A Fuzzy Trade-Off Ranking Method for Multi-Criteria Decision-Making
The aim of this paper is to present a trade-off ranking method in a fuzzy multi-criteria decision-making environment. The triangular fuzzy numbers are used to represent the imprecise numerical quantities in the criteria values of each alternative and the weight of each criterion. A fuzzy trade-off ranking method is developed to rank alternatives in the fuzzy multi-criteria decision-making problem with conflicting criteria. The trade-off ranking method tackles this type of multi-criteria
... ti-criteria problems by giving the least compromise solution as the best option. The proposed method for the fuzzy decision-making problems is compared against two other fuzzy decision-making approaches, fuzzy Technique for Order Preference by Similarity to the Ideal Solution (TOPSIS) and fuzzy VlseKriterijuska Optimizacija I Komoromisno Resenje (VIKOR), used for ranking alternatives. Axioms 2018, 7, 1 2 of 21 the decision criteria in the AHP are supposed to be independent from each another. This technique has been used in a number of publications (e.g.,     ). The Multi-Attribute Utility Theory (MAUT) by Keeney and Raiffa  represents a classical approach in MCDM analysis. This is a structured methodology based on the utility axioms introduced by von Neumann and Morgenstern. In the algorithm, a utility value is assigned to each action whilst quantifying all individual preferences. Some examples of the use of MAUT in the decision-making can be found in  . In turn, the Elimination and Choice Expressing Reality (ELECTRE) by Roy is an outranking approach that is used to discard unacceptable alternatives. This approach was modified in PROMETHEE (Preference Ranking Organization Method for Enrichment Evaluations) by Brans and Vincke  . PROMETHEE exists in three versions: the PROMETHEE I (partial ranking), the PROMETHEE II (complete ranking) and the PROMETHEE-GAIA (geometrical analysis for interactive aid). Several authors applied the outranking techniques in MCDM problems (e.g.,    ). The genetic algorithm (GA) has also been used for MCDM problems (e.g.,      ). It is widely used thanks to its universal nature. A problem with GA is that it generates a large number of solutions that are mostly redundant. Wang and Yang  used the particle swarm optimization (PSO) to determine a ranking for MCDM problems. Particle swarm is capable of improving the search ability of GA thanks to its better convergence to the Pareto frontier. However, as noted in , PSO requires significant computational time. There is a number of techniques related to the ranking of available alternatives that are presented by the Pareto solutions. In all these techniques, the ranking is based on a metric introduced in the criteria space. The Technique for Order Preference by Similarity to the Ideal Solution (TOPSIS) was first proposed by Hwang and Yoon  . The original TOPSIS method presumed priori weights for criteria to be specified by the Decision-Maker. The TOPSIS approach is based on an individual evaluation score that depends on the distances from the alternative to the ideal and anti-ideal solutions. This type of evaluation is obviously the best for the non-conflicting multi-criteria problems, where the alternative that is the closest to the ideal solution is also the farthest from the anti-ideal solution. However, in a conflicting multi-criteria problem, such an assumption cannot always be realized. This drawback in TOPSIS is addressed in a few papers    . It was Kao  who practically suggested to measure the distance in L 1 norm instead of L 2 -norm implemented in the conventional TOPSIS method to overcome the inconsistency problem. TOPSIS has been widely used in MCDM due to its simplicity (e.g.,     ). The VlseKriterijuska Optimizacija I Komoromisno Resenje (VIKOR) algorithm proposed by Opricovic  is based on the compromise programming with weights to be prescribed to the performances by the Decision-Maker. Such weights are subjective and depend on how adequately such quantitative characteristics reflect the individual preferences of the Decision-Maker. The real-life design is usually related to the inevitable uncertainties in the input data, parameters, etc. The uncertainty in the MCDM problem includes the imprecision of criteria values, vagueness in the importance of criteria (weights) and dealing with qualitative, linguistic or incomplete information. The concept of fuzziness, first introduced by Zadeh , is proved to be an efficient tool to include the uncertainties in the MCDM problems. Numerous fuzzy MCDM methods have been developed, including         . They utilize the fuzzy numbers in the formulation of their fuzzy MCDM methods. There are two ways used in solving the fuzzy MCDM problems  . One way to solve the fuzzy MCDM problem is based on utilizing the fuzzy MCDM method [44, 45] . Another way is reduced to the defuzzification of the fuzzy MCDM problem and solving it by a conventional MCDM method  . The defuzzification process converts the fuzzy numbers into crisp values. In both ways, the defuzzification process is essential, since the MCDM solution must provide a crisp result.