Closeness coefficient based nonlinear programming method for interval-valued intuitionistic fuzzy

Applied Soft Computing 11(2011)3402–3418

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Applied Soft

Computing

j o u r n a l h o m e p a g e :w w w.e l s e v i e r.c o m /l o c a t e /a s o

Closeness coef?cient based nonlinear programming method for interval-valued intuitionistic fuzzy multiattribute decision making with incomplete preference information

Deng-Feng Li ?

School of Management,Fuzhou University,No.2,Xueyuan Road,Daxue New District,Fuzhou District,Fuzhou,Fujian 350108,China

a r t i c l e i n f o Article history:

Received 3November 2009Received in revised form 18September 2010

Accepted 3January 2011

Available online 11January 2011Keywords:

Fuzzy multiattribute decision making Interval-valued intuitionistic fuzzy set Mathematical programming Uncertainty Preference

Closeness coef?cient

a b s t r a c t

The aim of this paper is to develop a closeness coef?cient based nonlinear programming method for solving multiattribute decision making problems in which ratings of alternatives on attributes are expressed using interval-valued intuitionistic fuzzy (IVIF)sets and preference information on attributes is incomplete.In this methodology,nonlinear programming models are constructed on the concept of the closeness coef?cient,which is de?ned as a ratio of the square of the weighted Euclidean distance between an alternative and the IVIF negative ideal solution (IVIFNIS)to the sum of the squares of the weighted Euclidean distances between the alternative and the IVIF positive ideal solution (IVIFPIS)as well as the IVIFNIS.Simpler nonlinear programming models are deduced to calculate closeness intuitionistic fuzzy sets of alternatives to the IVIFPIS,which are used to estimate the optimal degrees of membership and hereby generate ranking order of the alternatives.The derived auxiliary nonlinear programming models are shown to be ?exible with different information structures and decision environments.The proposed method is validated and compared with other methods.A real example is examined to demonstrate applicability of the proposed method in this paper.

1.Introduction

Fuzzy multiattribute decision making (MADM)problems are wide spread in real life situations.A fuzzy MADA problem is to choose the best solutions (i.e.,alternatives)from or rank feasible alternatives on multiple attributes by using the fuzzy set introduced by Zadeh [38].The fuzzy set has been successfully applied to many practical decision making problems [1,7,9,16,21].In reality,however,it may not be easy to identify exact value for the membership degree of an element to a given set.In this case,a range of values may be a more appropriate measurement to accommodate the uncertainty,imprecision or vagueness.As such,the fuzzy set was extended to the interval-valued fuzzy (IVF)set [39]characterized by an interval-valued membership function,which approximates the “real”but unknown membership degree of an element to the given set.

The fuzzy set uses only a membership function,which assigns to each element of the universe of discourse a number from the unit interval [0,1]to indicate degree of belongingness to the fuzzy set under consideration.Degree of non-belongingness is just automatically the complement to 1of the membership degree to the fuzzy set.However,a human being who expresses the degree of membership of a given element in a fuzzy set very often does not express corresponding degree of non-membership as the complement to 1.Thus in 1983,Atanassov [2]introduced the concept of the intuitionistic fuzzy (IF)set,which is characterized by two functions expressing the degree of belongingness and the degree of non-belongingness,respectively.The IF set seems to be accurately to uncertainty quanti?cation and provides the opportunity to precisely model the problem based on the existing knowledge and observations [6,10,22,23,27,28].It is worthwhile to notice that the IF set and the IVF set are mathematically equivalent although their interpretive settings and motivations are quite different [11,12].The former starts from the idea of evaluating degrees of membership and non-membership independently,while the latter captures the idea of ill-known membership degree [11,13].Recently,Dubois et al.[11]pointed out that “there is a terminological clash between Atanassov’s ‘intuitionistic fuzzy sets’and what is currently understood as intuitionistic logic [31].They differ both by their motivations and their underlying mathematical structure.Calling the Atanassov’s theory intuitionistic leads to a

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D.-F.Li/Applied Soft Computing11(2011)3402–34183403 misunderstanding”.But here we are not involved in this discussion and still use the term“intuitionistic fuzzy”(IF)set in the sense of Atanassov[3,4].

Since the IF set was introduced by Atanassov[2],many researchers have paid great attention to discussion on possible applications of the IF set to the practical situations[33–36].In many real decision situations,however,it may be dif?cult to determine exact values for the membership and non-membership degrees of an element to a given set.Thus,the IF set was further extended to develop the interval-valued IF(IVIF)set in the spirit of the ordinary IVF set[5].The IVIF set,which is characterized by a membership function and a non-membership function whose values are intervals rather than real numbers,seems to suitably describe MADM problems in which satisfaction(or membership)and non-satisfaction(or non-membership)degrees of alternatives on attributes cannot be expressed with exact numerical values[37].As far as we know,there is less investigation on how to solve MADM problems in which ratings of alternatives on attributes are expressed using IVIF sets and preference information on attributes is incomplete.In this paper,a new nonlinear programming method for solving the above MADM problems is developed on the concept of the closeness coef?cient,which is de?ned as a ratio of the square of the weighted Euclidean distance between an alternative and the IVIF negative ideal solution(IVIFNIS)to the sum of the squares of the weighted Euclidean distances between the alternative and the IVIF positive ideal solution(IVIFPIS)as well as the IVIFNIS.The closeness coef?cients of alternatives to the IVIFPIS may be regarded as continuous and monotonic functions with respect to the membership and non-membership degrees of alternatives on attributes.As a result,the closeness coef?cients of alternatives to the IVIFPIS are IF sets,which are called the closeness IF sets for short in this paper.Auxiliary nonlinear programming models are constructed to estimate the closeness IF sets,which are used to estimate the optimal degrees of membership and hereby generate the ranking order of the alternatives.

The paper is organized as follows.Section2brie?y reviews the concept of the IVIF set and operations.Section3formulates MADM problems in which ratings of alternatives on attributes are expressed with IVIF sets and preference information on the attributes is incomplete.Section4establishes auxiliary nonlinear programming models based on the concept of the closeness coef?cient and hereby develops a new methodology for solving the above MADM problems.Section5discusses some extensions or specializations of the derived auxiliary nonlinear programming models through relaxing or imposing some constraints/assumptions,which show that the proposed models and hereby the developed methodology are of universality and practicability.A real example of the investment company selection problem is analyzed in detailed in Section6.Conclusion and remarks are given in Section7.

2.Interval-valued intuitionistic fuzzy sets and operations

An IVIF set B in a?nite set X={x1,x2,...,x m}may be mathematically represented as B={ x i,? B(x i),? B(x i) |x i∈X},where? B(x i)?[0,1] and? B(x i)?[0,1]are interval-valued degrees of membership and non-membership of an element x i∈X to B,which are more or less independent from each other[5].The only requirement is that the sum of upper bounds of the interval-valued degrees is not greater than 1,i.e.,0≤sup{? B(x i)}+sup{? B(x i)}≤1.

For convenience,lower and upper bounds of? B(x i)and? B(x i)are denoted by l

B (x i), u

(x i), l

(x i)and u

(x i),respectively.Thus,the

IVIF set B may be concisely expressed as B={ x i,[ l

B (x i), u

(x i)],[ l

(x i), u

(x i)] |x i∈X},where0≤ u B(x i)+ u B(x i)≤1.

An IVIF index of an element x i∈X is expressed as B(x i)=[1? u B(x i)? u B(x i),1? l B(x i)? l B(x i)],which gives the range of indetermi-nacy membership degree of the element x i to the set B.

Let B and D be two IVIF sets in the set X.According to the set-theoretical operations,relations and operations between IVIF sets are stipulated as follows[3,5]:

(1)Inclusion:B?D if and only if l

B (x i)≤ l

(x i), u

(x i)≤ u

(x i), l

(x i)≥ l

(x i)and u

(x i)≥ u

(x i)for every x i∈X.

(2)Equality:B=D if and only if l

B (x i)= l

(x i), u

(x i)= u

(x i), l

(x i)= l

(x i)and u

(x i)= u

(x i)for every x i∈X.

(3)Intersection:B∩D={ x i,[min{ l

B (x i), l

(x i)},min{ u

(x i), u

(x i)}],[max{ l

(x i), l

(x i)},max{ u

(x i), u

(x i)}] |x i∈X}.

(4)Union:B∪D={ x i,[max{ l

B (x i), l

(x i)},max{ u

(x i), u

(x i)}],[min{ l

(x i), l

(x i)},min{ u

(x i), u

(x i)}] |x i∈X}.

(5)Complementation:ˉB={ x i,[ l

B (x i), u

(x i)],[ l

(x i), u

(x i)] |x i∈X}.

It is easily derived from the above relations and operations(1)–(4)that[3]

(B∩D)?B?(B∪D),(B∩D)?D?(B∪D).(1)

IF l

B (x i)= u

(x i)and l

(x i)= u

(x i)for each element x i∈X,then the IVIF set B={ x j,[ l B(x i), u B(x i)],[ l B(x i), u B(x i)] |x i∈X}is reduced

to an IF set,denoted by{ x i, B(x i), B(x i) |x i∈X},where B(x i)= l B(x i)= u B(x i)and B(x i)= l B(x i)= u B(x i).Accordingly,the IF set was

a special case of the IVIF set[3,5].

