Manufacturing Lot Sizing with Backordering, Scrap, and Random Breakdown Occurring in Inventory-Stacking Period

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1 nd WSEAS In. Conf. on CIRCUIS, SYSEMS, SIGNAL and ELECOMMUNICAIONS (CISS'08)Acapulco, Mexico, January 5-7, 008 Manufacurin Lo Sizin wi ackorderin, Scrap, and Random reakdown Occurrin in Invenory-Sackin eriod SINGA WANG CHIU a YUAN-SHYI EER CHIU b a Deparmen of usiness Adminisraion, Caoyan Universiy of ecnoloy 68 Jifon Eas Road, Wufon, aicun, aiwan b Deparmen of Indusrial Enineerin and Manaemen, Caoyan Universiy of ecnoloy swan@mail.cyu.edu.w p:// Absrac: - is paper is concerned wi deerminaion of opimal lo size for an economic manufacurin quaniy model wi backorderin, scrap and breakdown occurrin in invenory-sackin period. Generaion of defecive iems and random breakdown of producion equipmen are ineviable in mos real-life manufacurin sysems. o cope wi e socasic macine failures, producion planners pracically calculae e mean ime beween failures (MF) and esablis e robus plan accordinly, in erms of opimal lo size a minimizes oal producion-invenory coss for suc an unreliable sysem. Random scrap rae is considered in is sudy, and breakdown is assumed o occur in invenory sackin period. Maemaical modelin and analysis is used and e renewal reward eorem is employed o cope wi e variable cycle len. An opimal manufacurin lo size a minimizes e lon-run averae coss for suc an imperfec sysem is derived. Numerical example is provided o demonsrae is pracical usaes. Key-Words: - Manufacurin Sysems, Macine breakdown, Lo size, ackorderin, Scrap Inroducion e economic manufacurin quaniy (EMQ) model (also known as economic producion quaniy (EQ) model) implicily assumes a iems produced are of perfec qualiy. u in real-life producion sysems, eneraion of defecive iems is ineviable. Hence, sudies ave been carried ou o enance e classic EMQ model by addressin e issue of imperfecion qualiy iems produced [-6]. Sock-ou siuaions may occur due o e excess demand. Someimes, ese soraes can be backordered and saisfied a a fuure ime, ence e overall producion-invenory coss can be reduced sinificanly [7-0]. Random breakdown of producion equipmen is anoer common and ineviable reliabiliy facors a rouble e producion planners and praciioners mos. o effecively manae and conrol e disrupion and minimize overall producion coss, become e primary ask of mos manufacurin firms. I is no wonder a deerminin opimal lo-size (or producion upime) for sysems wi macine failures as received aenion from researcers in recen decades [-8]. Groenevel e al. [5] sudied wo invenory conrol policies a deal wi macine failures. e firs one assumes a e producion of e inerruped lo is no resumed (called no resumpion - NR policy) afer a breakdown. e second policy considers a e producion of e inerruped lo will be immediaely resumed (called abor/ resume (AR) policy) afer e breakdown is fixed and if e curren on-and invenory falls below a cerain resold level. e repair ime is assumed o be neliible and e effecs of macine breakdowns and correcive mainenance on e economic lo sizin decisions are invesiaed. Ciu e al. [] invesiaed opimal run ime for EQ model wi scrap, rework and random breakdown. ey proposed and proved eorems on condiional convexiy of e ineraed cos funcion and on bounds of e producion run ime. en, an opimal run ime was locaed by e use of e bisecion meod based on e inermediae value eorem. is paper deerminaes e opimal manufacurin lo size for EQ model wi backloin, scrap, and random breakdown occurrin in invenory-sackin period. Maemaical Modelin Consider a a manufacurin process may randomly produce x porion of defecive iems a a rae d. All imperfec qualiy iems are assumed no repairable, are reaed as scrap. e producion rae is muc larer an e demand rae and e ISSN: ae 97 ISN:

2 producion rae of scrap iems d can be expressed as dx. Furer, soraes are backordered, ey will be saisfied firs wen e nex replenismen producion cycle beins. Accordin o mean ime beween failures (MF) daa, a random macine breakdown may ake place randomly in e invenory-sackin period (refer o Fiure ). e abor/resume (AR) invenory conrol policy is adoped in is sudy and under suc policy, wen a breakdown akes place, e macine is under correcive mainenance immediaely, and e repair ime is assumed o be consan. e inerruped lo will be resumed ri afer e macine is resored. Cos parameers include uni producion cos C, uni oldin cos, seup cos K, disposal cos per scrap iem C S, uni sorae/backordered cos b, and cos for repairin and resorin macine M. Addiional noaion is lised as follows. H H H d r ime Fi.: On-and invenory of perfec qualiy iems in EQ model wi scrap and breakdown occurrin in invenory-sackin period e opimal producion ime (i.e. producion upime) o be deermined for e proposed EQ model, producion ime before a random breakdown occurs, r ime required for repairin and resorin e macine, ime required for fillin backorder quaniy, ime required for deplein all available perfec qualiy on-and iems, sorae permied ime, H level of on-and invenory wen macine breakdown occurs, H level of on-and invenory wen macine is repaired and resored, H e maximum level of on-and invenory for eac producion cycle, Q producion lo size for eac cycle, maximum backorder level per cycle, e producion cycle len, C(,)oal producion-invenory coss per cycle, CU(,)oal producion-invenory coss per uni ime (e.. annual), E[CU(,)]e expeced oal producioninvenory coss per uni ime. e producion rae of perfec qualiy iems mus always be reaer an or equal o e sum of e demand rae and e producion rae of defecive iems d. Hence, e followin condiion mus old: (-d-)>0 or (-x-/)>0. Since denoes producion ime before a breakdown akin place in e invenory-sack period and le e consan macine repair ime r. e followin derivaion procedure is similar o wa was used by prior sudies [9-0]. From Fiure, one can obain e followin: H d () were dx. ( ) H H H () r ( ) ( ) H H + d () Q () (5) r H (6) (7) d (8) oal scrap iems produced durin producion upime are (see Fiure ): d ( ) d + d d r d ime Fi.: On-and invenory of scrap iems in EQ model wi scrap and breakdown occurrin in invenory-sackin period

3 nd WSEAS In. Conf. on CIRCUIS, SYSEMS, SIGNAL and ELECOMMUNICAIONS (CISS'08)Acapulco, Mexico, January 5-7, 008 d x Q (9) oal producion-invenory cos per cycle C(,) is: C(, ) K+ C ( ) + Cs ( x) + M H H+ H H + H H + + r + + ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) d d r + ( ) + b ( ) + ( ) (0) Subsiuin all relaed parameers from Eqs. () o (9) in Eq. (0), one obains C (,) as follows. C(, ) C + K+ M + CS x ( x) ( x) + ( x+ x ) + + x x b( x) + + ( + ) ( ) + ( x ) ( x ) x () e producion cycle len is no consan due o e assumpion of random scrap rae and a uniformly disribued breakdown is assumed o occur in e invenory sackin ime. us, o ake randomness of scrap and breakdown ino accoun, one