Summary 741- A manager has formulated the following LP problems. Use graphical
methods to find the optimal solutions. (In each, all variables are
nonnegative.)
a. Maximize: 10x 15y, subject to: 2x 5y 40 and 6x 3y 48.
b. Minimize: .75x y, subject to: x .5y 10 and x y 16. - Consider an LP problem in which a firm produces multiple goods (A
and B) using two inputs (X and Y) in limited supply. Suppose a
technological advance increases the amount of good A that can be
produced per unit of input X. How will this change the feasible region?
How will this affect the quantities of the goods produced in the profit-
maximizing solution to the LP problem? (To answer these questions, be
sure to graph the two resource constraints.) - An athlete carefully watches her intake of calcium, protein, and calories.
Her breakfast diet consists mainly of milk and cereal, whose prices and
nutrient contents appear in the following table:
Milk (1 oz.) Cereal (1 oz.)
Calcium 2 2
Protein 2 6
Calories 6 2
Price $.10 $.15She seeks a diet that supplies at least 50 units of calcium, 90 units of
protein, and 66 calories at minimum cost.
a. Formulate, graph, and solve this decision problem. What is the
minimum cost of meeting the nutrient requirements?
b. Calculate and provide an economic interpretation of the shadow
price associated with calcium.- A firm produces tires by two separate processes that require different
quantities of capital (K), labor (L), and raw materials (M). Process 1
requires one unit of K, four units of L, and two units of M to produce a
tire yielding a $4 profit. Process 2 requires one unit of K, two units of L,
and four units of M to produce a tire yielding a $6 profit. The available
supply of capital is 10; of labor, 32; and of raw materials, 32.
a. Formulate and solve (by graphing) the firm’s profit-maximization
problem.
b. Find the shadow prices of raw materials and labor. - Consider again the investment problem that opened the chapter.
a. Suppose the portfolio manager limits the portfolio to treasury bills
and treasury bonds. Using a graph, find the proportions of each type
of bond that maximize expected return subject to the risk and
maturity constraints.
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