3.MADM problems with IVIF sets and incomplete preference information

The discussed problems in this paper mainly are a type of MADM problems which have limited numbers of feasible alternatives evaluated on multiple attributes[1,6,8,18,19,21,22,32,37].Mathematically assume that a MADM problem has m alternatives x i(i=1,2,..., m)evaluated on n attributes a j(j=1,2,...,n).Denote the alternative set and the attribute set by X={x1,x2,...,x m}and A={a1,a2,...,a n}, respectively.Without loss of generality,assume that m≥2and n≥2are integers(otherwise the decision problems are much easy to be dealt with)[1,6,8,18,19,32,37].

3.1.Interval-valued intuitionistic fuzzy decision matrix

Sometimes available information is not suf?cient for the exact de?nition of membership and non-membership degrees for certain elements.There may be some ranges of hesitation degrees in the degree of membership and the degree of non-membership.In other words,the membership and non-membership degrees are given by using ranges of values rather than numerical values[8,32–34,36].In view that there are many real life situations where due to insuf?ciency in information availability,IVIF sets are appropriate to deal with

3404 D.-F.Li /Applied Soft Computing 11(2011)3402–3418

such problems.That is,according to experience and knowledge,the decision maker may believe that the membership/satisfaction degree ij of an alternative x i ∈X on an attribute a j ∈A is at least l ij whereas at most u ij ,where 0≤ l ij ≤ u ij ≤https://www.360docs.net/doc/c116711203.html,ly,there is an uncertainty in the membership degree ij of x i on a j ,which is estimated as a closed subinterval [ l ij , u ij

]of the interval [0,1].Likewise,the decision maker may think that the non-membership/non-satisfaction degree ij of x i on a j is at least l ij

whereas at most u ij ,where 0≤ l ij ≤ u ij

≤1.The non-membership degree of x i on a j is not always sure.There is an uncertainty in determination of ij ,which is evaluated as a subinterval [ l ij

, u ij ]of the interval [0,1].Here, u ij and u ij are required to satisfy the following condition:0≤ u ij + u ij ≤1.Then,ratings of the alternatives x i on the attributes a j can be expressed using IVIF sets,B ij ={ x i ,[ l ij , u ij ],[ l ij , u ij ] },usually denoted by B ij = [ l ij , u ij ],[ l ij , u ij

] for short.Thus,a MADM problem with IVIF sets can be concisely expressed in the interval-valued matrix format as follows:

M =( [ l ij , u ij

],[ l ij , u

ij ] )m ×n

??? [ l 11, u 11],[ l 11

, u 11] [ l 12, u 12],[ l 12

, u 12

] ···

[ l 1n , u 1n ],[ l 1

n , u 1n

] [ l 21, u 21],[ l 21, u 21

] [ l

22, u 22],[ l 22, u 22

] ···

[ l 2n , u 2n ],[ l 2n , u 2n

] ...

...

···...

[ l m 1, u m 1],[ l m

1, u m 1

] [ l m 2, u m 2],[ l m

2, u m 2] ··· [ l mn , u mn ],[ l mn , u mn ]

???,(2)

which is referred to as an IVIF decision matrix used to represent the MADM problem with IVIF sets.

Let B i =(B ij )1×n =( [ l

ij , u ij ],[ l ij , u ij ] )1×n

represent an IVIF set row vector of n -dimension.Often x i may be interchangeably used with B i .

3.2.Preference information structures of attribute importance

Usually different attributes may have different importance [8,34].Suppose that ωj (j =1,2,...,n )are weights of the attributes a j ∈A ,

which satisfy the following normalization conditions:ωj ∈[0,1](j =1,2,...,n )and n

j =1ωj

=1.Let ω=(ωj )n ×1represent a column vector of n -dimension.A set of all weight vectors is denoted by 0,i.e.,

0={ω=(ωj )n ×1|ωj ∈[0,1](j =1,2,...,m ),

n j =1

ωj =1}.

In real decision process,the decision maker may pay more attention to importance of some attributes,i.e.,specify some preference

relations on weights of attributes according to his/her knowledge,experience and judgment.Such information of attribute weights is https://www.360docs.net/doc/c116711203.html,ually incomplete information of attribute weights can be obtained according to partial preference relations on weights given by the decision maker and has several different forms of structures.Summarizing earlier research [26,32],these incomplete weight information structures may be expressed in the following ?ve basic relations among attribute weights,which are denoted by subsets s (s =1,2,3,4,5)of weight vectors in 0,respectively.

(1)A weak ranking: 1={ω∈ 0|ωt ≥ωj for all t ∈T 1and j ∈J 1},where T 1and J 1are two disjoint subsets of the subscript index set M ={1,2,...,n }of all attributes.Thus, 1is a set of all weight vectors in 0with the property that the weight of an attribute in the set T 1is greater than or equal to that of an attribute in the set J 1.

(2)A strict ranking: 2={ω∈ 0|ˇtj ≥ωt ?ωj ≥?tj for all t ∈T 2and j ∈J 2},where ?tj >0and ˇtj >0are constants,satisfying ˇtj >?tj ;T 2and J 2are two disjoint subsets of M .Thus, 2is a set of all weight vectors in 0with the property that the weight of an attribute in the set T 2is greater than or equal to that of an attribute in the set J 2but their difference does not exceed some range,i.e.,a closed interval [?tj ,ˇtj ].

(3)A ranking with multiples: 3={ω∈ 0|ωt ≥ tj ωj for all t ∈T 3and j ∈J 3},where tj >0is a constant;T 3and J 3are two disjoint subsets of M .Thus, 3is a set of all weight vectors in 0with the property that the weight of an attribute in the set T 3is greater than or equal to multiple of that of an attribute in the set J 3.

(4)An interval form: 4={ω∈ 0| j ≥ωj ≥áj for all j ∈J 4},where j >0and áj >0are constants,satisfying j >áj ;J 4is a subset of M .Thus, 4is a set of all weight vectors in 0with the property that the weight of an attribute in the set J 4does not exceed some range,i.e.,a closed interval [áj , j ].

(5)A ranking of differences: 5={ω∈ 0|ωt ?ωj ≥ωk ?ωs for all t ∈T 5,j ∈J 5,k ∈K 5and l ∈L 5},where T 5,J 5,K 5and L 5are four disjoint subsets of M .Thus, 5is a set of all weight vectors in 0with the property that the difference between weights of attributes in the sets T 5and J 5is greater than or equal to that of attributes in the sets K 5and L 5.

In reality,usually the preference information structure of attribute importance may consist of several sets of the above basic sets s (s =1,2,3,4,5).For example,the decision maker may provide a preference information structure expressed as follows [22,32]:

={ω∈ 0|0.15≤ω1≤0.55,0.2≤ω2≤0.65,0.1≤ω3≤0.35,ω2≥1.2ω1,0.02≤ω2?ω3≤0.45},

which may be decomposed into the following three basic subsets:

2={ω∈ 0|0.02≤ω2?ω3≤0.45},

3={ω∈ 0|ω2≥1.2ω1}and 4={ω∈

0|0.15≤ω1≤0.55,0.2≤ω2≤0.65,0.1≤ω3≤0.35},where 0={ω=(ωj )3×1|ω1+ω2+ω3=1,ωj ∈[0,1](j =1,2,3)}.In

other words,the information structure consists of the above three sets 2, 3and

4.Models and method for MADM problems with IVIF sets and incomplete preference information 4.1.Determination of the IVIFPIS and the IVIFNIS

In order to de?ne the concept of the closeness coef?cients,we need to determine the reference points [1,19,21,22],i.e.,the IVIFPIS and the IVIFNIS,denoted by x +and x ?,respectively.Determining the ratings of x +and x ?is a key problem.

As stated earlier,0≤ l ij ≤ u ij ≤1and 0≤ l ij ≤ u ij ≤1since all ratings B ij = [ l ij , u ij ],[ l ij , u ij

] are IVIF sets.According to Eq.(1),the membership and non-membership degrees of the IVIFPIS x +on the attribute a j ∈A may be chosen as 1and 0,respectively.Then,the rating

D.-F.Li /Applied Soft Computing 11(2011)3402–34183405

of x +on a j may be expressed as an IVIF set,{ x +,[1,1],[0,0] },usually denoted by [1,1],[0,0] for short.It is noticing that [1,1],[0,0] is essentially a degenerated IVIF set,i.e.,an IF set 1,0 .Thus,all ratings of x +on all attributes can be concisely expressed in the IVIF

vector as ( [ l +j , u +j ],[ l +j , u +

j ] )1×n

=( [1,1],[0,0] )1×n .Namely,x +is the alternative that the decision maker completely satis?es with respect to all the attributes a j (j =1,2,...,n ).

