Hesitant information sets and application in group decision making

doi:10.1016/j.asoc.2018.10.047

Applied Soft Computing

Volume 75, February 2019, Pages 120-129

https://doi.org/10.1016/j.asoc.2018.10.047 Get rights and content

Highlights

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New datastructure named hesitant information set (HIS).
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Useful in representing an agent’s confusion in evaluation of a data-item.
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Operations and properties for HIS.
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Extensions of HIS.
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Real Application in group multi attribute decision making.

Abstract

The recent information set theory provides a useful mechanism to represent an agent’s perceived information values. However, often a decision-maker (DM) considers multiple evaluations for the same information source value. To this end, we extend the recent information set as hesitant information set (HIS). It gives the multiple perceived information values, corresponding to an information source value. In the context of multi-attribute decision making, HIS represents a set of different possible subjective utilities that an agent may perceive as an evaluation of an alternative-attribute pair. The basic operations, and properties of HIS are investigated. A few information measures based on HIS are presented. Besides many illustrative examples, a real application in group multi attribute decision making problem is included.

Introduction

The fuzzy set theory [1] has gained popularity as an uncertainty representation theory to deal with the vagueness (or fuzziness). It represents in-exactness or ill-definedness about an information source value. Since its inception, several extensions of the fuzzy set theory have appeared in the literature, such as intuitionistic fuzzy set (IFS) [2], [3], type-2 fuzzy set [4], [5], interval valued fuzzy set (IVFS) [6], [7], interval valued fuzzy relations [8], vague fuzzy set [9], neutrosophic fuzzy set [10], and hesitant fuzzy set (HFS) [11]. All these extensions make use of the concept of membership function that maps an information source value to a crisp value in the interval $[0, 1]$ , giving its partial degree of association (belongingness) to a vague concept.

Fuzzy set provides a convenient method to depict vagueness. However, an agent inevitably encounters some hesitancy or doubt in arriving at a membership grade. There is no way in the conventional fuzzy set to incorporate this ambiguity associated with a membership function. To address this, the fuzzy set variants like Type 2 fuzzy set, IFS, IVFS, and HFS have been proposed in the literature. Type 2 fuzzy set addresses this problem by having the membership grade itself as fuzzy. Type- $n$ is a generalization of type-2 fuzzy set with each membership function as type $n - 1$ fuzzy set. IFS includes a hesitancy component, besides the usual membership and non-membership components in the fuzzy membership function.

IVFS has the membership grade in the form of an interval rather than a point estimate. A few uncertainty representation methods based on IVFS are presented in [8], [7], and a few based on IFS are proposed in [12], [13]. A method to quantify the lack of knowledge of experts in arriving at the membership grade, on the basis of the ignorance function [14], is developed in [15]. Similarly, the concept of interval-valued fuzzy relation is proposed in [8]. These concepts are interesting, but most of them address either of the two facets of uncertainty: (i) fuzziness (vagueness) or (ii) non-specificity (ambiguity), but not both at the same time.

In this regard, HFS [11] appears to be most suitable. It gives a set of multiple membership grades for a single information source value. That is, HFS allows to discretely include all the membership grades that an agent considers for an element. It has noticed a lot of attention in the recent times [16], [17] due to its ease of application and simplicity, in MADM applications, in particular. However, it suffers from the drawbacks of membership function.

In the context of MADM, the fuzzy concepts often being the linguistic expressions, are limited by the words in the daily language [18]. Human thinking and feelings that conjure the uncertainty in a given situation, have certainly more comprehensions and concepts than the words in the daily language. Therefore, it is often difficult to guarantee, unequivocally, a one to one mapping between the uncertainty perceived and its depiction by means of a membership function. Moreover, a membership grade is always relative without an explicit consideration of the associated information source value. Resultantly, the interpretation of a membership grade depends on the onlooker agent. For instance, an agent’s evaluation of a person as highly knowledgable conveys only a little information and it holds a different meaning for each interpreting agent.

Information set [19] addresses these shortcomings by giving a linked representation of an information source value and its evaluation (termed as agent), in terms of an entropy value, referred to as perceived information value that is uniquely interpretable. However, the original information set lacks a provision to represent ambiguity that an agent encounters quite often in MADM.

In this work, we present hesitant information set (HIS) that combines the advantages of both the information and hesitant fuzzy sets to represent the dual aspects – fuzziness and non-specificity – of uncertainty. More specifically, the proposed structures give the subjective evaluations for the alternatives, and help to determine the best alternative while retaining the diversity in the evaluations. In contrast to an information set that gives a single perceived information value for an information source value, HIS gives a set of information values corresponding to a single information source value. The information source values in HIS can be attribute values, probability values, or in fact the (evaluating) agent values. We have shown the usefulness of the proposed structures in a group MADM problem.

