Evidential supplier selection based on interval data fusion
aa r X i v : . [ c s . A I] M a r Evidential supplier selection based on interval datafusion
Zichang He a , Wen Jiang a, ∗ a School of Electronics and Information, Northwestern Polytechnical University, Xi’an,Shaanxi, 710072, China
Abstract
Supplier selection is a typical multi-criteria decision making(MCDM) prob-lem and lots of uncertain information exist inevitably. To address this issue,a new method was proposed based on interval data fusion. Our method fol-lows the original way to generate classical basic probability assignment(BPA)determined by the distance among the evidences. However, the weights ofcriteria are kept as interval numbers to generate interval BPAs and do thefusion of interval BPAs. Finally, the order is ranked and the decision ismade according to the obtained interval BPAs. In this paper, a numericalexample of supplier selection is applied to verify the feasibility and validityof our method. The new method is presented aiming at solving multiple-criteria decision-making problems in which the weights of criteria or expertsare described in fuzzy data like linguistic terms or interval data.
Keywords:
Dempster-Shafer theory; Interval data; MCDM; Supplierselection; TOPSIS; Fuzzy data ∗ Corresponding author at: School of Electronics and Information, Northwestern Poly-technical University, Xi’an, Shaanxi 710072, China. Tel: (86-29)88431267. E-mail address:[email protected], [email protected]
Preprint submitted to Elsevier March 7, 2017 . Introduction
Multiple-criteria decision-making has wide application in dealing with thecomparison of multiple decisions. Because many decision-making projectslike supplier selection will inevitably include the consideration of evidencebased on several criteria[1], rather than on a preferred single criterion in realworld. An effective framework for decisions comparison based on the evalu-ation of multiple criteria is proposed in MCDM. It means that an optimaldecision will be made based on comprehensive consideration. Compared withthose approaches based on experience and intuition, MCDM is apparentlymore objective and reasonable[2–4].In realistic situation, selections often proceed under the environment oc-cupied with unknown and uncertain information. As the the complexityof the system grows, the uncertainty of the problems and the fuzziness ofhuman’s thinking constantly increase accordingly. Hence, It is difficult forpeople to judge and distribute the importance of each criterion in MCDM.Fuzzy sets theory introduced by Zadeh is a good approach to settle with theuncertain information[5–7]. And it has wide application in MCDM[8–11].Decision making and optimization under uncertain environment is heavilystudied[12, 13]. Besides fuzzy sets theory, D-S evidence theory introducedby Dempster and Shafe[14, 15] plays an important role in making decisionsunder uncertain environment[16–18]. Due to the efficiency modeling andfusion of information, evidence theory is widely used[19–21]. D-S evidencetheory is also a powerful tool to deal with MCDM problems[22–26].As an effective tool to deal with the fuzzy data, interval number has beenwidely applied in MCDM. Chen and Chen-Tung(2000) defined a preferencerelation between each pair of plant locations based on the interval analysis2nd proposed ranking method to determine the ranking order of all candi-date location[27]. Some paper like Janhanshanloo et.al(2006)[28], Yue andZhongliang(2011)[29] and Liu et.al(2013)[30] proposed their methods to gen-erate weights of criteria based on interval data. And Deng et.al(2011)[31]converted interval number to a crisp weight based on distance function andproposed a method to combine D-S evidence theory and fuzzy set theory toaddress MCDM problems.In this paper, we propose a new method to solve MCDM problems basedon the interval data fusion. In our method, interval data is retained duringfusion process, which has some appreciable properties. First, interval datareflects the concrete and detailed information of the objects in a great extent.Additionally, it has some practical applications in some specific situations.For example, when the system requires the extremely high degree of accuracywe can only employ the lower limiting value and abandon the rest informa-tion to make decisions. Second, the weights of criteria or the informationsources are allowed to be modeled as fuzzy number. Because we can con-vert the fuzzy description into interval data based on fuzzy set theory(FST).This property is quite useful in that not only quantitative data but alsothe qualitative representation is widely used in the practical decision-makingproblems. Third, fusing evidence based on interval data conforms with theuniversal cognition. Crisp data is a special form of interval data(like 0.5 canbe seen as [0.5, 0.5]) in a way. In other words, our method is a generalizedone of Deng et.al(2011)[31].The rest of this paper is organized as follows. The preliminaries of the basictheory employed are briefly presented in Section 2. And then our new fusionmethod based on interval data is proposed in Section 3. Section 4 takes anumerical example of supplier selection to show the efficiency of the method.3inally, the paper is concluded in Section 5.
2. Preliminaries
In this section, some preliminaries such as interval number, fuzzy set the-ory(FST), Dempster-Shafer theory(DST) and Pignistic probability transfor-mation(PPT) are briefly introduced.
