Proceedngs of the 0 Internatonal Conference on Industral Engneerng and Operatons Management Kuala Lumpur, Malaysa, January 4, 0 Cyclc Schedulng n a Job shop wth Multple Assembly Frms Tetsuya Kana and Koch Naade Department of Industral Engneerng and Management Nagoya Insttute of Technology, Nagoya 466-8555, JAPAN Abstract Ths paper deals wth a ob shop n a part producton system wth multple assembly frms. The part manufacturer produces multple types of products n a ob shop. The cyclc schedule s repeatedly appled. From the nventory of processed parts, assembly factores tae multple types of fnshed products wth ther own fxed ntervals. Ths paper frst gves an effectve numercal procedure whch gves optmal schedules and cycle tme mnmzng the nventory and setup costs of the part factory when the recept ntervals to the assembly factores are gven. Then the optmal schedules, cycle tme and recept ntervals of assembly frms mnmzng the total average cost n a supply chan are derved, whch s the sum of nventory and setup costs n the part factory and the transportaton costs and nventory costs n the assembly frms. Ths schedule s compared wth the other ndvdual optmal schedule and ntervals and ts superorty on performance s shown. Keywords Job shop, Cyclc schedulng, EOQ, Lot szng, Supply Chan Optmzaton.. Introducton Small amounts of producton for many nds of products have come to be requested and ts effcent producton method has become mportant. Schedulng and lot szng problems n a factory are the bass of such optmzaton. The economc order quantty (Economc Order Quantty: EOQ) s well-nown as a method of mnmzng the nventory and order costs of the factory. In many cases, the assembly frms decde the nterval and the amount of orders to the parts plant based on the economc order quantty. On the other hand, the form of a cyclc schedulng that repeats the producton of lots wth a constant t s accordng to the demand for the post-processng s often taen, and the producton schedule of the parts plant has receved the restrcton to the parts plant where parts are suppled by the recept nterval of two or more assembly halls n the downstream. The cyclc schedulng problem n the ob shop envronment has been taen up n Ouennche and Boctor[] and Matsushta and Naade[]. Naade et al. [3] have treated the case that the fnal product of each nd s receved at a constant nterval tme. Trab et al. [4] treated a flow shop problem wth cyclc schedulng. Ouyang and Zhu[5] developed an economc lot schedulng n manufacturng/remanufacturng systems. Chatfeld [6] consdered an economc lot schedulng problem n a sngle machne by the genetc lot schedulng procedure. In decades, the supply chan optmzaton becomes also mportant, under whch total costs n the part producton and the assemble factory are mnmzed. Recently, Clausen and Ju [7] have consdered a sngle machne lot szng and schedulng problem n a part factory wth an assembly factory usng the parts, whch dscusses optmal schedulng mnmzng the total long-run average nventory and transportaton costs n a supply chan. In ths paper, the ob shop s consdered n a part producton factory wth multple assembly factores, whch receve parts. The part manufacturer produces multple types of products n a ob shop. The cyclc schedule s repeatedly appled. From the nventory of processed parts, assembly factores tae multple types of fnshed products n ther own fxed ntervals. An effectve numercal procedure s developed for dervng optmal schedules and cycle tme mnmzng the nventory and setup costs of the part factory when the recept ntervals are gven. Numercal examples show the performance of the algorthm. The optmal schedules, cycle tme and recept ntervals of assembly factores are also derved mnmzng the total average cost n a supply chan. Ths schedule s compared wth the other ndvdual optmal schedule and ntervals and ts superorty on performance s shown. 556