In the same way,the membership and non-membership degrees of the IVIFNIS x ?on the attribute a j ∈A may be chosen as 0and 1,respectively.The rating of x ?on a j may be expressed as an IVIF set,{ x ?,[0,0],[1,1] },usually denoted by [0,0],[1,1] for short.Obviously, [0,0],[1,1] is essentially a degenerated IVIF set,i.e.,an IF set 0,1 .Thus,all ratings of x ?on all the attributes can be concisely expressed in

the IVIF vector as ( [ l ?j , u ?j ],[ l ?j , u ?

j ] )1×n

=( [0,0],[1,1] )1×n .That is to say,x ?is the alternative that the decision maker completely dissatis?es with respect to all the attributes a j (j =1,2,...,n ).

It is easily seen that x ?is no other than the complement (or negation)of x +[3].4.2.The weighted Euclidean distance between alternatives

In practice,IVIF sets B ij = [ l ij , u ij ],[ l ij , u ij ] may be mathematically interpreted as follows.The interval [ l ij , u ij

]of B ij means that the membership degree ij of the alternative x i ∈X on the attribute a j ∈A may take any value between l

ij and u ij ,i.e., ij ∈[ l ij , u ij

].Likewise,[ l ij , u ij ]means that the non-membership degree ij of x i ∈X on a j may take any value between l ij and u ij ,i.e., ij ∈[ l ij , u ij

].Thus,in order to compare alternatives x i (i =1,2,...,m ),distance measures can be used to measure differences between an alternative x i and the IVIFPIS x +as well as the IVIFNIS x ?.Here the weighted Euclidean distances between x i and x +as well as x ?are de?ned as follows [3,7]:

d (x i ,x +)=

n j =1

{[ωj (1? ij )]2+(ωj ij )2},

(3)

and

d (x i ,x ?)=

n j =1

{(ωj ij )2+[ωj (1? ij )]2}.(4)

It is worthwhile to notice that there are various distance measures between IF (or IVIF)sets.Atanassov [3]and Burillo and Bustince [7]

de?ned the distances using only membership and membership degrees.Grzegorzewski [17]proposed some distance measures based on Hausdorff metric.Szmidt and Kacprzyk [29,30]proposed the distance measures from a three-dimension geometrical representation of the IF set and showed that the distance measures should be calculated by taking into account all three parameters,including the membership degree,non-membership degree and hesitation degree.Szmidt and Kacprzyk [30]claimed that their approach ensures that the distances for fuzzy sets and IF sets can be easily compared since it re?ects distances in three-dimensional space,while distances due to Atanassov [3]are the orthogonal projections of the real distances.

We de?ne the weighted Euclidean distances (i.e.,Eqs.(3)and (4))due to the following consideration.The chosen distance measure ensures that the de?ned closeness coef?cients of alternatives to the IVIFPIS are continuous and monotonic functions of the variables

ij ∈[ l ij , u ij ]and ij ∈[ l ij

, u ij

],respectively,which are of use and importance in deducing simpler auxiliary nonlinear programming models for calculating the closeness IF sets (see Sections 4.3and 4.4).4.3.Closeness functions and monotonicity

In a similar way to the concept of closeness coef?cients in the TOPSIS [19],the closeness coef?cient of an alternative x i ∈X with respect to the IVIFPIS x +is de?ned as follows:

C i (( ij )m ×n ,( ij )m ×n ,(ωj )n ×1)=

(d (x i ,x ?))

(d (x i ,x ?))2

+(d (x i ,x +))

,(5)

which is a ratio of the square of the weighted Euclidean distance between x i and x ?to the sum of the squares of the weighted Euclidean

distances between x i and x +as well as x ?,where ( ij )m ×n and ( ij )m ×n represent matrices of m ×n ,satisfying ij ∈[ l ij , u ij ]and ij ∈[ l ij

, u ij

];ω=(ωj )n ×1is the weight vector in the preference information structure de?ned as above.

It is noticing that the closeness coef?cient de?ned by Eq.(5)is slightly different from that in the TOPSIS [19].The reason is that the de?ned closeness coef?cient ensures that the constructed nonlinear programming models in this paper are easier solved and cost less computation time than the nonlinear programming models derived from the concept of the closeness coef?cient [19]in which the objective functions are ratios of the square roots for the sums of the squares that are very dif?cult to be computed.

Obviously,0≤(d (x i ,x ?))2≤(d (x i ,x ?))2+(d (x i ,x +))2

.Hence,it directly follows that

0≤C i (( ij )m ×n ,( ij )m ×n ,(ωj )n ×1)≤1.

(6)

According to Eqs.(3)and (4),C i (( ij )m ×n ,( ij )m ×n ,(ωj )n ×1)may be explicitly written out as follows:

C i (( ij )m ×n ,( ij )m ×n ,(ωj )n ×1)=

j =1

{(ωj ij )2+[ωj (1? ij )]2}

j =1

{(ωj ij )2+[ωj (1? ij )]2}+ n

j =1

{[ωj (1? ij )]2+(ωj ij )2}.(7)

Obviously,C i (( ij )m ×n ,( ij )m ×n ,(ωj )n ×1)is a continuous function of (2m +1)n variables,including ij ∈[ l ij , u ij ], ij ∈[ l ij , u ij

](i =1,2,...,m ;j =1,2,...,n )and (ωj )n ×1∈ .Now,we prove the monotonicity of the aforementioned function C i (( ij )m ×n ,( ij )m ×n ,(ωj )n ×1).

3406 D.-F.Li/Applied Soft Computing11(2011)3402–3418

Partial differentials of C i(( ij)

m×n ,( ij)

m×n

,(ωj)

n×1

)(i=1,2,...,m)with respect to the variables ij(j=1,2,...,n)are computed as

follows:

?C i(( ij)

m×n ,( ij)

m×n

,(ωj)

n×1

)

? ij

2(ωj)2[ ij(d(x i,x+))2+(1? ij)(d(x i,x?))2]

[((d(x i,x?))2+(d(x i,x+))2]2

respectively.Since ij≥0and1? ij≥0,it directly follows that

?C i(( ij)

m×n ,( ij)

m×n

,(ωj)

n×1

)

? ij

≥0.

It is worthwhile to notice that ij=0and1? ij=0are not valid instantaneously.Especially,ifωj/=0then

?C i(( ij)

m×n ,( ij)

m×n

,(ωj)

n×1

)

? ij

>0.

Therefore,C i(( ij)

m×n ,( ij)

m×n

,(ωj)

n×1

)(i=1,2,...,m)are monotonic and non-decreasing functions of the variables ij∈[ l ij, u ij](j=1,2,

...,n).

Similarly,partial differentials of C i(( ij)

m×n ,( ij)

m×n

,(ωj)

n×1

)(i=1,2,...,m)with respect to the variables ij(j=1,2,...,n)are computed

as follows:

?C i(( ij)

m×n ,( ij)

m×n

,(ωj)

n×1

)

? ij

2(ωj)2[(1? ij)(d(x i,x+))2+ ij(d(x i,x?))2]

[((d(x i,x?))2+(d(x i,x+))2]2

respectively.Noticing that ij≥0and1? ij≥0because of ij∈[ l ij, u ij]?[0,1],it directly follows that

?C i(( ij)

m×n ,( ij)

m×n

,(ωj)

n×1

)

? ij

≤0.

Obviously, ij=0and1? ij=0are not valid instantaneously.Especially,ifωj/=0then

?C i(( ij)

m×n ,( ij)

m×n

,(ωj)

n×1

)

? ij

<0.

Therefore,C i(( ij)

m×n ,( ij)

m×n

,(ωj)

n×1

)(i=1,2,...,m)are monotonic and non-increasing functions of the variables ij∈[ l ij, u ij](j=1,2,

...,n).

4.4.Closeness IF sets

Obviously,[ l

ij , u

]and[ l

, u

]are closed and bounded subintervals of the unit interval[0,1].Then,the continuous functions

C i(( ij)

m×n ,( ij)

m×n

,(ωj)

n×1

)are bounded.In other words,values of each continuous function C i(( ij)

m×n

,( ij)

m×n

,(ωj)

n×1

)lie in some

range of the interval[0,1]when the variables ij, ij andωj take all values in the intervals[ l

ij , u

],[ l

, u

]and[0,1],respectively.This

range is essentially a closed and bounded subinterval of the interval[0,1],denoted by[C l

i ,C u

].It is easily derived from Eq.(6)and the

notation of[C l

i ,C u

]as well as the property of the continuous function that

0≤C l

i ≤C i(( ij)

m×n

,( ij)

m×n

,(ωj)

n×1

)≤C u

≤1.(8)

for all ij∈[ l ij, u ij], ij∈[ l ij, u ij](i=1,2,...,m;j=1,2···,n)and(ωj)n×1∈ .Namely,the closeness coef?cient of the alternative x i to the

IVIFPIS x+is an IVF set[C l

i ,C u

]of the interval[0,1][39].

