A fuzzy multi-criteria decision-making model by associating technique for order preference by similarity to ideal solution with relative preference relation
Introduction
Decision-making is one of important issues for enterprises because the issue is to find an optimal alternative from a number of feasible alternatives. Further, decision-making with several evaluation criteria is named multi-criteria decision-making (MCDM) [1], [2], [3], [4], [5], [7], [8], [9], [11], [13], [14], [16], [17], [18], [19], [21], [22], [23], [25], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [44], [45], [46], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61], [62]. A MCDM model is commonly expressed in matrix format as follows.andwhere are feasible alternatives, are evaluation criteria, is the evaluation rating of on , and is the weight of .
MCDM problems are practically classified into two categories. One is classical MCDM problems such as [25], [29], [34]. The other is fuzzy multi-criteria decision-making (FMCDM) problems such as [3], [5], [8], [9]. In the classical MCDM problems, evaluation ratings and criteria weights on certain environment are expressed by crisp values. In the FMCDM problems, evaluation ratings and criteria weights are assessed on imprecision, subjectivity or vagueness, so the ratings and weights are often presented by linguistic terms [20], [26] and then transformed into fuzzy numbers [67], [69], [70]. For instance, the evaluation ratings may be described by very poor (VP), poor (P), medium poor (MP), fair (F), medium good (MG), good (G) and very good (VG), and the criteria weights are denoted by very low (VL), low (L), medium (M), high (H) and very high (VH). Since the linguistic terms above can be transformed into fuzzy numbers, the evaluation is viewed as a FMCDM problem.
Many past researchers applied classical MCDM methods under fuzzy environment [3], [4], [5], [7], [8], [9], [11], [13], [16], [17], [22], [23], [27], [28], [30], [31], [33], [35], [36], [37], [38], [39], [40], [41], [42], [44], [45], [46], [48], [49], [50], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61], [62] to solve FMCDM problems. The approaches are commonly classified into two categories which are defuzzification and fuzzy extension. Defuzzification often lost lots of messages, so Chen [9], Liang [42], Raj and Kumar [49], Wang et al. [62] supposed that classical MCDM methods, such as technique for order preference by similarity to ideal solution (TOPSIS) [29], should be extended under fuzzy environment, i.e. fuzzy extension of TOPSIS [7], [8], [9], [11], [22], [30], [35], [36], [37], [38], [45], [46], [50], [53], [55], [56], [57], [58], [60], [62]. TOPSIS proposed by Hwang and Yoon is one of well-known classical MCDM methods and often generalized under fuzzy environment into fuzzy extension of TOPSIS. The underlying logic of TOPSIS is to define ideal solution and anti-ideal solution. The ideal solution is a solution that maximizes benefit criteria and minimizes cost criteria, whereas the anti-ideal solution is a solution that maximizes cost criteria and minimizes benefit criteria. In short, the ideal solution is composed of all best values on criteria, and the anti-ideal solution consists of all worst values on criteria. Thus the optimal alternative has the shortest distance to the ideal solution and the farthest distance to the anti-ideal solution for all feasible alternatives.
In numerous extensions of TOPSIS, approaches of Chen [9], Liang [42], Raj and Kumar [49], Wang et al. [62] are useful for FMCDM. However, there were some drawbacks in their works. For instance, Liang [42], Raj and Kumar [49] utilized maximizing and minimizing sets [10] to rank evaluated values which were trapezoidal fuzzy numbers multiplied and then added. Since two trapezoidal fuzzy numbers multiplied would not be a trapezoidal fuzzy number, the raking for the type of fuzzy numbers was complex and difficult. Moreover, distance values from two varied alternatives to ideal solution (or anti-ideal solution) might be indiscernible, when intersections of the two alternatives and the ideal (or anti-ideal) solution on all criteria were null. Chen [9] used (0, 0, 0) and (1, 1, 1) to be respectively the worst and best values of alternatives on criteria. The two values might far from away minimum and maximum values in FMCDM problems, so (0, 0, 0) and (1, 1, 1) did not stand for the worst and best values of alternatives on criteria. Besides, multiplying ratings by weights (i.e. weighted ratings) were presented by triangular fuzzy numbers in Chen’s method as ratings and weights were triangular fuzzy numbers. However, a weighted rating being product of two triangular fuzzy numbers was not a triangular fuzzy number. Practically, these problems can be solved by defuzzification or fuzzy preference relation. Through previous description, messages are often lost on defuzzification. Additionally, fuzzy operation by preference relation [4], [6], [33], [34], [36], [41], [43], [44], [47], [54], [63], [64], [65], [66], [68] is a hard work because fuzzy numbers are pair-wise compared under fuzzy environment. Thus we propose a FMCDM model based on TOPSIS and relative preference relation to resolve these ties. The relative preference relation is an improvement of fuzzy preference relation, and does not pair-wise compare fuzzy numbers under fuzzy environment.
