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FUZZY BELIEF STRUCTURE BASED VIKOR METHOD: AN APPLICATION FOR RANKING DELAY CAUSES OF TEHRAN METRO SYSTEM BY FMEA CRITERIA

FUZZY BELIEF STRUCTURE BASED VIKOR METHOD: AN APPLICATION FOR RANKING DELAY CAUSES OF TEHRAN... TRANSPORT ISSN 1648-4142 / eISSN 1648-3480 2016 Volume 31(1): 108–118 doi:10.3846/16484142.2016.1133454 FUZZY BELIEF STRUCTURE BASED VIKOR METHOD: AN APPLICATION FOR RANKING DELAY CAUSES OF TEHRAN METRO SYSTEM BY FMEA CRITERIA 1 2 Seyed Hossein Razavi Hajiagha , Shide Sadat Hashemi , 3 4 Yousef Mohammadi , Edmundas Kazimieras Zavadskas 1, 3 Dept of Management, Khatam Institute of Higher Education, Tehran, Iran Saramadan Andisheh Avina Co., Tehran, Iran Dept of Construction Technology and Management, Vilnius Gediminas Technical University, Vilnius, Lithuania Submitted 31 August 2015; resubmitted 29 November 2015; accepted 29 November 2015 Abstract. Public transport is a critical part of civilization in this decade. The amount of money invested and the criticality of transferring people in an acceptable time and without any conflict made it a challenging problem for managers, especially in metropolises. Absolutely, making effective decisions in this area requires considering different aspects. Waiting time is a key criterion in apprising quality of public transport. In this paper, a real world case study of ranking causes of delay in Tehran (Iran) metro system is solved by developing multi attribute group decision-making VIšeKriterijumska Optimizacija I KOmpromisno Rešenje (in Serbian, VIKOR) method under uncertainty, where this uncertainty is captured by Fuzzy Belief Structures (FBS). The obtained results are then compared with a previously pro- posed Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method with FBSs. The results show that human related issues, along with the problems related to line and transportation system are the most important causes of delay. The obtained results of the problem seem acceptable for decision makers. Keywords: group decision-making; VIKOR; fuzzy belief structure; center of gravity; failure mode and effect analysis. Introduction (UN 2010). As Bainster (1996), and Gifford and Steg (2007) believed, transport holds major economic and Finding an ideal decision is the pursuit of many scholars social importance in improving expected standards of and practitioners in different fields (Tzeng, Huang 2011). consumption and quality of life. Multiple Criteria Decision-Making (MCDM) is a field Considering the above challenges, transporta- of operations research, which deals with evaluating and tion system is one of the implicational areas of MADM choosing the best alternative based on several criteria. (Bagočius et al. 2014; Elevli 2014; Šateikienė et al. 2015). Yoon and Hwang (1995) classified MCDM problems An MADM problem can be defined as follow (Zimmer - into two main categories: Multiple Attribute Decision- man 1987): Suppose that we have a nonempty and finite Making (MADM) and Multiple Objective Decision- set of decision alternatives, that their desirability will be Making (MODM), the former applies in evaluation judged according to a finite set of goals, attributes or type problems while the latter is suitable for design type criteria. The aim of MADM is to determine an optimal problems (Simon (1977) classified decision problems alternative having the highest degree of desirability in into selection and design problems). respect to all relevant goals. e Th aim of MCDM methods is to provide a logical e Th real world decision-making process is a com- and scientific framework of dealing with risky decisions. mon problem for employing the uncertainty phenom- One of the main challenges of 21st century is urban- ena. In fact, the required knowledge to formalize a deci- ization growth (Martine 2005). As stated by UN, today sion-making problem are usually subjected to uncertain- 54% of the world’s population lives in urban areas (UN ty. Dubois and Prade (1982), and Klir (1987) introduced 2015). Considering this growing urbanization phenom- ambiguity and vagueness as two types of uncertainty in ena, transport plays a crucial role in urban management Corresponding author: Seyed Hossein Razavi Hajiagha E-mail: [email protected] Copyright © 2016 Vilnius Gediminas Technical University (VGTU) Press http://www.tandfonline.com/TRAN Transport, 2016, 31(1): 108–118 real-world problems. While ambiguity refers to unspe- Yang et al. (2006) introduced the FBS where evaluation cific choice between alternatives, vagueness deals with grades are expressed as fuzzy numbers. In fact, an FBS is a combination of fuzzy set theory and the evidence com- situations where there are not any sharp boundaries bination rule of the Dempster–Shafer theory and there- among some domain of interest (Inuiguchi et al. 2000). fore is a powerful way of dealing with uncertainty. Jiang This undeniable uncertainty is widely known and et al. (2011) applied the FBS model to extend the TOP- accepted in decision-making, especially  – in MADM SIS method for group decision-making. Vahdani et  al. problems. Presence and acceptance of uncertainty re- (2014) also applied the above TOPSIS and FBS method quires a formal framework to be analyzed. Fuzzy set the- in the Failure Mode and Effect Analysis (FMEA) for ory, introduced by Zadeh (1965), is one of the common ranking the risk causes. frameworks in handling uncertainties (Liu, Lin 2006). e a Th im of this paper is to propose an extension of Bellman and Zadeh (1970) initially applied the concept VIKOR method under FBS. VIKOR method is a com- of fuzzy sets in decision-making problems. Fuzzy sets promise based method for ranking alternatives (Oprico- assign a membership value to each elements of a set. vic 1998). This method is applied for MADM problems In ordinal fuzzy sets, this membership values are exact with irrelevant and conflicting attributes (Opricovic, numbers. Some scholars criticized the ordinal fuzzy sets Tzeng 2004). As the founder of VIKOR method, Opri- due to crispness of its membership values (Grattan- covic extended the fuzzy VIKOR method and employed Guinness 1976). Therefore, some extensions are given it in some applications (Opricovic 2007, 2011). Vah- to the concept of membership functions or values. dani et  al. (2010) proposed the interval type-2 VIKOR Zadeh (1975) introduced type-2 fuzzy sets where method (they applied the interval-valued fuzzy sets with membership function itself is a fuzzy set. As an exten- a similar definition of interval type-2 fuzzy sets). The sion, interval type-2 fuzzy sets considered membership method is also developed in intuitionistic fuzzy (Devi functions as closed intervals (Liang, Mendel 2000). 2011) and interval valued intuitionistic fuzzy (Park et al. Simplicity of operations over intervals caused more in- 2011) environments. e Th idea of VIKOR method is also teresting about application of interval type-2 fuzzy sets extended to solve multi-objective linear programming in MADM. Wang et al. (2012) have found the interval problems (Razavi Hajiagha et  al. 2014). In this paper, type-2 fuzzy sets a very useful mean to depict the in- it is supposed that a group of experts participate in de- formation in decision-making process. They used it for cision-making process, who expresses their evaluations a group decision-making problem in order to calculate in decision matrix based on FBS models, i.e. they de- the attributes weights and aggregated decision matrix. termine their belief degrees’ regard to fuzzy evaluation grades of each alternatives with respect to each criteria. Chen (2013) used the interval type-2 trapezoidal fuzzy numbers to determine the alternatives ratings and the Considering the advantages of FBS in modeling data uncertainty by aggregating fuzzy evaluation grades and importance of various criteria. Baležentis and Zeng belief degrees and combining these advantages with the (2013) extended the MULTIMOORA method based VIKOR method’s advantage in determining a compro- upon fuzzy type-2 sets and generalized interval-valued mise solution of a decision-making problem could have trapezoidal fuzzy numbers. This paper was proposed to made the FBS-VIKOR method as an appealing method aggregate the group decision-making in human resource of solving Multiple Attribute Group Decision-Making management. The application of type-2 fuzzy numbers is (MAGDM) problems. illustrated in Maldonado et al. (2014), which proposed a The rest of paper is organized as follows. The design of multi objective genetic optimization problem. VIKOR method is briefly introduced in Section 1. Then, Atanassov (1986) extended the ordinal fuzzy sets by the required concepts of FBS models are overviewed in attending a non-membership value v beyond the clas- Section 2. The proposed method of VIKOR under FBSs sic membership value μ, and called the corresponding is explained in Section 3. A real world case study is then set as Intuitionistic Fuzzy Set (IFS). In a classic fuzzy solved by using the proposed method in Section 4. Fi- v 1−µ v+µ≤1 set , while in an IFS, . In an ordinal nally, the paper is concluded. IFS, the μ and v values are determined as crisp num- bers. Latter, Atanassov and Gargov (1989) developed 1. VIKOR IFS to Interval-Valued Intuitionistic Fuzzy Sets (IVIFSs) A decision-making problem can be formulated in the where membership and non-membership degrees are form of a decision matrix consist of the following ele- expressed as closed intervals. There is a planetary of re- ments (Yu 1990): searches done over extending MADM techniques under – the set of alternatives A AA , ,,… A ; { } IFS and IVIFS environments (Zhou et  al. 2013; Razavi 12 m – the set of criteria X XX , ,,… X ; { } Hajiagha et  al. 2015; Zavadskas et  al. 2015; Tan et  al. 12 n  2014; Chen 2015; Hashemi et al. 2016). – the outcome or decision matrix Dx = , which ij  Fuzzy Belief Structure (FBS) is another extension element x represents performance of alternative ij of ordinal fuzzy sets. Initially, the Belief Structure (BS) A based on the criterion X , for each i = 1, 2, …, i j introduced by Yang and Singh (1994), and Yang and Sen m and j = 1, 2, …, n; (1994) as an evidential reasoning approach for solving – the vector W ww , ,,… w , where w ≥ 0, j = { } 12 n j MADM problems. In a FBS model, the linguistic vari- 1, 2, …, n illustrates the weight or importance of ables of evaluation grades are considered as crisp values. criterion X . = S. H. Razavi Hajiagha et al. Fuzzy belief structure based VIKOR method: an application for ranking ... An MCDM method to rank a set of alternatives – alternative j and j , where QQ = if C is not 1 2 j2 [2] 2 based on conflicting criteria so-called VIKOR is intro- satisfied; duced to be employed in practical problems. This meth- – alternatives jj ,,…, j if C is not satisfied, al- 1 2 k 1 od is based on closeness to the Ideal solution. Presume ternative j is determined by Q −Q  DQ for k [k] [1] a set of m alternatives and n criteria, where the rating of maximum k, where QQ = . jk [] k each alternative A against to a criterion X is denoted as i j f . This method includes the following steps: 2. Fuzzy Belief Structure (FBS) ij + − Step 1. Determine the best f and worst f rating for BS is a distributed assessment scheme where belief de- i i all criteria as follow: grees are attained to different evaluation grades. Suppose that the evaluation grade of criteria consist a standard ff = max ; (1) j ij with N grades H ,H ,, … H and assume that is pre- { } − 12 N ff = min . (2) j ij ferred to H . A BS for the assessment of criterion c i k+1 can be represented as the following distribution: In this regard, the ideal and anti-ideal candidates + + + −− − are respectively as ff , ,..., f and f , f ,..., f . ( ) ( ) S c H ,β= ,n 1,2,,N , (8) 12 n 12 n ( ) ( ) { } nn Step 2. Calculate the average S ∈ 0,1 and the worst where: b is the belief degree of the grade H in the n n i  group R ∈ 0,1 scores for the alternative A : i i  evaluation, that β ≥ 0 , n = 1, 2, …, N and β ≤1. n n j=1 ff − ( ) j ij Sw = ; (3) Eq. (8) means that the criterion c is assessed with ij + − ff − j=1 the belief degree b at the grade H . If β= 1 , the dis- ( ) jj ∑ n n n j=1   ff − tribution is complete and if β ≤1  – it is incomplete. ( ) j ij ∑ n   Rw = max , (4) j=1 i j   + −   ff − ( ) jj   e e Th valuation grade consists of {H ,H ,, … H } 12 N where: w , j  = 1, 2, …, n indicates the relative impor- crisp number in ordinal BS. However, the ambiguity tance weights of the criterion j, which experts have de- and vagueness of real-world problems required evalua- termined. tion grades to be represented by fuzzy numbers. In this case, the fuzzy evaluation grades deal with fuzziness or Step 3. Calculate the Q for i = 1, 2, …, m as: vagueness and the belief degrees handled incomplete- ++ SS −− R R ( ) ( ) ness or ignorance. ii Qv + 1− v , (5) ( ) An important concept of FBS is to measure the −+ − + S −− S R R ( ) ( ) belief distance. Jiang et al. (2011) defined the distance where: between to FBS S and S as: 1 2 S = minS ; i d SS ,, d B B ( ) ( ) BS 12 BS 1 2 SS = max ; (6)  T  +  BB −− S BB . (9) RR = min ; ( ) ( )  12 12  i 2   RR = max . (7) That B and B are the belief degree distributions 1 2   of S and S , respectively. Also, Ss = represents the Here v indicates the weight of maximum group 1 2 ij   utility. It can be determined as: similarity matrix between fuzzy evaluations grades. If the v 0.5  – selected by majority; utilities of each fuzzy evaluation grade are represented by – v = 0.5  – consensus; nnnn trapezoidal fuzzy numbers, i.e. U H = uuuu ,,, , ( ) ( ) n 1 23 4 v 0.5  – veto. n  = 1, 2, …, N. Then, the similarity between j j j j Step 4. Sort the S, R and Q values by ascending order i i i i H = uu , ,, u u and H = uuuu ,,, is calcu- ( ) i 12 3 4 j ( 1 2 3 4) to rank each on. There would be 3 lists of ranking il- lated as: lustrated as: S , R and Q . uu − [0] [0] [0] ∑ kk k=1 s 1− , (10) ij Step 5. Indicate the smallest Q value related to alterna- i 4 tive j as a compromise solution if: 1 n  u ∈ 0,1 where: 01 ≤ s ≤ , since  , k  = 1, …, 4. If H ij n k  – C   – the alternative j has an acceptable advan- 1 1 nn is a triangular fuzzy number, then uu = and the 1 23 tage: Q −Q ≥ DQ , where DQ= and [2] [1] similarity measure is measured as the same. Jiang et al. m−1 m – the number of alternatives; (2011) proved that the distance measure in Eq. (9) is – C   – the alternative j is stable within the deci- 2 1 located between 0 and 1, and is a symmetric measure. sion-making process  – it is also the best ranked This measure will be used in the proposed VIKOR based in S or Q . [0] [0] on FBS model. If one of these conditions is not satisfied, then the solution is as follow: == = Transport, 2016, 31(1): 108–118 3. VIKOR Method Based on FBSs where: d d k As discussed earlier, in this paper, the multi-attribute a ww+− 1 β ; 1, ∏ ∑ n ij) ( k k n=1 VIKOR method is developed when decision-making k=1 information is given based on the FBS models. At the dk a= 1−β w ; ∏ ∑ next subsections, the fundamentals of the proposed al- 2, ( k n ij) n=1 k=1 gorithm are explained. Then, an algorithmic scheme is NN K presented to summarize these explanations.  dk d k µ= w β +− 1 w β −  ∑∑ ∏ kk n,, ij n ij  nn 11 k=1  3.1. Problem Definition and Data Preparation −1 K N e co Th nsidered problem is a multi-attribute group deci-  d k Nw −11− β  . (13) ( )  ∏ ∑ k n,ij sion-making problem. Suppose that a group of K deci-  k=1 n=1  sion makers, denoted by DM , k = 1, 2, …, K are gath- ered to evaluate a set of alternatives A AA , ,,… A { } 12 m 3.3. Normalizing the Aggregated Decision Matrix based on a set of criteria X XX , ,,… X . Each de- { } 12 n If the FBS models are incomplete individually, then ob- cision maker has a specific importance according to his/ tained aggregated FBS model will be incomplete. There- her role in decision-making process. Suppose that deci- fore, a normalizing stage is proposed to complete the sion makers importance are determined in the form of aggregated FBS models. DK 12 W ww , ,,… w a vector . In addition, the criteria { } In this paper, the normalization is done weight vector W ww , ,,… w can be determined { } 12 n based on the method of Jiang et al. (2011). Let using methods like simple rating or group Analytic Hi- ij sH= ,β= ,n 1,2,…,N be an incomplete FBS. ( ) erarchy Process (AHP) (Yoon, Hwang 1995; Saaty 1988). ij { n n } Each decision maker individually completed his/ her en, Th N BS peak points are defined as: kk  decision matrix D = Sx , k  = 1, 2, …, K, where ( ) , ij m SP s =H ,β= , n 1,2,…,N  k ( ) {( ) } ij nn Sx is an FBS model like: ( ) ij m = 1, 2, …, N, (14) kk Sx =H ,β= ,n 1,2,…,N , (10) ( ) {( ) } ij n n,ij with: where: β represents decision maker k’s assurance β +β , ; mn =  nH n,ij β= (15) that the performance of alternative A , i = 1, 2, …, m is β , m≠ n, at evaluation grade H , n  = 1, 2, …, N in criterion X , n j ij where: β=1− β is the degree of ignorance for in- jn = 1,2,.., β= 1 ∑ . If , then the kth decision maker’s Hn ∑ n,ij n=1 n=1 complete s . Then, the center of gravity of FBS model s ij ij evaluation over A alternative in X criterion is complete i j is defined as: and if β < 1 , it is incomplete. n,ij SP s ( ) ∑ ij n=1 m=1 SC s ( ) ij 3.2. Constructing the Aggregated Decision Matrix  As an important step in group decision-making, it is re-   β  quired to aggregate the individual decision matrices in n   m=1 an aggregated one. This aggregation is carried out using H , ,nN 1,2,…, . (16)   the evidential reasoning approach (Yang, Xu 2002; Yang     et al. 2006). Since the individual decision matrices D ,   k  = 1, 2, …, K are fuzzy belief matrices, the aggregated Then, the complete SC s is used instead of in- ( ) decision matrix D will be a fuzzy belief matrix. It can be ij complete s in the pooled decision matrix X. expressed as: ij 1 2 K   3.4. Constructing the Fuzzy Belief Distance Matrix D Sx D⊕ D ⊕…⊕ D , (11) ( ) ij   Considering the decreasing nature of evaluation grades, where: Sx represents the aggregated fuzzy belief + ( ) ij i.e. HH > , the positive ideal f and negative ideal nn +1 j performance of alternative A in criterion X . The FBS − i j f , FBS models for each criterion C , j = 1, 2, …, n are model Sx =H ,β= ,n 1,2,…,N is constructed defined as follows: ( ) ( ) ij { n n,ij } based on FBS models Sx , k = 1, 2, …, K. Wang et al. ( ) (17) ij fH = ( ,1),(H ,0),,(H ,0) { } j NN−11 (2006) proposed an analytic relation between the indi- and: vidual belief degrees β , k  = 1, 2, …, K with the ag- n,ij gregated belief degree of β , as: fH = ,0 , H ,0 ,, H ,1 . (18) {( ) ( ) ( )} n,ij j NN−11 µ⋅ aa − ( ) ij 1 2 β = , (12) Now, the S and R must be found applying Eqs (3) n i i  and (4), respectively. To compute these values, the dis- 11 −µ⋅ − w  ( )   k=1 = = == == = S. H. Razavi Hajiagha et al. Fuzzy belief structure based VIKOR method: an application for ranking ... tance between FBS models are used. The S is calculated S = minS ; as follow by considering Eqs (3) and (9): SS = max (25) n i d ff , ( ) BS j ij and: Sw = . (19) ij ∑ + − + d ff , RR = min ; j=1 ( ) BS j j RR = max . (26) The R is also defined as:  d ff , Calculate the Q for i  = 1, 2, …, m using Eq. (5), ( ) BS j ij i  Rw = max . (20) where the v multiplier is interpreted as in classic VIKOR i j  + −  d ff , + + ( ) BS j j method. Applying the common values of S   = R   = 0,  – – and S   = R  = 1 in Eq. (5), the following relation is ob- Denominators of both Eqs (19) and (20) are fixed tained: values, since the ideal and negative ideal solutions are similar, in the sense of FBS models, for all the criteria. Q= vS+ 1− v R . (27) ( ) ii i This fixed denominator is determined as below using Sort the S, R, and Q values by ascending order to Eq. (9) as: 1/2 rank alternatives. There would be 3 lists of ranking il-  1 + − + − + − d ff , = BB −− S BB . (21) lustrated as: S , R and Q . Aer det ft ermining these ( ) ( ) ( ) BS j j  j j j j [0] [0] [0]  lists, by using step 5 of the VIKOR method in Section 1, e f Th ollowing relation is obtained for all values of the final ranking list will be obtained. j, j = 1, 2, …, n by applying the positive ideal and nega- tive ideal belief degrees in Eqs (17) and (18) in Eq. (21): 3.6. Schematic Algorithm 1/2 In this regard, the proposed methodology is illustrated   + − d ff , = S + S −+ S S . (22) (( ) ( )) ( ) BS j j  11 NN N1 NN  in Fig. 1.   