. Model. Flow of Products The model treated n ths paper s llustrated n Fg... A Job shop s n a part factory and each type of tems are processed n machnes wth a fxed order. Fnshed products processed at ther machnes are put n nventory places for assembly factores. They are receved wth a fxed tme nterval o by assembly factory. In the assembly factory tems are processed contnuously and the nventory decreases n tme. In ths study, a cyclc schedule s appled n a fnte horzon H. Contnuous determnstc demand s assumed to assemble tems, and the demand must be satsfed n assembly frms, and the assembler s demand for parts must also be satsfed n the nventory places of the part factory. The cycle tme s gven by a fnte horzon H dvded by some postve nteger, so the cycle tmet s gven by T H for some postve nteger. We also assume that n each cycle each nd of tems are produced as a batch at a tme. Assembly factory produces assembly tems, and t consumes part at a determnstc rate ntervals are also assumed by o H wth nteger. r,. The part recept tem tem 3 tem machne Part Factory machne machne 3 machne 4 Fnshed tems Item Item Item 3 recept o o o o 3 o 3 o 3 Assy Fac Assy Fac Assy Fac 3 Fgure : Flow of products The cost ncurred n the part factory conssts of a holdng cost for wor-n-process, an nventory cost for the fnshed tems, a setup cost for the machne. The cost ncurred n the assembly factory ncludes the transportaton cost and nventory cost. 3. Formulaton 3. Notatons Gven notatons are the followngs: m : the number of machnes, M {,, m} : a set of machnes, n : the number of tems, N {,, n} : a set of nds of tems, l : the number of assembly factores, L {,, l} : a set of assembly factores, m : the number of machnes n whch tem s processed, n : the number of tems whch are processed at machne, P( ) : a set of tems whch are processed at machne, (, ) : the ndex of the machne whch processes tem n the th order, r, : the demand rate of tem at assembly factory, r r,, p : a producton rate of tem, at machne, c,,,, : a setup cost from tem to tem at machne, L s : a setup tme from tem to tem at machne, h, : a holdng cost rate of tem at machne, h : an nventory cost rate per unt tem of tem, v, : a transportaton cost of tem to assembly factory, t, : carryng tme of tem from machne to the next machne, H : the length of fnte plannng horzon, ê : the frst recevng epoch of assembly factory. 557
Next we gve the varable determned by fxng and ( L ). : the number of producton cycles n horzon H, : the number of recepts by assembly factory n horzon H, T : the length of one producton cycle tme, o : the nterval between successve recepts by assembly factory,, : the processng tme of tem at machne n one cycle, gven by, r T p, max a; both o a and T a are ntegers.,, e : the remander obtaned when ê s dvded byt. Next we show the decson varables for a schedulng problem under fxed cycle tme and recept ntervals., f tem s operated after processng tem at machne b,, 0 otherwse f the last tem s and the frst tem s n one cycle at machne x,, 0 otherwse d, : the epoch when machne starts processng of tem,, : an nteger valuable mang the dfference between recevng epoch of factory and arrvng epoch at the nventory of fnshed tem less than or equal to T, : an nteger valuable mang the dfference between recevng epoch of factory and arrvng epoch at the, nventory of fnshed tem less than or equal to 3. Formulaton. The model studed n ths paper can be formulated nto the followng program. Mnmze m c,, b,, h, (, ) r d, (, ), (, ) d, (, ), (, ) T M P( ) P ( ), N h r, (, m ) N T o v h r, d, (, m ), (, ), (, ),, m t m e T L N, o r, h, () L N o L N subect to d, (, ), (, ) t, (, ) d,, (, ),..., m, N, () d,,,,, (,,,, ) s d M b x,, P( ),, M, (3) d,, s,, d, T,, P( ),, M, (4) 0 d m m t m e T ( ),, (, ), (, ), (, ),, N, L, (5) b,,, P( ),, b,,, P( ), M, (6) P( ),,, P( ) P ( ),,, {0,},,, (, ) 0 N, P ( ), b, x,,, x, M,,, P( ),, M, (7) b, x {0, },, P( ),, M, (8) d,,..., m, (9) max {0,,...,[ ]}, {0,,,..., T }, N, L. (0),,, Obectve functon gves a cost rate of the supply chan at unt tme. The frst term of the obectve functon s the sum of setup costs at all machnes. The second s the sum of nventory costs on wor-n-processes at all machnes. The thrd term s the total nventory costs of fnshed products after the fnal processng to the nventory place. The fourth s the total cost of nventory on fnshed cost at all nventory places, whch has been dscussed n [] and [3]. The ffth s the transportaton cost, and the last s the total nventory costs at all recpents.. 558
