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Unformatted text preview: 11 Transportation and assignment problems In this section we study two classical and very useful linear programming mod- els: the “transportation problem” and a special case, the “assignment problem.” Incidentally, we shall see an important class of problems that exhibit massive degeneracy. We begin with a simple example of a transportation problem, mod- ified from one in the seminal book on linear programming . We consider a company with plants in Seattle and San Diego, with capacities 350 and 625 cases per week respectively. The company must satisfy demand in New York, Chicago and Topeka of 325, 300, and 275 cases respectively. Trans- portation costs (in dollars per case) are as follows. New York Chicago Topeka Seattle 225 153 162 San Diego 225 162 126 We must decide how many cases to send on each route in order to satisfy the demand at minimum total transportation cost. In the general transportation problem, we have m “origins,” labeled i = 1 , 2 , . . . , m : origin i has a supply of s i units. We also have n “destinations,” labeled j = 1 , 2 , . . . , n : destination j has demand d j units. To transport one unit from origin i to destination j costs c ij , and our aim is to satisfy all the demands using the available supplies, at minimum total transportation cost. For the general transportation problem to be feasible, the total supply must be at least as large as the total demand. In fact, we do not really restrict the model if we make the following assumption: (11.1) total supply m X i =1 s i = total demand n X j =1 d j . If, in our problem, the total supply strictly exceeded the total demand, we could simply introduce an extra “dummy” destination. The demand at this destina- tion is the difference between supply and demand, and the transportation cost from any origin to this destination is zero. Clearly, this new problem is equiva- lent to the original problem, and it satisfies assumption (11.1). Henceforth, we therefore assume that condition (11.1) holds. We introduce variables x ij that measure the number of units we transport from origin i to destination j . Our transportation problem then becomes the following linear program. minimize m X i =1 n X j =1 c ij x ij subject to n X j =1 x ij = s i ( i = 1 , 2 , . . . , m ) m X i =1 x ij = d j ( j = 1 , 2 , . . . , n ) x ≥ ....
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