# Extension of Axiomatic Design Principles for Multicriteria Decision Making Problems in Intuitionistic Fuzzy Environment

Authors: Ming Li
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

Axiomatic Design (AD) principles have been used to resolve the multicriteria decision making (MCDM) problems in engineering. With respect to MCDM problems in intuitionistic fuzzy environment, in which the criteria values take the form of intuitionistic fuzzy numbers, a new MCDM method is developed. Firstly, the approach proposed by Chen is extended to aggregate the decision makers’ opinions in intuitionistic fuzzy environment. Secondly, membership common area and nonmembership common area are derived from the membership probability density function and the nonmembership probability density function, respectively. Then the membership information content and nonmembership information content are obtained based on the basic ideal of axiomatic design principles. Afterwards, the score functionS and accuracy function H in intuitionistic fuzzy sets are extended with the information content to compare the alternatives. The alternatives that have the lowest values of functions of S and H are the best. Finally, a numerical example is used to illustrate the availability of the proposed method.


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## Body

## 1. Introduction

Decision making problems are widespread in engineering such as in construction engineering [1], industrial engineering and manufacturing systems [2–6], computer engineering [7, 8], chemical engineering [9, 10], aerospace mechanical engineering [11], and bioengineering [12]. In decision making problems, it is often to consider the evidence based on several criteria rather than on a single criterion. Moreover, with the increasing complexity of engineering and socioeconomic environment, it is more and more difficult for a single decision maker to consider all relevant aspects of a problem [13]. Therefore, complex decision problems often have to be conducted by a group of experts with the integration of their knowledge and experiences [14]. A lot of work has been done on these complex multicriteria decision making problems and many methods are proposed to deal with them [15–20].Axiomatic design (AD) principles are developed to form systematic scientific basis for designers, especially in the product design and software design, and are widely used to solve many design problems. It is a methodology to describe design objects and a set of axioms to evaluate relations between intended functions and means by which they are achieved [21]. AD principles allow for the selection of not only the best alternative within a set of criteria but also the most appropriate alternative [22]. Recently, the studies aiming at solving multicriteria decision making problems based on AD principles have been proposed. For example, AD principle has been used to resolve the flexible manufacturing system configuration selection [23], equipment selection [24], transportation company selection [22], manufacture technologies selection [16], and shipyards selection [25] problems.In the application of axiomatic design principles for MCDM problems, many aspects in engineering cannot be evaluated in a quantitative form but rather in a qualitative way; that is, with vague or imprecise knowledge [26], especially in the early stage of engineering, the data available is often limited and inaccurate [27]. Decisions have to be made in circumstances with vague, imprecise and uncertain information. In this condition, it is more suitable for decision makers to provide their preferences by means of linguistic variables instead of numerical ones regarding the uncertain knowledge they have about the problem. Therefore, fuzzy sets are proposed to cope with the linguistic evaluating information [22]. The core of fuzzy sets is the determination of the membership [28]. However, in many situations, the available information is not sufficient for the exact definition of degree of membership for a certain element. There may be some hesitation degree between the membership and the nonmembership degrees. Thus due to insufficiency in information availability, Boran et al. and Li [29, 30] introduced the concept of intuitionistic fuzzy sets (IFSs), which is a generalization of the concept of fuzzy sets. IFSs are characterized by a membership function and a nonmembership function and are more suitable for dealing with fuzziness and uncertainty [31]. Due to its capability of accommodating hesitation in human decision processes, IFSs have been widely used to tackle imprecise and uncertain decision information in decision making [32–37].In this paper, axiomatic design principles are extended to solve the group decision making problems in intuitionistic fuzzy environment. To do that, the remainder of this paper is set out as follows. In the next section, we introduce some basic concepts of intuitionistic fuzzy set theory and axiomatic design principles. In the third section, firstly, the axiomatic design principles are extended in intuitionistic fuzzy environment. Afterwards, we develop the MCDM method based on the extended axiomatic design principles in intuitionistic fuzzy environment. In Section4, we give an example to illustrate the availability of the proposed method. In the final section, we conclude the paper and give some remarks.


