An integrated method for cognitive complex multiple experts multiple criteria decision making based on ELECTRE III with weighted Borda rule
Introduction
Multiple Criteria Decision Making (MCDM), which involves a set of finite alternatives being evaluated over several criteria with different importance, is popular in all walks of our life. MCDM methods are useful in finding reasonable decision results and thus have been widely applied in solving real-life problems in different areas such as beneficiation of coal [6], waste recycling [16], reverse logistics [2], and sustainable housing affordability evaluation [33]. The ELECTRE (ELimination Et Choix Traduisant la REalité in French, which means elimination and choice expressing reality) method is one of the most popular MCDM methods. The original ELECTRE method, which is now named ELECTRE I, was designed by Roy [37] to overcome the defect of the weighted sum method concerning the compensation effect in handling selection problems. Due to apparent advantages of the ELECTRE I over the weighted sum-based methods, the ELECTRE method soon attracted much attention of scholars and has been further extended into plentiful derivations, including the ELECTRE II [39], ELECTRE III [38], ELECTRE IV [41] and ELECTRE TRI [40]. The ELECTRE II, ELECTRE III, and ELECTRE IV were designed to rank alternatives while the ELECTRE TRI was proposed to solve assignment problems. Regarding the ranking methods, the ELECTRE II establishes the concordance, discordance and indifference sets to capture the outranking relations between alternatives; the ELECTRE III method takes into account the ambiguous and uncertain information; the ELECTRE IV is designed for the case that the weights of criteria cannot be obtained. For details about the characteristics of ELECTRE methods, please refer to the survey paper written by Govindan and Jepsen [18].
The ELECTRE III plays a prominent role in the family of ELECTRE methods since it introduces the preference threshold, indifference threshold and veto threshold to represent the inherent uncertainty of experts’ judgments. It has been applied successfully in solving MCDM problems such as managing energy systems [34], evaluating the service quality of international airports [28], selecting vendors [30], and ranking universities [15]. In addition, it has also been extended into different contexts, such as the fuzzy ELECTRE III [30] and intuitionistic fuzzy ELECTRE III [48,49]. However, these extensions can only be used to tackle quantitative MCDM problems but not qualitative MCDM problems with cognitive complex linguistic information.
Hesitant Fuzzy Linguistic Term Set (HFLTS) [36], as a generalized linguistic representation model, has turned out to be a useful tool to represent cognitive complex information in qualitative decision-making problems [22]. With the context-free grammar and transformation function [36], people can use linguistic expressions, such as “between good and very good” or “at least good”, to flexibly and comprehensively express their cognitive complex linguistic preferences. These linguistic expressions align closely with people's thought and cognition. It is a big step forward to the traditional Computing With Words (CWW) model [51], given that the traditional CWW model is based on singleton linguistic terms. Investigating MCDM methods with hesitant fuzzy linguistic information is critical for solving qualitative MCDM problems with complex cognitive linguistic information. However, most existing hesitant fuzzy linguistic MCDM methods [17,20,23,24,48] were based on the weighted sum. In this regard, unreasonable results may be produced since the weighted sum may produce the compensation effect that a big value may enlarge the average value of a set of small values. We observe that Liao et al. [26] investigated the hesitant fuzzy linguistic MCDM method based on the ELECTRE II.
Given that ELECTRE III is efficient in handing imprecision information and eliminating compensation effect resulting from outliers, it is needed to investigate the ELECTRE III model for MCDM problems in which the cognitive complex linguistic information is represented by HFLTSs.
In MCDM problems, different criteria may have different weights. How to determine the weights of criteria is an important topic in the ELECTRE III method. If the number of criteria is too large or there is no significant difference between criteria, people usually set the weights of criteria equal. Liu and Zhang [27] researched an entropy-based method to derive the weights of criteria in the ELECTRE III. This method is efficient only when experts are not able to provide any preference information about the importance of criteria. It is too objective to be consistent with the subjective property of the ELECTRE methods. Figueira and Roy [13] modified the SRF (a weight-determining method originally proposed by Simo and then improved by Roy and Figueira) method to deduce the weights of criteria subjectively. It should be noted that the correlations among criteria should not be ignored when determining the weights of criteria; however, the SRF method only takes into account subjective weights but the objective interaction effects among criteria [5] are ignored. To obtain reasonable weights of criteria, a combinative weight-determining method is proposed in this paper by considering subjective weights and objective weights simultaneously.
In addition, due to the limitation of individual's knowledge, experience and time, it is common to form a decision committee to find reasonable and convincing solution based on the comprehensive judgments of multiple experts. As a hot topic in decision analysis, Multiple Experts Multiple Criteria Decision Making (MEMCDM) has obtained many achievements in different fields, such as the research output evaluation [20], water resource management [31], and freight forwarder selection in airlines [11]. However, a number of experts would cause complexity in exploring the final solution of the group. Within the MEMCDM framework, two aggregation approaches are usually used: one is to aggregate the different judgments of individual experts first and then derive the group decision based on the collective information; the other is to yield the individual ranking associated to each expert and then fuse the results of all experts by some aggregation methods. The former would produce compensation effect when aggregating individual judgments by weighted sum methods. Thus, this paper uses the weighted Borda rule [29] to aggregate individual rankings and obtain the final ranking of alternatives.
