SOFT COMPUTING BASED ON A MODIFIED MCDM APPROACH UNDER INTUITIONISTIC FUZZY SETS

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1 Iranan Journal of Fuzzy Systems Vol. 14, No. 1, (2017 pp SOFT COMPUTING BASED ON A MODIFIED MCDM APPROACH UNDER INTUITIONISTIC FUZZY SETS M. R. SHAHRIARI Abstract. The current study set to extend a new VIKOR method as a compromse rankng approach to solve multple crtera decson-makng (MCDM problems through ntutonstc fuzzy analyss. Usng compromse method n MCDM problems contrbutes to the selecton of an alternatve as close as possble to the postve deal soluton and far away from the negatve deal soluton, concurrently. Usng Atanassov ntutonstc fuzzy sets (A-IFSs may smultaneously express the degree of membershp and non-membershp to decson makers (DMs to descrbe uncertan stuatons n decson-makng problems. The proposed ntutonstc fuzzy VIKOR ndcates the degree of satsfacton and dssatsfacton of each alternatve wth respect to each crteron and the relatve mportance of each crteron, respectvely, by degrees of membershp and non-membershp. Thus, the ratngs for the mportance of crtera, DMs, and alternatves are n lngustc varables and expressed n ntutonstc fuzzy numbers. Usng IFS aggregaton operators and wth respect to subjectve judgment and objectve nformaton, the most sutable alternatve s ndcated among potental alternatves. Moreover, practcal examples llustrate the procedure of the proposed method. 1. Introducton Multple crtera decson-makng (MCDM method s a wdely used decsonmakng method [7, 23, 31]. The typcal MCDM problem s concerned wth the rankng order of decson alternatves accordng to dfferent conflctng crtera. In practce, t s not possble for alternatves to concurrently satsfy all crtera [13, 46, 3, 22]. In fact, f one alternatve obtans a good score wth respect to one crteron, t s least lkely that n other crtera the same good score s acheved. Alternatves possess ther own strengths regardng dfferent crtera that may not be consstent [2, 38, 39, 20, 48]. Dfferent MCDM methods work based on Pareto optmalty concept. The VIKOR (Vlse Krterjumska Optmzacja I Kompromsno Resenje n Serban method by Oprcovc [27] s one and was developed for mult-attrbute optmzatons of complex systems; t s a compromse rankng approach for MCDM problems. Ths method concentrates on the rankng and selectng from a set of alternatves n the presence of conflctng crtera, ntroducng the mult-crtera rankng ndex accordng to the partcular measure of closeness to the deal soluton [6]. Compared to Receved: February 2016; Revsed: June 2016; Accepted: August 2016 Keywords and phrases: Multple crtera decson makng (MCDM, Decson makers (DMs, Atanassov ntutonstc fuzzy sets (A-IFSs, Intutonstc fuzzy numbers.

2 24 M. R. Shahrar other MCDM methods (e.g., TOPSIS, PROMETHEE, and ELECTRE, VIKOR has ts advantages [29, 30]. VIKOR determnes a compromse soluton accepted by DMs snce t provdes a maxmum group utlty for the majorty and a mnmum of ndvdual regret for the opponent. MCDM problems have an uncertanty because all MCDM methods operate accordng to DMs subjectve preferences and judgments. They utlze lngustc varables by ther knowledge and experence whch contan uncertanty n real-lfe stuatons snce t s dffcult to apply precse numbers to descrbe DMs assessments. On the other hand, conflctng crtera n MCDM problems cause further uncertanty. Hence, usng decson approaches that can handle these uncertantes may be helpful n havng realstc and reasonable judgments [25, 21, 42, 14, 40]. Turnng ther subjectve perceptons of a fuzzy concept, n most real-lfe cases, such as mportance or excellence, nto crsp numbers may get the DMs nto trouble [43, 37, 35, 34]. Thus, to solve MCDM problems n fuzzy envronments, a number of MCDM methods usng the concept of fuzzy sets have been presented [9, 11, 12, 19, 53, 47, 41, 32, 26]. The noton of ntutonstc fuzzy sets (IFSs was ntated by Atanassov [4] characterzed by a membershp functon and non-membershp functon and s the generalzaton of the concept of fuzzy sets [6], characterzed only by a membershp functon and more sutable for dealng wth vagueness compared wth fuzzy sets. The ablty of descrbng agreement, dsagreement, and hestancy values n the Atanassov ntutonstc fuzzy set (A-IFS s a powerful tool to deal wth vagueness [24, 36, 16]. It has receved hgh attenton n varous research felds, especally many complex decson-makng problems n whch decson nformaton s provded by a DM and s often mprecse or uncertan due to tme pressure, lack of data, or DMs lmted attenton and nformaton processng capabltes. Snce VIKOR s a hghly powerful MCDM method, more attenton has been attracted by extensons on ths method and ts applcatons under fuzzy envronments. Chen and Wang [10] provded a ratonal and systematc process to develop the best alternatve and compromse soluton under each selecton crteron. Chang [8] ntroduced a modfed VIKOR method for multple crtera analyss to avod numercal dffcultes n solvng problems va the tradtonal VIKOR method. To mprove servce qualty among domestc arlnes n Tawan, James [18] used a modfed VIKOR method. Jahan et al. [16] developed a verson of VIKOR method, coverng all types of crtera wth emphass on compromse soluton. Sanaye et al. [2] assessed the ratngs and weghts expressed by trapezodal or trangular fuzzy numbers usng lngustc values. Aghajan et al. [1] presented an evaluaton model based on determnstc data, fuzzy numbers, nterval numbers, and lngustc terms, and then they appled a combnaton of analytc herarchy process (AHP and entropy method for attrbutes weghtng n ther proposed MCDM method. Crstbal [33] appled VIKOR to select a renewable energy project correspondng to the renewable energy plan launched by Spansh government and combned t wth AHP method for weghtng the mportance of dfferent crtera. Vahdan et al. [45] appled a VIKOR method to solve fuzzy group decson-makng problems. Developng two new ndces was smultaneously taken nto account to consder the relatve dstance of alternatves from postve and negatve deal solutons. A new ndex was

