Optimization-3Q

6.1 A dietitian is planning a menu that consists 3 main foods: food A, B, and C. Each

ounce of food A contains 3 units of fat, 1 unit of carbohydrate, 4 units of protein, 2

units of cholesterol, 1 unit of Vitamin B and 6 units of vitamin D. Each ounce of food B

contains 6 units of fat, 2 unit of carbohydrate, 8 units of protein, 3 units of cholesterol,

3 units of Vitamin B and 3 units of vitamin D. Each ounce of food C contains 2 units of

fat, 5 unit of carbohydrate, 2 units of protein, 1 unit of cholesterol, 2 units of Vitamin

B and 5 units of vitamin D. The dietitian wants the meal to provide at least 36 units

of carbohydrate, 48 units of protein, 18 units of Vitamin B and 30 units of vitamin D.

If the prices for food A, B and C are 15, 25 and 20 cents per ounce, respectively, then

how many ounces of each food should be served to minimize cost of the meal, minimize

the fat and cholesterol contained in the meal and satisfy the dietitian’s requirements?

6.2 Consider a refinery that produces three types of motor oil–Standard, Extra and Super.

The selling prices are $9.00,$13.00 and $19.00 per barrel, respectively. These oils can

be made from three basic ingredients-crude oil, paraffin and filler. The costs of the

ingredients are $19.00, $9.00 and $11.00 per barrel, respectively. Company engineers

have developed the following specifications for each oil:

– Standard – 60% paraffin, 40% filler

– Extra – at least 25% crude oil and no more than 45% paraffin

– Super – at least 50% crude oil and no more than 25% paraffin

The CO2 emissions of Standard, Extra and Super oils are 13.0, 11.8 and 8.0 units per

barrel. With a supply capacity of 110, 90,and 70 thousands barrels per week for crude

oil, paraffin and filler, what to blend in order to maximize profits as well as satisfying

the requirements of EPA to minimize the CO2 emissions of the all products of this

industry? Solve the problem using goal programing approach, if the goals are, profit

1

greater than or equal to $5 per barrel and CO2 emissions less than or equal to 10 units

per barrel.

6.3 Solve the following bi-objective NLP problem using (a) a weighting method and (b)

a constraint method with 20 parameters (wi and εi) [from Das and Dennis (1998),

SIAM. J. Optim., Vol. 8, 631-657].

min x

[ Z1(x) = x

2 1 + x

2 2 + x

2 3 + x

2 4 + x

2 5

Z2(x) = 3×1 + 2×2 − x33 + 0.01× (x4 − x5)3 ]

(1)

subject to

x1 + 2×2 − x3 − 0.5×4 + x5 = 2 (2) 4×1 − 2×2 + 0.8×3 + 0.6×4 + 0.5×25 = 0 (3) x21 + x

2 2 + x

2 3 + x

2 4 + x

2 5 ≤ 10 (4)

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