On Effect of Delayed Differentiation on a Multiproduct Vendor Buyer Integrated Inventory System with Rework

Transcription

1 Internatonal Journal of Appled Engneerng Research ISSN Volume 12, Number 19 (2017) pp Research Inda Publcatons. On Effect of Delayed Dfferentaton on a Multproduct Vendor Buyer Integrated Inventory System wth Rework Me-Fang Wu Assocate Professor, Department of Industral Engneerng & Systems Management, Feng Cha Unversty, achung 407, awan. ffany Chu PhD canddate, Department of Accountng and Informaton Systems, Rutgers Busness School, Rutgers Unversty, Newark, New Jersey 07102, USA. Hong-Dar n Professor, Department of Industral Engneerng and Management, Chaoyang Unversty of echnology, Wufong, achung 413, awan. Snga Wang Chu* Professor, Department of Busness Admnstraton, Chaoyang Unversty of echnology, Wufong, achung 413, awan. *Correspondence emal Orcd ID: X Abstract Effect of delayed dfferentaton on a multproduct vendor-buyer ntegrated nventory system wth rework was nvestgated [1], usng mathematcal modelng and dfferental calculus. he optmal soluton to ths multproduct problem was derved to mnmze overall system cost and shorten fabrcaton cycle tme, and effects of varaton n system parameters to optmal soluton and to the system were explored. hs paper uses an algebrac approach n leu of dfferental calculus n [1] to reexamne the problem and demonstrate that the optmal soluton s obtanable usng ths alternatve approach. he objectve of ths paper s to provde a smplfed method for managers/practtoners n producton plannng and control, wth a means to effectvely resolve the problem wthout dervatves. Keywords: Operatons research, multproduct fabrcaton system, delayed dfferentaton, vendor-buyer system, algebrac approach, mult-delvery, rework INRODUCION Managers of producton plannng constantly seek ways to ncrease machne utlzaton, assure qualty of products, and reduce relevant operatng costs. he most economcal batch sze was frst ntroduced by aft [2] to mnmze total producton-nventory cost, wheren some smple assumptons of hs model nclude perfect producton process, fabrcaton of sngle product, and contnuous end tem ssung polcy. However, producton of defectve tems s nevtable, due to varous unexpected or uncontrollable factors. Many studes have been performed to address dfferent aspects of mperfect producton stuatons and ther subsequent matters [3-12]. Certan defectve/nonconformng features of products can sometmes be reworked, so they can keep hold of the acceptable qualty [13-18].Also, n real supply-chan systems, the dstrbuton of end products are often handled by dverse dscontnuous mult-shpment polces [19-25]. In order to ncrease machne utlzaton, fabrcaton of multproduct on a sngle machne can be an effectve operatng strategy. Durng past decades, studes relatng to varous aspects of mult- product fabrcaton systems have been extensvely explored [26-32]. Further, when multple products sharng a common part, the delayed dfferentaton strategy s often consdered wth the am of shortenng producton cycle length and reducng total relevant system costs [33-38]. Chu et al. [1] derved the optmal producton-shpment solutons and nvestgated the effect of delayed dfferentaton on a multproduct vendor-buyer ntegrated nventory system wth rework, usng mathematcal modelng and conventonal dfferental calculus method. Unlke the needs of usng dfferental calculus, Grubbstrom and Erdem [39] ntroduced an algebrac approach to solve a specfc economc order quantty (EOQ) problem wthout usng dervatves. A few studes employed the same/smlar approach to resolve varous aspects of lot szng problems n fabrcaton and/or vendor-buyer ntegrated supply-chan systems [40-42]. kewse, ths paper extends such a smplfed approach to reexamne the multproduct problem descrbed n [1]. and demonstrate that the optmal solutons can be derved wthout dervatves. 8144