It is easily derived from Eq.(8)that C l

i +(1?C u

)=1+(C l

?C u

)≤1.Hence,C l

and C u

satisfy the following conditions:0≤C l

≤1,

0≤1?C u

i ≤1and C l

+(1?C u

)≤1.According to the de?nition of the IF set[2,3],the IVF set[C l

,C u

]may be equivalently expressed as

an IF set[3,5,11–13],C i= C l i,1?C u i ,which means that the membership/closeness and non-membership/non-closeness degrees of the

alternative x i∈X to the IVIFPIS x+are C l i and1?C u i,respectively. C

i =C u

?C l

re?ects that there is an uncertainty on the closeness degree

of x i∈X to x+.Thus,IF sets C i= C l i,1?C u i can be used to rank the alternatives x i(i=1,2,...,m).Therefore,how to calculate IF sets C i(i=1, 2,...,m)is a key problem.

4.5.Nonlinear programming models for estimating closeness IF sets

As stated earlier,C l

i and C u

are the lower and upper bounds of C i(( ij)

m×n

,( ij)

m×n

,(ωj)

n×1

),which are continuous and bounded functions

of the variables ij∈[ l ij, u ij], ij∈[ l ij, u ij](i=1,2,...,m;j=1,2,...,n)and(ωj)n×1∈ .Combining with Eq.(7),C l i and C u i can be captured through solving the nonlinear programming models constructed as follows:

C l i =min{C i(( ij)

m×n

,( ij)

m×n

,(ωj)

n×1

)}

s.t.?

( ij)

m×n

∈?

( ij)

m×n

∈?

(ωj)

n×1

∈

(9)

D.-F.Li/Applied Soft Computing11(2011)3402–34183407 and

C u i =max{C i(( ij)

m×n

,( ij)

m×n

,(ωj)

n×1

)}

s.t.?

( ij)

m×n

∈?

( ij)

m×n

∈?

(ωj)

n×1

∈ ,

,(10)

respectively,where? ={( ij)m×n| l ij≤ ij≤ u ij(i=1,2,...,m;j=1,2,...,n)}and? ={( ij)m×n| l ij≤ ij≤ u ij(i= 1,2,...,m;j=1,2,...,n)}indicate all possible values of membership and non-membership degrees of m alternatives on n attributes, respectively; is the preference information structure given by the decision maker a priori.

Obviously,the nonlinear programming model(i.e.,Eq.(9)or(10))has(2m+1)n variables unknown need to be determined,including ij∈[ l ij, u ij], ij∈[ l ij, u ij](i=1,2,...,m;j=1,2,...,n)and(ωj)n×1∈ .The number of the variables unknown,denoted by N=(2m+1)n for short,is a strict monotonic increasing function of m and n.N will increase remarkably if either m or n becomes great.For example,N=21 if m=3and n=3,whereas N=66if m=5and n=6.In other words,N may be very great even if m and n are small.Thus,solving Eqs.(9)and (10)is very dif?cult and costs a long computation time in that there are N variables unknown need to be determined.Therefore,we need to establish an ef?cient algorithm for solving Eqs.(9)and(10)through reducing the number of the variables unknown.

According to the monotonicity of C i(( ij)

m×n ,( ij)

m×n

,(ωj)

n×1

)as stated earlier,it easily follows that C i(( ij)

m×n

,( ij)

m×n

,(ωj)

n×1

)

reaches its minimum at the lower bounds l

ij of the intervals[ l

, u

]and the upper bounds u

of the intervals[ l

, u

]instantaneously.

Thereby,Eq.(9)can be further simpli?ed as follows:

C l i =min{C i(( l

)

m×n

,( u

)

m×n

,(ωj)

n×1

)}

s.t.(ωj)

n×1∈ .

(11)

Obviously,Eq.(11)is a nonlinear programming model with n variables unknown,includingωj(j=1,2,...,n).The number n of the variables unknown in Eq.(11)is greatly smaller than the number N=(2m+1)n of the variables unknown in Eq.(9).In other words,solving Eq.(11)can reduce2mn variables unknown by contrast to solving Eq.(9).Without doubt,solving Eq.(11)is much easier than solving Eq.

(9)in that there are only n variables unknown need to be determined when Eq.(11)is solved whereas there are N=(2m+1)n variables unknown need to be determined when Eq.(9)is solved.

In the same analysis to Eq.(9),C i(( ij)

m×n ,( ij)

m×n

,(ωj)

n×1

)reaches its maximum at the upper bounds u

of the intervals[ l

, u

]and

the lower bounds l

ij of the intervals[ l

, u

]instantaneously.Hence,Eq.(10)can be further simpli?ed as follows:

C u i =max{C i(( u

)

m×n

,( l

)

m×n

,(ωj)

n×1

)}

s.t.(ωj)

n×1∈ .

(12)

It is easy to see that Eq.(12)is a nonlinear programming model with n variables unknown,includingωj(j=1,2,...,n).The number n of the variables unknown in Eq.(12)is greatly smaller than the number N=(2m+1)n of the variables unknown in Eq.(10).Thus,solving Eq.

(12)can reduce2mn variables unknown by contrast to solving Eq.(10).Therefore,solving Eq.(12)is much easier than solving Eq.(10)in that there are only n variables unknown need to be determined when Eq.(12)is solved whereas there are N=(2m+1)n variables unknown need to be determined when Eq.(10)is solved.

Using existing nonlinear programming methods[24,25,40]and some heuristic and meta-heuristic approaches such as Tabu Search, Simulated Annealing,Genetic Algorithms,Evolution Programming(EP),Evolution Strategies(ES)and Particle Swarm Optimization(PSO) as well as Ant Colony Optimization(ACO)[8,14,15,20],Eqs.(11)and(12)are solved and hereby closeness IF sets C i= C l i,1?C u i are computed for the alternatives x i(i=1,2,...,m).The ranking order of the alternatives x i(i=1,2,...,m)is generated through comparing the closeness IF sets C i.Therefore,how to compare/rank the closeness IF sets C i is a key problem.

4.6.Inclusion comparison probability of the closeness IF sets

To make comparison between alternatives,we de?ne a binary relation on the set of the alternatives,X.Notation“x i x k”means that the alternative x i is not worse than x k.Let p(x i x k)represent the probability of the event“x i x k”.

As stated earlier,the closeness IF sets of the alternatives x i and x k are C i and C k,respectively.Thus,“x i x k”may be corre-spondingly expressed as“C i?C k”,i.e., C l i,1?C u i ? C l k,1?C u k ,which is equivalently written as C l i≥C l k and C u i≥C u k according to the inclusion relation of the IVIF sets in Section2.We may regard“C i?C k”as an IF event from a probability viewpoint.This idea was?rstly put forward by Szmidt and Kacprzyk[29]when they studied the entropy in the context of IF events.Thus, comparing two alternatives becomes computing the probability of the inclusion comparison event of corresponding closeness IF sets.

The probability of the IF event“C i?C k”is denoted by p(C i?C k),which is called the inclusion comparison probability of the IF sets C i and C k.Sometimes p(C i?C k)is called the inclusion comparison probability for short.Obviously,p(x i x k)=p(C i?C k)since they are no more than using two different semantics to describe the same relation on two alternatives.Here,the inclusion comparison probability of C i and

C k is de?ned as follows:

p(x i x k)=p(C i?C k)=max

1?max

C u

?C l

i C k

(13)

where C i= C l i,1?C u i ,C k= C l k,1?C u k , C

i =C u

?C l

and C

=C u

?C l

. C

and C

are hesitation degrees of the alternatives x i and x k,

respectively.

The inclusion comparison probability p(C i?C k)has some useful and important properties,which are summarized in Theorem1.

3408 D.-F.Li/Applied Soft Computing11(2011)3402–3418

Theorem1.Let C i= C l i,1?C u i ,C k= C l k,1?C u k and C t= C l t,1?C u t be three IF sets.Then

(P1)0≤p(C i?C k)≤1;

(P2)p(C i?C k)=0if C u i≤C l k;

(P3)p(C i?C k)=1if C l i≥C u k;

(P4)Complementary:p(C i?C k)+p(C i?C k)=1;

(P5)p(C i?C k)=p(C i?C k)=1/2if p(C i?C k)=p(C i?C k);

(P6)Transitivity:p(C i?C t)≥1/2if p(C i?C k)≥1/2and p(C k?C t)≥1/2.

Proof.See Appendix I.

4.7.Determination of optimal membership degrees for alternatives

Using Eq.(13)and Theorem1,inclusion comparison probabilities of pair-wise alternatives in X are obtained and concisely expressed in the matrix format as follows:

P=(p ik)m×m,

where

p ik=p(x i x k)=p(C i?C k)(i,k=1,2,...,m).(14) According to(P1)and(P4)in Theorem1,it is easy to see that

0≤p ik≤1,p ik+p ki=1(i,k=1,2,...,m),

which implies that P is a fuzzy complementary judgment https://www.360docs.net/doc/c116711203.html,ing Theorem2in Appendix II(see Appendix II for details),we can determine optimal degrees of membership for the alternatives x i(i=1,2,...,m)as follows:

?i=

m(m?1)

k=1

p ik+

.(15)

Then,the ranking order of all alternatives x i(i=1,2,...,m)is generated according to the descending order of the values?i.

