks automatic electric ones. The shop can use at most MaxElectricity units of electricity and MaxGas units of gas. How many of each type of jack should the shop purchase to maximize the amount of cars processed every hour?

A railway company is planning the infrastructure for the city and is considering two types of transportation, trains and trams. Each train can transport PeoplePerTrain people comfortably per hour and each tram can transport PeoplePerTram people comfortably per hour. Since trains take longer to build, the number of trams must be at least MinTramToTrainRatio times the number of trains. The company wants to transport at least MinTotalPeople people per hour and aims to minimize the total number of transportation units.

An automotive company is comparing NumCatalysts different catalysts. The process with catalyst i requires PlatinumRequired[i] units of platinum and PalladiumRequired[i] units of palladium and can perform the conversion to carbon dioxide at a rate of ConversionRate[i] units per second. There are TotalPlatinumAvailable units of platinum and TotalPalladiumAvailable units of palladium available. How many of each catalyst should be used to maximize the amount converted into carbon dioxide?

There are two processes, J and P, to extract metal from ores. Process J extracts MetalExtractedJ units of metal using WaterUsedJ units of water and produces PollutionProducedJ units of pollution. Process P extracts MetalExtractedP units of metal using WaterUsedP units of water and produces PollutionProducedP units of pollution. The objective is to determine the number of times each process should be performed to maximize the total metal extracted, subject to the constraints that the total water used does not exceed MaxWater units and the total pollution produced does not exceed MaxPollution units.

An airport needs to install either escalators or elevators to transport people. Escalators have a transport capacity of EscalatorCapacity and take up EscalatorSpace units of space. Elevators have a transport capacity of ElevatorCapacity and take up ElevatorSpace units of space. The total transport capacity of all installed escalators and elevators must be at least RequiredCapacity. Additionally, the number of escalators must be at least EscalatorElevatorRatio times the number of elevators, and at least MinElevators elevators must be used. The goal is to minimize the total units of space taken by escalators and elevators.

A research group is making fabric and plastic using two methods, method A and method B. Method A produces FabricUnitsA units of fabric and PlasticUnitsA units of plastic per hour. Method B produces FabricUnitsB units of fabric and PlasticUnitsB units of plastic per hour. Method A requires SpecialElementA units of a special element while method B requires SpecialElementB units of the same special element. The research group has available TotalSpecialElement units of the special element and must make at least MinFabric units of fabric and MinPlastic units of plastic. How many of each method should be executed to minimize the total time needed?

Due to an accident, at least MinLocals locals must be moved across a lake. They can either be transported over the lake by a kayak or a motorboat. Kayaks can transport KayakCapacity people per trip and motorboats can transport MotorboatCapacity people per trip. Kayaks take KayakTime minutes per trip whereas motorboats take MotorboatTime minutes per trip. Due to the limited number of motorboats available to the locals, there can be at most MaxMotorboatTrips motorboat trips and at least MinKayakTripProportion of the total trips should be by kayak. How many of each transportation method should be used to minimize the total amount of time needed to transport all the locals?

A researcher is outsourcing annotations and has two options: a specialized third-party or a common third-party annotation company. The specialized company can annotate at a rate of SpecializedRate images per hour whereas the common company can annotate at a rate of CommonRate images per hour. However, the specialized company charges SpecializedCost per hour and the common company charges CommonCost per hour. The researcher has deadlines to meet and must complete a dataset of at least TotalImages images. They also have some special images that only the specialized company can annotate. Therefore, at least a third of the TotalImages must be allocated to the specialized company.

A golf course is organizing an event and can transport guests using either golf carts or pull carts. Each golf cart can transport GuestsPerGolfCart guests, while each pull cart can transport GuestsPerPullCart guest. However, due to spatial constraints, at most MaxGolfCartPercentage of all carts can be golf carts. The goal is to transport at least MinGuests guests while minimizing the total number of carts used.

