A Weighted UTASTAR Method for the Multiple Criteria Decision Making with Interval Numbers

Transcription

1 3rd Internatonal Conference on Management Scence and Management Innovaton MSMI 2016) A Weghted UTASTAR Method for the Multple Crtera Decon Makng wth Interval Number Wen-Tao Xong Jng Cheng School of Mathematc and Stattc, Hube ngneerng Unverty, Xaogan, Chna -mal: xong_2009@foxmalcom School of School of conomc and Management, Hube ngneerng Unverty, Xaogan, Chna -mal: chengng2003@126com opton evaluated on multple crtera, whch can handle both crp and fuzzy evaluaton data hanfar et al [7] ued the UTASTAR model n fuzzy envronment for electng the uppler Bede, the weght nformaton can be not neglected n ome MCDM problem Conderng the weght of all crtera, th paper extend the tradtonal UTASTAR approach and propoe a WUTASTAR method for the MCDM problem wth nterval nformaton, where the performance value of alternatve n the reference et are exact, wherea the one of alternatve under evaluaton are nterval number The remander of th paper organzed a follow Secton 2 brefly revew the arthmetc operaton of nterval number and the concept of degree of poblty Secton 3 propoe a WUTASTAR approach for the MCDM problem wth nterval uncertan nformaton, where the performance of the alternatve under evaluaton expreed by nterval number In ecton 4, an example ued to llutrate the procedure of propoed WUTASTAR method Fnally, ecton 5 conclude the paper Abtract Th paper preent a weghted UTASTAR WUTASTAR) method to extend the conventonal UTASTAR approach for the multple crtera decon makngmcdm) problem wth uncertan nterval nformaton In th method, the crtera weght of all crtera are frtly ued to normalze the performance value of all alternatve Subequently, two lnear programmng model are contructed to determne the margnal utlty functon of each crteron baed on the preference preorder n the reference et Fnally, the overall utlty nterval of each alternatve under evaluaton wth nterval nformaton calculated and the rankng obtaned baed on the degree of poblty between par of alternatve In the end of th paper, an example for evaluatng uppler ued to llutrate the procedure of the propoed WUTASTAR approach Keyword- UTASTAR method; multple crtera decon makng; nterval number I INTRODUCTION The UTASTAR method[1] an mproved veron of the orgnal UTAutlty addtve) approach[2], whch can nfer one or more pecewe lnear utlty functon ung the lnear programmng technque Dfferent from the UTA approach, the UTASTAR method utlzed the dfference between the margnal utlte of two ucceve value of each crteron a unknown varable Bede, t ued two error ntead of one error to adut the overall utlty of alternatve accordng the addtve formula After the UTASTAR approach wa ntroduced n 1985, everal dfferent applcaton were preented n lterature For ntance, Matorak and Sko [3] mplemented the UTASTAR method to ae 192 therapeutc categore for nvetment purpoe n the Greek pharmaceutcal market Grgoroud et al [4] ued the UTASTAR method to aggregate the margnal performance of Key Performance Indcator Kraadak et al [5] appled the UTASTAR method a a mean of determnng crtera average weght per cluter of tudent for the mult-crtera cluterng approach However, ome real-lfe problem may nvolve uncertan data n the evaluaton proce It hard for the decon maker to etmate ther preference wth an exact cale Many reearcher have propoed fuzzy UTASTAR approach to ort the alternatve n the uncertan tuaton Patnotak et al [6] preented the UTASTAR method to nfer fuzzy utlty functon from a partal preorder of 2016 The author - Publhed by Atlant Pre II TH BASIC CONCPT AND OPRATIONS OF INTRVAL NUMBRS x a nonnegatve nterval number, f x [ x, x] {a 0 x a x}, where x and x are the crp real number pecally, x a nonnegatve real number f x x Defnton 1[8] For convenence, all the nterval number n th paper are nonnegatve nterval number In order to compare and rank the nterval number, the arthmetc operaton and the degree of poblty are gven a below Defnton 2[8] Let x [ x, x], y [ y, y] be two dfferent nterval number and k 0, then x y [ x, x] [ y, y] [ x y, x y] ; 191 x y [ x y, x y] 1) kx [k x, k x] pecally, kx 0 f k 0 2)

