تفاوت بین تفسیر مدیریتی و ریاضی از نتایج تجزیه و تحلیل حساسیت در برنامه ریزی خطی

This paper shows that managerial questions are not answered satisfactorily with the mathematical interpretation of sensitivity analysis when the solution of a linear programing model is degenerate. Most of the commercially available software packages provide sensitivity results about the optimality of a basis and not about the optimality of the values of the decision variables. The misunderstanding of the shadow price and the validity range information provided by a simplex based computer program may lead to wrong decision with considerable financial losses and strategic consequences. The paper classifies the most important types of sensitivity information, graphically illustrates degeneracy, and demonstrates its effect on sensitivity analysis. A production planning example is provided to show the possibility of faulty production management decisions when sensitivity results are not understood correctly. Finally the recommendations for the users of linear programing models and for software developers are provided.

Linear programing (LP) is one of the most extensively used operations research technique in production and operations management [1]. As a result of the development of computer technology and the rapid evolution of user friendly LP software, every operation manager can run an LP software easily and quickly on a laptop computer. Although solving LP models is now accessible for everybody, the interpretation of the results requires a lot of skill. Most of the management science and OR textbooks pay special attention to sensitivity analysis, and the problems of degeneracy, but sensitivity analysis under degeneracy is rarely discussed. Commercially available software do not give enough information to the user about the existence and the consequences of these, very common, “special cases”. In practice, managers very frequently misinterpret the LP results which may lead to erroneous decisions and to important financial and/or strategic disadvantages. Several papers have addressed this issue. Evans and Baker draw the attention to the consequences of the misinterpretation of sensitivity analysis results in management. They illustrate their point with a simple example and list some published cases in which the erroneous interpretation of sensitivity analysis results is obvious [2]. Aucamp and Steinberg [3] also warn that shadow price analysis is incorrect in many textbooks, and that the shadow price is not equal to “the optimal solution of the dual problem” when the obtained optimal solution is degenerate. They present some examples of shadow price calculations by commercial packages. Akgül [4] refines the shadow price definition of Aucamp and Steinberg, and introduces the negative and positive shadow prices for the increase and the decrease of the RHS elements. Greenberg [5] shows that very frequently practical LP models have a netform structure, and netform structures are always degenerate. He illustrates sensitivity analysis of netform type models by one of the Midterm Energy Market Model of the U.S. Department of Energy. Gal [6] summarizes most of the critics concerning sensitivity analysis of LP models and highlights some important research directions. Rubin and Wagner [7] illustrates the traps of the interpretation of LP results by using the industry cost curve model in a tutorial type paper written for managers and instructors. Jansen et al. [8] explain the effect of degeneracy on sensitivity analysis by using a transportation model, and present the shortcomings of the most frequently used LP packages. They also show how complete, correct sensitivity analysis can be done. Wendell [9] and [10] also pays special attention to the correct and practically useful calculation of sensitivity information. The biggest problem is not that operations researchers are unaware of the difficulties of sensitivity analysis. This issue is discussed thoroughly in the scientific literature (see for example [6], [10] and [11]) and a complete, mathematically correct treatment of sensitivity analysis is presented by Jansen et al. [8], and by Roos et al. [12]. Practice, however, shows that the problem is not widely known among the LP users, and the available commercial software packages are not helpful in recognizing the difficulties. The main objective of this paper is to explain the difference between the managerial questions and the traditional mathematical interpretation of sensitivity analysis. In the first part of the paper basic definitions are introduced, the most important types of sensitivity information are classified, and degenerate LP solutions are illustrated graphically. In the second part a production planning problem is used to demonstrate the consequences of incorrect interpretations of the provided sensitivity information. Finally, some recommendations are made for both practitioners and software developers.

The main objective of the paper is to show that implementation of sensitivity analysis in commercial packages, and managerial interpretation of sensitivity analysis of linear programming models are different. The sensitivity information given by the simplex based commercial packages tell the user in what range the data can vary to keep the obtained optimal basis optimal, and how the current optimal basis solution changes as a function of the problem data. When an optimal solution of an LP model is degenerate then there are several optimal basis providing the same optimal value, and possibly all optimal basis provide different sensitivity results. These results are mathematically correct, but their information content is either incomplete or irrelevant from the management decision point of view. Management wants to know either the sensitivity information concerning activities in an optimal solution (Type II sensitivity), or the sensitivity information concerning the objective function (Type III sensitivity). The situation is a little different in case of solvers based on the IPM, because these solvers provide strictly complementary solutions. These optimal solutions are not basis solutions. Therefore Type I sensitivity has no relevance. However using efficient optimal basis identification techniques, an optimal basis can be produced when needed. On the other hand, Type II and Type III information coincide in this case. But to distinguish among Type II and Type III sensitivity is important for decision making purposes, specially when a non-strictly complementary optimal solution is implemented. Both the graphical solution of the small LP model and the logical solution of the production planning model have illustrated the existence of the three type of sensitivities. Users should be careful when sensitivity results of an LP package are used for management decisions. Almost all practical size problems are degenerate, and the sensitivity information depends on the basis found by the computer program. Different software may give different results to the same model. Sometimes the goodness of the sensitivity output can be checked by simple logic, but in most of the cases there is no direct way of evaluating the results. Linear programming will probably stay one of the most popular operations research tool used in practice. The development of computer technology brought nearer this tool to inexperienced users. The interpretation of the sensitivity output of the currently available software packages is difficult and contains several traps because they provide only Type I sensitivity information. Software producers have a lot of possibility to help avoid the presented problems and to serve better the users in the management area.