SUPPLEMENT TO CHAPTER EIGHT
THE TRANSPORTATION MODEL
After completing this supplement you should be able to:
Describe the nature of a transportation problem.
Solve transportation problems manually and interpret the results. SUPPLEMENT OUTLINE
Obtaining an Initial Solution
The Intuitive Lowest-Cost Approach
Testing for Optimality
Evaluating Empty Cells: The Stepping-Stone Method
Evaluating Empty Cells: The MODI Method
Obtaining an Improved Solution
Unequal Supply and Demand
Summary of Procedure
Discussion and Review Questions
The Transportation problem involves finding the lowest-cost plan for distributing stocks of goods or supplies from multiple origins to multiple destinations that demand the goods. For instance, a firm might have three factories, all of which are capable of producing identical units of the same product, and four warehouses that stock or demand those products, as depicted in Figure 1. The transportation model can be used to determine how to allocate the supplies available from the various factories to the warehouses that stock or demand those goods, in such a way that total shipping cost is minimized. Usually, analysis of the problem will produce a shipping plan that pertains to a certain period of time (day, week), although once the plan is established, it will generally not change unless one or more of the parameters of the problem (supply, demand, unit shipping cost) changes.
The transportation model starts with the development of a feasible solution, which is then sequentially tested and improved until an optimal solution is obtained. The description of the technique on the following pages focuses on each of the major steps in the process in this order: 1.
Obtaining an initial solution.
Testing the solution for optimality.
Improving sub optimal solutions.
OBTAINING AN INITIAL SOLUTION
To begin the process, it is necessary to develop a feasible distribution plan. A number of different methods are available for obtaining such a plan. The discussion here will focus on the intuitive approach, a heuristic approach that yields an initial solution that is often optimal or near optimal. The Intuitive Lowest-Cost Approach
With the intuitive approach, cell allocations are made according to cell cost, beginning with the lowest cost. The procedure involves these steps: 1.
Identify the cell with the lowest cost.
Allocate as many units as possible to that cell, and cross out the row or column (or both) that is exhausted by this. 3.
Find the cells with the next lowest cost from among the feasible cells. 4.
Repeat steps (2) and (3) until all units have been allocated. Cell 1-D has the lowest cost ($1) (see Table 2). The factory 1 supply is 100, and the warehouse D demand is 160. Therefore, the most we can allocate to this cell is 100 units. Since the supply of 100 is exhausted, we cross out the costs in the first row along with the supply of 100. In addition, we must adjust the column total to reflect the allocation, which leaves 60 units of unallocated demand in column D. The next lowest cost is $3 in cell 2-B. Allocating 90 units to this cell exhausts the column total and leaves 110 units for the supply of factory 2 (see Table 3). Also, we cross out the costs for column B. The next lowest cost (that is not crossed out) is the $5 in cell 3-D. Allocating 60 units to this cell exhausts the column total and leaves 90 units for row 3. We now cross out the costs in column D (see Table 4). At this point, there is a tie for the next lowest cell cost: cell 3-A and cell 2-C each has a cost of $8. Break such a tie arbitrarily. Suppose we choose cell 3-A. The demand is 80 units, and the remaining supply is 90 units in the row. The smaller of these is 80, so that amount is placed in cell 3-A (see Table 5). This exhausts the column, so we...
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