2 Defnton 3 [8] Let x [ x, x] and y [ y, y] be two dfferent nterval number, then the degree of poblty of P x y ) defned a: max{0, x y} max{0, x y} P x y) x x y y Smlarly, the degree of poblty of P y x ) can be defned n the ame way That max{0, y x} max{0, y x} P y x) x x y y III 1 P x y) TH PROPOSD WIGHTD UTASTAR APPROACH Aume that a MCDM problem cont of m crteron G g1, g2,, gm and n alternatve A a1, a2,, an Let w be the weght value of the th crteron g, m atfyng w 1 1 Wthout lo of generalty, all the crtera are condered a beneft crtera that am at maxmzaton Furthermore, uppoe that the et of alternatve dvded nto two part: the reference et A a1, a2,, a t and the et of alternatve under evaluaton B A\ A at 1, at 2,, an In the reference et, the alternatve are ranked fully or partally baed on the DM preference The performance value of each alternatve a under the crteron g x, whch exact However, the one of ak number [ x, xk ], 1,2,, m k 3) 4) B uncertan, denoted by the nterval Accordng to the tradtonal UTASTAR method, a new approach called a WUTASTAR preented a below The method utlze the weght nformaton of all crtera and can deal wth the MCDM problem wth nterval number Step 1 For each crp value x of the alternatve a A and the nterval number [ k, k ] x x of the alternatve ak B, the normalzed value y, y and k y k are calculated a: x xk xk y, y, y k k, max{ x, xk} max{ x, xk} max{ x, xk} 1,2,, t; k t 1, t 2,, n; 1,2,, m 5) Step 2 Conderng the mportance of all crtera, the weghted normalzed value z and nterval number [ z, z k ] are calculated a below k z w y, z w y, zk w y, k k k 1,2,, t; k t 1, t 2,, n; 1,2,, m 6) where w the weght of the crteron m w 1 1 g, atfyng Step 3 Let g max max { z}, max { zk} and 1,2,, t k t1, t2,, n g mn mn { z }, mn { z } k be the bet and the 1,2,, t k t1, t2,, n wort level of the crteron g, repectvely And the crteron evaluaton cale nterval [ g, g ] dvded nto 1 equal nterval 1 2 [, ] g g, 2 3 [ g, g] 1,,[ g, g ] Thu, the breakpont can be calculated ung the followng formula for each crteron g 1 g g g g ), 1,2,, 1 Step 4 Ung the lnear nterpolaton, the margnal utlty of the weghted normalzed performance value z of alternatve a A on the th crteron can be approxmated 1 for z [ g, g ] a: z g u z u g u g u g ) 1 ) ) )), 1 g g where u g ) the utlty value of breakpont 7) 8) g on the crteron g Baed on the tradtonal UTASTAR method, 1 let u g ) u g ) 0, 1,2,, 1 for each crteron g, whch atfe the followng contran u g m, 2 1 ), 1,2,, m T Step 5Let 1, 2,, 1, ) and h 1,1,,1, z g,0,0,,0) 1 g g T, for z g g, 1,2,, m, 1 [, ] z g where h the 1 dmenon vector and 1 g g 192

3 n the 1 row Thu, we can expre the global utlty of any a A ung the followng addtve formula: 9) m m T 1 1 U a ) u z ) h Generally, there may be the devaton ung the addtve formula to etmate the utlty functon of decon maker Thu, the underetmate error and overetmate error are ntroduced a below U a ) U a ), wth, 0 Furthermore, for every par of the ucceve alternatve belongng to a, a 1 A, the devaton of them computed a: a, a ) U a ) U a ) U a) U a 1) 1 1 Step 6 In mult-attrbute utlty theory, the utlty functon U a ) compatble, f a, a 1) f a a 1, and a, a 1) 0 f a ~ a 1 for