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Transportation problems

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Transportation Problem Set

1. Consider a transportation problem with 4 sources and 3 demand points.
(a) Write the min-cost model in full (without using ∑ notation).
(b) Draw the network.
(c) Construct the standard data table.

2. Consider a transportation problem for which , where is a handling cost at node i. (This can be unloading/loading charge from/to a barge or railroad car.) Prove that if total supply equals total demand, every feasible shipping policy is optimal.

3. Prove or give a counter-example to each of the following.
(a) If assignment costs are unique, the optimal assignment is unique.
(b) If there is excess supply and all shipping costs are positive, every optimal shipping policy ships only the amount of total demand (i.e., no demand is over-satisfied).

4. A 400-meter relay requires 4 different swimmers who each swim 100 meters of backstroke, breaststroke, butterfly, and freestyle. A coach has 5 swimmers whose expected times (in seconds) are given in the following table.
backstroke breaststroke butterfly freestyle
Chris 65 73 63 57
Charlie 67 70 65 58
Francis 68 72 69 55
Jackie 67 75 70 59
Terry 71 69 75 57

(a) How should the coach assign swimmers to the relay so as to minimize their total time?
(b) Consider the swimmer not assignment to any event. How much must each of the 4 event-times decrease for that swimmer to be assigned to an event? Who is replaced?
(c) How valuable would a new swimmer be with the following times:
backstroke breaststroke butterfly freestyle
No Name 66 72 68 57

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Solution Summary

This shows how to work with a transportation problem: write the min-cost model, draw the network, and construct the data table.

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