Definition 2.1. (Interval number) An interval number ˜ a is defined as ˜ a =[ a L , a U ] = { x | a L ≤ x ≤ a U } where a L is the lower limiting value and a U isthe upper limiting value while x ∈ [0 , . Let ˜ a and ˜ b be two arbitrary positive closed interval numbers. The basicalgorithm of interval number is given as follows[32]:˜ a + ˜ b = [ a L + b L , a U + b U ] (1)˜ a × ˜ b = [ a L b L , a U b U ] (2)˜ a ÷ ˜ b = [ a L b L , a U b U ] (3) k ˜ a = [ ka L , ka U ] (4)4˜ a = [ 1 a U , a L ] (5)for ˜ a and ˜ b , let norm (cid:13)(cid:13)(cid:13) ˜ a − ˜ b (cid:13)(cid:13)(cid:13) = (cid:12)(cid:12) a L − b L (cid:12)(cid:12) + (cid:12)(cid:12) a U − b U (cid:12)(cid:12) be the so-called distancebetween the interval number ˜ a and ˜ b . Apparently, the larger (cid:13)(cid:13)(cid:13) ˜ a − ˜ b (cid:13)(cid:13)(cid:13) is, themore ˜ a and ˜ b differ. Especially, interval number ˜ a equals to ˜ b completelywhen (cid:13)(cid:13)(cid:13) ˜ a − ˜ b (cid:13)(cid:13)(cid:13) = 0. Fuzzy set Introduced by Zadeh is an extension of classic set[33]. It is anefficient tool to model linguistic variables.
A fuzzy set is any set that allows its members to have different grades ofmembership in the interval [0,1]. It consists of two components: a set and amembership function associated with it.
Definition 2.2. (Fuzzy set). Let X be a collection of objects denoted gener-ally by x, a fuzzy subset of X ˜ A is a set of ordered pairs[34]: ˜ A = { ( x, µ ˜ A ( x ) | x ∈ X ) } (6) µ ˜ A ( x ) is called the membership function (generalized characteristic function)which maps X to the membership space M. Its range is the subset of nonneg-ative real members whose supreme is finite. Definition 2.3. (Triangular fuzzy number). A fuzzy number is a fuzzy sub-set of X. And a triangular fuzzy number ˜ A can be defined by a triplet (a,b,c) hown in Fig. 1. Its membership function is defined as[35] µ ˜ A ( x ) = , x < a x − ab − a , a ≤ x ≤ b c − xc − b , b ≤ x ≤ c , x > c (7) Figure 1: A triangular fuzzy number
Linguistic variable is a variable with linguistic words or sentences in a naturallanguage[36]. It is widely used in practical life and it is one of the most clas-sical fuzzy information. When dealing with situations which are too complexor ill-defined to be accurately described in conventional quantitative expres-sions, it’s convenient and reasonable to do a qualitative description. Gener-ally, each linguistic variable corresponds to a fuzzy set. For example, theselinguistic variables can be expressed in positive triangular fuzzy number[37]as Table 1.Virtually, the concrete models used to represent the linguistic items are flexi-ble and changeable. To apply which kind of represent method depends on therealistic application systems and the domain experts’ opinions. In MCDM6 able 1: Linguistic variables for importance in triangular fuzzy number[37]Terms triangular fuzzy numberVery low (VL) (0, 0.1, 0.3)Low (L) (0.1, 0.3, 0.5)Medium (M) (0.3, 0.5, 0.7)High (H) (0.5, 0.7, 0.9)Very high (VH) (0.7, 0.9, 1.0) problems like supplier selection in Section 4, our method adopted the methodwhich converts the linguistic variable into interval data.