In the ELECTRE III method, the ranking result of alternatives associated to each expert is obtained based on the intersections of alternatives’ ascending pre-orders and descending pre-orders produced by the distillation algorithm. Aggregating pre-orders by intersections would cause preferential, indifferent and incomparable relations. Taking the incomparable relation as an example, if alternative A1 outranks alternative A2 in one pre-order and A2 outranks A1 in another pre-order, the relation between alternatives A1 and A2 is incomparable by intersections. The incomparable relations of alternatives resulted from intersections make it difficult to rank alternatives for the group. In this sense, when the number of experts is great in an MEMCDM problem, the different rankings of alternatives corresponding to different experts may lead to lots of incomparable relations of alternatives, which reduces the value of the final conclusion regarding the ranking of alternatives. Hence, it is not applicable to use the intersections to aggregate all individual experts’ rankings results within the MEMCDM framework when the number of experts is large. A feasible technique to integrate experts’ rankings of alternatives but reduce the incomparable relations of alternatives for the expert group is the Borda rule [29], an election method originated from Arrow's social choice model. Borda rule is based on the ranking orders of alternatives. It is consistent with the outputs of the ELECTRE III method in which the ranking orders of alternatives associated with each expert are obtained. The Borda rule is valid and easy to execute in MEMCDM problems especially when the rankings of alternatives corresponding to the experts contradict each other. Considering that different experts may own different importance, the weighted Borda rule is adopted in this paper to aggregate the ranking results of individual experts obtained by the hesitant linguistic ELECTRE III method.
It is noted that the conflict may exist between different experts. To measure the acceptance degree of the collective ranking result corresponding to each expert and the group, some ordinal consensus measures based on the weighted Borda counts are defined. In this way, an understandable ranking conclusion can be obtained, together with the ordinal consensus degree of the expert group, revealing the reliability of the ranking result.
Brief speaking, this paper dedicates to proposing an integrated method for cognitive complex MEMCDM problems based on the hesitant fuzzy linguistic ELECTRE III (HFL-ELECTRE III) and the weighted Borda rule. The HFL-ELECTRE III method uses the HFLEs to capture cognitive complex information, including crisp values, interval values, singleton linguistic terms and linguistic expressions, of multiple experts in decision-making problems. In this way, the application field of the ELECTRE III method is broadened. This paper intends to obtain the following novel achievements:
- (1)
After conceptualizing the cognitive complex MEMCDM problems, we introduce an interesting score function of HFLEs, based on which, the concordance and discordance indices of pairwise alternatives are calculated. Then, the outranking relations between alternatives are obtained.
- (2)
A combinative weight-determining method is introduced, which involves both subjective and objective opinions about the importance of criteria.
- (3)
We use the weighted Borda rule instead of the intersections of ranking orders of alternatives to aggregate the rankings of alternatives corresponding to individual experts.
- (4)
Some ordinal consensus measures are proposed to identify the reliability of the collective ranking result.
- (5)
For the facility of application, we give a detailed procedure of the integrated HFL-ELECTRE III method with weighted Borda rule to solve cognitive complex MEMCDM problems. The merits of this method are clearly illustrated.
Hospital ranking is significant for patients, doctors and hospital managers [46]. A reasonable ranking of hospitals is beneficial for patients to choose a suitable hospital for healthcare. Additionally, the hospital ranking is necessary for doctors to know the rank of the hospital where they work, and the top hospitals should use sophisticated treatment techniques in their fields. A good ranking also has reference values for hospital managers to know the advantages and disadvantages compared with other hospitals from the ranking list. In this paper, we apply the proposed integrated HFL-ELECTRE III method to a case study concerning the hospital ranking in China. The case study validates the efficiency of the proposed method and also has some insights to hospital management.
The remainder of this paper is organized as follows: Section 2 describes the cognitive complex MEMCDM problems. Section 3 addresses the integrated hesitant linguistic ELECTRE III method with the weighted Borda rule. Section 4 implements the proposed method to hospital ranking in China, and validates the method by comparative discussions. Section 5 ends the paper with conclusions.
Section snippets
Conceptualization of the cognitive complex multiple experts multiple criteria decision-making problems from management perspective
To start our study, we first conceptualize cognitive complex MEMCDM problems that will be addressed in this paper. Rules to transform the quantitative information in the forms of crisp values or interval values, and the qualitative information in terms of singleton linguistic terms or complex linguistic expressions, to HFLTSs are described in detail.
For an MEMCDM problem, let be a set of experts, be a set of alternatives, and be a set of
An integrated hesitant linguistic ELECTRE III method with weighted Borda rule for cognitive complex multiple criteria group decision-making problems
This section aims to propose a novel method to solve cognitive complex MEMCDM problems with both complex quantitative and qualitative information as described in Section 2 based on the ELECTRE III method and the weighted Borda rule.
Illustrative example: hospital ranking in China
In this section, the integrated hesitant linguistic ELECTRE III method is implemented to obtain the ranking of hospitals in China. Comparative analyses are given to show the validation and effectiveness of the proposed method.
Conclusions and future research directions
This paper developed an integrated hesitant fuzzy linguistic ELECTRE III method with the weighted Borda rule to handle cognitive complex MEMCDM problems. The originality and contributions of this paper can be justified as follows:
- (1)
We, for the first time, proposed the framework of the cognitive complex MEMCDM and unified the cognitive information, including crisp values, interval values, singleton linguistic terms and linguistic expressions, by HFLEs with six transformation rules. The hesitant
Acknowledgment
The work was supported by the National Natural Science Foundation of China (71771156), the 2019 Sichuan Planning Project of Social Science (No. SC18A007), and the 2019 Soft Science Project of Sichuan Science and Technology Department (No. 2019JDR0141).
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