section10_11 - 10 The transportation problem In this...

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Unformatted text preview: 10 The transportation problem In this section we study a classical and very useful linear programming model: the transportation problem. We begin with a simple example of a trans- portation problem, taken from the seminal book on linear programming [3]. We consider a company with plants in Seattle and San Diego, with ca- pacities 350 and 600 cases per week respectively. The company must satisfy demand in New York, Chicago and Topeka of 325, 300, and 275 cases respec- tively. Transportation costs (in dollars per case) between plants and demand points 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: (10.1) 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 destination is the difference between supply and demand, and the trans- portation cost from any origin to this destination is zero. Clearly, this new problem is equivalent to the original problem, and it satisfies assumption (10.1). Henceforth, we therefore assume that condition (10.1) 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 55 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 . The objective function is the transportation cost, added up over all pairs of origins and destinations. The first set of constraints (for i = 1 , 2 , . . . , m ) ensures that all the supply at origin i is used up exactly. The second set of constraints (for j = 1 , 2 , . . . , n ) ensures that all the demand at destination j is satisfied exactly. An AMPL model capturing this problem is the following file transp.mod from [4]. set ORIG; # origins set DEST; # destinations param supply {ORIG} >= 0; # amounts available at origins param demand {DEST} >= 0; # amounts required at destinations check: sum {i in ORIG} supply[i] = sum {j in DEST} demand[j]; param cost {ORIG,DEST} >= 0; # shipment costs per unit var Trans {ORIG,DEST} >= 0; # units to be shipped minimize total_cost: sum {i in ORIG, j in DEST} cost[i,j] * Trans[i,j];...
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This note was uploaded on 01/09/2009 for the course ORIE 320 taught by Professor Bland during the Fall '07 term at Cornell University (Engineering School).

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section10_11 - 10 The transportation problem In this...

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