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that have the potential of bringing high returns typically also carry high risks of losing money. In a service industry, customer satisfaction and the cost of providing service are two conflicting criteria that would be useful to consider. <span>In our daily lives, we usually weigh multiple criteria implicitly and we 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 mul

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in 1979 titled: "MCDM – If not a Roman Numeral, then What?" MCDM is concerned with structuring and solving decision and planning problems involving multiple criteria. The purpose is to support decision makers facing such problems. <span>Typically, there does not exist a unique optimal solution for such problems and it is necessary to use decision maker’s 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"

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riteria. 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 maker’s preferences to differentiate between solutions. <span>"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 t

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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. <span>The concept of an optimal solution is often replaced by the set of nondominated solutions. A nondominated solution has the property that it is not possible to move away from it to any other solution without sacrificing in at least one criterion. Therefore, it makes sense for the decision maker to choose a solution from the nondominated set. Otherwise, he could do better in terms of some or all of the criteria, and not do worse i

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ut sacrificing in at least one criterion. Therefore, it makes sense for the decision maker to choose a solution from the nondominated set. Otherwise, he could do better in terms of some or all of the criteria, and not do worse in any of them. <span>Generally, however, the set of nondominated solutions is too large to be presented to the decision maker for his final choice. Hence we need tools that help the decision maker focus on his 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

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ws upon knowledge in many fields including: Mathematics Behavioral decision theory Economics Computer technology Software engineering Information systems A typology[edit] There are different classifications of MCDM problems and methods. <span>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 represent

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ariables 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. <span>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 t

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m 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. <span>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

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rences 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. <span>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, [8] and Köksalan and Sagala, 1995 [9] ) and design

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preferences." These methods have been well-developed for both the multiple criteria evaluation (see for example Geoffrion, Dyer and Feinberg, 1972, [8] and Köksalan and Sagala, 1995 [9] ) and design problems (see Steuer, 1986 [10] ). <span>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 [11] ). When the mathematical programming models contain integer variables, the design problems become harder to solve. Multiobjective Com

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ve. Multiobjective Combinatorial Optimization (MOCO) constitutes a special category of such problems posing substantial computational difficulty (see Ehrgott and Gandibleux, [12] 2002, for a review). Representations and definitions[edit] <span>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 speci

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ulting problem is called a Multiple Criteria Evaluation problem. If Q 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 <span>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 mak

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olution is nondominated so long as it is not inferior to any other available solution in all the considered criteria. Definition 2. x* ∈ X is efficient if there does not exist another x ∈ X such that f(x) ≥ f(x*) and f(x) ≠ f(x*) . <span>If an MCDM problem represents a decision situation well, then the most preferred solution of a DM has to be an efficient solution in the decision space, and its image is a nondominated point in the criterion space. Following definitions are also important. Definition 3. q* ∈ Q is weakly nondominated if there does not exist another q ∈ Q such that q > q* . Definition 4. x* ∈ X is weakl

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en used (Charnes and Cooper, 1961 [22] ). Fuzzy-set theorists Fuzzy sets were introduced by Zadeh (1965) [23] as an extension of the classical notion of sets. This idea is used in many MCDM algorithms to model and solve fuzzy problems. <span>Multiattribute utility theorists Multiattribute utility or value functions are elicited and used to identify the most preferred alternative or to rank order the alternatives. Elaborate interview techniques, which exist for eliciting linear additive utility functions and multiplicative nonlinear utility functions, are used (Keeney and Raiffa, 1976 [24] ). French school The French school focuses on decision aiding, in particular the ELECTRE family of outranking methods that originated in France during t