Dempster-Shafer theory is a mathematical theory of evidence which is usedto combine separate pieces of information(evidence) to calculate the beliefprobability of an event. In a D-S theory reasoning scheme, the set of possiblehypotheses are collectively called the frame of discernment Θ, defined asfollows[15]: Θ= { H , H , H , . . . . . . , H n } where n is the number of exclusive and exhaustive elements in the set. Formthe frame of discernment Θ, let P ( θ ) denote the power set composed withthe 2 N propositions A of Θ: P ( θ ) = {∅ , { H } , { H } , . . . . . . { H n } , { H ∪ H } , { H ∪ H } , . . . . . . θ } where ∅ denotes the empty set. Then a mass function m is defined asm (cid:0) θ (cid:1) ∈ [0 ,
1] to distribute the belief across the frame meeting the followingconditions: m ( ∅ ) = 0 and P A ⊆ θ m ( A ) = 17nder these circumstances, the beliefs of the evidence source can only beassigned to non-empty hypotheses and must sum to 1. When the belief isassigned to one hypothesis, the more elements the hypothesis contains, theless information it offers. Especially, a hypothesis containing all the elementsmeans nothing is informative essentially. For the algorithm designed to accessevidence, the most significant ability is to combine the evidence from multiplesources. And the crucial process of combining two pieces of evidence fromindependent sources is fulfilled with the following equation called Dempster’scombination rule: m ( A ) = P ∀ x,y : x ∩ y = A m ( X ) · m ( Y )1 − K (8)with K = X ∀ X,Y : X ∩ Y = ∅ m ( X ) · m ( Y ) (9)where m ( A ) is the new belief for the hypothesis A yielded from the originalevidence m and m . Apparently, Eq.(8) can only be applied when K = 0. K is called the conflict coefficient and 1 − K is a constant coefficient used tonormalize the combined evidence. And all combined evidence whose intersec-tion is not the hypothesis of interest A is represented by K. Its value revealsthe degree of the confliction between the two original evidence, K = 0 meansthe consistence of the belief assignment, whereas K = 1 means the completecontradiction .Likewise, when the evidence is from j different sources, the rule can be ex-pressed as: m ( A ) = P ∀ X,Y ...Z : X ∩ Y ... ∩ Z = A m ( X ) · m ( Y ) · · · m j ( Z )1 − K (10)with K = X ∀ X,Y ...Z : X ∩ Y ... ∩ Z = ∅ m ( X ) · m ( Y ) · · · m j ( Z ) (11)8 .4. Pignistic probability transformation Virtually, two levels are classified to describe the beliefs: one is the credallevel where belief is entertained. And the other one is the pignistic levelwhere beliefs are feasible to make decisions[38]. The term ”pignistic” pro-posed by Smets is originated from the word pignus, meaning ’bet’ in Latin.Pignistic probability has a wide application on decision-making. Principle ofinsufficient reason is used to assign the basic probability of multiple-elementset to singleton set. In other word, a belief interval is distributed into thecrisp ones determined as: bet ( A i ) = X A i ⊆ A k m ( A k ) | A k | (12)where | A k | denotes the number of elements in the set called the cardinality.Eq.(12) is also called as Pignistic Probability Transformation(PPT).
3. Proposed method
In this section, our new method based on interval data fusion is proposed.In general, a basic MCDM problem can be modeled as follows: For a certainproblem, there is a committee of k decision-makers { D , D , D , . . . . . . , D k } to evaluate it. Each decision maker holds m alternatives { A , A , A , . . . . . . , A m } .And for each alternative, n criteria { C , C , C , . . . . . . , C n } are in considera-tion to make decisions(usually the same criterion is shared). The followingis a succinct model proposed by Hwang and Yoon[39] to express MCDM in9 matrix format. C C . . . C n D k = A A ... A m r r . . . r n r r . . . r n ... ... . . . ... r m r m . . . r mn where r mn is the rating of alternative A m with respect to criteria C n whichis usually described crisply or fuzzily. In our method, r mn is allowed to be acrisp number or in the form of an interval data. For now, the facing problemis how to acquire r mn .In the practical, the final aim is often to rank the alternatives and make thebest selection. Accordingly, the final scores of every alternative are not caredtoo much. Considering that, Hwang and Yoon proposed TOPSIS(Techniquefor Order Preference by Similarity to Ideal Solution) to solve MCDM[40].The principle is that the chosen alternative should have the shortest distancefrom the positive ideal solution and the farthest distance from the positiveideal solution. Based on TOPSIS, Deng et al.[31] proposed a new methodusing FST together with DST. In that method, the ideal solution , negativeideal solution is determined and the distances of an alternative between themare determined. Then the classical BPA is generated to describe how closebetween both the alternative to ideal solution and to negative ideal solution.In the classical TOPSIS, the performance ratings and the weights of thecriteria are given as crisp values. Hence, Deng et al.[31] changed the fuzzyMCDM problem into a crisp one via using the distance function. However,his method only average the lower limitation and the upper limitation of theinterval. The new crisp weights are generated according to the average in theessence. It means that one criterion holds the weight of [0 . , .
9] measures the10ame as another one holds [0 . , . a , a smaller result of a U − a L represents that the informationabout the criterion is more clear when the sum of a L and a U is constant.Whereas the above, it’s rational for us to allocate a larger crisp weight to thecriterion which weighs [0 . , .
6] than the one holding [0 . , .
9] . To improveDeng et al.’s method[31], we retain the interval data in the fusion procedureand generate the interval BPA. Based on the TOPSIS, the elements of ourinterval BPA are { IS(ideal solution) } , { NS(negative solution) } and { IS,NS } ,of which IS,NS is the frame of discernment. The following is the example ofthe interval BPA for one certain alternative m ( { IS } ) = [ a L , a U ] m ( { N S } ) = [ b L , b U ] m ( { IS, N S } ) = [ c L , c U ]It means that:1) The hypothesis ”the alternative is an ideal solution” is upheld with beliefdegree from a L to a U .2) The hypothesis ”the alternative is a negative ideal solution” is upheld withbelief degree from b L to b U .3) The hypothesis ”the alternative is perceived as a discernment, namely itis likely to be an ideal solution or a negative solution” is upheld with beliefdegree from c L to c U .It is worth mentioning that m ( { IS, N S } ) = 1 − m ( { IS } ) − m ( { N S } ).Hence, it’s easy to know that c L = 1 − a U − b U and c U = 1 − a L − b L .11or the focal element ( { IS } , { N S } , { IS, N S } ), there is another way to ex-press interval BPA as { [ a L , b L , c L ] , [ a U , b U , c U ] } . When making a decisionbased on interval BPA, we can fuse the BPA consisting of the lower limita-tion and the one consisting of the upper limitation into a classical BPA byfusing the left part and right part with 8. In the ultimate, PPT is used tocompare the BPA of IS. In accordance with the notion mentioned above, ournew method can be stated step by step as follows: Step 1.
