A queung theory approach to compare ob shop and lean work cell consderng transportatons ohammad oshref-javad Department of Industral and Systems Engneerng Auburn Unversty Auburn, Alabama- 36849, USA Abstract Whle parts transportaton s one of the maor non-value added actons n factores, t s usually gnored by managers. In ths paper, a queung theory approach s appled to compare ob shop and lean workstatons. In addton to transportaton tmes, batchng, processng, batch watng and delay tmes are also consdered. The throughput tme (TPT) and work n process (WIP) are evaluated for lean and batch producng workstatons. The results show that the transportaton tme should be consdered to obtan the real throughput tme and WIP of the system. Keywords Lean producton, Queung odel, Transportaton, Batchng, Job shop. Introducton In today s compettve busness envronment, companes should desgn effcent and productve manufacturng systems to survve. One of the mportant ssues n desgnng effcent manufacturng systems s reducng varous types of wastes, such as materals, processng, moton, etc. Transportaton s one of the maor types of waste n factory. A large porton of transportaton s parts transportatons; parts (work-n-process nventores) are moved from one machne to another machne to be processed. Thus, part transpratons, as a bg waste n the factory, should be reduced to make the manufacturng system more effectve. Job shop s a type of manufacturng approach whch was bascally ntroduced around 840s []. achnes are placed n functonal layout; smlar machnes are placed n one area. Parts are moved from one machne to another machne n batches. On the other hand, the lean approach was ntroduced n 960s. Each part s moved to the next machne after t was processed on the current machne and nspected by the operator. Ths s one of the desgn rules of lean workstatons: move one-check one- move one on (O-CO-OO) []. ultfunctonal operator(s) are assgned to cells to perform operatons n one-pece-flow and very short transportaton tmes. Hopp and Spearman [] were among the frst authors worked on batch producton system usng queung theory. Curry and Feldman [3] extended models to consder setup tmes and batch move model. In ther models, authors consdered transportaton tme neglgble. That s, batches are moved to the next workstaton nstantaneously. Svasamy and Pukazhens [4] developed the formulas for dscrete tme bulk servce queue wth accessble batch. The authors assumed ndvdual arrval of batches wth no transpraton tme. Concernng the lterature, authors have gnored transportaton n batch producton system. One of the reasons may support batchng s that less transportaton s performed snce movements are done for each batch nstead of each ndvdual part. In ths paper we use queung theory to compare batch and lean workstatons consderng part transportaton tmes. Throughput tme, work-n-process, and transportaton tme n batch producton system and lean work cell wll be compared. The rest of the paper s organzed as follows. Secton represents problem descrpton and assumptons. Secton 3 provdes mathematcal formulaton of the problem. An llustratve example s gven n secton 4 and conclusons are dscussed n secton 5.. Problem descrpton and assumptons Ths paper compares two workstatons, batch n ob shop and lean. In ob shop, parts are processed on machnes whch are located n the factory based on functonal layouts. Parts are accumulated n batches. Ready batches are
. oshref-javad moved to other sectons by transportaton vehcles, say AGV or forklft. The problem wll be formulated under the followng assumptons: We formulate the problem for sngle product workstaton. It s assumed that the forklft (AGV) follows a route n functonal layout. When the forklft arrves at a workstaton, f the batch s ready, the forklft carres t to the next workstaton; otherwse, contnues ts path. The forklft can carry one batch at a tme. For smplfcaton, the pckup tme and drop tme of the batches by the forklft s assumed neglgble. On the other hand, lean work-cells are usually U-shaped controlled by mult-functonal operators and part movements tmes are small. Transportaton tme n the lean work-cell s the sum of operator s walk tmes. 3. Problem formulaton The followng notaton has been used to formulate the problem:,,k Indces of workstatons,,, Number of workstatons n ob shop E[T a (B)] Average nter-arrval tmes of batches to workstaton C ( B ) Squared coeffcent of varaton of batch nter-arrval tmes at workstaton k Batch sze n workstaton E[S ] Average servce tme of workstaton C () I Squared coeffcent of varaton of ndvdual servce tme of workstaton s C e Squared coeffcent of varaton of batch nter-departures tme from transportaton workstaton C ar ( B ) Squared coeffcent of varaton of batch nter-arrval tme to transportaton workstaton E[R ] Average transportaton tme to workstaton C r u Squared coeffcent of varaton of ndvdual transportaton tme to workstaton Utlzaton of workstaton µ Utlzaton of transportaton workstaton r Relatve arrval rate to transportaton workstaton TPT B Throughput tme n ob shop TPT L Throughput tme n lean workstaton DTR Transportaton delay tme at workstaton BT Batch processng tme and watng at workstaton TR Total transportaton tme and delay to workstaton T achnng tme on machne The goal s to calculate and compare the throughput tme, transportaton tme, and work n process n batch producton system and lean work cell. 3. Throughput tme n ob shop Fgure shows the throughput for a machne n the batchng system. Total tme ncludes batch formng tme, transportaton watng tme, transportaton tme, batch watng tme and processng watng tme. Curry and Feldman [3] proposed the followng formula for throughput tme of each ndvdual part n batch producton workstaton: C k + C ( I) u ( k ) ( k ) BT ( )( ) ET [ ( S)] ET [ ( S)] ET [ ( S)] ET [ ( S)] e s = + + + u where u a k E[ S] =. C e s the squared coeffcent of varaton of batch nter-departure tmes from the batch- ET [ ] move transportaton workstaton to workstaton. The batch s transported to workstaton. Curry and Feldman [3] consdered batch transpraton nstantaneously. After transportaton, batches wat n the queue to start processng at ()
. oshref-javad the sngle unt server workstaton (Batch watng). Snce the server s a sngle unt server, parts are processed ndvdually. So other parts n the batch have to wat. The next term n the formula adds the processng tme on machne to total throughput tme. Fnally, the last term s the average tme for batch formng; Processed parts are accumulated n the batch. Fgure : Batchng process and transportaton n ob shop Transportaton tme conssts of delay tme due to unavalablty of the forklft (AGV) and batch transportaton by the forklft. Delay tme depends on the poston of the forklft when the batch s ready for transportaton at the prevous workstaton and the average transportaton tmes between each two workstatons. If we assume the current workstaton s workstaton, thus: ER [ ] DTR = ( ) ( [ k] [ ]) E R + E R = k= () ER [ ] k = k calculates the transportaton delay tme. The transportaton tme s unformly dstrbuted along the way. Thus, total transportaton tme s: TR = DTR + E[ R ] (3) Therefore, the total throughput tme at workstaton s: TPT = TR + BT (4) Fnally, total throughput tme of the part followng workstatons stpt B = TPT. Note that the sum of travels between each par of workstatons must be less than batch producton tme n any of the workstaton so that the system can reach steady state ( k = = ER [ ] K. ES [ ], for=,..., ); otherwse, the system k wll not get to steady state. To determne the squared coeffcent of varaton of nter-departures tme from the forklft (.e. nter-arrval tmes to each workstaton), the transportaton n each workstaton s consdered as a batch move model (.e. oven). Thus, we have to types of workstatons: one s the machne workstaton and the other one s the transportaton workstaton (forklft). Therefore, the followng formula s used to calculate the SCV of nter-departure tme from the forklft [5]: C ( μ ) C + μ C (5) e ar r 3
. oshref-javad where C ar s the squared coeffcent of varaton of nter-arrval tmes to transportaton workstaton n terms of batches. Snce the forklft s followng the route and repeats t, so C ar s obtaned usng recursve equatons, one equaton for each transportaton workstaton. Fgure shows the transportaton system whch s a recurrent system composed of four workstatons. The system shows that the forklft repeats ths route. Four recursve equatons are used to determne the SCV of nter-departure tme from forklft. Furthermore, μ s determned wth respect to the transportaton tme n each staton. We consder the transportaton system as a CONWIP system n whch WIP s set to. The utlzaton of each workstaton n CONWIP system s [3]: μ ( w) = wre. [ R] = rtpt. ( w). (6) where w s the number of nventory n CONWIP system and TPT s the throughput tme n transportaton workstaton. Snce w=, TPT = E[R ] accordng to the followng formula [3]: TPT w ER ER TPT w rtpt. ( w ) ( w ). r E[ R] ( Cr ) ( ) = [ ] + [ ]. ( ) + =. (7) The relatve arrval rate n ths system s for =,,. Therefore, μ = ER [ ] ER [ ]. (8) Fgure : Recurrent transpraton system: recursve equatons are used to determne the departng SCV of the supposed transportaton workstatons. 3. Throughput tme n lean workstaton Lean work cell s composed of a number of workstatons located n U-shaped cell. The approach to calculate the cycle tme and throughput tme s dfferent from what we used n ob shop. Throughput tme n a lean work cell s based on cycle tme. Cycle tme s a perod of tme durng whch one part s produced [6]. Cycle tme s the sum of manual tme and walk tme: NCT = ( anual tme + Walk tme) (9) = NCT must be greater than any of the machnng tme (T ) n the cell; one of the desgn rules proposed by [6]. Also, NCT should be slghtly less than takt tme to respond to the customer demands. The number of nventory n the cell (Sotck-on-hand, SOH) equals the sum of workstatons and de-couplers: SOH = ( workstaton + decoupler) (0) = 4
