Research ArticleA Combined Weighting Method Based on Hybrid ofInterval Evidence Fusion and Random Sampling
Ying Yan1 and Bin Suo2
1School of Economics and Management, Southwest University of Science and Technology, Mianyang 621010, China2Institute of Electronic Engineering, China Academy of Engineering Physics, Mianyang 621900, China
Due to the complexity of system and lack of expertise, epistemic uncertainties may present in the experts’ judgment on theimportance of certain indices during group decision-making. A novel combination weighting method is proposed to solve theindex weighting problemwhen various uncertainties are present in expert comments. Based on the idea of evidence theory, varioustypes of uncertain evaluation information are uniformly expressed through interval evidence structures. Similarity matrix betweeninterval evidences is constructed, and expert’s information is fused. Comment grades are quantified using the interval number,and cumulative probability function for evaluating the importance of indices is constructed based on the fused information.Finally, index weights are obtained by Monte Carlo random sampling. The method can process expert’s information with varyingdegrees of uncertainties, which possesses good compatibility. Difficulty in effectively fusing high-conflict group decision-makinginformation and large information loss after fusion is avertible.Original expert judgments are retained rather objectively throughoutthe processing procedure. Cumulative probability function constructing and random sampling processes do not require any humanintervention or judgment. It can be implemented by computer programs easily, thus having an apparent advantage in evaluationpractices of fairly huge index systems.
1. Introduction
The establishment of index system and the determination ofcorresponding index weights are very significant during theappraisal processes of risk assessment, system evaluation, andeconomic efficiency evaluation. Considering the importantrole of weight during the appraisal process that reflects theeffect of the index on the previous level decision-makingelements, the accuracy of evaluation and the correctness ofdecision-making are all based on it. The index weightingmethod can be classified into subjective weighting method,objective weighting method, and combination weightingmethod due to the diversity of data sources.
Subjective weighting refers to the method in which theimportance of indices was assessed according to the quali-tative judgment of the decision-makers based on their pro-fessional knowledge and experience. And the index weights
of these judgments were quantified through some specificstandard. Common subjective weighting methods includecyclic scoring, AHP, FAHP, and binomial coefficient method.AHP is the most widely used one among them. However,it is difficult to achieve construction or examination ofconsistent judgmentmatrices for AHP.The consistency checkof multiexpert group weighting problems remains difficultdespite the fact that the difficulty in constructing consistentjudgment matrices can be solved by a group order relationanalysis for AHP. In general, subjective weighting methodshave not advanced sufficiently.
The weights were determined according to the rela-tionship between the original data in objective weight-ing method. Typical objective weighting methods includeentropy method, multiobjective programming, and princi-pal component analysis. Compared with subjective weight-ing methods, objective weighting methods own better
HindawiDiscrete Dynamics in Nature and SocietyVolume 2017, Article ID 8751683, 8 pageshttps://doi.org/10.1155/2017/8751683
Not at all important Unimportant Moderately important Important Very important
Expert 1 √ √ √0.6 0.3 0.1
Expert 2 √ √[0.1, 0.3] [0.6, 0.9]Expert 3 √
1.0Weights of different indices can be obtained by synthesizing evaluation opinions of𝐿 number of experts.
mathematical bases but worse interpretability. Thus, theirconclusions are inconsistent with the actual importance ofindices at times [1].