Determine the ideal solution and negative ideal solution. Andgenerate the classical BPA of each performance based on the distance betweenIS and NS.
Step 2.
Convert the criteria’s weights including crisp data(0.5 can be seenas [0.5,0.5]) and linguistic items into an interval number. And then discountthe classical BPAs using the interval data to generate the interval BPA ofeach performance. Combine the interval BPAs of each criterion to get onecomprehensive evaluation of an alternative.
Step 3.
Convert the decision makers’ weights including crisp data and lin-guistic items into interval numbers. And then discount the interval BPAs ofcombined performance (obtained in Step 2) using the interval data to gener-ate the interval BPA of each performance. Combine the interval BPAs of alldecision makers’ to get the performance of each alternative.
Step 4.
Combine the the left part and the right part of the interval BPAsto get the final performance of each alternative.
Step 5.
Compare and rank the order of decisions based on PPT and makethe best decision. 12 . Numerical example
Supplier selection is a typical MCDM problem where lots of fuzzy informationexist. In reality, although managers claims that the quality is the most im-portant attribute for a supplier, they actually choose suppliers based largelyon cost and delivery performance[41]. To identify the availability of our newmethod, the numerical example used in paper[31] is adopted in this section.The initial condition, such as the classical BPA of each performance and theweights of each criterion as well as the weights of experts are shown in Table2.As shown in Table 2, there are four criteria, including product late delivery,cost, risk factor and suppliers’ service performance detailed as following:C1: Product late delivery. The delivery process can reflect the service abilityof a supplier. It is considered to investigate whether the supplier can supplystable and constant appreciation serve for the enterprise.C2: Cost. A good price measures quite a lot in reducing cost and increasingthe competitive force.C3: Risk factor. If we want to make long-term cooperation with a sup-plier, then we must take its risk factor (political factor, economic factor, thereputation, etc.) into account.C4: Supplier’s service performance. Service performance means the sustain-ing promotion of the product and service(e.g. product quality acceptancelevel, technological support, information process), which is deemed as thecore factor. 13 able 2: Data of supplier selection in Deng et.al(2011)
Performance C C C C { IS } , { NS } , { IS, NS } ) ( { IS } , { NS } , { IS, NS } ) ( { IS } , { NS } , { IS, NS } ) ( { IS } , { NS } , { IS, NS } )DM1 Weights [0.20,0.35] [0.30,0.55] [0.05,0.30] [0.25,0.50][0.20,0.45] Supplier1 (0.60,0.20,0.20) (0.6429,0.0714,0.2857) (0.60,0.20,0.20) (0.60,0.20,0.20)Supplier2 (0.60,0.20,0.20) (0.6429,0.0714,0.2857 (0.50,0.50,0) (0.50,0.50,0)Supplier3 (0.50,0.50,0) (0.50,0.50,0) (0.60,0.20,0.20) (0.6667,0,0.3333)Supplier4 (0.66667,0,0.3333) (0.6667,0,0.3333) (0.50,0.50,0) (0.50,0.50,0)Supplier5 (0,0.6667,0.3333) (0,0.6667,0.3333) (0,0.6667,0.3333) (0,0.6667,0.3333)Supplier6 (0.20,0.60,0.20) (0.0714,0.6429,0.2857) (0.6667,0,0.3333) (0,0.6667,0.3333)DM2 Weights [0.25,0.45] [0.20,0.55] [0.05,0.3] [0.20,0.60][0.35,0.55] Supplier1 (0.60,0.20,0.20) (0.6429,0.0714,0.2857) (0.50,0.50,0) (0.60,0.20,0.20)Supplier2 (0.60,0.20,0.20) (0.6429,0.0714,0.2857) (0,0.6667,0.3333) (0.50,0.50,0)Supplier3 (0.50,0.50,0) (0.50,0.50,0) (0.6667,0,0.3333) (0.6667,0,0.3333)Supplier4 (0.66667,0,0.3333) (0.6667,0,0.3333) (0.50,0.50,0) (0.50,0.50,0)Supplier5 (0,0.6667,0.3333) (0,0.6667,0.3333) (0,0.6667,0.3333) (0.20,0.60,0.20)Supplier6 (0.20,0.60,0.20) (0.0714,0.6429,0.2857) (0.6667,0,0.3333) (0,0.6667,0.3333)DM3 Weights [0.20,0.55] [0.20,0.70] [0.10,0.40] [0.20,0.60][0.70,0.95] Supplier1 (0.60,0.20,0.20) (0.6429,0.0714,0.2857) (0.5714,0.2857,0.1429) (0.6667,0,0.3333)Supplier2 (0.60,0.20,0.20) (0.6429,0.0714,0.2857 (0.6667,0,0.3333) (0.2857,0.5714,0.1429)Supplier3 (0.50,0.50,0) (0.50,0.50,0) (0.6667,0,0.3333) (0.6667,0,0.3333)Supplier4 (0.66667,0,0.3333) (0.6667,0,0.3333) (0.5714,0.2857,0.1429) (0.6667,0,0.3333)Supplier5 (0,0.6667,0.3333) (0,0.6667,0.3333) (0,0.6667,0.3333) (0.2857,0.5714,0.1429)Supplier6 (0.20,0.60,0.20) (0.0714,0.6429,0.2857) (0.6667,0,0.3333) (0,0.6667,0.3333) t should be noticed that the weights are ready interval data. If they are de-scribed in the fuzzy linguistic items, we can also convert them into intervaldata. Table 3 is an example in which the linguistic items and their accord-ing interval data differ in different situations. It is one of the remarkableadvantages of our method. And the criterion of the value of interval numberdepends on the experts’ opinions. Table 3: Convert linguistic variables into the interval dataTerms Interval dataVery low (VL) [0, 0.3]Low (L) [0.1, 0.5]Medium (M) [0.3, 0.7]High (H) [0.5, 0.9]Very high (VH) [0.7, 1.0]
Before applying our method, a flow chart(Figure 2) is shown to summarize thewhole procedure of applying our method in the supplier selection problem.Based on it, the detailed processes will be illustrated step by step in thefollowing.
Step 1.
Determine the ideal solution and negative ideal solution. Andgenerate the classical BPA of each performance based on the distance betweenIS and NS.Since the classical BPA of performance is already known in Table 2, we willimplement the following steps of our method to these data in order.
Step 2.
Convert the criteria’s weights including crisp data(0.5 can be seenas [0.5,0.5]) and linguistic items into an interval number. And then discountthe classical BPA using the interval data to generate the interval BPA ofeach performance. Combine the interval BPA of each criterion to get one15 igure 2: Supplier selection based on interval data fusion comprehensive evaluation of an alternative.In this situation, after simple data processing we can put the weights of16riteria into use directly, since all the original weights are in the form ofinterval data. What we only need to do is to normalize the interval numberwith the following equation: W = h a L , a U i. a max (13)where a max is the largest number among all the limitation values of the in-tervals.For example, the new weights of DM1’s four criteria are as following respec-tively: W C = [ 0 . , .
35 ] . .
70 = h . , . i W C = [ 0 . , .
55 ] . .
70 = h . , . i W C = [ 0 . , .
30 ] . .
70 = h . , . i W C = [ 0 . , .
50 ] . .
70 = h . , . i let the new weight be W = [ W min , W max ], then we can determine the intervalBPA as m ( { IS } ) = (cid:2) a L W min , a U W max (cid:3) (14) m ( { N S } ) = (cid:2) b L W min , b U W max (cid:3) (15) m ( { IS, N S } ) = (cid:2) − a L W min − b L W min , − a U W max − b U W max (cid:3) (16)17o the integrated one is expressed as: m ( { IS } , { N S } , { IS, N S } ) = (cid:16) h a L W min , a U W max i , h b L W min , b U W max i , h − a L W min − b L W min , − a U W max − b U W max i(cid:17) (17)or m ( { IS } , { N S } , { IS, N S } ) = (cid:16) m left , m right (cid:17) (18)with m left = h a L W min , b L W min , − a L W min − b L W min i m right = h a U W max , b U W max , − a U W max − b U W max i Let us take DM1’s evaluation to C1 of supplier1 as an example:m C [ IS ] = h . × . , . × . i = h . , . i m C [ N S ] = h . × . , . × . i = h . , . i m C [ IS, N S ] = h − . × . − . × . , − . × . − . × . i = h . , . i By using the Eq.(17), the rest interval BPA of each performance is listed inTable 4.Now all the preparation before fusion is completed. Takes decision maker1’sevaluation to supplier1 as an example to illustrate the procedure of combiningthe interval BPA(Figure 3).As the flow chart reveals, an interval BPA is equal to two classical BPAsgroups which consists of the left part and the right part of the interval re-spectively. Then the four BPAs consisting of the left part are fused togetherbased on DST, so do the other groups. The newly obtained two BPAs are18 able 4: Generating the interval BPAs of four criteria respectively.
