Observation of experts attitudes through multi-criteria decision-making

Abstract

Keywords: Human Attitude; Hesitant Fuzzy Soft Set; Hesitant Fuzzy Bonferroni Mean; MCDM

Introduction

Fuzzy set theory was instigated by Zadeh [29] which provides a suitable framework to model several problems with uncertain and ambiguous data. It is identified by a function defined on a set and takes values in the interval [0,1]. The value of a function determines the degree of membership of an element in the universe of discourse. There are certain situations where fuzzy set theory cannot be effectively employed. To overcome difficulties arising

to model the problems that could not be modeled with the environment of fuzzy set theory, Molodtsov [13] introduced soft set which is a set associated with a set of parameters. Maji et al. [15] combined the concepts of soft sets and fuzzy sets and defined fuzzy soft sets (FSS). FSS theory is more applicable in intelligent systems, identification problems, pattern recognition, optimization, and control theory. Roy and Maji [19] successfully applied the fuzzy soft set theory to decision-making problems.

*Corresponding author. *E-mail address: asmamahmood@gcuf.edu.pk This paper was recommended for publication in revised form by Regional Editor Ahmet Selim Dalkilic Published by Yıldız Technical University Press, İstanbul, Turkey Copyright 2021, Yıldız Technical University. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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Hesitancy is one of the most important factors which stops us from taking decisions at the right time. So it is very important to incorporate a hesitancy attitude to model decision-making processes. Hesitant fuzzy set (HFS) is a generalization of a fuzzy set which was introduced by Torra [22]. HFS can be more accurately reflect the people, hesitancy in stating their preferences about objects as compared to the fuzzy set. Xia and Xu [26] presented a series of aggregation operators for the elements of HFS. In recent times, some applications of HFS were represented by Faruk and Serif [12,16]. Hesitant fuzzy multi-criteria decision-making (MCDM) method was presented by Hu et al. [9]. Beg and Rashid [2] used HFS to model 2-tuple linguistic information. Some applications of the hesitant fuzzy soft set (HFSS) in MCDM were studied by Wang et al. [24]. MCDM facilitates the decision-making process when the situation of selecting the best alternative is complicated. Some of the methods are surveyed by Aruldoss et al. [1]. A multi-criteria decision analysis method called TOPSIS, an abbreviation of “The technique for order of preference by similarity to ideal solution” is developed by Hwang and Yoon [8]. Simplicity, rationality, intelligibility and good computational efficiency are some advantages of TOPSIS. TOPSIS is based on distances between two sets or elements and the obtained solution is close to the ideal situation and far from the worst situation. Xu and Xia [25] presented some basic distance and similarity measures for hesitant fuzzy sets. Chen [5] extended the TOPSIS for solving multi criteria decision making problems in the framework of fuzzy sets (see also, [3]). Moreover, this multi-criteria method is utilized by many researchers [7,6,18,10] to solve the decision-making problems based on the fuzzy soft set, intuitionistic fuzzy soft set and hesitant fuzzy soft set theories. Another MCDM method is VIKOR which is also based on “closeness to the ideal”, and is the compromised ranking method. A comparative analysis of VIKOR and TOPSIS is presented by Opricovic and Tzeng [17]. The multi-criteria aggregation function is introduced by Yager [28] which is initially defined as Bonferroni mean (BM) which is a mean type aggregation operator. BM facilitates to cater inter-related arguments of individuals in the group decision-making process. Hesitant fuzzy Bonferroni mean (HFBM) is a BM that is calculated for hesitant fuzzy elements as defined by Zhu et al. [30]. By using BM and weighted Bonferroni mean (WBM) based on FSS, Beg et al. [4] found the optimum fuzzy soft constants (OFSCs). By using OFSCs, they developed a system of fuzzy soft differential equations and discussed different cases observed after the decision to analyze the human attitude. These different cases arise according to satisfaction or dissatisfaction, encouragement or discouragement of other people for the decisions. Solutions of system of linear differential equations, their stability, phase portraits have been discussed by Strogatz [21]. He also presented a dynamic model of love which was also discussed by Sprott [20]. A variety of research articles is published on decision-making approaches that rank a finite set

of alternatives. However, no work except [4,11,23,14] has been done on what happens after a decision has been taken. There is a gap between the development of a system of linear equations and MCDM which is carried through in this research by utilizing TOPSIS and VIKOR. For this purpose, hesitant fuzzy soft matrices are considered as initial data to rank the alternatives. Then a system of differential equations is developed by using the obtained OFSCs to analyze the human attitude. PRELIMINAries Let us recall some basic definitions and known results. Definition 1 [9] Let U be a universe of discourse. The hesitant fuzzy set A on U is identified by a function hA on U that returns a subset of [0,1]. In the sequel, by hesitant fuzzy set, we mean a discrete hesitant fuzzy set where each hA(x) is represented as a set of finite values, {x1, x2, ... , xn}. The following is an example of discrete hesitant fuzzy set. Example 1 A bank manager intimates five different options to a customer who wants to open a bank account. Let U = {u1,u2,u3,u4,u5} be a universe of discourse, where u1, u2, u3, u4 and u5 represent saving account, basic checking account, interest bearing checking account, money-market deposit account and certificate of deposit, respectively. A hesitant fuzzy set h over U can be represented as:

represents that there are three possible degrees of membership 0.2, 0.3 and 0.5 for saving account which can be interpreted as some hesitancy. Definition 2 [24] Let H(U) be the set of all hesitant fuzzy sets in U and A⊆E (a set of parameters). A pair (F,A) is called a hesitant fuzzy soft set over U, where F is a mapping given by F:A H(U). Example 2 Let U={u1, u2, u3, u4, u5} be the same set as given in above example and a set E={e1, e2, e3} of parameters representing emergency cash, interest rate and credit card facility, respectively. Then a hesitant fuzzy soft set can be described as

