Generation of efficient solutions in Multiobjective Mathematical Programming problems using GAMS. Effective implementation of the ε-constraint method

According to the most widely accepted classification the Multiobjective Mathematical Programming (MMP) methods can be classified as a priori, interactive and a posteriori, according to the decision stage in which the decision maker expresses his/her preferences. Although the a priori methods are the most popular, the interactive and the a-posteriori methods convey much more information to the decision maker. Especially, the aposteriori (or generation) methods inform the decision maker about the whole context of the decision alternatives before his/her final decision. However, the generation methods are the less popular due to their computational effort and the lack of widely available software. The basic step towards further penetration of the generation methods in MMP applications, is to provide appropriate codes for Mathematical Programming (MP) solvers that are widely used by people in engineering, economics, agriculture etc. The present work is an effort to effectively implement the ε-constraint method for producing the efficient solutions in a MMP. We propose a variation of the method (augmented ε-constraint method-AUGMECON) that produces only efficient solutions (no weakly efficient solutions) and also avoids redundant iterations as it can perform early exit from the relevant loops (that lead to infeasible solutions), accelerating the whole process. Finally, we implement the method in an adjustable GAMS model using an example from the energy sector, describing in detail the necessary code. 1. Multiobjective Mathematical Programming and efficient solutions The solution of Mathematical Programming (MP) problems with only one objective function is a straightforward task. The output is the optimal solution and all the relevant information about the values of the decision variables, shadow prices etc. In Multiobjective Mathematical Programming (MMP) there are more than one objective functions and there is no single optimal solution that simultaneously optimizes all the objective functions. In these cases the decision makers are looking for the “most preferred” solution. In MMP the concept of optimality is replaced with that of efficiency or Pareto optimality. The efficient (or Pareto optimal, nondominated, non-inferior) solutions are the solutions that cannot be improved in one objective function without deteriorating their performance in at least one of the rest. The mathematical definition of the efficient solution is the following (without loss of generality assume that all the objective functions fi, i=1...p are for maximization): A feasible solution x of a MMP problem is efficient if there is no other feasible solution x’ such as fi(x’) ≥ fi(x) for every i=1, 2, ...,p with at least one strict inequality. Every efficient solution corresponds to a nondominated or non-improvable vector in the criterion space. If we replace the condition fi(x’) ≥ fi(x) with fi(x’) > fi(x) we obtain the weakly efficient solutions. Weakly efficient solutions are not usually pursued in MMP because they may be dominated by other efficient solutions. The rational decision maker is looking for the most preferred solution among the efficient solutions of the MMP. In the absence of any other information, none of these solutions can be said to be better than the other. Usually a decision maker is needed to provide additional preference information and to identify the “most preferred” solution. 2. Classification of the MMP methods According to Hwang and Masud (1979) the methods for solving MMP problems can be classified into three categories according to the phase in which the decision maker involves in the decision making process expressing his/her preferences: The a priori methods, the interactive methods and the generation or a posteriori methods. In a priori methods the decision maker expresses his/her preferences before the solution process (e.g. setting goals or weights for the objective functions). The criticism about the a priori methods is that it is very difficult for the decision maker to know beforehand and to be able to accurately quantify (either by means of goals or weights) his/her preferences. In the interactive methods phases of dialogue with the decision maker are interchanged with phases of calculation and the process usually converges after a few iterations to the most preferred solution. The decision maker progressively drives the search with his answers towards the most preferred solution. The drawback is that he never sees the whole picture (the set of efficient solutions) or an approximation of it. Hence, the most preferred solution is “most preferred” in relation to what he/she has seen and compare so far. In a posteriori methods (or generation methods) the efficient solutions of the problem (all of them or a sufficient representation) are generated and then the decision maker involves, in order to select among them, the most preferred one (see e.g. the interactive filtering process proposed by Steuer, 1986). 3. Generation methods The generation methods are the less popular due to their computational effort (the calculation of the efficient solutions is usually a time consuming process) and the lack of widely available software. However, they have some significant advantages. The solution process is divided into two phases: First, the generation of the efficient solutions and subsequently the involvement of the decision maker when all the information is on the table. Hence, they are favourable whenever the decision maker is hardly available and the interaction with him is difficult, because he is involved only in the second phase, having at hand all the possible alternatives (the efficient solutions of the MMP). Besides, the fact that none of the potential solutions has been left undiscovered, reinforces the decision maker’s confidence on the final decision. For special kind of MMP problems (mostly linear problems) of small and medium size, there are also methods that produce the entire efficient set (see e.g. Steuer 1986, Mavrotas 1998, Miettinen 1999). Here we will focus on the general case, where relatively large MMP problems can be tackled. In general, the most widely used generation methods are the weighting method and the ε-constraint method. These methods are used to provide a representative subset of the efficient set. problems. Assume the following MMP: max (f1(x), f2(x), . . . , fp(x)) st