New Entropy Weight-Based TOPSIS for Evaluation of Multi-objective Job-Shop Scheduling Solutions

Transcription

1 Eropy Wegh-Baed TOPSIS for Evaluao of Mul-obecve Job-Shop Schedulg Soluo J Wag, J Che, T u 2, George uag 2, Y F Zhag, S D Su Key Laboraory of Coeporary Deg ad Iegraed Maufacurg Techology, Mry of Educao, Norhweer Polyechcal Uvery, X'a, Shaax, PRCha 2 Depare of Idural ad Maufacurg Sye Egeerg, The Uvery of og Kog, og Kog, PRCha wagq@wpueduc Abrac - Facg wh a obaed e of Pareo oluo o ul-obecve ob-hop chedulg proble, core ue how o evaluae he be-coproe oluo fro hee o-doaed oluo for deco aker TOPSIS a a ul-crera deco akg ehod roduced o he evaluao of hee Pareo oluo The radoal Eropy wegh approach ued o calculae he wegh of each crero fal o fahfully rafor he forao fro he Eropy value o Eropy wegh, ad a abrup chage of Eropy wegh are whle facg wh a very lgh chage of Eropy value whe hee Eropy value are cloe o To addre he, a ew raforao forula fro he Eropy value o Eropy wegh propoed o o oly herage he orgally good perforace bu alo ake up he exg defcecy Fally, he reul of a lluraed exaple how ha he feably ad accuracy of he propoed approach Keyword - Mul-obecve ob-hop chedulg, Pareo oluo, Eropy, TOPSIS, Mul-crera deco akg I INTRODUCTION Mul-obecve ob-hop chedulg proble coder everal obecve ulaeouly whe a chedule geeraed Geerally, hee obecve clude: Te-oreed obecve, uch a Makepa, delvery dae, copleed e, lead e ad arde; 2 Cooreed obecve, uch a oal capal occupao of work proce (WIP ad arde quay of ob, ad reource ulzao; 3 Throughpu-oreed obecve accordg o he hroughpu accoug (TA of heory of cora (TOC Thee dffere obecve ay o alway be coe wh each oher For exaple, coderg ecooc bach polcy, ache ulzao wll be creaed ad he uafacory Makepa of each ob gh be curred For he ul-obecve opzao proble of JSP, uly approach ralae ul-obecve ob hop chedulg proble o gle-obecve proble coderg wegh of each obecve ug a uly fuco or weghg fuco Noe ha he wegh of each obecve hould be kow advace Furherore, oly oe oluo obaed a he be oluo o uve ad uable for he accurae, or ucera, or chageable preferece of wegh for each obecve O he oher had, ore ad ore ude adop he Pareo opaly approach o ulaeouly opze hee ulple obecve ad oba a group of Pareo opal oluo Pareo-opaly approach gve up he effor o oe opal oluo eeg for all obecve ad rafer her effor o eek all effce or odoaed oluo for all obecve whou provdg he preferece of each obecve A e of o-doaed oluo ha cover he rade-off aog hee ulobecve are he o-called Pareo froer Thee oluo provde deeper gh o he deco aker (DM ha hoe of gle-obecve proble For olvg he ul-obecve JSP, oe reearcher preeed effecve hybrd approach, cludg he hybrd approach of evoluoary algorh (EA ad fuzzy logc [], hybrd approach of a coloy opzao (ACO ad ulaed aealg (SA [2], hybrd approach of abu earch ad geec algorh [3] Copared wh he popular reearche whch focu o obag Pareo opal oluo, few reearche cocerae o he evaluao of Pareo opal oluo Th udy adop TOPSIS (he echque for preferece by lary o deal oluo [4] o evaluae he Pareo oluo of ul-obecve JSP owever, he radoal Eropy wegh approach baed o Shao Eropy fal o fahfully rafor he forao fro he Eropy value o Eropy wegh, ad a abrup chage of Eropy wegh are whle facg wh a very lgh chage of Eropy value whe hee Eropy value are cloe o To addre h proble, h reearch, a ew raforao forula fro he Eropy value o Eropy wegh propoed ad proved o prove he raforao perforace II MULTI-OBJECTIVE JOB SOP SCEDULING PROBLEM The ul-obecve ob hop chedulg proble co of a fe e W of w ob o be proceed o a fe e E of e ache Each ob W k u be proceed o every ache oce ad co of a e of e operao{ O, k, O2, k,, O h, k}, h =,2,, e, k =,2,, w, whch have o be cheduled a predeered order called a precedece cora Each operao o be proceed for a uerruped proceg perod of p hk, The purpoe of chedulg o fd a age of /2/$ IEEE 464