Similarly, d ff , can be found by using ( ) BS j ij 4. Case Study Eq.  (9) and implying the belief degrees of f and f , ij In this regard, the proposed algorithm is applied in a respectively. By Computing distances for each alternative problem of public transportation. with each criterion, the following distance matrix can be Tehran, the capital of Iran, with a population of + + constructed, where d ff , is illustrated with d : ( ) BS j ij ij more than 14 million, is 17th largest metropolitan of the world. According to formal reports, more than 18 ++ +  dd  d 11 12 1n million intra-urban trips are taken in a working day  ++ + dd  d  of Tehran. This magnitude of trips highlighted the im - 21 22 2n D= . (23)  portance of intra-urban transformation system. Tehran     metro consist of 5 lines and about 90 stations around the + + +  dd  d m12 m mn  city, covers more than 3 million daily trips in Tehran and therefore it plays an important role in Tehran’s transfor- Then, the matrix D is normalized by dividing it mation system. Fig. 2 displays the Tehran’s metro map.  to d, and the normalized distance matrix ND= nd , ij e Th Metro system plays an important role in Teh-  ij + ran’s intra-urban transportation system. Considering a where nd = , i = 1, 2, …, m; j = 1, 2, …, n . At the ij d uniform traveling rate in its 12-hour working day, a one- next stage, the weighted normalize fuzzy belief distance minute delay in this system means that 70 person-hour  matrix WND= wnd is computed by multiplying working time of people will be lost. In addition, this delay ij  caused traffic in metro stations, which its handling will each element’s of matrix ND in its corresponding crite- ++ be a challenging task. Therefore, it is necessary to find rion importance, i.e. wnd = w d , i  = 1, 2, …, m; j  = ij j ij the most important causes of metro delays in Tehran. 1, 2, …, n. To this end, the FMEA approach is used. FMEA determines the most important risks based on Risk Pri- 3.5. Applying VIKOR Method ority Number (RPN) index. RPN is defined as RPN  = Since the weighted normalized fuzzy belief distance ma- Occurrence⋅Sverity⋅Detection. In fact, the more impor- trix is obtained, now the VIKOR method can be applied tant risk is one that has higher severity; its occurrence is to solve the MAGDM problem based on FBSs. First, the high, or its detection is more difficult. The classic FMEA values of S , i = 1, 2, …, m is computed for alternatives as: has some weaknesses as noted by Vahdani et al. (2014): – traditional FMEA neglects the relative impor- . (23) S = wnd ( ) i ∑ ij tance among occurrence, severity, and detection; j=1 – different values of occurrence, severity, and de- en, t Th he values of R , i = 1, 2, …, m are calculated: tection may produce similar RPN values; (24) R = max wnd ( ) – the traditional FMEA neglect to human/expert i ij knowledge. defining: Transport, 2016, 31(1): 108–118 Fig. 1. The flowchart of proposed FBS-VIKOR method for MAGDM Fig. 2. Tehran metro map To avoid these weaknesses, the FMEA is combined A team including 5 experts from Tehran Urban and Suburban Railway Operation Co is formed to find the with FBS based VIKOR method to evaluate the impor- most important delay causes in Tehran metro system. tance of delay causes. In fact, FMEA acts to define the critical criteria for ranking the causes of delay. Considering the previous reports and documents, the team identified 11 factors as the main causes of delays: S. H. Razavi Hajiagha et al. Fuzzy belief structure based VIKOR method: an application for ranking ... – A   – the transportation system (including the Table 2. FBS individual decision matrices train propulsion, brake system, pneumatic sys- Alter- Ex- tem, wagons door, computer accessories, and Severity Occurrence Detection natives perts technical facilities); E (0.70, 0.20, 0.10) (0.20, 0.60, 0.20) (0.30, 0.40, 0.30) – A  – power systems including the high pressure, (0.80, 0.05 , 0.05) (0.75, 0.05 , 0.05) (0.10, 0.40, 0.10) distribution channel, transfer lines, etc.; (0.80, 0.20, 0.00) (0.80, 0.10, 0.10) (0.80, 0.15, 0.05) A E 1 3 – A  – facilities including electrical and mechanical (0.10, 0.65, 0.25) (0.80, 0.10, 0.05) (0.15, 0.60, 0.15) facilities; E (0.20, 0.30, 0.50) (0.80, 0.10, 0.10) (0.60, 0.20, 0.10) – A   – control and sign unit involving the switch E (0.60, 0.25, 0.15) (0.70, 0.15, 0.15) (0.10, 0.20, 0.60) machine, rail circuit, interlocking, and software 1 (0.00, 0.05, 0.20) (0.90, 0.00, 0.00) (0.01, 0.01, 0.10) and hard ware of traffic control and trains safety 2 (0.90, 0.10, 0.00) (0.80, 0.10, 0.10) (0.10, 0.10, 0.80) A E system; 2 3 (0.85, 0.05, 0.00) (0.80, 0.05, 0.05) (0.05, 0.05, 0.30) – A  – telecommunications; E (0.40, 0.20, 0.10) (0.80, 0.10, 0.10) (0.10, 0.10, 0.80) – A  – line including the railway and switch; (0.20, 0.20, 0.50) (0.10, 0.60, 0.20) (0.10, 0.10, 0.80) – A  – buildings and stations; (0.05, 0.05, 0.50) (0.90, 0.05, 0.05) (0.01, 0.01, 0.20) – A  – passengers and other human factors; E (0.70, 0.15, 0.15) (0.70, 0.20, 0.10) (0.20, 0.20, 0.60) – A  – unexpected events such as fire, smoke natu- A E 9 3 3 ral disasters, railway failure, and others; (0.75, 0.10, 0.15) (0.10, 0.50, 0.10) (0.00, 0.10, 0.30) – A   – leadership including the train leadership, E (0.80, 0.10, 0.10) (0.30, 0.20, 0.10) (0.10, 0.10, 0.10) supervisor of traffic control center, and mainte- (0.60, 0.15, 0.15) (0.50, 0.30, 0.20) (0.75, 0.15, 0.05) nance; (0.05, 0.50, 0.05) (0.80, 0.050, 0.05) (0.05, 0.60, 0.05) – A   – traffic management in a way to indicate (0.50, 0.40, 0.10) (0.30, 0.20, 0.50) (0.30, 0.30, 0.30) 11 A E 4 3 how to utilize the system. E (0.50, 0.20, 0.30) (0.70, 0.20, 0.10) (0.8, 0.15, 0.05) es Th e causes are evaluated by participating experts E (0.50, 0.30, 0.20) (0.80, 0.10, 0.10) (0.7, 0.2, 0.1) based on three criteria: (0.80, 0.10, 0.10) (0.05, 0.10, 0.85) (0.05, 0.1, 0.85) – Severity: occurring this cause, how much severe, (0.90, 0.05, 0.05) (0.05, 0.05, 0.60) (0.01, 0.01, 0.20) based on delay time, will be the delays caused (0.80, 0.10, 0.10) (0.10, 0.10, 0.80) (0.10, 0.20, 0.70) A E 5 3 by it? E (0.8, 0.00, 0.00) (0.05, 0.05, 0.10) (0.05, 0.05, 0.3) – Occurrence: how much probable is its occur- (0.60, 0.30, 0.10) (0.80, 0.10, 0.10) (0.70, 0.20, 0.10) rence? (0.70, 0.20, 0.10) (0.15, 0.25, 0.60) (0.15, 0.25, 0.60) – Detection: how much is it possible to detect the (0.80, 0.00, 0.00) (0.80, 0.01, 0.01) (0.01, 0.01, 0.1) considered cause before its occurrence? (0.95, 0.05, 0.00) (0.50, 0.40, 0.10) (0.40, 0.30, 0.30) A E 6 3 Using a group AHP method, prior to constructing E (0.90, 0.00, 0.10) (0.80, 0.15, 0.05) (0.70, 0.20, 0.00) individual matrices, the importance of criteria are de- (0.90, 0.10, 0.00) (0.90, 0.10, 0.00) (0.50, 0.20, 0.00) termined as w  = 0.31; w  = 0.41; w  = severity occurrence detection (0.85, 0.05, 0.10) (0.05, 0.10, 0.85) (0.05, 0.10, 0.85) 0.28 and also W  = (0.2, 0.2, 0.2, 0.2, 0.2). (0.70, 0.10, 0.10) (0.10, 0.10, 0.60) (0.40, 0.10, 0.50) An FBS scale is also developed for assessing each (0.80, 0.15, 0.05) (0.20, 0.20, 0.60) (0.30, 0.00, 0.07) A E 7 3 delay cause about these criteria, as illustrated in Table 1. E (0.80, 0.10, 0.10) (0.10, 0.40, 0.10) (0.10, 0.40, 0.30) Decision makers express their beliefs about how percent (0.70, 0.10, 0.00) (0.60, 0.20, 0.00) (0.30, 0.20, 0.10) they believe that, the severity, occurrence, or detection (0.60, 0.15, 0.25) (0.70, 0.15, 0.15) (0.70, 0.15, 0.15) of a delay cause is hard, moderate, and/or simple. (0.80, 0.05, 0.05) (0.05, 0.50, 0.05) (0.95, 0.03, 0.02) Considering this FBS scale, individual fuzzy belief A E (0.95, 0.00, 0.05) (0.60, 0.30, 0.10) (0.70, 0.20, 0.10) decision matrices including the evaluation of experts are 8 3 E (0.85, 0.05, 0.10) (0.90, 0.05, 0.05) (0.95, 0.05, 0.00) shown in Table 2. 4 (0.50, 0.30, 0.20) (0.90, 0.10, 0.00) (0.90, 0.10, 0.00) The next step is to form an aggregated decision 5 (0.10, 0.10, 0.80) (0.80, 0.20, 0.00) (0.00, 0.05, 0.95) matrix, constructing individual decision matrices, by (0.02, 0.02, 0.30) (0.90, 0.05, 0.05) (0.00, 0.02, 0.10) applying Eqs (12) and (13), and decision makers weight E vector W . This matrix D is formed as follows (Table 3). A E (0.20, 0.10, 0.70) (0.97, 0.02, 0.01) (0.15, 0.15, 0.70) 9 3 (0.70, 0.30, 0.00) (0.85, 0.10, 0.05) (0.05, 0.05, 0.20) Considering Table 3, except for Sx and Sx , E ( ) ( ) 91 83 all other aggregated FBS models are incomplete, there- (0.10, 0.20, 0.70) (1.00, 0.00, 0.00) (0.10, 0.20, 0.40) fore, a normalizing stage is necessary. Applying Eqs (15) (0.40, 0.15, 0.45) (0.60, 0.20, 0.10) (0.40, 0.20, 0.20) and (16), the normalized FBS decision matrix will be (0.80, 0.05, 0.05) (0.00, 0.05, 0.30) (0.05, 0.05, 0.30) obtained as illustrated in Table 4. A E (0.80, 0.10, 0.10) (0.30, 0.00, 0.70) (0.20, 0.20, 0.60) 10 3 (0.60, 0.15, 0.15) (0.20, 0.40, 0.10) (0.50, 0.30, 0.20) Table 1. FBS scale used for delay causes evaluation (0.00, 0.20, 0.70) (1.00, 0.00, 0.00) (0.60, 0.20, 0.20) (0.50, 0.30, 0.20) (0.70, 0.20, 0.10) (0.10, 0.10, 0.80) Evaluation grade Linguistic term Utility 1 E (0.70, 0.15, 0.05) (0.70, 0.15, 0.15 ) (0.10, 0.25, 0.55) H severe/hard (0.5, 0.7, 0.9) A E (0.90, 0.10, 0.00) (0.40, 0.50, 0.10) (0.30, 0.00, 0.70) 11 3 H moderate (0.3, 0.5, 0.7) 2 (0.10, 0.10, 0.30) (0.85, 0.10, 0.05) (0.40, 0.10, 0.30) H weak/simple (0.1, 0.3, 0.5) (0.10, 0.10, 0.80) (0.40, 0.20, 0.10) (0.30, 0.20, 0.00) 5 Transport, 2016, 31(1): 108–118 Table 3. Aggregated decision matrix Alter- Severity Occurrence Detection natives (0.5525, 0.2679, 0.1631) (0.7252, 0.1612, 0.0830) (0.4110, 0.3597, 0.1318) (0.6284, 0.1213, 0.0778) (0.8519, 0.0600, 0.0600) (0.0691. 