DECISION SCIENCES INSTITUTE Optimal Location of Repair Stations in a Garments Production Line

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1 Hossin & Srer DECISION SCIENCES INSTITUTE Optiml Lotion o Repir Sttions in Grments Proution Line M Shhrir Jhn Hossin Louisin Stte University Emil: msjhossin1@gmil.om Bhb R. Srer Louisin Stte University Emil: bsrer@lsu.eu ABSTRACT In mnuturing system the eetive items etete t the inspetion sttion, re generlly srppe or repire t regulr eet-reting worsttion or t eite repir sttion o the line. In this reserh sewing line with n inspetion sttion lote t the en, is onsiere to me eisions onerning these issues. A unit ost untion is evelope or lterntive eisions on eh type o eet. To minimize the unit proution ost, rtionl mixe integer nonliner progrmming (MINLP) problem is ormulte n solve optimlly. The test problem is reletive o some sewing lines o reyme grments inustry. KEYWORDS: Rewor/repir, Deet, Frtionl progrmming, Optimiztion, Grments proution. INTRODUCTION Inspetion n repir re two importnt prts o qulity ontrol. In inustries qulity inspetions re one in orer to ientiy nononormities. Deetive items re oten sent b to the worsttion where the eet ours. This ommon repir proeure sometimes reues line eiieny. At the sme time i eetive items re repire t eite repir sttion it s ixe ost or tht repir sttion. On the other hn ll eetive items re not repirble; some eetive items re srppe inurring wste o mteril n other relte osts. Thereore, it is essentil to te n optiml eision on eetive items whether they shoul be srppe, or repire t regulr on-line worsttion(s) or t eite o-line repir-sttion(s) in orer to ensure minimum ost o proution per unit n optimum number o o-line repir-sttions to enhne throughput rte. Aoring to Amerin Apprel n Footwer Assoition (AAFA, 2012), pprel n textile inustries re the worl s thir lrgest inustry setor ter uto inustries, n re goo exmples o imperet seril proution lines. Due to high lbor ost most low-ost grments in Ameri, Austrli or Europen ountries re importe rom the ountries lie Chin, Ini, Bnglesh or Vietnm where humn lbor is vilble with omprtively low ost. Grments qulity is vitl tor to pture new mret. Lot o wors hve been one to improve the proutivity, line eiieny o pprel inustries n eventully to reue the ost o proution. Islm et l. (2014) presente se stuy or obtining n optiml lyout esign in n pprel inustry by pproprite line blning. On the other hn, Bhir (2011) isusse bout ssembly line blning by simultion. Severl reserh re oun on optimizing qulity inspetion n repir or reuing ost o qulity or

2 Hossin & Srer inresing the net proit. Rviv (2013) presente n lgorithm or mximizing the expete proit rom n unrelible seril proution line in whih nononorming items re sent b or repir ter going through the inspetion sttions. None o these stuy isusses bout oline repir ilities or eetive items. Builing suitble rmewor or ting opertionl eision on eite repir sttions still requires extensive reserh s this involves onliting objetives. Consiering this t this reserh ouses in eveloping moel to minimize the unit ost o proution to etermine suitble number n lotion o eite o-line repir sttions. In orer to el with this objetive, ollowing steps re perorme in this reserh: () To etermine proution ost per unit s untion o yle time, proportion o eets, ixe n vrible ost, whih re eventully epenent on the lotion o o-line repir sttions. (b)to in the optimum number n lotion o o-line repir sttion(s) tht minimizes the proution ost per unit. The outomes o this reserh provie n esily exeutble rmewor to etermine the requirement o repir ilities or eling with non-onorming items whih re ientiie t the stge o qulity inspetion in the sewing line. PROBLEM DESCRIPTION Assume grments proution line with N sewing sttions rrnge sequentilly to perorm sequene o opertions neee to omplete grments prout. Eh worsttion perorms ierent type o opertions, ompleting t lest one opertion t tht sttion. A prout is lssiie s eetive orresponing to the mjor soure (worsttion) o eet. An inspetion sttion is lote t the en o the line. This inspetion sttion ientiies n seprtes nononorming items bse on the type n origin o the eet(s). Proportion o eh type o eetive items re estimte rom historil experienes. Some worsttion my not proue ny eets t ll n the probbility o eet or tht worsttion is ssume to be zero. I eetive items re repire then number o onorming items inreses. Agin i the eetive items re sent b to the regulr on-line sewing sttion or repir, the regulr proution my be hmpere. The problem thus is to eie whether the eetive item shoul be sent b to originl sewing sttion or repir or there shoul be eite repir-sttion so tht regulr proution is not interrupte, or the eetive grments shoul be srppe. During ming this orretive eision, proution ost shoul be ept t minimum level. The problem esribe here is speii sitution whih n be ormulte bse on some ssumptions: () A blne grments proution line with no prllel worsttion is onsiere. (b) An inspetion sttion is lote t the en o the line with negligible inspetion error. () Any prtiulr eetive item oes not hol multiple types o nononormity t time. () Any in o eets n be repire n repire items ssume to be onorming prouts. (e) Repir ost is onstnt or prtiulr eet repire t prtiulr worsttion. () Srppe items re vlueless. (g) There is no spe limittion in the shop loor.