## 2. Preliminaries

### 2.1. Intuitionistic Fuzzy Sets

IFSs are an extension of the classical fuzzy set theory and are fit to deal with vagueness and uncertainty.Definition 1.
IFSA in a finite set X can be written as [29, 30]:
(1)A={{x,μA(x),νA(x)}∣x∈X}
which is characterized by a membership function μA(x) and a nonmembership function νA(x) where μA(x),νA(x):X→[0,1] with the condition 0≤μA(x)+νA(x)≤1. A third parameter of A is πA(x), known as the intuitionistic fuzzy index or hesitation degree of whether x belongs to A or not as
(2)πA(x)=1-μA(x)-νA(x).
It is obviously seen that for eachx∈X, we have
(3)0≤πA(x)≤1.
WhenμA(x)=1-νA(x) for all elements, the traditional fuzzy set concept is obtained [20]. Therefor fuzzy sets are the special case of IFSs.
The score functionS of an intuitionistic fuzzy number can be represented as follows [38]:
(4)S=μA(x)-νA(x),S∈[-1,1].
And the accuracy functionH of an intuitionistic fuzzy number can be represented as follows [39]:
(5)H=μA(x)+νA(x),H∈[0,1].Definition 2.
A standard triangular intuitionistic fuzzy number (STIFN) [40] A in X is represented by A=([x1,x2,x3];μA;νA) as shown in Figure 1.Figure 1
A standard triangular intuitionistic fuzzy number.Definition 3 (arithmetic operations on intuitionistic fuzzy numbers).
For two STIFNs,A=([x1,x2,x3];μA;νA) and B=([y1,y2,y3];μB;νB) with μA≠μB,νA≠νB, for A>0,B>0 and λ>0, the arithmetic operation is defined as follows [40–42]:
(6)A+B=([x1+y1,x2+y2,x3+y3];min(μA,μB);max(νA,νB)),A-B=([x1-y1,x2-y2,x3-y3];min(μA,μB);max(νA,νB)),A×B=([x1×y1,x2×y2,x3×y3];min(μA,μB);max(νA,νB)),AB=([x1y1,x2y2,x3y3];min(μA,μB);max(νA,νB)[x1y1，x2y2，x3y3]),λA=([λx1,λx2,λx3];1-(1-μA)λ;vAλ).Definition 4 (hamming distance on intuitionistic fuzzy numbers [42]).
The Hamming distance between intuitionistic fuzzy numbersA and B is calculated as
(7)d(A,B)=18(|(1+μA-νA)·x1-(1+μB-νB)·x2|+2×|(1+μA-νA)·x1-(1+μB-νB)·x2|+|(1+μA-νA)·x1-(1+μB-νB)·x2|).Definition 5 (the order relation on intuitionistic fuzzy numbers).
The order relation between two intuitionistic fuzzy numbers based on the score functionS and the accuracy function H is defined as follows [43].(i)
IfSA<SB, then A is worse than B.(ii)
IfSA>SB, then A is better than B.(iii)
IfSA = SB, then
(1)
ifHA=HB, then A and B are equal;(2)
ifHA<HB, then A is worse than B;(3)
ifHA>HB, then A is better than B.


### 2.2. Axiomatic Design Principles

The most important concepts in AD principles are independence axiom and information axiom [21, 44]. The independence axiom states that the independence of functional requirements (FRs) must be maintained. Functional requirements refer to the minimum set of independent requirements that characterizes the design goals. The information axiom states that the design that has the smallest information content is the best design among the designs that satisfy the independence axiom.The information axiom facilitates the selection of proper alternatives [21]. It is symbolized by the information content that is related to the probability of satisfying the design goals. The information content (Ii) is given by
(8)Ii=log21pi,
where pi is the probability of achieving a given FR.If there is more than one FR, the information content is the sum of all the probabilities, which can be calculated as follows:(9)Isystem=-log2(∏i=1mpi)=-∑i=1mlog2pi=-∑i=1mlog21pi.The probability of success is given by what a designer wishes to achieve in terms of design range and what the capacity of the system in terms of system range [27, 44]. The intersection area of the design range and the system range is the common area where the acceptable solution exists, as shown in Figure 2.Figure 2
The common area of system and design ranges.Therefore, in the case of uniform probability distribution functionpi can be written as [44]
(10)pi=CommonAreaSystemDesign.Thus, the information content is equal to [27]
(11)Ii=log2SystemDesignCommonArea.


## 2.1. Intuitionistic Fuzzy Sets

IFSs are an extension of the classical fuzzy set theory and are fit to deal with vagueness and uncertainty.Definition 1.
IFSA in a finite set X can be written as [29, 30]:
(1)A={{x,μA(x),νA(x)}∣x∈X}
which is characterized by a membership function μA(x) and a nonmembership function νA(x) where μA(x),νA(x):X→[0,1] with the condition 0≤μA(x)+νA(x)≤1. A third parameter of A is πA(x), known as the intuitionistic fuzzy index or hesitation degree of whether x belongs to A or not as
(2)πA(x)=1-μA(x)-νA(x).
It is obviously seen that for eachx∈X, we have
(3)0≤πA(x)≤1.
WhenμA(x)=1-νA(x) for all elements, the traditional fuzzy set concept is obtained [20]. Therefor fuzzy sets are the special case of IFSs.
The score functionS of an intuitionistic fuzzy number can be represented as follows [38]:
(4)S=μA(x)-νA(x),S∈[-1,1].
And the accuracy functionH of an intuitionistic fuzzy number can be represented as follows [39]:
(5)H=μA(x)+νA(x),H∈[0,1].Definition 2.
A standard triangular intuitionistic fuzzy number (STIFN) [40] A in X is represented by A=([x1,x2,x3];μA;νA) as shown in Figure 1.Figure 1
A standard triangular intuitionistic fuzzy number.Definition 3 (arithmetic operations on intuitionistic fuzzy numbers).
For two STIFNs,A=([x1,x2,x3];μA;νA) and B=([y1,y2,y3];μB;νB) with μA≠μB,νA≠νB, for A>0,B>0 and λ>0, the arithmetic operation is defined as follows [40–42]:
(6)A+B=([x1+y1,x2+y2,x3+y3];min(μA,μB);max(νA,νB)),A-B=([x1-y1,x2-y2,x3-y3];min(μA,μB);max(νA,νB)),A×B=([x1×y1,x2×y2,x3×y3];min(μA,μB);max(νA,νB)),AB=([x1y1,x2y2,x3y3];min(μA,μB);max(νA,νB)[x1y1，x2y2，x3y3]),λA=([λx1,λx2,λx3];1-(1-μA)λ;vAλ).Definition 4 (hamming distance on intuitionistic fuzzy numbers [42]).
The Hamming distance between intuitionistic fuzzy numbersA and B is calculated as
(7)d(A,B)=18(|(1+μA-νA)·x1-(1+μB-νB)·x2|+2×|(1+μA-νA)·x1-(1+μB-νB)·x2|+|(1+μA-νA)·x1-(1+μB-νB)·x2|).Definition 5 (the order relation on intuitionistic fuzzy numbers).
The order relation between two intuitionistic fuzzy numbers based on the score functionS and the accuracy function H is defined as follows [43].(i)
IfSA<SB, then A is worse than B.(ii)
IfSA>SB, then A is better than B.(iii)
IfSA = SB, then
(1)
ifHA=HB, then A and B are equal;(2)
ifHA<HB, then A is worse than B;(3)
ifHA>HB, then A is better than B.