2 operao o e lo o a ache order o opze all he ulple obecve Le T be he e of chedulg obecve, T = { T, T2,, T,, T} The ypcal obecve clude zao of copleed e of ob, zao of axu lead/arde e, ad zao of produco co The ul-obecve JSP odeled a follow f ( θ = ( T, T2,, T ( T = (ax Ck k T2 = (ax dk Ck k T3 = ( Cac + Cdyac g ξ ( θ 0, ξ =, 2,, ζ where C k deoe copleo e of ob k, dk deoe he due dae of ob k, Cac deoe he ac co, deoe he dyac co, ζ deoe he oal Cdyac quae of cora, g ξ ( θ deoe he ξ -h cora, uch a e cora of releae dae, due dae, capacy cora ad precedece cora Afer opzg uder he obecve e T ad cora e g( θ, we ca ga he Pareo oluo e S = {, 2,, } The he core ue becoe how o evaluae hee oluo ad oupu he preferece * * order S, S = { ( < (2 < < ( } Ⅲ TOPSIS BASED ON ENTROPY WEIGT TOPSIS a clacal ul-arbue deco aaly ehod o rak a fe uber of alerave baed o he cocep ha he choe alerave hould have he hore geoerc dace fro he pove deal oluo ad he loge geoerc dace fro he egave deal oluo [4] We adop TOPSIS o ake deco aog he e of Pareo oluo S = {, 2,, } o he ba of crera The dealed ep are decrbed a follow ( Coruc he deco arx The deco arx X fored by Pareo oluo ad crera X = ( x =,2,,, =,2,, (2 where x a crp value dcag he perforace rag of -h alerave wh regard o -h crero (2 Noralze he deco arx For he purpoe of coparably aog crera, he raw deco arx ha o be oralzed We chooe vecor oralzed ehod, ad rafor he deco arx o oralzed deco arx R = ( r 2 = r = x x (3 (3 Calculae he copreheve wegh of each crero We egrae ubecve facor ad obecve facor o calculae he copreheve wegh of each crero Le be he copreheve wegh of crero, =, he a he wegh vecor of crera deoed a = (, 2,,,, = ( b a ( b a (4 = where b he ubecve wegh of crero, whch repree he DM preferece for obecve or crera ad ca be obaed by AP or exper evaluag ehod a he obecve wegh, whch oba accordg o Shao Eropy ehod ad calculaed a follow a = ( ( (5 = [ p l p ] (6 l = p = x x (7 = where 0 If p = 0, aued ha p l p = 0, =,2,,, =,2,, (4 Calculae he weghed oralzed deco arx The weghed oralzed value v calculaed a v = r where he wegh of crero V = ( v = ( r (8 (5 Deere he deal ad egave-deal oluo A = {(ax v J,( v J M} = { v, v,, v,, v } (9 A = {( v J,(ax v J M} = { v, v,, v,, v } ( where J deoe he e of beef crera, ad J repree he e of co crera For he beef crera, he DM wa o have a axu value aog he alerave Whle for he co crera, he DM wa o have a u value aog alerave A + ad A are deal ad egave-deal oluo ha are vrual oluo wh opal or wor evaluao value wh regard o each crero repecvely (6 Calculae he eparao eaure ug Eucldea dace The eparao of each alerave fro he deal oluo( ad egave-deal oluo( S gve repecvely + S ( ( S = v v 2 ( (2 = S = v v (7 Calculae he relave cloee o he deal oluo C = S ( S + S, 0< < (3 + C 465

3 where C he relave cloee of he alerave wh repec o A + Apparely, f = A +, C =, larly = A, C = 0 S + 0, S 0, C [0,] (8 Rak preferece by C Accordg o he decedg order of C, A e of Pareo alerave ca be raked, he larger C ea beer alerave The raked oluo alerave e decrbed by S = { ( < 2 < < ( } Ⅳ NEW TRANSFROMATION FORMULA OF ENTROPY WEIGT A Tradoal Traforao Forula of Eropy Wegh The radoal Eropy wegh approach ued o calculae he wegh of each crero hrough forula (5-7, owever, he raforao forula fal o fahfully rafor he forao fro he Eropy value o Eropy wegh, ad a abrup chage of Eropy wegh are whle facg wh a very lgh chage of Eropy value whe Eropy value, =,2,, Eve f he Eropy value ha a lgh chage, he Eropy wegh a would geerae gfca