4.8.Process and algorithm of the proposed methodology

In this subsection,we summarize the algorithm and process of the closeness coef?cient based nonlinear programming method for solving MADM problems with IVIF sets and incomplete preference information as follows.

Step1:Identify the evaluation attributes and alternatives.

Step2:Pool the decision maker’s opinion to get ratings of alternatives on attributes,i.e.,the IVIF decision matrix M=

( [ l

ij , u

],[ l

, u

] )

m×n

Step3:Pool the decision maker’s opinion to get a speci?c preference information structure on attributes,i.e.,the set .

Step4:Construct auxiliary nonlinear programming models for each alternative x i∈X using Eqs.(11)and(12).

Step5:Solve the auxiliary nonlinear programming models using the nonlinear programming methods and obtain closeness IF sets C i=

C l

i ,1?C u

of alternatives x i∈X to the IVIFPIS x+.

Step6:Construct the inclusion comparison probability matrix P=(p ik)m×m by pair-wise comparison of all the alternatives x i(i=1,2,..., m)using Eqs.(13)and(14).

Step7:Compute optimal membership degrees?i of the alternatives x i(i=1,2,...,m)using Eq.(15).

Step8:Determine the best alternative from the set X and generate the ranking order of the alternatives x i(i=1,2,...,m)according to the descending order of all optimal membership degrees?i.

5.Discussions on generalizations or specializations of the auxiliary nonlinear programming models

The derived auxiliary nonlinear programming models(i.e.,Eqs.(11)and(12))are extremely?exible compared to the existing analysis methods[1,19,21,22,32,33,37]so that many additional features or constraints/requirements can be built into these basic models.The list given below is not intended to be exhaustive,but suggests the kind of generalizations/extensions or specializations that are possible.

(a)Eqs.(11)and(12)are easily extended to the situations in which the IVIFPIS and the IVIFNIS are not?xed a priori.As stated earlier,x+

and x?in Eqs.(11)and(12)are?xed,i.e.,their IVIF sets( [ l+

j , u+

],[ l+

, u+

] )

1×n

=( [1,1],[0,0] )1×n and( [ l?

, u?

],[ l?

, u?

] )

1×n

( [0,0],[1,1] )1×n are constants.In other words,the decision maker could de?ne a?xed IVIFPIS and a?xed IVIFNIS,which are?xed reference points.Thus,the obtained closeness IF sets and ranking order of the alternatives could not change if the number of the alternatives is https://www.360docs.net/doc/c116711203.html,ly,inclusion or exclusion of alternatives could not affect the ranking order of the newly set of alternatives.

In some situations,however,the decision maker could not de?ne a?xed IVIFPIS and a?xed IVIFNIS.For example,the deci-

sion maker could de?ne an IVIFPIS?x+and an IVIFPIS?x?,whose vectors of IVIF sets are denoted by( [ l+

j , u+

],[ l+

, u+

] )

1×n

( [g+

j ,g+

],[b+

,b+

] )

1×n

and( [ l?

, u?

],[ l?

, u?

] )

1×n

=( [g?

,g?

],[b?

,b?

] )

1×n

,where g+

=max{ u

|i=1,2,...,m},b+

=min{ l

|i=

D.-F.Li/Applied Soft Computing11(2011)3402–34183409

1,2,...,m},g?

j =min{ l

|i=1,2,...,m}and b?

=max{ u

|i=1,2,...,m}.According to the inclusion relation of the IVIF sets in Section

2,it is easily seen that

[g?

j ,g?

],[b?

,b?

] ? [ l

, u

],[ l

, u

] ? [g+

,g+

],[b+

,b+

] (j=1,2,...,n),

where( [ l

j , u

],[ l

, u

] )

1×n

,( [g+

,g+

],[b+

,b+

] )

1×n

and( [g?

,g?

],[b?

,b?

] )

1×n

are the IVIF sets of the alternative x i,the IVIFPIS?x+and

the IVIFNIS?x?on a j,respectively.In this case,we de?ne the weighted Euclidean distances d(x i,?x+)and d(x i,?x?)between x i and?x+as well as?x?in a similar way to Eqs.(3)and(4).The closeness coef?cient of the alternative x i to?x+is explicitly de?ned as follows:

?C i (( ij)

m×n

,( ij)

m×n

,(ωj)

n×1

j=1

{[ωj( ij?g?

)]2+[ωj(b?j? ij)]2}

j=1

{[ωj( ij?g?

)]2+[ωj(b?j? ij)]2}+

j=1

{[ωj(g+

? ij)]2+[ωj( ij?b+

)]2}

.(16)

In a similar analysis to Eqs.(11)and(12),it easily follows that?C l

i and?C u

of the closeness IF sets ?C l

,1??C u

of alternatives x i∈X to?x+

can be captured through solving the nonlinear programming models constructed as follows:

?C u i =max{?C i(( u

)

m×n

,( l

)

m×n

,(ωj)

n×1

)}

s.t.(ωj)

n×1∈

(17)

and

?C l i =min{?C i(( l

)

m×n

,( u

)

m×n

,(ωj)

n×1

)}

s.t.(ωj)

n×1∈ ,

(18)

respectively.However,the closeness IF sets and ranking order of alternatives obtained by Eqs.(17)and(18)stand only for the given set of alternatives.Inclusion or exclusion of alternatives could affect the ranking order of the newly set of alternatives.

(b)If weights of attributes are known a priori,then Eqs.(11)and(12)are still used provided we dispose of all the constraints(ωj)

n×1

∈

and regard all variablesωj as known constants.Essentially, C l

i and C u

of the closeness IF sets C i= C l i,1? C u i of alternatives x i to x+are

easily obtained through directly computing:

C l i =C i(( l

)

m×n

,( u

)

m×n

,(ωj)

n×1

j=1

{(ωj l

)2+[ωj(1? u ij)]2}

j=1

{(ωj l

)2+[ωj(1? u ij)]2}+

j=1

{[ωj(1? l

)]2+(ωj u ij)2}

(19)

and

C u i =C i(( u

)

m×n

,( l

)

m×n

,(ωj)

n×1

j=1

{(ωj u

)2+[ωj(1? l ij)]2}

j=1

{(ωj u

)2+[ωj(1? l ij)]2}+

j=1

{[ωj(1? u

)]2+(ωj l ij)2}

,(20)

respectively.

(c)The distance measure in Eqs.(11)and(12)is easily replaced with the weighted Hamming distance.As stated earlier,in Eqs.(11) and(12),the weighted Euclidean distances(i.e.,Eqs.(3)and(4))are used to measure differences between alternatives and the IVIFPIS x+ as well as the IVIFNIS x?.If the weighted Hamming distance is utilized in place of the weighted Euclidean distance,then the closeness coef?cient of an alternative x i∈X to x+is written as follows:

C1 i (( ij)

m×n

,( ij)

m×n

,(ωj)

n×1

j=1

[(ωj ij)+ωj(1? ij)]

j=1

[(ωj ij)+ωj(1? ij)]+

j=1

[ωj(1? ij)+(ωj ij)]

which implies that

C1 i (( ij)

m×n

,( ij)

m×n

,(ωj)

n×1

,( j)

n×1

j=1

ωj( ij+1? ij)

.(21)

In a similar way to Eqs.(11)and(12),it easily follows that C1l

i and C1u

of the closeness IF sets C1

= C1l

,1?C1u

of the alternatives x i∈X

to the IVIFPIS x+are obtained through solving the linear programming models as follows:

C1l i =min{C1

(( l

)

m×n

,( u

)

m×n

,(ωj)

n×1

)}

s.t.(ωj)

n×1∈

(22)

and

C1u i =max{C1

(( u

)

m×n

,( l

)

m×n

,(ωj)

n×1

)}

s.t.(ωj)

n×1∈ ,

(23)

respectively,which are easily solved using the Simplex Method for linear programming[25].

6.A real example of the investment company selection problem and comparison analysis of the computation results

In this section,an example adapted from[37]for a MADM problem of alternatives is used as a demonstration of the application of the proposed methodology in a realistic scenario,as well as a validation of the effectiveness of the proposed methodology in this paper.

3410 D.-F.Li /Applied Soft Computing 11(2011)3402–3418

6.1.Description of the speci?c investment company selection problem

The investment company selection problem was ?rstly discussed under linguistic information environment by Herrera and Herrera-Viedma [18].Its IVIF version was extended by Ye [37].In order to make comparison analysis easily,a real example from [37]is directly chosen as follows.

There is a panel with the following four possible companies (i.e.,alternatives)to invest the money:(1)x 1is a car company;(2)x 2is a food company;(3)x 3is a computer company;(4)x 4is an arms company.The investment company must take a decision according to the following three attributes/criteria:(1)a 1is the risk analysis;(2)a 2is the growth analysis;(3)a 3is the environmental impact analysis.The four possible alternatives (i.e.,companies)are evaluated using the IVIF sets by the decision maker under the above three attributes,as listed in the following IVIF matrix,which is directly taken from [37]:

M =?