2 Internatonal Journal of Appled Engneerng Research ISSN Volume 12, Number 19 (2017) pp Research Inda Publcatons. HE PROBEM, MODEING AND PROPOSED MEHOD he problem as n [1] s a mult-product sngle-machne vendor buyer ntegrated system wth features of delayed dfferentaton and rework process. Annual product demand s λ for dfferent products (where = 1, 2,, ) and these products share a common part. wo-stage fabrcaton process s used, wheren the frst stage only produces common parts, at a rate of P 1,0, and upon completon of stage 1, dfferent lots of common parts are made ready for the producton n the second stage, where dfferent end products are fabrcated at a rate of P 1, (where = 1, 2,, ) n sequence under the common producton cycle tme polcy (Fgure 1). he objectves are to ncrease utlzaton, shorten producton cycle tme, and mnmze total producton- nventory-delvery costs. reparable, thru rework processes at a rate of P 2, (where = 0, 1, 2,, ) n the end of regular producton processes n each stage (see Fgures 1 and 2). No shortages are allowed, so we have (P 1, d 1, λ ) > 0 for = 0, 1, 2,,. Fgure 2: Inventory status of nonconformng common parts and nonconformng end products n the proposed multproduct vendor-buyer ntegrated system [1] In the end of rework process t 2,, fxed quantty n nstallments of the fnshed lot are dstrbuted to buyers, at a fxed nterval of tme n delvery tme t 3, (see Fg. 1). Addtonal notaton used n ths study ncludes the followng (where = 1, 2,, stands for dfferent products n stage 2; and = 0 represents the common part n stage 1): Q fabrcaton lot sze for product, K fabrcaton setup cost for product, C unt producton cost for product, C R, unt reworkng cost for product, h 1, unt holdng cost for product, h 2, unt holdng cost per reworked tem for product, Fgure 1: Inventory status of common parts and end products n the proposed multproduct vendor-buyer ntegrated system [1] Durng the producton processes n each stage, x porton of nonconformng tems may randomly be produced at a rate of d 1, (where d 1, = P 1, x and = 0, 1, 2,, ; wth = 0 denotes t s n stage 1). All nonconformng tems are assumed to be h 3, unt holdng cost for stocks stored at buyer s sde, h 4, unt holdng cost for safety stocks stored at producer s sde, K 1, fxed delvery cost per shpment for product, C, unt delvery cost for product, t 1, fabrcaton uptme for product, t 2, rework tme for product, t 3, delvery tme for product, common producton cycle length - the decson varable, 8145

3 Internatonal Journal of Appled Engneerng Research ISSN Volume 12, Number 19 (2017) pp Research Inda Publcatons. n number of fxed quantty nstallments of fnshed lot to be dstrbuted to buyers n each cycle, the other decson varable, α completon rate of common part as compared to the fnshed product, H nventory level of common part at the tme for end product, H 1, nventory level of perfect qualty tems n the end of regular producton, H 2, nventory level of perfect qualty tems n the end of rework process, t n, a fxed nterval of tme between two consecutve delveres n t 3,, I(t) on-hand nventory level of perfect qualty tems at tme t, I d(t) on-hand nventory level of defectve tems at tme t, I the left-over number of fnshed tems of product n each t n, at buyer s sde, D number of fnshed tems of product to be dstrbuted to customer n each shpment, C(, n) total relevant system cost per cycle, E[] the expected common producton cycle length, E[C(, n)] the expected total relevant system cost per cycle, E[CU(, n)] the expected total relevant system cost per unt tme. From Fgure 1, we observe the common producton cycle tme and lot