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on the decision-makers, risk preferences. Optimists add the maximum value and pessimists add the minimum value. Definition 4 [4] Let l,m be two natural numbers and xi ≥ 0 where i ∈ {1,2,…,n} then Bonferroni mean Bl,m is defined as

represents that a value {0.2,0.5} is assigned to saving account based on the criteria of emergency cash. The hesitant fuzzy soft set given above can be written in matrix form or tabular form as shown in Table 1

Table 1. Representation of Hesitant Fuzzy Soft Set u1 u2 u3 u4 u5

To apply the decision-making methods while taking HFSS as initial data, it is essential to define distance and similarity measures. Some distance measures between two hesitant fuzzy sets (HFSs) have been defined by Xu and Xia [25]. Hesitant weighted hamming distance between two hesitant fuzzy sets is defined as: Definition 3 [25] Let M and N be two HFSs on the universe X={x1, x2, ... , xn} and the weight of each element xi∈X is wi, where i∈{1, 2, ... , n}, wi∈[ 0,1] and then the hesitant weighted hamming distance between two hesitant fuzzy sets is given by

is called a score function of h, where #h is the number of elements in h. Thus a score function of a hesitant fuzzy element gives an average value of all elements in h. Consistent with [27], some operational laws for any three hesitant fuzzy elements h,h_{1}, h_{2} and a scalar λ are defined as:

Definition 7 [30] Let h1, h2, ... , hn be hesitant fuzzy elements and l,m be two natural numbers, then hesitant fuzzy Bonferroni mean HFBM is defined as

Definition 5 [4] Let l, m be two natural numbers and xi ≥ 0 (i = 1,2,…,n) and wi(≥ 0) be the weights for xi with the condi, then weighted Bonferroni mean (WBM) tion is defined as

cannot be calculated when . In this case, a shorter one is extended and a value is added several times in it. This value depends

Definition 8 [30] Let h1, h2, ... , hn be hesitant fuzzy elements with weight vector W=[w1, w2, ... , wn] for which

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and l,m two natural numbers, then weighted hesitant fuzzy Bonferroni mean WHFBM is defined as

Definition 9 [3] Hesitant fuzzy soft sets can be aggregated by a matrix X, where

and Beg et al. [4] considered the following system of linear fuzzy soft differential equations

An attitude analyzer method through hesitant fuzzy Bonferroni mean This method consists of the following steps: Step 1: Let A = {A1, A2,…, An} be a set of objects/alternatives and B = {B1, B2,…, Bn} be a set of attributes/criteria. Decision advisors present their opinions in the form of m × n hesitant fuzzy soft sets D1, D2,…, Dp. Step 2: Calculate Bonferroni hesitant fuzzy soft set (BHFSS) Bm × n by applying definition 7 on D1, D2,…, Dp and then calculate B'm × n by using definition 6 whose entries are the score functions of respective entries of Bm × n. Step 3: Consider a weight vector W = {w1, w2, ... , wn} . Then weighted such wn > 0 for j=1,2,...,n and Bonferroni fuzzy soft set (WBFSS) Cm×1 = [ci,1], is computed where ci,1, i=1,2,...,m are calculated by using definition 8 for ith entries of the set B'm × n. Step 4: Optimal fuzzy soft constants (i,j=1,2) are chosen from ci,1, i=1,2,...,m. If two persons select same alternative, then

If two persons select different alternatives, then (1) where P1 and P2 are two variables which represent the attitude of two persons after taking a decision at the time t. While dP1/dt and dP1/dt represent the change in persons, attitudes after some time due to that decision and (i,j=1,2) are optimum fuzzy soft constants (OFSCs) taken as signed fuzzy numbers which denote the influence on ith person of his internal feelings and jth person,s feelings. Positive and negative sign is assigned to according to encouragement or discouragement of a person about a decision. Stability of system (1) depends upon eigen values of the matrix

Analysis of the Future Attitudes of Experts Based on Fuzzy Soft Differential Equations In this section, we develop the algorithms to analyze the human attitude of two persons after taking a decision, which are based on the following MCDM methods • Bonferroni mean • TOPSIS • VIKOR where hesitant fuzzy soft sets are considered as initial data.

