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## Alternatives

## Criteria / Attributes

## Decision Maker (DM) Preferences

## Decision Matrix

Miller G.,(1956). The magical number seven, plus or minus two: some limits on our capacity for processing information, Psychological Review 63 (2): 81-97 Zeleny M.,(2011). Multiple Criteria Decision Making (MCDM): From Paradigm Lost to Paradigm Regained? Journal of Multi-Criteria Decision Analysis 18: 77-89

Although the Multi-Criteria Decision Making (MCDM) models may be widely diverse, a typical MCDM model usually includes the following components:

Suppose that we have several options for solving the problem, ways to achieve goals, actions or objects, among which should be made a choice. These variants are usually called alternatives. We will deal with the class of decision problems with a finite number of choices (with a discrete set of alternatives) that are known and can be explicitly listed at the beginning of decision-making process. This class of decision problems commonly referred as Multi-Attribute Decision Making (MADM) problems.

During the process of decision-making, alternatives are evaluated by a set of criteria (usually conflicting), which measure how satisfactorily each alternative achieves the goals. Attributes represent the different dimensions from which the alternatives can be viewed. This may be characteristics, factors, qualities, performance and parameters of alternatives. In cases when the number of criteria / attributes is large, they can be arranged in a hierarchical manner.

The presence of conflicting criteria and a limited number of possible alternatives usually causes that the best values of criteria can't be reached simultaneously. This in turn leads to the need of choosing an alternative which would provide some reasonable trade-off for the conflicting criteria. Making sensible trade-offs is one of the most substantial and difficult challenges in MCDM. According to M.Zeleny (Zeleny, 2011): "No trade-offs - no Decision Making - and vice versa. The existence of trade-offs is a necessary and sufficient condition for Decision Making". Usually, the considered criteria aren't equivalent, i.e. one of them are more important for the decision-maker, than others. And it is a basis for the achievement of a compromise. Therefore, the DM's preference structure has to be specified and incorporated into a decision model.

The performance values for available alternatives against the considered criteria usually represent in a matrix format. The performance values are derived from the judgments acquired from a DM or a consulted expert. The judgments can be also derived from other data provided by non-experts, for example, opinions taken from potential customers or stakeholders.

Miller G.,(1956). The magical number seven, plus or minus two: some limits on our capacity for processing information, Psychological Review 63 (2): 81-97 Zeleny M.,(2011). Multiple Criteria Decision Making (MCDM): From Paradigm Lost to Paradigm Regained? Journal of Multi-Criteria Decision Analysis 18: 77-89

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## Multi-Criteria Decision Making Methods

**WSM** - Weighted Sum Model ( Simple Additive Weighting (SAW), Scoring Method (SM))

Fishburn, P.C.,(1967). Additive Utilities with Incomplete Product Set: Applications to Priorities and Assignments, Operations Research Society of America (ORSA) Publication, Baltimore, MD.

**WPM** - Weighted Product Model

Miller, D.W.; M.K., Starr (1969). Executive Decisions and Operations Research. Englewood Cliffs, NJ, U.S.A.: Prentice-Hall, Inc

**AHP** - Analytic Hierarchy Process

Saaty, T.L.,(1980). The Analytic Hierarchy Process, McGraw-Hill International, New York, NY.

**ANP** - Analytic Network Process

Saaty, Thomas L., (1996). Decision Making with Dependence and Feedback: The Analytic Network Process. Pittsburgh, Pennsylvania: RWS Publications. ISBN 0-9620317-9-8.

**ELECTRE** - Elimination and Et Choice Translating Reality

Roy, Bernard (1991). The Outranking Approach and The Foundations of Electre Methods. Theory and Decision 31 (1):49-73 (1991)

**TOPSIS** - Technique for Order Preference by Similarly to Ideal Solution

Hwang, C.L.; Yoon, K. (1981). Multiple Attribute Decision Making: Methods and Applications. New York: Springer-Verlag.

**VIKOR ** - Multi-criteria optimization and compromise solution

Opricovic, S.;Tzeng, G. H.,(2004). Compromise Solution by MCDM Methods: A Comparative Analysis of VIKOR and TOPSIS. European Journal of Operational Research, 156, pp. 445-455.

**PROMETHEE** - Preference Ranking Organization Method for Enrichment Evaluation

Brans,J.P.; Vincke,P.,(1985). "A preference ranking organisation method: The PROMETHEE method for MCDM". Management Science.

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## Technique we use

In our applications, we use MCDM methods which enable to find non-dominated (efficient, Pareto-optimal) solutions that correspond to decision maker's preferences. Particular attention is paid to ensuring that in the course of the decision-making the effective decisions corresponding to some values of preferences of the decision-maker weren't unreasonably excluded from the consideration that, unfortunately, takes place in some known MCDM methods.