A smoothie shop produces two different types of smoothies: acai berry smoothies and banana chocolate smoothies. It takes AcaiBerriesPerAcaiSmoothie units of acai berries and WaterPerAcaiSmoothie units of water to make one acai berry smoothie. It takes BananaChocolatePerBananaSmoothie units of banana chocolate and WaterPerBananaSmoothie units of water to make one banana chocolate smoothie. The number of banana chocolate smoothies made must be more than the number of acai berry smoothies made. Additionally, at least MinAcaiFraction of the total smoothies made must be acai berry smoothies. The smoothie shop has TotalAcaiBerries units of acai berries and TotalBananaChocolate units of banana chocolate. The objective is to determine the number of each type of smoothie to make in order to minimize the total amount of water used.

A handmade sports equipment manufacturing company makes two types of sports equipment: basketballs and footballs. Basketballs require BasketballMaterial units of materials and BasketballHours hours to make, whereas footballs require FootballMaterial units of materials and FootballHours hours to make. The manufacturing company has TotalMaterial units of materials available and their workers can work for at most TotalHours hours. Since basketballs sell better, there must be at least BasketballToFootballRatio times as many basketballs as footballs, but the manufacturing company would like at least MinFootballs footballs. How many of each type of sports equipment should the manufacturing company make to maximize the total number of sports equipment produced?

A researcher must perform two types of experiments, in-vivo and ex-vivo. In-vivo experiments require PrepTimeInVivo minutes to prepare and ExecTimeInVivo minutes to execute. Ex-vivo experiments require PrepTimeExVivo minutes to prepare and ExecTimeExVivo minutes to execute. In-vivo experiments give the researcher RadiationInVivo units of radiation while ex-vivo experiments give the researcher RadiationExVivo units of radiation. At most MaxPrepTime minutes can be used for preparation and MaxExecTime minutes for execution of the experiments. How many of each experiment should be scheduled by the researcher to minimize the total radiation received by the researcher?

A village delivers mail to nearby villages either by runners or canoers. Runners can carry RunnerCapacity bags of mail each time and take RunnerTime hours. Canoers can carry CanoerCapacity bags of mail each time and take CanoerTime hours. At most CanoeProportion of deliveries can be by canoe. Additionally, the village can spare at most TotalHours total hours and at least MinRunners individual runners must be used. How many of each way to deliver must be used to maximize the total amount of mail that can be delivered?

A woman on a diet needs to consume two types of meal preps. Each meal prep either provides ProteinPerSmoothie units of protein and CaloriesPerSmoothie calories, or ProteinPerBar units of protein and CaloriesPerBar calories. The woman must consume a number of protein bars equal to RatioBarsToSmoothies times the number of smoothies. She can consume at most MaxCalories calories and aims to maximize her protein intake by choosing the appropriate quantities of meal preps.

A firm operates NumFactories factories, each producing anti-itch injections and topical cream per hour. Every hour, each factory requires a certain number of plastic units to manufacture the packaging for these products. The total available plastic units are limited, and the firm has to meet minimum production requirements for both anti-itch injections and topical cream. The objective is to determine how many hours to run each factory to minimize the total operational time while satisfying the plastic usage constraint and the production requirements for both products.

A hospital prepares batches of medication patches and anti-biotic creams. Each medication patch requires TimePerMedPatch minutes to prepare and MaterialsPerMedPatch units of materials. Each anti-biotic cream requires TimePerCream minutes to prepare and MaterialsPerCream units of materials. Since anti-biotic creams are used more often, there must be at least MinCreamPatchRatio times as many anti-biotic creams as medication patches. Due to storage reasons, the hospital can make at most MaxTotalBatches batches of medication patches and anti-biotic creams in total. The hospital has available TotalAvailableMinutes minutes of staff to spare and TotalAvailableMaterials units of materials. If each batch of medication patches can treat PeoplePerMedPatch people and each batch of anti-biotic cream can treat PeoplePerCream people, how many batches of each should be made to maximize the number of people that can be treated?

