CASE: CUTTING CAFETERIA COST
Problem Statement
?Special dish Casserole served at All University
Cafeteria every Thursday ?Cafeteria manager Maria Gonzalez aims at cost reduction ?Cost reduction to be achieved by reducing the cost of two main ingredients viz., potato and green beans ?Constraints: Casserole must contain 180 g of protein, 80 mg of iron and 1,050 mg of vitamin C
Problem Statement
?Minimum 10kg of casserole to be prepared
?No upper limit on quantity of casserole prepared
?Nutritional values of potato and green beans are
given as
Potato Protein Iron Vitamin C 1.5 gm per 100gm 0.3 mg per 100 gm 12 mg per 100 gm Green Beans 5.67 gm per 10 ounces 3.402 mg per 10 ounces 28.35 mg per 10 ounces
? 1 ounce = 28.35 gm
Problem Statement
?Cost of Potato = $ 0.4 / pound and cost of Green
Beans = $ 1.0 / pound ?Weight of Potato : Weight of Green Beans = 6:5 in casserole ?Conversion Table
Lbs 1 0.0625 16 1 Ounce gm 453.6 28.35
Question 1
Determine the amount of potatoes and green beans Maria should purchase each week for the casserole to minimize the ingredient costs while meeting nutritional, taste, and demand requirements.
1) Amount of Potatoes and Green Beans
? Decision Variable:
x1 be quantity of Potatoes in gm x2 be quantity of Green beans in gm ? Objective Function: Minimize Z = 0.000882 x1 + 0.002205 x2 ? Subject to: x1 + x2 ? 10000 (Total quantity more than 10 kg) x1 ? 5454.54 and x2 ? 4545.45 (Ratio constraint) 0.015 x1 + 0.020 x2 ? 180 (Protein quantity) 0.003 x1 + 0.012 x2 ? 80 (Iron quantity) 0.120 x1 + 0.100 x2 ? 1050 (Vitamin C quantity) x1 / x2 ? 1.2 (Potatoes and Green beans in ratio 6:5) x1, x2 ? 0 (Non negative constraint)
Amount of Potatoes and Green Beans
?Solving the above equation
?Quantity of Potato = 6153.85 gm ?Quantity of Green Beans = 5128.21 gm ?Minimum cost = $16.73
Question 2
Maria is not very concerned about the taste of the casserole; she is only concerned about meeting nutritional requirements and cutting costs. She therefore forces Edson to change the recipe to allow for only at least a one to two ratio in the weight of potatoes to green beans. Given the new recipe, determine the amount of potatoes and green beans Maria should purchase each week.
2) Potato to Green Beans ratio 1:2
? Decision Variable:
x1 be quantity of Potatoes in gm x2 be quantity of Green beans in gm ? Objective Function: Minimize Z = 0.000882 x1 + 0.002205 x2 ? Subject to: x1 + x2 ? 10000 (Total quantity more than 10 kg) x1 ? 3333.33 and x2 ? 6666.67 (Ratio Constraint) 0.015 x1 + 0.020 x2 ? 180 (Protein quantity) 0.003 x1 + 0.012 x2 ? 80 (Iron quantity) 0.120 x1 + 0.100 x2 ? 1050 (Vitamin C quantity) x1 / x2 ? 0.5 (Potatoes and Green beans in ratio 1:2) x1, x2 ? 0 (Non negative constraint)
Potato to Green Beans ratio 1:2
?Solving the above equation
?Quantity of Potato = 3333.33 gm
?Quantity of Green Beans = 6666.67 gm ?Minimum cost = $17.6
Question 3
Maria decides to lower the iron requirement to 65 mg since she determines that the other ingredients, such as the onions and cream of mushroom soup, also provide iron. Determine the amount of potatoes and green beans Maria should purchase each week given this new iron requirement.
3) Lowering Iron content to 65 mg
? Decision Variable:
x1 be quantity of Potatoes in gm x2 be quantity of Green beans in gm ? Objective Function: Minimize Z = 0.000882 x1 + 0.002205 x2 ? Constraint: x1 + x2 ? 10000 (Total quantity more than 10 kg) x1 ? 5454.54 and x2 ? 4545.45 (Ratio Constraint) 0.015 x1 + 0.020 x2 ? 180 (Protein quantity) 0.003 x1 + 0.012 x2 ? 65 (Iron quantity) 0.120 x1 + 0.100 x2 ? 1050 (Vitamin C quantity) x1 / x2 ? 1.2 (Potatoes and Green beans in ratio 6:5) x1, x2 ? 0 (Non negative constraint)
Lowering Iron content to 65 mg
?Solving the above equation
?Quantity of Potato = 5684.21 gm
?Quantity of Green Beans = 4736.82 gm ?Minimum cost = $15.46
Question 4
Maria learns that the wholesaler has a surplus of green beans and is therefore selling the green beans for a lower price of $0.50 per lb. Using the same iron requirement from part (3) and the new price of green beans, determine the amount of potatoes and green beans Maria should purchase each week.
