International Journal of Computational Intelligence Systems

Volume 12, Issue 1, November 2018, Pages 172 - 182

Multi-adjoint based group decision-making under an intuitionistic fuzzy information system

Authors
Meishe Liang1, 2, liangmeishe@163.com, Jusheng Mi1, mijsh@263.net, Tao Feng3, fengtao_new@163.com, Bin Xie4, xiebin_hebtu@126.com
1College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, 050024, P.R.China
2Department of Scientific Development and School-Business Cooperation, Shijiazhuang University of Applied Technology, Shijiazhuang, 050081, P.R.China
3College of Science, Hebei University of Science and Technology, Shijiazhuang, 050018, P.R.China
4Information and Technology College, Hebei Normal University, Shijiazhuang, 050024, P.R.China
Received 10 July 2018, Accepted 28 October 2018, Available Online 12 November 2018.
DOI
https://doi.org/10.2991/ijcis.12.1.172How to use a DOI?
Keywords
Multi-criteria Group Decision Making; Group Decision Making; Intuitionistic Fuzzy Sets; Evidence Theory
Abstract

The construction of belief intervals is crucial for decision-making in multi-attribute group information integration. Based on multi-adjoint and evidence theory, an approach to multi-criteria group decision-making(MCGDM) in intuitionistic fuzzy information system is proposed. First, the upper and lower approximations of alternatives are calculated by multi-adjoint operators under the correlation matrices which were given by different experts. After that the belief and plausibility functions are gained by intuitionistic fuzzy probability formulas. Second, the belief intervals of alternatives are acquired by combining all experts’ evidence. Then the alternatives are ranked by comparing the belief intervals. Finally, the effectiveness of the method is verified by an application of business transaction. Compared with the existing model, the method introduced in this paper is more effective and accurate.

Copyright
© 2018, the Authors. Published by Atlantis Press.
Open Access
This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

1. Introduction

Decision-making(DM), which is an important form of human thinking and cognitive activity, widely exists in every field of human social life. Multi-criteria group decision making (MCGDM) 1,2 is to find the optimal object by ranking a given set of alternatives according to a group of decision makers. During the process of MCGDM, firstly, preference of alternatives is provided by each decision maker according to the values of multi-criteria; secondly, all preference relations are comprehensively considered and a satisfactory solution is presented for the decision makers. Fuzzy decision is the most common research field of MCGDM.

Fuzzy set theory was initiated by Zadeh 3 in 1965. After that, Atanassov 4 proposed intuitionistic fuzzy set (IFS) theory. The membership and the non-membership are used to describe the relationship between the objects and attributes, which makes the hesitant degree to be clearly described. IFSs depict more exquisite, and the membership degree, the non-membership degree and the hesitation degree are implied in the decision-making behavior to accept, reject, and hesitation. Following this, many scholars applied the theory to the area of decision making field 5,6, and propose many different intuitionistic fuzzy decision-making models 7,8,9.

A large body of pioneering works has focused on the problems of DM based on fuzzy set theory, which has been mentioned in the literature 10,11. These studies are mainly focused on two aspects. The first is the construction of aggregation operators which combined the information carried by attributes together. Considering the relation among attributes, Yager 12 proposed a power average operator. Zhou et al 13 extended the power average operator. After that, Zhou and Chen 14, Xu and Wang 15 applied power average operator to deal with linguistic information. Other related works on the construction of aggregation operators have been reported in 16,17. The second is the weights of decision makers, since different decision maker play different role in the decision group. Xu and Cai 18 determined the weights of experts according to nonlinear optimization models. Wan 19 established a model to calculate weighting vector on account of the similarity degree matrix of decision makers’ judgement. Liang and Wei 20 introduced a new similarity measure by the relationship between entropy and similarity measure to derive the relative importance weight of experts. More recently, Ren and Wei 21 developed a correctional score function of dual hesitant fuzzy elements to solve multi-attribute decision-making problems in which the attributes were in different priority levels. Considering the attribute values are hybrid types, Jin 22 introduced certitude structure to tackle with such MAGDM problems. Zhu 23 defined t-norms and t-conorms products to construct ANDf p f s decision making and ORf p f s decision making methods. Zhang and Guo 24 introduced some formulaes to compute priority weights which helped decision makings to express their preference flexibly. Khalid and Beg 25 mainly focused on incomplete hesitant fuzzy preference relations in group decision making. By defining a hesitant upper bound condition, this model promises to estimate missing information that is expressible and without voiding the originality of information provided by the expert. Based on the I2LGA operator, Liu and Chen 26 proposed a new method for multi-attribute group decision making.

These methods mentioned above are important for the study of MCGDM, and have been used in many organizational decisions. Despite these achievement, there are still some certain decision problems which are not taken into consideration yet. In many real decision making of business transactions, for example, a man want to buy a house or an CEO want to choose an optimal project for his(her) company. There is only one decision maker for this MD problem. In fact, it is difficult for decision maker to rank all objects and make a clear decision after the attributes value of the decision objects are known. There are two reasons. Firstly, the decision maker may have a poor understanding of attributes and is not clear about the intrinsic relationship between them. Secondly, it is difficult to get any reasonable and reliable combination results by all attributes. Thus, the man may seek help from the property consultants and the CEO may consult the board of directors. We denote the property consultants and the board of directors as consulting experts. The problem can be described as following:

Suppose that there are several objects O = {o1,o2, ⋯ ,on}. Let AT = {a1,a2, ⋯ ,am} is a set of attributes or decision parameters. The IFV is applied to describe the value of an object with an attribute as shown in Table 1. The decision maker wants to rank the objects completely and finds the optimal object by AT.