Step 5: Develop a system of fuzzy soft differential equations as defined in [4] and check its stability to determine the next viewpoint of two persons. In the following example, we discuss one of the cases discussed in [21] and draw phase portrait and analyze the human attitude of two persons after taking a decision. Example 3 There are five different tiles and sanitary packages A1, A2, A3, A4, and A5 in a store. Mr. Ali and Mr. Amir have to choose one of them for the construction of their houses. For this purpose they hire three contractors who give their feedback in connection with three attributes: B1: beauty, B2: long lasting, B3: strength. We have to analyze their future attitude due to their decisions. Step 1: The fuzzy soft sets representing the experts, opinions are given in Table 2. Step 2: Using definition 7 with l=m=1, BHFSS B5 × 3 is calculated as shown in Table 3. and score functions of respective entries of B5 × 3 are shown in Table 4. Calculation of the element of first row and first column of B is as follows:

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Step 3: Let W1 = [(0.4 0.3 0.3)] and W2=[(0.7 0.2 0.1)] be two weight vectors for Mr. Ali and Mr. Amir respectively conforming to their personal choices. Then using WBM with l = m = 1 for B1, B2, B3 in B'5×3, WBFSM is calculated for Mr. Ali as:

Note that WHFBM(A5) < WHFBM(A2 ) = WHFBM(A1) < WHFBM(A4) < WHFBM(A3), that is, 0.1456 < 0.2340 < 0.2424 < 0.1456 < 0.2736. Now, WBFSM for Mr. Amir is calculated as:

Note that, WHFBM(A5) < WHFBM(A1) < WHFBM(A2) = WHFBM(A3) < WHFBM(A4) that is, 0.1183 < 0.1865 < 0.1881 < 0.2121 < 0.2137. Therefore A3 is the best option for Mr. Ali and A4 is the best option for Mr. Amir.

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Step 4: Since both persons select different stores so they may not be satisfied with each other’s decision. Note that Step 5: Finally consider the following system of fuzzy soft differential equations;

Figure 1. Phase Portrait for P1 P2-plane and phase portrait for differential equations (2).

one of h1, h2, h3 has two elements, then BM(h1, h2, h3) consists of nine elements. It is difficult to handle nine elements for further computation. And if two of h1, h2, h3 have two elements, then BM(h1, h2, h3) consists of ninety elements. Then it is more difficult to handle ninety elements so we take the score function for those hesitant fuzzy elements for further computations. An attitude analyzer method through TOPSIS This method is composed of the following steps: Step 1: Decision advisors present their opinions in the form of hesitant fuzzy soft sets. Step 2: Find the collective decision set X by the aggregation formula defined in definition 9. Performance of alternative Ai with respect to attribute Bi is denoted as xij in an aggregated set X. Step 3: To find the positive ideal solution and negative ideal solution, we have to find respective induced fuzzy soft sets. Induced fuzzy soft set is found by computing score functions of all hesitant fuzzy elements using definition 6. The values in the matrices presented in step 1 having maximum or minimum values in respective induced fuzzy soft sets are positive ideal solutions (PIS) and negative ideal solutions (NIS) respectively. Step 4: Consider the weight vectors W1 and W2 for two persons according to their personal choices. Construct positive ideal separation matrix D+ and negative ideal separation matrix D- by the formulae

where distance formula is used as defined in definition 3. Step 5: Calculate the relative closeness (RC) of each alternative to the ideal solution as follows:

Figure 1 and Figure 2 show that Mr. Ali and Mr. Amir disagree with each others, feelings over the time t. Remark When BM is applied to h1, h2, h3 then BM(h1, h2, h3) consists of only one element if h1, h2, h3 all have one element. If

Step 6: Rank all the alternatives according to the closeness coefficients, the alternative having the greater value RC(Ai) is better.

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Step 7: Find optimal fuzzy soft constants from RC(Ai) and derive the system of linear fuzzy soft differential equations as defined (1) to study the feelings of people with respect to time.

An attitude analyzer method through VIKOR This method comprises the following steps: Step 1 - Step 4: Same as specified in the previous method (section 3.2). Step 5: Calculate Si, Ri and Qi i=1,2,...,m by the following formulae:

and v is the weight of the strategy of “the majority of criteria”, here v = 0.5 Step 6: An alternative x' is selected as a compromised solution if Q' is minimum and x' is graded the best if the following two conditions are satisfied: 1.

2. Alternative x' must also be the best ranked by S or/and R.

Step 7: Find OFSCs from Qi and develop the system of linear fuzzy soft differential equations. Example 4 Assume that there are five industries to invest the money namely; A1: textile industry, A2: food industries, A3: cement industry, A4: hotel and tourism, A5: Sugar industry. Mr. Ali and Mr. Amir have to choose the best one. They hire three experts D1, D2 and D3 for their valuable suggestions on the basis of four parameters: B1- future growth, B2- tax problems, B3- quality, B4- risk issues. Our objective is to analyze the attitude of two persons after some time due to their decisions which is carried through in three ways. Through TOPSIS Step 1: Three experts represent the alternatives with respect to the attributes in hesitant fuzzy soft set as shown in Tables 5-7. Step 2: Aggregate all the decision matrices into the collective decision matrix as shown in Table 8. Description of one of the elements of Table 8 is as follows:

Conflict Of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethics

There are no ethical issues with the publication of this manuscript.

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