3 Soft Computng Based on a Modfed MCDM Approach Under Intutonstc Fuzzy Sets 25 presented to dstngush alternatves n the assessment process by combnng the effect of majorty of crtera weghts and ndvdual regrets weghts; hence, they extended the method for an ntutonstc fuzzy envronment. Wu and Yu [49] extended VIKOR for MCDM problems wth ratngs of alternatves expressed usng ntutonstc fuzzy sets, and the weghts of the crtera are completely unknown. They establshed an entropy weght model to determne the weghts wth respect to a set of crtera represented by IFSs. Accordng to the exstng lterature, most studes on VIKOR are under the envronment of determnstc data or tradtonal fuzzy numbers; however, because of ts extent, t seems to be almost unknown research feld n solvng MCDM problems. The purpose of present study s to extend a new VIKOR method for an ntutonstc fuzzy envronment. The current paper s organzed as follows: In the followng, the VIKOR method s brefly ntroduced. Secton 3 contans prelmnares related to IFSs. The proposed VIKOR method s represented n secton 4. In secton 5, an llustratve example of the proposed method s presented. The paper s concluded n secton VIKOR Method Snce Oprcovc and Tzeng [29, 28] developed VIKOR method, dfferent researches have been focused on t. VIKOR was extended n dfferent ways n tradtonal fuzzy or crsp envronment. One of the most applcable extensons used n ths study s based on Vahdan et al. [45] approach for the fuzzy envronment. The procedure by a group of DMs s as follows [45]: The MCDM problem contanng crsp and fuzzy numbers can be expressed n the matrx format for each DM as: x 11 x 1(k 1 x 1k x 1n x m1 x m(k 1 x mk x mn Step 1: Establsh a group of L DMs. A group of L DMs (l = 1, 2,..., L s assumed, responsble for evaluatng m alternatves wth respect to each n crteron. Step 2: Defne and descrbe a set of relevant crtera. Step 3: Obtan the ratng of a potental alternatve versus each selected crteron for each DM. Step 4: Aggregate the ratngs of alternatves versus each subjectve crteron ( x j and fuzzy weghts of selected crtera ( w j. Step 5: Compute the normalzed decson matrx. Vector normalzaton s appled to calculate r j and r j. Step 6: Construct the fuzzy weghted normalzed decson matrx Ṽ = [ṽ j] m n. The fuzzy weghted normalzed decson matrx s computed through multplyng each matrx column by the fuzzy weght ( w j, assgned by parwse comparsons of elements. Thus, ṽ j = w j r j, = 1, 2,..., m; j = 1, 2,..., k 1 (1 ṽ j = w j r j, = 1, 2,..., m; j = k, k + 1,..., n (2

