Research Article: Some linguistic neutrosophic Hamy mean operators and their application to multi-attribute group decision making

Date Published: March 7, 2018

Publisher: Public Library of Science

Author(s): Peide Liu, Xinli You, Fei Li.


Linguistic neutrosophic numbers (LNNs) can easily describe the incomplete and indeterminate information by the truth, indeterminacy, and falsity linguistic variables (LVs), and the Hamy mean (HM) operator is a good tool to deal with multiple attribute group decision making (MAGDM) problems because it can capture the interrelationship among the multi-input arguments. Motivated by these ideas, we develop linguistic neutrosophic HM (LNHM) operator and weighted linguistic neutrosophic HM (WLNHM) operator. Some desirable properties and special cases of two operators are discussed in detail. Furthermore, considering the situation in which the decision makers (DMs) can’t give the suitable weight of each attribute directly from various reasons, we propose the concept of entropy for linguistic neutrosophic set (LNS) to obtain the attribute weight vector objectively, and then the method for MAGDM problems with LNNs is proposed, and some examples are used to illustrate the effectiveness and superiority of the proposed method by comparing with the existing methods.

Partial Text

Nowadays, the multi-attribute decision-making (MADM) or MAGDM is widely existed in various fields [1–5], and to obtain accurate evaluation information is one of premises for DMs to make rational and feasible decision. However, in real-world situation, there are a variety of limitations, such as too much redundant data, uncertainty and complexity of the decision-making environment, difficulties of exploiting information etc. Therefore, it is a concerned topic in decision-making theoretical field about how to describe the attribute values of alternatives and reduce information loss. In qualitative environment, decision information can be usually estimated by linguistic terms (LTs) rather than exact numerical values due to universal uncertainty and the vagueness of human judgement. Zadeh [6] firstly introduced the notion of LVs). Later, Herrera and Herrera-Viedma [7, 8] proposed a linguistic assessments consensus model and further developed the steps of linguistic decision analysis. Xu [9] proposed a linguistic hybrid arithmetic average operator to solve MAGDM problems. However, these methods based on the LVs can only reflect the truth/membership degree. Then, Chen et al. [10] proposed the linguistic intuitionistic fuzzy number (LIFN) which takes the form of γ = (sα,sβ), where sα and sβ represent the truth/membership and falsity/non-membership degrees used by LVs based on the given LT set (LTS). It is obvious that the LIFN can describe more complex linguistic information than LVs. Based on the LIFN, some scholars [11, 12] proposed some improved aggregation operators for LIFNs, and applied them to MADM or MAGDM problems.

In this section, based on the operational laws of LNNs, we shall explore the HM operator to deal with LNNs and develop LNHM operator and WLNHM operator, and then we also discuss some properties and some special cases of these new operators.

Entropy is a useful tool to measure uncertainty in a set, including fuzzy set (FS), intuitionistic fuzzy set (IFS) and vague set etc. Here the LNS is characterized by handling uncertain information with truth-membership function, indeterminacy-membership function and falsity-membership function, respectively. As a consequence, it’s necessary to further define the entropy of LNS. Zadeh [38] first introduced the entropy of FS to measure fuzziness in 1965. Later De Luca-Termini [41] axiomatized the non-probabilistic entropy. Based on above studies, the entropy E of a fuzzy set A should satisfy the following axioms:

For a MAGDM with LNNs, let A = {A1,A2,….,Am} be a set of alternatives, D = {D1,D2,….,Dt} be the set of DMs and λ = (λ1,λ2,….,λt)T be the weight vector of DMs Dh(h=1,2,….,t), λh ∈ [0,1],h = 1,2,….,t and ∑h=1tλh=1. Let C = {C1,C2,….,Cn}be the set of attributes and there are two kinds of attributes, i.e., the benefit attributes and the cost attributes. Here we assume the weight vector of the attributes is unknown. If the hth (h = 1,2,….,t) DM provides the evaluation of the alternative Ai(i = 1,2,….,m) about the attribute Cj(j = 1,2,….,n) based on the LTS, such as s = {s0 = extremely poor,s1 = very poor,s2 = poor,s3 = slightly poor,s4 = medium,s5 = slightly better,s6 = good, s7 = very good, s8 = perfect}, by the form of a LNN xijh=(lαijh,lβijh,lγijh) for αijh,βijh,γijh∈[0,t] (h = 1,2,….,t;i = 1,2,….,m;j = 1,2,….,n). Therefore, we can obtain the hth LNN decision matrix Xh=[xijh]m×n. Based on these information, the proposed MAGDM method can be presented as follows:

In the following, an illustrative example about the selection of investment alternatives from [22] is provided to show the advantages of the proposed method.

In this paper, we propose the LNNHM, WLNNHM operators. Then we investigate some desirable properties and further discuss their special cases when the parameter takes different values. Further, we define the entropy of LNS and apply it to determinate weights. Based on the WLNNHM operator and entropy weight measure, we develop a novel MAGDM method with LNNs. We demonstrate the feasibility and advantages of proposed method by comparing with the existing methods [13,21,22]. In the future research, we shall further develop other methods with LNNs, such as TODIM and VIKOR of LNNs, and apply them to handle MADM or MAGDM problems, especially when we need to consider incomplete, indeterminate and inconsistent information in the problems. On the other hand, we can develop the potential applications of the proposed method to different domains.




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