An aggregation method for solving group multi-criteria decision-making problems with single-valued neutrosophic sets
Graphical abstract
Introduction
Multi-criteria decision-making (MCDM) is a collection of tools and methods used to solve problems with multiple and often conflicting criteria. Group decision-making is the process of making a decision based on feedback from more than one decision maker (DM). Group MCDM (GMCDM) is a complex process involving multiple criteria and multiple DMs. This complexity is amplified when the process involves qualitative and quantitative judgments on the potential alternatives with respect to the relevant criteria. These judgments are often vague and contradictory, and significantly complicate the construction of knowledge-based rules and the establishment of decision support procedures. Vagueness can occur under the following six circumstances: (1) the words that are used in antecedents and consequents of evaluation rules can mean different things to different people [1,2]; (2) consequences obtained by polling a group of experts are often different for the same rule or statement because the experts are not necessarily in agreement [3,4]; (3) decision groups are often heterogeneous due to the different extent of its members’ expertise, knowledge and experience [5,6]; (4) expert estimates of criteria importance or performance of alternatives with respect to intangible parameters are not always consistent [7]; (5) information provided by individuals is usually incomplete or ill-defined [8,9]; and, (6) DMs are not always confident about the correctness of their own reasoning [10].
Fundamentally, uncertainty is an attribute of information [11]. There are two main types of uncertainties: external and internal. The external (or stochastic) uncertainty implies that the events or statements are well defined, but the state of the system or environmental conditions lying beyond the control of the DM might not be known completely. The internal uncertainty (or fuzziness) refers to the vagueness concerning the description of the semantic meaning of events, phenomena, or statements themselves, including uncertainties about DM preferences, imprecise judgments and ambiguity of information [12,13]. In this regard, Zadeh [14, p. 28] wrote: “As the complexity of a system increases, our ability to make precise and yet significant statements about its behavior diminishes until a threshold is reached when precision and significance (relevance) become almost mutually exclusive characteristics.” Therefore, precise quantitative analysis is not likely to have much relevance in problems which involve humans either as individuals or in groups.
The presence of multiple vague measures in GMCDM has continued to challenge researchers and the problems associated with finding a comprehensive approach to modeling ambiguous information has still not been adequately resolved. In the sense of Ackoff [15], the problem of having ill-defined goals, ill-defined procedures or ill-defined data is a mess. Several theories have emerged during the last 50 years that generalize traditional probability theory and are more appropriate for not-probabilistic information formats in which evidence about uncertainty appears. These include Chiquet’s theory of capacities, random set theory, evidence theory, possibility theory, Walley’s theory of imprecise probabilities, fuzzy set (FS) theory, rough set theory, intuitionistic fuzzy set (IFS) theory and neutrosophic set (NS) theory, among others [[16], [17], [18], [19], [20], [21]].
The most commonly used methodology for representing and manipulating imprecise and uncertain information in multi-criteria decision systems is the theory of FSs. However, while focusing on the membership grade (i.e., truthfulness or possibility) of vague parameters or events, FSs fail to consider falsity and indeterminacy magnitudes of measured responses. In practical terms, the problem of projecting multi-source and multivariate group decision uncertainty using mathematical models remains intractable in terms of FSs. In the late 90 s Atanassov [17] introduced and developed the idea of IFSs, intuitionistic logic and intuitionistic algebra allowing for more complex mental constructs and semantic uncertainties. In addition to the membership grade, IFSs consider non-membership levels. However, IFSs cannot handle all uncertainty cases, particularly paradoxes. NSs are the cutting-edge concept first introduced by Smarandache [20] in the late 90 s and developed in the 21st century. NSs generalize FSs and IFSs. NSs and, in particular, single-valued NSs are characterized by three independent membership magnitudes, namely, falsity, truth and indeterminacy. Such a formulation allows to model the most general cases of ambiguity, including paradoxes.
This paper proposes a new approach to represent multi-source uncertainty of estimates provided by various domain experts in MCDM problems, and a methodology to integrate these measures within one decision support procedure.
Most of the existing studies on neutrosophic approaches to GMCDM problems focus on the development of aggregation operators to be applied to neutrosophic decision matrices in order to obtain group estimates of criteria and alternatives. At the same time, the truth, falsity and indeterminacy levels used to represent the uncertainty inherent to DMs’ judgments are not given an explicit interpretation. These levels are usually treated as abstract triads of (non-standard) reals without highlighting their role as reliability measures or specifying the variables that they depend on.
The artificiality and routineness deriving from an overuse of aggregation operators together with the tendency to overlook a concrete interpretation for neutrosophic values within a given GMCDM problem represent a gap in the literature that need to be consider in order to investigate ways to effectively improve the applicability of decision-making processes.
The proposed GMCDM approach aims at increasing reliability, coherence and dependability of the final outcome by accounting for three different and independent reliability measures that can affect DMs’ estimates, namely, DMs’ credibility, inconsistency inherent to DMs’ evaluation processes, and DMs’ confidence in their own evaluation abilities. In order to do so, an assessment procedure for overall priorities of criteria and alternatives is developed using the technology of single-valued NSs.
More precisely, given a committee of heterogeneous DMs/experts who must evaluate the performance of a set of alternatives with respect to criteria, we deal with the following problem.