0.0890, 0.6086) A (0.5450, 0.1097, 0.2658) (0.4538, 0.3133, 0.1032) (0.0866, 0.1070, 0.4815) A (0.4615, 0.3073, 0.1510) (0.6704, 0.1483, 0.1655) (0.5654, 0.2650, 0.0976) (0.8340, 0.0846, 0.0533) (0.2047, 0.0756, 0.5511) (0.1848, 0.1137, 0.4875) (0.8948, 0.0524, 0.0283) (0.6827, 0.1608, 0.1284) (0.3899, 0.2020, 0.2062) (0.8269, 0.0772, 0.0536) (0.2036, 0.1940, 0.4712) (0.2205, 0.1447, 0.5391) A (0.7955, 0.0862, 0.1036) (0.6914, 0.1902, 0.0585) (0.8844, 0.0773, 0.0382) A (0.2159, 0.1366, 0.5461) (0.9350, 0.0503, 0.0147) (0.0593, 0.0939, 0.5615) (0.5573, 0.1173, 0.2762) (0.4704, 0.1219, 0.2391) (0.3745, 0.1896, 0.3083) (0.5002, 0.1425, 0.2611) (0.6635, 0.2061, 0.0847) (0.2328, 0.1190, 0.5211) Table 4. Normalized FBS decision matrix Alter- Severity Occurrence Detection natives A (0.5580, 0.2734, 0.1686) (0.7354, 0.1714, 0.0932) (0.4435, 0.3922, 0.1643) (0.6860, 0.1788, 0.1353) (0.8613, 0.0694, 0.0694 ) (0.1469. 0.1668, 0.6864) (0.5715, 0.1362, 0.2923) (0.4970, 0.3565, 0.1464) (0.1949, 0.2153, 0.5898) A (0.4882, 0.3340, 0.1777) (0.6757, 0.1536, 0.1707) (0.5894, 0.2890, 0.1216) (0.8434, 0.0940, 0.0626) (0.2609, 0.1318, 0.6073) (0.2561, 0.1850, 0.5589) (0.9036, 0.0591, 0.0372) (0.6921, 0.1701, 0.1378) (0.4572, 0.2693, 0.2735) (0.8410, 0.0913, 0.0677) (0.2474, 0.2377, 0.5149) (0.2524, 0.1766, 0.5710) (0.8004, 0.0911, 0.1085) (0.7114, 0.2101, 0.0785) (0.8844, 0.0773, 0.0382) A (0.2497, 0.1704, 0.5799) (0.9350, 0.0503, 0.0147) (0.1544, 0.189, 0.6566) (0.5736, 0.1337, 0.2926) (0.5266, 0.1781, 0.2953) (0.4170, 0.2321, 0.3508) (0.5323, 0.1746, 0.2931) (0.6787, 0.2214, 0.1000) (0.2752, 0.1614, 0.5635) + − The normalized FBS decision matrix is used to Considering d ff , = 0.6325 , the normal- ( ) BS j j ized distance matrix ND is found by dividing elements of compute the distance matrix D by Eq. (23). Consider- + − D into dS f , f . e Th obtained matrix is formed as: ing the fuzzy utilities of evaluation grades, the similarity ( ) B jj matrix is first calculated: 0.3345 0.1984 0.4103     1 0.8 0.6 0.2418 0.1097 0.7743     S= 0.8 1 0.8     0.3668 0.3704 0.7057     0.6 0.8 1 0.3831 0.2592 0.3028       0.1192 0.6764 0.6579 e di Th stance matrix D is computed as below by us-   0.0731 0.2385 0.4298 ing S: ND=     0.2116 0.1255 0.2595 0.1222 0.6448 0.6652     0.1529 0.0694 0.4897 0.1606 0.2115 0.0861       0.2320 0.2343 0.4463 0.6706 0.0471 0.7570     0.2423 0.1639 0.1915   0.3657 0.3945 0.4811       0.0754 0.4278 0.4161 0.3903 0.2379 0.6492       0.0462 0.1509 0.2718 D=   Constructing the matrix ND, it remains to find the   0.0773 0.4078 0.4207 weighted normalized decision matrix WND, by multi-   plying each element of ND at its associated attributes 0.1016 0.1338 0.0544   weight. Then, the S and R values for each alternative i i 0.4241 0.0298 0.4788   is determined by Eqs (23) and (24). Defining S   =   0.2313 0.2495 0.3040 + – – R   = 0, and S   = R   = 1, the Q , i  = 1, 2, …, 11 values   0.2469 0.1505 0.4106     are computed using Eq. (27), for different values of v. S. H. Razavi Hajiagha et al. Fuzzy belief structure based VIKOR method: an application for ranking ... Table 5. S , R and Q values for different v i i i Alternative S R i i 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 A 0.2999 0.1149 2 3 3 3 3 3 3 3 3 3 3 A 0.3367 0.2168 9 8 7 7 6 5 5 5 5 5 5 A 0.4632 0.1976 7 7 8 8 9 9 9 9 9 9 9 A 0.3098 0.1188 3 4 4 4 4 4 4 4 4 4 4 A 0.4985 0.2773 11 11 11 11 11 11 11 11 11 11 11 A 0.2408 0.1203 4 2 2 2 2 2 2 2 2 2 2 A 0.4885 0.2644 10 10 10 10 10 10 10 10 10 10 10 A 0.1606 0.0867 1 1 1 1 1 1 1 1 1 1 1 A 0.4392 0.2120 8 9 9 9 8 8 8 8 8 8 8 A 0.4098 0.1618 5 5 5 5 5 6 6 7 7 7 7 A 0.4003 0.1818 6 6 6 6 7 7 7 6 6 6 6 These va lues are computed and ranked at Table 5. Ac- Conclusions cording to this table, there are not any paradoxical rank- Multi attribute group decision-making problems are ing in different values of v and just some minor differ - a set of widely used procedures to choose the best al- ences occur for these values. Using the mean of ranks as ternative or to rank a set of alternatives based on a set the aggregating methods (Yoon, Hwang 1995), the final of different and usually contrasting attributes. Vincke ranking of delay causes is presented in Table 6, in the (1992) believed that this family of problems is usually FBS-VIKOR column. difficult to solve. This difficulty is a result of nonexist- According to the results of Table 6, the three most ence of a global optimal solution for these problems. important delay causes include passengers and other hu- On the other hand, oen de ft cision makers do not have man factors, line including the railway and switch, the complete information regard to required information for transportation system (including the train propulsion, decision-making problems. This feature of uncertainty brake system, pneumatic system, wagons door, comput- is formulated under different frameworks. One of the er accessories, and technical facilities). Therefore, some recent and strength frameworks of dealing with uncer- training and informative programs should be planned to tainty is FBS. This well-defined approach constituted of resolve human related issues. In addition, it seems nec- an evaluation grade that is represented as fuzzy grades essary to design and deploy an improved maintenance along with a belief degree assigned to each evaluation program for solving the problems related to line and grade. In FBS, decision makers most specify their be- transportation system. In this table, the results obtained lief degree about each alternative performance regard to by solving the problem using FBS-TOPSIS method of each attribute in every evaluation grade. Combination Jiang et al. (2011) and Vahdani et al. (2014) is also pre- of fuzzy set theory and the evidence combination rule sented. As it is clear from Table 5 results, except for 6th of the Dempster–Shafer theory made FBS as a powerful and 7th ranks, all other alternatives obtained similar way of portraying human uncertainty. ranks in both methods. The Spearman’s rank correlation In this paper, the VIKOR as an accepted MAGDM between both methods is about 0.97. method is extended under the condition that decision makers express their judgments about alternatives per- Table 6. Final ranking and comparison formance regard to attributes in the form of FBS models. Using evidential reasoning approach, the individual FBS Rank Alternatives decision matrices of decision makers are aggregated in a FBS-VIKOR FBS-TOPSIS single one. Then, the center of gravity method is applied A 3 3 to normalize the aggregated decision matrix. Distances A 6 5 of alternatives from these ideals are computed following the VIKOR method procedure by defining the positive A 9 9 ideal and negative ideal FBS models. A 4 4 e m Th ain advantages of the proposed method can A 11 11 be stated as: A 2 2 – providing a flexible framework for experts to A 10 10 state their judgment in a linguistic evaluation A 1 1 grade along with belief degrees; – provide a method of group decision-making by A 8 8 aggregating different experts’ opinion by consid- A 5 7 ering their relative importance and weights; A 7 6 – obtain a consensus based result for the decision- Transport, 2016, 31(1): 108–118 making process which can be applied in different Elevli, B. 2014. Logistics freight center locations decision by using fuzzy-PROMETHEE, Transport 29(4): 412–418. transportation decision-making problems and http://dx.doi.org/10.3846/16484142.2014.983966 other areas. Grattan-Guinness, I. 1976. Fuzzy membership mapped onto An application of the proposed method is shown intervals and many-valued quantities, Mathematical Logic in ranking the causes of delay in Tehran metro system. Quarterly 22(1): 149–160. Ranking criteria are defined according to FMEA ap- http://dx.doi.org/10.1002/malq.19760220120 proach. Then, five experts’ opinion are gathered in the Gifford, R.; Steg, L. 2007. The impact of automobile traffic on form of FBS model and the proposed FBS-based VIKOR quality of life, in T. Gärling, L. Steg (Eds.). re Th ats from method is applied. The obtained results showed that the Car Traffic to the Quality of Urban Life: Problems, Causes, most important delay causes are human related issues Solutions, 33–52. and line and transportation system. Finding the most Hashemi, S. S.; Razavi Hajiagha, S. H.; Zavadskas, E. K.; important delay causes, a set of corrective actions can be Amoozad Mahdiraji, H. 2016. Multicriteria group decision designed to resolve the undesirable consequences. Com- making with ELECTRE III method based on interval-val- ued intuitionistic fuzzy information, Applied Mathematical parison of the results with a previously proposed meth- Modelling 40(2): 1554–1564 od represented a high consistency among the methods http://dx.doi.org/10.1016/j.apm.2015.08.011 as a strengthening fact about the proposed FBS-VIKOR Inuiguchi,  M.; Ramík, J. 2000. Possibilistic linear program- method. Future researches in this area can be directed ming: a brief review of fuzzy mathematical programming toward extension of other decision-making methods, and a comparison with stochastic programming in portfo- include outranking methods, under FBS information. lio selection problem, Fuzzy Sets and Systems 111(1): 3–28. In addition, researchers can concentrate on extension of http://dx.doi.org/10.1016/S0165-0114(98)00449-7 mathematical operations of FBSs. Jiang, J.; Chen, Y.-W.; Chen, Y.-W.; Yang, K.-W. 2011. 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FUZZY BELIEF STRUCTURE BASED VIKOR METHOD: AN APPLICATION FOR RANKING DELAY CAUSES OF TEHRAN METRO SYSTEM BY FMEA CRITERIA