Objects a1 a2 am
o1 µ11,ν11 µ12,ν12 µ1m,ν1m
o2 µ21,ν21 µ22,ν22 µ2m,ν2m
on µm1,νm1 µm2,νm2 µmm,νmm
Table 1.

The IF information system.

Zhao 27 proposed an approach of multi-criteria ranking based on intuitionistic fuzzy soft set to deal with decision making problems above. In this paper, several principles were proposed for solving the MCDM completely ranking problem and also getting the only one optimal selection. However, the results obtained by this method may be contrary to our intuitive judgment.

To overcome this shortcoming, A group of consulting experts is introduced to provide advice for decision maker. Because of the work experience, the professional field are different, each expert may provide different viewpoint in multi-criteria which is described by the intuitionistic fuzzy relation matrix Dk. And we propose an approach to group decision-making in intuitionistic fuzzy information system. By consulting some experts, and using adjoint triple and evidence theory, decision maker can rank all objects and make a clear decision based on individual preferences. The contribution of this paper are listed as follows:

  1. (1)

    The multi-adjoint theory and D-S theory are applied to intuitionistic fuzzy group decision, which can effectively enhance the ability of the model to deal with intuitionistic fuzzy decision-making problem.

  2. (2)

    From the intuitionistic fuzzy relation matrix Dk, the weight of each attribute is obtained according to the kth expert, which effectively avoid the error caused by a given weight.

The remainder of this paper is structured as follows. In Section 2, we recall preliminaries from intuitionistic fuzzy sets, and D-S theory. In Section 3, we introduce the multi-adjoint intuitionistic fuzzy rough sets, after that the believe function and plausibility function based on intuitionistic fuzzy probability are calculated. Next, in Section 4, we present intuitionistic fuzzy MCGDM method. An application of business transaction on housing purchase for using the proposed method is demonstrated, and we also compare this model with other methods. Finally, we conclude the paper with a summary and give an outlook for further research.

2. Preliminaries

As a generalization of fuzzy set, since considering the support, oppose and hesitation of the three aspects of information, IFS is more flexible and practical in dealing with ambiguity and uncertainty information systems. In this section, we briefly introduce the basic notions and definitions of IFSs theory and D-S theory.

2.1. IFSs and IF approximation space

Definition 1. (Atanassov 1986) 4

Let U be a nonempty objects set. An IFS A has the form A = {〈µA(x), νA(x)〉 : xU}, where the mappings µA : U → [0,1] and νA : U → [0,1] denote the degree of membership and nonmembership of each element xU to the set A (namely µA(x) and νA(x)) respectively, and 0 ≤ µA(x) + νA(x) ≤ 1 for each xU. 1 − µA(x) − νA(x) can be interpreted as the hesitancy degree of x to A.

The family of all IFS subsets of U is denoted by IFS(U). We call A(x) = 〈µA(x), νA(x)〉 an intuitionistic fuzzy value(IFV). Especially, for any AP(U), if xA, then A(x) = 〈1,0〉, if xA, then A(x) = 〈0,1〉. Let BelA(x) = µA(x), PlA(x) = 1 − νA(x), then the intuitionistic fuzzy set A can also be written as A = {BIA(x) : xU}, in which BIA(x) = [BelA(x),PlA(x)] expresses the belief interval of x to A.

Distance measure is an important uncertainty measurements in IFS theory.

Definition 2.

Let U be a finite and nonempty set, A,BIFS(U), the distance between A and B is defined as D(A,B)=1|U|xU[|μA(x)μB(x)|+|νA(x)νB(x)|4+max{|μA(x)μB(x)|,|νA(x)νB(x)|}2].

The intuitionistic fuzzy binary relation is an intuitionistic subset on U × U. IR is defined as IR = {〈µIR(x,y), νIR(x,y)〉 : (x,y) ∈ U × U}, where µIR : U × U → [0,1], νIR : U × U → [0,1], and satisfied the condition 0 ≤ µIR(x,y) + νIR(x,y) ≤ 1, ∀(x,y) ∈ U × U. We denote all intuitionistic fuzzy binary relation sets on U as IFR(U × U). For IRIFR(U × U), if IR(x,x) = 〈1,0〉 is satisfied for all xU, then IR is reflexive; if IR(x,y) = IR(x,y) is satisfied for all (x,y) ∈ U × U, then IR is symmetric.

Definition 3. (Wu & Zhou 2008) 28

Let U be a finite and nonempty set, IR is an intuitionistic fuzzy binary relation on U × U. The ordered pair (U,IR) is called intuitionistic fuzzy approximation space.

