Computer Modeling in Engineering & Sciences

DOI: 10.32604/cmes.2021.016758

ARTICLE

Weighted Parameterized Correlation Coefficients of Indeterminacy Fuzzy Multisets and Their Multicriteria Group Decision Making Method with Different Decision Risks

Cheng Du1 and Jun Ye2,*

1Beijing Urban Construction Group Co., Ltd., Beijing, 100088, China
2School of Civil and Environmental Engineering, Ningbo University, Ningbo, 315211, China
*Corresponding Author: Jun Ye. Email: yejun1@nbu.edu.cn
Received: 24 March 2021; Accepted: 08 May 2021

1  Introduction

Fuzzy sets (FSs) and fuzzy multisets (FMs) are vital mathematical tools in the expression and processing of fuzzy information since there are uncertainty and vagueness in many real-life problems. In the fuzzy theory, the FS presented by Zadeh [1] is depicted by a degree of membership, which usually contains almost one occurrence of each element in FS. Since then, fuzzy sets have wildly applied in various areas [2–9]. As an extension of FS, Yager [10] and Miyamoto [11] proposed the concept of fuzzy bag/FM, where each element in a universe set can be repeated more than once with the different and/or identical membership values. Then, FMs have been used for various applications in many areas [12–15]. Recently, El-Azab et al. [16] proposed a correlation coefficient of FMs as the extension of a correlation coefficient of FSs [17] and gave its application in the setting of FMs. Further, Das [18] put forward weighted fuzzy soft multiset (WFSM) and introduced an adjustable approach to the WFSMs-based decision-making method in uncertain environment. Furthermore, an interval-valued fuzzy multiset (IvFM) [19] was presented as the extension of FMs, where each element in a universe set can be repeated more than once with the different and/or identical membership values. In the hesitant fuzzy environment, then, a hesitant fuzzy set (HFS) and an interval-valued hesitant fuzzy set (IvHFS) [20] were represented by a group of different fuzzy values, but they cannot reflect the same fuzzy values in a group of hesitant fuzzy values. Under indeterminate and inconsistent environments, the neutrosophic multisets with the same or different neutrosophic components were proposed based on truth, falsity and indeterminacy membership functions and used for the applications in physical processes [21,22].

Due to the vagueness and indeterminacy of human cognition/judgments in complicated real problems, the human judgment may contain the hybrid information of the partial determinacy and partial indeterminacy. However, FMs and fuzzy soft multisets cannot represent the indeterminate fuzzy sequences, while a neutrosophic/indeterminate number (NN/IN) [23–25] is composed of its determinate term d and its indeterminate term uI with indeterminacy I ∈ [inf I, sup I] and denoted as z = d + uI for d, u ∈ ℜ. In indeterminate problems, since NN can be considered as a changeable interval number/value regarding a changeable range/value of I ∈ [inf I, sup I], it can flexibly and conveniently describe the hybrid information of the partial determinacy and partial indeterminacy, and also easily indicate a family of interval numbers (zi = [di + uiinf I, di + uisup I] for i = 1, 2, …, q) depending on a group of specified indeterminate ranges of I ∈ [inf I, sup I], which shows its highlighting advantage in the indeterminate information expression. Therefore, NNs have been wildly used in many areas, such as rock mechanics [26,27], decision making [28], fault diagnosis [29], and linear and nonlinear programming [30,31].