Constrants show the followngs. The constrant () s on the transportaton epochs between two successve machnes, and (3) s on the setup tmes, where M ' s constant and has a bg value. Constrant (4) means that processng for one tem at each machne s once n each cycle, and (5) means that the dfference between the recevng epochs and the arrvng epochs at machnes s less than, whch decdes the average cost on the fnshed products, whch are dscussed n []. Constrants (6) to (0) show the constrants on values that decson varables tae. 3.3 Formulaton under Fxed Cycle Tme and Recept Intervals The formulaton descrbed n secton 3. ncludes the non-lnear obectve functon and constrants. Thus we fx the cycle tme T and tme nterval between successve recevng epoch o, that also maes the varables and, fxed, and the problem n secton 3. becomes a mxed nteger lnear program. Varable has an upper bound for the problem havng solutons. Snce the sum of total processng and setup tmes s less than or equal to the cycle tme. Let the order of processng at machne denoted by and ( ) denote the tem whch s processed n the th order at machne n one cycle. Then must satsfy r H p ( ), P mn. n mn s ( ), ( ), s ( n ), (), If the cycle tme T s fxed, then M n (3) can be replaced by T, and constrant (4) can be removed. 3.4 Economc Order Quanttes on the Assembly Frms () When the assembly factores decde the nter-recept ntervals o ( )n advance, then they decde the values by * usng EOQ, The optmal nter-recept nterval o at assembly factory s gven by * o h N r, v N,. Note that ths value does not usually tae the value obtaned by dvdng the plannng horzon H by some nteger, * and so the optmal nterval n practce s o or * o tang the smaller average cost of assembly factory. 4. Numercal Algorthms Under the tme ntervals between successve recepts fxed, the algorthm for fndng the optmal cycle tme and the schedule n the ob shop are dscussed. Snce dervng optmal solutons for all possble cycle tmes leads to too many computaton tmes, we use the upper and lower bounds on the optmal average costs for each cycle tme T H for nteger, and we solve these bounds effcently for several necessary, whch maes the necessary computaton tme small. We have tested several types of bounds, and we use the followng lower and upper bounds. As lower bounds LP( ) : an obectve value for the optmal soluton of lnear relaxaton problem, LB ( ) : an obectve value for the optmal soluton of the problem wth the cost at the nventory of fnshed tems at the recept space estmated as the smallest value. As upper bounds UB ( ) : an obectve value for the optmal soluton of the problem wth the cost at the nventory of fnshed tems at the recept space estmated as the largest value, UB ( ) : the obectve value for the optmal soluton of the problem wth the schedule as the one n optmal soluton obtaned by dervng UB ( ). 559
UB ( ) and LB ( ) can be obtaned wth a few tme by computng the soluton on UB ( ). We also denote the optmal obectve value by SOL( ). The numercal algorthm s gven for dervng optmal cycle tme and schedule when the tme ntervals between successve recepts are gven. Step : The upper bound on, denoted by, s computed by (). Let X {,..., }. max max Step : For all Step 3: For X, derve LP( ) and set LB( ) by ths value. X whch has mnmal LB( ), compute UB ( ), UB ( ) and LB ( ). Set the smaller upper bound and the greater lower bound as UB( ) and LB( ), respectvely. If there s ˆ such UB( ) LB( ˆ ) then ˆ s removed from X, and f for some UB( ) LB( ), then s excluded from X. Then for X whch has the second mnmal LB( ) the bounds are computed n the smlar way, and contnue the procedure untl for all elements n X bounds are computed. Step 4: For X whch has mnmal UB( ) derve SOL( ) by the orgnal problem wth the constrant that obectve functon s greater than or equal to LB( ) and less than or equal to UB( ). If SOL( ) LB( ˆ ) for some ˆ, then ˆ s excluded from X. Then for X whch has second mnmal UB( ), SOL( ) s computed n the smlar way, and contnue the procedure untl SOL( ) are computed.for all elements n X Step 5: Output SOL( ) whch attans mnmum among X. 5. Numercal Experments 5. Computaton Tmes of the Algorthm The numercal experments are conducted n the case where a ob shop has 5 machnes and 5 to 9 tems. The tme nterval between successve recepts are set by () n secton 3.4. In Table, the results of the numbers of solvng problems and computaton tmes are shown. In ths example, 0 experments have been conducted for each number of tems on PC wth Core Quad CPU (.83GHz), and the averages of ther results are gven. For ndvdual combnatoral problems appearng n the algorthm software Xpress 7 s used. From ths table, the numbers on dervng bounds on