2.2. D-S theory

Question assumptions are all possible outcomes which can be discerned by people on a subject. If these assumptions are mutually exclusive and complete with all possible descriptions of the problem, then these assumption collections Θ are called the recognition framework. The power set of Θ is denoted by 2Θ.

Definition 4. (Shafer 1976) 29

A set function m : 2Θ → [0,1] is referred to as a basic probability assignment or mass distribution, if it satisfies the following axioms:

  1. (1)

    m(∅) = 0

  2. (2)

    A⊆Θm(A) = 1.

If m(A) ≠ 0, then subset A is called the focal element of m. The value of m(A) represents the degree of belief that a specific element of Θ belongs to A. All focal elements of m is denoted as M, the pair (M,m) is called a belief structure on Θ.

Definition 5. (Shafer 1976) 29

Let m : 2Θ → [0,1] be a basic belief assignment function. ∀X ⊆ Θ, the belief function (Bel : m → [0,1]) and plausibility function (Pl : m → [0,1]) of X are respectively defined as:

Bel(X)=YXm(Y);Pl(X)=YXm(Y).

The belief interval of X is defined as Bl(X) = [Bel(X),Pl(X)].

X,Y ⊆ Θ, The belief intervals of X,Y are Bl(X) = [Bel(X),Pl(X)] and Bl(Y) = [Bel(Y),Pl(Y)] respectively. If Bel(X) > Bel(Y), then Bl(X) > Bl(Y), denoted as X > Y. If Bel(X) = Bel(Y) and Pl(X) > Pl(Y), then Bl(X) > Bl(Y), denoted as X > Y. If Bel(X) = Bel(Y) and Pl(X) = Pl(Y), then Bl(X) = Bl(Y), denoted as X = Y.

Definition 6. (Shafer 1976) 29

Let Bel1,Bel2, ⋯ ,Beln be belief functions of Θ. Then combined evidence can be calculated by orthogonal sum m = m1m2 ⊕ ⋯ ⊕ mn for fusing independent information sources mi. The orthogonal sum is associative and commutative; it is defined in Dempster’ s rule of combination:

m(A)=A1A2An=Am1(A1)m2(A2)mn(An)1N,
in which
N=A1A2An=m1(A1)m2(A2)mn(An).

3. Multi-adjoint IFRSs and the belief interval

In this section, construction of the belief interval has been proposed in intuitionistic fuzzy information system. First, multi-adjoint theory has been introduced in intuitionistic fuzzy information system, the lower and the upper approximations of each object according to expert are obtained. Second, the belief function and plausibility function are calculated by intuitionistic fuzzy probability for each object.

3.1. Multi-adjoint IFRSs

Now we mainly introduce the definition of multi-adjoint intuitionistic fuzzy rough sets (short for Multi-adjoint IFRSs) in this subsection.

Definition 7. (Eugenia, Medina & Ramirez 2013)30

Let (P1, ≤1), (P2, ≤2), (P3, ≤3) be three posets. The pair (&, ↙,↖) is called an adjoint triple with respect to P1,P2,P3, if the three mappings & : P1 × P2P3, ↙: P3 × P2P1, ↖: P3 × P1P2 satisfy,

  • x1 zy iff x&y3 z iff y2 zx, for any xP1,yP2,zP3;

  • & is order-preserving on both arguments;

  • ↙,↖ are order-preserving on the first argumentand order-reserving on the second argument.

There are some special examples of adjoint triples:

  • Gödel adjoint triple:

    x&Gy=min{x,y},zGx={1,ifxzz,otherwise;

  • Product adjoint triple:

    x&Py=x·y,zGx=min{1,z/x};

  • Łukasiewicz adjoint triple:

    x&Ly=max{0,x+y1},zLx=min{1,1x+z}.

The above adjoint pairs formed by a t-norm and its residuated implications, which can be seen as degenerate examples of general adjoint triples, and in this three adjoint pairs ↖G=↙G, ↖P=↙P, ↖L=↙L, since &G, &P, &L are commutative.

Definition 8.

Given an intuitionistic fuzzy approximation space (U,IR), ∀AIFS(U), the lower and the upper approximations of A are defined respectively as

IR¯(A)={μIR¯(A)(x),νIR¯(A)(x):xU};
IR¯(A)={μIR¯(A)(x),νIR¯(A)(x):xU},
where
μIR¯(A)(x)=inf{μA(y)xyμIR(x,y):yU},νIR¯(A)(x)=sup{(1νIR(x,y))&xyνA(y):yU};μIR¯(A)(x)=sup{μIR(x,y)&xyμA(y):yU},νIR¯(A)(x)=inf{νA(y)xy(1νIR(x,y)):yU}.

Definition 9.

For all AIFS(U), the pair (IR¯(A),IR¯(A)) is called a multi-adjoint intuitionistic fuzzy rough set of A with respect to IR.

An important feature of the presented framework is that the user may represent explicit preferences among the objects and acquire different upper and lower approximations in an intuitionistic fuzzy decision system, by associating different family adjoint triples. In next section, the probability of approximations will be calculated to construct the belief interval which will be used in the decision system.

