Linear programming with Triangular Intuitionistic Fuzzy Number

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1 EUSFLAT-LFA 2011 July 2011 Ax-les-Bans, France Lnear programmng wth Trangular Intutonstc Fuzzy Number Dpt Dubey 1 Aparna Mehra 2 1 Department of Mathematcs, Indan Insttute of Technology, Hauz Khas, New Delh , Inda 2 Department of Mathematcs, Indan Insttute of Technology, Hauz Khas, New Delh , Inda Abstract Ths paper presents an approach based on value and ambguty ndexes defned n [1] to solve lnear programmng problems wth data as trangular ntutonstc fuzzy numbers. Keywords: Trangular ntutonstc fuzzy number, Fuzzy lnear programmng, Rankng method. 1. Introducton Decson makng problems exhbt some level of mprecsons and vagueness n estmaton of model parameters. Such phenomena has been very well captured through fuzzy sets n modelng these problems. Applcatons of fuzzy set theory n decson makng and n partcular to optmzaton problems have been extensvely studed ever snce the ntroducton of fuzzy sets by Zadeh [2]. As a result, a large volume of research has appeared n ths drecton (please see, [3, 4, 5]. The most common concept used n almost all these studes s rankng of fuzzy numbers. Rankng of fuzzy numbers s an mportant ssue n the study of fuzzy set theory. In order to rank fuzzy numbers, one fuzzy number needs to be compared wth the others, but t s dffcult to determne clearly whch of them s larger or smaller. Numerous methods have been proposed n lterature to rank fuzzy numbers (for example, please see, [6, 7, 8, 9, 10, 11, 12], and references theren. Also, dfferent methods satsfy dfferent desrable crtera. Many of these methods are based on the area measurement wth the ntegral value about the membershp functon of fuzzy numbers. Notably, one thng s clear that there exsts no unquely best method for comparng fuzzy numbers. Recent years have wtnessed a growng nterest n the study of decson makng problems wth ntutonstc fuzzy sets/numbers (for example, see, [1, 13, 14, 15, 16]. The ntutonstc fuzzy set (IFS s an extenson of fuzzy set (FS where the degree of nonmembershp denotng the non-belongngness to a set s explctly specfed along wth the degree of membershp of belongngness to the set. Unlke the FS where the non-membershp degree s taken as one mnus the membershp degree, n IFS, the membershp and nonmembershp degrees are more or less ndependent and related only by that the sum of two degrees must not exceed one. Another notable extenson of FS s the nterval-valued fuzzy set (IVFS [17], whch s characterzed by an nterval-valued membershp functon. In [18], Atanassov and Gorgov, and later Deschrver and Kerre [19] proved that the two concepts of IFS and IVFS are somorphc to each other. Vrtually, they can be used n mathematcally equvalent sense. On the front of rankng ntutonstc fuzzy numbers (IFNs, some work has been reported n the lterature [1, 13, 20, 21, 22, 23, 24]. Grzegorzewsk [20] defned two famles of metrcs n the space of IFNs and proposed a rankng method for IFNs based on these metrcs. Mtchell [21] extended the natural orderng of real numbers to trangular ntutonstc fuzzy numbers (TIFNs by adoptng a statstcal vew pont and nterpretng each ntutonstc fuzzy number (IFN as ensemble of ordnary fuzzy numbers. L [13] proposed a rankng order relaton of TIFN usng lexcographc technque. Earler, Nayagam et al. [22] ntroduced TIFNs of specal type and descrbed a method to rank them. Although ther rankng method appears to be attractve, the very defnton of TIFN seems unrealstc. Ths s because the trangular non-membershp functon s defned to geometrcally behave n an dentcal manner as the membershp functon. Su [24] nvestgated the