4) Lower price of Green Beans ($0.50/lb)
? Decision Variable:
x1 be quantity of Potatoes in gm x2 be quantity of Green beans purchased in gm ? Objective Function: Minimize Z = 0.000882 x1 + 0.001102 x2 ? Constraint: x1 + x2 ? 10000 (Total quantity more than 10 kg) x1 ? 5454.54 and x2 ? 4545.45 (Ratio Constraint) 0.015 x1 + 0.020 x2 ? 180 (Protein quantity) 0.003 x1 + 0.012 x2 ? 65 (Iron quantity) 0.120 x1 + 0.100 x2 ? 1050 (Vitamin C quantity) x1 / x2 ? 1.2 (Potatoes and Green beans in ratio 6:5) x1, x2 ? 0 (Non negative constraint)
Lower price of Green Beans ($0.50/lb)
?Solving the above equation
?Quantity of Potato = 5684.21 gm
?Quantity of Green Beans = 4736.82 gm ?Minimum cost = $10.24
Question 5
Maria decides that she wants to purchase lima beans instead of green beans since lima beans are less expensive and provide a greater amount of protein and iron than green beans. Maria again wields her absolute power and forces Edson to change the recipe to include lima beans instead of green beans. Maria knows she can purchase lima beans for $0.60 per lb from the wholesaler. She also knows that lima beans contain 22.68 g of protein per 10 ounces of lima beans, 6.804 mg of iron per 10 ounces of lima beans, and no vitamin C. Using the new cost and nutritional content of lima beans, determine the amount of potatoes and lima beans Maria should purchase each week to minimize the ingredient costs while meeting nutritional, taste, and demand requirements. The nutritional requirements include the reduced iron requirement from part (3).
5) Green Beans replaced by Lima Beans
? Decision Variables:
x1 be quantity of Potatoes in gm x2 be quantity of Lima beans in gm ? Objective Function: Minimize Z = 0.000882 x1 + 0.001323 x2 ? Constraints: x1 + x2 ? 10000 (Total quantity should be more than 10 kg) x1 ? 5454.54 and x2 ? 4545.45 (Ratio Constraint) 0.015 x1 + 0.080 x2 ? 180 (Protein quantity) 0.003 x1 + 0.024 x2 ? 65 (Iron quantity) 0.120 x1 ? 1050 (Vitamin C quantity) x1 / x2 ? 1.2 (Potatoes and Lima beans in ratio 6:5) x1, x2 ? 0 (Non negative constraint)
Green Beans replaced by Lima Beans
?Solving the above equation
?Quantity of Potato = 8750 gm ?Quantity of Lima Beans = 7291.67 gm ?Minimum cost = $17.36
Question 6
Will Edson be happy with the solution in part (5)? Why or why not? Ans: The only concern for Edson is to maintain the taste of Casserole by maintaining the ratio of Potatoes : Green Beans to 6:5, but since we have replaced Green beans by Lima beans; it can be highly probable that Edson would not be happy.
Question 7
An All-State student task force meets during Body Awareness Week and determines that All-State University’s nutritional requirements for iron are too lax and that those for vitamin C are too stringent. The task force urges the university to adopt a policy that requires each serving of an entrée to contain at least 120 mg of iron and at least 500 mg of vitamin C. Using potatoes and lima beans as the ingredients for the dish and using the new nutritional requirements, determine the amount of potatoes and lima beans Maria should purchase each week.
