An innovative tool for selecting MCDA method tailored to the decision problem.

Wątróbski, J., Jankowski, J., Ziemba, P., Karczmarczyk, A., & Zioło, M. (2018). Generalised framework for multi-criteria method selection. Omega.
https://doi.org/10.1016/j.omega.2018.07.004.

Wątróbski, Jarosław, et al. "Generalised framework for multi-criteria method selection: Rule set database and exemplary decision support system implementation blueprints." Data in brief 22 (2019): 639.

Open access, https://www.sciencedirect.com/science/article/pii/S0305048317308563

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The method allows the hierarchizing of a decision problem. Criterial evaluations of variants are set with reference to other variants in pairwise comparison matrices, with the use of nine-degree scale describing an advantage of one variant over another. Weights of criteria are set in a similar way. Next, the values of pairwise comparison matrices are aggregated to criterial preference vectors, then these are additively aggregated to a single synthesized criterion, on the basis of which a ranking of variants is determined.

The ANP method is a generalization of the AHP. Instead of hierarchizing a decision problem, the method allows constructing a net model, in which connections between criteria and variants, variant-criterion feedback or horizontal relations between individual criterion can occur. In this method, preference aggregation is based on a Markov chain and is carried out in a so-called Supermatrix.

TOPSIS (Technique for Order Performance by Similarity to Ideal Solution) is a method based on the concept of representing all the variants, along with the positive ideal solution (PIS) and the negative ideal solution (NIS) as points on an Euclidean space. The ranking of the variants is obtained based on the relative distances of the solutions from the reference PIS and NIS points. The best variant should have the smallest distance from the PIS and biggest distance from NIS.

The ANP method is a generalization of the AHP. Instead of hierarchizing a decision problem, the method allows constructing a net model, in which connections between criteria and variants, variant-criterion feedback or horizontal relations between individual criterion can occur. In this method, preference aggregation is based on a Markov chain and is carried out in a so-called Supermatrix.

Qualitative measurements are used in order to represent preferences on the ordinal scale. To compare variants with regard to criteria, five relations preferences are used, from indifference to very strong preference. Similarly, weights of criteria are expressed on a five-degree quality scale. Next, a preference graph is constructed using outranking relation, as it is the case inElectre I, for example.

Presentations are required for each criterion, triangle fuzzy numbers which determine degrees of variant affiliations to individual linguistic values describing individual criteria. Next, on the basis of values of vertexes of individual fuzzy numbers characteristic variants are generated. These variants are compared pairwise by a decision-maker and their model ranking is generated.

It deals with the selection problematics. Preferences in Electre I methods are modelled by using binary outranking relations, and the method can be used when the criteria have been coded in numerical scales. The foundations of the method’s algorithm are concordance and discordance indexes. Next, the best variants are those which are not outranked by any others.

The method deals with the ranking problematics. The calculation algorithm is almost the same as in Electre I. However, in Electre II, one can distinguish between a strong and a weak preference relation. Computation procedures consist of four steps: partitioning set of variants, building a complete pre-order, determining a complete pre-order and defining the partial pre-order.

It is based on pseudo-criteria (indifference, preference and veto thresholds are determined), instead of true criteria. After determining a decision-maker’s preferences, concordance and discordance indexes are carried out, and the final ranking of variants is determined on the basis of results of distillation procedures.

Methodologically, it resembles Electre I and also deals with the selection problematics. The differences between the methods lie in the fact that in Electre IS pseudo-criteria (indifference, preference and veto thresholds) are used.

Similarly to Electre III, with regard to the use of pseudo-criteria. However, in Electre IV, one does not define weights of criteria, therefore, all criteria are equal.

It deals with sorting problematics, but it uses pseudo-criteria. The method is very similar to Electre III in terms of procedures. In Electre Tri, decision variants are compared with variant profiles. The profiles are “artificial variants” limiting individual quality classes. The profiles are defined by a decision-maker, while determining the values of thresholds and weights of criteria.

It allows both quantitative and qualitative criteria with the use of two separate domination measurements to be taken into consideration. Subsequent measurements are aggregated into one value describing the performance of a variant. Therefore, it is essential to apply a function which brings quantitative and qualitative measurements to the same level, and allows presenting them on a common scale. Weights of criteria are expressed in a quantitative way in this method.