the obectve functons or SOL( ) are much smaller than the number of dervng LP( ). The proposed method n secton 4 decreases computaton tmes for dervng optmal cycle tmes and ob shop schedules. tems solvng LP( ) Computaton Tmes Table : Computaton tmes dervng UB ( ), UB ( ), SOL( ) LB ( ) Computaton tmes solvng Computaton tmes Total Computaton Tmes 5 73.95 0.35.05 0..70 0.47.07 6 63.80 0.36.60 0.50.70 0.90.76 7 54.0 0.37.60.0.0 3.96 5.35 8 4.35 0.35.45 4.46.05 05.7 9.98 9 3.50 0.33 3.60 936.06.70 759.5 3695.90 5. Comparson on Recept Intervals Optmal schedules and cycle tme n a ob shop are calculated for several combnatons on the tme ntervals between successve recepts (by changng ' ). Here t s assumed that H =40, and ' =0, 0, 30 or 40. Ths means that the plannng horzon s 40 hours (0 days) and the numbers of recepts are once, twce, three tmes or four tmes n one day. The results are gven n the case where there are 5 machnes, 5 tems and 3 assembly factores. The same performance s also evaluated n the case that ther ntervals are computed by EOQ (n ths case, the correspondng s (7,, 7)). ' 560
From the table, even among the optmal ntervals restrcted to combnatons of 4 cases ( ' =0, 0, 30 or 40), optmal ntervals (0, 0, 0) gve mnmal total cost n a supply chan. Note that n the case (30, 0, 0), whch s the combnaton of the nearest ntervals to the EOQ ntervals (7,, 7), the total cost n assembly factores s smaller than that under (0, 0, 0), but the total cost n the part factory s much greater. The combnaton (0, 0, 0) mnmzes the total cost n a part factory, but n ths case the total cost n assembly factores s much greater. Thus, by settng approprately the recept ntervals to the assembly factores and cycle tme n a part factory, the total cost n a supply chan s mnmzed. Table : Comparson among multple combnatons on recept ntervals ' Total costs n assembly factores Optmal Total cost n the part factores Total costs (7,,7) 73.8 7 379..0 (0,0,0) 957.0 0 787.4 744.4 (0,0,0) 747.6 0 98.4 666.0 (30,0,0) 74.7 0 07.8 85.5 6. Concluson In ths study a producton and nventory system consstng of a ob shop n a part factory and multple assembly factores are developed. The cyclc schedule s repeatedly appled. From the nventory of processed parts, assemblers tae multple types of fnshed products to ther own factores n ther own fxed ntervals. An effectve procedure s developed for dervng optmal schedules and cycle tme mnmzng the nventory and setup costs of the part factory when the recept ntervals are gven. Then the optmal schedules, cycle tme and recept ntervals of assembly factores are derved mnmzng the total average cost n a supply chan. Numercal examples show the effectveness of the procedure and the superorty of derved supply-chan optmal schedule, cycle tme and recept ntervals to the ndvdual optmal schedules. A numercal algorthm for dervng near-optmal schedules and cycle tmes n practcal scale problems should be developed and t s left for future research. Acnowledgement Ths research s supported by JSPS Grand-n-Ad for Scentfc Research (C) 5045. References. Ouenncche. J. and Boctor, F., 998. Sequencng, Lot Szng and Schedulng of Several Products n Job Shops: the Common Cycle Case Approach, Internatonal Journal of Producton Research, 36(4), 5-40.. Matsushta, S. and Naade, K., 000. A Cost Mnmzaton Problem n Job Shop wth a Common Cycle, Journal of the Japan Socety of Logstcs Systems, (), 63-74 (n Japanese). 3. Naade, K., Ishhara, K. and Matsushta, S., 003. ``Job Shop Schedulng wth a Common Cycle and an Average Cost Crteron" Proceedngs of the 7th Internatonal Conference on Producton Research, August 3-6, Blacsburg, Vrgna USA, R-(00). 4. Torab, S. A., Ghom, S.M.T.F. and Karm B, 006. A Hybrd Genetc Algorthm for the Fnte Horzon Economc Lot and Delvery Schedulng n Supply Chans, European Journal of Operatonal Research, 73(), 73-89. 5. Ouyang, H. and Zhu, X., 008. An Economc Lot Schedulng Problem for Manufacturng and Remanufacturng, Proceedngs of IEEE Conference on Cybernetcs and Intellgent Systems, Chengdu, 7-75. 6. Chatfeld, D.C., 007. The Economc Lot Schedulng Problem: A Pure Genetc Search Approach, Computers and Operatons Research, 34(0), 865-88. 7. Clausen, J. and Ju, S. 006. A Hybrd Algorthm for Solvng the Economc Lot and Delvery Schedulng Problem n the Common Cycle Case, European Journal of Operatonal Research, 75(), 4-50. 56