1. A priori determination of the weighting coefficients of the importance of decision criteria.

2. Determination the performance values of the alternatives in terms of each decision criterion.

3. Determination of the ideal (utopian) and the anti-utopian points corresponding to the best and worst values by each criterion separately for all feasible alternatives.

4. Normalizing the criteria and the Decision Matrix by means of monotonic transformation.

5. Calculation for each alternative of distance to the utopian point according to the weighted metrics of L1 (Manhattan metric), L2 (Euclidean metric) and L∞ (Tchebycheff metric)

6. Rank the alternatives, sorting by the weighted distance from the utopian point.

7. Initially, take as a compromise decision the best-ranked alternative ( which is calculated with the help of metric L∞ )

8. A more detailed additional analysis of the compromise decision and the several next rating (most promising) alternatives (including sensitivity analysis to the variations of the weights of the importance and performance values of the alternatives variations).

9. The decision maker decides then to accept compromise decision or not. If yes, the process ends with the implementation of the current compromise decision. Otherwise, the model (weights of importance, alternatives and attributes values) has to be modified and selected another compromise decision.

Yu, P., (1973). A Class of Solutions for Group Decision Problems, Management Science, 19(8), 936-946.

Zeleny, M., (1973). Compromise Programming, in, Multiple Criteria Decision Making, University of South Carolina Press, Columbia.

Volkovich,V., (1989). Problems of Multiobjective Mathematical Programming and the Algorithms of their Solution, Working Paper WP-89-057, International Institute for Applied Systems Analysis, Austria.

Wierzbicki, A.P., (1998). Reference Point Methods in Vector Optimization and Decision Support. IIASA Interim Report IR-98-017.

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## Coming Soon!

In his life, a person constantly faced with situations where he has to make a choice between several variants, evaluating them from different perspectives. Sometimes a person makes a choice, without much thought, but sometimes it can be quite difficult, primarily because of the complexity of the examination and comparison of several characteristics of the variants simultaneously (Miller, 1956). Often in such cases, people are trying to develop simplified choice strategies to alleviate the decision-making problem. Unfortunately, such strategies can lead to significant errors. In such cases, using Multi-Criteria Decision Making (MCDM) techniques is highly recommendable.

... read moreThere is a wide range of the methods available to the solution of MCDM problems in various applications. Only some of them:

**WSM** - Weighted Sum Model ( Simple Additive Weighting (SAW), Scoring Method (SM))

Fishburn, P.C.,(1967).Additive Utilities with Incomplete Product Set: Applications to Priorities and Assignments, Operations Research Society of America (ORSA) Publication, Baltimore, MD.

**WPM** - Weighted Product Model

Miller, D.W.; M.K. Starr, (1969). Executive Decisions and Operations Research. Englewood Cliffs, NJ, U.S.A.: Prentice-Hall, Inc

**AHP** - Analytic Hierarchy Process

Saaty, T.L.,(1980).The Analytic Hierarchy Process, McGraw-Hill International, New York, NY.

**ANP** - Analytic Network Process

Saaty, Thomas L., (1996). Decision Making with Dependence and Feedback: The Analytic Network Process. Pittsburgh, Pennsylvania: RWS Publications. ISBN 0-9620317-9-8.

**ELECTRE** - Elimination and Et Choice Translating Reality

Roy, Bernard, (1991). The Outranking Approach and The Foundations of Electre Methods. Theory and Decision 31 (1):49-73 (1991)

**TOPSIS** - Technique for Order Preference by Similarly to Ideal Solution

Hwang, C.L.; Yoon, K. (1981). Multiple Attribute Decision Making: Methods and Applications. New York: Springer-Verlag.

**VIKOR ** - Multi-criteria optimization and compromise solution

Opricovic, S., and Tzeng, G. H.,(2004).Compromise Solution by MCDM Methods: A Comparative Analysis of VIKOR and TOPSIS. European Journal of Operational Research, 156, pp. 445-455.

**PROMETHEE** - Preference Ranking Organization Method for Enrichment Evaluation

Brans,J.P.; and Vincke,P.,(1985). A preference ranking organisation method: The PROMETHEE method for MCDM. Management Science.

The choice of the most suitable MCDM method for the concrete app significantly depends on the solved task. Since there is no method that would have exceeded all others in all indicators, the most rational, in our opinion, is integration in one technique several well-proven MCDM methods. In our technique for weighting decision criteria and obtaining information on the performance values of the alternatives in terms of each decision criterion, we generally use Direct Criteria Weighting methods or AHP method (Saati, 1986) and its later modifications. Non-dominated (Pareto-optimal) decisions and the rating of feasible alternatives are defined on the basis of MCDM distance-based methods and their analogues (Yu,1973), (Zeleny,1973), (Volkovich, 1989), (Wierzbicki,1998).

... read more