4 26 M. R. Shahrar If the supports of trangular fuzzy numbers do not belong to the nterval [0, 1], scalng has to transform them back nto the nterval. Step 7: Determne the postve deal and negatve deal solutons. A and A values are defned as: A = n(v1, v 2,..., v k 1, ṽ k, ṽ k+1,..., ṽ n {( ( } = max ṽ j j J, mn ṽ j j J = 1, 2,..., m (3 Where A = (v 1, v 2,..., v k 1, ṽ k, ṽ k+1,..., ṽ n {( ( } = mn ṽ j j J, max ṽ j j J = 1, 2,..., m ( max ṽ j = ( mn ṽ j = max mn v j1, max v j1, mn v j2, max v j3 v j2, mn v j3 J = j = 1, 2,..., n j s assocated wth beneft crteron J = j = 1, 2,..., n j s assocated wth cost crteron Step 8: Construct deal separaton matrx (D and ant-deal separaton matrx (D. Step 9: Compute h, I, ζ and ξ values for j = 1, 2,..., n as follows: n h = w jd j (5 j=1 I = max w j jd j (6 n ζ = w j d j j=1 (7 ξ = max w j d j j (8 Where w j s obtaned usng the defuzzfcaton method. Step 10: Compute the values of τ and η ndces. I I + I I + f h = h + h h + τ = h h + f I = I + ( h h + ( I I + h h + v + I I + (1 v otherwse (9 and ξ ξ + ξ ξ + f ζ = ζ + ζ ζ + ζ ζ + f ξ = ξ + η = ( ξ ξ + ξ ξ + γ + (4 ( ζ ζ + ζ ζ + (1 γ otherwse (10

5 Soft Computng Based on a Modfed MCDM Approach Under Intutonstc Fuzzy Sets 27 where { h + = mn h h = max h, { I + = mn I I = max I { ξ + = max ξ ξ = mn ξ, { ζ + = mn ζ ζ = max ζ v and γ are regarded as a weght for strategy of the majorty of the crtera, whereas (1 v and (1 γ are ndvdual regret weghts. v and γ values of all wthn the range of 0 to 1 and these strateges can be compromsed by v = 0.5 and γ = 0.5. Step 11: Calculate collectve ndex (CI. CI s calculated by: CI = τ + 1 η (B + Φ (11 Where the second term refers to all for whch η > 0 whle Φ refers to all for whch η = 0 and Φ = (mn η (B mnj wj. Step 12: Rank the preference order. The best satsfactory alternatve can be determned accordng to preference rank order of τ and η. The mnmum value of CI ndcates the better performance for alternatve. 3. Prelmnares on Intutonstc Fuzzy Sets As a generalzaton of fuzzy set, ntutonstc fuzzy sets (IFSs assgn to each element a membershp degree and a non-membershp degree; therefore, t s powerful n dealng wth uncertanty, mprecson, and vagueness. Atanassov [4] defned IFSs as follows: Defnton 3.1. Let a set X be fxed, the concept of IFS A on X as follows: A = {< x, µ A (x, A (x > x X} (12 Where functons µ A (x and v A (x denote the degrees of membershp and nonmembershp of the element to the set A, respectvely, where µ A : X [0, 1] and v A : X [0, 1] denote, respectvely, the membershp functon and non-membershp functon of A wth the condton 0 µ A (x + v A (x 1 for any x X. For each IFS A n X, π A (x = 1 µ A (x A (x (13 s called the degree of ndetermnacy of x to A, or the degree of hestancy of x to A. For the convenence of computaton, Xu and Yager [51] called α = (µ α, α, π α an ntutonstc fuzzy number (IFN, where µ α [0, 1], α [0, 1], µ α (x + α (x 1 (14 π α = 1 µ α v α (15 For an IFN α = (µ α, α, π α, f the value µ α gets larger and the value v α gets smaller, IFN ncreases, and thus from (4, α + = (1, 0, 0 and α = (0, 1, 0 are the largest and smallest IFNs, respectvely.

6 28 M. R. Shahrar Defnton 3.2. Xu and Yager [51] and Xu [52] ntroduced some basc operatons for two IFNs α 1 and α 2 : 1 α 1 α 2 = (µ α1 + µ α2 µ α1 µ α2, v α1 v α2 ; 2 α 1 α 2 = (µ α1 µ α2, v α1 + v α2 v α1 v α2 ; 3 λα 1 = (1 (1 µ α1 λ, α1 λ, λ > 0; 4 α λ 1 = (µ α1 λ, 1 (1 v α1 λ, λ > 0; 5 α 1 α 2 = (mn{µ α1, µ α2 }, max{ α1, α2 }; 6 α 1 α 2 = (max{µ α1, µ α2 }, mn{ α1, α2 }; Where = max and = mn. Xu and Yager [51] ntroduced a devaton factor s(α = µ α α, whch s called a score of α.h(α = µ α + α s called the accuracy degree of IFN α. Xu and Yager [51] gave an order relaton between two IFNs α 1 and α 2 based on the score functon s and the accuracy functon h. Defnton 3.3. If S(α 1 < S(α 2, then α 1 < α 2; If S(α 1 = S(α 2, then If h(α 1 < h(α 2, then α 1 = α 2; If h(α 1 < h(α 2, then α 1 < α Proposed Decson Method wth the Intutonstc Fuzzy Informaton Assume a commttee of LDMs (D 1, D 2,..., D L, and for each DM, the MCDM problem s represented by an ntutonstc fuzzy decson matrx R as follows: x 11 x 1n R =..... x m1 x mn (16 Let A be a set of alternatves and C a set of crtera, where A = {A 1, A 2,..., A m } and C = {C 1, C 2,..., C n }. Assume that the characterstcs of the alternatve A are presented by IFS as A = {(C 1, µ 1, 1, (C 2, µ 2, 2,..., (C n, µ n, n }, = 1, 2,..., m, where µ j ndcates the degree to whch alternatve A satsfes crteron C j ; j ndcates the degree to whch alternatve A does not satsfy crteron C j ( j, j, = 1, 2,..., m; j = 1, 2,..., n. The steps of the proposed VIKOR method for solvng the MCDM problems are descrbed as follows: Step 1: A group of L DMs (l = 1, 2,..., L s assumed responsble for evaluatng alternatves wth respect to each n crteron. The relatve mportance of DMs s evaluated n lngustc terms represented by IFNs. The weght of an IFND k = [µ k, k, π k ] whch rates the kth DM s computed by Boran et al. [6]: ( ( µk µ k + π k µ k + v k λ k = ( ( L µk k=1 µ k + π k (17 µ k + v k

