Mathematical Model of a Transportation Problem
m … number of sources
n … number of destinations
ai … capacity of i-th source (in tons, pounds, liters, etc) bj … demand of j-th destination (in tons, pounds, liters, etc.) cij … cost coefficients of material shipping (unit shipping cost) between i-th source and j-th destination (in $ or as a distance in kilometers, miles, etc.) xij … amount of material shipped between i-th source and j-th destination (in tons, pounds, liters etc.) 7. The Transportation Problem
There is a type of linear programming problem that may be solved using a simplified version of the simplex technique called transportation method. Because of its major application in solving problems involving several product sources and several destinations of products, this type of problem is frequently called the transportation problem. It gets its name from its application to problems involving transporting products from several sources to several destinations. Although the formation can be used to represent more general assignment and scheduling problems as well as transportation and distribution problems. The two common objectives of such problems are either (1) minimize the cost of shipping munits to n destinations or (2) maximize the profit of shipping m units to n destinations. Let us assume there are m sources supplying n destinations. Source capacities, destinations requirements and costs of material shipping from each source to each destination are given constantly. The transportation problem can be described using following linear programming mathematical model and usually it appears in a transportation tableau. There are three general steps in solving transportation problems. We will now discuss each one in the context of a simple example. Suppose one company has four factories supplying four warehouses and its management wants to determine the minimum-cost shipping schedule for its weekly output of chests. Factory supply, warehouse demands, and shipping costs per one chest (unit) are shown in Table 7.1
Table 7.1 ”Data for Transportation Problem”
At first, it is necessary to prepare an initial feasible solution, which may be done in several different ways; the only requirement is that the destination needs be met within the constraints of source supply. 7.1 The Transportation Matrix
The transportation matrix for this example appears in Table 7.2, where supply availability at each factory is shown in the far right column and the warehouse demands are shown in the bottom row. The unit shipping costs are shown in the small boxes within the cells (see transportation tableau – at the initiation of solving all cells are empty). It is important at this step to make sure that the total supply availabilities and total demand requirements are equal. Often there is an excess supply or demand. In such situations, for the transportation method to work, a dummy warehouse or factory must be added. Procedurally, this involves inserting an extra row (for an additional factory) or an extra column (for an ad warehouse). The amount of supply or demand required by the ”dummy” equals the difference between the row and column totals. In this case there is:
Total factory supply … 51
Total warehouse requirements … 52
This involves inserting an extra row - an additional factory. The amount of supply by the dummy equals the difference between the row and column totals. In this case there is 52 – 51 = 1. The cost figures in each cell of the dummy row would be set at zero so any units sent there would not incur a transportation cost. Theoretically, this adjustment is equivalent to the simplex procedure of inserting a slack variable in a constraint inequality to convert it to an equation, and, as in the simplex, the cost of the dummy would be zero in the objective function.
Table 7.2 "Transportation Matrix for Chests Problem With an Additional Factory (Dummy)"
7.2 Initial Feasible Solution
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