A company in the desert can transport goods to rural cities either by camel caravans or desert trucks. A camel caravan can deliver GoodsCamelCaravan units of goods per trip and takes HoursCamelCaravan hours. A desert truck can deliver GoodsDesertTruck units per trip and takes HoursDesertTruck hours. The company prefers to have more camel caravans than desert trucks. If the company needs to deliver TotalGoods units of goods, how many of each transportation method should be used to minimize the total number of hours required?

Both chemical A and chemical B need to be added to a mixer for making bread. One unit of chemical A takes TimeA to be effective while one unit of chemical B takes TimeB to be effective. Because chemical A can be dangerous, there has to be at most MaxProportionAtoB as much chemical A as chemical B in the mixer. If there has to be at least MinUnitsA units of chemical A in the mixer and at least MinTotalChemicals units of total chemicals in the mixer, how many units of each should be added to minimize the total time it takes for the mixed bread to be ready?

A senior home has snacks of spinach and soybeans to increase their intake of iron. A cup of spinach contains FibreInSpinach units of fibre and IronInSpinach mg of iron. A cup of soybeans contains FibreInSoybeans units of fibre and IronInSoybeans mg of iron. The number of cups of spinach must exceed the number of cups of soybeans due to other nutritional requirements. Additionally, a cup of spinach contains CaloriesInSpinach calories while a cup of soybeans contains CaloriesInSoybeans calories. The elderly need in total at least TotalFibreRequired units of fibre and TotalIronRequired mg of iron. How many cups of each should the nurses schedule to maximize the seniors’ total caloric intake?

A keyboard manufacturer makes mechanical and standard keyboards. Mechanical keyboards are becoming more popular and thus the manufacturer aims to have MechanicalToStandardRatio times as many mechanical keyboards as standard keyboards. A mechanical keyboard costs PlasticPerMechanicalKeyboard units of plastic and SolderPerMechanicalKeyboard units of solder whereas a standard keyboard costs PlasticPerStandardKeyboard units of plastic and SolderPerStandardKeyboard units of solder. There are still customers that prefer a less noisy alternative. Therefore, there must be at least MinStandardKeyboards standard keyboards. If the company has available TotalPlastic units of plastic and TotalSolder units of solder, how many of each type should be manufactured to maximize the total number of keyboards?

A dessert shop needs to decide the quantities of two desserts, matcha ice cream and orange sorbet, to minimize the total amount of flavouring used. Each order of matcha ice cream requires FlavouringMatcha units of flavouring and IceCreamMatcha units of ice cream. Each order of orange sorbet requires FlavouringOrange units of flavouring and WaterOrange units of water. The shop has AvailableIceCream units of ice cream and AvailableWater units of water. At least MinMatchaPercentage percent of the total desserts made must be matcha ice cream. Additionally, the number of orders of orange sorbet must exceed the number of orders of matcha ice cream.

 Consider a set P described by linear inequality constraints, that is:
 P = {x \in R^N | a_i^T x <= b_i , i = 1, ..., m}.

where a_i \in R^N and b_i \in R
 A ball with center y in R^N and radius r is defined as the set of all points within (Euclidean) distance r from y.
We are interested in finding a ball with the largest possible radius, which is entirely contained within the set P (The center of such a ball is called the Chebychev center of P.

 A quantity y is known to depend on another quantity x. A set of corresponding values has been collected for x and y and is presented. The k-th y value takes y_{k} and the k-th x value takes x_{k}.
Fit the ‘best’ straight line y = bx + a where the objective is to minimize the maximum deviation of all the observed values of y from the value predicted by the linear relationship.

A quantity y is known to depend on another quantity x. A set of corresponding values has been collected for x and y and is presented. The k-th y value takes y_{k} and the k-th x value takes x_{k}.
Fit the ‘best’ quadratic curve y = c * x^2 + b * x + a to this set of data points. The objective is to minimize the sum of absolute deviations of each observed value of y from the value predicted by the quadratic relationship.