a, a A 1, where the notaton a a 1 mean that a prefer to a 1 and the notaton a ~ a 1 denote that a ndfferent to a a potve number to enure the trct nequalty 1 a, a 1) 0 In order to make the margnal utlty functon be a content a poble wth the DM preference, the um of all error varable hould be mnmzed Thu, a lnear programmng model contructed a below mn f [ ] aa a, a 1) f a a 1 t, a A, a, a 1) 0 f a ~ a 1 m 1 1, 11, 0, 0,,, and a A 10) where,, are the unknown varable, and an arbtrarly mall potve number, can be fxed a 0 Step 7 Unfortunately, the optmal oluton of the above lnear programmng uually non-unque Thu, the potoptmalty analy needed to determne the fnal margnal utlty functon Several model have been preented n lterature to elect an optmal oluton Here, we ue the followng MP2 model from Beuthe and Scannella [9] to determne the fnal margnal utlty functon where max t [ ] f, 11) aa all the contrant n 10) f the optmal value of the LP n Step 6 and and are unknown varable Step 8 After the margnal utlty functon obtaned ung the lnear programmng model 10) and 11), we can further calculate the utlty nterval of each alternatve under evaluaton For each alternatve ak B, The utlty nterval calculated ung the equaton 9) baed on the pont of nterval number [ z, z k ] k Step 9 Ung the formula 3) and 4), the degree of poblty of utlty nterval obtaned n Step 8 calculated, and then the alternatve under evaluaton are ranked accordng to the degree of poblty That, f the degree of poblty of P a a 1) 05, t condered a the alternatve a prefer the alternatve a 1, denoted by a P a a ) 1 IV a 1 AN ILLUSTRATD XAMPL In th ecton, the example adapted from Karande and Chakraborty [10] to llutrate the propoed WUTASTAR method In ther example, fve crtera were condered about a uppler electon problem, whch were technologcal capablty TC), conformance qualty CQ), conflct reoluton CR), relatonhp cloene RC) and proftablty of uppler PS), repectvely All thoe crtera were expreed n percentage value and were beneft crtera The weght of all crtera were w TC =03410, w CQ =02900, w CR =01627, w RC =01073 and w PS =00990, repectvely Suppoe that all the uppler are condered a dfferent alternatve Fve uppler were evaluated n the reference et and ther rankng known, denoted a a1 a3 a2 a5 a 4 Table 1 preent the performance of fve alternatve or uppler wth repect to all the condered crtera Dfferent from Karande and Chakraborty [10], fve another uppler aumed and need further to ranked n th ecton baed on the known rankng n the reference et Ther performance value are denoted a nterval number and hown n Table II 193

4 Suppler TABL I TH PRFORMANC VALUS OF ALTRNATIVS IN TH RFRNC ST Weght Interval TC CQ CR RC PS a a a a a TABL II TH PRFORMANC INTRVALS OF TH ALTRNATIVS UNDR VALUATION Suppler TC CQ CR RC PS a 6 [5337, 5671] [6055, 8238] [4140, 4206] [5008, 5665] [3364, 3882] a 7 [4926, 6293] [5991, 7401] [5458, 6578] [4776, 6264] [3459, 5108] a 8 [5221, 6287] [7379, 8733] [3849, 5175] [5620, 5801] [3609, 3775] a 9 [5529, 6611] [6062, 8102] [3869, 6045] [5527, 5893] [4375, 4616] a 10 [5065, 5909] [7123, 8553] [4797, 6166] [6047, 6490] [3373, 4222] TABL III WIGHTD NORMALIZD OF ALL ALTRNATIVS UNDR TH CONSIDRD CRITRIA Suppler TC CQ CR RC PS a a a a a a 6 [02681, 02849] [02008, 02732] [01007, 01023] [00801, 00906] [00587, 00677] a 7 [02475, 03162] [01987, 02454] [01328, 01600] [00764, 01002] [00603, 00891] a 8 [02623, 03159] [02447, 02896] [00936, 01259] [00899, 00928] [00629, 00658] a 9 [02778, 03322] [02010, 02687] [00941, 01471] [00884, 00943] [00763,00805] a 10 [02545, 02969] [02362, 02836] [01167, 01500] [00967, 01038] [00588, 00736] Utlzng the preented WUTASTAR method, the weghted