7) Iron ? 120 mg and Vitamin C ? 500 mg
? Decision Variables:
x1 be quantity of Potatoes in gm x2 be quantity of Lima beans in gm ? Objective Function: Minimize Z = 0.000882 x1 + 0.001323 x2 ? Constraints: x1 + x2 ? 10000 (Total quantity more than 10 kg) x1 ? 5454.54 and x2 ? 4545.45 (Ratio Constraint) 0.015 x1 + 0.080 x2 ? 180 (Protein quantity) 0.003 x1 + 0.024 x2 ? 120 (Iron quantity) 0.120 x1 ? 500 (Vitamin C quantity) x1 / x2 ? 1.2 (Potatoes and Lima beans in ratio 6:5) x1, x2 ? 0 (Non negative constraint)
Iron ? 120 mg and Vitamin C ? 500 mg
?Solving the above equation
?Quantity of Potato = 5454.54 gm ?Quantity of Lima Beans = 4545.45 gm ?Minimum cost = $10.82
doc_470310644.pptx
Problem Statement
?Special dish Casserole served at All University
Cafeteria every Thursday ?Cafeteria manager Maria Gonzalez aims at cost reduction ?Cost reduction to be achieved by reducing the cost of two main ingredients viz., potato and green beans ?Constraints: Casserole must contain 180 g of protein, 80 mg of iron and 1,050 mg of vitamin C
Problem Statement
?Minimum 10kg of casserole to be prepared
?No upper limit on quantity of casserole prepared
?Nutritional values of potato and green beans are
given as
Potato Protein Iron Vitamin C 1.5 gm per 100gm 0.3 mg per 100 gm 12 mg per 100 gm Green Beans 5.67 gm per 10 ounces 3.402 mg per 10 ounces 28.35 mg per 10 ounces
? 1 ounce = 28.35 gm
Problem Statement
?Cost of Potato = $ 0.4 / pound and cost of Green
Beans = $ 1.0 / pound ?Weight of Potato : Weight of Green Beans = 6:5 in casserole ?Conversion Table
Lbs 1 0.0625 16 1 Ounce gm 453.6 28.35
Question 1
Determine the amount of potatoes and green beans Maria should purchase each week for the casserole to minimize the ingredient costs while meeting nutritional, taste, and demand requirements.
1) Amount of Potatoes and Green Beans
? Decision Variable:
x1 be quantity of Potatoes in gm x2 be quantity of Green beans in gm ? Objective Function: Minimize Z = 0.000882 x1 + 0.002205 x2 ? Subject to: x1 + x2 ? 10000 (Total quantity more than 10 kg) x1 ? 5454.54 and x2 ? 4545.45 (Ratio constraint) 0.015 x1 + 0.020 x2 ? 180 (Protein quantity) 0.003 x1 + 0.012 x2 ? 80 (Iron quantity) 0.120 x1 + 0.100 x2 ? 1050 (Vitamin C quantity) x1 / x2 ? 1.2 (Potatoes and Green beans in ratio 6:5) x1, x2 ? 0 (Non negative constraint)
Amount of Potatoes and Green Beans
?Solving the above equation
?Quantity of Potato = 6153.85 gm ?Quantity of Green Beans = 5128.21 gm ?Minimum cost = $16.73
Question 2
Maria is not very concerned about the taste of the casserole; she is only concerned about meeting nutritional requirements and cutting costs. She therefore forces Edson to change the recipe to allow for only at least a one to two ratio in the weight of potatoes to green beans. Given the new recipe, determine the amount of potatoes and green beans Maria should purchase each week.
2) Potato to Green Beans ratio 1:2
? Decision Variable:
x1 be quantity of Potatoes in gm x2 be quantity of Green beans in gm ? Objective Function: Minimize Z = 0.000882 x1 + 0.002205 x2 ? Subject to: x1 + x2 ? 10000 (Total quantity more than 10 kg) x1 ? 3333.33 and x2 ? 6666.67 (Ratio Constraint) 0.015 x1 + 0.020 x2 ? 180 (Protein quantity) 0.003 x1 + 0.012 x2 ? 80 (Iron quantity) 0.120 x1 + 0.100 x2 ? 1050 (Vitamin C quantity) x1 / x2 ? 0.5 (Potatoes and Green beans in ratio 1:2) x1, x2 ? 0 (Non negative constraint)
Potato to Green Beans ratio 1:2
?Solving the above equation
?Quantity of Potato = 3333.33 gm
?Quantity of Green Beans = 6666.67 gm ?Minimum cost = $17.6
Question 3
Maria decides to lower the iron requirement to 65 mg since she determines that the other ingredients, such as the onions and cream of mushroom soup, also provide iron. Determine the amount of potatoes and green beans Maria should purchase each week given this new iron requirement.
3) Lowering Iron content to 65 mg
? Decision Variable:
x1 be quantity of Potatoes in gm x2 be quantity of Green beans in gm ? Objective Function: Minimize Z = 0.000882 x1 + 0.002205 x2 ? Constraint: x1 + x2 ? 10000 (Total quantity more than 10 kg) x1 ? 5454.54 and x2 ? 4545.45 (Ratio Constraint) 0.015 x1 + 0.020 x2 ? 180 (Protein quantity) 0.003 x1 + 0.012 x2 ? 65 (Iron quantity) 0.120 x1 + 0.100 x2 ? 1050 (Vitamin C quantity) x1 / x2 ? 1.2 (Potatoes and Green beans in ratio 6:5) x1, x2 ? 0 (Non negative constraint)
Lowering Iron content to 65 mg
?Solving the above equation
?Quantity of Potato = 5684.21 gm
?Quantity of Green Beans = 4736.82 gm ?Minimum cost = $15.46
Question 4
Maria learns that the wholesaler has a surplus of green beans and is therefore selling the green beans for a lower price of $0.50 per lb. Using the same iron requirement from part (3) and the new price of green beans, determine the amount of potatoes and green beans Maria should purchase each week.