For the sake of clarity, mathematical preliminaries are expressed in Section 2. The relative preference relation on fuzzy numbers is presented in Section 3. In Section 4, a FMCDM model based on TOPSIS and relation preference relation is proposed. A numerical example constructed on the FMCDM model is illustrated in Section 5. Finally, feasibility and rationality of the proposed model is demonstrated in Section 6.
Section snippets
Mathematical preliminaries
In this section, we view notions of fuzzy sets and fuzzy numbers [67], [69], [70] expressed as follows. Definition 2.1 Let U be a universe set. A fuzzy set A of U is defined by a membership function , where , , indicates the degree of x in A. Definition 2.2 of the fuzzy set A is a crisp set . Definition 2.3 Support of the fuzzy set A is a crisp set . Definition 2.4 The fuzzy set A of U is normal iff . Definition 2.5 The fuzzy set A of U is convex iff , , where
A relative preference relation on fuzzy numbers
Herein, concerning definitions of the relative preference relation on fuzzy numbers [67], [69], [70] are defined as follows. Definition 3.1 Let and denote two triangular fuzzy numbers. The addition of A and B by extension principle is Definition 3.2 The multiplication of a triangular fuzzy number A and a crisp value t is Definition 3.3 Let A and B be two triangular fuzzy numbers, where and . A fuzzy preference
A FMCDM model constructed on TOPSIS and relative preference relation
In this section, a FMCDM model constructed on TOPSIS and relative preference relation is derived to solve FMCDM problems. To demonstrate the FMCDM model, we present computation steps of TOPSIS below.
Step 1: Identify a decision matrix for a giving MCDM problem.
The decision matrix is as similar as that described in section 1.
Step 2: Normalize the decision matrix.
Let be normalized value of in the decision matrix, where ; .
Step 3: Derive weighted decision
A numerical example of site selection for building factory
Assume that an enterprise desires to select a site for building a new factory. To evaluate the location problem, four experts , and are employed in consult council for site selection. They evaluate three feasible sites and based on six criteria including climate condition , regional demand , expansion possibility , transportation availability , labor force and investment cost . In addition to investment cost (unit: million dollars) is a cost
Feasibility and rationality demonstration for the proposed model
To demonstrate the result rationality in Section 5, we use following computation for verifying to be the optimal site. Herein, multiplication [42], [49], [67], [69], [70] of two triangular fuzzy numbers by extension principle is first utilized in the site selection problem as follows. Definition 6.1 Let and be two triangular fuzzy numbers. The multiplication of and is defined aswhere . The membership
Conclusions
In this paper, we propose a FMCDM model based on TOPSIS and relative preference relation to solve FMCDM problems. The relative preference relation is revised form fuzzy preference relation. It has strength of fuzzy preference relation, but no weakness of fuzzy preference relation. Generally, operation complexity of fuzzy preference relation is high due to fuzzy comparison on pair-wise. Through previous descriptions, it is obvious that pair-wise comparison derived by the relative preference
Acknowledgements
This research work was partially supported by the National Science Council of the Republic of China under Grant No. NSC 99-2410-H-346-003-.
References (70)
- et al.
A state-of the-art survey of TOPSIS applications
Expert Syst. Appl.
(2012) - et al.
Multi-attribute decision analysis with fuzzy pairwise comparisons
Fuzzy Sets Syst.
(1989) - et al.
A new incomplete preference relations based approach to quality function deployment
Inform. Sci.
(2012) - et al.
A novel hybrid MCDM approach based on fuzzy DEMATEL, fuzzy ANP and fuzzy TOPSIS to evaluate green suppliers
Expert Syst. Appl.
(2012) - et al.
A survey analysis of service quality for domestic airlines
Euro. J. Operat. Res.
(2002) Extensions to the TOPSIS for group decision-making under fuzzy environment
Fuzzy Sets Syst.
(2000)Ranking fuzzy numbers with maximizing set and minimizing set
Fuzzy Sets Syst.
(1985)Comparative analysis of SAW and TOPSIS based on interval-valued fuzzy sets: discussions on score functions and weight constraints
Expert Syst. Appl.
(2012)- et al.
An OWA-TOPSIS method for multiple criteria decision analysis
Expert Syst. Appl.
(2011) The canonical representation of multiplication operation on triangular fuzzy numbers
Comput. Math. Appl.
(2003)
A fuzzy MCDM method for solving marine transshipment container port selection problems
Appl. Math. Comput.