 A number of power stations are committed to meeting the following electricity load demands over a day. Period t of a day has demand demand_{t}. There are several types of generating unit available: the number of k-th type units is num_{k}. Type k generator has to work between a minimum level minlevel_{k} and a maximum level maxlevel_{k}.  There is an hourly cost of running each type k generator at minimum level runcost_{k}.  In addition, there is an extra hourly cost extracost_{k} for each megawatt at which a type k unit is operated above the minimum level.  Starting up a type k generator also involves a cost startcost_{k}.
Which generators should be working in which periods of the day to minimize total cost?

 We suppose that some products can be produced, each of them passing through several stages. Necessary time to produce 1 unit of product k is produce_time_{k, s} hours for stage s. Availability in working hours is time_{s} for stage s.  The unitary profits for product k is profit_{k}.
Determine the quantities to be produced such that all the production constraints are satisfied and the profit is maximized.

 The knapsack problem has a significant place in the study of integer programming models with binary variables.  In the knapsack problem, one needs to pack a set of items in a container. Each item k has value value_{k} and size size_{k}. And the container has maximum capacity C.  If the total size of the items exceeds the capacity, we can't pack them all.  In that case, the problem is to choose a subset of the items of maximum total value that will fit in the container.
Maximize the total value of the packed items

 We assume that there are available at the market different foods and that the food k sells at a price price_{k} per unit.  In addition there are several basic nutritional ingredients and, to achieve a balanced diet, each individual must receive at least demand_{m} units of nutrient m per day.  Finally, we assume that each unit of food k contains nutri_{k, m} units of nutrition m
We want to minimize the total prices of the foods to purchase.

 Consider the problem of operating a warehouse, by buying and selling the stock of a certain commodity, in order to maximize profit over a certain length of time, say several periods. The warehouse has a fixed capacity, and there is a cost holding_cost per unit for holding stock for one period. The selling price, price_{n}$, of the commodity is known to fluctuate over a number of time periods.  In any period n the price holding for purchase is cost_{n}. The warehouse is originally empty and is required to be empty at the end of the last period.
Determine the amount of buying and selling in each period to maximize the profit over the months

 A manufacturer wishes to produce alloy_quant lb of an alloy that consists of several kinds of metals. And by weight, the alloy we want contains a target quantitytarget_{m} of metal metal_{m}. Several other kinds of alloys are available at various prices.  These alloys can be melted to get the alloy we want. Alloy k has ratio_{k, m} ratio of metal_{m} and is sold at price of price_k. Ratio ratio_{k, m} ranges between 0 and 1.0 and should be properly treated in optimization
The manufacturer wishes to find the amounts of the various alloys needed and to determine the cheapest combination.

 A small firm specializes in making some types of spare automobile parts.  Each part is has to go through several shops  The required worker-hours for part k in shop s is time_{k, s}. The profit of part k is profit_{k}. The capacity of shop s is capacity_{s} working hours.
Formulate the problem of determining the quantities of each spare part to be made during the month so as to maximize profit.

 A small computer manufacturing company forecasts the demand over the next n-th month to be demand_{n}. These demands must be satisfied and can not be shorted. In any month it can produce at most max_regular_amount units, using regular production, at a cost of cost_regular dollars per unit.  By using overtime, it can produce additional units at cost_overtime dollars per unit.  The firm can store units from month to month at a cost of store_cost dollars per unit per month.
Formulate the problem of determining the production schedule that minimizes the cost.

 To operate a cafeteria, the manager must ensure on-call duty based on the statistics on the required staff.  In the future days, the number of desired employees on day n is denoted with num_{n}. Each employee works n_working_days days in a row and then has n_resting_days days off.
Calculate the minimum number of employees to hire

 An Italian transport company must send empty containers from its depots to ports.  The number of containers available in the depot i is denoted with numdepot_i.  Container requirement in port j is numport_{j}.  Transport of containers is by barges.  Each barge can only contain two containers and the cost of transport (per barge) is proportional to the distance traveled (price is price Euros/km). The distances between depot i and port j is distance_{i, j}.
Minimize the total transportation cost.