Problem:
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Assumption: Let experts’ estimates of the objects (I alternatives and J criteria) be affected by three diverse and independent factors: first, the experts have different credibility (i.e., voting power); second, the local priorities that are derived using relative comparison judgments are characterized by an inconsistency or an error measure; third, due to the lack of information or scarce experience, some experts do not feel confident about their own judgments.
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Question: How can these uncertainty metrics be incorporated into a coherent ranking model to increase dependability of the group decision outcome? That is, how can multi-group multi-person expert judgments affected by these uncertainty metrics be coherently formalized and synthesized to yield reliable overall rankings of the criteria and alternatives?
Until recently, modeling and handling independent multi-source uncertainties inherent to a single information unit was challenging due to the lack of appropriate formal tools. With the development of the NS and single-valued NS concepts, the problem of simultaneously handling different ambiguity indicators of one variable can be resolved by converting the values of the indicators into the truth-, falsity- and indeterminacy-membership grades of the corresponding variable. That is, the reliability of any estimate provided by one of the expert (i.e., the importance of a criterion or the performance of an alternative with respect to a criterion) can be expressed by a triad of independent magnitudes, , where represents the expert’s credibility, the inconsistencies/errors intrinsic to the expert’s evaluation process and the expert’s confidence in his/her own ability and experience to evaluate the importance of the criteria and the performance of the alternatives.
After interpreting triads of reliability measures as neutrosophic values and group estimates as single-valued NSs, a deneutrosophication process is designed to synthesize crisp values representative of group priorities which are, in turn, used to estimate the overall performance of the alternatives.
It must be noted that the proposed formulation assumes independency among the alternatives’ performances, i.e., synergy effects do not occur with respect to the alternatives’ joint performance. Moreover, non-linear dependencies among criteria, in terms of their importance for the achievement of the overall problem objective, are not considered.
Finally, an illustrative example is provided to show how taking into account multi-source uncertainty indicators inherent to the experts’ evaluations may deeply affect the results obtained in a standard fuzzy environment even in the case of very simple ranking problems.
The remainder of this paper is structured as follows. Section 2 offers a literature review focusing the recent applications of FSs, IFSs and NSs to GMCDM problems. Section 3 outlines the key features of the proposed NS-based GMCDM approach highlighting its differences and advantages with respect to the existing models as well as some theoretical and practical implications. Section 4 provides the necessary technical preliminaries and notations relative to FSs, IFSs, NSs, and the basic structure of a conventional GMCDM approach. Section 5 presents the aggregation and synthesizing mechanisms proposed for solving GMCDM problems with single-valued NSs. Section 6 presents an illustrative example to demonstrate the applicability of the proposed method. Finally, Section 7 draws the conclusion and some future research directions.
Section snippets
Applications of FSs/IFSs/NSs to GMCDM in the current literature
Complex problem solving is associated with the gathering of interest groups or experts to discuss the critical issues, such as for conflict resolution, for planning and design, for policy formation or for plan brainstorming [22,23]. Multi-criteria decision-making methods are an important set of tools for addressing challenging business decisions since they allow managers to better proceed in the face of uncertainty, complexity, and conflicting objectives [24]. Group multi-criteria decision
Key features of the proposed NS-based GMCDM approach: theoretical and practical implications
Our approach differs from those presented in the existing literature mainly in two ways.
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Neutrosophic values are not used to construct the comparison matrices. The comparison matrices can be obtained through any crisp relative measurement method [[73], [74], [75], [76]]. Neutrosophic values are introduced only after each DM has evaluated the local priorities to simultaneously handle three key different ambiguity indicators, namely, the credibility of the DM, the inconsistency
Technical preliminaries and notations
In this section, the foundations and differences among the most significant formal theories dealing with vague and imprecise information (primarily of the “internal” type) in humanistic decision-making systems are discussed. Afterwards, the conventional GMCDM approach to the problem of ranking a set of alternatives with respect to a set of selected evaluation criteria is outlined.
The proposed NS-based GMCDM approach
As highlighted in Subsection 1.1 and Section 3, we focus on defining a procedure that relates the single DM’s priorities obtained through any methodology based on a comparison principle to his/her own levels of credibility, inconsistency and confidence. After being interpreted as single-valued NSs, the group levels are reduced to representative crisp values through an ad hoc deneutrosophication process. These values are, in turn, used to estimate the overall performance of the alternatives.
Fig.
An illustrative example
Let four domain experts with unequal voting powers be responsible for the assessment of three alternatives with respect to five criteria (, , ). First, each expert builds pairwise comparison matrices as shown in Eq. (2) and applies the right eigenvalue method to reveal the weights of the criteria () and the priorities of the alternatives with respect to each criterion (). The consistency ratio (respectively, ) is calculated for each comparison
Conclusion and future research directions
We have proposed a novel method to handle multi-source uncertainty measures reflecting the reliability of experts’ assessments in GMCDM problems based on single-valued NSs.
NSs are characterized by three independent membership magnitudes (falsity, truth and indeterminacy) and can be applied to model situations characterized by complex uncertainty, including those leading to paradoxical results.
Most of the studies on neutrosophic approaches to GMCDM problems that have been presented over the last
Acknowledgement
The authors would like to thank the anonymous reviewers, the associate editor and the editor for their insightful comments and suggestions.
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