Application of FMCDM model to selecting the hub location in the marine transportation: a case study in southeastern Asia
Math. Comput. Modell.
The application of TOPSIS method in selecting fixed seismic shelter for evacuation in cities
Syst. Eng. Proc.
A direct interval extension of TOPSIS method
Expert Syst. Appl.
An approach to generalization of fuzzy TOPSIS method
Inform. Sci.
Decision making in a fuzzy environment and its application to multicriteria power engineering problems
Nonlin. Anal.: Hybrid Syst.
Performance evaluation for airlines including the consideration of financial ratios
J. Air Trans. Manage.
A model of consensus in group decision making under linguistic assessments
Fuzzy Sets Syst.
Fuzzy Rasch model in TOPSIS: a new approach for generating fuzzy numbers to assess the competitiveness of the tourism industries in Asian countries
Tour. Manage.
Emergency alternative evaluation under group decision makers: a method of incorporating DS/AHP with extended TOPSIS
Expert Syst. Appl.
Group decision making and consensus under fuzzy preferences and fuzzy majority
Fuzzy Sets Syst.
Support managers’ selection using an extension of fuzzy TOPSIS
Expert Syst. Appl.
Solving multi-period project selection problems with fuzzy goal programming based on TOPSIS and a fuzzy preference relation
Inform. Sci.
Prioritizing the best sites for treated wastewater instream use in an urban watershed using fuzzy TOPSIS
Resour., Conserv. Recycl.
Fuzzy TOPSIS for group decision making: a case study for accidents with oil spill in the sea
Expert Syst. Appl.
Fuzzy MCDM based on ideal and anti-ideal concepts
Euro. J. Operat. Res.
A new method of obtaining the priority weights from an interval fuzzy preference relation
Inform. Sci.
Approaches to collect decision making with fuzzy preference relations
Fuzzy Sets Syst.
Organizational strategy development in distribution channel management using fuzzy AHP and hierarchical fuzzy TOPSIS
Expert Syst. Appl.
Extension of the TOPSIS method for decision making problems under interval-valued intuitionistic fuzzy environment
Appl. Math. Modell.
A dynamic consensus scheme based on a nonreciprocal fuzzy preference relation modeling
Inform. Sci.
Evaluation model of business intelligence for enterprise systems using fuzzy TOPSIS
Expert Syst. Appl.
Mathematical analysis of fuel cell strategic technologies development solutions in the automotive industry by the TOPSIS multi-criteria decision making method
Int. J. Hydro. Energy
Implementation of a hybrid fuzzy system as a decision support process: a FAHP–FMCDM–FIS composition
Expert Syst. Appl.
A multi-criteria interval-valued intuitionistic fuzzy group decision making with Choquet integral-based TOPSIS
Expert Syst. Appl.
Cited by (51)
Bi-objective Pareto optimization for clustering-based hierarchical power control in a large-scale PV power plant
2023, Sustainable Energy Technologies and AssessmentsA spherical fuzzy methodology integrating maximizing deviation and TOPSIS methods
2021, Engineering Applications of Artificial IntelligenceFORA: An OWO based framework for finding outliers in web usage mining
2019, Information FusionPrediction OF CI engine performance, emission and combustion parameters using fish oil as a biodiesel by fuzzy-GA
2019, EnergyCitation Excerpt :Many researchers have proposed the TOPSIS to solve the Multi Criteria Decision Making problem [70–79]. Wang (2014) suggested TOPSIS to find relative preference relation [80]. Chang (2016) proposed TOPSIS for the quality evaluation of the diesel engine [81].
An integrated decision model of restoring technologies selection for engine remanufacturing practice
2019, Journal of Cleaner ProductionCitation Excerpt :To this end, here we conduct the life cycle assessment (LCA) and cost analysis for the remanufacturing restoring technologies and their technical parameters are also considered separately in order to make an informed decision. As a typical decision-making approach, Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) has been applied to building energy performance, selection of remanufacturing designs alternatives, and materials selection (Wang, 2015; Wang and Chan, 2013; Yang et al., 2017; Wang, 2014). However, these studies covered limited criteria or failed to address the fuzziness of final results.
Enhancing decision-making flexibility by introducing a new last aggregation evaluating approach based on multi-criteria group decision making and Pythagorean fuzzy sets
2017, Applied Soft Computing JournalCitation Excerpt :Decision makers (DMs)’ preferences are the main factors of this process. The MCDM approach often has some common characteristics and objectives: first, the alternatives that are to be assessed; second, the criteria that are employed to rank candidates; third, opinions that show the assessment of a candidate’s expected score on the criteria; and fourth, criteria weights which depict the relative importance of each criterion in comparison with other criteria [5,32,35,36,39]. Uncertainty is a very important aspect of any MCDM process.