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Disciplina: MCDA - Multi-Critério de Apoio à Decisão 

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Curso Completo (Apostila)

Artigos Básicos para o Estudo da MCDA


Departamento de Engenharia de produção e Sistemas

Laboratório de MCDA - LabMCDA

Coordenor: Prof. Dr. Leonardo Esslin - 1998


Multi-Critério de Apoio à Decisão (MCDA - Multiple-Criteria Decision Analysis)

From Wikipedia

MCDM" redirects here. For the use in cosmology, see meta-cold dark matter.


Multiple-criteria decision-making or multiple-criteria decision analysis (MCDA) is a sub-discipline of operations research (or Wikwpédia: operations research) that explicitly considers multiple criteria in decision-making environments. Whether in our daily lives or in professional settings, there are typically multiple conflicting criteria that need to be evaluated in making decisions. Cost or price is usually one of the main criteria. Some measure of quality is typically another criterion that is 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 to have the cheapest car to be the most comfortable and the safest. In portfolio management, we are interested in getting high returns but at the same time reducing our risks. Again, the stocks 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.

Plot of two criteria when maximizing return and minimizing risk in financial portfolios (Pareto-optimal points in red).

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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 multiple parties who are deeply affected from 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,[1][2] have been developed.


  • 1 Foundations, concepts, definitions

    • 1.1 A typology

    • 1.2 Representations and definitions

    • 1.3 Generating nondominated solutions

    • 1.4 Solving MCDM problems

    • 1.5 MCDM methods

  • 2 See also

  • 3 References

  • 4 Further reading

Foundations, Concepts, Definitions

MCDM or MCDA are well-known acronyms for multiple-criteria decision-making and multiple-criteria decision analysis. Stanley Zionts wrote an article 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. 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" 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 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 in any of them. 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 Group on MCDA,[3] and INFORMS Section on MCDM.[4] For a history see: Köksalan, Wallenius and Zionts (2011).[5] MCDM draws upon knowledge in many fields including:

  • Mathematics

  • Behavioral decision theory

  • 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.[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).

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

"max" q

subject to

q ∈ Q

MCDAM Methods

The following MCDM methods are available, many of which are implemented by specialized decision-making software:[1][2]

Further reading

  • Maliene, V. (2011). "Specialised property valuation: Multiple criteria decision analysis". Journal of Retail & Leisure Property 9 (5): 443–50. doi:10.1057/rlp.2011.7.

  • Mulliner E, Smallbone K, Maliene V (2013). "An assessment of sustainable housing affordability using a multiple criteria decision making method". Omega 41 (2): 270–79.doi:10.1016/

  • Maliene, V. et al (2002). "Application of a new multiple criteria analysis method in the valuation of propert y" Arquivo".PDF"   . FIG XXII International Congress: pp. 19–26.

  • A Brief History prepared by Steuer and Zionts

  • Malakooti, B. (2013). Operations and Production Systems with Multiple Objectives. John Wiley & Sons.




OBSERVAÇÃO: São no total 10 Artigos disponibilizados sob o Formato ".PDF     ".

Curso Completo (Apostila)

Artigos Básicos para o Estudo da MCDA


Departamento de Engenharia de Produção e Sistemas

Laboratório de MCDA - LabMCDA

Coordenor: Prof. Dr. Leonardo Esslin - 1998


Exemplo Prático:

Exemplo Prático - Avaliar o Sucesso Profissional de um Professor numa Universidade Particular e para Identificar Ações Alternativas. Formato ".PDF"     .


Neste Trabalho de nossa autoria, são apresentados todos os passos dentro desta Disciplina MCDA. Pode ser visto, por exemplo, como Construir o Mapa Cognitivo para o Problema em questão, até se chegar ao uso da Escala MacBeth. E os Resultados discutidos!



Vídeo Aula:

Estamos preparando uma Vídeo-Aula baseada neste Trabalho Prático (Realizado quando do Doutorado que fizemos em 1998. Aguardem!

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