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TRANSPORT ISSN 1648-4142 / eISSN 1648-3480 2016 Volume 31(1): 108–118 doi:10.3846/16484142.2016.1133454 FUZZY BELIEF STRUCTURE BASED VIKOR METHOD: AN APPLICATION FOR RANKING DELAY CAUSES OF TEHRAN METRO SYSTEM BY FMEA CRITERIA 1 2 Seyed Hossein Razavi Hajiagha , Shide Sadat Hashemi , 3 4 Yousef Mohammadi , Edmundas Kazimieras Zavadskas 1, 3 Dept of Management, Khatam Institute of Higher Education, Tehran, Iran Saramadan Andisheh Avina Co., Tehran, Iran Dept of Construction Technology and Management, Vilnius Gediminas Technical University, Vilnius, Lithuania Submitted 31 August 2015; resubmitted 29 November 2015; accepted 29 November 2015 Abstract. Public transport is a critical part of civilization in this decade. The amount of money invested and the criticality of transferring people in an acceptable time and without any conflict made it a challenging problem for managers, especially in metropolises. Absolutely, making effective decisions in this area requires considering different aspects. Waiting time is a key criterion in apprising quality of public transport. In this paper, a real world case study of ranking causes of delay in Tehran (Iran) metro system is solved by developing multi attribute group decision-making VIšeKriterijumska Optimizacija I KOmpromisno Rešenje (in Serbian, VIKOR) method under uncertainty, where this uncertainty is captured by Fuzzy Belief Structures (FBS). The obtained results are then compared with a previously pro- posed Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method with FBSs. The results show that human related issues, along with the problems related to line and transportation system are the most important causes of delay. The obtained results of the problem seem acceptable for decision makers. Keywords: group decision-making; VIKOR; fuzzy belief structure; center of gravity; failure mode and effect analysis. Introduction (UN 2010). As Bainster (1996), and Gifford and Steg (2007) believed, transport holds major economic and Finding an ideal decision is the pursuit of many scholars social importance in improving expected standards of and practitioners in different fields (Tzeng, Huang 2011). consumption and quality of life. Multiple Criteria Decision-Making (MCDM) is a field Considering the above challenges, transporta- of operations research, which deals with evaluating and tion system is one of the implicational areas of MADM choosing the best alternative based on several criteria. (Bagočius et al. 2014; Elevli 2014; Šateikienė et al. 2015). Yoon and Hwang (1995) classified MCDM problems An MADM problem can be defined as follow (Zimmer - into two main categories: Multiple Attribute Decision- man 1987): Suppose that we have a nonempty and finite Making (MADM) and Multiple Objective Decision- set of decision alternatives, that their desirability will be Making (MODM), the former applies in evaluation judged according to a finite set of goals, attributes or type problems while the latter is suitable for design type criteria. The aim of MADM is to determine an optimal problems (Simon (1977) classified decision problems alternative having the highest degree of desirability in into selection and design problems). respect to all relevant goals. e Th aim of MCDM methods is to provide a logical e Th real world decision-making process is a com- and scientific framework of dealing with risky decisions. mon problem for employing the uncertainty phenom- One of the main challenges of 21st century is urban- ena. In fact, the required knowledge to formalize a deci- ization growth (Martine 2005). As stated by UN, today sion-making problem are usually subjected to uncertain- 54% of the world’s population lives in urban areas (UN ty. Dubois and Prade (1982), and Klir (1987) introduced 2015). Considering this growing urbanization phenom- ambiguity and vagueness as two types of uncertainty in ena, transport plays a crucial role in urban management Corresponding author: Seyed Hossein Razavi Hajiagha E-mail: [email protected] Copyright © 2016 Vilnius Gediminas Technical University (VGTU) Press http://www.tandfonline.com/TRAN Transport, 2016, 31(1): 108–118 real-world problems. While ambiguity refers to unspe- Yang et al. (2006) introduced the FBS where evaluation cific choice between alternatives, vagueness deals with grades are expressed as fuzzy numbers. In fact, an FBS is a combination of fuzzy set theory and the evidence com- situations where there are not any sharp boundaries bination rule of the Dempster–Shafer theory and there- among some domain of interest (Inuiguchi et al. 2000). fore is a powerful way of dealing with uncertainty. Jiang This undeniable uncertainty is widely known and et al. (2011) applied the FBS model to extend the TOP- accepted in decision-making, especially  – in MADM SIS method for group decision-making. Vahdani et  al. problems. Presence and acceptance of uncertainty re- (2014) also applied the above TOPSIS and FBS method quires a formal framework to be analyzed. Fuzzy set the- in the Failure Mode and Effect Analysis (FMEA) for ory, introduced by Zadeh (1965), is one of the common ranking the risk causes. frameworks in handling uncertainties (Liu, Lin 2006). e a Th im of this paper is to propose an extension of Bellman and Zadeh (1970) initially applied the concept VIKOR method under FBS. VIKOR method is a com- of fuzzy sets in decision-making problems. Fuzzy sets promise based method for ranking alternatives (Oprico- assign a membership value to each elements of a set. vic 1998). This method is applied for MADM problems In ordinal fuzzy sets, this membership values are exact with irrelevant and conflicting attributes (Opricovic, numbers. Some scholars criticized the ordinal fuzzy sets Tzeng 2004). As the founder of VIKOR method, Opri- due to crispness of its membership values (Grattan- covic extended the fuzzy VIKOR method and employed Guinness 1976). Therefore, some extensions are given it in some applications (Opricovic 2007, 2011). Vah- to the concept of membership functions or values. dani et  al. (2010) proposed the interval type-2 VIKOR Zadeh (1975) introduced type-2 fuzzy sets where method (they applied the interval-valued fuzzy sets with membership function itself is a fuzzy set. As an exten- a similar definition of interval type-2 fuzzy sets). The sion, interval type-2 fuzzy sets considered membership method is also developed in intuitionistic fuzzy (Devi functions as closed intervals (Liang, Mendel 2000). 2011) and interval valued intuitionistic fuzzy (Park et al. Simplicity of operations over intervals caused more in- 2011) environments. e Th idea of VIKOR method is also teresting about application of interval type-2 fuzzy sets extended to solve multi-objective linear programming in MADM. Wang et al. (2012) have found the interval problems (Razavi Hajiagha et  al. 2014). In this paper, type-2 fuzzy sets a very useful mean to depict the in- it is supposed that a group of experts participate in de- formation in decision-making process. They used it for cision-making process, who expresses their evaluations a group decision-making problem in order to calculate in decision matrix based on FBS models, i.e. they de- the attributes weights and aggregated decision matrix. termine their belief degrees’ regard to fuzzy evaluation grades of each alternatives with respect to each criteria. Chen (2013) used the interval type-2 trapezoidal fuzzy numbers to determine the alternatives ratings and the Considering the advantages of FBS in modeling data uncertainty by aggregating fuzzy evaluation grades and importance of various criteria. Baležentis and Zeng belief degrees and combining these advantages with the (2013) extended the MULTIMOORA method based VIKOR method’s advantage in determining a compro- upon fuzzy type-2 sets and generalized interval-valued mise solution of a decision-making problem could have trapezoidal fuzzy numbers. This paper was proposed to made the FBS-VIKOR method as an appealing method aggregate the group decision-making in human resource of solving Multiple Attribute Group Decision-Making management. The application of type-2 fuzzy numbers is (MAGDM) problems. illustrated in Maldonado et al. (2014), which proposed a The rest of paper is organized as follows. The design of multi objective genetic optimization problem. VIKOR method is briefly introduced in Section 1. Then, Atanassov (1986) extended the ordinal fuzzy sets by the required concepts of FBS models are overviewed in attending a non-membership value v beyond the clas- Section 2. The proposed method of VIKOR under FBSs sic membership value μ, and called the corresponding is explained in Section 3. A real world case study is then set as Intuitionistic Fuzzy Set (IFS). In a classic fuzzy solved by using the proposed method in Section 4. Fi- v 1−µ v+µ≤1 set , while in an IFS, . In an ordinal nally, the paper is concluded. IFS, the μ and v values are determined as crisp num- bers. Latter, Atanassov and Gargov (1989) developed 1. VIKOR IFS to Interval-Valued Intuitionistic Fuzzy Sets (IVIFSs) A decision-making problem can be formulated in the where membership and non-membership degrees are form of a decision matrix consist of the following ele- expressed as closed intervals. There is a planetary of re- ments (Yu 1990): searches done over extending MADM techniques under – the set of alternatives A AA , ,,… A ; { } IFS and IVIFS environments (Zhou et  al. 2013; Razavi 12 m – the set of criteria X XX , ,,… X ; { } Hajiagha et  al. 2015; Zavadskas et  al. 2015; Tan et  al. 12 n  2014; Chen 2015; Hashemi et al. 2016). – the outcome or decision matrix Dx = , which ij  Fuzzy Belief Structure (FBS) is another