# Question: Northeastern Airlines is considering the purchase of new long medium

Northeastern Airlines is considering the purchase of new long-, medium-, and short-range jet passenger airplanes. The purchase price would be $67 million for each long-range plane, $50 million for each medium-range plane, and $35 million for each short-range plane. The board of directors has authorized a maximum commitment of $1.5 billion for these purchases. Regardless of which airplanes are purchased, air travel of all distances is expected to be sufficiently large that these planes would be utilized at essentially maximum capacity. It is estimated that the net annual profit (after capital recovery costs are subtracted) would be $4.2 million per long-range plane, $3 million per medium-range plane, and $2.3 million per short-range plane.

It is predicted that enough trained pilots will be available to the company to crew 30 new airplanes. If only short-range planes were purchased, the maintenance facilities would be able to handle 40 new planes. However, each medium-range plane is equivalent to 11/3 short-range planes, and each long-range plane is equivalent to 12/3 short-range planes in terms of their use of the maintenance facilities. The information given here was obtained by a preliminary analysis of the problem. A more detailed analysis will be conducted subsequently. However, using the preceding data as a first approximation, management wishes to know how many planes of each type should be purchased to maximize profit.

(a) Formulate an IP model for this problem.

(b) Use the computer to solve this problem.

(c) Use a binary representation of the variables to reformulate the IP model in part (a) as a BIP problem.

It is predicted that enough trained pilots will be available to the company to crew 30 new airplanes. If only short-range planes were purchased, the maintenance facilities would be able to handle 40 new planes. However, each medium-range plane is equivalent to 11/3 short-range planes, and each long-range plane is equivalent to 12/3 short-range planes in terms of their use of the maintenance facilities. The information given here was obtained by a preliminary analysis of the problem. A more detailed analysis will be conducted subsequently. However, using the preceding data as a first approximation, management wishes to know how many planes of each type should be purchased to maximize profit.

(a) Formulate an IP model for this problem.

(b) Use the computer to solve this problem.

(c) Use a binary representation of the variables to reformulate the IP model in part (a) as a BIP problem.

## Relevant Questions

Consider the two-variable IP example discussed in Sec. 12.5 and illustrated in Fig. 12.3. (a) Use a binary representation of the variables to reformulate this model as a BIP problem. Consider the following special type of shortest-path problem (see Sec. 10.3) where the nodes are in columns and the only paths considered always move forward one column at a time. The numbers along the links represent ...Consider the following IP problem: Maximize Z = 5x1 + x2, Subject to and x1 ≥ 0, x2 ≥ 0 x1, x2 are integers. (a) Solve this problem graphically. Use the BIP branch-and-bound algorithm presented in Sec. 12.6 to solve the following problem interactively: Maximize Z = 5x1 + 5x2 + 8x3 – 2x4 – 4x5, Subject to and xj is binary, for j = 1, 2, . . . , 5. The board of directors of General Wheels Co. is considering six large capital investments. Each investment can be made only once. These investments differ in the estimated long-run profit (net present value) that they will ...Post your question