A fuzzy version of the AHP method, in which criterial evaluations of variants are determined, with regard to other variants in pairwise comparison matrices, with the use of a fuzzy scale.

A fuzzy version of the AHP method, in which criterial evaluations of variants are determined, with regard to other variants in pairwise comparison matrices, with the use of a fuzzy scale.

The fuzzy TOPSIS method, similarly to its crisp version, is based on the concept of representing the positive ideal solution (PIS), negative ideal solution (NIS) and all variants on an Euclidean space. However, in this variant of the method, the values of decision attributes are represented by triangular or trapezoidal fuzzy numbers. The distance between the alternatives and PIS and NIS are calculated as a sum of distances between two fuzzy numbers representing each criterion separately.

A fuzzy version of the ANP method, in which criterial evaluations of variants are determined with regards to other variants in pairwise comparison matrices, with the use of a fuzzy scale.

A fuzzy version of the ANP method, in which criterial evaluations of variants are determined with regards to other variants in pairwise comparison matrices, with the use of a fuzzy scale.

The fuzzy TOPSIS method, similarly to its crisp version, is based on the concept of representing the positive ideal solution (PIS), negative ideal solution (NIS) and all variants on an Euclidean space. However, in this variant of the method, the values of decision attributes are represented by triangular or trapezoidal fuzzy numbers. The distance between the alternatives and PIS and NIS are calculated as a sum of distances between two fuzzy numbers representing each criterion separately.

In the method, variants fulfilling individual criteria are chosen. However, all criteria have the same weight. In the method, for extracting the maximum value of the attribute one chooses variants maximally fulfilling one of the criteria.

A fuzzy version of the Promethee I method. Weights and criterial variant evaluations are presented by means of fuzzy numbers.

A fuzzy version of the Promethee II method. Weights and criterial variant evaluations are presented by means of fuzzy numbers.

A fuzzy version of the SAW method, in which weights and criterial variant evaluations are presented by means of triangle or trapezoid fuzzy numbers.

The fuzzy TOPSIS method, similarly to its crisp version, is based on the concept of representing the positive ideal solution (PIS), negative ideal solution (NIS) and all variants on an Euclidean space. However, in this variant of the method, the values of decision attributes are represented by triangular or trapezoidal fuzzy numbers. The distance between the alternatives and PIS and NIS are calculated as a sum of distances between two fuzzy numbers representing each criterion separately.

A fuzzy version of the VIKOR method, in which criterial variant evaluations are presented by means of triangle fuzzy numbers.

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IDRA (Intercriteria Decision Rule Approach) uses a mixed utility function which considers weights and compromises between criteria. On this basis, preference indexes depicting binary relations are determined. Aggregation of such indexes is based on the assumption that each piece of information about mutual substitutions between criteria constitutes a decision rule. A decision-maker assigns a different reliability to such a rule. Next, these reliabilities are used when calculating an aggregated preference index.

It assumes both quantitative and qualitative assessments of variants, which are relative to subsequent criteria and ordinal criterion weights. In the next steps, variants are considered with relations to criteria ranking from the most to the least important one. During each step, a set of variants is reduced in such a way that there only variants are left, and these are, to a considerable degree, considered in a given step of a criterion.

Individual variants are here compared in a pairwise comparison matrix with the use of a seven-degree qualitative scale. Next, the comparison results are transformed into an interval scale. Weights of criteria, which are transferred from the qualitative scale to a quantitative one, and normalized to 100 (percentage scale), are determined in a similar way. Criterial preferences of variants are additively aggregated as a weighted average.

It uses qualitative variant evaluations and quantitative, normalized to 1, weights of criteria. Variants’ evaluations, which are relative to each criterion, are also normalized in the way where the best variant receives 1, the worst – 0, and other variants are given in-between values. For each pair of criteria, another matrix containing preference indexes is determined. All such matrices are aggregated into a global matrix, from which next preference rankings are constructed.

In MAUT, the most important is an assumption that a decision-maker’s preferences can be expressed by means of an analytical global utility function, while taking into consideration all considered criteria. Knowledge of this function makes it possible to obtain a set, ranked in terms of ‘optimality,’ of decision variants.

MAVT is very similar to MAUT, but it uses the value function. In other words, in MAUT a ranking of variants is determined with the use of the utility function in the additive form, whereas in MAVT a ranking is obtained with the use of the multiplicative value function.