extension element x represents performance of alternative ij of ordinal fuzzy sets. Initially, the Belief Structure (BS) A based on the criterion X , for each i = 1, 2, …, i j introduced by Yang and Singh (1994), and Yang and Sen m and j = 1, 2, …, n; (1994) as an evidential reasoning approach for solving – the vector W ww , ,,… w , where w ≥ 0, j = { } 12 n j MADM problems. In a FBS model, the linguistic vari- 1, 2, …, n illustrates the weight or importance of ables of evaluation grades are considered as crisp values. criterion X . = S. H. Razavi Hajiagha et al. Fuzzy belief structure based VIKOR method: an application for ranking ... An MCDM method to rank a set of alternatives – alternative j and j , where QQ = if C is not 1 2 j2 [2] 2 based on conflicting criteria so-called VIKOR is intro- satisfied; duced to be employed in practical problems. This meth- – alternatives jj ,,…, j if C is not satisfied, al- 1 2 k 1 od is based on closeness to the Ideal solution. Presume ternative j is determined by Q −Q  DQ for k [k] [1] a set of m alternatives and n criteria, where the rating of maximum k, where QQ = . jk [] k each alternative A against to a criterion X is denoted as i j f . This method includes the following steps: 2. Fuzzy Belief Structure (FBS) ij + − Step 1. Determine the best f and worst f rating for BS is a distributed assessment scheme where belief de- i i all criteria as follow: grees are attained to different evaluation grades. Suppose that the evaluation grade of criteria consist a standard ff = max ; (1) j ij with N grades H ,H ,, … H and assume that is pre- { } − 12 N ff = min . (2) j ij ferred to H . A BS for the assessment of criterion c i k+1 can be represented as the following distribution: In this regard, the ideal and anti-ideal candidates + + + −− − are respectively as ff , ,..., f and f , f ,..., f . ( ) ( ) S c H ,β= ,n 1,2,,N , (8) 12 n 12 n ( ) ( ) { } nn Step 2. Calculate the average S ∈ 0,1 and the worst where: b is the belief degree of the grade H in the n n i  group R ∈ 0,1 scores for the alternative A : i i  evaluation, that β ≥ 0 , n = 1, 2, …, N and β ≤1. n n j=1 ff − ( ) j ij Sw = ; (3) Eq. (8) means that the criterion c is assessed with ij + − ff − j=1 the belief degree b at the grade H . If β= 1 , the dis- ( ) jj ∑ n n n j=1   ff − tribution is complete and if β ≤1  – it is incomplete. ( ) j ij ∑ n   Rw = max , (4) j=1 i j   + −   ff − ( ) jj   e e Th valuation grade consists of {H ,H ,, … H } 12 N where: w , j  = 1, 2, …, n indicates the relative impor- crisp number in ordinal BS. However, the ambiguity tance weights of the criterion j, which experts have de- and vagueness of real-world problems required evalua- termined. tion grades to be represented by fuzzy numbers. In this case, the fuzzy evaluation grades deal with fuzziness or Step 3. Calculate the Q for i = 1, 2, …, m as: vagueness and the belief degrees handled incomplete- ++ SS −− R R ( ) ( ) ness or ignorance. ii Qv + 1− v , (5) ( ) An important concept of FBS is to measure the −+ − + S −− S R R ( ) ( ) belief distance. Jiang et al. (2011) defined the distance where: between to FBS S and S as: 1 2 S = minS ; i d SS ,, d B B ( ) ( ) BS 12 BS 1 2 SS = max ; (6)  T  +  BB −− S BB . (9) RR = min ; ( ) ( )  12 12  i 2   RR = max . (7) That B and B are the belief degree distributions 1 2   of S and S , respectively. Also, Ss = represents the Here v indicates the weight of maximum group 1 2 ij   utility. It can be determined as: similarity matrix between fuzzy evaluations grades. If the v 0.5  – selected by majority; utilities of each fuzzy evaluation grade are represented by – v = 0.5  – consensus; nnnn trapezoidal fuzzy numbers, i.e. U H = uuuu ,,, , ( ) ( ) n 1 23 4 v 0.5  – veto. n  = 1, 2, …, N. Then, the similarity between j j j j Step 4. Sort the S, R and Q values by ascending order i i i i H = uu , ,, u u and H = uuuu ,,, is calcu- ( ) i 12 3 4 j ( 1 2 3 4) to rank each on. There would be 3 lists of ranking il- lated as: lustrated as: S , R and Q . uu − [0] [0] [0] ∑ kk k=1 s 1− , (10) ij Step 5. Indicate the smallest Q value related to alterna- i 4 tive j as a compromise solution if: 1 n  u ∈ 0,1 where: 01 ≤ s ≤ , since  , k  = 1, …, 4. If H ij n k  – C   – the alternative j has an acceptable advan- 1 1 nn is a triangular fuzzy number, then uu = and the 1 23 tage: Q −Q ≥ DQ , where DQ= and [2] [1] similarity measure is measured as the same. Jiang et al. m−1 m – the number of alternatives; (2011) proved that the distance measure in Eq. (9) is – C   – the alternative j is stable within the deci- 2 1 located between 0 and 1, and is a symmetric measure. sion-making process  – it is also the best ranked This measure will be used in the proposed VIKOR based in S or Q . [0] [0] on FBS model. If one of these conditions is not satisfied, then the solution is as follow: == = Transport, 2016, 31(1): 108–118 3. VIKOR Method Based on FBSs where: d d k As discussed earlier, in this paper, the multi-attribute a ww+− 1 β ; 1, ∏ ∑ n ij) ( k k n=1 VIKOR method is developed when decision-making k=1 information is given based on the FBS models. At the dk a= 1−β w ; ∏ ∑ next subsections, the fundamentals of the proposed al- 2, ( k n ij) n=1 k=1 gorithm are explained. Then, an algorithmic scheme is NN K presented to summarize these explanations.  dk d k µ= w β +− 1 w β −  ∑∑ ∏ kk n,, ij n ij  nn 11 k=1  3.1. Problem Definition and Data Preparation −1 K N e co Th nsidered problem is a multi-attribute group deci-  d k Nw −11− β  . (13) ( )  ∏ ∑ k n,ij sion-making problem. Suppose that a group of K deci-  k=1 n=1  sion makers, denoted by DM , k = 1, 2, …, K are gath- ered to evaluate a set of alternatives A AA , ,,… A { } 12 m 3.3. Normalizing the Aggregated Decision Matrix based on a set of criteria X XX , ,,… X . Each de- { } 12 n If the FBS models are incomplete individually, then ob- cision maker has a specific importance according to his/ tained aggregated FBS model will be incomplete. There- her role in decision-making process. Suppose that deci- fore, a normalizing stage is proposed to complete the sion makers importance are determined in the form of aggregated FBS models. DK 12 W ww , ,,… w a vector . In addition, the criteria { } In this paper, the normalization is done weight vector W ww , ,,… w can be determined { } 12 n based on the method of Jiang et al. (2011). Let using methods like simple rating or group Analytic Hi- ij sH= ,β= ,n 1,2,…,N be an incomplete FBS. ( ) erarchy Process (AHP) (Yoon, Hwang 1995; Saaty 1988). ij { n n } Each decision maker individually completed his/ her en, Th N BS peak points are defined as: kk  decision matrix D = Sx , k  = 1, 2, …, K, where ( ) , ij m SP s =H ,β= , n 1,2,…,N  k ( ) {( ) } ij nn Sx is an FBS model like: ( ) ij m = 1, 2, …, N, (14) kk Sx =H ,β= ,n 1,2,…,N , (10) ( ) {( ) } ij n n,ij with: where: β represents decision maker k’s assurance β +β , ; mn =  nH n,ij β= (15) that the performance of alternative A , i = 1, 2, …, m is β , m≠ n, at evaluation grade H , n  = 1, 2, …, N in criterion X , n j ij where: β=1− β is the degree of ignorance for in- jn = 1,2,.., β= 1 ∑ . If , then the kth decision maker’s Hn ∑ n,ij n=1 n=1 complete s . Then, the center of gravity of FBS model s ij ij evaluation over A alternative in X criterion is complete i j is defined as: and if β < 1 , it is incomplete. n,ij SP s ( ) ∑ ij n=1 m=1 SC s ( ) ij 3.2. Constructing the Aggregated Decision Matrix  As an important step in group decision-making, it is re-   β  quired to aggregate the individual decision matrices in n   m=1 an aggregated one. This aggregation is carried out using H , ,nN 1,2,…, . (16)   the evidential reasoning approach (Yang, Xu 2002; Yang     et al. 2006). Since the individual decision matrices D ,   k  = 1, 2, …, K are fuzzy belief matrices, the aggregated Then, the complete SC s is used instead of in- ( ) decision matrix D will be a fuzzy belief matrix. It can be ij complete s in the pooled decision matrix X. expressed as: ij 1 2 K   3.4. Constructing the Fuzzy Belief Distance Matrix D Sx D⊕ D ⊕…⊕ D , (11) ( ) ij   Considering the decreasing nature of evaluation grades, where: Sx represents the aggregated fuzzy belief + ( ) ij i.e. HH > , the positive ideal f and negative ideal nn +1 j performance of alternative A in criterion X . The FBS − i j f , FBS models for each criterion C , j = 1, 2, …, n are model Sx =H ,β= ,n 1,2,…,N is constructed defined as follows: ( ) ( ) ij { n n,ij } based on FBS models Sx , k = 1, 2, …, K. Wang et al. ( ) (17) ij fH = ( ,1),(H ,0),,(H ,0) { } j NN−11 (2006) proposed an analytic relation between the indi- and: vidual belief degrees β , k  = 1, 2, …, K with the ag- n,ij gregated belief degree of β , as: fH = ,0 , H ,0 ,, H ,1 . (18) {( ) ( ) ( )} n,ij j NN−11 µ⋅ aa − ( ) ij 1 2 β = , (12) Now, the S and R must be found applying Eqs (3) n i i  and (4), respectively. To compute these values, the dis- 11 −µ⋅ − w  ( )   k=1 = = == == = S. H. Razavi Hajiagha et al. Fuzzy belief structure based VIKOR method: an application for ranking ... tance between FBS models are used. The S is calculated S = minS ; as follow by considering Eqs (3) and (9): SS = max (25) n i d ff , ( ) BS j ij and: Sw = . (19) ij ∑ + − + d ff , RR = min ; j=1 ( ) BS j j RR = max . (26) The R is also defined as:  d ff , Calculate the Q for i  = 1, 2, …, m using Eq. (5), ( ) BS j ij i  Rw = max . (20) where the v multiplier is interpreted as in classic VIKOR i j  + −  d ff , + + ( ) BS j j method. Applying the common values of S   = R   = 0,  – – and S   = R  = 1 in Eq. (5), the following relation is ob- Denominators of both Eqs (19) and (20) are fixed tained: values, since the ideal and negative ideal solutions are similar, in the sense of FBS models, for all the criteria. Q= vS+ 1− v R . (27) ( ) ii i This fixed denominator is determined as below using Sort the S, R, and Q values by ascending order to Eq. (9) as: 1/2 rank alternatives. There would be 3 lists of ranking il-  1 + − + − + − d ff , = BB −− S BB . (21) lustrated as: S , R and Q . Aer det ft ermining these ( ) ( ) ( ) BS j j  j j j j [0] [0] [0]  lists, by using step 5 of the VIKOR method in Section 1, e f Th ollowing relation is obtained for all values of the final ranking list will be obtained. j, j = 1, 2, …, n by applying the positive ideal and nega- tive ideal belief degrees in Eqs (17) and (18) in Eq. (21): 3.6. Schematic Algorithm 1/2 In this regard, the proposed methodology is illustrated   + − d ff , = S + S −+ S S . (22) (( ) ( )) ( ) BS j j  11 NN N1 NN  in Fig. 1.   