sgned dstance method for rankng nterval valued fuzzy numbers, whch for trangular fuzzy numbers becomes analogous to the centrod method. Very recently, Neh [23] put forward a new orderng method for IFNs n whch two characterstc values for IFNs are defned by the ntegral of the nverse fuzzy membershp and non-membershp functons multpled by the grade wth powered parameter. Almost parallel, L [1] ntroduced a new defnton of the TIFN whch has an appealng and logcally reasonable nterpretaton. He defned two concepts of the value and the ambguty of a TIFN smlar to those for a fuzzy number ntroduced by Delgado et al. [25]. These are then used to defne the value ndex and the ambguty ndex for TIFN. A rato rankng method s developed for orderng TIFNs. Ths double-ndexed approach s found to be more robust and effectve than any sngle-ndex approaches for rankng IFNs. Furthermore, the method also takes nto consderaton a parameter λ [0, 1] whch may reflect the subectve atttude of the decson maker. We shall be elaboratng on these aspects n the sectons to follow. In ths paper we frst defned a TIFN whch s more general than the one defned n [1, 13]. We extend the defntons of the value and the ambguty ndex gven by L [1] to the newly defned TIFNs. Our man am has The authors - Publshed by Atlants Press 563

2 been to research a meanngful approach to handle lnear programmng problems (LPPs wth data as ntutonstc fuzzy numbers. The bref descrpton of the paper s as follows. In secton 2, besdes certan basc defntons, we also present a bref overvew of the value and the ambguty ndex for TIFNs, and gve a new rankng functon. In Secton 3, usng the new rankng functon, a method s proposed to solve lnear programmng problems wth TIFNs. Secton 4 presents some llustratve examples. The paper s summarzed n Secton 5. 1 uã ν µ a ν a µ a ā µ ā ν 2. Prelmnares We quote few a defntons and propertes of trangular ntutonstc fuzzy numbers (TIFNs relevant to the present work. Defnton 1 [26] An ntutonstc fuzzy set (IFS ã assgns to each element x of the unverse X a membershp degree µã(x [0, 1] and a non-membershp degree νã(x [0, 1] such that µã(x + νã(x 1. An IFS ã s mathematcally represented as { x, µã(x, νã(x x X}. The value πã(x = 1 µã(x νã(x s called the degree of hestancy or the ntutonstc ndex of x to ã. In ths work, X = R. The next few concepts are taken from [1]. Defnton 2 A TIFN ã = {(a µ, a, ā µ ;, (a ν, a, ā ν ; uã} s an IFS n R, whose membershp and non-membershp functons are respectvely defned as follows: (x a µ a a µ a µ x < a x = a µã(x = (ā µ x ā µ a < x ā µ a 0 otherwse, a x + uã(x a ν a a ν a ν x < a uã x = a νã(x = x a + uã(ā ν x ā ν a < x ā ν a 1 otherwse. The values and uã respectvely represent the maxmum degree of the membershp and the nonmembershp such that 0 1, 0 uã 1 and 0 + uã 1. The same s depcted n Fgure 1. Observe that the way TIFN s defned here slghtly dffer from the one defned n [1]. Here, we put forward the dea that µã(x = 0 does not mean that νã(x = 1. Observe that for x [a ν, a µ ] and x [ā µ, ā ν ], t s µã(x = 0, νã(x < 1. In a smlar way to the arthmetc operatons of the trangular fuzzy numbers (TFNs and the TIFNs, the arthmetc operatons over the above descrbed TIFNs are defned as follows. Fgure 1: Trangular Intutonstc Fuzzy number (TIFN Defnton 3 Let ã = {(a µ, a, ā µ ;, (a ν, a, ā ν ; uã} and b = {(b µ, b, b µ ; w b, (b ν, b, b ν ; u b} be two TIFNs and k be a real number. Then ã + b = {(a µ + b µ, a + b, ā µ + b µ ; mn{, w b}, (a ν + b ν, a + b, ā ν + b ν ; max{uã, u b}}. { {(ka µ, ka, kā µ ;, (ka ν, ka, kā ν ; uã} k > 0 kã = {(kā µ, ka, ka µ ;, (kā ν, ka, ka ν ; uã} k < 0. We shall be usng value and ambguty ndexes defned n [1]. Followng on