## 2.2. Axiomatic Design Principles

The most important concepts in AD principles are independence axiom and information axiom [21, 44]. The independence axiom states that the independence of functional requirements (FRs) must be maintained. Functional requirements refer to the minimum set of independent requirements that characterizes the design goals. The information axiom states that the design that has the smallest information content is the best design among the designs that satisfy the independence axiom.The information axiom facilitates the selection of proper alternatives [21]. It is symbolized by the information content that is related to the probability of satisfying the design goals. The information content (Ii) is given by
(8)Ii=log21pi,
where pi is the probability of achieving a given FR.If there is more than one FR, the information content is the sum of all the probabilities, which can be calculated as follows:(9)Isystem=-log2(∏i=1mpi)=-∑i=1mlog2pi=-∑i=1mlog21pi.The probability of success is given by what a designer wishes to achieve in terms of design range and what the capacity of the system in terms of system range [27, 44]. The intersection area of the design range and the system range is the common area where the acceptable solution exists, as shown in Figure 2.Figure 2
The common area of system and design ranges.Therefore, in the case of uniform probability distribution functionpi can be written as [44]
(10)pi=CommonAreaSystemDesign.Thus, the information content is equal to [27]
(11)Ii=log2SystemDesignCommonArea.


## 3. The Method for Multicriteria Decision Making Problems Based on the Extended AD Principles in Intuitionistic Fuzzy Environment

### 3.1. The Extension of AD Principles in Intuitionistic Fuzzy Environment

The criteria values are considered as linguistic variable in intuitionistic fuzzy environment. We have incomplete information about the system and design ranges. The system and design ranges for a certain criterion will be expressed by using “membership,” “nonmembership” and “hesitation degree.”There are two kinds of functions of intuitionistic fuzzy numbers, which are membership function and nonmembership function. Correspondingly, we can get two kinds of probability density function in the crisp case. Therefore, the intersection areas of membership functions and nonmembership function of intuitionistic fuzzy numbers can be obtained, respectively, as shown in Figure3.Figure 3
The common areas of system and design ranges of intuitionistic fuzzy numbers.The membership information contentIM and nonmembership information content IN are equal to
(12)IM=log2MembershipSystemDesignMembershipCommonArea,IN=log2NonmembershipSystemDesignNonmembershipCommonArea.In intuitionistic fuzzy environments, the score functionS function and the accuracy function H are used to compare the intuitionistic fuzzy values. In the study, we extend the functions S and H with the information content in axiomatic design environment. We define the following score function SIC and accuracy function HIC of the information content, which are calculated as
(13)SIC=IM-IN,HIC=IM+IN.


### 3.2. The Method for Multicriteria Decision Making Problems Based on the Extended AD Principles in Intuitionistic Fuzzy Environment