To overcome the shortcomings of the subjective andobjective weighting methods, in recent years, methods whichcombine the subjective and objective weighting have beendeveloping rapidly. These methods include modified AHPand the entropy method [2], maximum entropy weighting[3], geometric operator weighting [4], and weighting methodbased on maximizing the difference in evaluation results[5]. But in general, most of the computing methods arecomplicated, and some of them also can hardly reflectdifferent emphasis of experts placed on different indices.Thus, they are still imperfect. A variancemaximization-basedcombination weighting method was proposed by Sun andBao [6]. It determined the weights of group decision-makinginformation by using AHP and rough set theory respectively.Then the comprehensive weights were obtained by variancemaximization of the two previous weights. The method isrational to some extent. However, construction of judgmentmatrix and consistency check for group decision-makinginformation remains difficult in case of highly conflictingopinions. A structure entropy method was proposed byCheng [7]. In this manner, the qualitative “typical rank-ing” was replaced by a quantitative membership functionthrough entropy function before postprocessing. Althoughthe method can achieve good results, it is difficult for expertsto reasonably rank the importance of numerous indices forthe relatively large index system. Three different secondarysorting methods were used to rank evaluation indices; ratio-nal ranking would be obtained by Kendall test when theprevious ranking is contradictory [8]. Combined weights ofindices were obtained ultimately by calculating the standarddeviations of evaluation indices. The method can effectivelyreduce the inconsistencies in importance ranking. Neverthe-less, the subjective determination of reasonable ranking andthe importance ratios of indices can still exist. In addition,due to the complexity of system and lack of expertise,epistemic uncertainties may occur in judgment of groupdecision-making for the importance of certain indices [9].For example, theymay be unable to fully determinewhether acertain index is “important” or “very important”. Currently,researches on index weighting determination during expertevaluation under epistemic uncertainties are still insufficient.
Accordingly, in order to solve the above problems, varioustypes of uncertain information of expert comments during
group decision-making were fully considered in the paper. Aunified expression is proposed for this uncertain informationbased on the reconstruction of interval evidence structure.And multisource information is fused according to thesimilarity of interval evidences. Then, cumulative probabilityfunction is constructed based on this to evaluate the impor-tance of indices. Finally, index weights are determined byMonte Carlo random sampling.
2. Problem Description
Assume that the number of experts involved in evaluatinga certain project is 𝐿. Experts set was denoted as E ={𝑒1, 𝑒2, . . . , 𝑒𝐿}; the number of indices involved in the projectis 𝑀. The index set was denoted as A = {𝑎1, 𝑎2, . . . , 𝑎𝑀}.In general, it was difficult for experts to directly determinethe weight of each index. And evaluating the importancedegree of indices by comment grades as “not at all important,”“unimportant,” “moderately important,” “important,” and“very important” was comparatively easy. Comment gradeset was denoted as H = {𝐻1, 𝐻2, 𝐻3, 𝐻4, 𝐻5}. Evaluationopinions of expert 𝑒𝑖 (𝑖 = 1, 2, . . . , 𝐿)’s on index 𝑎𝑘 (𝑘 =1, 2, . . . ,𝑀) were
where 𝑝𝑖,𝑗(𝑎𝑘) was the credibility of 𝑒𝑖’s evaluation for index𝑎𝑘 by comment grade 𝐻𝑗. Considering the epistemic uncer-tainties, experts were allowed to select 1 to 3 comment gradesfor the same index in the questionnaire. 𝑝𝑖,𝑗(𝑎𝑘) could beeither point value or interval number. Table 1 is an exampleof questionnaires. Obviously, 𝑝𝑖,𝑗(𝑎𝑘) satisfied the followingequation:
0 ≤ 𝑝𝑖,𝑗 (𝑎𝑘) ≤ 1,min𝑁∑𝑗=1
𝑝𝑖,𝑗 (𝑎𝑘) ≤ 1,max𝑁∑𝑗=1
𝑝𝑖,𝑗 (𝑎𝑘) ≥ 1.(2)
3. Novel Weighting Method
Due to the epistemic uncertainties of experts and inevitableinconsistencies in different experts’ opinions on the same
Discrete Dynamics in Nature and Society 3
index, how to rationally fuse expert information was a verycritical issue. Evidence theory is a potent method to fusemultisource uncertain information, which has been widelyused in the fields of data fusion, artificial intelligence, expertsystem, and decision analysis [10–12]. To get more reasonableand reliable fusion results, the expression form and ideaof interval evidence theory were utilized to fuse expert’scomments in this paper. Finally, the index weights weredetermined by random sampling of fused information.