the lnes of L [1] we compute the value ndex and the ambguty ndex for the modfed TIFNs (Defnton 2. We here skp the detaled but straghtforward workng of how the two ndexes have been evolved, and present only the fnal formulas for them. Defnton 4 Let ã = {(a µ, a, ā µ ;, (a ν, a, ā ν ; uã} be a TIFN. Then the value and the ambguty of a ã are gven as follows. ( The value of the membershp functon of ã s V µ (ã = (aµ + 4a + ā µ, 6 whle the value of the non-membershp functon s V ν (ã = (aν + 4a + ā ν (1 uã. 6 ( The ambguty of the membershp functon of ã s A µ (ã = (āµ a µ, 3 whle the ambguty of the non-membershp functon of ã s A ν (ã = (āν a ν (1 uã. 3 Obvously, A µ (ã A ν (ã. Further, V µ (ã, V ν (ã, A µ (ã and A ν (ã have some useful propertes whch are summarzed below. Proposton 1 Let ã = {(a µ, a, ā µ ;, (a ν, a, ā ν ; uã} and b = {(b µ, b, b µ ; w b, (b ν, b, b ν ; u b} be two TIFN s and k 1, k 2 be nonnegatve real numbers. Then 564

3 ( V µ (ã ( V µ (k 1 ã + k 2 b = mn{, w b} k 1 + V µ ( b w b ( V ν (ã ( V ν (k 1 ã + k 2 b = mn{1 uã, 1 u b} k 1 1 uã+ V ν ( b 1 u b ( A µ (k 1 ã + k 2 b = mn{, w b} A µ ( b w b (v A ν (k 1 ã + k 2 b = mn{1 uã, 1 u b} A ν ( b 1 u b ( k 1 A µ (ã ( + k 1 A ν (ã 1 uã+ Assumpton 1 From now onwards we consder those TIFN ã that satsfy the condton V µ (ã V ν (ã. Defnton 5 Let ã = {(a µ, a, ā µ ;, (a ν, a, ā ν ; uã} be a TIFN. Then the value ndex and the ambguty ndex of ã are respectvely defned as follows V (ã, λ = V µ (ã + λ(v ν (ã V µ (ã and A(ã, λ = A ν (ã λ(a ν (ã A µ (ã, where λ [0, 1] s a weght whch represents the decson maker s (DM preference nformaton. It allows flexblty to ncorporate the subectve atttude of DM n the model. λ [0, 1/2 shows the pessmstc behavor whle λ (1/2, 1] ndcates optmstc behavor of the DM, and λ = 1/2 can be nterpreted as an ndfferent atttude of the DM. Note that for any λ, A(ã, λ 0. We say that a TIFN ã s non-negatve f V (ã, λ 0. Defne F (ã, λ = V (ã, λ A(ã, λ. (1 For a predefned value of λ [0, 1], we defne a new rankng (orderng relaton for TIFNs ã and b as follows ã b f and only f F (ã, λ F ( b, λ. It s reasonable to beleve that, more s the value and lesser s the ambguty of the TIFN the larger s the TIFN. And also, λ > 0 represents the DM s preference nformaton. In vew of these facts t makes sense to assume V µ (ã V ν (ã. Thereby the proposed rankng works for a subset of the set of TIFNs whch satsfes assumpton 1. Obvously, the rankng order depends on the atttude parameter λ. It can easly be seen that the proposed rankng functon F satsfes propertes A 1, A 2, A 3, A 5 and A 6 of [11] desred to be satsfed by any reasonable rankng functon. For the sake of completeness, we lst these propertes below. Let S be the set of fuzzy quanttes, and M be an orderng approach. A 1. For an arbtrary fnte subset A of S and ã A, ã b by M on A. A 2. For an arbtrary fnte subset A of S and (ã, b A 2, ã b and b ã by M on A, we should have ã b by M on A. A 3. For an arbtrary fnte subset A of S and (ã, b, c A 3 ; ã b and b c by M on A, we should have ã c by M on A. A 4. For an arbtrary fnte subset A of S and (ã, b A 2, nf supp(ã >sup supp( b, we should have ã b by M on A. A 5. Let S and S be two arbtrary fnte sets of fuzzy quanttes n whch M can be appled and ã and b are n S S. We obtan the rankng order ã b by M on S ff ã b by M on S. A 6. Let ã, b, ã + c, b + c be elements of S. If ã b by M on {ã, b}, then ã + b b + c by M on {ã + c, b + c}. The proposed F does not satsfy A 4 because for two fuzzy numbers ã and b f nf supp(ã >sup supp( b then V (ã, λ V ( b, λ but we can not say anythng about A(ã, λ and A( b, λ. Therefore, there s a possblty that wth our rankng approach, ã b. For nstance, take