7 Soft Computng Based on a Modfed MCDM Approach Under Intutonstc Fuzzy Sets 29 and L λ k = 1 k=1 Step 2: Defne all objectve and subjectve crtera wth beneft or cost types and based on these crtera, dentfy the alternatves. ( Step 3: Determne the alternatves ratng matrx R (k = r (k j for each DM versus each crteron va lngustc terms. Step 4: Construct an aggregated ntutonstc fuzzy decson matrx R = (r j m n and aggregate the weght of each crteron w j based on DMs opnons applyng IFWA operator proposed by Xu [50]. ( r j =IF W A λ r (1 j, r(2 j,..., r(l j =λ 1 r (1 j λ 2 r (2 j... λ L r (L j [ ] = 1 Π L k=1 (1 µ(k j λ k, Π L k=1 (v(k j λ k, Π L k=1 (1 µ(k j λ k Π L k=1 (v(k j λ k Where λ = {λ 1, λ 2..., λ L } s the weght of each DM and L k=1 λ k = 1, λ k [0, 1]. Here r j = (µ r (x j, r (x j, π r (x j, = 1, 2,..., m; j = 1, 2,..., n. Let w (k j = [µ (k j, (k j Then, the weght of each crteron s calculated as follows: w j = IF W A λ ( w (1 [ = 1 Π L k=1 w = [w 1, w 2,..., w j ] (18, π (k j ] be an IFN assgned to crteron C j by the kth DM. j, w (2 j,..., w (L j = λ 1 w (1 j λ 2 w (2 j Where w j = (µ wj, wj, π wj, j = 1, 2,..., n. λ L w (L j ] (1 µ(k j λ k, Π L k=1 ((k j k, Π L k=1 (1 µ(k j λ k Π L k=1 ((k j k Step 5: Construct the fuzzy weghted decson matrx V = W R usng relaton (2 of Defnton 2. Step 6: Determne the postve deal and negatve deal solutons represented by A and A, respectvely. The largest and smallest IFNs are α + = (1, 0, 0 and α = (0, 1, 0, respectvely. For beneft crteron, DM desres to have a maxmum value, and for cost crteron, desres mnmum value among the alternatves; therefore, A ndcates the most preferable alternatves, and A ndcates least preferable alternatves. Accordng to equatons (3 and (4 and relatons (5 and (6 of Defnton 2, the postve deal soluton and negatve deal soluton are descrbed as follows: {( A = (max j {( A = (mn (max µ j j J, (mn j j J } j J = 1, 2,..., m µ j j J, (max j j J } (mn j j J = 1, 2,..., m &(mn µ j j, &(max µ j j, (19 (20 (21

8 30 M. R. Shahrar Where: J = j = 1, 2,..., n j s assocated wth beneft corteron J = j = 1, 2,..., n j s assocated wth cost corteron Step 7: Construct D and D are deal and non-deal separaton matrxes, respectvely. d(v 11, v1 d(v 1n, vn D =.. d(v m1, v1 d(v mn, vn (22 D = d(v 11, v 1 d(v 1n, v n.. d(v m1, v1 d(v mn, vn Where d(a, B s the defnton of dstance accordng to Grzegorewsk [15] and defned as follows: { } d(a, B = max µ A (x µ B (x, A (x B (x. (24 Step 8: Defuzzfy w usng the score functon accordng to Bandyopadhyay et al. [5] and compute h, I, ζ and ξ values usng equatons (5 to (8. Step 9: Compute τ and η usng equatons (9 and (10. Step 10: Calculate collectve ndex (CI usng equaton (11. Step 11: Rank the preference order. The best satsfactory alternatve can be determned accordng to τ and η preference rank order. The mnmum value of CI ndcates the better performance for alternatve. (23 5. Applcaton of the Proposed Method n Solvng Problems 5.1. Illustratve example one. In ths sub-secton, a numercal example llustrates how the proposed VIKOR method can be used under uncertanty. Suppose that a company manufacturng tractor components desres to renew the manufacturng system. Ths company should evaluate and select the most approprate one among the avalable alternatve flexble manufacturng systems (FMSs to produce a group of products. A 1, A 2, A 3, A 4, and A 5 are fve alternatve FMSs, determned and evaluated by a commttee of three DMs aganst fve selected crtera. The functonal requrements (FRs that must be satsfed by an FMS are provded as follows regardng the characterstc of the product group manufactured by a company [44]: C 1 : Qualty of results C 2 : Ease of use C 3 : Compettve C 4 : Adaptablty C 5 : Expandablty