3.1. Interval Evidence Theory. Interval evidence theory isestablished based on the frame of discernment. The frameof discernment is defined as the set containing all possibleresults for a specific problem. It is generally a nonempty setand represented by Θ.Definition 1. Interval basic probability assignment (IBPA)[12–15]: let𝑚 be a mapping from the power set to the intervalnumber [0, 1]; that is, 𝑚: 2Θ → [0, 1], for Θ’s 𝑛 number ofsubsets 𝐴 𝑖 (𝑖 = 1, 2, . . . , 𝑛); IBPA was
𝑚(𝐴 𝑖) = [𝑎𝑖, 𝑏𝑖] , (3)
where 0 ≤ 𝑎𝑖 ≤ 𝑏𝑖 ≤ 1. If IBPA satisfied simultaneously thefollowing conditions
𝑎𝑖 ≤ 𝑚 (𝐴 𝑖) ≤ 𝑏𝑖,𝑛∑𝑖=1
𝑎𝑖 ≤ 1,𝑛∑𝑖=1
𝑏𝑖 ≥ 1,𝑚 (𝐻) = 0, ∀𝐻 ∉ Θ
(4)
then𝑚 was called a valid IBPA.
Definition 2. Normalization principle [15, 16]: if 𝑚 satisfied(4), which was a valid IBPA, and 𝑚(𝐴 𝑖) = [𝑎𝑖, 𝑏𝑖] (where0 ≤ 𝑎𝑖 ≤ 𝑏𝑖 ≤ 1) and if 𝑎𝑖 and 𝑏𝑖 satisfied simultaneouslythe following conditions
𝑛∑𝑗=1
𝑏𝑗 − (𝑏𝑖 − 𝑎𝑖) ≥ 1,𝑛∑𝑗=1
𝑎𝑗 + (𝑏𝑖 − 𝑎𝑖) ≥ 1,(𝑖, 𝑗 = 1, 2, . . . , 𝑛)
(5)
then𝑚 was called normalized IBPA.If𝑚 was a valid IBPA but not normalized, normalization
would be needed according to the formula below in order tonarrow the interval width and reduce redundancy:
max[[𝑎𝑖, 1 −𝑛∑𝑗=1 𝑗 =1
𝑏𝑗]] ≤ 𝑚 (𝐴 𝑖)
≤ min[[𝑏𝑖, 1 −𝑛∑𝑗=1 𝑗 =1
𝑎𝑗]] .(6)
Definition 3. Dempster’s rule of interval evidence combina-tion [15, 16]: wvidences from different information sourcescould be fused into more reliable, accurate information usingthe rule of evidence combination. As the most classic evi-dence combination rule, Dempster’s rule could be extendedto the field of interval evidence. Let 𝑚1 and 𝑚2 be the valid,normalized IBPA under the same frame of discernment Θ,where
As indicted in (9), traversal of all point values BPA(basic probability assignment) satisfying constraints in theIBPA interval was needed during the combination of intervalevidences to form fused point value sets. Final combinationresults could be obtained by taking the maximum andminimum values in these point value sets.
4 Discrete Dynamics in Nature and Society
3.2. Fusion of GroupDecision-Making Information. 𝐿 numberof experts’ comments on indices 𝑎𝑘 (𝑘 = 1, 2, . . . ,𝑀) in acertain project could be written as
Inconsistencies in different experts’ opinions on index 𝑎𝑘were inevitable. Thus, conflict handling during the opinionsfusion process was very critical. Based on the principle thatthe greater the conflict with group opinions, the lower thecredibility, expert opinions should be fused after assign-ment of certain weights to obtain reliable results. Similaritybetween evidences reflects conflicting opinions from anotheraspect. Thus, similarity between expert opinions was calcu-lated first. Weights of expert comments were then calculatedbased on the calculated similarity.
Let the frame of discernment for expert evaluationopinions be Θ and IBPA of experts 𝑒𝑛 (1 ≤ 𝑛 ≤ 𝐿) and𝑒𝑟 (1 ≤ 𝑟 ≤ 𝐿)’s comments on index 𝑎𝑘 be 𝑚𝑘𝑛 and 𝑚𝑘𝑟 ,respectively. Then the Euclidean distance between experts 𝑒𝑛and 𝑒𝑟’s comments on index 𝑎𝑘 was defined as
Equation (18) was precisely the final fusion results of expertevaluation information.
