Optimal Production

Essay

Optimal Production. Ozark Telephone, Inc. (OTI) is a small telephone company offering local dial-tone service to its franchised areas in rural southeastern Missouri. A new office park development site is being planned within OTI's territory and John Sample, a network engineer, has to maximize the conversation capacity per line under cost and technology constraints using both traditional copper-wire lines and new fiber-optic lines.
OTI wants to gradually move into the all-digital communication environment possible with fiber-optics, so a company policy has been adopted specifying that at least 3 fiber-optic lines be employed for every 2 copper lines on new installations. To minimize the need to quickly retrain its linemen, OTI wants at least 30% of new telephone lines installed to be copper. No existing telephone facilities run to the development site, and OTI must use its own facilities to carry the traffic (it cannot lease capacity from any other local telephone company). Finally, current costs and technologies dictate that 1 fiber line can carry the equivalent of 5 copper lines at the same cost to OTI. That is, if one copper line can carry one telephone conversation, fiber optic lines can carry five conversations at no cost penalty. Sample's objective is to maximize the capacity per line of the transmissions facilities being built to carry traffic to/from the office park.
A. Using the inequality form of the constraint conditions, set up and interpret the linear programming problem Sample would use to determine the optimal percentage of copper and fiber-optic lines. Also formulate the problem using the equality form of the constraint conditions.
B. With a graph, determine the optimal solution; check your solution algebraically. Fully interpret solution values.
C. Holding all else equal, how much would the capacity of fiber optic lines have to fall to alter the optimal construction mix determined in part B?
D. Calculate the opportunity cost, measured in terms of conversation capacity per line, of OTI's 30% copper line constraint.