Since FMs, HFSs, IvFMs, and IvHFSs cannot express indeterminate fuzzy sequences by the indeterminate membership values in group decision making (GDM) problems with indeterminate information, various measure algorithms of FMs, HFSs, IvFMs, and IvHFSs cannot also deal with such GDM problems with the indeterminate fuzzy sequences and indeterminate decision risks of decision makers because indeterminate decision-making problems need to be solved by an indeterminate decision making method. If NNs/INs are introduced into fuzzy sequences in FMs and IvFMs, we can produce indeterminate/NN fuzzy sequences from the different and/or identical indeterminate membership values with some indeterminate ranges and construct an indeterminate fuzzy multiset (IFM) so as to present the parameterized correlation coefficients (PCCs) of IFMs by a parameterized de-neutrosophication method of IFM for handling indeterminate GDM problems with decision makers’ different decision risks. Motivated by these new ideas, this study proposes IFM, where each element in a universe set can be repeated more than once with the different and/or identical indeterminate membership values, and then introduces two PCCs of IFMs based on the de-neutrosophication of transforming IFMs into the parameterized fuzzy multisets (PFMs) by a parameter λ ∈ [0, 1] (i.e., the parameterized de-neutrosophication method) and their GDM approach with decision makers’ different decision risks (small, moderate and large risks for λ = 0, 0.5, 1) to solve indeterminate GDM problems in IFM setting.

However, main contributions of this study are summarized as follows:

1.    The proposed IFM contains various families of FMs, HFSs, IvFMs, and IvHFSs depending on the indeterminate ranges and values of I and provides an expression form of the different and/or identical indeterminate membership values that HFS and IvHFS cannot express.

2.    The proposed two PCCs of IFMs based on the parameterized de-neutrosophication method can provide effective mathematical models for solving indeterminate decision-making problems because the correlation coefficient of FMs cannot deal with indeterminate decision-making issues.

3.    The developed GDM method based on the two PCCs can solve indeterminate GDM problems, such as the selecting problem of slope design schemes (SDSs) for an open pit mine with decision makers’ different decision risks in the actual indeterminate application and show its highlighting advantages of the practicability and flexibility in IFM setting.

This paper is composed of the following parts. Section 2 introduces preliminaries of FMs. Section 3 proposes IFM and two PCCs of IFMs based on the de-neutrosophication of transforming IFMs into PFMs by a parameter λ ∈ [0, 1]. Section 4 develops a GDM method using PCCs of IFMs along with decision makers’ different decision risks (small, moderate and large risks for λ = 0, 0.5, 1) in IFM setting. In Section 5, the developed GDM method is used in a GDM example on a selection problem of SDSs for an open pit mine with decision makers’ different decision risks to demonstrate the usability and flexibility of the developed GDM method in IFM setting. Conclusions and further research are contained in Section 6.

2  Preliminaries of FMs

In FM/fuzzy bag presented by Yager [10], each element yj in a universe set Y = {y1, y2, …, ys} can be repeated more than once with the different and/or identical membership values. Then, a FM B is denoted as B={⟨yj,bB(yj)⟩|yj∈Y}, where bB(yj) is the non-increasing membership sequence defined by (bB1(yj),bB2(yj),…,bBqj(yj)) such as bB1(yj)≥bB2(yj)≥,…,≥bBqj(yj) for bBk(yj)∈[0,1] and yj ∈ Y (k = 1, 2, …, qj; j = 1, 2, …, s). For convenience, each component ⟨yj,bB(yj)⟩ in B is simply denoted as each fuzzy sequence bj=(bj1,bj2,…,bjqj).

Let two FMs be B1 = {b11, b12, …, b1s} and B2 = {b21, b22, …, b2s}, where b1j=(b1j1,b1j2,…,b1jqj) and b2j=(b2j1,b2j2,…,b2jqj) (j = 1, 2, …, s) are two groups of fuzzy sequences with the weight vector η = (η1, η2, …, ηs) regarding the importance of b1j and b2j. Based on the correlation coefficient of FMs [16], the weighted correlation coefficient of FMs B1 and B2 is introduced as follows:

Nw(B1,B2)=Rw(B1,B2)Rw(B1,B1)Rw(B2,B2)                          =∑j=1sηj{ b1j1b2j1+b1j2b2j2+…+b1jqjb2jqj }∑j=1sηj{ (b1j1)2+(b1j2)2+…+(b1jqj)2 }×ηj∑j=1s{ (b2j1)2+(b2j2)2+…+(b2jqj)2 } (1)

where Rw(B1,B2)=∑j=1sηj{b1j1b2j1+b1j2b2j2+…+b1jqjb2jqj} is the weighted correlation between B1 and B2 and Rw(B1,B1)=∑j=1sηj{(b1j1)2+(b1j2)2+…+(b1jqj)2} and Rw(B2,B2)=∑j=1sηj{(b2j1)2+(b2j2)2+…+(b2jqj)2} are the weighted informational energy of B1 and B2, respectively.

Based on the properties of the correlation coefficient between FMs [16], the weighted correlation coefficient Nw(B1, B2) should imply the following properties:

(i) Nw(B1, B2) = Nw(B2, B1);

(ii) Nw(B1, B2) = 1 iff B1 = B2;

(iii) Nw(B1, B2) ∈ [0, 1].

3  IFMs and Their PCCs

Based on the hybrid concept of FMs [16] and NNs [23–25], this section presents the concept of IFM, the relations of IFMs, and the de-neutrosophication approach of IFM with a parameter λ ∈ [0, 1], and then proposes two PCCs of IFMs.

Definition 3.1. Set Y = {y1, y2, …, ys} as a universe set. An IFM M over Y is presented by the following mathematical symbol:

Z={⟨yj,zZ(yj,I)⟩|yj∈Y,I∈[infI,supI]} (2)

where zZ(yj,I) is the non-increasing indeterminate membership sequence defined by zZ2(yj,I),…,zZqj(yj,I)), (zZ1(yj,I) with the condition zZk+1(yj,I)=djk+1+ujk+1I⊆zZk(yj,I)=djk+ujkI⊆[0,1] (k = 1, 2, …, qj; j = 1, 2, …, s) for I ∈ [inf I, sup I], djk, ujk ∈ ℜ and yj ∈ Y. In the IFM Z, each element yj in Y may occur more than once with the different and/or identical indeterminate membership values.

For instance, Z1={⟨y1,(0.8+2I,0.7+I)⟩,⟨y2,(0.7+2.5I,0.7+2.5I)⟩,⟨y3,(0.6+1.5I)⟩} for I ∈ [–0.1, 0.1] in the universe set Y = {y1, y2, y3} is an IFM.

Especially when qj = 1 (j = 1, 2, …, s) in Z, the IFM Z is reduced to the indeterminate FS. Then, FM/IvFM is a special case of IFM with a specified value I = inf I = sup I or a specified interval value I = [inf I, sup I] since IFM contains a family of IFMs/FMs corresponding to a group of specified indeterminate ranges/values of I.

Then, each component ⟨yj,(zZ1(yj,I),zZ2(yj,I),…,zZqj(yj,I))⟩ (j = 1, 2, …, s) in Z is simply denoted as zj(I)=(zj1(I),zj2(I),…,zjqj(I)), which is called an indeterminate/NN fuzzy sequence (NNFS). Clearly, the NNFS zj(I) is either a single-valued fuzzy sequence or an interval-valued fuzzy sequence depending on zjk(I)=djk+ujkI∈[0,1] or zjk(I)=[djk+ujkinfI,djk+ujksupI]⊆[0,1] (k = 1, 2, …, qj; j = 1, 2, …, s) in zj(I).