[0.4,0.5],[0.3,0.4] [0.4,0.6],[0.2,0.4] [0.1,0.3],[0.5,0.6] [0.6,0.7],[0.2,0.3] [0.6,0.7],[0.2,0.3] [0.4,0.7],[0.1,0.2] [0.3,0.6],[0.3,0.4] [0.5,0.6],[0.3,0.4] [0.5,0.6],[0.1,0.3] [0.7,0.8],[0.1,0.2]

[0.6,0.7],[0.1,0.3] [0.3,0.4],[0.1,0.2]

(24)

Assume that the decision maker provides a preference information structure on the attributes a 1,a 2and a 3,which is expressed with the set as follows:

={ω∈ 0|0.15≤ω1≤0.55,0.2≤ω2≤0.65,0.1≤ω3≤0.35,ω2≥1.2ω1,0.02≤ω2?ω3≤0.45}.

(25)

The meaning of the set is interpreted in Section 3.2.The problem is how to rank the four companies (i.e.,alternatives)x i (i =1,2,3,4)on all the three attributes and choose the best https://www.360docs.net/doc/c116711203.html,putation results obtained by the developed methodology

Using the developed method in this paper,the aforementioned speci?c investment company selection problem is solved according to the algorithm and process as stated in Section 4.8as follows.

Step 1:As stated earlier,it is easy to see that there are three evaluation attributes,including a 1(the risk analysis),a 2(the growth analysis)

and a 3(the environmental impact analysis),and four evaluation alternatives (i.e.,companies),including x 1(car company),x 2(food company),x 3(computer company)and x 4(arms company).

Step 2:Using the Statistics and Knowledge Engineering,ratings of the alternatives on the three attributes are given by the domain

expert/decision maker.Here the IVIF decision matrix M =( [ l ij , u ij ],[ l ij , u ij ] )4×3

is given by Eq.(24),which is directly taken from [37].

Step 3:According to the experience,knowledge and preference,the decision maker may give a speci?c preference information structure

on the attributes a 1,a 2and a 3,where the is given by Eq.(25)as stated earlier.

Step 4:According to Eqs.(11)and (12),two auxiliary nonlinear programming models can be constructed for the alternative x 1as follows:

C l 1

=min

(0.4ω1)2+(0.4ω2)2+(0.1ω3)2+(0.6ω1)2+(0.6ω2)2+(0.4ω3)

2(0.4ω1)2

+(0.4ω2)2

+(0.1ω3)2

+(0.6ω1)2

+(0.6ω2)2

+(0.4ω3)2

+(0.6ω1)2

+(0.6ω2)2

+(0.9ω3)2

+(0.4ω1)2

+(0.4ω2)2

+(0.6ω3)2

s.t.(ω1,ω2,ω3)T

∈

(26)

and

C u 1=max

(0.5ω1)2+(0.6ω2)2+(0.3ω3)2+(0.7ω1)2+(0.8ω2)2+(0.5ω3)

2(0.5ω1)2

+(0.6ω2)2

+(0.3ω3)2

+(0.7ω1)2

+(0.8ω2)2

+(0.5ω3)2

+(0.5ω1)2

+(0.4ω2)2

+(0.7ω3)2

+(0.3ω1)2

+(0.2ω2)2

+(0.5ω3)

s.t.(ω1,ω2,ω3)T

∈

(27)

respectively.

Step 5:Using existing nonlinear programming methods [24,25,40],optimal objective function values of the above two nonlinear programming models (i.e.,Eqs.(26)and (27))are obtained as follows:

C l 1=0.2533,

C u

1=0.4028,

respectively,i.e.,the closeness IF set of the alternative x 1is C 1= 0.2533,0.5972 .

In the same way,i.e.,repeating Steps 4and 5for the alternatives x i (i =2,3,4),we can construct the auxiliary nonlinear programming models for the alternatives x i ,which are similar to Eqs.(26)and (27).Solving these nonlinear programming models,the closeness IF sets of the alternatives x i (i =2,3,4)are obtained as follows:

C 2= 0.4778,0.2891 ,

C 3= 0.4432,0.3862 ,

C 4= 0.4249,0.4843

respectively.

Step 6:Using Eq.(13),the inclusion comparison probability of the IF sets C 2and C 3can be calculated as follows:

p (C 2?C 3)=max

1?max

0.6138?0.4778

(0.7109?0.4778)+(0.6138?0.4432)

,0 ,0

=0.6631.

According to (P4)in Theorem 1,it easily follows that

p (C 2?C 3)=1?0.6631=0.3369.

Using Eq.(14),we obtain

p 23=p (x 2 x 3)=p (C 2?C 3)=0.6631

D.-F.Li /Applied Soft Computing 11(2011)3402–34183411

and

p 32=p (x 3 x 2)=p (C 3?C 2)=p (C 2?C 3)=0.3369.

In the same way,the inclusion comparison probability of the IF sets C 2and C 4can be obtained as follows:p (C 2?C 4)=max

1?max

0.5157?0.4778

(0.7109?0.4778)+(0.5157?0.4249)

,0 ,0

=0.8830.

According to (P4)in Theorem 1,it easily follows that

p (C 2?C 4)=1?0.8830=0.1170.

Hence,according to Eq.(14),we have

p 24=p (x 2 x 4)=p (C 2?C 4)=0.8830

and

p 42=p (x 4 x 2)=p (C 4?C 2)=p (C 2?C 4)=0.3369.

Likewise,the inclusion comparison probabilities of all other IF sets can be obtained and expressed in the matrix format as follows:

P =?

0.5000

10.50.66310.883010.33690.50.72261

0.11700.27740.5

(28)

Step 7:Using Eq.(15)with Eq.(28)and m =4,the optimal membership degrees of the alternatives x i (i =1,2,3,4)are computed as

follows:

?1=

m (m ?1)

k =1

p 1k +m 2?1

=14(4?1)

(0.5+0+0+0)+42?1

=0.1250,

?2=

m (m ?1)

k =1

p 2k +m 2?1

14(4?1)

(1+0.5+0.6631+0.8830)+42?1

=0.3372,?3=

m (m ?1)

k =1

p 3k +m 2

=14(4?1)

(1+0.3369+0.5+0.7226)+42?1

=0.2966

and

?4=

m (m ?1)

k =1

p 4k +m 2

=14(4?1)

(1+0.1170+0.2774+0.5)+42?1

=0.2412,

respectively.

Step 8:It is easy to see that the decreasing order of the optimal membership degrees of the alternatives x i (i =1,2,3,4)is obtained as follows:

?2>?3>?4>?1.

Hence,the ranking order of the four alternatives (i.e.,companies)is generated as follows:

x 2 x 3 x 4 x 1

and the best alternative is x 2,i.e.,the food https://www.360docs.net/doc/c116711203.html,parison analysis of the obtained results

To conduct comparison analysis,the proposed methodology in this paper is applied to solve other example of the investment company selection problem as stated in Section 6.1,which is a special case of the example in Section 6.2,i.e.,weights of the attributes a 1,a 2and a 3are constants known.Assume that ω1=0.35,ω2=0.25and ω3=0.40,which were given by Ye [37].

As stated in Section 5,the proposed methodology is still applicable to the MADM problems with weights known a priori .Using Eqs.(19)and (20)with ω1=0.35,ω2=0.25and ω3=0.40,we directly compute

C u 1

=(0.5ω1)2+(0.6ω2)2+(0.3ω3)2+(0.7ω1)2+(0.8ω2)2+(0.5ω3)

2(0.5ω1)2

+(0.6ω2)2

+(0.3ω3)2

+(0.7ω1)2

+(0.8ω2)2

+(0.5ω3)2

+(0.5ω1)2

+(0.4ω2)2

+(0.7ω3)2

+(0.3ω1)2

+(0.2ω2)2

+(0.5ω3)

=0.5231

and

C l 1=

(0.4ω1)2+(0.4ω2)2+(0.1ω3)2+(0.6ω1)2+(0.6ω2)2+(0.4ω3)

2(0.4ω1)2

+(0.4ω2)2

+(0.1ω3)2

+(0.6ω1)2

+(0.6ω2)2

+(0.4ω3)2

+(0.6ω1)2

+(0.6ω2)2

+(0.9ω3)2

+(0.4ω1)2

+(0.4ω2)2

+(0.6ω3)

=0.3989.

3412 D.-F.Li /Applied Soft Computing 11(2011)3402–3418

Hence,the closeness IF set of the alternative x 1is

C 1= 0.3989,0.4769 .

In a similar way,the closeness IF sets of the alternatives x i (i =2,3,4)are obtained as follows:

C 2= 0.6166,0.2366 ,

C 3= 0.5352,0.3003 ,

C 4= 0.6195,0.2937

respectively.

Noticing that the fact C l 2=0.6166>C u 1=0.5231,according to (P2)and (P4)in Theorem 1,it easily follows that p ( C 1?

