Problems

g.

This solution is optimal. Are there multiple optimal solutions? Why?

h.

If there are multiple optimal solutions, identify the alternate solutions.

A-49

21. A wood products firm in Oregon plants three types of trees—white pines, spruce, and ponderosa pines—to produce pulp for paper products and wood for lumber. The company wants to plant enough acres of each type of tree to produce at least 27 tons of pulp and 30 tons of lumber. The company has developed the following linear programming model to determine the number of acres of white pines (x_{1}), spruce (x_{2}), and ponderosa pines (x_{3}) to plant to minimize cost:

minimize Z = 120x_{1 }+ 40x_{2 }+ 240x_{3 }(cost, $) subject to

( p u l p , t o n s ) ( l u m b e r , t o n s ) x 1 , x 2 , x 3 Ú 0 2 x 1 + 6 x 2 + 3 x 3 Ú 3 0 4 x 1 + x 2 + 3 x 3 Ú 2 7

Solve this model using the simplex method.

22. A baby products firm produces a strained baby food containing liver and milk, each of which con- tribute protein and iron to the baby food. Each jar of baby food must have 36 milligrams of protein and 50 milligrams of iron. The company has developed the following linear programming model to determine the number of ounces of liver (x_{1}) and milk (x_{2}) to include in each jar of baby food to meet the requirements for protein and iron at the minimum cost:

minimize Z = 0.05x_{1 }+ 0.10x_{2 }(cost, $) subject to

( p r o t e i n , m g ) ( i r o n , m g ) x 1 , x 2 Ú 0 5 x 1 + 5 x 2 Ú 5 0 6 x 1 + 2 x 2 Ú 3 6

Solve this model using the simplex method.

23.

Solve the linear programming model in Problem 22 graphically, and identify the points on the graph that correspond to each simplex tableau.

24.

The Cookie Monster Store at South Acres Mall makes three types of cookies—chocolate chip, pecan chip, and pecan sandies. Three primary ingredients are chocolate chips, pecans, and sugar. The store has 120 pounds of chocolate chips, 40 pounds of pecans, and 300 pounds of sugar. The following linear programming model has been developed for determining the number of batches of chocolate chip cookies (x

_{1}), pecan chip cookies (x_{2}), and pecan sandies (x_{3}) to make to maxi- mize profit:

maximize Z = 10x_{1 }+ 12x_{2 }+ 7x_{3 }(profit, $) subject to

( s u g a r , l b . ) ( c h o c o l a t e c h i p s , l b . ) ( p e c a n s , l b . ) x 1 , x 2 , x 3 Ú 0 x 1 + 2 x 3 … 4 0 1 0 x 1 + 5 x 2 … 1 2 0 2 0 x 1 + 1 5 x 2 + 1 0 x 3 … 3 0 0

Solve this model using the simplex method.