two fuzzy numbers ã = (9, 10, 20; 1 and b = (8.7, 8.8, 8.9; 1. Then nf supp(ã >sup supp( b, but by usng the above proposed rankng functon F we get ã b. In contnuaton, we would lke to add that the proposed rankng has an obvous advantage over some other sngle-ndex rankng. For nstance, f we wsh to rank ã = {(0.2, 0.5, 0.8; 1, (0.2, 0.5, 0.8; 0} and b = {(0.35, 0.5, 0.65; 1, (0.35, 0.5, 0.65; 0}, then they turn out to be equal by any rankng method suggested n [20, 21, 13, 24]. However, usng the proposed rankng approach, we can easly say that ã b. It s mportant to observe the followng. For a TIFN ã, V µ (ã V ν (ã by assumpton1. Consequently, for 0 λ 1 λ 2 1, V (ã, λ 1 A(ã, λ 1 V (ã, λ 2 A(ã, λ 2,.e., F (ã, λ 1 F (ã, λ Lnear Programmng wth Trangular Intutonstc Fuzzy Numbers The purpose of ths secton s to study a class of fuzzy lnear programmng problems n whch the data parameters are TIFNs. Consder the followng lnear programmng problem (IFLP max subect to c ã b, = 1,..., m 0, = 1,..., n, where, c = {(c µ, c, c µ ; w c, (c ν, c, c ν ; u c }, b = {(b µ, b, b µ ; w b, (b ν, b, b ν ; u b } and ã = {(a µ, a, ā µ ; w ã, (a ν, a, ā ν ; u ã }, = 1,..., m, = 1,..., n, are TIFN s. 565

4 Usng the rankng functon F, for a predefned λ [0, 1], (IFLP s equvalent to the followng crsp optmzaton problem. (COP λ max F ( subect to c, λ ( n F ã, λ F ( b, λ, = 1,..., m 0, = 1,..., n. Further, usng (1 and Proposton 1, problem (COP λ s equvalent to the followng lnear programmng problem. (CLP λ subect to max (1 λ mn {w c } (1 λ mn {1 u c } +λ mn {1 u c } λ mn {w c } (1 λ mn { } V µ ( c A µ ( c V µ (ã (1 λ mn {1 uã } +λ mn {1 uã } λ mn { } w c V ν ( c 1 u c w c V ν (ã 1 uã A µ (ã A ν (ã 1 uã A ν ( c 1 u c (1 λ(v µ (b A ν (b + λ(v ν (b A µ (b = 1,..., m 0 = 1,..., n. Here we assume that the DM s ratonal enough to provde the ntutonstc fuzzy data such that problem (CLP λ remans bounded and feasble for at least one choce of λ. For λ = 1, (CLP λ reduce to max subect to mn {1 u c } mn {1 uã } mn { } V ν ( c 1 u c mn {w c } V ν (ã 1 uã A µ (ã A µ ( c w c (V ν (b A µ (b, 0, = 1,..., n. = 1,..., m If we take λ = 0 n (CLP λ, we get the followng optmzaton problem max subect to mn { } mn {w c } V µ ( c V µ (ã mn {1 uã } w c mn {1 u c } A ν ( c 1 u c A ν (ã 1 uã (V ν (b A µ (b = 1,..., m 0, = 1,..., n. For the sake of observaton, consder a partcular stuaton when only c, = 1,..., n, are TIFNs and the rest of the data parameters are crsp numbers n a lnear program. (IFOBLP max subect to c =0 a b, = 1,..., m 0, = 1,..., n. Usng the proposed rankng functon, (IFOBLP s equvalent to the followng crsp problem. (COBLP λ max F ( subect to c, λ a b, = 1,..., m 0, = 1,..., n. It s worth to notce that f x 0, x λ 1 and x 1 are optmal solutons of (COLP 0, (COLP λ1, 0 < λ 1 < 1, and (COLP 1, respectvely. Then F ( n c x 0, 0 < F ( n c x ( n λ 1, λ 1 < F c x 1, 1. In other words, the optmal obectve value wll ncreasng when the DM gradually moves from pessmstc atttude (λ = 0 to optmstc atttude (λ = 1. Thus, the rankng orderng s consstent wth our common sense of decson makng that, f the feasble set of the problem remans the same then, the optmstc vewpont should always yeld a better obectve value than the one we get wth pessmstc thnkng. 4. Numercal Illustraton We present few examples to depct the workng of the proposed rankng technque for lnear programmng problem wheren the data s specfed as satsfyng assumpton 1. Example 1 Consder the followng ntutonstc fuzzy lnear program 566