3.2. The belief function and plausibility function based on IF probability

According to literature (Feng, Zhang & Mi) 31, a novel probability of an A-IF set is defined by using the (α,β) level set. And now we just show the conclusion as the following definition will be used later.

Definition 10. (Feng, Zhang & Mi) 31

Let U be a nonempty finite set, (U,P(U),P) is a probability space. ∀AIFS(U) the probability measure P* of A is defined as

P*(A)=xU((1νA(x))2(1μA(x)νA(x))2)P(x).

It is obviously that P*(∅) = 0P*(U) = 1 .

If (U,P(U),P) is a probability space, IRIFR(U × U) is a reflexive and symmetric intuitionistic fuzzy binary relation, (U,IR) is an intuitionistic fuzzy approximate space mentioned in Definition 3. Let IR(x,y) = IRS(x)(y), IRS(x) is an intuitionistic fuzzy set on U.

Definition 11. (Feng, Zhang & Mi) 31

Let U be a nonempty finite set, a set function m : IFS(U) → [0,1] is referred to as a basic probability assignment (also called mass function) if it satisfies axioms (M1) and (M2)

  • (M1) m(∅) = 0;

  • (M2) ∑AIFS(U) m(A) = 1.

For AIFS(U) with m(A) ≠ 0 is referred to as a focal element of m. We denote all focal elements of m by M. The pair (M,m) is called an IF belief structure.

Theorem 1. (Feng, Zhang & Mi) 31

Let U = {x1,x2, ⋯ ,xn} be a nonempty and finite universe of discourse, (U,IR) a reflective and symmetric intuitionistic fuzzy approximation space.AIFS(U)we define

mIR(A)={x:A=IRS(x)P*(1x),ifAG;0,otherwise.

Then mIR is a mass function.

The probability measures of IR¯(A) and IR¯(A) have the following expressions.

Definition 12.

Suppose U be a nonempty and finite universe of discourse, (U,IR) a reflective and symmetric intuitionistic fuzzy approximation space. P is a probability measure on U, ∀AIFS(U),

BelIR(A)=P*(IR¯(A));
PlIR(A)=P*(IR¯(A)).

Therefore, we have the following results.

Theorem 2.

Suppose U be a nonempty and finite universe of discourse, (U,IR) a reflective and symmetric intuitionistic fuzzy approximation space. P is a probability measure on U,AIFS(U),

BelIR(A)=XGmIR(X)((1νIR¯(XA)(x))2(1μIR¯(XA)(x)νIR¯(XA)(x))2)PlIR(A)=XGmIR(X)((1νIR¯(XA)(x))2(1μIR¯(XA)(x)νIR¯(XA)(x))2).
where
μIR¯(XA)(x)=inf{μA(y)xyμX(y):yU},νIR¯(XA)(x)=sup{(1νX(y))&xyνA(y):yU};μIR¯(XA)(x)=sup{μX(y)&xyμA(y):yU},νIR¯(XA)(x)=inf{νA(y)xy(1νX(y)):yU}.

Proof.

BelIR(A)=P*(IR¯(A))=xUP(x)((1νIR¯(A)(x))2(1μIR¯(A)(x)νIR¯(A)(x))2)=RS(x)G(y:RS(y)=RS(x)P(1y))((1νIR¯(A)(x))2(1μIR¯(A)(x)νIR¯(A)(x))2)=XGmIR(X)((1sup{(1νX(y))&xyνA(y):yU})2(1inf{μA(y)xyμX(y):yU}sup{(1νX(y))&xyνA(y):yU})2)=XGmIR(X)((1νIR¯(XA)(x))2(1μIR¯(XA)(x)νIR¯(XA)(x))2)..

Similarly, we get PlIR(A)=P*(IR¯(A))=xUP(x)((1νIR¯(A)(x))2(1μIR¯(A)(x)νIR¯(A)(x))2)=RS(x)G(y:RS(y)=RS(x)P(1y))((1νIR¯(A)(x))2(1μIR¯(A)(x)νIR¯(A)(x))2)=XGmIR(X)((1inf{νA(y)xy(1νX(y)):yU})2(1sup{μX(y)&xyμA(y):yU}inf{νA(y)xy(1νX(y)):yU})2)=XGmIR(X)((1νIR¯(XA)(x))2(1μIR¯(XA)(x)νIR¯(XA)(x))2)..

Theorem 3.

Suppose U be a nonempty and finite universe of discourse, IRIFR(U × U) a reflective and symmetric intuitionistic fuzzy binary relation, the following properties hold:

  1. 1.

    BelIR(∅) = PlIR(∅) = 0, BelIR(U) = PlIR(U) = 1;

  2. 2.

    BelIR(A) ≤ P*(A) ≤ PlIR(A),∀AIFS(U);

  3. 3.

    BelIR(A) ≤ BelIR(B),PlIR(A) ≤ PlIR(B), ∀A,BIFS(U),AB.

Proof.

  1. (1)

    According to Definition 8 and 10, we have BelIR(∅) = PlIR(∅) = P*(∅) = 0, BelIR(U) = PlIR(U) = P*(U) = 0;

  2. (2)

    BelIR(A)=P*(IR¯(A))P*(A)P*(IR¯(A))=PlIR(A)

  3. (3)

    A,BIFS(U), if AB, then BelIR(A)=P*(IR¯(A))P*(IR¯(B))=BelIR(B). Similarly, we have PlIR(A) ≤ PlIR(B).