chage For exaple, here are 4 Eropy value wh lgh dfferece, = (09999,09998,09997,09996 If ug he radoal Eropy forula (5, he Eropy wegh a : a = (0,02,03,04 Obvouly, he Eropy value ha alo he ae value, whle he Eropy wegh ha a vble dfferece Therefore he reul obvouly ureaoable ad here ex defcecy he radoal Eropy forula (5 I order o llurae he proble, furher aheacal proof gve a follow For ay wo crera ad, he correpodg Eropy wegh ad Eropy value calculaed by radoal Eropy forula a ad a, ad, repecvely ere 0, 0,, {,2,,, }, O he codo of we aue < Le ε ad c be pove cro varable, ε, c (0,, he = ε ad = c c ε = ( ( = c 0,he 0 < ε < c ad ε c Therefore 0< ε c < ad ε c Le a a = ( ( = ( ( = ε c, he a a γ,where γ alo a cro pove varable Therefore he cocluo afey draw ha whe, 0 < < <, he correpodg a Eropy wegh rao apparely aboral, lke << a a B Traforao Forula of Eropy Wegh I he paper, a ew raforao forula of Eropy wegh propoed ad gve a follow = (2 (2 (4 a Proof he u of all crera wegh u be kep o he ew Eropy forula (4 = [(2 (2 ] = a 2 he characerc ha he larger Eropy value hould reflec aller Eropy wegh u be hered Owg o ( =,2,, a defe value, = u be a defe value Becaue he relaohp 0 ex, cera ha he relaohp 0 a hold Therefore a he oooe decreag fuco wh he gle paraeer 3 he Eropy wegh wll o chage abruply whe all Eropy value chage whe Eropy value cloe o,,, {,2,,, }, ad are wo evaluao crera The correpodg Eropy wegh are a ad a repecvely, whle correpodg Eropy value are ad whe, alo we aue < The = ε, ε a cro pove varable, ε (0, a ε = (2 (2 = = a 2 2 ε Sce 0, (2 [,2], hu 0 2 Therefore, a a, ha a a A he proof eoed above, he ew Eropy wegh approach able o avod he proble ha Eropy wegh chaged abruply whe Eropy value Therefore he ew Eropy forula ore reaoable ha he radoal oe C Coparo aog Dffere Traforao Forula To verfy he fahfully raforao ably fro he Eropy value o Eropy wegh, he furher coparo of Eropy wegh accordg o dffere Eropy value ug radoal forula, a reved forula (Zhou e al 2007[5] ad he propoed ew forula eed ad gve Table I 466

4 TABLE I COMPARISON OF ENTROPY WEIGTS TRANSFORMATION CORRESONDING TO DIFFERENT ENTROPY VALUES Cae 05 0 Trado al forula (5 Eropy Wegh Zhou e al (2007 forula (4 Trado al forula (5 Eropy Wegh Zhou e al (2007 forula (4 Trado al forula (5 Eropy Wegh Zhou e al (2007 forula ( Fro Table I, ca be ee ha whe, he radoal forula o reaoable owg o he abrup chage of Eropy wegh For exaple, whe Eropy value vecor are (09999, 09998, 09997, (09990, 09980, ad (09000, 08000, 07000, her Eropy wegh are he ae: (0667, 03333, Furherore, hee lar Eropy valve lead o a large chage of he correpodg Eropy wegh Bu, ug he ew raforao forula, he ew Eropy wegh vecor becoe (03334, 03333, 03333, (03330, 03333, ad (03056, 03333, 036 repecvely The coparo wh he radoal approach ad he reved forula [5] are lluraed Fg Eropy wegh Eropy wegh Eropy wegh cloe o Eropy value cloe o Eropy value cloe o Eropy value Tradoal forula(5 Zhou e al(2007 forula(4 Fg Eropy wegh coparo Apparely, he ew raforao forula ha wo advaage: whe, he ew Eropy wegh ever chage abruply; 2 whle 05 or 0, he Eropy wegh have he ae chage edecy ad o gfca dfferece Furherore, fro he Table I ad Fg, he ew Eropy wegh approach alo uperor o he reved approach of [5] Ⅴ CASE STUDY The exaple fro Wu e al (2006 [6], whch ulple obecve hybrd geec algorh ued o olve ul-obecve flexble ob-hop chedulg proble (FJSP ad he a Pareo e of 20 chedulg oluo obaed S = {, 2,, 20}, here = 20 The deco obecve e T={ Makepa, 2 axu lead/arde e, 3produco co, 4WIP orage co, 5 overall equpe workload, 6 axu equpe workload}, whch ea = 6 The Pareo oluo e preeed Table Ⅱ TABLE Ⅱ PARETO SOLUTIONS OF FJSP FROM WU ET AL (2006 T S (h 2(h 3( 4( 5(h 6(h ( Coruc ad oralze he deco arx The 20 6 deco arx of he 20 Pareo oluo ad 6 crera preeed by forula (2 ad oralzed by forula (3 (2 Calculae wegh of each crero The Eropy value of each crero obaed ug forula (6 ad forula (7 = (09996, 09466, 09997, 09862, 09999, The radoal raforao forula (5 geerae a abrup chage of Eropy wegh The he reved raforao forula Zhou e al [5] elae he abrup chage bu he dcrao of Eropy wegh o 467

