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1.A price-taking firm sells S units of its product at the market price of p. Management's

policy is to replace defective units at no additional charge, on the first-come, first-served

basis, while replacement units are available. Because management does not want to risk

making the same mistake twice, it produces the units that it sells to the market on one

machine. Moreover, it produces the replacement units, denoted X, on a second, higherquality machine. The fixed cost associated with operating both machines, the variable cost,

and replacement cost are given is the short-run cost function C(X) = 100 + 20S + 30X.

The exact probability that a unit will be defective is r. Acting out of caution, however,

management always underestimate the reliability of its product. Nonetheless, it imposes

the condition that X r.S. The demand for the firm's product is given by S(r) = 10000e-0.2r.

Hence the decision problem is to maximize the net profit function P(X):

2Suppose the only foods available in your local store are potatoes and steak. The decision

about how much of each food to buy is to made entirely on dietary and economic

considerations. We have the nutritional and cost information in the following table:

per unit of potatoes per unit of steak minimum

requirement

units of carbohydrates 3 1 8

units of vitamins 4 3 1 9

units of proteins 1 3 7

unit cost 25 50

a. Formulate the problem in terms of linear inequalities and an objective function.

b. Solve the problem geometrically.

c. Explain how the 2:1 cost ratio (steak to potatoes) dictates that the solution must be

where you said it is.

d. Find a cost ratio that would move the optimal solution to a different choice of

numbers of food units, but that would still require buying both steak and potatoes.

e. Find a cost ratio that would dictate buying only one of the two foods in order to

minimize cost.

NB/explain the meaning of each constraint formulated in each case and every step taken to arriving to the constraint answer in each

FORMULATE AN LPP FOR THE FOLLOWING

Bryant's Pizza, Inc. is a producer of frozen pizza products. The company makes a net income of $1.00 for each regular pizza and $1.50 for each deluxe pizza produced. The firm currently has 150 pounds of dough mix and 50 pounds of topping mix. Each regular pizza uses 1 pound of dough mix and 4 ounces (16 ounces= 1 pound) of topping mix. Each deluxe pizza uses 1 pound of dough mix and 8 ounces of topping mix. Based on the past demand per week, Bryant can sell at least 50 regular pizzas and at least 25 deluxe pizzas. The problem is to determine the number of regular and deluxe pizzas the company should make to maximize net income. Formulate this problem as an LP problem.