It is easy to verify that BelIR and PlIR degenerate into the crisp belief and plausibility functions when the belief structure (M,m) and A are crisp. Thus BelIR and PlIR are measures of belief and plausibility functions on IFS(U). Convert belief function and plausibility function into Mass functions, we have:

m(yes)=BelIR,m(no)=1PlIR,m(yes,no)=PlIRBelIR.

4. The construction of multi-adjoint IF group decision-making method

To make our model easier to understand and realize, we suppose that O = {o1,o2, ⋯ ,on} is a set of decision objects. A man wants to choose the best one from O. There are m attributes to take into consideration, the attribute set AT = {a1,a2, ⋯ ,am}. For each object ol,ln, there is an intuitionistic fuzzy set of oi on AT denoted as Al = {〈µj(ol),νj(ol)〉 : j = 1,2, ⋯ ,m}. There is an advisory group consisting of K experts Mk(k = 1,2, ⋯ ,K) each expert gets the intuitionistic fuzzy relation matrix Dk according to the attributes.

(μ11k,ν11kμ12k,ν12kμ1mk,ν1mkμ21k,ν21kμ22k,ν22kμ2mk,ν2mkμm1k,νm1kμm2k,νm2kμmmk,νmmk)

Where μijk, νijk denote the degree to which the expert k considers the attribute i to be relevant and irrelevant with the attribute jand 0μijk+νijk1, for ∀i, j ∈ {1,2, ⋯m}, k ∈ {1,2, ⋯K}. satisfies the properties

μiik,νiik=1,0,forim;μijk,νijk=μjik,νjik,fori,jm.

It is clear that (AT,Dk) is an intuitionistic fuzzy approximation space. Before group decision-making information integration, a hypothesis must be proposed firstly.

4.1. Group decision-making information integration

Before group decision-making information integration, a hypothesis must be proposed firstly.

Assume

The sources of information in decision-making system are mostly reliable. That means the intuitionistic fuzzy relation matrices given by most experts are reliable. Otherwise, the system has lapsed and the group decision-making information integration is meaningless.

Let yes express supportive of the expert Mk to a decision object; No indicate appositive of expert Mk to a decision objects; (yes, no) mean cannot judge (or hesitate).

  • Step 1 From Definition 8, the lower and upper approximations according to expert are obtained. They are Dk¯(ol) and Dk¯(ol), which indicate the degrees of certainty and probability of the expert Mk with regard to decision object ol on each attribute. The calculation is as follows

    μ¯ik(ol)=inf{μj(ol)ijμijk:jm};ν¯ik(ol)=sup{(1νijk)&ijνj(ol):jm};μ¯ik(ol)=sup{μijk&ijμj(ol):jm};ν¯ik(ol)=inf{vj(ol)ij(1νijk):jm}.

  • Step 2 From Definition 12, the belief function BelDk(ol)=P*(Dk¯(ol)) and plausibility function PlDk(ol)=P*(Dk¯(ol)) of the expert Mk with regard to decision object ol are calculated. The weight of ai in Dk can be used as the probability of the attribute ai under expert Mk. The calculation is as follows

    Let Dk be the intuitionistic fuzzy relation matrix given by expert Mk. μijk, νijk denote the degree to which the expert k considers the attribute i to be relevant and irrelevant with the attribute j. Then the function sijk=μijkνijk is denoted as the relevant score of the attribute i to the attribute j by expert k.

    The weight of ai under expert Mk is

    Pk(ai)=j=1msiji=1m(j=1msij).

    Pk(ai) reflects the degree to which attribute ai is recognized by expert Mk. The higher the correlation with other attributes, the more important of attribute ai, and the bigger weight of ai. Therefore, the weights of attributes based on correlation are reasonable.

  • Step 3 Convert the belief function and the plausibility function to the Mass function. Let

    mlk(yes)=BelDk(ol),mlk(no)=1PlDk(ol),mlk(yes,no)=PlDk(ol)BelDk(ol).

  • Step 4 Due to the interference of different factors, some experts give evidence that may be larger in conflict with other experts, and the rule of evidence combination does not recognize strong evidence nor the weak evidence. To get a reasonable and reliable evidence combination results, it is necessary to analyze the conflict of evidence before combination. We use distance measure to modify the evidence, the bigger the distance, the smaller the degree of support and the bigger the discount factor. According to definition 2, the distance between any two evidences can be calculated. Let Δ = (ϑij)K × K be the distance matrix of objects, discount factor of evidences is defined as:

    ωk=1k=1Kϑij1min{k=1Kϑij:i=1,2K}.

    The revised evidence indicates that:

    ml*k(yes)=ωkmlk(yes),ml*k(no)=ωkmlk(no),ml*k(yes,no)=1ml*k(yes)ml*k(no).

  • Step 5 Combine all experts evidence for each decision object. The calculation is as follow

    ml(A)=A1A2AKml*1(A1)ml*2(A2)ml*K(AK)1N.
    in which N=A1A2AKml*1(A1)ml*2(A2)ml*K(AK), and A,A1,A2, ⋯ ,AK ∈ {yes,no,(yes,no)}.

  • Step 6 Compares the belief interval of each decision object and makes the optimal decision.

4.2. Instance analyses on business transaction

Many business transactions 32 are made through evaluation a set of potential trading alternatives. In such a decision making problem, decision maker needs to rank the potential alternatives or select an optimal one based on the evaluation information associated a set of criteria. Before decision making, a group of consultant experts is asked to provide their advise.

As one of the basic condition of living and an important component part of household property, housing has been highly valued by Chinese people. Many families even take out two or three generations of savings to buy a house. Therefore, how to buy a satisfactory house is very important to an individual or a family. In this process, the decision maker first collects all kinds of information about the purchase of houses such as the quality of the property service, location etc. If the advantages and disadvantages of the alternative houses are obvious, the decision maker can make a direct choice. Otherwise, he needs to consult property consultants, and then make decisions based on his own needs. Now we verify the validity of the model with the examples of housing transaction.

Example 1

Suppose O = {o1,o2,o3,o4,o5,o6} represent six alternative houses for sale, AT = {a1,a2,a3,a4} be a set of attributes such as the location of the house, the quality of the property service, the building structure and the average price. The intuitionistic fuzzy sets of the six houses according to AT are shown in Table 2.

Objects a1 a2 a3 am
o1 〈0.80,0.20〉 〈0.78,0.18〉 〈0.69,0.28〉 〈0.78,0.12〉
o2 〈0.76,0.23〉 〈0.78,0.20〉 〈0.70,0.28〉 〈0.75,0.18〉
o3 〈0.80,0.10〉 〈0.60,0.24〉 〈0.75,0.20〉 〈0.70,0.25〉
o4 〈0.75,0.20〉 〈0.82,0.15〉 〈0.55,0.28〉 〈0.80,0.18〉
o5 〈0.65,0.28〉 〈0.74,0.20〉 〈0.70,0.15〉 〈0.75,0.20〉
o6 〈0.80,0.10〉 〈0.60,0.24〉 〈0.75,0.15〉 〈0.70,0.25〉
Table 2.

The IFSs of the houses according to AT.

P = {M1,M2,M3,M4} is a advisory group consists of four property consultants. The four members evaluate the attributes with their personal experience and information. Their evaluate intuitionistic fuzzy relation matrix about attribute set AT are shown in Table 3. Decision maker should rank houses based on individual preferences and specialists’ opinions. And the MD process by means of the proposed method is described as follows.

a1 a2 a3 a4
D1 a1 〈1.00,0.00〉 〈0.73,0.25〉 〈0.58,0.35〉 〈0.53,0.44〉
a2 〈0.73,0.25〉 〈1.00,0.00〉 〈0.55,0.40〉 〈0.25,0.61〉
a3 〈0.58,0.35〉 〈0.55,0.40〉 〈1.00,0.00〉 〈0.55,0.39〉
a4 〈0.53,0.44〉 〈0.25,0.61〉 〈0.55,0.39〉 〈1.00,0.00〉

D2 a1 〈1.00,0.00〉 〈0.93,0.06〉 〈0.69,0.27〉 〈0.71,0.26〉
a2 〈0.93,0.06〉 〈1.00,0.00〉 〈0.76,0.20〉 〈0.64,0.31〉
a3 〈0.69,0.27〉 〈0.76,0.20〉 〈1.00,0.00〉 〈0.86,0.13〉
a4 〈0.71,0.26〉 〈0.64,0.31〉 〈0.86,0.13〉 〈1.00,0.00〉

D3 a1 〈1.00,0.00〉 〈0.75,0.25〉 〈0.59,0.39〉 〈0.65,0.30〉
a2 〈0.75,0.25〉 〈1.00,0.00〉 〈0.74,0.24〉 〈0.80,0.19〉
a3 〈0.59,0.39〉 〈0.74,0.24〉 〈1.00,0.00〉 〈0.89,0.11〉
a4 〈0.65,0.30〉 〈0.80,0.19〉 〈0.89,0.11〉 〈1.00,0.00〉

D4 a1 〈1.00,0.00〉 〈0.93,0.06〉 〈0.80,0.10〉 〈0.66,0.27〉
a2 〈0.93,0.06〉 〈1.00,0.00〉 〈0.73,0.23〉 〈0.44,0.43〉
a3 〈0.80,0.10〉 〈0.73,0.23〉 〈1.00,0.00〉 〈0.63,0.33〉
a4 〈0.66,0.27〉 〈0.44,0.43〉 〈0.63,0.33〉 〈1.00,0.00〉
Table 3.

The IF relation matrixes.

  • Step 1 Let τ : ai × ajproduct. According to Definition 8, the the lower and upper approximations of alternatives w.r.t experts are obtained.

  • Step 2 According to formulas (6)we can easily get the weight of ai under expert Mk, as is shown in Table 4. According to the formulas (3) to (5) the belief function and plausibility function of each house in the intuitionistic fuzzy relation matrix given by the specialists are calculated.