It was noteworthy that, during the information fusionprocess, we did not use the approachmentioned in literatures[14–16] that the weighted average evidences were recombined𝐿−1 times applyingDempster’s rule.Thiswas because that theself-combination 𝐿 − 1 times of weighted average evidenceswould induce apparent “focusing” effect which means thefusion results would converge fast towards to focal elementsof larger BPA.The fast convergencewas helpful to target iden-tification and fusion decision. However, it was unsuitable forobjective description of multiple experts’ evaluation opinionsand results in the high probability of biased results due tothe strong subjective tendency and unreasonable explanationof the operation. For example, suppose that an index wascommented by 10 experts and weighted average of expertgroup’s opinions was
m = [0.42 0.52 0.06 0 0] . (19)
After recombining 9 times of evidence byDempster’s rule, thecombined expert opinions could be obtained as
m = [0.18 0.82 0 0 0] . (20)
Discrete Dynamics in Nature and Society 5
It can be seen that focusing effect was very apparent aftercombination. After weighted averaging operation withoutthe recombination, the confidence probability of the indexwhich was displayed as “very important” or “unimportant”was high. However, after 𝐿 − 1 times of combination,the confidence probability of “very important” was almostcompletely abandoned, while the confidence probability of“unimportant” was greatly enhanced. Obviously, the resultsafter 9 times of self-combination were far deviated fromthe original determinations of experts, which would affectthe subsequent judgments greatly.Therefore, Dempster’s rulewas not used in this paper for self-combination of weightedaverage evidences during the information fusion. We didthis way aiming to reflect the experts’ original judgmentsobjectively and to utilize known information to the utmost.This could lay a solid foundation for the subsequent objectiveweighting of indices.
3.3. Random Weight Calculation Method Based on MonteCarlo. Interval numbers [0, 5] were divided into five contin-uous intervals
which correspond to the 5 comment grades of the commentsetH and can transform qualitative comments into quantita-tive values. Clearly
𝑚𝑘 (𝑉𝑗) = ��𝑘 (𝐻𝑗) (𝑗 = 1, 2, . . . , 5) . (22)
Based on the fusion results of the expert group’s opinions,the probability density function (PDF) for importance ofindex 𝑎𝑘 was constructed as
𝑓𝑘 (𝑥) = 𝑁∑𝑗=1
𝛿 (𝑥 | 𝑉𝑗)𝑚𝑘 (𝑉𝑗) , (23)
where
𝛿 (𝑥 | 𝑉𝑗) = {{{1 if 𝑥 ∈ 𝑉𝑗0 if 𝑥 ∉ 𝑉𝑗, 1 ≤ 𝑗 ≤ 𝑁. (24)
Notably, 𝑚𝑘(𝑉𝑗) in (23) was the interval number, so thecumulative probability function for importance of index 𝑎𝑘was in the form of probability envelope. Its lower bound was
CDF𝑘 (𝑥) = ∫𝑥0inf (𝑓𝑘 (𝑡)) 𝑑𝑡. (25)
And the upper bound was
CDF𝑘 (𝑥) = ∫𝑥0sup (𝑓𝑘 (𝑡)) 𝑑𝑡. (26)
Mean of the upper and lower bounds was taken as theestimated value of CDF standing for the importance of index𝑎𝑘:
𝐹 (𝑥) = CDF𝑘 (𝑥) + CDF𝑘 (𝑥)2 . (27)
Meanwhile, it was noted that 𝑚𝑘(𝑉𝑗) was IBPA. Equa-tions (25) and (26) must satisfy max(CDF𝑘(𝑥)) ≤ 1 andmax(CDF𝑘(𝑥)) ≤ 1, so there might be the situation in whichmax(𝐹(𝑥)) = 1. Thus, normalization of (27) was needed:
𝐹 (𝑥) = 𝐹 (𝑥)max (𝐹 (𝑥)) . (28)
Inverse function of (23) was sought
𝑥𝑘𝑙 = 𝐹−1𝑘 (𝑃) . (29)
By Monte Carlo simulation, 𝑁 number of probability valuesbetween [0, 1] was randomly generated, and 𝑁 number ofsamples 𝑥𝑘𝑙 was obtained by sampling. Average importance𝑢𝑘 of index 𝑎𝑘 could be obtained by
𝑢𝑘 = ∑𝑁𝑙=1 𝑥𝑘𝑙𝑁 . (30)
After the normalization of 𝑢𝑘, the weight of index 𝑎𝑘 could beobtained as
𝜛𝑘 = 𝑢𝑘∑𝑀𝑘=1 𝑢𝑘 . (31)
4. Numerical Examples
Ahome appliancemanufacturer planned to choose a logisticscompany from 10 candidates as its third-party logisticsservice provider. The evaluation and choice were carried outfrom four respective aspects: assets and solvency, profitabil-ity, service capability, and innovative development ability.During the evaluation, importance of each index, i.e., indexweight, should be determined first. Thus, five peer expertswere invited by company A to score the importance of fourindices based on the template inTable 1.The results are shownin Table 2.