szes as follows: Q t1, t2, t3, for 0, 1, 2,, (1) Q for 1, 2,, (2) Q Q (3) he prerequste condton of the proposed system [1] s t1,0 t2,0 t1, t2, 1 or 1 E[ x ] 1 E[ x ] 0 Q0 Q P1,0 P 2,0 1 P1, P 2, (4) he followng total relevant system cost per cycle C(, n) conssts of fabrcaton setup cost; varable fabrcaton cost; rework cost; and holdng costs for reworked tems, safety stock, and perfect qualty stocks n both stages; fxed and varable delvery costs; and holdng costs for stocks stored at buyers sde n stage two [1]: C, n d1,0t1,0 K0 C0Q0 CR,0x0Q0 h2,0 t2,0 0 x0q0 2 H1,0t1,0 H2,0 H1,0 d1,0t1,0 h1,0 t2,0 t1,0 H t1 t t2, K CQ CR, xq nk1, C, Q h2, t2, 2 Q H1, t1, H 2, H 1, n 1 d1, t1, h t t H t t n 2 nd I tn, nn 1 ni t1, t2, h3, Itn, xq , 1, 2, 2, 3, 1, Substtutng relevant system parameters [1] n Eq. (5) along wth and usng the expected value of x (to take the randomness of defectve rate nto account), and wth extra dervatons we obtan E[CU(, n)] as follows [1]: K0 E CU, n E C, n / E C00 CR,00E[ x0] z0 where 1 (5) 2 K nk 1, h1, 1, C C R, E [ x ] C, 2, 2 n h2, E[ x ] h3, 1 E[ x ] 1, E[ x ] 2 2 P1, n (6) Ex h1, E x 0 0 h2,00 E[ x0 ] 2 P1, z0 ; Ex [ ] h1,0 j 00 E[ x0 ] 1 P1, P 2, 1 j1 1 1 Ex [ ] ; and 1, 2, P1, he proposed algebrac method 2 1 E[ x] 1 E[ x ] for 1, 2,, P1, It can be seen that the decson varables n E[CU(, n)] (.e., Eq. (6)) are n the forms of, -1, n -1, and n -1. et ω 0, ω 1, ω 2, ω 3, and ω 4 stand for the followng: 0 C00 CR,00E[ x0 ] 1 0 C CR, E[ x ] C, 1 K K (8) 1 2 1, 1 K (9) (7) 8146

4 Internatonal Journal of Appled Engneerng Research ISSN Volume 12, Number 19 (2017) pp Research Inda Publcatons h 1, h2, E[ x ] h3, 1 Ex [ ] 3 z0 2, E[ x ] P 1, (10) 2 2 h1, h3, 4 1, 1, (11) Substtute ω nto Eq. (6), E[CU(, n)] becomes the followng: 1 1 1, E CU n n n We can rearrange Eq. (12) as follows:, (12) E CU n n n (13) It s noted that E[CU(, n)] wll be mnmzed f the second and thrd terms of Eq. (13) equal to zeros. hat s and 1 (14) n (15) Substtute ω nto Eq. (15), we obtan the followng: * n 2 K0 K h3, h1, 1, h 1, h2, E[ x ] h3, 1 Ex [ ] K1, z0 2, E[ x ] P1, P 2, (16) Once n s found, E[CU(, n)] can be reconsdered as a functon wth sngle decson varable. So, we rearrange Eq. (12) as follows: 1 1, E CU n 0 1 2n 3 4n or, 2 E CU n 0 1 2n 3 4n 2 n n (17) (18) It s noted that E[CU(, n)] can be mnmzed f the second term of Eq. (19) equals to zero. herefore, we have the followng: * 1 2n n Substtute ω nto Eq. (19), we obtan the followng: (19) 0 1, * h 1, 1, h2, E[ x ] h3, 1 Ex [ ] 1, z0 2, E x 1 2 n 2 2 P1, n K K nk [ ] (20) he resultng optmal solutons (equatons (16) and (20) to the problem are dentcal to what were obtaned n [1]. he smlar procedure for seekng an nteger value for n can be appled by usng n - and n + as shown n [1], and the effects of varaton n system parameters to the optmal soluton and to ths specfc system can be fully explored accordngly. CONCUSIONS Chu et al. [1] derved the optmal soluton and nvestgated the effect of delayed dfferentaton on a multproduct vendor-buyer ntegrated nventory system wth rework usng mathematcal modelng along wth dfferental calculus. Unlke the needs of usng dfferental calculus, an algebrac approach s employed n ths paper to reexamne ther proposed problem [1] and show that optmal solutons are obtanable by a smplfed method. We provde a means for managers/ practtoners n the feld of producton plannng and control (who may only have basc algebra background) to successfully resolve the problem wthout dervatves. REFERENCES [1] Chu, Y-S.P., Kuo, J-S., Chu, S.W., Hseh, Y Effect of delayed dfferentaton on a mult product vendor buyer ntegrated nventory system wth rework. Advances n Producton Engneerng & Management, 11(4), pp [2] aft, E.W he most economcal producton lot. Iron Age, 101, pp [3] Shh, W Optmal nventory polces when stock-outs result from defectve products. Internatonal Journal of Producton Research, 18(6), pp [4] Beleck,., Kumar, P.R., Optmalty of zero-nventory polces for unrelable producton faclty. Operatons Research, 36, pp [5] Abboud, N.E A smple approxmaton of the EMQ model wth Posson machne falures. Producton Plannng and Control, 8(4), pp [6] Chelb, A., Rezg, N Analyss of a producton/nventory system wth randomly falng 8147