3 Hossin & Srer MODEL FORMULATION Beore eveloping the moel some nottions or system prmeters, vribles n perormne mesures re neee to be eine. All nottions re eine in the ollowing sub-setions. System Prmeters F C C = Fixe ost or the sewing line ($/hour), F C = Fixe ost or regulr worsttion ($/hour), = Fixe ost e or eite o-line repir sttion or opertion ($/hour), = Vrible ost o sewing inishe item i no item is repire ($/unit), repiring t eite o-line repir sttion ($/unit), on-line worsttion ($/unit), N = Totl number o worsttions, t worsttion, P t r = Vrible ost o = Vrible ost o repiring t regulr p =Proportion o eetives o = Proessing time o opertion t regulr worsttion (hours), R t = Repir time o eetive item t regulr worsttion (hours) n T = Cyle time when no repir wor is one (hours). Intermeite Vribles R T = Totl repir time o n item t regulr worsttion (hours), when repir wors re one (hours), n TC = Totl ost ($/hour). T e = Eetive yle time Vribles n Perormne Mesures u = Unit ost ($/unit), opertion n eet proue t worsttion. x = (0,1), the number o eite o-line repir sttions or y = 0-1 binry vrible initing eision on srpping (0) or repir (1) or Now, the proution ost per unit o goo item hs to be reue, whih is the prime objetive o this reserh. Thus the optimiztion problem n be ormulte s rtionl mixe-integer nonliner progrm s written below. Problem Z MINLP N N N N N F r Min u C C x Te p x p y 1 x 1 p p y (1) Subjet to 1,2,..., N x y 0 (1) P R, 0,1, 0,1 T t p t y 1 x 0 (1b) e T T x y (1) e

4 Hossin & Srer Problem Z MINLP is rtionl mixe-integer nonliner progrmming problem (MINLP) or whih the solution is not immeite. Sine the urrent problem Z MINLP nnot it to n existing problem, series o trnsormtions s menble to the requirement is one here (Li, 1994; Chng, 2001; Hossin & Srer, 2016) to hieve the solution gol. Ater severl trnsormtion n letting 1 N N w 1 p 1 p 1 y b, Te w w,,, n x y w w Problem is trnsorme to mixe-integer liner y w w e progrmming problem (MILP) s Problem Z MILP : Z MINLP x w b w x w N N N N F b r e r Min u w C w C w p w p w p w Subjet to 1 N N e p 1 w p 1 1 w n 1,2,..., N: x w (2) (2) y 0 (2b) b e w t w t w 0 (2) Tw b w 0 (2) n 0 w Mx (2e) w b x M w w b x M n 0 w Mx (2) w x M w w x M e n 0 w My (2g) w y M w e w y M w x y M w w x y M, w N 1 1, x 0,1, y 0,1 b 1 w 1 p, w T where M is very lrge number. The inl version o problem or N number o worsttions, mong whih w, e w, w re positive vlues. Given T, Mx n 0 w My (2h), (2i) C F x,,, C, ZMILP hs totl o 2+6N vribles y re 0-1 binry integers, n, r be solve by mixe-integer brnh n boun metho. A CASE STUDY IN GARMENTS INDUSTRY, w, w p, t P n t R the problem b, w, ZMILP n The pprel inustry is o gret importne to the eonomy in terms o tre, employment, investment n revenue ll over the worl. On experiening suh n enevor in reyme grments (RMG) inustry, 5 ierent 7-worsttion sewing lines re onsiere tht mnutures ertin esign o brne T-shirts. Seprte istint opertions re being one in eh o the worsttions. The opertions ollow sequene s shown in Figure 1.