C 2)=0and p ( C 1?

C 2)=1.

Using Eq.(13),we get

p (

C 2?

C 3)=max 1?max

0.6997?0.6166

(0.7634?0.6166)+(0.6997?0.5352)

,0,0

=0.7331.

Hence,it follows that

p (

C 2?

C 3)=1?0.7331=0.2669,

according to (P4)in Theorem https://www.360docs.net/doc/c116711203.html,ing Eq.(14),we have

p 23=p (x 2 x 3)=p (

C 2?

C 3)=0.7331

and

p 32=p (x 3 x 2)=p (

C 3?

C 2)=p (

C 2?

C 3)=0.2669.

Likewise,Using Eqs.(13)and (14)and Theorem 1as well as in the same way to Step 6in Section 6.2,the inclusion comparison probability matrixcan be obtained as follows:

P =?

0.5

000

10.50.73310.616010.26690.50.31911

0.38400.68090.5

Using Eq.(15)and in the same way to Step 7in Section 6.2,optimal degrees of membership for the alternatives x i (i =1,2,3,4)can be calculated as follows:

?1=0.1250,

?2=0.3208,

?3=0.2572,

?4=0.2971

respectively.Then,the best alternative is x 2and the ranking order of the four alternatives is generated as follows:

x 2 x 4 x 3 x 1,

which is the same as that obtained by using the IVIF weighted average operator and a new accuracy function [37].

It is noticing that although the decision results obtained by the decision making method based on a new accuracy function [37]and the developed methodology in this paper are the same,the decision principles and processes are very different.On the one hand,the former simply summed the weighted average values of the IVIF sets of the alternatives x i (i =1,2,3,4)on the attributes a j (j =1,2,3)using the set-theoretical operations and ranked the alternatives based on a new accuracy function (i.e.,a defuzzi?cation method).The latter takes into consideration not only both the IVIFPIS x +and the IVIFNIS x ?but also the relative importance of the squares of the weighted Euclidean distances between x i and x +as well as x ?.On the other hand,the former is only applicable to the MADM problems in which weights of attributes are already known a priori .The latter can solve the MADM problems with different preference information structures between completely known and completely unknown as stated earlier.Moreover,the latter introduced the concepts of the pair-wise inclusion comparison probability and the optimal degree of membership for the alternatives which provide a new ranking method for IVIF sets by taking into consideration preference comparison information of all pair-wise alternatives rather than single alternative information.7.Conclusions

Based on the concept of the closeness coef?cient which is similar to that in the TOPSIS [19],auxiliary nonlinear programming models are constructed to solve MADM problems in which membership/satisfaction and non-membership/non-satisfaction degrees of alternatives on attributes are expressed using IVIF sets and preference information of attribute importance is incomplete.Simpler nonlinear programming models are deduced to calculate the closeness IF sets of alternatives with respect to the IVIFPIS,which are used to estimate optimal degrees of membership and hereby generate the ranking order of alternatives.Mathematically the proposed ranking method of the IVIF sets in this paper is more rational and reliable than that based on the score and accuracy functions since the former takes into consideration all pair-wise comparison information of the discussed alternatives rather than information of single alternative.Furthermore,the derived auxiliary nonlinear programming models (i.e.,Eqs.(11)and (12))in this paper is extremely ?exible since some extensions or specializations are easily implemented by relaxing or imposing some speci?c features,constraints,assumptions or requirements.These extensions or specializations of the derived auxiliary nonlinear programming models can further increase the range of their applicability or simplify/decrease their solving process and computation amount.It is shown from aspects of theory analysis and speci?c application that the derived auxiliary nonlinear programming models and hereby the developed methodology are applicable to MADM problems with IVIF sets and different preference information https://www.360docs.net/doc/c116711203.html,ly,the developed models and method in this paper are of universality and practicability.

As stated earlier,the IF set and IVF set are two special cases of the IVIF set.Therefore,the proposed methodology in this paper can be applied to MADM problems with either IF sets or IVF sets and incomplete information.

D.-F.Li /Applied Soft Computing 11(2011)3402–34183413

Although the investment company selection problem is chosen to be an illustrating example,the proposed methodology in this paper may be applicable to some similar decision problems such as engineering management,environment and ?ood control as well as supply chain.

It is easy to see that constructing IVIF sets (i.e.,extracting interval-valued degrees of membership and non-membership)is a key problem of applying the proposed methodology to practical decision situations.Generating methods of IVIF sets or extracting methods of interval-valued degrees of membership and non-membership will be investigated in near future.Moreover,it is worthwhile to investigate other effective and practical methods for the derived auxiliary nonlinear programming models (i.e.,Eqs.(11)and (12))including heuristic and meta-heuristic approaches in the future.Acknowledgments

This research was sponsored by the Natural Science Foundation of China (Nos.70871117and 70902041)and the Humanities and Social Sciences research project of the Ministry of Education of China (08JC630072).Appendix A.Appendix I Proof of Theorem 1

In this proof,we will verify the six properties (P1)–(P6)of the inclusion comparison probability aforementioned in Theorem 1.(P1)It is obvious that

max

C u k

?C l i

C i + C k

≥0.

Hence,it easily follows that

1?max

C u k

?C l i

C i + C k

≤1.

Then,we get

max

1?max

C u k

?C l i

C i + C k

≤1.

Thereby,it is directly derived from Eq.(13)that 0≤p (C i ?C k )≤1,i.e.,(P1)is valid.

(P2)It is easily derived from the de?nition of the IF sets C i = C l i ,1?C u i and C k = C l k

,1?C u k that both C l i ≤C u i and C l k ≤C u k

,i.e.,C u i ?C l i ≥0and C u k

?C l k ≥https://www.360docs.net/doc/c116711203.html,bining these with the given assumption:C u i ≤C l k ,i.e.,C u i ?C l k

≤0,we have 0≤ C i + C k =(C u i ?C l i )+(C u k

?C l k )=(C u k ?C l i )+(C u i ?C l k )≤C u

k ?C l i ,which directly implies that

C u k

?C l i

C i + C k

≥1.

Then,we get

max

C u k

?C l i

C i C k ,0

C u k

?C l i

C i C k

≥1,

i.e.,

1?max

C u k

?C l i

C i C k

≤0.

Hereby,we have

max

1?max

C u k

?C l i

C i + C k

=0.

Thus,it is proven that p (C i ?C k )=0according to Eq.(13),i.e.,(P2)is valid.

(P3)Obviously, C i + C k ≥https://www.360docs.net/doc/c116711203.html,bining this with the given assumption:C l i ≥C u k

,i.e.,C u k ?C l i

≤0,it directly follows that C u k

?C l i

C i + C k

≤0.

3414 D.-F.Li /Applied Soft Computing 11(2011)3402–3418

Then,max

C u k ?C l i C i

+ C

=0.Hence,we have

1?max

C u k

?C l i

C i + C k

=1.

It easily follows that

max 1?max

C u k

?C l i

C i + C k

=1.

Thus,it is proven that p (C i ?C k )=1according to Eq.(13),i.e.,(P3)is valid.

(P4)Analyzing the IF sets C i = C l i ,1?C u i and C k = C l k

,1?C u k

,it is easily seen that there are the following four relations for the values C l i ,C u k

,C l k and C u k :(a)C l i ≤C l k and C u i ≤C u k ;(b)C l i ≤C l k and C u i ≥C u k ;(c)C l i ≥C l k and C u i ≤C u k ;(d)C l i ≥C l k and C u i ≥C u k

.In the following,we will verify (P4)according to the aforementioned cases (a)–(d).

Case (a):It is directly derived from the de?nition of the IF set C i = C l i ,1?C u i that C l i ≤C u i .Combining this with the given assumption:C u i ≤C u k

,it follows that C l i ≤C u k ,i.e.,C u k ?C l i

≥0.Two cases should be further taken into consideration.(C1)If C l k

≤C u i ,i.e.,C u i ?C l k

≥0,then C i + C k =(C u i ?C l i )+(C u k

?C l k )=(C u k ?C l i )+(C u i ?C l k )≥C u

k ?C l i ≥0,which implies that

0≤

C u k

?C l i

C i + C k

≤1.

Hence,it directly follows that

0≤max

C u k

?C l i

C i + C k

C u k

?C l i

C i + C k

≤1,

i.e.,

0≤1?max

C u k

?C l i

C i + C k ,0

=1?

C u k

?C l i

C i + C k

≤1.

Thereby,we have

max 1?max

C u k

?C l i

C i + C k ,0

=1?max

C u k

?C l i

C i + C k ,0

=1?

C u k

?C l i

C i + C k

According to Eq.(13),we get

p (C i ?C k )=1?

C u k

?C l i

C i + C k

Similarly,it is proven that

p (C i ?C k )=1?

C u i ?C l k

C i C k

Combining the aforementioned two equalities with C i =C u i ?C l i and C k =C u k

?C l k

,we have p (C i ?C k )+p (C i ?C k )=

C u k

?C l i

C i + C k

C u i ?C l k

C i + C k

=2?