Similarly, d ff , can be found by using ( ) BS j ij 4. Case Study Eq.  (9) and implying the belief degrees of f and f , ij In this regard, the proposed algorithm is applied in a respectively. By Computing distances for each alternative problem of public transportation. with each criterion, the following distance matrix can be Tehran, the capital of Iran, with a population of + + constructed, where d ff , is illustrated with d : ( ) BS j ij ij more than 14 million, is 17th largest metropolitan of the world. According to formal reports, more than 18 ++ +  dd  d 11 12 1n million intra-urban trips are taken in a working day  ++ + dd  d  of Tehran. This magnitude of trips highlighted the im - 21 22 2n D= . (23)  portance of intra-urban transformation system. Tehran     metro consist of 5 lines and about 90 stations around the + + +  dd  d m12 m mn  city, covers more than 3 million daily trips in Tehran and therefore it plays an important role in Tehran’s transfor- Then, the matrix D is normalized by dividing it mation system. Fig. 2 displays the Tehran’s metro map.  to d, and the normalized distance matrix ND= nd , ij e Th Metro system plays an important role in Teh-  ij + ran’s intra-urban transportation system. Considering a where nd = , i = 1, 2, …, m; j = 1, 2, …, n . At the ij d uniform traveling rate in its 12-hour working day, a one- next stage, the weighted normalize fuzzy belief distance minute delay in this system means that 70 person-hour  matrix WND= wnd is computed by multiplying working time of people will be lost. In addition, this delay ij  caused traffic in metro stations, which its handling will each element’s of matrix ND in its corresponding crite- ++ be a challenging task. Therefore, it is necessary to find rion importance, i.e. wnd = w d , i  = 1, 2, …, m; j  = ij j ij the most important causes of metro delays in Tehran. 1, 2, …, n. To this end, the FMEA approach is used. FMEA determines the most important risks based on Risk Pri- 3.5. Applying VIKOR Method ority Number (RPN) index. RPN is defined as RPN  = Since the weighted normalized fuzzy belief distance ma- Occurrence⋅Sverity⋅Detection. In fact, the more impor- trix is obtained, now the VIKOR method can be applied tant risk is one that has higher severity; its occurrence is to solve the MAGDM problem based on FBSs. First, the high, or its detection is more difficult. The classic FMEA values of S , i = 1, 2, …, m is computed for alternatives as: has some weaknesses as noted by Vahdani et al. (2014): – traditional FMEA neglects the relative impor- . (23) S = wnd ( ) i ∑ ij tance among occurrence, severity, and detection; j=1 – different values of occurrence, severity, and de- en, t Th he values of R , i = 1, 2, …, m are calculated: tection may produce similar RPN values; (24) R = max wnd ( ) – the traditional FMEA neglect to human/expert i ij knowledge. defining: Transport, 2016, 31(1): 108–118 Fig. 1. The flowchart of proposed FBS-VIKOR method for MAGDM Fig. 2. Tehran metro map To avoid these weaknesses, the FMEA is combined A team including 5 experts from Tehran Urban and Suburban Railway Operation Co is formed to find the with FBS based VIKOR method to evaluate the impor- most important delay causes in Tehran metro system. tance of delay causes. In fact, FMEA acts to define the critical criteria for ranking the causes of delay. Considering the previous reports and documents, the team identified 11 factors as the main causes of delays: S. H. Razavi Hajiagha et al. Fuzzy belief structure based VIKOR method: an application for ranking ... – A   – the transportation system (including the Table 2. FBS individual decision matrices train propulsion, brake system, pneumatic sys- Alter- Ex- tem, wagons door, computer accessories, and Severity Occurrence Detection natives perts technical facilities); E (0.70, 0.20, 0.10) (0.20, 0.60, 0.20) (0.30, 0.40, 0.30) – A  – power systems including the high pressure, (0.80, 0.05 , 0.05) (0.75, 0.05 , 0.05) (0.10, 0.40, 0.10) distribution channel, transfer lines, etc.; (0.80, 0.20, 0.00) (0.80, 0.10, 0.10) (0.80, 0.15, 0.05) A E 1 3 – A  – facilities including electrical and mechanical (0.10, 0.65, 0.25) (0.80, 0.10, 0.05) (0.15, 0.60, 0.15) facilities; E (0.20, 0.30, 0.50) (0.80, 0.10, 0.10) (0.60, 0.20, 0.10) – A   – control and sign unit involving the switch E (0.60, 0.25, 0.15) (0.70, 0.15, 0.15) (0.10, 0.20, 0.60) machine, rail circuit, interlocking, and software 1 (0.00, 0.05, 0.20) (0.90, 0.00, 0.00) (0.01, 0.01, 0.10) and hard ware of traffic control and trains safety 2 (0.90, 0.10, 0.00) (0.80, 0.10, 0.10) (0.10, 0.10, 0.80) A E system; 2 3 (0.85, 0.05, 0.00) (0.80, 0.05, 0.05) (0.05, 0.05, 0.30) – A  – telecommunications; E (0.40, 0.20, 0.10) (0.80, 0.10, 0.10) (0.10, 0.10, 0.80) – A  – line including the railway and switch; (0.20, 0.20, 0.50) (0.10, 0.60, 0.20) (0.10, 0.10, 0.80) – A  – buildings and stations; (0.05, 0.05, 0.50) (0.90, 0.05, 0.05) (0.01, 0.01, 0.20) – A  – passengers and other human factors; E (0.70, 0.15, 0.15) (0.70, 0.20, 0.10) (0.20, 0.20, 0.60) – A  – unexpected events such as fire, smoke natu- A E 9 3 3 ral disasters, railway failure, and others; (0.75, 0.10, 0.15) (0.10, 0.50, 0.10) (0.00, 0.10, 0.30) – A   – leadership including the train leadership, E (0.80, 0.10, 0.10) (0.30, 0.20, 0.10) (0.10, 0.10, 0.10) supervisor of traffic control center, and mainte- (0.60, 0.15, 0.15) (0.50, 0.30, 0.20) (0.75, 0.15, 0.05) nance; (0.05, 0.50, 0.05) (0.80, 0.050, 0.05) (0.05, 0.60, 0.05) – A   – traffic management in a way to indicate (0.50, 0.40, 0.10) (0.30, 0.20, 0.50) (0.30, 0.30, 0.30) 11 A E 4 3 how to utilize the system. E (0.50, 0.20, 0.30) (0.70, 0.20, 0.10) (0.8, 0.15, 0.05) es Th e causes are evaluated by participating experts E (0.50, 0.30, 0.20) (0.80, 0.10, 0.10) (0.7, 0.2, 0.1) based on three criteria: (0.80, 0.10, 0.10) (0.05, 0.10, 0.85) (0.05, 0.1, 0.85) – Severity: occurring this cause, how much severe, (0.90, 0.05, 0.05) (0.05, 0.05, 0.60) (0.01, 0.01, 0.20) based on delay time, will be the delays caused (0.80, 0.10, 0.10) (0.10, 0.10, 0.80) (0.10, 0.20, 0.70) A E 5 3 by it? E (0.8, 0.00, 0.00) (0.05, 0.05, 0.10) (0.05, 0.05, 0.3) – Occurrence: how much probable is its occur- (0.60, 0.30, 0.10) (0.80, 0.10, 0.10) (0.70, 0.20, 0.10) rence? (0.70, 0.20, 0.10) (0.15, 0.25, 0.60) (0.15, 0.25, 0.60) – Detection: how much is it possible to detect the (0.80, 0.00, 0.00) (0.80, 0.01, 0.01) (0.01, 0.01, 0.1) considered cause before its occurrence? (0.95, 0.05, 0.00) (0.50, 0.40, 0.10) (0.40, 0.30, 0.30) A E 6 3 Using a group AHP method, prior to constructing E (0.90, 0.00, 0.10) (0.80, 0.15, 0.05) (0.70, 0.20, 0.00) individual matrices, the importance of criteria are de- (0.90, 0.10, 0.00) (0.90, 0.10, 0.00) (0.50, 0.20, 0.00) termined as w  = 0.31; w  = 0.41; w  = severity occurrence detection (0.85, 0.05, 0.10) (0.05, 0.10, 0.85) (0.05, 0.10, 0.85) 0.28 and also W  = (0.2, 0.2, 0.2, 0.2, 0.2). (0.70, 0.10, 0.10) (0.10, 0.10, 0.60) (0.40, 0.10, 0.50) An FBS scale is also developed for assessing each (0.80, 0.15, 0.05) (0.20, 0.20, 0.60) (0.30, 0.00, 0.07) A E 7 3 delay cause about these criteria, as illustrated in Table 1. E (0.80, 0.10, 0.10) (0.10, 0.40, 0.10) (0.10, 0.40, 0.30) Decision makers express their beliefs about how percent (0.70, 0.10, 0.00) (0.60, 0.20, 0.00) (0.30, 0.20, 0.10) they believe that, the severity, occurrence, or detection (0.60, 0.15, 0.25) (0.70, 0.15, 0.15) (0.70, 0.15, 0.15) of a delay cause is hard, moderate, and/or simple. (0.80, 0.05, 0.05) (0.05, 0.50, 0.05) (0.95, 0.03, 0.02) Considering this FBS scale, individual fuzzy belief A E (0.95, 0.00, 0.05) (0.60, 0.30, 0.10) (0.70, 0.20, 0.10) decision matrices including the evaluation of experts are 8 3 E (0.85, 0.05, 0.10) (0.90, 0.05, 0.05) (0.95, 0.05, 0.00) shown in Table 2. 4 (0.50, 0.30, 0.20) (0.90, 0.10, 0.00) (0.90, 0.10, 0.00) The next step is to form an aggregated decision 5 (0.10, 0.10, 0.80) (0.80, 0.20, 0.00) (0.00, 0.05, 0.95) matrix, constructing individual decision matrices, by (0.02, 0.02, 0.30) (0.90, 0.05, 0.05) (0.00, 0.02, 0.10) applying Eqs (12) and (13), and decision makers weight E vector W . This matrix D is formed as follows (Table 3). A E (0.20, 0.10, 0.70) (0.97, 0.02, 0.01) (0.15, 0.15, 0.70) 9 3 (0.70, 0.30, 0.00) (0.85, 0.10, 0.05) (0.05, 0.05, 0.20) Considering Table 3, except for Sx and Sx , E ( ) ( ) 91 83 all other aggregated FBS models are incomplete, there- (0.10, 0.20, 0.70) (1.00, 0.00, 0.00) (0.10, 0.20, 0.40) fore, a normalizing stage is necessary. Applying Eqs (15) (0.40, 0.15, 0.45) (0.60, 0.20, 0.10) (0.40, 0.20, 0.20) and (16), the normalized FBS decision matrix will be (0.80, 0.05, 0.05) (0.00, 0.05, 0.30) (0.05, 0.05, 0.30) obtained as illustrated in Table 4. A E (0.80, 0.10, 0.10) (0.30, 0.00, 0.70) (0.20, 0.20, 0.60) 10 3 (0.60, 0.15, 0.15) (0.20, 0.40, 0.10) (0.50, 0.30, 0.20) Table 1. FBS scale used for delay causes evaluation (0.00, 0.20, 0.70) (1.00, 0.00, 0.00) (0.60, 0.20, 0.20) (0.50, 0.30, 0.20) (0.70, 0.20, 0.10) (0.10, 0.10, 0.80) Evaluation grade Linguistic term Utility 1 E (0.70, 0.15, 0.05) (0.70, 0.15, 0.15 ) (0.10, 0.25, 0.55) H severe/hard (0.5, 0.7, 0.9) A E (0.90, 0.10, 0.00) (0.40, 0.50, 0.10) (0.30, 0.00, 0.70) 11 3 H moderate (0.3, 0.5, 0.7) 2 (0.10, 0.10, 0.30) (0.85, 0.10, 0.05) (0.40, 0.10, 0.30) H weak/simple (0.1, 0.3, 0.5) (0.10, 0.10, 0.80) (0.40, 0.20, 0.10) (0.30, 0.20, 0.00) 5 Transport, 2016, 31(1): 108–118 Table 3. Aggregated decision matrix Alter- Severity Occurrence Detection natives (0.5525, 0.2679, 0.1631) (0.7252, 0.1612, 0.0830) (0.4110, 0.3597, 0.1318) (0.6284, 0.1213, 0.0778) (0.8519, 0.0600, 0.0600) (0.0691. 