normalzed performance value of all alternatve are hown n Table 3 The mot and the leat preferred value for each crtera are ealy obtaned baed on Table 3, whch 03410, 02900, 01627, 01073, and 02390, 01935, 00933, 00654, 00509, repectvely The range of each crteron dvded nto ome equal nterval and the number of nterval gven n the lat row n Table 1 Thu, the breakpont of each crteron are calculated ung the formula 7) Subequently, the followng lnear programmng model etablhed accordng to the preference relaton n the reference et and the model 10) 5 mn f 1 t , , , , , 11, 12, 13, 14, 21, 22, 23, 31, 32, 33, 41, 42, 51, 52,, 0, 1, 2,3, 4,5 194

5 where the value of and are fxed a and 0, repectvely The lnear programmng olved ung Lngo oftware The optmal obectve value f 0 However, the optmal oluton non-unque Therefore, the potoptmalty analy needed Accordng to the model 11), the optmal reult of th lnear programmng are obtaned a: , 11 0, 12 0, 13 0, 14 0, , 22 0, 23 0, 31 0, 32 0, 33 0, , , , 52 0 Ung thee value, the utlty nterval are further calculated for all alternatve under evaluaton a Ua 6 )=[02176, 06633], Ua 7 )=[01904, 08826], Ua 8 )=[06113, 06836], Ua 9 )=[04213, 07843], Ua 10 )=[06898, 09319] Baed on the degree of poblty, the fnal rankng acheved a: a10 a8 a9 a7 a 6 V CONCLUSION The orgnal UTASTAR method dealt wth the decon makng problem wth crp value Th paper extend the orgnal UTASTAR method from two apect On one hand, the weght of each crteron condered n ntal data All the performance value are weghted normalzed, where the cot crteron and beneft crteron can be handled n a unfed framework On the other hand, the alternatve under evaluaton are meaured wth uncertan nterval number Utlzng the propoed WUTASTAR method, the utlty nterval number are obtaned to rank all the alternatve ung the degree of poblty If the performance value crp, the preented WUTASTAR method can be ued to compare the alternatve under evaluaton traghtly ACKNOWLDGMNT Th work upported by Scentfc Reearch Program Funded by Hube Provncal ducaton Department No Q ) and Natonal Natural Scence Foundaton of Chna No ) RFRNCS [1] Y Sko, D Yannacopoulo, UTASTAR: An ordnal regreon method for buldng addtve value functon, Invetgacao Operatonal, vol 5, no 1, pp 39-53, 1985 [2] Jacquet-Lagreze, J Sko, Aeng a et of addtve utlty functon for multcrtera decon-makng, the UTA method, uropean ournal of operatonal reearch, vol 10, no 2, pp , 1982 [3] K Matorak, Sko, Value focued pharmaceutcal trategy determnaton wth multcrtera decon analy technque, Omega vol 59, Part A, pp 84 96, 2015 [4] Grgoroud, Orfanoudak, C Zopound, Strategc performance meaurement n a healthcare organaton: A multple crtera approach baed on balanced corecard, Omega, vol 40, no 1, pp , 2012 [5] Kraadak, K Lakotak, NF Matatn, Student behavour n peer aement: a mult-crtera cluterng approach, uropean Journal of ngneerng ducaton, vol 39, no 3, pp , 2014 [6] I Patnotak, D Apotolou, G Mentza, Fuzzy UTASTAR: A method for dcoverng utlty functon from fuzzy data, xpert Sytem wth Applcaton: An Internatonal Journal, vol 38, no 12, pp , 2011 [7] M hanfar, AT hlagh, MA Keramat, J Nazem, The Development of UTASTAR Method n Fuzzy nvronment for Suppler Selecton, uropean Journal of Scentfc Reearch, vol 108, no 3, pp , 2013 [8] Z Yue, An extended TOPSIS for determnng weght of decon maker wth nterval number, Knowledge-Baed Sytem, vol 24, no 1, pp , 2011 [9] M Beuthe, G Scannella, Comparatve analy of UTA multcrtera method, uropean Journal of Operatonal Reearch, vol 130, no 2, pp , 2001 [10] P Karande, S Chakraborty, Suppler Selecton Ung Weghted Utlty Addtve Method, Journal of The Inttuton of ngneer Inda): Sere C, vol 96, pp1-10,