Performance C C C C { IS } , { NS } , { IS, NS } ) ( { IS } , { NS } , { IS, NS } ) ( { IS } , { NS } , { IS, NS } ) ( { IS } , { NS } , { IS, NS } )DM1 Weights [0.2857,0.5] [0.4286,0.7857] [0.0714,0.4286] [0.3571,0.7143]Supplier1 ([0.1714,0.3],[0.0571, ([0.2755,0.5051],[0.0306, ([0.0428,0.2572],[0.0143, ([0.2143,0.4286],[0.0714,0.1],[0.7715,0.6]) 0.0561],[0.6939,0.4388]) 0.0857],[0.9429,0.6571]) 0.1429],[0.7143,0.4285])Supplier2 ([0.1714,0.3],[0.0571, ([0.2755,0.5051],[0.0306, ([0.0357,0.2143],[0.0357, ([0.1785,0.3572],[0.1785,0.1],[0.7715,0.6]) 0.0561],[0.6939,0.4388]) 0.2143],[0.9286,0.5714]) 0.3572],[0.6430,0.2856])Supplier3 ([0.1429,0.25],[0.1429, ([0.1429,0.3928],[,0.1429, ([0.0428,0.2572],[0.0143, ([0.2381,0.4762],[0,0.25],[0.7142,0.5]) 0.3928],[0.7242,0.2144]) 0.0857],[0.9429,0.6571]) 0],[0.7619,0.5238])Supplier4 ([0.1905,0.3333],[0, ([0.2857,0.5238],[0, ([0.0357,0.2143],[0.0357, ([0.1785,0.3572],[0.1785,0],[0.8095,0.6667]) 0],[0.7143,0.4762]) 0.2143],[0.9286,0.5714]) 0.3572],[0,0])Supplier5 ([0,0],[0.1905, ([0,0],[0.2857, ([0,0],[0.0476, ([0,0],[0.2381,0.3333],[0.8095,0.6667]) 0.5238],[0.7143,0.4762]) 0.2857],[0.9524,0.7143]) 0.4762],[0.7619,0.5238])Supplier6 ([0.0571 0.1],[0.0714, ([0.0306 0.0561],[0.2755, ([0.0476 0.2857],[0, ([0 0],[0.2381,0.3],[0.7715,0.6]) 0.5051],[0.6939,0.4388]) 0],[0.9524,0.7143]) 0.4762],[0.7619,0.5238])DM2 Weights [0.3571,0.6428] [0.2857,0.7857] [0.0714,0.4286] [0.2857,0.8571]Supplier1 ([0.2143,0.3857],[0.0714, ([0.1837,0.5051],[0.0204, ([0.0357,0.2143],[0.0357, ([0.1714,0.5143],[0.0571,0.1286],[0.7143,0.4857]) 0.0561],[0.7957,0.4388]) 0.2143],[0.9286,0.5714]) 0.1714],[0.7715,0.3143])Supplier2 ([0.2143 0.3857],[0.0714, ([0.1837 0.5051],[0.0204, ([0 0],[0.0476, ([0.1429 0.4285],[0.1429,0.1286],[0.7143,0.4857]) 0.0561],[0.7957,0.4388]) 0.2857],[0.9524,0.7143]) 0.4285],[0.7142,0.1430])Supplier3 ([0.1785,0.3214],[0.1785, ([0.1429,0.3928],[0.1429, ([0.0476,0.2857],[0, ([0.1905,0.5714],[0,0.3214],[0.6430,0.3572]) 0.3928],[0.7142,0.2144]) 0],[0.9524,0.7143]) 0],[0.8095,0.4286])Supplier4 ([0.2381,0.4286],[0, ([0.1905,0.5238],[0, ([0.0357,0.2143],[0.0357, ([0.1429,0.4285],[0.1429,0],[0.7619,0.5714]) 0],[0.8095,0.4762]) 0.2143],[0.9286,0.5714]) 0.4285],[0.7142,0.1430])Supplier5 ([0,0],[0.2381, ([0,0],[0.1905, ([0,0],[0.0476, ([0.0571,0.1714],[0.1714,0.4286],[0.7619,0.5714]) 0.5238],[0.8095,0.4762]) 0.2857],[0.9524,0.7143]) 0.5143],[0.7715,0.3143])Supplier6 ([0.0714,0.1286],[0.2143, ([0.0204,0.0561],[0.1837, ([0.0476,0.2857],[0, ([0,0],[0.1905,0.3857],[0.7143,0.4857]) 0.5031],[0.7957,0.4388]) 0],[0.9524,0.7143]) 0.5714],[0.8095,0.4286])DM3 Weights [0.2857,0.7857] [0.2857,1] [0.1429,0.5714] [0.2857,0.8571]Supplier1 ([0.1714,0.4714],[0.0571, ([0.1837,0.6429],[0.0204, ([0.0817,0.3265],[0.0408, ([0.1905,0.5714],[0,0.1571],[0.7715 0.3715]) 0.0714],[0.7957,0.2857]) 0.1632],[0.8775 0.5103]) 0],[0.8095,0.4286])Supplier2 ([0.1714,0.4717],[0.0571, ([0.1837,0.6429],[0.0204, ([0.0952,0.3810],[0, ([0.0816,0.2449],[0.1632,0.1571],[0.7715,0.3715]) 0.0714],[0.7957,0.2857]) 0],[0.9048,0.6190]) 0.4897],[0.7552,0.2654])Supplier3 ([0.1429,0.3928],[0.1429, ([0.1429 