9 Soft Computng Based on a Modfed MCDM Approach Under Intutonstc Fuzzy Sets 31 The mportance weghts of three DMs and fve crtera are defned n Table 1 usng the followng lngustc terms. The performance ratngs of the alternatves are characterzed by the followng lngustc terms wth respect to the crtera: Extreme Good (EG/Extreme Hgh (EH, Very Good (VG/Very Hgh (VH, Good (G/Hgh (H, Medum Good (MG/Medum Hgh (MH, Far (F/Medum (M, Medum poor (MP/Medum Low (ML, Poor (P/Low (L, Very Poor (VP/Very Low (VL, Extreme Poor (EP/Extreme Low (EL (Table 2. In ths example, Lngustc varables Interval valued fuzzy numbers Very Poor (VP (0.90, 0.10, 0.00 Poor (P (0.75, 0.20, 0.05 Moderately Poor (MP (0.50, 0.45, 0.05 Far (F (0.35, 0.60, 0.05 Moderately Good (MG (0.25, 0.60, 0.15 Good (G (0.10, 0.78, 0.12 Very Good (VG (0.10, 0.90, 0.00 Table 1. Lngustc Varables for Ratng the Importance of Crtera and the DMs Lngustc varables Intutonstc fuzzy numbers Extreme good (EG/Extreme hgh (EH (1.00, 0.00, 0.00 Very verygood (VVG/ Very very hgh (VVH (0.90, 0.10, 0.00 Very good (VG/Very hgh (VH (0.80, 0.10, 0.10 Good(G/Hgh(H (0.70, 0.20, 0.10 Medum good (MG/medum hgh (MH (0.60, 0.30, 0.10 Far (F/medum (M (0.50, 0.40, 0.10 Medum bad(mb/medum low (ML (0.40, 0.50, 0.10 Bad (B/Low (L (0.25, 0.60, 0.15 Very bad (VB/Very low (VL (0.10, 0.75, 0.15 Very very bad (VVb/ Very very low (VVL (0.10, 0.90,0.00 Table 2. Lngustc Varables for the Ratng of Alternatves the DMs mportance s not equal; hence, ther relatve mportance s gven n Table 3, contanng the DMs weghts calculated by equaton (17. The alternatves ratngs wth respect to each crteron descrbed by the DMs are llustrated n Table 4 usng lngustc ratng varables (Steps 1 to 3. In step DM 1 DM 3 DM 2 Lngustc terms Important medum Very mportant Intutonstc fuzzy (0.75, 0.2, 0.05 (0.50, 0.45, 0.05 (0.90, 0.10, 0.00 number Weght Table 3. The Relatve Importance of the DMs

10 32 M. R. Shahrar Crtera Alternatves Decson makers DM1 DM2 DM3 A 1 H H MH A 2 VG G VG C 1 A 3 VG VG VG A 4 VH VH H A 5 F F MG A 1 MG MG G A 2 MB MB MB C 2 A 3 VVG VG VG A 4 VVG VG VG A 5 MB F F A 1 G G VG A 2 VG G VG C 3 A 3 VG G G A 4 VG G G A 5 G MG MG A 1 H H H A 2 MB F MB C 4 A 3 VH H H A 4 H MH MH A 5 M MH M A 1 MG MG MG A 2 MH MH M C 5 A 3 VG G G A 4 G G F A 5 MB F MB Table 4. Ratngs of the Alternatves 4, the ntutonstc fuzzy ratngs of each alternatve are descrbed by the DMs wth respect to each crteron and aggregated usng equaton (18 to construct decson matrx (Tables 5 and 6. The mportance weght of each crteron s calculated by equaton (19 (Table 7 and 8 applyng the selected crtera weghts, evaluated by the DMs va lngustc varables and ther respectve IFNs (Table 1. C 1 C 2 C 3 A 1 (0.663, 0.236, (0.644, 0.254, (0.746, 0.151, A 2 (0.708, 0.118, (0.400, 0.500, (0.780, 0.118, A 3 (0.800, 0.100, (0.844, 0.100, (0.740, 0.156, A 4 (0.764, 0.133, (0.844, 0.100, (0.740, 0.156, A 5 (0.543, 0.356, (0.466, 0.433, (0.639, 0.260, Table 5. Aggregated Intutonstc Fuzzy Decson Matrx After constructng the ntutonstc fuzzy weghted decson matrx (Step 5, the postve deal soluton and negatve deal soluton are determned for each crteron (Tables 9 and 10 (Step 6. Afterwards, the deal separaton matrx D and nondeal separaton matrx D are constructed (Step 7. Bandyopadhyay et al. [5]