    Experts Pk(a1) Pk(a2) Pk(a3) Pk(a4)
    M1 0.33 0.23 0.28 0.16
    M2 0.26 0.26 0.25 0.23
    M3 0.21 0.27 0.25 0.28
    M4 0.28 0.29 0.26 0.17
    Table 4.

    The weight of attributes.

  • Step 3 Translate them into mass functionsas shown in Table 5.

    Experts o1 o2 o3 o4 o5 o6
    M1 〈0.65,0.33,0.02〉 〈0.59,0.41,0.00〉 〈0.62,0.28,0.11〉 〈0.61,0.37,0.02〉 〈0.60,0.36,0.04〉 〈0.64,0.27,0.09〉
    M2 〈0.61,0.29,0.11〉 〈0.57,0.36,0.07〉 〈0.58,0.24,0.18〉 〈0.57,0.31,0.11〉 〈0.57,0.34,0.09〉 〈0.58,0.24,0.18〉
    M3 〈0.61,0.28,0.10〉 〈0.57,0.37,0.06〉 〈0.59,0.28,0.12〉 〈0.57,0.33,0.10〉 〈0.61,0.35,0.04〉 〈0.59,0.28,0.13〉
    M4 〈0.63,0.32,0.05〉 〈0.59,0.39,0.02〉 〈0.58,0.25,0.17〉 〈0.59,0.33,0.08〉 〈0.57,0.35,0.08〉 〈0.59,0.25,0.16〉
    Table 5.

    The mass function of each house in Example 1.

  • Step 4–5 Revise the evidence above. Combine all the experts’ evidence, we get the belief interval for each house:

    BI(o1)=[0.947,0.948],BI(o2)=[0.846,0.847],BI(o3)=[0.958,0.963],BI(o4)=[0.904,0.905],BI(o5)=[0.889,0.889],BI(o6)=[0.961,0.956].

  • Step 6 Compare the belief interval of each house, we know that o6 > o3 > o1 > o4 > o5 > o2. Therefore, the consumer should buy the house o6.

Example 2

In the example above, we only used Product adjoint pair to calculate the intuitionistic fuzzy upper and lower approximations. Assumes that the main consideration of the consumer is good education resources and community environment when buying a house. The consumer insists that the location of the house, the quality of the property service are more important than other factors. Thus, let

τaiaj={Łukasiewicze,ifi{1,2};Producttriple,otherwise.

Because the Product adjoint triple implication results in lower values and has more influence on the infimum in the lower approximation. According to the formulas (1) to (6), the belief function and plausibility function of each house in the intuitionistic fuzzy relation matrix given by the specialists are calculated. Translate them into mass functions, as shown in Table 6.

Experts o1 o2 o3 o4 o5 o6
M1 〈0.64,0.36,0.00〉 〈0.59,0.41,0.00〉 〈0.62,0.31,0.07〉 〈0.61,0.38,0.01〉 〈0.61,0.36,0.02〉 〈0.64,0.29,0.07〉
M2 〈0.65,0.32,0.03〉 〈0.59,0.39,0.02〉 〈0.58,0.30,0.11〉 〈0.59,0.35,0.06〉 〈0.60,0.35,0.06〉 〈0.50,0.28,0.11〉
M3 〈0.60,0.31,0.04〉 〈0.60,0.39,0.01〉 〈0.60,0.34,0.06〉 〈0.60,0.36,0.04〉 〈0.62,0.36,0.01〉 〈0.62,0.32,0.06〉
M4 〈0.57,0.35,0.01〉 〈0.59,0.39,0.02〉 〈0.58,0.29,0.13〉 〈0.60,0.35,0.05〉 〈0.59,0.35,0.06〉 〈0.60,0.27,0.16〉
Table 6.

The mass function of each house in Example 2.

Calculate the discount factor of each specialist and revise the evidence above. Then using the same method to combine all the experts’ evidence, we get the belief interval for each house:

BI(o1)=[0.934,0.934],BI(o2)=[0.837,0.837],BI(o3)=[0.931,0.932],BI(o4)=[0.885,0.885],BI(o5)=[0.893,0.893],BI(o6)=[0.952,0.953].

Comparing the belief interval of each house, we know that o6 > o1 > o3 > o5 > o4 > o2. Therefore, the consumer should buy the house o6.

If the consumer puts the dwelling Environment quality in the first place, the quality of the property service and the building structure will be considered more than other factors. Thus, let

τaiaj={Łukasiewicze,ifi{2,3};Producttriple,otherwise.

Similarly, we get the following results o6 > o3 > o1 > o4 > o2 > o5. Therefore, the consumer should buy the house o6.

4.3. Comparisons with other methods

In this subsection, we will compare the model with other literatures. In literature 27, the main MCDM ranking process is Max{choice-value} - Min{hesitation} - Max{score}. Now let us take a simulation.

Example 3

We use five different pairs of level value to rank the objects. For example, the L(0.70,0.30)-level, the L(mid,mid)-level, the L(top,bot)-level, the L(top,top)-level, the L(bot,bot)-level. Compute the level soft set L(s,t), as is shown in Table 7.