The rest of the paper is organized as follows. Section 2 gives the necessary concepts to build the background. The concept of hesitant information set is introduced in Section 3. The basic operations and properties of the proposed hesitant information set are investigated in Section 4. In Section 5, a few information measures based on the proposed HIS are formulated. Section 5 gives applications of hesitant information set in a group multi attribute decision making problem. Section 6 concludes the paper with an outlook on the future work.

Section snippets

Background

Let $U$ denote an universal set of objects. A fuzzy set [1] corresponding to a vague concept $X$ , defined over $U$ , is denoted by $F_{X}$ . It is characterized by a membership function for $X$ , denoted by $μ_{X}$ , representing a mapping: $μ_{X} : U \to [0, 1]$ The membership degree or grade of an object $u \in U$ in $X$ is denoted by $μ_{X} (u)$ , and the fuzzy set $F_{X}$ is represented as: $F_{X} = {(μ_{X} (u) ∕ u) : u \in U, μ_{X} (u) \in [0, 1]}$ The concept of fuzzy set is extended as intuitionistic fuzzy set (IFS) [2], [3] to represent the hesitancy that an agent faces

Background

The term information, as coined by Shannon, refers to a measure of uncertainty. Shannon’s entropy is one such measure that quantifies the probabilistic uncertainty from the probability values that are directly available. However, in the case of fuzzy domain, a membership function is required to capture the underlying possibility distribution of the information source values. Hence, the role of agent assumes importance in the possibilistic domain. At times, it is difficult for an agent to define

Extension principle

An extension principle is proposed for extending the conventional operations to the class of HISs. We consider an operator $O$ and develop a general extension principle to extend $O$ on a set of HISs.

Let $O$ be a function $O : {[0, 1]}^{n} \to [0, 1]$ . Let $ℋ$ denote a set of $n$ HISs $ℋ = {H_{1} (u), \dots, H_{n} (u)}$ , then, the extension of $O$ on $ℋ$ is given by: $O_{ℋ} = \cup_{γ \in {H_{1} (u) \times \dots \times H_{n} (u)}} {O (γ)}$ where, $H_{i} (u) = {S_{i}^{(1)} (u), \dots, S_{i}^{(N)} (u)}$ .

Example 4.1

Let $H_{1} (u) = {0.4, 0.2}$ , $H_{2} (u) = {0.1}$ , and $H_{3} (u) = {0.5}$ be the HISs in $ℋ = {H_{1} (u), H_{2} (u), H_{3} (u)}$ with $n = 3$ . If $O$ is an

Information measures for hesitant information sets

HIS gives the primary evaluation of an information source value in terms of the information values. In this section, we develop a few information measures that provide a second hand evaluation of an information source value, and also of the agent himself by the virtue of the corresponding information values.

Definition 5.1 Information Source Measure

For $H_{M} (u) = {S_{e}^{(1)} (u), \dots, S_{e}^{(N)} (u)}$ , it is given as: $Γ_{M} (u) = - \sum_{k} I_{e} (u) log (S_{e}^{(k)} (u)), k = 1, \dots N; a n d$ where $S_{e}^{(k)} (u) = I_{e} (u) g_{e}^{(k)} (u)$ , and $k$ is a pointer to the various elements of $H_{M} (u)$ .

Note: - $log (0)$ is

Applications in group multi attribute decision making

In general, the human decision making process is often characterized by the uncertainty of vagueness, and imperfect and incomplete knowledge of the DMs. As a result, a DM sometimes has multiple evaluations for an alternative. Hence, decisions are often made by a group in order to minimize the errors of judgment by a single DM. For instance, in a supplier selection process, various bidders are evaluated against multiple attributes, often, by multiple DMs. The essence of group MADM approach is to

Conclusions and future work

Hesitant information set (HIS) provides a convenient data structure to represent the various evaluations of an information source value, for which it gives multiple information values that are the values as perceived by an agent or a group of agents. It is inspired from the real world difficulty of an agent to represent the imprecision precisely. It does not suffer from the shortcomings associated with the membership function and comprises of the possible information (entropy) values that an

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Hesitant information sets and application in group decision making

Highlights

Abstract

Introduction

Section snippets

Background

Background

Extension principle

Information measures for hesitant information sets

Applications in group multi attribute decision making

Conclusions and future work

Inf. Control

Fuzzy Sets and Systems

Inform Control

Inform. Sci.

Fuzzy Sets and Systems

Expert Syst. Appl.

European J. Oper. Res.

Internat. J. Approx. Reason.

European J. Oper. Res.

Inform. Sci.

Expert Syst. Appl.

Knowl.-Based Syst.

Appl. Soft Comput.

Inform. Sci.

Procedia Comput. Sci.

Inform. Sci.