5 Internatonal Journal of Appled Engneerng Research ISSN Volume 12, Number 19 (2017) pp Research Inda Publcatons. producton unt subjected to a mnmum requred avalablty level. Internatonal Journal of Producton Economcs, 99(1-2), pp [7] Wu, M.-F., Chu, Y.-S.P., Sung, P.-C Optmzaton of a mult-product EPQ model wth scrap and an mproved mult-delvery polcy. Journal of Engneerng Research, 2(4), pp [8] Kundu, S., Chakrabart, An ntegrated mult-stage supply chan nventory model wth mperfect producton process. Internatonal Journal of Industral Engneerng Computatons, 6(4), pp [9] Boorla, S.M., Howard,.J Producton montorng system for understandng product robustness. Advances n Producton Engneerng and Management, 11(3), pp [10] Setawan, R A game theory approach n vendor-buyer probablstc nventory system wth mperfect qualty, nspecton error, mnmum servce level constrant and partal backorderng. Far East Journal of Mathematcal Scences, 100(10), pp [11] Snaga, S., Pertw,.S., Ardan,., Zuhr Inventory smulaton optmzaton under non statonary demand. Internatonal Journal of Appled Engneerng Research, 11(1), pp [12] Zhang, D., Zhang, Y., Yu, M A machnng process orented modelng approach for relablty optmzaton of falure-prone manufacturng systems. Journal of Engneerng Research, 4(3), pp [13] Yum, B.J., McDowell, E.D Optmal Inspecton Polces n a Seral Producton System ncludng scrap, rework and repar: An MIP approach. Internatonal Journal of Producton Research, 25(10), pp [14] Jamal, A.M.M., Sarker, B.R., Mondal, S Optmal manufacturng batch sze wth rework process at a sngle-stage producton system. Computers and Industral Engneerng, 47(1), pp [15] Chu, Y-S.P., Sung, P-C., Chu, S.W., and Chou, C Mathematcal modelng of a mult- product EMQ model wth an enhanced end tems ssung polcy and falures n rework. SprngerPlus, 4(1), art. no. 679, pp [16] Jndal, P., Solank, A Integrated supply chan nventory model wth qualty mprovement nvolvng controllable lead tme and backorder prce dscount. Internatonal Journal of Industral Engneerng Computatons, 7(3), pp [17] Jawla, P., Sngh, S.R Mult-tem economc producton quantty model for mperfect tems wth multple producton setups and rework under the effect of preservaton technology and learnng envronment. Internatonal Journal of Industral Engneerng Computatons, 7(4), pp [18] Chu, Y-S.P., Chang, K-W., Chu, S.W., Song, M-S Smultaneous determnaton of producton and shpment decsons for a mult-product nventory system wth a rework process. Advances n Producton Engneerng & Management, 11(2), pp [19] Sarker, B.R., Parja, G.R An optmal batch sze for a producton system operatng under a fxed-quantty, perodc delvery polcy. Journal of the Operatonal Research Socety, 45(8), pp [20] Hoque M.A., Goyal S.K A heurstc soluton procedure for an ntegrated nventory system under controllable lead-tme wth equal or unequal szed batch shpments between a vendor and a buyer. Internatonal Journal of Producton Economcs, 102(2), pp [21] Sngh, N., Vash, B., Sngh, S.R Analyss of three level supply chan of nventory wth deteroraton for mult-tems. Internatonal Journal of Industral Engneerng Computatons, 5(3), pp [22] Kwok, J.J.M., ee, D.