As can be seen from Table 2, experts’ evaluation opinionswere largely conflicting. The evaluation information con-tained multiple types of information such as precise infor-mation, epistemic uncertain information, and multi-intervalprobability information. During the processing, it is neededto express different types of information communicationsas interval evidence structure. For example, 𝐻3(1) couldbe expressed as 𝐻3([1, 1]); 𝐻2(0.2) and 𝐻3(0.8) could beexpressed as𝐻2([0.2, 0.2]),𝐻3([0.8, 0.8]).
Firstly, each evaluation opinion in Table 2 was checkedfor whether it was normalized IBPA. For example, for “assetsand solvency” index, expert 𝑒2’s opinion did not satisfy (3);thus normalization was needed following (4). Normalizationresults of Table 2 are shown in Table 3.
Afterwards, the evaluation opinions of five experts werefused. For the evaluation index “assets and solvency,”
6 Discrete Dynamics in Nature and Society
Table 2: Evaluation opinions of experts on the importance of various indices.
Assets andsolvency
Profitability Service capability Innovative development ability
CDF for its importance is shown in Figure 4.Average degree of importance of four indices could be
obtained by Monte Carlo random sampling with a samplingnumber of𝑁 = 2000 times:
𝑢1 = 2.06,𝑢2 = 3.17,𝑢3 = 3.69,𝑢4 = 1.35.(40)
After normalization, weights of four indices “assets andsolvency,” “profitability,” “service capability,” and “innovativedevelopment ability” could be obtained, respectively. Theywere as follows:
𝜛1 = 0.20,𝜛2 = 0.31,
𝜛3 = 0.36,𝜛4 = 0.13.
(41)
5. Conclusions
Index weighting is one of the key issues in group decision-making. Due to the complexity of the index system, expertstend to give index importance evaluation based on theirown knowledge and understanding in actual group decision-making process. Multi-interval probability values were evenassigned occasionally. In this paper, index weighting problemin such cases was studied. Various uncertain information isuniformly expressed through interval evidence structure byfully considering expert information during group decision-making. Then, probability information of multi-interval wasfused. The importance of indices is constructed based on the
8 Discrete Dynamics in Nature and Society
fused information by CDF evaluating. Finally, index weightsare determined by Monte Carlo random sampling.
The method proposed herein has the following advan-tages:
(1) It could give the weighting of group decision-making information with various degrees of uncer-tainties, such as precise evaluation information (e.g.,𝐻3(1)), epistemic uncertain evaluation information(e.g., 𝐻4(0.7), 𝐻5(0.3)), and multi-interval prob-ability evaluation information (e.g., 𝐻3([0.3, 0.5]),𝐻4([0.4, 0.7])), which possesses good compatibility.
(2) The method draws on the idea and expressionof interval evidence theory. The similarities ofgroup decision-making information were specifiedby Euclidean distance between interval vectors.Weighted averagemethodwas utilized to fuse expert’sinformation directly. Thus, the “one-vote veto” or“focusing effect” problem that arose during com-bination highly conflicting evidence fusion process,difficulties in constructing, and testing consistentjudgment matrices by AHP could all be avoided.
(3) Digitization and visualization of the fused groupdecision-making information can be achieved byCDF based on importance of indices constructedaccording to the information fused by interval evi-dence theory. The subsequent random sampling pro-cess is implemented entirely by computer programs.It is simple and efficient. It owns apparent advantagesfor the evaluation of systems with the large index andcomplex targets.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Scientific Research Foun-dation of Southwest University of Science and Technologyunder Grant no. 12sx7106.
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