0.0890, 0.6086) A (0.5450, 0.1097, 0.2658) (0.4538, 0.3133, 0.1032) (0.0866, 0.1070, 0.4815) A (0.4615, 0.3073, 0.1510) (0.6704, 0.1483, 0.1655) (0.5654, 0.2650, 0.0976) (0.8340, 0.0846, 0.0533) (0.2047, 0.0756, 0.5511) (0.1848, 0.1137, 0.4875) (0.8948, 0.0524, 0.0283) (0.6827, 0.1608, 0.1284) (0.3899, 0.2020, 0.2062) (0.8269, 0.0772, 0.0536) (0.2036, 0.1940, 0.4712) (0.2205, 0.1447, 0.5391) A (0.7955, 0.0862, 0.1036) (0.6914, 0.1902, 0.0585) (0.8844, 0.0773, 0.0382) A (0.2159, 0.1366, 0.5461) (0.9350, 0.0503, 0.0147) (0.0593, 0.0939, 0.5615) (0.5573, 0.1173, 0.2762) (0.4704, 0.1219, 0.2391) (0.3745, 0.1896, 0.3083) (0.5002, 0.1425, 0.2611) (0.6635, 0.2061, 0.0847) (0.2328, 0.1190, 0.5211) Table 4. Normalized FBS decision matrix Alter- Severity Occurrence Detection natives A (0.5580, 0.2734, 0.1686) (0.7354, 0.1714, 0.0932) (0.4435, 0.3922, 0.1643) (0.6860, 0.1788, 0.1353) (0.8613, 0.0694, 0.0694 ) (0.1469. 0.1668, 0.6864) (0.5715, 0.1362, 0.2923) (0.4970, 0.3565, 0.1464) (0.1949, 0.2153, 0.5898) A (0.4882, 0.3340, 0.1777) (0.6757, 0.1536, 0.1707) (0.5894, 0.2890, 0.1216) (0.8434, 0.0940, 0.0626) (0.2609, 0.1318, 0.6073) (0.2561, 0.1850, 0.5589) (0.9036, 0.0591, 0.0372) (0.6921, 0.1701, 0.1378) (0.4572, 0.2693, 0.2735) (0.8410, 0.0913, 0.0677) (0.2474, 0.2377, 0.5149) (0.2524, 0.1766, 0.5710) (0.8004, 0.0911, 0.1085) (0.7114, 0.2101, 0.0785) (0.8844, 0.0773, 0.0382) A (0.2497, 0.1704, 0.5799) (0.9350, 0.0503, 0.0147) (0.1544, 0.189, 0.6566) (0.5736, 0.1337, 0.2926) (0.5266, 0.1781, 0.2953) (0.4170, 0.2321, 0.3508) (0.5323, 0.1746, 0.2931) (0.6787, 0.2214, 0.1000) (0.2752, 0.1614, 0.5635) + − The normalized FBS decision matrix is used to Considering d ff , = 0.6325 , the normal- ( ) BS j j ized distance matrix ND is found by dividing elements of compute the distance matrix D by Eq. (23). Consider- + − D into dS f , f . e Th obtained matrix is formed as: ing the fuzzy utilities of evaluation grades, the similarity ( ) B jj matrix is first calculated: 0.3345 0.1984 0.4103     1 0.8 0.6 0.2418 0.1097 0.7743     S= 0.8 1 0.8     0.3668 0.3704 0.7057     0.6 0.8 1 0.3831 0.2592 0.3028       0.1192 0.6764 0.6579 e di Th stance matrix D is computed as below by us-   0.0731 0.2385 0.4298 ing S: ND=     0.2116 0.1255 0.2595 0.1222 0.6448 0.6652     0.1529 0.0694 0.4897 0.1606 0.2115 0.0861       0.2320 0.2343 0.4463 0.6706 0.0471 0.7570     0.2423 0.1639 0.1915   0.3657 0.3945 0.4811       0.0754 0.4278 0.4161 0.3903 0.2379 0.6492       0.0462 0.1509 0.2718 D=   Constructing the matrix ND, it remains to find the   0.0773 0.4078 0.4207 weighted normalized decision matrix WND, by multi-   plying each element of ND at its associated attributes 0.1016 0.1338 0.0544   weight. Then, the S and R values for each alternative i i 0.4241 0.0298 0.4788   is determined by Eqs (23) and (24). Defining S   =   0.2313 0.2495 0.3040 + – – R   = 0, and S   = R   = 1, the Q , i  = 1, 2, …, 11 values   0.2469 0.1505 0.4106     are computed using Eq. (27), for different values of v. S. H. Razavi Hajiagha et al. Fuzzy belief structure based VIKOR method: an application for ranking ... Table 5. S , R and Q values for different v i i i Alternative S R i i 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 A 0.2999 0.1149 2 3 3 3 3 3 3 3 3 3 3 A 0.3367 0.2168 9 8 7 7 6 5 5 5 5 5 5 A 0.4632 0.1976 7 7 8 8 9 9 9 9 9 9 9 A 0.3098 0.1188 3 4 4 4 4 4 4 4 4 4 4 A 0.4985 0.2773 11 11 11 11 11 11 11 11 11 11 11 A 0.2408 0.1203 4 2 2 2 2 2 2 2 2 2 2 A 0.4885 0.2644 10 10 10 10 10 10 10 10 10 10 10 A 0.1606 0.0867 1 1 1 1 1 1 1 1 1 1 1 A 0.4392 0.2120 8 9 9 9 8 8 8 8 8 8 8 A 0.4098 0.1618 5 5 5 5 5 6 6 7 7 7 7 A 0.4003 0.1818 6 6 6 6 7 7 7 6 6 6 6 These va lues are computed and ranked at Table 5. Ac- Conclusions cording to this table, there are not any paradoxical rank- Multi attribute group decision-making problems are ing in different values of v and just some minor differ - a set of widely used procedures to choose the best al- ences occur for these values. Using the mean of ranks as ternative or to rank a set of alternatives based on a set the aggregating methods (Yoon, Hwang 1995), the final of different and usually contrasting attributes. Vincke ranking of delay causes is presented in Table 6, in the (1992) believed that this family of problems is usually FBS-VIKOR column. difficult to solve. This difficulty is a result of nonexist- According to the results of Table 6, the three most ence of a global optimal solution for these problems. important delay causes include passengers and other hu- On the other hand, oen de ft cision makers do not have man factors, line including the railway and switch, the complete information regard to required information for transportation system (including the train propulsion, decision-making problems. This feature of uncertainty brake system, pneumatic system, wagons door, comput- is formulated under different frameworks. One of the er accessories, and technical facilities). Therefore, some recent and strength frameworks of dealing with uncer- training and informative programs should be planned to tainty is FBS. This well-defined approach constituted of resolve human related issues. In addition, it seems nec- an evaluation grade that is represented as fuzzy grades essary to design and deploy an improved maintenance along with a belief degree assigned to each evaluation program for solving the problems related to line and grade. In FBS, decision makers most specify their be- transportation system. In this table, the results obtained lief degree about each alternative performance regard to by solving the problem using FBS-TOPSIS method of each attribute in every evaluation grade. Combination Jiang et al. (2011) and Vahdani et al. (2014) is also pre- of fuzzy set theory and the evidence combination rule sented. As it is clear from Table 5 results, except for 6th of the Dempster–Shafer theory made FBS as a powerful and 7th ranks, all other alternatives obtained similar way of portraying human uncertainty. ranks in both methods. The Spearman’s rank correlation In this paper, the VIKOR as an accepted MAGDM between both methods is about 0.97. method is extended under the condition that decision makers express their judgments about alternatives per- Table 6. Final ranking and comparison formance regard to attributes in the form of FBS models. Using evidential reasoning approach, the individual FBS Rank Alternatives decision matrices of decision makers are aggregated in a FBS-VIKOR FBS-TOPSIS single one. Then, the center of gravity method is applied A 3 3 to normalize the aggregated decision matrix. Distances A 6 5 of alternatives from these ideals are computed following the VIKOR method procedure by defining the positive A 9 9 ideal and negative ideal FBS models. A 4 4 e m Th ain advantages of the proposed method can A 11 11 be stated as: A 2 2 – providing a flexible framework for experts to A 10 10 state their judgment in a linguistic evaluation A 1 1 grade along with belief degrees; – provide a method of group decision-making by A 8 8 aggregating different experts’ opinion by consid- A 5 7 ering their relative importance and weights; A 7 6 – obtain a consensus based result for the decision- Transport, 2016, 31(1): 108–118 making process which can be applied in different Elevli, B. 2014. Logistics freight center locations decision by using fuzzy-PROMETHEE, Transport 29(4): 412–418. transportation decision-making problems and http://dx.doi.org/10.3846/16484142.2014.983966 other areas. Grattan-Guinness, I. 1976. Fuzzy membership mapped onto An application of the proposed method is shown intervals and many-valued quantities, Mathematical Logic in ranking the causes of delay in Tehran metro system. Quarterly 22(1): 149–160. Ranking criteria are defined according to FMEA ap- http://dx.doi.org/10.1002/malq.19760220120 proach. Then, five experts’ opinion are gathered in the Gifford, R.; Steg, L. 2007. The impact of automobile traffic on form of FBS model and the proposed FBS-based VIKOR quality of life, in T. Gärling, L. Steg (Eds.). re Th ats from method is applied. The obtained results showed that the Car Traffic to the Quality of Urban Life: Problems, Causes, most important delay causes are human related issues Solutions, 33–52. and line and transportation system. Finding the most Hashemi, S. S.; Razavi Hajiagha, S. H.; Zavadskas, E. K.; important delay causes, a set of corrective actions can be Amoozad Mahdiraji, H. 2016. Multicriteria group decision designed to resolve the undesirable consequences. Com- making with ELECTRE III method based on interval-val- ued intuitionistic fuzzy information, Applied Mathematical parison of the results with a previously proposed meth- Modelling 40(2): 1554–1564 od represented a high consistency among the methods http://dx.doi.org/10.1016/j.apm.2015.08.011 as a strengthening fact about the proposed FBS-VIKOR Inuiguchi,  M.; Ramík, J. 2000. Possibilistic linear program- method. Future researches in this area can be directed ming: a brief review of fuzzy mathematical programming toward extension of other decision-making methods, and a comparison with stochastic programming in portfo- include outranking methods, under FBS information. lio selection problem, Fuzzy Sets and Systems 111(1): 3–28. In addition, researchers can concentrate on extension of http://dx.doi.org/10.1016/S0165-0114(98)00449-7 mathematical operations of FBSs. Jiang, J.; Chen, Y.-W.; Chen, Y.-W.; Yang, K.-W. 2011. 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