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Exercise 1: Alabama Atlantic is a lumber company that has three sources of wood and five
markets to be supplied. The annual availability of wood at sources 1, 2, and 3 is 15, 20, and 15
million board feet, respectively. The amount that can be sold annually at markets 1, 2, 3, 4, and 5
is 11, 12, 9, 10, and 8 million board feet, respectively. In the past the company has shipped the
wood by train. However, because shipping costs have been increasing, the alternative of using
ships to make some of the deliveries is being investigated. This alternative would require the
company to invest in some ships. Except for these investment costs, the shipping costs in thousands
of dollars per million board feet by rail and by water (when feasible) would be the following for
Unit Cost by Rail ($1000’s) Market
Unit Cost by Ship ($1000’s) Market
The capital investment (in thousands of dollars) in ships required for each million board feet to be
transported annually by ship along each route is given as follows:
Investment for Ships ($1000’s) Market
Considering the expected useful life of the ships and the time value of money, the equivalent
uniform annual cost of these investments is one-tenth the amount given in the table. The objective
is to determine the overall shipping plan that minimizes the total equivalent uniform annual cost
(including shipping costs).
You are the head of the OR team that has been assigned the task of determining this shipping plan
for each of the following three options.
Option 1: Continue shipping exclusively by rail.
Option 2: Switch to shipping exclusively by water (except where only rail is feasible).
Option 3: Ship by either rail or water, depending on which is less expensive for the route.
Present your results for each option. Compare.
Finally, consider the fact that these results are based on current shipping and investment costs, so
the decision on the option to adopt now should consider management’s projection of how these
costs are likely to change in the future. For each option, describe a scenario of future cost changes
that would justify adopting that option now.
Exercise 5: Northeastern Airlines is considering the purchase of new long-, medium-, and shortrange 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 1 3 short-range planes, and each long-range plane is
equivalent to 1 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. Solve this problem using the branch-and-bound algorithm (you can use either Excel or
python to help solve LP problems).
Exercise 3: Consider the two-variable integer program:
max 9𝑥1 + 5𝑥2
4𝑥1 + 9𝑥2 ≤ 35
𝑥1 ≤ 6
𝑥1 − 3𝑥2 ≥ 1
3𝑥1 + 2𝑥2 ≤ 19
𝑥1 , 𝑥2 ∈ 𝐼 +
Solve by branch-and-bound graphically and algebraically.
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