11 Soft Computng Based on a Modfed MCDM Approach Under Intutonstc Fuzzy Sets 33 C 4 C 5 A 1 (0.700, 0.200, (0.600, 0.300, A 2 (0.425, 0.474, (0.562, 0.337, A 3 (0.740, 0.156, (0.780, 0.118, A 4 (0.639, 0.260, (0.631, 0.265, A 5 (0.526, 0.374, (0.425, 0.474, Table 6. Aggregated Intutonstc Fuzzy Decson Matrx C 1 C 2 C 3 DM 1 M MH VH DM 2 M H VH DM 3 VL MH H Aggregated (0.258, 0.707, (0.576, 0.371, (0.855, 0.133, ntutonstc weght Table 7. Weghts and Aggregated Intutonstc Weghts of Crtera C 4 C 5 DM 1 VL MH DM 2 VL MH DM 3 M M Aggregated (0.211, 0.763, (0.444, 0.506, ntutonstc weght Table 8. Weghts and Aggregated Intutonstc Weghts of Crtera showed that, accordng to the score functon, µ j s usable as a defuzzfed IFN, lke w j (µ j, j π j ; accordng to above,h, I, ζ, and ξ values are computed usng equatons (5 to (8 (Step 8. Then, τ and η values are calculated usng equatons (9 and (10, where the weght values of v and γ are assumed 0.5 (Step 9. Fnally, the collectve ndex (CI of all alternatves s calculated usng equaton (11, and the FMSs are ranked accordng to the results (Steps 10 and 11. All calculated ndces values are provded n Table 11. Accordng to the results, the rankng orders of fve alternatves are A 3, A 4, A 5, A 1, and A Illustratve Example Two. In order to further clarfy the proposed method, another example s presented for the contractor selecton n a constructon ndustry by Vahdan et al. [45]. After pre-evaluaton, three contractors remaned as alternatves for further evaluaton. In order to evaluate alternatve contractors, a commttee composed of three DMs (DM 1, DM 2, and DM 3 havng equal degrees of mportance was formed because of the same backgrounds and experence. Ffteen crtera (C, = 1, 2,..., 15 are selected (Steps 1 and 2.

12 34 M. R. Shahrar C 1 C 2 C 3 A 1 (0.171, 0.776, (0.371, 0.531, (0.637, 0.263, A 2 (0.201, 0.742, (0.230, 0.686, (0.667, 0.235, A 3 (0.207, 0.737, (0.486, 0.434, (0.633, 0.268, A 4 (0.197, 0.746, (0.486, 0.434, (0.633, 0.268, A 5 (0.140, 0.812, (0.269, 0.643, (0.546, 0.358, Postve deal (0.207, 0.737, (0.486, 0.434, (0.667, 0.235, soluton (A Negatve deal (0.140, 0.812, (0.230, 0.686, (0.546, 0.358, soluton(a Table 9. Intutonstc Fuzzy Weghted Decson Matrx C 4 C 5 A 1 (0.148, 0.811, (0.266, 0.654, A 2 (0.090, 0.876, (0.249, 0.672, A 3 (0.156, 0.800, (0.346, 0.564, A 4 (0.135, 0.825, (0.280, 0.637, A 5 (0.111, 0.852, (0.189, 0.740, Postve deal (0.156, 0.800, (0.346, 0.564, soluton (A Negatve deal (0.090, 0.876, (0.189, 0.740, soluton (A Table 10. Intutonstc Fuzzy Weghted Decson Matrx Indces values Alternatve h I ζ ξ τ η CI Fnal rankng A A A A A Table 11. Indces Values and CIby the Proposed Method for the Illustratve Exampleone The weghts of these crtera obtaned by the DMs accordng to lngustc terms are defned n Table 1, and ther aggregated ntutonstc weghts obtaned by equaton (19 are shown n Table 12. The alternatves ratngs wth respect to each crteron represented by the DMs n lngustc terms are gven n Table 2 and Table 13 (Step3. The ntutonstc fuzzy decson matrx s calculated n Step 4 usng equaton (18 and s llustrated n Table 13. The weghted decson matrx s constructed (Step 5. Subsequently, the postve deal soluton and negatve deal soluton are determned, and the deal and non-deal separaton matrxes are constructed (Steps 6 and 7 (Table 14.