It is based on the assumption that the productivity of a variant is as good as its strongest attribute. In this method, therefore, a variant characterized by the best value of a criterion (max) is chosen, with regard to which this variant is the strongest (max).

It is based on the assumption that the productivity of a variant is as good as its weakest attribute. In this method, therefore, a variant characterized by the best value of a criterion (max) is chosen, with regard to which this variant is the weakest (min).

A fuzzy version of the Maximin method, in which one operates on linguistic values.

In MELCHIOR pseudo-criteria (indifference and preference thresholds) are used. This method is similar to Electre IV in terms of calculation; however, in MELCHIOR an ordinal relation is set between criteria. Preference aggregation takes place by testing conditions of concordance and lack of discordance.

In the method, variants satisfactorily fulfilling individual criteria on the basis of fuzzy (linguistic) values are chosen. All criteria have the same weights. In order to extract the maximum value of the attribute, one may choose variants maximally fulfilling one of the criteria, on the basis of fuzzy (linguistic) values.

As far as computation is concerned, NAIADE I is similar to Promethee, since variant ranking is determined on the basis of input and output preference flows. Nevertheless, when comparing variants there are six preference relations, which are based on trapezoid fuzzy numbers (apart from indifference of variants, one can distinguish weak and strong preference). In NAIADE I method, weights of criteria are not defined.

Akin to NAIADE I, preference relations are defined on the basis of trapezoid fuzzy numbers. As in Promethee II, it allows a complete ranking of variants on the basis of net preference flows to be determined.

It requires presenting variant assessment and a ranking of criteria on an ordinal scale. Next, with the use of the distance function a complete order of variants, with regard to subsequent criteria is determined, in which preference and indifference situations are acceptable. In the last step, rankings are aggregated into a global ranking, thus allowing an incomparability situation as well.

For every considered pair of criteria, the method allows distinguishing between a compensating (active) criterion and a compensated (passive) one. Separating active and passive compensation effect makes it possible to indicate compensation asymmetry. After analysing the compensation, a construction of binary indexes, based on the evaluation of the degree of active and passive preferences, is performed. On this basis preference, indifference or incomparability relations between variants are determined.

PAMSSEM I is a combination of Electre III, NAIADE I and Promethee I. Akin to NAIADE I, PAMSSEM I makes it possible to use fuzzy evaluations of variants. On the other hand, as in Electre III, there is a preference aggregation with the use of concordance and discordance indexes. Indifference, preference and veto thresholds are also used. Finally, as in Promethee I, a final ranking of variants is obtained, with the use of input and output preference flows.

A computational procedure is similar to PAMSSEM I; however, as in Promethee II, net preference flows are determined.

It is based on MAPPAC and to a great extent; its course is identical to the MAPPAC algorithm. The only difference appears on the stage of aggregation of preferences from matrices of preference indexes determined for individual pairs of criteria. On this stage, for each individual matrix, a frequency matrix of partial rankings is determined. On their basis, a frequency matrix of a global ranking is calculated. Then, on the basis of this frequency matrix, preference aggregation into global rankings takes place.

The Promethee method allows for determining a synthetic ranking of variants. Depending on implementation, the method can operate on real criteria or pseudo-criteria. Input and output preference flows are determined, on the basis of which one can create a partial Promethee I ranking can be created.

In Promethee II, on the basis of input and output preference flows, the values of net preference flows for individual variants are calculated. On this basis, a complete ranking of variants is obtained.

It allows use of qualitative evaluation variants, as well as quantitative and qualitative weights of criteria. Next permutations of rankings of variants are generated, and for every permutation; for every ranking, a concordance and discordance index is determined, firstly for individual ranking at the level of single criteria, then with relation to all possible rankings. As a result, an ordinal ranking with the best values of the concordance and discordance index is selected.

It is based on the analysis of variants’ concordance. A concordance matrix is determined, then three-valued pieces of information with positive/negative domination or variant indifference with regard to each criterion. Next, the probability of dominance for each compared pair of variants is determined, and on this basis, a variant order is obtained.

It requires giving quantitative evaluations of variants and weights. The evaluations of variants with regard to individual criteria should be proportionally normalized to the highest evaluation regarding each of the criteria. Preference aggregation comes down to determining a product of weights of a criterion, and the evaluation of a variant regarding this criterion. Next, all such products for a given variant are added up.

