A Diet Problem

Many LP formulations (such as Example 2 and the following diet problem) arise from situations in which a decision maker wants to minimize the cost of meeting a set of requirements.

My diet requires that all the food I eat come from one of the four “basic food 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 cheese-cake. Each brownie costs 50¢, each scoop of chocolate ice cream costs 20¢, each bottle of cola costs 30¢, and each piece of pineapple cheesecake costs 80¢. Each day, I must in-gest at least 500 calories, 6 oz of chocolate, 10 oz of sugar, and 8 oz of fat. The nutri-tional content per unit of each food is shown in Table 2. Formulate a linear programming model that can be used to satisfy my daily nutritional requirements at minimum cost.

Solution As always, we begin by determining the decisions that must be made by the decision maker: how much of each type of food should be eaten daily. Thus, we define the deci-sion variables:

x1 number of brownies eaten daily

x₂ number of scoops of chocolate ice cream eaten daily x3 bottles of cola drunk daily

x₄ pieces of pineapple cheesecake eaten daily

My objective is to minimize the cost of my diet. The total cost of any diet may be deter-mined from the following relation: (total cost of diet)  (cost of brownies)  (cost of ice cream)  (cost of cola)  (cost of cheesecake). To evaluate the total cost of a diet, note that, for example,

Cost of cola 



s f

t

cola

  

Applying this to the other three foods, we have (in cents)

Total cost of diet  50x1 20x2 30x3 80x4

Thus, the objective function is

min z 50x1 20x2 30x3 80x4

The decision variables must satisfy the following four constraints:

Constraint 1 Daily calorie intake must be at least 500 calories.

Constraint 2 Daily chocolate intake must be at least 6 oz.

Constraint 3 Daily sugar intake must be at least 10 oz.

Constraint 4 Daily fat intake must be at least 8 oz.

bottles of cola drunk Diet Problem

E X A M P L E 6

T A B L E 2 Nutritional Values for Diet

Type of Food Calories Chocolate (Ounces) Sugar (Ounces) Fat (Ounces)

Brownie 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

To express Constraint 1 in terms of the decision variables, note that (daily calorie intake)  (calories in brownies)  (calories in chocolate ice cream)  (calories in cola)  (calories in pineapple cheesecake).

The calories in the brownies consumed can be determined from Calories in brownies 



c r a o

lo w

r n ie

i s

e

  

Applying similar reasoning to the other three foods shows that

Daily calorie intake  400x1 200x2 150x3 500x4

Constraint 1 may be expressed by

400x1 200x2 150x3 500x4 500 (Calorie constraint) (21) Constraint 2 may be expressed by

3x1 2x2 6 (Chocolate constraint) (22)

Constraint 3 may be expressed by

2x1 2x2 4x3 4x4 10 (Sugar constraint) (23) Constraint 4 may be expressed by

2x1 4x2 x3 5x4 8 (Fat constraint) (24)

Finally, the sign restrictions xi 0 (i  1, 2, 3, 4) must hold.

Combining the objective function, constraints (21)–(24), and the sign restrictions yields the following:

min z 50x1 20x2 30x3 80x4

s.t. 400x1 200x2 150x3 500x4 500 (Calorie constraint) ⁽²¹⁾ s.t. 403x1 2x2 150x3 500x4 6 (Chocolate constraint) (22) s.t. 402x1 2x2 4x3 4x4 10 (Sugar constraint) ⁽²³⁾ s.t. 402x1 4x2 x3 5x4 8 (Fat constraint) (24)

s.t. 40xi 0 (i  1, 2, 3, 4) (Sign restrictions)

The optimal solution to this LP is x1 x4  0, x2  3, x3  1, z  90. Thus, the minimum-cost diet incurs a daily cost of 90¢ by eating three scoops of chocolate ice cream and drinking one bottle of cola. The optimal z-value may be obtained by substituting the optimal value of the decision variables into the objective function. This yields a total cost of z 3(20)  1(30)  90¢. The optimal diet provides

200(3)  150(1)  750 calories 2(3)  6 oz of chocolate 2(3)  4(1)  10 oz of sugar 4(3)  1(1)  13 oz of fat

Thus, the chocolate and sugar constraints are binding, but the calories and fat constraints are nonbinding.

A version of the diet problem with a more realistic list of foods and nutritional require-ments was one of the first LPs to be solved by computer. Stigler (1945) proposed a diet

brownies eaten

problem in which 77 types of food were available and 10 nutritional requirements (vitamin A, vitamin C, and so on) had to be satisfied. When solved by computer, the optimal solu-tion yielded a diet consisting of corn meal, wheat flour, evaporated milk, peanut butter, lard, beef, liver, potatoes, spinach, and cabbage. Although such a diet is clearly high in vital nu-trients, few people would be satisfied with it because it does not seem to meet a minimum standard of tastiness (and Stigler required that the same diet be eaten each day). The opti-mal solution to any LP model will reflect only those aspects of reality that are captured by the objective function and constraints. Stigler’s (and our) formulation of the diet problem did not reflect people’s desire for a tasty and varied diet. Integer programming has been used to plan institutional menus for a weekly or monthly period.^†Menu-planning models do con-tain constraints that reflect tastiness and variety requirements.

P R O B L E M S

Group A

1 There are three factories on the Momiss River (1, 2, and 3). Each emits two types of pollutants (1 and 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 0.10 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 0.20 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 0.40 ton and the amount of pollutant 2 by 0.30 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 least 40 tons. Formulate an LP that will minimize the cost of reducing pollution by the desired amounts. Do you think that the LP assumptions (Proportionality, Additivity, Divisibility, and Certainty) are reasonable for this problem?

2^‡ U.S. Labs manufactures mechanical heart valves from the heart valves of pigs. Different heart operations require valves of different sizes. U.S. Labs purchases pig valves from three different suppliers. The cost and size mix of the valves purchased from each supplier are given in Table 3.

Each month, U.S. Labs places one order with each supplier.

At least 500 large, 300 medium, and 300 small valves must be purchased each month. Because of limited availability of pig valves, at most 700 valves per month can be purchased from each supplier. Formulate an LP that can be used to minimize the cost of acquiring the needed valves.

3 Peg and Al Fundy have a limited food budget, so Peg is trying to feed the family as cheaply as possible. However, she still wants to make sure her family members meet their daily nutritional requirements. Peg can buy two foods. Food

1 sells for $7 per pound, and each pound contains 3 units of vitamin A and 1 unit of vitamin C. Food 2 sells for $1 per pound, and each pound contains 1 unit of each vitamin.

Each day, the family needs at least 12 units of vitamin A and 6 units of vitamin C.

a Verify that Peg should purchase 12 units of food 2 each day and thus oversatisfy the vitamin C requirement by 6 units.

b Al has put his foot down and demanded that Peg ful-fill the family’s daily nutritional requirement exactly by obtaining precisely 12 units of vitamin A and 6 units of vitamin C. The optimal solution to the new problem will involve ingesting less vitamin C, but it will be more ex-pensive. Why?

4 Goldilocks needs to find at least 12 lb of gold and at least 18 lb of silver to pay the monthly rent. There are two mines in which Goldilocks can find gold and silver. Each day that Goldilocks spends in mine 1, she finds 2 lb of gold and 2 lb of silver. Each day that Goldilocks spends in mine 2, she finds 1 lb of gold and 3 lb of silver. Formulate an LP to help Goldilocks meet her requirements while spending as little time as possible in the mines. Graphically solve the LP.

T A B L E 3

Cost Percent Percent Percent

Supplier Per Value ($) Large Medium Small

1 5 40 40 20

2 4 30 35 35

3 3 20 20 60

†Balintfy (1976).

‡Based on Hilal and Erickson (1981).