13 Soft Computng Based on a Modfed MCDM Approach Under Intutonstc Fuzzy Sets 35 Crtera Descrpton Decson makers DM 1 DM 2 DM 3 C 1 Tender prce H MH VH C 2 Fnancal statement VH H H C 3 Fnancal references H MH MH C 4 Falure to have contract completed H VH VH C 5 Cost overruns MH H MH C 6 Delay VH MH H C 7 Qualty MH M H C 8 Scale M MH MH C 9 Type H MH H C 10 Experence VH VH H C 11 Physcal resources H H MH C 12 Human resources H VH MH C 13 Current workload M ML ML C 14 Past clent/contractor relatonshp MH MH M C 15 Safety performance M MH ML Table 12. Weghts and Aggregated Intutonstc Weghts of Crtera Crtera Alternatves Decson makers DM1 DM2 DM3 A 1 VVH G VVH C 1 A 2 G MG MG A 3 MG G MG A 1 G G G C 2 A 2 MG MG L A 3 LG MG MG A 1 VVH G VVH C 3 A 2 VGHH MG MG A 3 MG G MG A 1 VVH VVH G C 4 A 2 G MG G A 3 MG MG MG A 1 G G MG C 5 A 2 L L VL A 3 VL VL VL A 1 L L VL C 6 A 2 VL L L A 3 VL VL VL A 1 G MG G C 7 A 2 MG G G A 3 L MG MG A 1 EG EG VVH C 8 A 2 VVH VVH MG A 3 MG MG VVH A 1 MG L L C 9 A 2 L VL L A 3 MG MG VL A 1 L L VL C 10 A 2 L VL L A 3 VVL L VVL A 1 VVH VVH G C 11 A 2 MG MG G A 3 G MG MG A 1 G G G C 12 A 2 L MG MG A 3 MG L MG A 1 G MG VG C 13 A 2 MG G MG A 3 L G G A 1 EG VVH EG C 14 A 2 VVH G VVH A 3 G G VVH A 1 G MG G C 15 A 2 MG G MG A 3 L L G Table 13. Ratngs of the Alternatves

14 36 M. R. Shahrar C 1 C 2 C 3 A 1 (0.856, 0.126, (0.700, 0.200, (0.856, 0.126, A 2 (0.637, 0.262, (0.507, 0.378, (0.748, 0.208, A 3 (0.637, 0.262, (0.507, 0.378, (0.637, 0.262, C 6 C 7 C 8 A 1 (0.203, 0.646, (0.670, 0.229, (1.000, 0.000, A 2 (0.203, 0.646, (0.670, 0.229, (0.841, 0.144, A 3 (0.100, 0.750, (0.507, 0.378, (0.748, 0.208, C 11 C 12 C 13 A 1 (0.856, 0.126, (0.700, 0.200, (0.670, 0.229, A 2 (0.637, 0.262, (0.507, 0.378, (0.637, 0.262, A 3 (0.637, , (0.507, 0.378, (0.593, 0.288, Table 14. Intutonstc Fuzzy Weghted Decson Matrx C 4 C 5 A 1 (0.856, 0.126, (0.670, 0.229, A 2 (0.670, 0.229, (0.203, 0.646, A 3 (0.600, 0.300, (0.100, 0.750, C 9 C 10 A 1 (0.392, 0.476, (0.203, 0.646, A 2 (0.203, 0.646, (0.153, 0.696, A 3 (0.476, 0.407, (0.153, 0.786, C 14 C 15 A 1 (1.000, 0.000, (0.670, 0.229, A 2 (0.856, 0.126, (0.637, 0.262, A 3 (0.792, 0.159, (0.447, 0.416, Table 15. Intutonstc Fuzzy Weghted Decson Matrx C 1 C 2 C 3 A 1 (0.435, 0.308, (0.499, 0.327, (0.435, 0.426, A 2 (0.324, 0.416, (0.361, (0.380, 0.480, A 3 (0.324, 0.416, (0.361, (0.324, 0.515, Postve deal soluton (A (0.435, 0.308, (0.499, 0.327, (0.435, 0.426, Negatve deal soluton (A (0.324, 0.416, (0.361, (0.324, 0.515, C 6 C 7 C 8 A 1 (0.129, 0.720, (0.217, 0.520, (0.324, 0.495, A 2 (0.129, 0.720, (0.217, 0.520, (0.273, 0.568, A 3 (0.064, 0.802, (0.164, 0.613, (0.242, 0.600, Postve deal soluton (A (0.129, 0.720, (0.217, 0.520, (0.324, 0.495, Negatve deal soluton (A (0.064, 0.802, (0.164, 0.613, (0.242, 0.600, C 11 C 12 C 13 A 1 (0.522, 0.355, (0.495, 0.366, (0.170, 0.692, A 2 (0.388, 0.455, (0.359, 0.507, (0.162, 0.705, A 3 (0.388, 0.455, (0.359, 0.507, (0.151, 0.715, Postve deal soluton (A (0.522, 0.355, (0.495, 0.366, (0.170, 0.692, Negatve deal soluton (A (0.388, 0.455, (0.359, 0.507, (0.151, 0.715, Table 16. Intutonstc Fuzzy Weghted Decson Matrx 5.3. Dscusson. Senstvty s recognzed on dfferent values of the majorty crtera,.e., v and γ; hence, the mentoned weghts values are changed smultaneously to perform senstvty analyss. The maxmal group utlty s 1; the maxmal regret s 0, and the combnaton of both s 0.5. To conduct senstvty analyss, v and γ are set by sx dfferent values between 0 and 1. Then, seven dfferent v and γ values are used n the proposed method to check the changes n the alternatves rankng n the gven example (Tables 19 and 20. In fact, the results of these changes could help the DMs by makng the assessment process easer and through determnng the prortes.