Definition 3.2. Set two NNFSs as z1j(I)=(z1j1(I),z1j2(I),…,z1jqj(I)) and z2j(I)=(z2j1(I),z2j2(I),…,z2jqj(I)) with NNs z1jk(I)=d1jk+u1jkI and z2jk(I)=d2jk+u2jkI for I ∈ [inf I, sup I] and d1jk,u1jk,d2jk,u2jk∈ℜ (k = 1, 2, …, qj; j = 1, 2, …, s). Then, their relations are defined below:

(1) z1j(I) ⊇ z2j(I) ⇔ z1jk(I)=d1jk+u1jkI⊇z2jk(I)=d2jk+u2jkI;

(2) z1j(I) = z2j(I) ⇔ z1j(I) ⊇ z2j(I) and z2j(I) ⊇ z1j(I);

(3) z1j(I)∪z2j(I)=(z1j1(I)∪z2j1(I),z1j2(I)∪z2j2(I),…,z1jqj(I)∪z2jqj(I));

(4) z1j(I)∩z2j(I)=(z1j1(I)∩z2j1(I),z1j2(I)∩z2j2(I),…,z1jqj(I)∩z2jqj(I));

(5) z1jC(I)=(1−z1jqj(I),1−z1jqj−1(I),…,1−z1j2(I),1−z1j1(I)) (the complement of z1j(I)).

For the de-neutrosophication of IFM, a parameter λ ∈ [0, 1] can be introduced to transform IFM into PFM with λ ∈ [0, 1] based on a parameterized de-neutrosophication method.

Definition 3.3. Let Z = {z1(I), z2(I), …, zs(I)} be an IFM, where zj(I)=(zj1(I),zj2(I),…,zjqj(I)) (j = 1, 2, …, s) for I ∈ [inf I, sup I] are NNFSs, and let λ ∈ [0, 1] be a parameter. Thus, PFM with λ ∈ [0, 1] is defined as Zλ = {z1(λ), z2(λ), …, zs(λ)}, where each parameterized NNFS with λ ∈ [0, 1] is represented as

zj(λ)=(zj1(λ),zj2(λ),…,zjqj(λ))=(dj1+uj1infI+λuj1(supI−infI),dj2+uj2infI+λuj2(supI−infI),…,djqj+ujqjinfI+λujqj(supI−infI)) (3)

Obviously, each zj(λ) (j = 1, 2, …, s) changes as the value of λ ∈ [0, 1] changes under a specifical range of I ∈ [inf I, sup I]. Thus, PFM with λ indicates a family of FMs corresponding to a group of values of λ ∈ [0, 1], while FM is only a special case of PFM for a specifical value of λ.

Based on the correlation coefficient of FMs [16], we propose two PCCs between two IFMs based on PFMs with λ ∈ [0, 1].

Definition 3.4. Let two IFMs be Z1 = {z11(I), z12(I), …, z1s(I)} and Z2 = {z21(I), z22(I), …, z2s(I)}, where z1j(I)=(z1j1(I),z1j2(I),…,z1jqj(I)) and z2j(I)=(z2j1(I),z2j2(I),…,z2jqj(I)) (j = 1, 2, …, s) for I ∈ [inf I, sup I] are two groups of NNFSs. Then the parameterized correlation of IFMs Z1 and Z2 based on PFMs with λ ∈ [0, 1] is proposed by:

Rλ(Z1,Z2)=∑j=1s{[d1j1+u1j1infI+λu1j1(supI−infI)][d2j1+u2j1infI+λu2j1(supI−infI)]+[d1j2+u1j2infI+λu1j2(supI−infI)][d2j2+u2j2infI+λu2j2(supI−infI)]+…+[d1jqj+u1jqjinfI+λu1jqj(supI−infI)][d2jqj+u2jqjinfI+λu2jqj(supI−infI)]} (4)

By applying Eq. (4), the parameterized correlations between Z1 and Z1 and between Z2 and Z2 are given as follows:

Rλ(Z1,Z1)=∑j=1s{[d1j1+u1j1infI+λu1j1(supI−infI)]2+[d1j2+u1j2infI+λu1j2(supI−infI)]2+…+[d1jqj+u1jqjinfI+λu1jqj(supI−infI)]2} (5)

Rλ(Z2,Z2)=∑j=1s{[d2j1+u2j1infI+λu2j1(supI−infI)]2+[d2j2+u2j2infI+λu2j2(supI−infI)]2+…+[d2jqj+u2jqjinfI+λu2jqj(supI−infI)]2} (6)

which are also named the parameterized informational energy of IFMs Z1 and Z2.

