# Identifies the feasible solution that maximizes

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identifies the feasible solution that maximizes profit, which is given approx- imately by x 1 = 20 . 4 and x 2 = 6 . 5. Note that this solution involves making use of the entire capacity available for metal stamping and engine assembly, but does not maximize use of capacity to assemble either cars or trucks. The optimal profit is over \$77 . 3 million per month, which exceeds by about \$10 million the profit associated with producing only cars.
2.1.2 Feeding an Army Suppose that two basic types of food are supplied to soldiers in an army: meats and potatoes. Each pound of meats costs \$1, while each pound of potatoes costs \$0 . 25. To minimize expenses, army officials consider serving only potatoes. However, there are some basic nutritional requirements that call for meats in a soldier’s diet. In particular, each soldier should get at least 400 grams of carbohydrates, 40 grams of dietary fiber, and 200 grams of protein in their daily diet. Nutrients offered per pound of each of the two types of food, as well as the daily requirements, are provided in the following table:
38 Nutrient Meats Potatoes Daily Requirement carbohydrates 40 grams 200 grams 400 grams dietary fiber 5 grams 40 grams 40 grams protein 100 grams 20 grams 200 grams Consider the problem of finding a minimal cost diet comprised of meats and potatoes that satisfies the nutritional requirements. Let x 1 and x 2 denote the number of pounds of meat and potatoes to be consumed daily. These quantities cannot be negative, so we have a constraint x 0. The nutritional requirements impose further constraints: 40 x 1 + 200 x 2 400 (carbohydrates) 5 x 1 + 40 x 2 40 (dietary fiber) 100 x 1 + 20 x 2 200 (protein) . The set of feasible solutions is illustrated in Figure 2.2(a). 5 10 5 10 meats (pounds) potatoes (pounds) feasible solutions protien carbohydrates dietary fiber 5 10 meats (pounds) ) s d n u o p ( s e o t a t o p minimal cost (a) (b) Figure 2.2: (a) Feasible solutions for an army diet. (b) Finding the solution that minimizes cost. In Figure 2.2(b), superimposed lines identify sets that lead to particular daily costs. Unlike the automobile manufacturing problem we considered in the previous section, we are now minimizing cost rather than maximizing profit. The optimal solution involves a diet that includes both meats and potatoes, and is given approximately by x 1 = 1 . 67 and x 2 = 1 . 67. The asso- ciated daily cost per soldier is about \$2 . 08. Note that the constraint brought
c Benjamin Van Roy 39 about by dietary fiber requirements does not affect the feasible region. This is because – based on our data – any serving of potatoes that offers sufficient carbohydrates will also offer sufficient dietary fibers. 2.1.3 Some Observations There are some interesting observations that one can make from the preceding examples and generalize to more complex linear optimization problems. In each case, the set of feasible solutions forms a polygon. By this we mean that the boundary of each is made up of a finite number of straight segments, forming corners where they connect. In the case of producing cars and trucks, this polygon is bounded.
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