15 Soft Computng Based on a Modfed MCDM Approach Under Intutonstc Fuzzy Sets 37 C 4 C 5 A 1 (0.606, 0.236, (0.341, 0.494, A 2 (0.474, 0.326, (0.103, 0.768, A 3 (0.425, 0.388, (0.051, 0.836, Postve deal soluton (A (0.606, 0.236, (0.341, 0.494, Negatve deal soluton (A (0.425, 0.388, (0.051, 0.836, C 9 C 10 A 1 (0.199, 0.613, (0.159, 0.691, A 2 (0.103, 0.739, (0.120, 0.735, A 3 (0.242, 0.563, (0.120, 0.813, Postve deal soluton (A (0.242, 0.563, (0.159, 0.691, Negatve deal soluton (A (0.103, 0.739, (0.120, 0.813, C 14 C 15 A 1 (0.381, 0.495, (0.217, 0.649, A 2 (0.326, 0.559, (0.206, 0.664, A 3 (0.302, 0.575, (0.145, 0.734, Postve deal soluton (A (0.381, 0.495, (0.217, 0.649, Negatve deal soluton (A (0.302, 0.575, (0.145, 0.734, Table 17. Intutonstc Fuzzy Weghted Decson Matrx Indces values Alternatve h I ζ ξ τ η CI Fnal rankng A A A Table 18. Indces Values and CI by the Proposed Method for the Illustratve Example Two v and γ values Alternatves CI Preference order rankng v = 0 and γ = 0 A A A A A v = 0.2 and γ = 0.2 A A A A A v = 0.4 and γ = 0.4 A A A A A v = 0.4 and γ = 0.5 A A A A A v = 0.6 and γ = 0.6 A A A A A v = 0.8 and γ = 0.8 A A A A A v = 1 and γ = 1 A A A A A Table 19. Dfferent Values of v and γ and Preference Order Rankng by the Proposed Method for Illustratve Example One

16 38 M. R. Shahrar v and γ values Alternatves CI Preference order rankng v = 0 and γ = 0 A A A v = 0.2 and γ = 0.2 A A A v = 0.4 and γ = 0.4 A A A v = 0.4 and γ = 0.5 A A A v = 0.6 and γ = 0.6 A A A v = 0.8 and γ = 0.8 A A A v = 1 and γ = 1 A A A Table 20. Dfferent Values of and γ and Preference Order Rankng by the Proposed Method for Illustratve Example One 6. Concludng The ntutonstc fuzzy VIKOR method focuses on assessng a set of alternatves under an uncertan envronment and rankng them wth respect to multple conflctng crtera. The mult-crtera decson makng problems vagueness has been expressed through ntutonstc fuzzy sets by a group of experts due to ts power n revealng DMs hestancy, assumng ther agreement and dsagreement. The proposed method gves the chance of utlzng more flexble way to deal wth real-lfe decson-makng problems and help the DMs to reach an acceptable compromse of the maxmum group utlty of the majorty and the mnmum of the ndvdual regret of the opponent. In the presented method, the relatve mportance of crtera, called ther weghts and the alternatves ratngs, can be descrbed by lngustc terms and then converted to IFNs. The ntutonstc fuzzy weghted averagng (IFWA operator s used to aggregate DMs judgments and weghts of crtera. The deal separaton matrx and ant-deal separaton matrx were constructed usng two IF-dstance measures. To consder the relatve dstance of alternatves from the postve deal soluton and negatve deal soluton, two ndces were used for rankng the alternatves based on the strategy of the majorty crtera and the ndvdual regret. Fnally, to hghlght the applcablty of the proposed method, two applcaton examples were used, and also dfferent weghts of the majorty crtera for the senstvty analyss purpose were changed n both flexble manufacturng systems (FMSs selecton and contractor selecton examples. The proposed method

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