A Typology

There are different classifications of MCDM problems and methods. A major distinction between MCDM problems is based on whether the solutions are explicitly or implicitly defined.

• Multiple-criteria evaluation problems: These problems consist of a finite number of alternatives, explicitly known in the beginning of the solution process. Each alternative is represented by its performance in multiple criteria. The problem may be defined as finding the best alternative for a decision maker (DM), or finding a set of good alternatives. One may also be interested in "sorting" or "classifying" alternatives. Sorting refers to placing alternatives in a set of preference-ordered classes (such as assigning credit-ratings to countries), and classifying refers to assigning alternatives to non-ordered sets (such as diagnosing patients based on their symptoms). Some of the MCDM methods in this category have been studied in a comparative manner in the book by Triantaphyllou on this subject, 2000.[6]

• Multiple-criteria design problems (multiple objective mathematical programming problems): In these problems, the alternatives are not explicitly known. An alternative (solution) can be found by solving a mathematical model. The number of alternatives is either infinite and not countable (when some variables are continuous) or typically very large if countable (when all variables are discrete).

Whether it is an evaluation problem or a design problem, preference information of DMs is required in order to differentiate between solutions. The solution methods for MCDM problems are commonly classified based on the timing of preference information obtained from the DM.

There are methods that require the DM’s preference information at the start of the process, transforming the problem into essentially a single criterion problem. These methods are said to operate by "prior articulation of preferences." Methods based on estimating a value function or using the concept of "outranking relations," analytical hierarchy process, and some decision rule-based methods try to solve multiple criteria evaluation problems utilizing prior articulation of preferences. Similarly, there are methods developed to solve multiple-criteria design problems using prior articulation of preferences by constructing a value function. Perhaps the most well-known of these methods is goal programming. Once the value function is constructed, the resulting single objective mathematical program is solved to obtain a preferred solution.

Some methods require preference information from the DM throughout the solution process. These are referred to as interactive methods or methods that require "progressive articulation of preferences." These methods have been well-developed for both the multiple criteria evaluation (see for example Geoffrion, Dyer and Feinberg, 1972,[7] and Köksalan and Sagala, 1995[8] ) and design problems (see Steuer, 1986[9]).

Multiple-criteria design problems typically require the solution of a series of mathematical programming models in order to reveal implicitly defined solutions. For these problems, a representation or approximation of "efficient solutions" may also be of interest. This category is referred to as "posterior articulation of preferences," implying that the DM’s involvement starts posterior to the explicit revelation of "interesting" solutions (see for example Karasakal and Köksalan, 2009[10]).

When the mathematical programming models contain integer variables, the design problems become harder to solve. Multiobjective Combinatorial Optimization (MOCO) constitutes a special category of such problems posing substantial computational difficulty (see Ehrgott and Gandibleux,[11] 2002, for a review).