(C u k

?C l k )+(C u i ?C l i

) C i + C k

=1,

i.e.,p (C i ?C k )+p (C i ?C k )=1.

(C2)If C l k ≥C u i

,then according to (P2)as stated earlier,it directly follows that p (C i ?C k )=0.Likewise,according to (P3)as above,it directly follows that p (C i ?C k )=1.Therefore,we have p (C i ?C k )+p (C i ?C k )=1.

Case (b)It directly follows from the de?nition of the IF set C k = C l k

,1?C u k that C l k ≤C u k

.Combining this with the given assumption:C u i ≥C u k

,it directly follows that C u i ?C l k ≥0.Furthermore,noticing the given assumption:C l i ≤C l k ,it follows that C l i ≤C u k ,i.e.,C u k ?C l i

≥0.Hence,we have

C i + C k =(C u i ?C l i )+(C u k ?C l k )=(C u k ?C l i )+(C u i ?C l k )≥C u

k ?C l i ≥0,

which implies that

0≤

C u k

?C l i

C i + C k

≤1.

D.-F.Li /Applied Soft Computing 11(2011)3402–34183415

Then,we have

0≤max

C u k

?C l i

C i + C k ,0

C u k

?C l i

C i + C k

≤1,

i.e.,

0≤1?max

C u k

?C l i

C i C k ,0

=1?

C u k

?C l i

C i C k

≤1.

Thereby,we have

max 1?max

C u k

?C l i

C i + C k ,0

=1?max

C u k

?C l i

C i + C k ,0

=1?

C u k

?C l i

C i + C k

Using Eq.(13),it is easily seen that

p (C i ?C k )=1?

C u k

?C l i

C i + C k

Similarly,it is proven that

p (C i ?C k )=1?

C u i ?C l k

C i + C k

Hence,we have

p (C i ?C k )+p (C i ?C k )=

C u k

?C l i

C i + C k +

C u i ?C l k

C i + C k

=2?

(C u k

?C l k )+(C u i ?C l i

) C i + C k

=1.

Namely,p (C i ?C k )+p (C i ?C k )=1.

Cases (c)and (d)can be proven in the same way to Cases (b)and (a),respectively.Thus,we prove that (P4)is valid.

(P5)If p (C i ?C k )=p (C i ?C k ),then according to (P4)as above,it easily follows that p (C i ?C k )=p (C i ?C k )=1/2,i.e.,(P5)is valid.(P6)According to the given assumption:p (C i ?C k )≥1/2and p (C k ?C t )≥1/2,we have

C u k

?C l i

(C u i ?C l i )+(C u k ?C l k

)≥

,C u k ?C l i ≥0

and

C u t

?C l k

(C u k

?C l k )+(C u t

?C l t

)≥

,C u t

?C l k ≥0.Simply computing,the above systems of inequalities can be rewritten as follows:

0≤C u k

?C l i ≤C u i ?C l

k and

0≤C u t

?C l k ≤C u k ?C l t ,respectively.Summing the above systems of inequalities,it follows that

0≤C u t

?C l i ≤C u i ?C l t .Hence,we obtain

0≤2(C u t

?C l i )≤(C u i ?C l t )+(C u t ?C l i )=(C u i ?C l i )+(C u t ?C l t )= C i + C t ,which implies that

0≤

C u t

?C l i

C i + C t

≤

.Then,we have

0≤max

C u t

?C l i

C i + C t ,0

C u t

?C l i

C i + C t

≤

.Hence,it easily follows that

1≥1?max

C u t

?C l i

C i + C t ,0

=1?

C u t

?C l i

C i + C t

≥

3416 D.-F.Li /Applied Soft Computing 11(2011)3402–3418

which implies that

1≥max {1?max {

C u t

?C l i

C i + C t

,0},0}=1?max {

C u t

?C l i

C i + C t

,0}≥

.According to Eq.(13),it is directly proven that p (C i ?C t )≥

,i.e.,(P6)is valid. Appendix B.Appendix II Proof of Theorem 2

In order to prove Theorem 2,we need to prove two conclusions,which are summarized in Lemmas 1and 2.Lemma 1.Assume that P is the inclusion comparison probability matrix given by Eq.(14).Then,P is a fuzzy complementary judgment

matrix.Proof.

According to (P1)and (P4)in Theorem 1,it directly follows that

0≤p ik ≤1,

p ik +p ki =1

(i,k =1,2,...,m ),

which are the conditions that a fuzzy complementary judgment matrix should satisfy [9,16].Thus,P is proven to be a fuzzy complementary judgment matrix.

Let p i =

m t =1

p it ,

(A1)

which is the sum of the inclusion comparison probabilities p it (t =1,2,...,m )in the i -th row.For a pair of p i and p k (i ,k =1,2,...,m ),the

linear transform is made as follows:

áik =

p i ?p k 2(m ?1)

2.(A2)

All values áik (i ,k =1,2,...,m )are concisely expressed in the matrix format as á=(áik )m ×m .

Lemma 2.Assume that P is the inclusion comparison probability matrix given by Eq.(14).The matrix áis obtained through using Eqs.

(A1)and (A2).Then,áis a fuzzy complementary and consistent judgment matrix.

Proof.We only need to prove that the judgment matrix áis fuzzy,complementary and additive transitive,respectively.

(C1)áis a fuzzy judgment matrix

According to Lemma 1as above,P is a fuzzy complementary judgment matrix,which implies that

0≤p ik ≤1,

p ii =0.5,p ik +p ki =1(i,k =1,2,...,m ).(A3)

It is easily seen that Eq.(A1)can be rewritten as follows:

p i =

m t =1,t /=i

p it +p ii

(i =1,2,...,m ).

Combining this with Eq.(A3),it easily follows that

0.5≤p i ≤(m ?1)+0.5(i =1,2,...,m ).

Hence,we have

?(m ?1)≤p i ?p k ≤m ?1

(i,k =1,2,...,m ),which implies that

12≤p i ?p k 2(m ?1)

≤12(i,k =1,2,...,m ).

Using Eq.(A2),it directly follows that

0≤áik ≤1

(i,k =1,2,...,m ),

(A4)

which is the condition that a fuzzy judgment matrix should satisfy [9,16].Thus,áis proven to be fuzzy.(C2)áis a complementary judgment matrix According to Eq.(A2),it directly follows that

áik +áki =

p i ?p

2(m ?1)+12 + p k ?p i

2(m ?1)

+12

(p i ?p k )+(p k ?p i )

2(m ?1)

+1=1

(i,k =1,2,...,m ),

D.-F.Li /Applied Soft Computing 11(2011)3402–34183417

i.e.,áik +áki =1(i ,k =1,2,...,m ),which is the condition that a complementary judgment matrix should satisfy.Hereby,áis proven to be complementary.

(C3)áis an additive transitive judgment matrix

According to Eq.(A2),it follows that for all i ,k =1,2,...,m ,

áit ?ákt

+12

= p i

t 2(m ?1)+12 ? p k ?p t 2(m ?1)+12

+12=(p i ?p t )?(p k ?p t )2(m ?1)+12=p i ?p k

2(m ?1)

+12=áik ,i.e.,áik =áit ?ákt +0.5(i ,k =1,2,...,m ).These equalities may be rewritten as follows:

(áik ?0.5)+(ákt ?0.5)=áit ?0.5(i,k =1,2,...,m ),

which is the condition that an additive transitive judgment matrix should satisfy [9,16].Thus,áis proven to be additive transitive.Proof of Theorem 2.We construct the matrix áby using Eqs.(A1)and (A2).According to Lemma 2,áis a fuzzy complementary and

consistent judgment matrix.In the following,we will determine optimal degrees of membership for alternatives x i (i =1,2,...,m )in a similar way to the weight determination method of eigenvectors.

Firstly,the sum of all values of each row in the matrix áis calculated as follows:

ái =

m t =1

áit

(i =1,2,...,m ).

Then,the normalized value of ái is computed as follows:

?i =á

m t =1át

(i =1,2,...,m ),

i.e.,

?i = m

t =1áit

m i =1

t =1áit

(i =1,2,...,m ).

According to Lemma 1,P is a fuzzy complementary judgment matrix.Then,we get

m t =1m j =1

p tj =

m 2

.Combining this with Eqs.(A1)and (A2),we have

?i =

t =1áit

1≤t

(áit +áti )+(m /2)=

m t =1

(p i ?p t )/[2(m ?1)]+1/2

[m (m ?1)/2]+(m /2)=

mp i ?

t =1p t

+m (m ?1)m 2(m ?1)

mp i ?

m t =1

j =1p tj

+m (m ?1)

m 2(m ?1)

=p i +(m /2)?1

m (m ?1)

k =1p ik

+(m /2)?1

m (m ?1)

Namely,

?i =

m (m ?1)

k =1

p ik +

m 2

(i =1,2,...,m ),

which are the optimal membership degrees of the alternatives x i .Thus,Theorem 2is proven. References

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