Operations Research:
1) Farmer Jones must determine how many acres of corn and wheat to plant this year. An acre of wheat yields 25 bushels of wheat and requires 10 hours of labor per week. An acre of corn yields 10 bushels of corn and requires 4 hours of labor per week. All wheat can be sold at $4 a bushel, and all corn can be sold at 3$ a bushel. Seven acres of land and 40 hours per week of labor are available. Government regulations require that at least 30 bushels of corn be produced during the current year. Let x1=number of acres of corn planted, and x2=number of acres of wheat planted. Using these decision variables, formulate an LP whose solution will tell farmer Jones how to maximize the total revenue from wheat and corn. Using the variables x1=number of bushels of corn produced and x2=number of bushels of wheat produced, reformulate farmer Jones LP.
2) My diet requires that all the food I eat come from one of the “basic four groups” (chocolate cake, ice cream, soda and cheesecake). At present , the following four foods are available for consumption: brownies, chocolate ice cream, cola, and pineapple cheesecake. Each brownie costs 50 cent, each scoop of chocolate ice cream costs 20 cent, each bottle of cola costs 30 cents, and each piece of pineapple cheesecake costs 80 cents. Each day, I must ingest at least 500 calories, 6 oz of chocolate, 10 oz of sugar , and 8 oz of fat. The nutritional content per unit of each food is shown in following table. Formulate a linear programming model that can be used to satisfy my daily nutritional requirements at minimum cost.
calories chocolate sugar fat
Brownies 400 3 2 2
chocolate ice cream ( 1 scoop) 200 2 2 4
Cola(1 bottle) 150 0 4 1
Pineapple cheesecake ( 1 piece) 500 0 4 5
3) An auto company manufactures cars and trucks. Each vehicle must be processed in the paint shop and body assembly shop. If the paint shop were only painting trucks, 40 per day could be painted. If the paint shop were only painting cars, 60 per day could be painted. If the body shop were only producing cars, it could process 50 per day. If the body shop were only producing trucks, it could process 50 per day. Each truck contributes 300$ to profit, and each car contributes 200$ to profit. Use linear programming to determine a daily production schedule that will maximize the company’s profit.
4) There are three factories on the Momiss river (1,2,3). Each emits two type of pollutants (1,2) into the river. If the waste from each factory is processed the pollution in the river can be reduced. It costs 15$ to process a ton of factory 1 waste , and each ton processed reduces the amount of pollutant 1 by 0.1 ton and the amount of pollutant 2 by 0.45 ton. It costs $10 to process a ton of factory 2 waste, and each ton processed will reduce the amount of pollutant 1 by 0.2 ton and the amount of pollutant 2 by 0.25 ton. It costs $20 to process a ton of factory 3 waste, and each ton processed will reduce the amount of pollutant 1 by 0.4 and the amount of pollutant 2 by 0.3 ton. The state wants to reduce the amount of pollutant 1 in the river by at least 30 tons and the amount of pollutant 2 in the river by at 40 tons. Formulate an LP that will minimize the cost of reducing pollution by the desired amounts.
5) Rylon corporation manufactures brute and chanelle perfumes. The raw material needed to manufacture each type of perfume can be purchased for 3$ per pound. Processing 1 lb of raw material requires 1 hour of laboratory time. Each pound of processed raw material yields 3 oz of regular brute perfume and 4 oz of regular chanelle perfume. Regular brute can be sold for 7$/oz and regular chanelle for 6$/oz. Rylon also has the option of further processing regular brute and regular chanelle to produce luxury brute, sold at 18$/oz, and luxury chanelle , sold at 14$/oz. each ounce of regular brute processed further requires an additional 3 hours of laboratory time and 4$ processing cost and yields 1 oz of luxury brute. each ounce of regular chanelle processed further requires an additional 2 hours of laboratory time and 4$ processing cost and yields 1 oz of luxury chanelle. Each year, Rylon has 6000 hours of laboratory time available and can purchase up to 4000lb of raw material. Formulate an LP that can be used to determine how Rylon can maximize profits. Assume that the cost of the laboratory hours is a fixed cost.