-Y Coopettve supply chan relatonshp model: Applcaton to the smartphone manufacturng network. PoS ONE, 10(7), art. no [23] RajKumar, N., Satheesh Kumar, R.M Automotve closed loop supply chan wth uncertanty. Internatonal Journal of Appled Engneerng Research, 10(55), pp [24] Khan, S.A.R., Qanl, D., Zhang, Y Usage of RFID technology n supply chan: Benefts and challenges. Internatonal Journal of Appled Engneerng Research, 11(5), pp [25] Setawan, R., ryanto A probablstc ntegrated vendor-buyer cooperatve nventory model wth mperfect qualty tems, controllable lead tme, uncertanty demands, nspecton error, shortage and backorderng allowed. Far East Journal of Mathematcal Scences, 99(1), pp [26] Federgruen, A., Katalan, Z Determnng producton schedules under base-stock polces n sngle faclty mult-tem producton systems. Operatons Research 46(6): [27] Rzk, N., Martel, A., D'Amours, S Mult-tem dynamc producton-dstrbuton plannng n process ndustres wth dvergent fnshng stages. Computers and Operatons Research, 33(12), pp [28] Chu, S.W., seng, C-., Wu, M-F., Sung, P-C

6 Internatonal Journal of Appled Engneerng Research ISSN Volume 12, Number 19 (2017) pp Research Inda Publcatons. Mult-tem EPQ model wth scrap, rework and mult-delvery usng common cycle polcy. Journal of Appled Research and echnology, 12(3), pp [29] Hamd, M., Farahmand, K., Reza Sajjad, S., Nygard, K.E A heurstc algorthm for a mult-product four-layer capactated locaton-routng problem. Internatonal Journal of Industral Engneerng Computatons, 5(1), pp [30] Fergany, H.A Probablstc mult-tem nventory model wth varyng mxture shortage cost under restrctons. SprngerPlus, 5(1), art. no [31] Chu, S.W., Chen, S.-W., Chang, C.-K., Chu, Y.-S.P Optmzaton of a mult-product ntra-supply chan system wth falure n rework. PoS ONE, 11(12), art. no. e [32] Zahed, Z., Ar Samadh,.M.A., Suprayog, S., Halm, A.H Integrated batch producton and mantenance schedulng for multple tems processed on a deteroratng machne to mnmze total producton and mantenance costs wth due date constrant. Internatonal Journal of Industral Engneerng Computatons 7(2): [33] Gerchak, Y., Heng, M An nventory model wth component commonalty. Operatons Research etters, 5(3), pp [34] Swamnathan, J.M., ayur, S.R Managng broader product lnes through delayed dfferentaton usng vanlla boxes. Management Scence, 44(12 Part 2), pp. S161-S172. [35] Graman, G.A., Magazne, M.J Implementaton ssues nfluencng the decson to adopt postponement. Internatonal Journal of Operatons and Producton Management, 26(10), pp [36] Kouvels, P., an, Z Flexble capacty nvestments and product mx: Optmal decsons and value of postponement optons. Producton and Operatons Management, 23(5), pp [37] Chu, Y-S.P., Hseh, Y-., Kuo, J-S. Chu, S.W A delayed dfferentaton mult- product FPR model wth scrap and a mult-delvery polcy I: Usng sngle-machne producton scheme. Internatonal Journal for Engneerng Modellng, 29(1-4), pp [38] Chu, S.W., Kuo, J-S., Chu, V., Chu, Y-S.P Cost mnmzaton for a mult-product fabrcaton-dstrbuton problem wth commonalty, postponement, and qualty assurance. Mathematcal and Computatonal Applcatons, 21(3), art. no. 38, pp [39] Grubbstrom, R.W., Erdem, A he EOQ wth backloggng derved wthout dervatves. Internatonal Journal of Producton Economcs, 59, pp [40] Chu, S.W., Sung, P.C., seng, C.., Chu, Y.S.P Mult-product FPR model wth rework and mult-shpment polcy resolved by algebrac approach. Journal of Scentfc and Industral Research, 74(10), pp [41] seng, C-., Wu, M-F., n, H-D., Chu, Y-S.P Solvng a vendor-buyer ntegrated problem wth rework and a specfc mult-delvery polcy by a two-phase algebrac approach. Economc Modellng, 36, pp [42] Chu, Y.-S.P., Wu, M.-F., Chu, S.W., Chang, H.-H A smplfed approach to the mult-tem economc producton quantty model wth scrap, rework, and mult-delvery. Journal of Appled Research and echnology, 13(4), pp