5 Hossin & Srer Figure 1: A brne T-shirt sewing line with 7 ierent worsttions 1 Shouler joint 3 L-Sie sem 5 Sleeve joint 7 Bottom hem Rw Mteril 2 R-Sie sem Mteril low 4 Collr joint 6 Sleeve hem Inspetion Sttion The tory bers ixe ost o proution $100 per hour or sewing line tht inlues pitl, lbor n ixe utility osts. Vrible ost o proution tht inlues mteril n vrible utility osts is liste s $3.10 or n item no mtter whether the item is eetive or not. All other neessry vrible osts n ixe osts involve with eh worsttion ( = 1, 2,,7) in the irst sewing line re estimte rom existing t n liste in Tble 1. This problem is ormulte s in orer to solve with brnh n boun metho. For 7-worsttion Z MILP problem there re 5+15(7) = 110 onstrints n 2+6(7) = 44 vribles mong whih 14 vribles re 0-1 binry integers. Thus there re 2 14 = 16,384 noes to be explore to in the optimum result, whih is big problem inee to elbortely emonstrte. W/S Tble 1: Fixe n vrible osts involve with ierent opertions Proportion o ($/hr) ($/unit) ($/unit) (se/unit) (se/unit) eets Opertion nme C r 1 Shouler Joint R-Sie sem L-Sie sem Ne Joint Sleeve Joint Sleeve hem Bottom hem A Three-Sttion Grments Sewing Line In orer to explin the solution proeure, 3-worsttion grments sewing line is onsiere with n inspetion sttion lote t the en o worsttion-3 or emonstrtive purposes. Let F C = $100/hour n = $3.10/unit. Also, s given in Tble 1, C = {1.70, 1.50, 2.20} ollrs/hour, r = {2.30, 3.85, 5.50} ollrs/t-shirt, P t R t = {0.44, 1.19, 1.21} ollrs/t-shirt, t P = {35, 43, 41}/3600 hour/t-shirt, t R = {20, 125, 100}/3600 hour/t-shirt p = {0.0040, , }, n T=43/3600 hour/t-shirt. A brnh n boun (B&B) metho is pplie on this 3-worsttion problem n illustrte in Figure 2. A totl o 19 noes were explore. The optimum result is oun t noe 10 with ( x, y ) Here 1 1 * * (, ) x y = {(0,1), (1,1), (0,0)} n the minimum unit ost is $4.37/T-shirt. = (0,1) inites tht the eetive items proue t irst worsttion shoul be p

6 Hossin & Srer repire t the regulr on-line worsttion-1. Similrly, ( x, y ) 2 2 = (1,1) inites tht eetive items proue t seon worsttion shoul be repire t seprte eite o-line repir sttion. Thus the eetive items proue t thir worsttion hve to be srppe. Figure 2: B&B results or 3-worsttion problem Ientil Sewing Lines with Vrying Output Qulity Five ierent ientil sewing lines, eh with 7 worsttions re mnuturing the sme T- shirts, but s mhine opertors re ierent or ierent lines, output qulity n throughput vries. Though ixe n vrible osts s well s proution n repir times re ssume to be the sme, eet probbilities re not so. Deet probbilities p, 1, 2,, 7 t orresponing opertions t ierent sewing lines re liste in Tble 2. All other t re the sme s shown in Tble 1 exept p. Results hve been summrize in Tble 2. Here (x,y) in Tble 2 inites the presribe optiml solutions with number o o-line repir sttions, unit proution ost n throughput rte or goo items in the lst three olumns, respetively. For sewing lines llowing no oline repir hve been nlyze s well. In this se lultions re one by onsiering very lrge vlues or ixe osts o oline repir sttions (i.e., ) in the existing problem setup suh tht the possibility o seleting eite repir worsttion is utomtilly neglete. The unit ost o proution n orresponing throughput rtes or goo items re lulte rom this problem setup (i.e., no oline repir) n note in the prenthesis in Tble 2 long with other optimum results.