4) Lower price of Green Beans ($0.50/lb)
? Decision Variable:
x1 be quantity of Potatoes in gm x2 be quantity of Green beans purchased in gm ? Objective Function: Minimize Z = 0.000882 x1 + 0.001102 x2 ? Constraint: x1 + x2 ? 10000 (Total quantity more than 10 kg) x1 ? 5454.54 and x2 ? 4545.45 (Ratio Constraint) 0.015 x1 + 0.020 x2 ? 180 (Protein quantity) 0.003 x1 + 0.012 x2 ? 65 (Iron quantity) 0.120 x1 + 0.100 x2 ? 1050 (Vitamin C quantity) x1 / x2 ? 1.2 (Potatoes and Green beans in ratio 6:5) x1, x2 ? 0 (Non negative constraint)
Lower price of Green Beans ($0.50/lb)
?Solving the above equation
?Quantity of Potato = 5684.21 gm
?Quantity of Green Beans = 4736.82 gm ?Minimum cost = $10.24
Question 5
Maria decides that she wants to purchase lima beans instead of green beans since lima beans are less expensive and provide a greater amount of protein and iron than green beans. Maria again wields her absolute power and forces Edson to change the recipe to include lima beans instead of green beans. Maria knows she can purchase lima beans for $0.60 per lb from the wholesaler. She also knows that lima beans contain 22.68 g of protein per 10 ounces of lima beans, 6.804 mg of iron per 10 ounces of lima beans, and no vitamin C. Using the new cost and nutritional content of lima beans, determine the amount of potatoes and lima beans Maria should purchase each week to minimize the ingredient costs while meeting nutritional, taste, and demand requirements. The nutritional requirements include the reduced iron requirement from part (3).
5) Green Beans replaced by Lima Beans
? Decision Variables:
x1 be quantity of Potatoes in gm x2 be quantity of Lima beans in gm ? Objective Function: Minimize Z = 0.000882 x1 + 0.001323 x2 ? Constraints: x1 + x2 ? 10000 (Total quantity should be more than 10 kg) x1 ? 5454.54 and x2 ? 4545.45 (Ratio Constraint) 0.015 x1 + 0.080 x2 ? 180 (Protein quantity) 0.003 x1 + 0.024 x2 ? 65 (Iron quantity) 0.120 x1 ? 1050 (Vitamin C quantity) x1 / x2 ? 1.2 (Potatoes and Lima beans in ratio 6:5) x1, x2 ? 0 (Non negative constraint)
Green Beans replaced by Lima Beans
?Solving the above equation
?Quantity of Potato = 8750 gm ?Quantity of Lima Beans = 7291.67 gm ?Minimum cost = $17.36
Question 6
Will Edson be happy with the solution in part (5)? Why or why not? Ans: The only concern for Edson is to maintain the taste of Casserole by maintaining the ratio of Potatoes : Green Beans to 6:5, but since we have replaced Green beans by Lima beans; it can be highly probable that Edson would not be happy.
Question 7
An All-State student task force meets during Body Awareness Week and determines that All-State University’s nutritional requirements for iron are too lax and that those for vitamin C are too stringent. The task force urges the university to adopt a policy that requires each serving of an entrée to contain at least 120 mg of iron and at least 500 mg of vitamin C. Using potatoes and lima beans as the ingredients for the dish and using the new nutritional requirements, determine the amount of potatoes and lima beans Maria should purchase each week.
7) Iron ? 120 mg and Vitamin C ? 500 mg
? Decision Variables:
x1 be quantity of Potatoes in gm x2 be quantity of Lima beans in gm ? Objective Function: Minimize Z = 0.000882 x1 + 0.001323 x2 ? Constraints: x1 + x2 ? 10000 (Total quantity more than 10 kg) x1 ? 5454.54 and x2 ? 4545.45 (Ratio Constraint) 0.015 x1 + 0.080 x2 ? 180 (Protein quantity) 0.003 x1 + 0.024 x2 ? 120 (Iron quantity) 0.120 x1 ? 500 (Vitamin C quantity) x1 / x2 ? 1.2 (Potatoes and Lima beans in ratio 6:5) x1, x2 ? 0 (Non negative constraint)
Iron ? 120 mg and Vitamin C ? 500 mg
?Solving the above equation
?Quantity of Potato = 5454.54 gm ?Quantity of Lima Beans = 4545.45 gm ?Minimum cost = $10.82
doc_470310644.pptx