 A division of an auto parts manufacturer produces P different parts using M different machines. Batch of 100 part p requires time_{m,p} hours on machine m. The division incurs a charge of cost_{m} per hour for using machine m. Machine m has an availability of up to available_{m} hours per month. The division sells part p in batches of 100 at price of price_{p} per batch. The division must produce at least min_batches_{p} batches of part p each month to fulfill a contract.
How many batches of each part should the manufacturer produce each month to maximize profit?

 Custom Tees is planning an online advertising campaign with A different ad types across two web companies. The company has set a goal of goal_young thousand clicks from visitors aged 18-25 and goal_old thousand clicks from visitors older than 25. The company has set a goal of goal_unique_young thousand unique clicks from visitors aged 18-25 and goal_unique_old thousand unique clicks from visitors older than 25. Ad type a has an estimated percentage of young_click_{a} clicks from the 18-25 age range. Ad type a has an estimated percentage of old_click_{a} clicks from older visitors. Ad type a has a cost of cost_{a} per 1000 clicks. Ad type a has a maximum allowable clicks of max_click_{a} in thousands. The estimated percentage of unique visitors for each ad type a is unique_click_{a}.
How many clicks from each ad type should Custom Tees purchase to minimize cost?

 A fine foods company produces gift baskets for a season that lasts W weeks. The demand for week w is demand_{w} and the demand must be met. Regular labor is available for regular_labor_{w} hours per week and costs $regular_cost/h. Overtime labor is available for overtime_labor_{w} hours per week and costs $overtime_cost/h. Each basket takes assembly_time hours of labor to assemble. The material cost for each basket is $material_cost and it sells for $selling_price. Unsold baskets from a week incur a $holding_cost holding cost.  No baskets are available at the start of week 1. Unsold baskets at the end of week W have a salvage value of $var{salvage_value} and do not incur a holding cost.
How much regular and overtime labor should be utilized each week to assemble the baskets to maximize the profit over the season?

 A division of an auto parts manufacturer produces P different parts using M different machines. Batch of 100 part p requires time_{m,p} hours on machine m. The division incurs a charge of cost_{m} per hour for using machine m. Machine m has an availability of up to available_{m} hours per month. Machine M and Machine M-1 can share availability. Individual availability conditions for Machine M and Machine M-1 can be disregarded. The division sells part p in batches of 100 at price of price_{p} per batch. The division must produce at least min_batches_{p} batches of part p each month to fulfill a contract.

 A division of an auto parts manufacturer produces P different parts using M different machines. Batch of 100 part p requires time_{m,p} hours on machine m. The division incurs a charge of cost_{m} per hour for using machine m. Machine m has an availability of up to available_{m} hours per month. The division sells part p in batches of 100 at price of price_{p} per batch. The division must produce at least min_batches_{p} batches of part p each month to fulfill a contract. Extra available time for machine m can be acquired at $extra_costs_{m}/hour.  Up to max_extra_{m} hours can be purchased for each machine m.

 A team of translators is being assem- bled for the summer Olympics to translate from English into the \{M} other languages. There is a pool of N translators available. Each translator i can translate into a certain set of languages and comes with a specific cost_i.
Select the combination of translators that ensures all required languages can be translated into, at the minimum possible cost.

A company has I projects to be done by consultants. They have a choice of J consultants. Each consultant may be assigned up to K projects.  Hiring consultant j incurs a fixed cost of f_j. Assigning project i to consultant j incurs an additional cost of c_{i,j}.
Determine the assignment of consultants to projects that minimizes the total cost.

The network faces an intrusion that requires immediate intervention. Interventions include isolating nodes or scanning nodes affected by the intruder. Each cluster of nodes must be treated with one intervention type consistently. Interventions can be processed either centrally or through distributed nodes. The processing times for each type of intervention are different across N clusters. The opportunity cost for processing is $central_cost/h for central and $distributed_cost/h for distributed processing. There is a maximum allotment of max_central hours for central processing and max_distributed hours for distributed processing to address the intrusion.
Determine the most cost-effective combination of interventions to halt the intruder's progress within the network.