Computational aspects and statistical applications of the transportation problem of linear programming

A special type of linear programming problem that arises quite frequently in practical application is the transportation problem. The transportation problem involves determining a shipping schedule that minimizes the total cost of shipment given that there are known fixed quantities of a commodity available for shipment at each origin, there are given quantities of a commodity required to be shipped at each destination, and the total shipments from all origins equal the total requirements of all destinations. It is assumed that the minimum cost of shipping a unit of commodity from any origin to any destination is known, and total shipping cost is obtained by taking the sum of individual cost;The algorithm used for finding an optimal solution requires that an initial basic feasible solution must first be determined. The algorithm improves this solution at each iteration, resulting in a total shipping cost that is less than or equal to the total cost in the previous iteration, until an optimal solution is found. There are different procedures for determining an initial feasible basis. Each of these procedures can give a different initial feasible basis that corresponds to varying total costs, and also takes varying amount of computational time in obtaining an initial feasible basis. A comparison of the various start procedures was conducted to determine the procedure that would give the best initial feasible basis, that is, one that is close to the optimum and does not use up a lot of CPU time in finding one. As a result of this study, it was determined that for large rectangular transportation problems the Large Amount Low Cost method would give the best initial feasible basis, and for square and not very rectangular problems, the Modified Minimum Row rule is best;The methods for solving transportation problems can be applied to solve some statistical problems. Two statistical problems that can be expressed in the form of a transportation problem are presented, namely, obtaining least absolute value estimates for the two-way classification model and the problem of controlled rounding. The application of the simple upper bounds procedure to obtain optimal solutions for these two problems is demonstrated through numerical examples