MCMP Problem

A general multiple criteria mathematical programming (MCMP) problem is given as follows ... 

Consider the general MCMP problem given in Equation 6.21. The assumption that there exists an optimal soluhon to the MCMP problem involving multiple criteria implies the existence of some preference ordering of the criteria by the DM. The goal programming (GP) formulation of the MCMP problem requires the DM to specify an acceptable level of achievement b for each criterion f, and specify a weight W (ordinal or cardinal) to be associated with the deviation between f, and bj Thus, the GP model of an MCMP problem becomes ... 

In this and the subsequent sections, we will focus on MCDM problems with infinite number of alternatives. In other words, the feasible alternatives are not known a priori but are represented by a set of mathematical (linear/nonlinear) constraints. These MCDM problems are called multicriteria mathematical programming (MCMP) problems. 

A solution to MCMP is called a superior solution if it is feasible and maximizes all the objectives simultaneously. In most MCMP problems, superior solutions do not exist as the objectives conflict with one another. 

NOTE One of the popular approaches to solving MCMP problems is to find an efficient solution that comes "as close as possible" to the ideal solution. We will discuss this approach later in Section A.7. 

For the MCMP problem given by Equation A.5, consider the following single objective optimization problem, called the P) problem. 

Warning Theorem 1 is only a sufficient condition and is not necessary. For example, there could be efficient solutions to MCMP which could not be obtained as optimal solutions to the problem. Such situations occur when the objective space is not a convex set. Flowever, for MCMP problems, where the objective functions and constraints are linear. Theorem A.1 is both necessary and sufficient. 

In MCMP problems, there are often an infinite number of efficient solutions and they are not comparable without the input from the DM. Hence, it is generally assumed that the DM has a real-valued preference function defined on the values of the objectives, but it is not known explicitly. With this assumption, the primary objective of the MCMP solution methods is to find the best compromise solution, which is an efficient solution that maximizes the DM s preference function. 

Interactive methods for MCMP problems rely on the progressive articulation of preferences by the DM. These approaches can be characterized by the following procedures ... 

Method of Global Criterion (Hwang and Masud 1979) and Compromise Programming (Zeleny 1982) fall under the class of MCMP methods that do not require any preference information from the DM. Consider the MCMP problem given by Equation 2.14. Let... 

Chapter 1 presents an overview of the MCDM models and solution methods. MCDM problems are categorized on the basis of whether (1) the alternatives are finite and known, which are called multi-criteria selection problems (MCSP) or (2) the alternatives are infinite and unknown, which are called multiple criteria mathematical programming (MCMP) problems. In this chapter, the most common methods that are available for solving both MCSP and MCMP problems are reviewed. 

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