Consequently, the two PCCs of IFMs Z1 and Z2 based on PFMs with λ ∈ [0, 1] are proposed by

N1λ(Z1,Z2)=Rλ(Z1,Z2)Rλ(Z1,Z1)Rλ(Z2,Z2)=∑j=1s{[d1j1+u1j1infI+λu1j1(supI−infI)][d2j1+u2j1infI+λu2j1(supI−infI)]+[d1j2+u1j2infI+λu1j2(supI−infI)][d2j2+u2j2infI+λu2j2(supI−infI)]+…+[d1jqj+u1jqjinfI+λu1jqj(supI−infI)][d2jqj+u2jqjinfI+λu2jqj(supI−infI)]}∑j=1s{[d1j1+u1j1infI+λu1j1(supI−infI)]2+[d1j2+u1j2infI+λu1j2(supI−infI)]2+…+[d1jqj+u1jqjinfI+λu1jqj(supI−infI)]2}×∑j=1s{[d2j1+u2j1infI+λu2j1(supI−infI)]2+[d2j2+u2j2infI+λu2j2(supI−infI)]2+…+[d2jqj+u2jqjinfI+λu2jqj(supI−infI)]2} (7)

N2λ(Z1,Z2)=Rλ(Z1,Z2)max{Rλ(Z1,Z1),Rλ(Z2,Z2)}=∑j=1s{[d1j1+u1j1infI+λu1j1(supI−infI)][d2j1+u2j1infI+λu2j1(supI−infI)]+[d1j2+u1j2infI+λu1j2(supI−infI)][d2j2+u2j2infI+λu2j2(supI−infI)]+…+[d1jqj+u1jqjinfI+λu1jqj(supI−infI)][d2jqj+u2jqjinfI+λu2jqj(supI−infI)]}max{∑j=1s{[d1j1+u1j1infI+λu1j1(supI−infI)]2+[d1j2+u1j2infI+λu1j2(supI−infI)]2+…+[d1jqj+u1jqjinfI+λu1jqj(supI−infI)]2},∑j=1s{[d2j1+u2j1infI+λu2j1(supI−infI)]2+[d2j2+u2j2infI+λu2j2(supI−infI)]2+…+[d2jqj+u2jqjinfI+λu2jqj(supI−infI)]2}} (8)

Regarding the properties of the correlation coefficient of FMs [16], the PCCs N1λ(Z1,Z2) and N2λ(Z1,Z2) for λ ∈ [0, 1] also imply the following properties:

(i) N1λ(Z1,Z2)=N1λ(Z2,Z1) and N2λ(Z1,Z2)=N2λ(Z2,Z1);

(ii) N1λ(Z1,Z2)=N2λ(Z1,Z2)=1 iff Z1 = Z2;

(iii) 0≤N1λ(Z1,Z2),N2λ(Z1,Z2)≤1.

Proof: Obviously, the properties (i) and (ii) are straightforward. Hence, we only verify the property (iii).

According to the Cauchy-Schwarz inequality (∑j=1s(ajbj))2≤∑j=1saj2∑j=1sbj2, there exists ∑j=1s(ajbj)≤∑j=1saj2∑j=1sbj2. Thus, we can give the following inequality for λ ∈ [0, 1]:

Hence, it is obvious that there also exists the following inequality for λ ∈ [0, 1]:

Thus, 0≤N1λ(Z1,Z2),N2λ(Z1,Z2)≤1 can hold according to the above inequalities.

Therefore, this proof is completed.

