Multiple-criteria decision-making (MCDM) or multiple-criteria decision analysis (MCDA) is a sub-discipline of operations research that explicitly evaluates multiple conflicting criteria in decision making (both in daily life and in settings such as business, government and medicine). It is also known as multi-attribute decision making (MADM), multiple attribute utility theory, multiple attribute value theory, multiple attribute preference theory, and multi-objective decision analysis.
Conflicting criteria are typical in evaluating options: cost or price is usually one of the main criteria, and some measure of quality is typically another criterion, easily in conflict with the cost. In purchasing a car, cost, comfort, safety, and fuel economy may be some of the main criteria we consider – it is unusual that the cheapest car is the most comfortable and the safest one. In portfolio management, managers are interested in getting high returns while simultaneously reducing risks; however, the stocks that have the potential of bringing high returns typically carry high risk of losing money. In a service industry, customer satisfaction and the cost of providing service are fundamental conflicting criteria.
In their daily lives, people usually weigh multiple criteria implicitly and may be comfortable with the consequences of such decisions that are made based on only intuition. On the other hand, when stakes are high, it is important to properly structure the problem and explicitly evaluate multiple criteria. In making the decision of whether to build a nuclear power plant or not, and where to build it, there are not only very complex issues involving multiple criteria, but there are also multiple parties who are deeply affected by the consequences.
Structuring complex problems well and considering multiple criteria explicitly leads to more informed and better decisions. There have been important advances in this field since the start of the modern multiple-criteria decision-making discipline in the early 1960s. A variety of approaches and methods, many implemented by specialized decision-making software,Weistroffer, HR, and Li, Y (2016). "Multiple criteria decision analysis software". Ch 29 in: Greco, S, Ehrgott, M and Figueira, J, eds, Multiple Criteria Decision Analysis: State of the Art Surveys Series, Springer: New York. have been developed for their application in an array of disciplines, ranging from politics and business to the environment and energy.
Foundations, concepts, definitions
MCDM or MCDA are acronyms for multiple-criteria decision-making and multiple-criteria decision analysis. Stanley Zionts helped popularizing the acronym with his 1979 article "MCDM – If not a Roman Numeral, then What?", intended for an entrepreneurial audience.
MCDM is concerned with structuring and solving decision and planning problems involving multiple criteria. The purpose is to support decision-makers facing such problems. Typically, there does not exist a unique optimal solution for such problems and it is necessary to use decision-makers' preferences to differentiate between solutions.
"Solving" can be interpreted in different ways. It could correspond to choosing the "best" alternative from a set of available alternatives (where "best" can be interpreted as "the most preferred alternative" of a decision-maker). Another interpretation of "solving" could be choosing a small set of good alternatives, or grouping alternatives into different preference sets. An extreme interpretation could be to find all "efficient" or "nondominated" alternatives (which we will define shortly).
The difficulty of the problem originates from the presence of more than one criterion. There is no longer a unique optimal solution to an MCDM problem that can be obtained without incorporating preference information. The concept of an optimal solution is often replaced by the set of nondominated solutions. A solution is called nondominated if it is not possible to improve it in any criterion without sacrificing it in another. Therefore, it makes sense for the decision-maker to choose a solution from the nondominated set. Otherwise, they could do better in terms of some or all of the criteria, and not do worse in any of them. Generally, however, the set of nondominated solutions is too large to be presented to the decision-maker for the final choice. Hence we need tools that help the decision-maker focus on the preferred solutions (or alternatives). Normally one has to "tradeoff" certain criteria for others.
MCDM has been an active area of research since the 1970s. There are several MCDM-related organizations including the International Society on Multi-criteria Decision Making, Euro Working Group on MCDA, and INFORMS Section on MCDM. For a history see: Köksalan, Wallenius and Zionts (2011). MCDM draws upon knowledge in many fields including:
Mathematics Decision analysis Economics Computer technology Software engineering Information systems
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. 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 finite or infinite (countable or not countable), but typically exponentially large (in the number of variables ranging over finite domains.)
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 rule-based decision 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,An Interactive Approach for Multi-Criterion Optimization, with an Application to the Operation of an Academic Department, A. M. Geoffrion, J. S. Dyer and A. Feinberg, Management Science, Vol. 19, No. 4, Application Series, Part 1 (Dec., 1972), pp. 357–368 Published by: INFORMS and Köksalan and Sagala, 1995 ) and design problems (see Steuer, 1986).
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).
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, 2002, for a review).
Representations and definitions
The MCDM problem can be represented in the criterion space or the decision space. Alternatively, if different criteria are combined by a weighted linear function, it is also possible to represent the problem in the weight space. Below are the demonstrations of the criterion and weight spaces as well as some formal definitions.
Criterion space representation
Let us assume that we evaluate solutions in a specific problem situation using several criteria. Let us further assume that more is better in each criterion. Then, among all possible solutions, we are ideally interested in those solutions that perform well in all considered criteria. However, it is unlikely to have a single solution that performs well in all considered criteria. Typically, some solutions perform well in some criteria and some perform well in others. Finding a way of trading off between criteria is one of the main endeavors in the MCDM literature.
Mathematically, the MCDM problem corresponding to the above arguments can be represented as
subject to
where is the vector of k criterion functions (objective functions) and is the feasible set, .
If is defined explicitly (by a set of alternatives), the resulting problem is called a multiple-criteria evaluation problem.
If is defined implicitly (by a set of constraints), the resulting problem is called a multiple-criteria design problem.
The quotation marks are used to indicate that the maximization of a vector is not a well-defined mathematical operation. This corresponds to the argument that we will have to find a way to resolve the trade-off between criteria (typically based on the preferences of a decision maker) when a solution that performs well in all criteria does not exist.
Decision space representation
The decision space corresponds to the set of possible decisions that are available to us. The criteria values will be consequences of the decisions we make. Hence, we can define a corresponding problem in the decision space. For example, in designing a product, we decide on the design parameters (decision variables) each of which affects the performance measures (criteria) with which we evaluate our product.
Mathematically, a multiple-criteria design problem can be represented in the decision space as follows:
where is the feasible set and is the decision variable vector of size n.
A well-developed special case is obtained when is a polyhedron defined by linear inequalities and equalities. If all the objective functions are linear in terms of the decision variables, this variation leads to multiple objective linear programming (MOLP), an important subclass of MCDM problems.
There are several definitions that are central in MCDM. Two closely related definitions are those of nondominance (defined based on the criterion space representation) and efficiency (defined based on the decision variable representation).