Criterial values of variants are calculated to a common internal scale. It is mathematically carried out by a decision-maker, and the value function used here (similarly to MAVT) using a range between the lowest and the highest values of variants for a given criterion. Variant evaluation here is not dependent on other ones. Moreover, the weights of criteria are determined by attributing explicit values to them, and not values relating to another criterion.

It is based on quantitative evaluations of variants and criteria weights. Furthermore, it allows applying real criteria and quasi-criteria and, in consequence, it uses the indifference and veto thresholds. As in Electre I and ARGUS, preference aggregation in this method is based on the analysis of concordance and discordance.

TOPSIS (Technique for Order Performance by Similarity to Ideal Solution) is a method based on the concept of representing all the variants, along with the positive ideal solution (PIS) and the negative ideal solution (NIS) as points on an Euclidean space. The ranking of the variants is obtained based on the relative distances of the solutions from the reference PIS and NIS points. The best variant should have the smallest distance from the PIS and biggest distance from NIS.

A decision-maker’s preferences are singled out from a reference set of variants. This set contains a list of sample decision options, which are ranked by the decision-maker in a ranking from the best to the worst one. In this ranking, a relation of preference or indifference can be used. Next, on the basis of the reference set fragmentary utility function for each criterion are created. Determining a global utility consists in adding values of all fragmentary utility functions for a given variant.

The VIKOR method consists in looking for a compromise, which is closest to an ideal solution. Firstly, the worst and the best values of individual criteria are determined. On the basis of the values, utility measures and regret measures for variants with regard to each criterion are calculated. Next, each variant’s minimal and maximum values, whose aggregation allows determining the position of a variant in the ranking.

The method allows the hierarchizing of a decision problem. Criterial evaluations of variants are set with reference to other variants in pairwise comparison matrices, with the use of nine-degree scale describing an advantage of one variant over another. Weights of criteria are set in a similar way. Next, the values of pairwise comparison matrices are aggregated to criterial preference vectors, then these are additively aggregated to a single synthesized criterion, on the basis of which a ranking of variants is determined.

A fuzzy version of the AHP method, in which criterial evaluations of variants are determined, with regard to other variants in pairwise comparison matrices, with the use of a fuzzy scale.

TOPSIS (Technique for Order Performance by Similarity to Ideal Solution) is a method based on the concept of representing all the variants, along with the positive ideal solution (PIS) and the negative ideal solution (NIS) as points on an Euclidean space. The ranking of the variants is obtained based on the relative distances of the solutions from the reference PIS and NIS points. The best variant should have the smallest distance from the PIS and biggest distance from NIS.

The method allows the hierarchizing of a decision problem. Criterial evaluations of variants are set with reference to other variants in pairwise comparison matrices, with the use of nine-degree scale describing an advantage of one variant over another. Weights of criteria are set in a similar way. Next, the values of pairwise comparison matrices are aggregated to criterial preference vectors, then these are additively aggregated to a single synthesized criterion, on the basis of which a ranking of variants is determined.

The VIKOR method consists in looking for a compromise, which is closest to an ideal solution. Firstly, the worst and the best values of individual criteria are determined. On the basis of the values, utility measures and regret measures for variants with regard to each criterion are calculated. Next, each variant’s minimal and maximum values, whose aggregation allows determining the position of a variant in the ranking.

The DEMATEL (Decision Making Trial and Evaluation Laboratory) method facilitates the decision making by providing a hierarchical structure of the criteria; however, contrary to the AHP method, it assumes that the elements of the structure are interdependent. A group of respondents is requested to evaluate the direct influence between any two factors on a 4-point scale, where 0 denotes no influence and 3 represents high influence. As a result of the method, a digraph showing casual relations among analyzed criteria is generated.

The REMBRANDT (Ratio Estimation in Magnitudes or deci-Bells to Rate Alternativews which are Non-Dominated) technique is a multiplicative version of the AHP method. Pairwise comparisons between the objects are performed by the decision-maker on a geometric scale (1/16, 1/4, 1, 4, 16) where 1 denotes indifference, 4 and 16 represent weak and strict preference for the base object over the second object. The results of the comparisons are then converted into an integer-valued gradation index. As a result, the irrelative importance of the objects is determined. Finally, a subjective rank ordering of the objects is performed by aggregation of the results.

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