0.5],[0.1429, ([0.0952,0.3810],[0, ([0.1905,0.5714],[0,0.3928],[0.7142 0.2144]) 0.5],[7142,0]) 0],[0.9048,0.6190]) 0],[0.8095,0.4286])Supplier4 ([0.1905,0.5238],[0, ([0.1905,0.6667],[0, ([0.0817,0.3265],[0.0408, ([0.1905,0.5714],[0,0],[0.8095 0.4762]) 0],[0.8095,0.3333]) 0.1632],[0.8775 0.5103]) 0],[0.8095,0.4286])Supplier5 ([0,0],[0.1905, ([0,0],[,0.1905, ([0,0],[0.0952, ([0.0816,0.2449],[0.1632,0.5238],[0.8095 0.4762]) 0.6667],[0.8095,0.3333]) 0.3810],[0.9048 0.6190]) 0.4897],[0.7552,0.2654])Supplier6 ([0.0571,0.1571],[0.1714, ([0.0204,0.0714],[0.1837, ([0.0952,0.3810],[0, ([0,0],[0.1905,0.4714],[0.7715 0.3715]) 0.1429],[0.7959,0.2857]) 0],[0.9048 0.6190]) 0.5714],[0.8095,0.4286]) igure 3: Combine the interval BPA of the classical properties and act as the left part and the right part of thenew interval BPA respectively. In the same way, we can get all the intervalBPAs which represent the comprehensive opinions of each supplier from eachexpert(Table 5). Step 3.
Convert the decision makers’ weights including crisp data and lin-guistic items into an interval number. And then discount the interval BPAof combined performance (obtained in Step 2) using the interval data to gen-erate the interval BPA of each performance. Combine the interval BPA of20 able 5: Fuse interval data to get comprehensive information.Performance The left part of interval BPA The right part of interval BPA( { IS } , { N S } , { IS, N S } ) ( { IS } , { N S } , { IS, N S } )DM1 Supplier1 (0.5133, 0.0980, 0.3887) (0.8009, 0.0987, 0.1004)Supplier2 (0.4596, 0.1772, 0.3632) (0.6881, 0.2353, 0.0766)Supplier3 (0.4003, 0.1895, 0.4102) (0.6817, 0.2520, 0.0663)Supplier4 (0.4938, 0.1246, 0.3815) (0.7447, 0.1782, 0.0771)Supplier5 ( 0, 0.5804, 0.4196) ( 0, 0.8812, 0.1188)Supplier6 (0.0734, 0.5131, 0.4135) (0.1203, 0.7369, 0.1428)DM2 Supplier1 (0.4502, 0.1128, 0.4370) (0.8108, 0.1268, 0.0624)Supplier2 (0.3929, 0.1800, 0.4271) (0.6438, 0.3116, 0.0446)Supplier3 (0.3920, 0.2114, 0.3966) (0.7549, 0.1995, 0.0456)Supplier4 (0.4502, 0.1086, 0.4412) (0.7774, 0.1821, 0.0405)Supplier5 (0.0343, 0.5015, 0.4642) (0.0387, 0.8905, 0.0708)Supplier6 (0.0854, 0.4518, 0.4628) (0.0996, 0.7475, 0.1529)DM3 Supplier1 (0.4722, 0.0699, 0.4579) (0.9206, 0.0456, 0.0338)Supplier2 (0.4015, 0.1829, 0.4155) (0.8124, 0.1506, 0.0370)Supplier3 (0.4015, 0.1829, 0.4155) (0.7903, 0.2097, 0)Supplier4 (0.5034, 0.0221, 0.4746) (0.9460, 0.0131, 0.0409)Supplier5 (0.0500, 0.4868, 0.4631) (0.0309, 0.9357, 0.0334)Supplier6 (0.1051, 0.4620, 0.4833) (0.0982, 0.8494, 0.0524) all decision makers’ to get the performance of each alternative.Step 3. is similar to Step 2. In other words, the process of Step 3 is nearly thesame as the last step essentially. It provides each interval BPA with anotherchance of applying Dempster-Shafer combination rule to be fused together.As a result, the BPA of ideal solution can be increased or reduced owing to21he proporty of DST, which will contribute to our decision making greatly.Firstly, Using the same method(using Eq.(13)), the new interval weights ofdecision makers’ reliability can also be obtained as follows: W DM = [ 0 . , .
