Multi-criteria decision-making ( MCDM ) or multi-criteria decision analysis ( MCDA ) is a sub-discipline of operations research That Explicitly multiple Evaluates conflicting criteria in decision making (both in daily life and in settings Such As business, government and medicine ). Conflicting criteria are the most cost-effective and cost-effective way to evaluate the cost effectiveness of a contract. 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 safest one. In portfolio management , we are interested in obtaining high returns of the losing money. In a service industry, customer satisfaction and the cost of providing services are fundamental conflicting criteria.
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 . [ Citation needed ] On the other hand, when stakes are high, it is important to explicitly evaluate multiple criteria. [ Citation needed ] 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 parts which are deeply affected by the consequences.
Structuring complex problems and considering multiple criteria explicitly leads to more informed and better decisions. There has been a lot of 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 ,   -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 well-known acronyms for multiple-criteria decision-making and multiple-criteria decision analysis ; Stanley Zionts helped popularize 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 decisions and planning problems involving multiple criteria. The purpose is to support decision-makers facing such problems. Typically, there is a unique optimal solution for such problems and it is necessary to use a 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” 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 “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 single 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 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, she / he could do better in terms of some or all of the criteria, and not to 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 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 on knowledge in many fields including:
- Decision analysis
- Computer Technology
- Software engineering
- Information systems
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 and placing refers to placing alternatives in a set of preference-ordered classes, 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 Triantaphyllou on this subject, 2000.
- Multiple-criteria design problems (multiple objective mathematical programming problems) : In these problems, the alternative is 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.
Whether it is an evaluation or a design problem, preference information of DMs is required in order to differentiate between solutions. The solution methods for MCDM problems are commonly based on the timing of preference information obtained from the DM.
DM’s preference information is the starting point 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 the concept of “outranking relations,” analytical hierarchy processes, and some decision-based methods. 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,
Some methods require preference from the DM throughout the solution process. These are referred to as “progressive articulation of preferences.” These methods-have-been well-developed for Both the multi-criteria evaluation (see for example Geoffrion, Dyer and Feinberg, 1972  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. Karasakal and Köksalan, 2009  ). In this paper, we present the results of this study .
When the mathematical programming models contain integer variables, the design problems become harder to solve. Multiobjective Combinatorial Optimization (MOCO), a computational complexity problem (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, it is also possible to represent the problem in the weight space. Below are some of the definitions.
Criterion space representation
Let us assume that we evaluate solutions in a specific situation using several criteria. Let us further assume that we are 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.
Solving MCDM problems
Different schools of thought have developed for solving MCDM problems (both of the design and evaluation type). For a bibliometric study showing their development over time, see Bragge, Korhonen, H. Wallenius and J. Wallenius . 
Multiple objective mathematical programming school
(1) Vector maximization : The purpose of vector maximization is to approximate the nondominated set; Originally developed for Multiple Objective Linear Programming Problems (Evans and Steuer, 1973,  Yu and Zeleny, 1975  ).
(2) Interactive programming : Phases of computation alternate with phases of decision-making (Benayoun et al, 1971;.  Geoffrion, Dyer and Feinberg, 1972;  Zionts and Wallenius, 1976;  Korhonen and Wallenius , 1988  ). No explicit knowledge of the DM’s value function is assumed.
Goal programming school
The purpose is to set apriori target values for goals, and to minimize weighted deviations from these goals. (Charnes and Cooper, 1961  ).
Fuzzy sets were introduced by Zadeh (1965)  as an extension of the classical notion of sets. This idea is used in many MCDM algorithms to model and solve fuzzy problems.
Multi-attribute utility theorists
Multi-attribute 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  ).
The French school in France, in particular the ELECTRE family of outranking methods that originated in France during the mid-1960s. The method was first proposed by Bernard Roy (Roy, 1968  ).
Evolutionary multiobjective optimization school (EMO)
EMO algorithms are the first step in the evolution of the genetic algorithm. The goal is to converge to a population of solutions which represents the nondominated set (Schaffer, 1984;  Srinivas and Deb, 1994  ). More recently, there are efforts to incorporate preference information into the solution process of EMO algorithms (see Deb and Köksalan, 2010  ).
Analytic hierarchy process (AHP)
The AHP first decomposes the decision problem into a hierarchy of subproblems. Then the decision-maker evaluates the relative importance of its various elements by pairwise comparisons. The AHP converts these evaluations to numerical values (weights or priorities), which are used to calculate a score for each alternative (Saaty, 1980  ). A consistency index measures the extent to which the decision-maker has been consistent in her responses.
The following MCDM methods are available, many of which are implemented by specialized decision-making software :  
- Aggregated Indices Randomization Method (AIRM)
- Analytic hierarchy process (AHP)
- Analytic network process (ANP)
- Best worst method (BWM)  
- Characteristic Objects METhod (COMET)  
- Choosing By Advantages (CBA)
- Data envelopment analysis
- Decision EXPERT (DEX)
- Disaggregation – Aggregation Approaches (UTA *, UTAII, UTADIS)
- Rough set (Rough set approach)
- Dominance-based rough set approach (DRSA)
- ELECTRE (Outranking)
- Evaluation Based on Distance from Average Solution (EDAS) 
- Evidence reasoning approach (ER)
- Goal programming (GP)
- Gray relational analysis (GRA)
- Inner product of vectors (IPV)
- Measuring Attractiveness by a Categorical Based Technical Evaluation (MACBETH)
- Multi-Attribute Global Inference of Quality (MAGIQ)
- Multi-attribute utility theory (MAUT)
- Multi-attribute value theory (MAVT)
- New Approach to Appraisal (NATA)
- Nonstructural Fuzzy Decision Support System (NSFDSS)
- Potentially All Pairwise RanKings of all possible Alternatives (PAPRIKA)
- PROMETHEE (Outranking)
- Stochastic Multicriteria Acceptability Analysis (SMAA)
- Superiority and inferiority ranking method (SIR method)
- Technique for the Order of Prioritization by Similarity to Ideal Solution (TOPSIS)
- Value analysis (VA)
- Value Engineering (VE)
- VIKOR method 
- Fuzzy VIKOR method 
- Weighted product model (WPM)
- Weighted sum model (WSM)
- Rembrandt method
- Architecture tradeoff analysis method
- Decision-making software
- Decision-making paradox
- Decisional balance sheet
- Multicriteria classification problems
- Rank reversals in decision-making
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