LetA={A1,A2,…,Am} be a discrete set of alternatives, C={C1,C2,…,Cn} the set of criteria,w={w1,w2,…,wn} the weight vector of criteria, where wj≥0, j=1,2,…,n, ∑j=1nwj=1, D={D1,D2,…,Dt} the set of decision makers, assume that the degree of importance of decision maker Di is ηi, where ηi∈[0,1] and ∑ν=1tην=1, and F={f^1,f^2,…,f^n} be a set of functional requirements (FRs); that is, the set of goals for the criteria, where f^j=(〈fj1,fj2,fj3〉;μj,νj)∈F takes the form of intuitionistic fuzzy number.Suppose thatR^k is the group decision making matrix, where r^ij(k)=(〈rij1(k),rij2(k),rij3(k)〉;μij(k),νij(k))∈R^k is a preference value, which takes the form of intuitionistic fuzzy number, given by the decision maker Dk∈D, for the alternative Ai∈A with respect to the criterion Cj∈C.The steps of the method for multicriteria decision making problems based on the extended axiomatic design principles in intuitionistic fuzzy environment are as follows.Step 1.
Transform the data into intuitionistic fuzzy numbers.
Preference values and functional requirements take the form of linguistic terms. Since linguistic terms are not mathematically operable, at first, they must be transformed into numbers.Step 2.
Aggregate the decision makers’ opinions.
In this step, the approach presented by Chen [45] is extended to aggregate the decision makers’ opinions in intuitionistic fuzzy environment. The steps are as follows.Step2.1. Calculate the degree of agreement. Formula (14) is used to calculate the degree of agreement S(R^i,R^j) of the opinions between each pair of decision makers Di and Dj, where S(R^i,R^j)∈[0,1], 1≤i≤t, 1≤j≤t, i≠j,
(14)S(R^i,R^j)=1m×n×∑p=1m∑q=1n(|(1+μpq(i)-νpq(i))∙rpq1(i)-(1+μpq(j)-νpq(j))∙rpq1(j)|+2×|(1+μpq(i)-νpq(i))∙rpq2(p)-(1+μpq(j)-νpq(j))∙rpq2(j)|+|(1+μpq(i)-νpq(i))∙rpq3(p)-(1+μpq(j)-νpq(j))∙rpq3(j)|1-18×(|(1+μpq(i)-νpq(i))·rpq1(i)-(1+μpq(j)-νpq(j))·rpq1(j)|+2×|(1+μpq(i)-νpq(i))·rpq2(p)-(1+μpq(j)-νpq(j))·rpq2(j)|+|(1+μpq(i)-νpq(i))·rpq3(p)-(1+μpq(j)-νpq(j))·rpq3(j)|)).Step2.2. Calculate the average degree of agreement A(Ei) of decision maker Di, where
(15)A(Ei)=1t-1∑j=1j≠itS(R^i,R^j).Step2.3. Calculate the relative degree of agreement RA(Ei) of decision maker Di, where
(16)RA(Ei)=A(Ei)∑i=1tA(Ei).Step2.4. Suppose that the importance weight of the decision makers and the agreement weight of the decision makers are y1 and y2, where y1∈[0,1] and y2∈[0,1]. The consensus degree coefficient C(Ei) of decision maker Di is calculated by
(17)C(Ei)=y1y1+y2*wi+y1y1+y2*RA(Ei).Step2.5. Aggregate the fuzzy opinions and the result is
(18)R~=C(E1)⊗R1⊕C(E2)⊗R2⋯C(En)⊗Rn=(〈∑k=1uC(Ek)×rij1(k),∑k=1uC(Ek)×rij2(k),∑k=1uC(Ek)×rij3(k)〉;mink(1-(1-μij(k)))C(Ek),maxk(νij(k))C(Ek)〈∑k=1uC(Ek)×rij1(k),∑k=1uC(Ek)×rij2(k)),
where, operators ⊗ and ⊕ are the intuitionistic fuzzy multiplication operator and the intuitionistic fuzzy addition operator, respectively.Step 3.
Calculate the membership information contentIijM.
For eachFRi, the membership information content IijM is calculated by (19)
(19)IijM={0ifr^ij1>f^j3orr^ij3<f^j1log2MembershipSystemDesignoff^jMembershipCommonAreaofr^ijandf^jifr^ij1≤f^j3andr^ij3≥f^j1,
where r^ij1 and r^ij3 are the lower and upper values of the alternative Ai on the criterion Ci, respectively, and f^j1 and f^j3 are the lower and upper values of FRi.Step 4.
Calculate the nonmembership information contentIijN.
For eachFRi, the nonmembership information content IijN is calculated by (20)
(20)IijN={0ifr^ij1>f^j3orr^ij3<f^j1log2NonmembershipSystemDesignoff^jNonmembershipCommonAreaofr^ijandf^jifr^ij1≤f^j3andr^ij3≥f^j1,
where r^ij1 andr^ij3 are the lower and upper values of the alternative Ai on the criterion Ci, respectively, and f^j1 and f^j3 are the lower and upper values of FRi.Step 5.
Calculate the value of score function of information content.
The value of score functionSiIC of the alternative Ai can be got as follows:
(21)SiIC=∑j=1nSijIC·wj=∑j=1n(IijM-IijN)·wj.Step 6.
Calculate the value of accuracy function of information content.
The value of accuracy functionHiIC of the alternative Ai can be got as follows:
(22)HiIC=∑j=1nHijIC·wj=∑j=1n(IijM+IijN)·wj.Step 7.
Select of the best alternative.
Since the values ofSIC and HIC are derived,the ranking order of all alternatives can be determined according to the following rules.(i)
IfSiIC<SjIC, then Ai is worse than Aj.(ii)
IfSiIC>SjIC, then Ai is better than Aj.(iii)
IfSiIC = SjIC, then
(1)
ifHiIC=HjIC, then Ai and Aj are equal;(2)
ifHiIC<HjIC, then Ai is worse than Aj;(3)
ifHiIC>HjIC, then Ai is better than Aj.