45 ] . .
95 = h . , . i W DM = [ 0 . , .
55 ] . .
95 = h . , . i W DM = [ 0 . , .
95 ] . .
95 = h . , i Then, we will use Eq.(18) to get the new interval BPAs. Next those intervalBPAs will be fused like Step 2. Still take supplier1 as an example and getthe result in Table 6.
Table 6: Fuse the three DMs’ evaluation of supplier1.The left part of interval BPA The right part of interval BPA( { IS } , { N S } , { IS, N S } ) ( { IS } , { N S } , { IS, N S } )DM1 (0.1080, 0.0206, 0.8714) (0.3795, 0.0468, 0.5737)DM2 (0.1659, 0.0416, 0.7925) (0.4694, 0.0734, 0.4572)DM3 (0.3479, 0.0515, 0.6006) (0.9206, 0.0456, 0.0338)Fusion result (0.4950, 0.0733, 0.4317) (0.8849, 0.1017, 0.0135) Table 6 reveals the the three evaluation of supplier1 offered by three experts.After the fusion, we get the final interval BPA which represents the mostoverall information about suppiler1. Using the same method, all the suppli-ers’ final interval BPA are obtained(Table 7). The rest steps are to comparethese interval BPAs and rank the order to make decision.22 able 7: Fuse the three DMs’ evaluation of each supplier.Performance The left part of interval BPA The right part of interval BPA( { IS } , { N S } , { IS, N S } ) ( { IS } , { N S } , { IS, N S } )Supplier1 (0.4950, 0.0733, 0.4317 (0.9696, 0.0201, 0.0103)Supplier2 (0.4106, 0.1516, 0.4378) (0.8849, 0.1017, 0.0135)Supplier3 (0.4019, 0.1878, 0.4104) (0.8851, 0.1149, 0)Supplier4 (0.5135, 0.0485, 0.4380) (0.9765, 0.0110, 0.0125)Supplier5 (0.0336, 0.5337, 0.4328) (0.0096, 0.9811, 0.0094)Supplier6 (0.0845, 0.4895, 0.4260) (0.0473, 0.9339, 0.0189) Step 4.
Combine the the lower part and the upper part of the interval BPAto get the final performance of each alternative.The greatest advantage of using interval data is that it can retain the originalinformation about the performance as much as possible during the fusionprocess. When it comes to rank the order of suppliers, a classical BPAconsisting of crisp number seems to be a more effective means. Hence, wewill combine the two parts of the interval BPA into one classical BPA inorder to select the best supplier.
Step 5.
Determine the final ranking order based on pignistic probabilitytransformation(PPT).The BPA of discernment ( m { IS, N S } ) has some effects on the accuracy ofmaking the best decision. To eliminate it, pignistic probability transforma-23ion is applied in our method. With the equation as Bet n { IS } = m n { IS } + m n { IS, N S } /2 (19)the final belief degrees of each supplier are showed in Table 8. And accord-ing to these data, the order is easily ranked as supplier 4 succ supplier 1 succ supplier 2 succ supplier 3 succ supplier 6 succ supplier 5. Apparently,supplier 4 is the best selection. It coincides with the results presented inpaper[31]. Furthermore, the final rank order is also coincided with the theleft part or the right part of interval BPA, which proves the feasibility andvalidity of our new method. Table 8: Convert the interval BPA back to classical BPAPerformance Fused results bet(IS) Final ranking orderSupplier1 (0.9833, 0.0119, 0.0048 0.9857 2Supplier2 (0.9177, 0.0752, 0.0072) 0.9213 3Supplier3 (0.9129, 0.0873, 0) 0.9129 4Supplier4 (0.9879, 0.0063, 0.0058) 0.9908 1Supplier5 (0.0050, 0.9910, 0.0042) 0.0071 6Supplier6 (0.0287, 0.9625, 0.0090) 0.0332 5
5. Conclusion
In reality, MCDM problem faces a mass of fuzzy information inevitably. Tohandle this problem, a new method is proposed based on interval data fu-sion. The fuzzy data is collected in the form of interval data in our method.24ompared with the original method, our method has remarkable superiorityin dealing with the fuzzy information. A supplier selection example is usedto illustrate the detailed procedures of our method and the result proves itscorrectness adequately. Our new method is worthy being taken into consider-ation when the fuzzy data grow rapidly as the system develops. Furthermore,our method holds quantities of opportunities to apply, especially in the fieldslike social, economy and so on.
6. Acknowledgement
The work is partially supported by National Natural Science Foundationof China (Grant No. 61671384), Natural Science Basic Research Plan inShaanxi Province of China (Program No. 2016JM6018), Aviation ScienceFoundation (Program No. 20165553036), the Fund of SAST (Program No.SAST2016083).