5 max 5x 1 + 3x 2 subect to 4x 1 + 3x x 1 + 3x 2 6 x 1, x 2 0, where c 1 = 5 = {(4, 5, 6; 3 4, (4, 5, 6.1; 1 4 }, c 2 = 3 = {(2.5, 3, 3.2; 1 2, (2, 3, 3.5; 1 4 }, a 11 = 4 = {(3.5, 4, 4.1; 1, (3, 4, 5; 0}, a 12 = 3 = {(2.5, 3, 3.5; 3 4, (2.4, 3, 3.6; 1 5 }, a 21 = 1 = {(0, 1, 2; 1, (0, 1, 2; 0}, a 22 = 3 = {(2.8, 3, 3.2; 3 4, (2.5, 3, 3.2; 1 6 }, b 1 = 12 = {(11, 12, 13; 1, (11, 12, 14; 0}, b 2 = 6 = {(5.5, 6, 7.5; 3 4, (5, 6, 8.1; 1 4 }. For λ = 1, usng the method descrbed n earler secton, the equvalent crsp formulaton s max x x 2 subect to 3.05x x x x The optmal soluton of the problem s x 1 = , x 2 = 0 wth optmal obectve value We next solve the gven program for λ = 0. Followng the drectons specfed n earler secton, we formulate the equvalent crsp model as follows. max 1.975x x 2 subect to x x x x The optmal soluton s x 1 = , x 2 = 0, wth optmal obectve value Next we apply our technque to the problems consdered by Su [24]. The frst problem s the one where only the obectve functon coeffcents are taken as IFNs. Example 2 Consder the ntutonstc lnear program where max 25x x 2 subect to 15x x x 1 + 6x x x c 1 = 25 = {(19, 25, 33; 0.9, (18, 25, 34; 1}, c 2 = 48 = {(44, 48, 54; 0.9, (43, 48, 56; 1}, Applyng the rankng functon F, the correspondng crsp lnear programs for λ = 1 and λ = 0 are respectvely gven as follows. max x x 2 subect to 15x x x 1 + 6x x x The optmal soluton s x 1 = 0, x 2 = 1500, wth optmal obectve value max x x 2 subect to 15x x x 1 + 6x x x The optmal soluton s x 1 = 0, x 2 = 1500, wth optmal obectve value Remark 1 Wth the ncrease n the ambguty of IFNs representng the cost coeffcents, we observe that the values of the cost coeffcents n the aforementoned crsp formulaton reduce whle, n the analogous formulaton n [24], they ncrease. Due to ths, the optmal value obtaned by Su [24] s hgher than the one obtaned heren for both cases. The other example n Su [24] has been of lnear program n whch the obectve functon coeffcents are crsp numbers whle the technology and resource coeffcents are TIFNs. Example 3 Consder the followng problem max 25x x 2 subect to 15x x x 1 + 6x x x x 1, x 2 0, a 11 = 15 = {(14, 15, 17; 0.9, (10, 15, 18; 0}, a 12 = 30 = {(25, 30, 34; 0.9, (23, 30, 38; 0}, a 21 = 24 = {(21, 24, 26; 0.9, (20, 24, 33; 0}, a 22 = 6 = {(4, 6, 8; 0.9, (2, 6, 11; 0}, a 31 = 17 = {(17, 21, 22; 0.9, (16, 21, 26; 0}, a 32 = 14 = {(12, 14, 19; 0.9, (8, 14, 22; 0}, b 1 = = {(44980, 45000, 45030; 0.9, (44970, 45000, 45070; 0}, b 2 = = {(23980, 24000, 24050; 0.9, (23940, 24000, 24060; 0} b 3 = = {(27990, 28000, 28030; 0.9, (27950, 28000, 28040; 0} Usng the approach of the paper, the assocated crsp lnear programs for λ = 1 and λ = 0 are respectvely as follows. max 25x x 2 subect to x x x x x x The optmal soluton s x 1 = , x 2 = , wth optmal obectve value max 25x x 2 subect to x x x x x x The optmal soluton s x 1 = , x 2 = , wth optmal obectve value

6 Remark 2 In the above example, t can easly be checked that the feasble set of the crsp optmzaton problem n [24] s contaned n the feasble set of the above descrbed crsp model for λ = 1, whch s further contaned n the feasble set descrbed by the crsp model for λ = 0. Ths leads to an mprovement n the optmal values of the obectve functon n our model of the example as compared to that of Su [24]. However, n general, we can not make any comparatve statement between the crsp model of an ntutonstc fuzzy lnear programmng problem proposed n ths paper and the one proposed by Su [24]. 