7 Hossin & Srer Sewing Line Tble 2: Deisions me regring on-line or o-line repir in 5 ierent sewing lines Proportion o eets n eision (x, y) It my be note tht eite oline repir worsttion provies ost no more thn tht obtine uner no-oline repir poliy. This oline repir poliy vntgeously provie no less throughput rte thn tht yiele rom the online repir system. Though ll the system prmetri t in Tble 1 remins unhnge or this problem exept, vrious optiml results re not the sme ue to the vrition in p. For exmple, there is no eite repir sttion require or sewing lines 2, wheres the number o eite o-line repir sttion require in sewing lines 1, 3, 4 n 5 re 1, 2, 2 n 3 respetively. CONCLUSION Deiing, whether the eetive items shoul be repire on-line t regulr worsttion(s) or oline t seprte eite repir sttion(s) or shoul be srppe inste o repir, is n importnt onern or ming the opertionl eision in line proution system. This reserh elt with loting repir sttion on or o the proution line wherein the inspetion is sttione t the en o min line. To esribe the implition o the presribe moel se o T-shirt sewing lines is presente in this rtile. It is observe tht, in generl, the optimum unit ost o proution is lower n throughput rte or goo items is higher or the ses where seprte o-line repir sttion is presribe. Though the problem investigte pertins to grments inustry, this reserh outome is beneiil n implementble to mny engineering proution systems, espeilly in isrete proution systems. Sine the grments inustry n other engineering inustries involve billions o ollrs o revenue n sles, smll improvement in those systems will liewise impt the setors with huge innil beneit to the mngement n onsumers. Consiering possibility o eteting nononormities within the proution line, epenent eets n/or the imperet inspetion ilities n be onsiere or extening the moel in uture. Reerenes Worsttions p (x, y) (0,1) (1,1) (0,0) (0,1) (0,0) (0,1) (0,1) p (x, y) (0,1) (0,1) (0,0) (0,1) (0,0) (0,1) (0,1) p (x, y) (0,1) (0,1) (1,1) (1,1) (0,0) (0,1) (0,1) p (x, y) (0,1) (0,1) (0,0) (1,1) (1,1) (0,1) (0,1) p (x, y) (0,1) (1,1) (0,0) (1,1) (0,0) (0,1) (1,1) p Number o eite repir sttion AAFA (2012). U.S. Apprel Imports Amerin Apprel & Footwer Assoition, 1601 North Kent Street, Suite 1200, Arlington, VA p Min. unit Proution ost ($/unit) (4.4880) (4.3805) (4.5342) (4.5209) (4.5605) Unit proution ost i ll repirs were one on line. Throughput rte o goo prouts i ll repirs were one on line. Throughput Rte or goo items (units/hour) ( ) ( ) ( ) ( ) ( )

8 Hossin & Srer Bhir, S.K. (2011). Assembly line blning in grment proution by simultion, Chpter 4 in Assembly Line-Theory n Prtie, eite by Wlemr Grzeh, ISBN: , publishe by InTeh, Jnez Trine 9, Rije, Croti, Chng, C.T. (2001). On the polynomil mixe 0-1 rtionl progrmming problems. Europen Journl o Opertionl Reserh, 131 (1), Hossin, M.S.J. & Srer, B.R. (2016) Optiml lotions o on-line n o-line rewor sttions in seril proution system, Interntionl Journl o Proution Reserh, 54(12): pp Islm, M.M., Mohiuin, H.M., Mehii, S.H., & Sib, N. (2014). An optiml lyout esign in n pprel inustry by pproprite line blning: A se stuy. Globl Journl o Reserhes in Engineering, 14 (5.1), Li, H.L. (1994). A globl pproh or generl 0-1 rtionl progrmming. Europen Journl o Opertionl Reserh, 73(3), Rviv, T. (2013). An eiient lgorithm or mximizing the expete proit rom seril proution line with inspetion sttions n rewor. OR Spetrum, 35(3),