When we consider the importance of each NNFS zkj (k = 1, 2; j = 1, 2, …, s) in Z1 and Z2, we specify the weight ηj ∈ [0, 1] of zkj with ∑j=1sηj=1. Thus, the weighted PCCs of IFMs Z1 and Z2 with λ ∈ [0, 1] can be presented by the following formulae:

Nw1λ(Z1,Z2)=∑j=1sηj{[d1j1+u1j1infI+λu1j1(supI−infI)][d2j1+u2j1infI+λu2j1(supI−infI)]+[d1j2+u1j2infI+λu1j2(supI−infI)][d2j2+u2j2infI+λu2j2(supI−infI)]+…+[d1jqj+u1jqjinfI+λu1jqj(supI−infI)][d2jqj+u2jqjinfI+λu2jqj(supI−infI)]}∑j=1sηj{[d1j1+u1j1infI+λu1j1(supI−infI)]2+[d1j2+u1j2infI+λu1j2(supI−infI)]2+…+[d1jqj+u1jqjinfI+λu1jqj(supI−infI)]2}×∑j=1sηj{[d2j1+u2j1infI+λu2j1(supI−infI)]2+[d2j2+u2j2infI+λu2j2(supI−infI)]2+…+[d2jqj+u2jqjinfI+λu2jqj(supI−infI)]2} (9)

Nw2λ(Z1,Z2)=∑j=1sηj{[d1j1+u1j1infI+λu1j1(supI−infI)][d2j1+u2j1infI+λu2j1(supI−infI)]+[d1j2+u1j2infI+λu1j2(supI−infI)][d2j2+u2j2infI+λu2j2(supI−infI)]+…+[d1jqj+u1jqjinfI+λu1jqj(supI−infI)][d2jqj+u2jqjinfI+λu2jqj(supI−infI)]}max{∑j=1sηj{[d1j1+u1j1infI+λu1j1(supI−infI)]2+[d1j2+u1j2infI+λu1j2(supI−infI)]2+…+[d1jqj+u1jqjinfI+λu1jqj(supI−infI)]2},∑j=1sηj{[d2j1+u2j1infI+λu2j1(supI−infI)]2+[d2j2+u2j2infI+λu2j2(supI−infI)]2+…+[d2jqj+u2jqjinfI+λu2jqj(supI−infI)]2}} (10)

Similarly, the weighted PCCs Nw1λ(Z1,Z2) and Nw2λ(Z1,Z2) with λ ∈ [0, 1] also imply the following properties:

(i) Nw1λ(Z1,Z2)=Nw1λ(Z2,Z1) and Nw2λ(Z1,Z2)=Nw2λ(Z2,Z1);

(ii) Nw1λ(Z1,Z2)=Nw2λ(Z1,Z2)=1 iff Z1 = Z2;

(iii) 0≤Nw1λ(Z1,Z2),Nw2λ(Z1,Z2)≤1.

By the similar proof way regarding the properties of the two PCCs, we can easily verify the above properties (i)–(iii).

4  GDM Approach Based on the Weighted PCCs in IFM Setting

Corresponding to the weighted PCCs of IFMs, we develop a GDM approach with decision makers’ different decision risks in IFM setting.

Regarding multicriteria GDM problems in indeterminate setting, suppose experts/designers preliminarily specify a set of p potential alternatives G = {g1, g2, …, gp}, which is evaluated by a set of s criteria C = {c1, c2, …, cs}. The weight vector of C is given as η = (η1, η2, …, ηs) to consider the importance of different criteria. Then, a group of q decision makers is invited to give the satisfactory assessments of an alternative gi (i = 1, 2, …, p) over criteria cj (j = 1, 2, …, s), which are expressed by the NNFSs zij(I)=(zij1(I),zij2(I),…,zjq(I))=(dij1+uij1I,dij2+uij2I,…,dijq+uijqI), (i = 1, 2, …, p; j = 1, 2, …, s) for zijk(I)=dijk+uijkI⊆[0,1] and I ∈ [inf I, sup I]. Thus, all NNFSs are constructed as their decision matrix Z=(zij(I))p×s in IFM setting.