5. Concludng Remarks We have defned a more general defnton of TIFN than the ones exstng n lterature. The value and the ambguty ndexes defned by [1] have been computed for these TIFNs. Thereafter, a rankng functon has been proposed keepng central thought that the same has to be used to solve a class of lnear programmng problems n whch the data parameters are TIFNs. The soluton methodology for such a class of lnear programs s llustrated through examples. In ths paper we have consdered the TIFNs whch satsfy assumpton 1. Indeed ths s a lmtaton of the present work. The task of developng a more effectve rankng method for a broader class of TIFNs whch can also be effectvely appled to solve lnear programmng problems wth ntutonstc fuzzy parameters s stll an open research ssue. Acknowledgements: The frst author would lke to thank the Natonal Board of Hgher Mathematcs (NBHM, Inda, for fnancal support for research. The authors are ndebted to the referees for ther valuable suggestons and observatons. References [1] D. F. L. A rato rankng method of trangular ntutonstc fuzzy numbers and ts applcaton to madm problems. Computer and Mathematcs wth Applcatons, 60: , [2] Lotf A. Zadeh. Fuzzy sets. Informaton and Control, 8: , [3] R. E. Bellman and L. A. Zadeh. Decson makng n fuzzy envronment. Management Scences, 17:B 141 B 164, [4] C. R. Bector and S. Chandra. Fuzzy Mathematcal Programmng and Fuzzy Matrx Games. Sprnger Verlag, [5] M. Inuguch, H. Ichhash, and Y. Kume. Relatonshps between modalty constraned programmng problems and varous fuzzy mathematcal programmng problems. Fuzzy Sets and Systems, 49: , [6] S. Abbasbandy. Rankng of fuzzy numbers, some recent and new formulas. In IFSA-EUSFLAT 2009, pages , [7] T. C. Chu and C. T. Tsao. Rankng fuzzy numbers wth an area between the centrod pont and orgnal pont. Computers Mathematcs and Applcatons, 43: , [8] M. Jmenez. Rankng fuzzy numbers through the comparson of ts expected ntervals. Internatonal Journal of Uncertanty, Fuzzness and Knowledge- Based Systems, 4: , [9] X. W. Lu and S. L. Han. Rankng fuzzy numbers wth preference weghtng functon expectatons. Computers and Mathematcs wth Applcatons, 49: , [10] H.R. Malek. Rankng functons and ther applcatons to fuzzy lnear programmng. Far East Journal of Mathematcal Scences, 4: , [11] X. Wang and E. E. Kerre. Reasonable propertes for the orderng of fuzzy quanttes (. Fuzzy Sets and Systems, 118: , [12] X. Wang and E. E. Kerre. Reasonable propertes for the orderng of fuzzy quanttes (. Fuzzy Sets and Systems, 118: , [13] J. X. Nan and D.-F. L. A lexcographc method for matrx games wth payoffs of trangular ntutonstc fuzzy numbers. Internatonal Journal of Computatonal Intellgence Systems, 3: , [14] D. F. L. Multattrbute decson makng models and methods usng ntutonstc fuzzy sets. Journal of Computer and System Scences, 70:73 85, [15] D. F. L. Extenson of the lnmap for multattrbute decson makng under atanassov s ntutonstc fuzzy envronment. Fuzzy Optmzaton and Decson Makng, 7:17 34, [16] D. F. L, G. H. Chen, and Z. G. Huang. Lnear programmng method for multattrbute decson makng usng f sets. Informaton Scences, 180: , [17] Lotf A. Zadeh. The concept of a lngustc varable and ts applcaton to approxmate reasonng. Informaton Scences, 8: , [18] K. T. Atanassov. Interval valued ntutonstc fuzzy sets. Fuzzy Sets and Systems, 31: , [19] G. Deschrver and E. E. Kerre. On the relatonshp between some extensons of fuzzy set theory. Fuzzy Sets and Systems, 133: , [20] P. Grzegorzewsk. Dstances and orderngs n a famly of ntutonstc fuzzy numbers. In Proceedngs of the Thrd Conference on Fuzzy Logc and Technology (Eusflat03, pages , [21] H. B. Mtchell. Rankng ntutonstc fuzzy numbers. Internatonal Journal of Uncertanty, Fuzzness and Knowledge-Based Systems, 12: , [22] V. L. Nayagam, G. Vankateshwar, and G. Svaraman. Rankng of ntutonstc fuzzy numbers. In 2008 IEEE Internatonal Conference on Fuzzy Systems, pages , [23] H. M. Neh. A new rankng method for ntutonstc fuzzy numbers. Internatonal Journal of Fuzzy 568

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