## 3.1. The Extension of AD Principles in Intuitionistic Fuzzy Environment

The criteria values are considered as linguistic variable in intuitionistic fuzzy environment. We have incomplete information about the system and design ranges. The system and design ranges for a certain criterion will be expressed by using “membership,” “nonmembership” and “hesitation degree.”There are two kinds of functions of intuitionistic fuzzy numbers, which are membership function and nonmembership function. Correspondingly, we can get two kinds of probability density function in the crisp case. Therefore, the intersection areas of membership functions and nonmembership function of intuitionistic fuzzy numbers can be obtained, respectively, as shown in Figure3.Figure 3
The common areas of system and design ranges of intuitionistic fuzzy numbers.The membership information contentIM and nonmembership information content IN are equal to
(12)IM=log2MembershipSystemDesignMembershipCommonArea,IN=log2NonmembershipSystemDesignNonmembershipCommonArea.In intuitionistic fuzzy environments, the score functionS function and the accuracy function H are used to compare the intuitionistic fuzzy values. In the study, we extend the functions S and H with the information content in axiomatic design environment. We define the following score function SIC and accuracy function HIC of the information content, which are calculated as
(13)SIC=IM-IN,HIC=IM+IN.


## 3.2. The Method for Multicriteria Decision Making Problems Based on the Extended AD Principles in Intuitionistic Fuzzy Environment

LetA={A1,A2,…,Am} be a discrete set of alternatives, C={C1,C2,…,Cn} the set of criteria,w={w1,w2,…,wn} the weight vector of criteria, where wj≥0, j=1,2,…,n, ∑j=1nwj=1, D={D1,D2,…,Dt} the set of decision makers, assume that the degree of importance of decision maker Di is ηi, where ηi∈[0,1] and ∑ν=1tην=1, and F={f^1,f^2,…,f^n} be a set of functional requirements (FRs); that is, the set of goals for the criteria, where f^j=(〈fj1,fj2,fj3〉;μj,νj)∈F takes the form of intuitionistic fuzzy number.Suppose thatR^k is the group decision making matrix, where r^ij(k)=(〈rij1(k),rij2(k),rij3(k)〉;μij(k),νij(k))∈R^k is a preference value, which takes the form of intuitionistic fuzzy number, given by the decision maker Dk∈D, for the alternative Ai∈A with respect to the criterion Cj∈C.The steps of the method for multicriteria decision making problems based on the extended axiomatic design principles in intuitionistic fuzzy environment are as follows.Step 1.
Transform the data into intuitionistic fuzzy numbers.
Preference values and functional requirements take the form of linguistic terms. Since linguistic terms are not mathematically operable, at first, they must be transformed into numbers.Step 2.
Aggregate the decision makers’ opinions.
In this step, the approach presented by Chen [45] is extended to aggregate the decision makers’ opinions in intuitionistic fuzzy environment. The steps are as follows.Step2.1. Calculate the degree of agreement. Formula (14) is used to calculate the degree of agreement S(R^i,R^j) of the opinions between each pair of decision makers Di and Dj, where S(R^i,R^j)∈[0,1], 1≤i≤t, 1≤j≤t, i≠j,
(14)S(R^i,R^j)=1m×n×∑p=1m∑q=1n(|(1+μpq(i)-νpq(i))∙rpq1(i)-(1+μpq(j)-νpq(j))∙rpq1(j)|+2×|(1+μpq(i)-νpq(i))∙rpq2(p)-(1+μpq(j)-νpq(j))∙rpq2(j)|+|(1+μpq(i)-νpq(i))∙rpq3(p)-(1+μpq(j)-νpq(j))∙rpq3(j)|1-18×(|(1+μpq(i)-νpq(i))·rpq1(i)-(1+μpq(j)-νpq(j))·rpq1(j)|+2×|(1+μpq(i)-νpq(i))·rpq2(p)-(1+μpq(j)-νpq(j))·rpq2(j)|+|(1+μpq(i)-νpq(i))·rpq3(p)-(1+μpq(j)-νpq(j))·rpq3(j)|)).Step2.2. Calculate the average degree of agreement A(Ei) of decision maker Di, where
(15)A(Ei)=1t-1∑j=1j≠itS(R^i,R^j).Step2.3. Calculate the relative degree of agreement RA(Ei) of decision maker Di, where
(16)RA(Ei)=A(Ei)∑i=1tA(Ei).Step2.4. Suppose that the importance weight of the decision makers and the agreement weight of the decision makers are y1 and y2, where y1∈[0,1] and y2∈[0,1]. The consensus degree coefficient C(Ei) of decision maker Di is calculated by
(17)C(Ei)=y1y1+y2*wi+y1y1+y2*RA(Ei).Step2.5. Aggregate the fuzzy opinions and the result is
(18)R~=C(E1)⊗R1⊕C(E2)⊗R2⋯C(En)⊗Rn=(〈∑k=1uC(Ek)×rij1(k),∑k=1uC(Ek)×rij2(k),∑k=1uC(Ek)×rij3(k)〉;mink(1-(1-μij(k)))C(Ek),maxk(νij(k))C(Ek)〈∑k=1uC(Ek)×rij1(k),∑k=1uC(Ek)×rij2(k)),
where, operators ⊗ and ⊕ are the intuitionistic fuzzy multiplication operator and the intuitionistic fuzzy addition operator, respectively.Step 3.
Calculate the membership information contentIijM.
For eachFRi, the membership information content IijM is calculated by (19)
(19)IijM={0ifr^ij1>f^j3orr^ij3<f^j1log2MembershipSystemDesignoff^jMembershipCommonAreaofr^ijandf^jifr^ij1≤f^j3andr^ij3≥f^j1,
where r^ij1 and r^ij3 are the lower and upper values of the alternative Ai on the criterion Ci, respectively, and f^j1 and f^j3 are the lower and upper values of FRi.Step 4.
Calculate the nonmembership information contentIijN.
For eachFRi, the nonmembership information content IijN is calculated by (20)
(20)IijN={0ifr^ij1>f^j3orr^ij3<f^j1log2NonmembershipSystemDesignoff^jNonmembershipCommonAreaofr^ijandf^jifr^ij1≤f^j3andr^ij3≥f^j1,
where r^ij1 andr^ij3 are the lower and upper values of the alternative Ai on the criterion Ci, respectively, and f^j1 and f^j3 are the lower and upper values of FRi.Step 5.
Calculate the value of score function of information content.
The value of score functionSiIC of the alternative Ai can be got as follows:
(21)SiIC=∑j=1nSijIC·wj=∑j=1n(IijM-IijN)·wj.Step 6.
Calculate the value of accuracy function of information content.
The value of accuracy functionHiIC of the alternative Ai can be got as follows:
(22)HiIC=∑j=1nHijIC·wj=∑j=1n(IijM+IijN)·wj.Step 7.
Select of the best alternative.
Since the values ofSIC and HIC are derived,the ranking order of all alternatives can be determined according to the following rules.(i)
IfSiIC<SjIC, then Ai is worse than Aj.(ii)
IfSiIC>SjIC, then Ai is better than Aj.(iii)
IfSiIC = SjIC, then
(1)
ifHiIC=HjIC, then Ai and Aj are equal;(2)
ifHiIC<HjIC, then Ai is worse than Aj;(3)
ifHiIC>HjIC, then Ai is better than Aj.


## 4. Illustrative Example

Knowledge management refers to the management of the knowledge distributed in the organization [46]. An aviation design institute is desired to select the fittest knowledge map design for knowledge management. Knowledge map refers to the document category with the concept of hierarchy employed to organize and find organizational knowledge [47, 48]. It is the key component in knowledge management system. After preevaluation, five knowledge map designs denoted by A={A1,A2,A3,A4,A5} remained as alternatives for further evaluation. The committee composed of three decision makers participates in this study. Because of the different backgrounds of the decision makers, the weight degrees of the three decision makers are 0.3, 0.3, and 0.4, respectively. The degrees of weight degrees of the decision makers and the weight of the relative degree of agreement of the decision makers are equal and the values are both 0.5. The decision makers take the decision according to the four criteria including objective (C1), familiar (C2), comprehensive (C3), and flexible (C4). The weights of the criteria have been determined and the values are 0.2, 0.3, 0.3, and 0.2, respectively. The decision makers use the standard triangle intuitionistic fuzzy terms in Table 1 to express their preferences.Table 1
Definitions of standard triangle intuitionistic fuzzy terms.
Definitely low (DL)
〈
[
(
0
,
0
,
0.1
)
;
0.10
,
0.90
]
〉
Very low (VL)
〈
[
(
0
,
0.1
,
0.25
)
;
0.25
,
0.75
]
〉
Low (L)
〈
[
(
0
,
0.3
,
0.45
)
;
0.40
,
0.55
]
〉
Medium (M)
〈
[
(
0.25
,
0.5
,
0.65
)
;
0.50
,
0.45
]
〉
High (H)
〈
[
(
0.45
,
0.7
,
0.8
)
;
0.60
,
0.30
]
〉
Very high (VH)
〈
[
(
0.55
,
0.9
,
0.95
)
;
0.75
,
0.10
]
〉
Definitely high (DH)
〈
[
(
0.85
,
1
,
1
)
;
0.90
,
0.10
]
〉The linguistic evaluating information of knowledge map designs given by the decision makers is shown in Table2. The functional requirements (FRs) are shown in Table 3.Table 2
The evaluation information given by the decision makers.
D
1
D
2
D
3
C
1
C
2
C
3
C
4
C
1
C
2
C
3
C
4
C
1
C
2
C
3
C
4
A
1
VL
L
VL
M
L
VL
VH
DH
M
VL
L
VH
A
2
L
M
M
M
VL
L
VL
M
H
VL
M
VL
A
3
DL
L
VL
H
H
VL
DH
L
VL
L
VH
VH
A
4
VL
H
M
VL
DL
DH
L
VH
H
VL
DL
M
A
5
DH
VL
H
M
L
VL
M
L
VL
L
L
LTable 3
The functional requirements of the criteria.
C
1
C
2
C
3
C
4
FRs
H
M
H
HThe first step is the transformation of the linguistic evaluation information and functional requirements into intuitionistic fuzzy numbers. Then they can be dealt with by the following steps.Step2. Aggregate the decision makers’ opinions.Step2.1. Calculate the degree of agreement as follows:(23)S(R^1,R^2)=14×5×∑i=14∑j=15((|(1+μij(1)-νij(1))·rij1(1)-(1+μij(2)-νij(2))·rij1(2)|+2×|(1+μij(1)-νij(1))·rij2(1)-(1+μij(2)-νij(2))·rij2(2)|+|(1+μij(1)-νij(p))·rij3(1)-(1+μij(2)-νij(2))·rij3(2)|)1-18×∑i=14∑j=15(×(|(1+μij(1)-νij(1))·rij1(1)-(1+μij(2)-νij(2))·rij1(2)|+2×|(1+μij(1)-νij(1))·rij2(1)-(1+μij(2)-νij(2))·rij2(2)|+|(1+μij(1)-νij(p))·rij3(1)-(1+μij(2)-νij(2))·rij3(2)|))=0.706,S(R^1,R^3)=0.741,S(R^3,R^2)=0.721.Step2.2. Calculate the average degree of agreement of decision makers as follows:
(24)A(E1)=13-1∑j=2j≠i3S(R^1,R^j)=0.724,A(E2)=13-1∑j=1j≠i3S(R^2,R^j)=0.713,A(E3)=13-1∑j=1j≠i3S(R^3,R^j)=0.731.Step2.3. Calculate the relative degree of agreement of decision makers as follows:
(25)RA(E1)=A(E1)∑i=1nA(Ei)=0.334,RA(E2)=A(E2)∑i=1nA(Ei)=0.329,RA(E3)=A(E3)∑i=1nA(Ei)=0.337.Step2.4. Calculate the consensus degree coefficient as follows:
(26)C(E1)=0.50.5+0.5*w1+0.50.5+0.5*RA(E1)=0.317,C(E2)=0.50.5+0.5*w2+0.50.5+0.5*RA(E2)=0.315,C(E3)=0.50.5+0.5*w3+0.50.5+0.5*RA(E31)=0.369.Step2.5. Aggregate the fuzzy opinions. The aggregated results are presented in Table 4.Table 4
The aggregated evaluation information of the alternatives.
C
1
C
2
C
3
C
4
A
1
〈
[
(
0.092
,
0.310
,
0.460
)
;
0.087
,
0.913
]
〉
〈
[
(
0.000
,
0.163
,
0.313
)
;
0.087
,
0.913
]
〉
〈
[
(
0.173
,
0.425
,
0.544
)
;
0.087
,
0.913
]
〉
〈
[
(
0.549
,
0.805
,
0.871
)
;
0.913
,
0.173
]
〉
A
2
〈
[
(
0.166
,
0.385
,
0.516
)
;
0.087
,
0.913
]
〉
〈
[
(
0.079
,
0.290
,
0.440
)
;
0.101
,
0.899
]
〉
〈
[
(
0.171
,
0.374
,
0.524
)
;
0.087
,
0.913
]
〉
〈
[
(
0.158
,
0.353
,
0.503
)
;
0.899
,
0.171
]
〉
A
3
〈
[
(
0.142
,
0.257
,
0.375
)
;
0.033
,
0.967
]
〉
〈
[
(
0.000
,
0.237
,
0.387
)
;
0.087
,
0.913
]
〉
〈
[
(
0.470
,
0.678
,
0.744
)
;
0.087
,
0.913
]
〉
〈
[
(
0.345
,
0.648
,
0.745
)
;
0.913
,
0.470
]
〉
A
4
〈
[
(
0.166
,
0.290
,
0.406
)
;
0.033
,
0.967
]
〉
〈
[
(
0.410
,
0.573
,
0.660
)
;
0.101
,
0.899
]
〉
〈
[
(
0.079
,
0.253
,
0.384
)
;
0.038
,
0.962
]
〉
〈
[
(
0.265
,
0.499
,
0.618
)
;
0.899
,
0.079
]
〉
A
5
〈
[
(
0.269
,
0.448
,
0.551
)
;
0.101
,
0.899
]
〉
〈
[
(
0.000
,
0.174
,
0.324
)
;
0.087
,
0.913
]
〉
〈
[
(
0.221
,
0.490
,
0.624
)
;
0.172
,
0.802
]
〉
〈
[
(
0.174
,
0.324
,
0.087
)
;
0.913
,
0.221
]
〉From the table we see that the values of alternativesA3 on criterion C1 and A4 on criteria C1 and C3 are beyond the scope of the functional requirements. Therefore, A3 and A4 are removed from the set of available alternatives.Step3. Calculate the membership information content IijM.The following calculation of membership information contentI21M of alternative A2 on criterion C1 is provided as an illustration:
(27)I21M=log2MembershipSystemDesignofA21MembershipCommonAreaofA21andFR1=log2((Areaoftriangle〈(0.166,0.385,0.516);0.087〉)×(IntersectionAreaoftriangles〈(0.166,0.385,0.516);0.087〉and〈(0.45,0.7,0.8);0.60〉)-1)=log20.0151490.001127=3.748.The calculation results of all the membership information contents are shown in Table5.Table 5
The membership information contents of the alternatives.
C
1
C
2
C
3
C
4
A
1
9.329
3.914
2.704
0.560
A
2
3.748
1.005
3.576
4.581
A
6
2.004
3.525
1.454
4.517Step4. Calculate the nonmembership information content IijN.The following calculation of nonmembership information contentI21Nof alternative A2 on criterionC1 is provided as an illustration:
(28)I21N=log2NonmembershipSystemDesignofA21NonmembershipCommonAreaofA21andFR1=log2((Areaoftriangle〈(0.166,0.385,0.516);0.913〉)×(IntersectionAreaoftriangles〈(0.166,0.385,0.516);0.913〉and〈(0.45,0.7,0.8);0.30〉)-1)=log20.159970.00223=6.162.The calculation results of all the nonmembership information contents are shown in Table6.Table 6
The non-membership information contents of the alternatives.
C
1
C
2
C
3
C
4
A
1
11.619
5.680
5.210
1.394
A
2
6.162
2.702
5.868
6.809
A
6
4.565
5.292
3.419
6.505Step5. Calculate the value of score function of information content.The value of score functionS2Ic of the alternative A2can be gotten as follows:
(29)S2IC=∑j=14S2jIC·wj=∑j=1n(I2jM-I2jN)·wj=(3.748-6.162)×0.2+(1.005-2.702)×0.2+(3.576-5.868)×0.2+(4.581-6.809)×0.2=-0.314.The values of all the score functions are shown in Table7.Table 7
The values of score functions.
A
1
A
2
A
5
0.168
−0.314
−0.309Step
6. Calculate the value of accuracy function of information content.The value of accuracy functionH2IC of the alternative A2 can be gotten as follows:
(30)H2IC=∑j=14H2jIC·wj=∑j=1n(I2jM+I2jN)·wj=(3.748+6.162)×0.2+(1.005+2.702)×0.2+(3.576+5.868)×0.2+(4.581+6.809)×0.2=1.043.The values of all the accuracy functions are shown in Table8.Table 8
The values of accuracy functions.
A
1
A
2
A
5
1.559
1.043
1.292Step7. Select of the best alternative. Ranking the alternatives in the ascending order of the SIC firstly, and then ranking the alternatives in the ascending order of HIC in the condition that SIC are equal. The final ranking of the alternatives is
(31)A1≻A5≻A2.From the final ranking, clearly, we see thatA1 is the fittest knowledge map design, followed by A5, while A2 is considered as least fit. Since the values of A3 and A4 on some criteria are beyond the scope of the functional requirements, the alternatives A3 and A4 are deemed as unfit.From the example we can see that(1) the preferences are expressed comprehensively with the membership degree, the nonmembership degree and hesitation degree in intuitionistic fuzzy environment; (2) the efficiency of the method is improved because of the movement of the unfit alternatives in the calculation.


## 5. Conclusions

In this paper, we make the extension of AD principles for multicriteria group decision making problems in intuitionistic fuzzy environment. After the transformation and the aggregation of the decision makers’ opinions, we use the membership probability density function and the nonmembership probability density function to calculate the membership common area and nonmembership common area. Then the membership information content and nonmembership information content are obtained based on the basic idea of axiomatic design principles. Afterwards, the score function and accuracy function are extended to compare the alternatives in intuitionistic fuzzy environment. Finally, the availability of the proposed method is validated by the numerical example.The proposed method has the following main advantages.(1)
The preferences are expressed more comprehensively for the fuzziness and uncertainties are characterized not only by the membership but also the nonmembership degrees in intuitionistic fuzzy environment.(2)
The efficiency of the decision making process is improved because of the remove of the unfit alternatives in the middle of calculation.(3)
Not only the best alternative within a set of criteria but also the most appropriate alternative can be selected because the axiomatic design principles allow functional requirements being not only extreme numbers but also middle numbers.Some distinguished contributions of this study are as follows(1)
The extension of axiomatic design principles in intuitionistic fuzzy environment implies that(1) the membership information content and nonmembership information content are proposed to calculate the gap between preference values of the alternative and function requirements on the criteria; (2) the score function and the accuracy function in intuitionistic fuzzy sets are extended based on the membership information content and nonmembership information content to compare the alternatives.(2)
The method for multicriteria decision making based on the extended axiomatic design principles in intuitionistic fuzzy environment is proposed to cope with the linguistic evaluation information. Moreover, in the method, the approach presented by Chen and Tan [38] is also extended to aggregate the decision makers’ linguistic opinions, which is an important reference for the aggregation of decision makers’ opinions in intuitionistic fuzzy environment.(3)
In the end of the paper, the numerical example can be a reference for the knowledge map design evaluation and selection.



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*Source: 813471-2013-04-03.xml*