14. decision making with interactions Ralph is now managing a firm with interdependent service centers.

Two such centers are involved, say, power and maintenance. For each such center, 80% of total output goes to manufacturing and 20% goes to the other service center. The variable cost of power is 10 per unit while the variable cost of maintenance is 15 per unit. Production requires 800 units of each service. Hence, 1, 000 (gross) units of each are produced at a total variable cost of 25, 000. At this point Ralph starts to consider an opportunity to purchase power (all or some) from an outside vendor, and the following questions are to be answered with this in mind.

(a) What is the cost per unit of power? (Hint: simultaneous equations are necessary, and 13.5417 is an important number.)

(b) Now formulate and solve a program to determine the minimum cost activity levels for power and maintenance in order to provide manufacturing with at least 800 units of each service. You can infer from the noted production plan that each unit of power requires .2 units of maintenance and vice versa. So the technical constraints in your program should be P −.2M ≥ 800 and M − .2P ≥ 800, where P denotes units of power and M denotes units of maintenance.

(c) Compare your cost per unit of power in

(a) above with the shadow price on the power constraint in

(b) above. Carefully explain why they differ, if the differ, or why they are the same if they are the same.

(d) Suppose the outside source will sell power at 12 per unit. Should this offer be accepted? If so, determine the total saving.

(e) Next formulate and solve a program related to that in

(b) above but with a third variable, x, units of power purchased from the outside source at 12 per unit.

(f) In part

(d) above you used a cost of internal power of 13.5417 to answer the sourcing question, but in the program in

(e) you used a cost of internal power of 10 per unit to answer the sourcing question. Carefully explain. (Hint: decision framing is at work.)