can use e renewal reward eorem in invenory cos analysis o cope wi variable cycle len and use ineraion of C(,) o deal wi e random breakdown appenin in invenory sackin ime. e expeced oal producion-invenory coss per uni ime can be calculaed as follows. E C(, ) f ( ) d 0 E CU(, ) E E C d 0 (, ) ( / ) ( E[ x] ) / E[ x] S E[ x] () E[ CU (, ) ] C + C E[ x] ( K+ M) Ex x E[ x] + ( b+ ) E + x + ( + ) E + x E[ x] () E x Le E0 ; E ; E ; x E E ; E E x x en Eq. () becomes: [ (, )] E[ x] ( K+ M) E CU CE + C E + E 0 S 0 + E E b E ( ) + E + ( + ) E + E E () 0 Convexiy of E[CU(,)] e opimal invenory operain policy can be obained by minimizin e expeced cos funcion. For e proof of convexiy of E[CU(,)], one can uilize e Hessian marix equaion [9] and verify e exisence of e followin: [ ] (, ) E CU(, ) E CU > E CU(, ) E CU(, ) 0 (5) E[CU(,)] is sricly convex only if Eq. (5) is saisfied, for all and differen from zero. y compuin all e elemens of e Hessian marix equaion, one obains: [ ] (, ) E CU(, ) ECU ( K+ M) E + E > 0 E CU(, ) E CU(, ) 0 (6) Eq. (6) is resulin posiive, because all parameers are posiive. Hence, E[CU(,)] is a sricly convex funcion. I follows a for opimal producion upime and maximal backorder level, one can differeniae E[CU(,)] wi respec o and wi respec o, and solve linear sysems of Eqs.(7) and (8) by sein ese parial derivaives equal o zero. [ (, )] ( ) E CU K M + E 0+ E0 ( b+ ) E E + E E E (7) ISSN: ae 99 ISN:

4 nd WSEAS In. Conf. on CIRCUIS, SYSEMS, SIGNAL and ELECOMMUNICAIONS (CISS'08)Acapulco, Mexico, January 5-7, 008 [ (, )] E CU E + ( b+ ) E + E + E 0 E ( K+ M) E + E ( b+ ) E E E + E b E 0 ( + ) 0 (8) (9) E (0) ( b+ ) E From Eq. () and Eqs. (9-0) one can obain e opimal lo-size Q and opimal backorder level as follows: Q E ( K+ M) E0+ E ( b+ ) E E E + E b E ( + ) 0 () produced x, follows a uniform disribuion over e inerval [0, 0.5]. Addiional parameers used are summarized as follows. K $0 for eac producion run, C S $.00 disposal cos for eac scrap iem, C $.00 per iem, M $500 repair cos for eac breakdown, $0.6 per iem per uni ime, b $0.8 per iem backordered per uni ime, 0.08 years, ime needed o repair and resore e macine. Applyin Eqs. (9), (), (), and (), one can obain e opimal producion ime 0.5 years, e opimal lo size Q,, e backorder,68 and E[CU(,)]$7,85.0. Effec of defecive rae x on e expeced cos E[CU(,)] is illusraed in Fiure. Q E () ( b+ ) E. Resuls and Verificaion Suppose a e breakdown facor is no considered, en e cos and ime for repairin failure macine M0 and 0, Eqs. (-) become e same equaions as were iven by Ciu and Ciu [9]: Q K + Ex x ( b+ ) E x / Q b+ x E x / () () Furer, suppose a e reular producion process produces no defecive iems, i.e. x 0, en Eqs. (-) become e same equaions as were presened by e classic EQ model wi backorderin permied [0-]: K b+ Q b Q ( b ) + (5) (6) Numerical Example Assume annual producion rae of a manufacured iem is 8,000 unis and demand of is iem is,000 unis per year. e percenae of random scrap iems Fi.: Effec of defecive rae x on E[CU(,)] Suppose e resul of is paper is no available, one can only use a closely relaed lo-size soluion iven by [9] for solvin suc an unreliable sysem and obainin Q,0 (or 0.5) and 78. luin is lo-size soluion ino Eq. (), one as E[CU(,)] $8,06.9. As a resul, i pays exra 9.99% on oal seup and oldin coss an e opimal producion- invenory coss obained by usin lo-size decisions of e presen sudy. 5 Conclusion Generaion of random scrap iems and socasic breakdown of producion equipmen are ineviable in mos real-life manufacurin sysems. is paper invesiaes eir effec on e lo-size decision of EQ model, so a e overall producion-invenory coss can be minimized. Since lile aenion was paid o e aforemenioned area, is paper inends o fill e ap. ISSN: ae 00 ISN:

