Abstract
In a fast changing global market, a manager is concerned with cost uncertainties of the cost matrix in transportation problems (TP) and assignment problems (AP).A time lag between the development and application of the model could cause cost parameters to assume different values when an optimal assignment is implemented. The manager might wish to determine the responsiveness of the current optimal solution to such uncertainties. A desirable tool is to construct a perturbation set (PS) of cost coeffcients which ensures the stability of an optimal solution under such uncertainties.The widely-used methods of solving the TP and AP are the stepping-stone (SS) method and the Hungarian method, respectively. Both methods fail to provide direct information to construct the needed PS. An added difficulty is that these problems might be highly pivotal degenerate. Therefore, the sensitivity results obtained via the available linear programming (LP) software might be misleading.We propose a unified pivotal solution algorithm for both TP and AP. The algorithm is free of pivotal degeneracy, which may cause cycling, and does not require any extra variables such as slack, surplus, or artificial variables used in dual and primal simplex. The algorithm permits higher-order assignment problems and side-constraints. Computational results comparing the proposed algorithm to the closely-related pivotal solution algorithm, the simplex, via the widely-used pack-age Lindo, are provided. The proposed algorithm has the advantage of being computationally practical, being easy to understand, and providing useful information for managers. The results empower the manager to assess and monitor various types of cost uncertainties encountered in real-life situations. Some illustrative numerical examples are also presented.
Highlights
The widely-used methods of solving transportation problems (TP) and assignment problems (AP) are the stepping-stone (SS) method and the Hungarian method, respectively
We have proposed a general-purpose uni ed algorithm to solve the classical TP and AP
It is computationally practical in the sense that it does not require any slack/surplus variables or any arti cial variables
Summary
The widely-used methods of solving transportation problems (TP) and assignment problems (AP) are the stepping-stone (SS) method and the Hungarian method, respectively. A small variation in the problem, such as the introduction of side-constraints, destroys the special structure and requires a new solution algorithm These algorithms obtain solution e ciency at the expense of managerial insight, as the nal solutions from these algorithms do not provide su cient information to perform post optimality analysis for TP and AP. Most linear programming books, including management science and operations research books, have extensive discussion of linear programming (LP) sensitivity analysis (SA) but remain silent about the SA and side-constraints for TP and AP Both methods, the SS and the Hungarian, lack the ability to test the validity of the current optimal solution with changes in cost parameters without resolving the problem.
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