O L(0.70,0.30) L(mid,mid) L(top,bot) L(top,top) L(bot,bot)
o1 4 2 0 1 1
o2 3 2 0 0 0
o3 3 2 1 2 1
o4 3 2 1 2 1
o5 3 3 0 0 1
o6 2 2 2 2 2
Table 7.

The choice-value of different level soft set L(s,t).

Rank the objects according to the choice-value from the largest to the smallest. The result is as following.

L(0.70,0.30):o2>o1=o3=o4=o5>o6;L(mid,mid):o5>o1=o2=o3=o4=o6;L(top,bot):o6>o3=o4>o1=o2=o5;L(top,top):o3=o4=o6>o1>o2=o5;L(bot,bot):o6>o1=o3=o4=o5>o2.

The alternatives cannot be sorted by the strict order from the choice-value. Thus, we have to continue to calculate the degree-hesitation of the objects, as is shown in Table 8.

O o1 o2 o3 o4 o5 o6
degree-hesitation 0.17 0.12 0.36 0.27 0.33 0.41
Table 8.

The degree-hesitation of objects.

Using the Min{hesitation} principle based on the first principle, the result is as follows.

L(0.70,0.30):o2>o1>o4>o5>o3>o6;L(mid,mid):o5>o2>o1>o4>o3>o6;L(top,bot):o6>o4>o3>o2>o1>o5;L(top,top):o4>o3>o6>o1>o2>o5;L(bot,bot):o6>o1>o4>o5>o3>o2.

Now, we rank the objects in a strict order completely in any level value. From the form of the results, both models can get a complete ordering of candidates, and also get the optimal object.

It is easy to see from the Table 3 that A3A6, we should get o6 > o3 in the whole order. However, we get o3 > o6 in complete ranking order on the level value of L(0.70,0.30), L(mid,mid), and L(top,top). And the intuitionistic fuzzy numbers of o1 is lager than o2 according to the first, second and forth attributes, and the third nearly the same. we should get o1 > o2 in the whole order. However, we get o2 > o1 in complete ranking order on the level value of L(0.70,0.30), L(mid,mid), and L(top,bot). Thus, the results obtained by the method in literature27 are not very accurate.

The reasons for the above problems are the personal limitations of the decision-makers. Therefore, it is necessary to consult relevant experts when making decisions. By consulting relevant experts, the decision-makers can understand the decision alternatives more deeply. From the intuitionistic fuzzy relation matrix Dk, we can get the weight of each attribute according to the kth expert. This method can effectively avoid the error caused by a given weight. Moreover, by using the different adjoint operators, the decision-maker can express different preferences among attributes or decision parameters. In other words, the research work in this paper is more accurate and generalized.

5. Conclusions and further study

With respect to the problem of multiple attribute group decision-making in intuitionistic fuzzy information system, multi-adjoint theory and D-S theory have been first used in this paper. An approach to group decision-making in intuitionistic fuzzy information system based on multi-adjoint and evidence theory is proposed. Firstly, using multi-adjoint intuitionistic fuzzy approximation operators and probability formula, we obtain the belief function and plausibility function of objects. Then, the D-S theory is used to combine all experts’ evidence to obtain the belief intervals. Ranking the objects by comparing the belief interval, the optimal scheme is acquired. Finally, the feasibility and effectiveness of this method is verified by an example. The multi-adjoint theory and D-S theory are applied to intuitionistic fuzzy group decision, which can effectively enhance the ability of the model to deal with intuitionistic fuzzy decision-making problem. At the same time, different use of adjoint triples increases the flexibility to deal with fuzzy decision-making problems.

In the future work, we shall continue to study the effects of general adjoint triples on solving the decision-making problem.

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (No. 61573127), Natural Science Foundation of Hebei province (No. A2018210120, A2018205103), Training funds for 333 Talents Project in Hebei Province (No. A2017002112).

References

1.F Meng and Q An, A new approach for group decision making method with hesitant fuzzy preference relation, Knowl.-Based Syst, Vol. 172, 2017, pp. 1-15.
19.S Wan, Determination of experts’ weights based on vague set for multi-sttribute group decision-making, Commum. Appl. Math. Comput, Vol. 24, No. 1, 2010, pp. 45-52.
29.G Shafer, A Mathematical Theory of Evidence, Princeton University Press, Princeton, 1976.
Journal
International Journal of Computational Intelligence Systems
Volume-Issue
12 - 1
Pages
172 - 182
Publication Date
2018/11/12
ISSN (Online)
1875-6883
ISSN (Print)
1875-6891
DOI
https://doi.org/10.2991/ijcis.12.1.172How to use a DOI?
Copyright
© 2018, the Authors. Published by Atlantis Press.
Open Access
This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - Meishe Liang
AU  - Jusheng Mi
AU  - Tao Feng
AU  - Bin Xie
PY  - 2018
DA  - 2018/11/12
TI  - Multi-adjoint based group decision-making under an intuitionistic fuzzy information system
JO  - International Journal of Computational Intelligence Systems
SP  - 172
EP  - 182
VL  - 12
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.12.1.172
DO  - https://doi.org/10.2991/ijcis.12.1.172
ID  - Liang2018
ER  -