5 nd WSEAS In. Conf. on CIRCUIS, SYSEMS, SIGNAL and ELECOMMUNICAIONS (CISS'08)Acapulco, Mexico, January 5-7, 008 Acknowledemens Auors ank Naional Science Council of aiwan for supporin is sudy under Gran #: NSC 95-6-H References: [] R.E. arlow and F. roscan, Maemaical eory of Reliabiliy. Jon Wiley and Sons, New York, 965. [] M. en-daya and M. Haria, Economic lo scedulin problem wi imperfec producion processes. Journal of Operaional Researc Sociey, Vol. 5, 000, pp [].C.E. Cen, An economic order quaniy model wi demand-dependen uni producion cos and imperfec producion processes. IIE ransacion, Vol., 99, pp. -8. []. K. Das, S. Sarkar, Opimal prevenive mainenance in a producion invenory sysem. IIE ransacions, Vol., 999, pp [5] H. L. Lee and M. J. Rosenbla, Simulaneous deerminaion of producion cycle and inspecion scedules in a producion sysem. Manaemen Science, Vol., 987, pp [6] M.J. Rosenbla and H. L. Lee, Economic producion cycles wi imperfec producion processes. IIE ransacion, Vol. 8, 986, pp [7] S.W. Ciu, Opimal replenismen policy for imperfec qualiy EMQ model wi rework and backloin. Applied Socasic Models in usiness and Indusry, Vol., 007, pp [8] S.W. Ciu and Y-S.. Ciu, Maemaical modelin for producion sysem wi backloin and failure in repair. Journal of Scienific & Indusrial Researc, Vol. 65, No. 6, 006, pp [9] Y-S.. Ciu and S.W. Ciu, A finie producion model wi random defecive rae and soraes allowed and backordered. Journal of Informaion & Opimizaion Sciences, Vol., 00, pp [0].A. Hayek and M. K. Salame, roducion lo sizin wi e reworkin of imperfec qualiy iems produced. roducion lannin and Conrol, Vol., 00, pp [] A. Arreola-Risa and G.A. DeCroix, Invenory manaemen under random supply disrupions and parial backorders. Naval Researc Loisics, Vol. 5, 998, pp [] M. er, M.J.M. osner and H. Zao, roducion-invenory sysems wi unreliable macines. Operaions Researc, Vol., 99, pp.-8. [] S.W. Ciu, S.L. Wan and Y-S.. Ciu, Deerminin e opimal run ime for EQ model wi scrap, rework, and socasic breakdowns. European Journal of Operaional Researc, Vol. 80, 007, pp [] K.J. Cun, ounds for producion lo sizin wi macine breakdowns. Compuers and Indusrial Enineerin, Vol., 997, pp.9-. [5] H. Groenevel, L. inelon and A. Seidmann, roducion lo sizin wi macine breakdowns. Manaemen Science, Vol. 8, 99, pp.0-. [6] C.H. Kim and Y. Hon, An exended EMQ model for a failure prone macine wi eneral lifeime disribuion. Inernaional Journal of roducion Economics, Vol. 9, 997, pp.5-. [7] H. Kun, A dynamic lo sizin model wi exponenial macine breakdowns. European Journal of Operaional Researc, Vol. 00, 997, pp [8] V. Makis and J. Fun, An EMQ model wi inspecions and random macine failures. Journal of Operaional Researc Sociey, Vol. 9, 998, pp [9] R.L. Rardin, Opimizaion in Operaions Researc. Inernaional Ediion, renice-hall, Inc., New Jersey, 79-7, 998. [0] F. S. Hillier and G. J. Lieberman, Inroducion o Operaions Researc. McGraw Hill Co. Inc. New York, 00. [] E.A. Silver, D. F. yke and R. eerson, Invenory Manaemen and roducion lannin and Scedulin. Jon Wiley & Sons, Inc., New York, 998. ISSN: ae 0 ISN: