A Diet ExampleBreathtakers, a health and fitness center, operates a morning fitness program for senior citizens. The program includes aerobic exercise, either swimming or step exercise, followed by a healthy breakfast in the dining room. Breathtakers' dietitian wants to develop a breakfast that will be high in calories, calcium, protein, and fiber, which are especially important to senior citizens, but low in fat and cholesterol. She also wants to minimize cost. She has selected the following possible food items, whose individual nutrient contributions and cost from which to develop a standard breakfast menu are shown in the following table:

Breakfast Food

Fat Cholesterol Iron Calcium Protein Fiber Calories (g) (mg) (mg ) 0 6 (mg) (g) (g) Cost

1. Bran cereal (cup) 2. Dry cereal (cup) 3. Oatmeal (cup) 4. Oat bran (cup) 5. Egg 6. Bacon (slice) 7. Orange 8. Milk2% (cup) 9. Orange juice (cup) 10. Wheat toast (slice)

90

0

20

3

5

$0.1 8 0.22 0.10 0.12 0.10 0.09 0.40 0.16 0.50

110 100 90 75 35 65 100 120

2 2 2 5 3 0 4 0

0 0 0 270 8 0 12 0

4 2 3 1 0 1 0 0

48 12 8 30 0 52 250 3

4 5 6 7 2 1 9 1

2 3 4 0 0 1 0 0

65

1

0

1

26

3

3

0.07

The dietitian wants the breakfast to include at least 420 calories, 5 milligrams of iron, 400 milligrams of calcium, 20 grams of protein, and 12 grams of fiber. Furthermore, she wants to limit fat to no more than 20 grams and cholesterol to 30 milligrams.

Decision VariablesThis problem includes 10 decision variables, representing the number of standard units of each food item that can be included in each breakfast: x1 = cups of bran cereal x2 = cups of dry cereal x3 = cups of oatmeal x4 = cups of oat bran x5 = eggs x6 = slices of bacon x7 = oranges x8 = cups of milk x9 = cups of orange juice x10 = slices of wheat toast

The Objective FunctionThe dietitian's objective is to minimize the cost of a breakfast. The total cost is the sum of the individual costs of each food item: Minimize Z= .18x1+ .22x2+ .10x3+ .12x4+ .10x5+ .09x6 + .40x7+ .16x8+ .50x9+ .07x10 >=420

Model ConstraintsThe constraints are the requirements for the nutrition items:

90x1+ 110x2+ 100x3+ 90x4+ 75x5+ 35x6 + 65x7+ 100x8+ 120x9+ 65x10 >=420 calories 2x2+ 2x3+ 2x4+ 5x5+ 3x6 + 4x8+ x10=400mgcalcium 3x1+ 4x2+ 5x3+ 6x4+ 7x5+ 2x6 + x7+ 9x8+ x9+ 3x10>=20g protein 5x1+ 2x2+ 3x3+ 4x4+ x7+ 3x10>=12g fiber

Model SummaryThe linear programming model for this problem can be summarized as follows: Minimize Z= .18x1+ .22x2+ .10x3+ .12x4+ .10x5+ .09x6 + .40x7+ .16x8+ .50x9+ .07x10 >=420 Subject to 2x2+ 2x3+ 2x4+ 5x5+ 3x6 + 4x8+ x10=400 3x1+ 4x2+ 5x3+ 6x4+ 7x5+ 2x6 + x7+ 9x8+ x9+ 3x10>=20 5x1+ 2x2+ 3x3+ 4x4+ + x7+ 3x10>=12

An Investment ExampleKathleen Allen, an individual investor, has $70,000 to divide among several investments. The alternative investments are municipal bonds with an 8.5% annual return, certificates of deposit with a 5% return, treasury bills with a 6.5% return, and a growth stock fund with a 13% annual return. The investments are all evaluated after 1 year. However, each

investment alternative has a different perceived risk to the investor; thus, it is advisable to diversify. Kathleen wants to know how much to invest in each alternative in order to maximize the return. The following guidelines have been established for diversifying the investments and lessening the risk perceived by the investor: 1. No more than 20% of the total investment should be in municipal bonds. 2. The amount invested in certificates of deposit should not exceed the amount invested in the other three alternatives. 3. At least 30% of the investment should be in treasury bills and certificates of deposit. 4. To be safe, more should be invested in CDs and treasury bills than in municipal bonds and the growth stock fund, by a ratio of at least 1.2 to 1. Kathleen wants to invest the entire $70,000.

Decision VariablesFour decision variables represent the monetary amount invested in each investment alternative: X1 = amount ($) invested in municipal bonds X2 = amount ($) invested in certificates of deposit X3 = amount ($) invested in treasury bills X4 = amount ($) invested in growth stock fund

The Objective FunctionThe objective of the investor is to maximize the total return from the investment in the four alternatives. The total return is the sum of the individual returns from each alternative. Thus, the objective function is expressed as

Maximize Z = $0.085x1 + 0.05x2 + 0.065x3 + 0.130x4 where

Z = total return from all investments 0.085x1 = return from the investment in municipal bonds 0.05x2 = return from the investment in certificates of deposit 0.065x3 = return from the investment in treasury bills 0.130x4 = return from the investment in growth stock fund

Model ConstraintsIn this problem the constraints are the guidelines established for diversifying the total investment. Each guideline is transformed into a mathematical constraint separately. The first guideline states that no more than 20% of the total investment should be in municipal bonds. The total investment is $70,000; 20% of $70,000 is $14,000. Thus, this constraint is X1