Chapter 4.2, Problem 27E

Set up and solve Exercises 23–29 by the simplex method.

APPLY IT Manufacturing Bicycles A manufacturer of bicycles builds racing, touring, and mountain models. The bicycles are made of both steel and aluminum. The company has available 91,800 units of steel and 42,000 units of aluminum. The racing, touring, and mountain models need 17, 27, and 34 units of steel, and 12, 21, and 15 units of aluminum, respectively. (See Exercise 28 in Section 4.1.)

(a) How many of each type of should be made in order to maximize profit if the company makes $8 per racing bike, $12 per touring bike, and $22 per mountain bike?

(b) What is the maximum possible profit?

(c) Does it require all of the available units of steel and aluminum to build the bicycles that produce the maximum profit? If not, how much of each material is left over? Compare any leftover to the value of the relevant slack variable.

(d) There are many unstated assumptions in the problem given above. Even if the mathematical solution is to make only one or two types of the bicycles, there may be demand for the type(s) not being made, which would create problems for the company. Discuss this and other difficulties that would arise in real situation.