5 lar Eropy value weakeed ere he ew raforao forula (4 fahfully work The correpodg coparo led Table Ⅲ Whe he ubecve wegh vecor e a b = (05, 02, 05, 02, 005, 025 by DM, he copreheve wegh obaed by ug forula (4: = (0480, 02078, 0480, 02000, 00493, (3 Eablh he weghed oralzed deco arx ad deere he deal ad egave-deal oluo A + = {00309, 00095, 0033, 0085, 0007, 00504} A = {00374, 00883, 00359, 00584, 005, 00600} (4 Calculae he eparao eaure S + ={00660, 00396, 00437, 00483, 00730, 0079, 0040, 00646, 00432, 00238, 00424, 00336, 00567, 00346, 00356, 00423, 00376, 0073, 00394, 0043} S ={00246, 00796, 00573, 00473, 00405, 00372, 0076, 00255, 00462, 00663, 00469, 00564, 00337, 00548, 00784, 00496, 00734, 00775, 00632, 00587} (5 Calculae relave cloee o he deal oluo ad rak preferece C = {0276, 06679, 05673, 04946, 0357, 03200, 06357, 02828, 0566, 07360, 05247, 06264, 03728, 0627, 06877, 05399, 0665, 0878, 0658, 05870} he raked order of oluo alerave Therefore he alerave 8 he be oe aog all he 20 oluo The coparo of hee hree raforao forula fro Eropy value o Eropy wegh led Table Ⅲ Noe ha he reul defed fro he propoed ew forula (4 dffere wh oher, whch exacly verfe he propoed approach feable ad proe TABLE Ⅲ COMPARISON OF TREE TRANSFORMATION FORMULAS FORM ENTROPY VALUE TO ENTROPY WEIGT Tradoal forula(5 Zhou e al (2007 forula(4 = (09996, 09466, 09997, 09862, 09999, a Rakg of oluo a ={0006,07797,00045, 0202,00009,00067} Zhou a ={659,0695,0659,0668, 0659,0659} a = {0649,0736,0648, 067,0648,0649} Ⅵ CONCLUSION For he evaluao of ul-obecve JSP chedulg Pareo oluo, a ul-crera deco akg ehod aed TOPSIS roduced To ake he evaluao ore accurae, boh ubecve ad obecve facor are codered owever, he radoal Eropy wegh ehod fal o fahfully rafor he forao fro he Eropy value o Eropy wegh, ad a abrup chage of Eropy wegh are whle facg wh a very lgh chage of Eropy value whe hee Eropy value are cloe o The pheoeo proved by he rc aheacal proof Furherore a ew Eropy wegh forula propoed o addre he proble Alo her characerc ad he correpodg provee are proofed Through evaluag he ul-obecve obhop chedulg oluo, he proved Eropy wegh ehod o oly ake up he exg defcecy whe hee Eropy value are cloe o, bu alo herage he orgally good perforace oher uao ACKNOWLEDGMENT Th reearch poored by Naoal Naural Scece Foudao of Cha (No , , ad Bac Reearch Foudao of Norhweer Polyechcal Uvery (No JC REFERENCES [] I Kace, S aad, ad P Bore, "Pareo-opaly approach for flexble ob-hop chedulg proble: hybrdzao of evoluoary algorh ad fuzzy logc," Maheac ad Copuer Sulao, vol60, o3-5, pp , 2002 [2] N Paah ad R Tavakkol-Moghadda, "Solvg a ulobecve ope hop chedulg proble by a ovel hybrd a coloy opzao," Exper Sye wh Applcao, vol38, o3, pp , 20 [3] G Vlco ad J-C Bllau, "A abu earch ad a geec algorh for olvg a bcrera geeral ob hop chedulg proble," Europea Joural of Operaoal Reearch, vol90, o2, pp 398-4, 2008 [4] C L wag ad K Yoo, Mulple arbue deco akg: Mehod ad applcao Berl, edelberg: Sprger, 98 [5] C Zhou, G Zhag, ad G L Wag, "Mulobecve deco akg approach baed o Eropy wegh for reervor flood corol operao," Shul Xuebao, vol38, o, pp 00-06, 2007 [6] X L Wu, S D Su, J J Yu, ad F Zhag, "Reearch o ul-obecve opzao for flexble ob hop chedulg," Copuer Iegraed Maufacurg Sye, vol2, o5, pp ,