. oshref-javad Therefore, the throughput tme s calculated usng the followng formula: TPTL = NCT SOH () TPT L represents the perod of tme when parts enters to the cell to the last operatons n the cell when the part s produced. Ths tme ncludes processng tmes, load/unload tmes, nspecton and transportaton n the cell. (a) (b) Fgure 3: anufacturng systems: (a) Lean workstaton, D-shaped staton represents de-coupler. (b) Job shop. 4. An Illustratve Example We show the comparson usng an example. In ths example a part s produced through four workstatons. The process starts n saw workstaton, and then goes to mllng, drllng and lathe workstatons, respectvely. Fgure 3 shows two manufacturng systems: (a) Lean work cell and (b) Job shop. The lean work-cell s staffed by one operator. We assume a typcal lean work-cell. Walk tmes and manual tmes are assumed to be 5 seconds and 5 seconds respectvely. oreover, NCT s greater than T. Thus, no manpulaton of machnes s needed to decrease the processng tmes. In ob shop system, a forklft s followng the route from workstaton to 4 and repeats ths route. The system s composed of four seral workstatons. Table gves the values of nput parameters. E[R ] have been calculated based on the dstance between workstatons and the movement velocty of the forklft. Note that there s no transportaton n workstaton snce parts are arrved n batches at workstaton by another transportaton system. Table : Input parameters and values for the llustratve problem. Parameter Value 4 E[T a (B)] 66.5m C ( B ) 0.004 {E(S ), E(S ), E(S 3 ), E(S 4 )} {50s,60s,55s,50s} { Cs() I, Cs() I, Cs3() I, Cs4() I } {0.,0.3,0.,0.} {E(R ), E(R ), E(R 3 ), E(R 4 )} {0s, 60s, 80s, 60s} { r r r3 r4 C, C, C, C } 0. K 50 The throughput rates of both systems are the same. Table gves the throughput tme of an ndvdual part n ob shop and lean work cell. There s a huge dfferent between throughput tme n two systems; 505.9 seconds n the ob shop and 400 seconds n the lean work cell. Each ndvdual part spends averagely 930 seconds n the system for transportaton, whch s the sum of transportaton tme and delay tme of the vehcle. Work n process n the lean work cell s the sum of the number of workstatons and de-couplers. Usng Lttle s law, we can obtan WIP B n the ob shop system. In addton, the transportaton tme n the lean work cell s much less than n the ob shop system due to short dstances between machnes n the work cell. 5
Table : Comparson of throughput tme, transportaton tme and work-n-process n ob shop and lean work cell. Throughput tme Transportatonn tme Work n process. oshref-javad Lean 400 s 0 s 5 Job shop 505.9 s 930 s 88.6 Fgure 4 llustrates the throughput tme of an ndvdual part n ob shop for dfferent batch szes. The total transportaton tme s the same for all batch szes. As the batch sze ncreases, the average batch formng and process watng tme are ncreased as well as batch watng tme. The reason for larger batch sze s to spread the transportaton, ncludng transportaton and delay tme, across the parts n the batch. Although average transportaton tme decreases as the batch sze s reduced, but the throughput tme s ncreasng due to sgnfcant ncrease n the batch formng tme and the process watng tme (un-batchng tme). 6000 4000 Throughput Tme 000 0000 8000 6000 4000 000 Processng Wat for process Batch watng Transportaton Formng 0 0 0 30 40 Batch sze 50 Fgure 4: Throughput tme n ob shop for dfferent batch szes 5. Conclusons One of the mportant ssues n manufacturng systems s reducng wastes. Transportaton s one of the maor wastes n manufacturng systems. In ths paper, we compared ob shop and lean workstatons consderng transportaton tmes. We utlzed queung models to calculate throughput tme n ob shop system. A stochastc transportaton system was consdered n whch a forklft (or AGV) follows a route from one workstaton to another workstaton. Also the throughput tme and work n process as performance measures were calculated for lean work cell. An llustratve example was used to show the comparson of two systems. The results show that the transportatonn tme s low n lean work cell due to short dstances between workstatons n the cell. On the other hand, transportaton n ob shop s composed of transportatonn tme and delay tme for the vehcle. Also, when the batch sze s decreased, the transportaton s almost the same, but the watng tmes n batch formng and un-batchng are reduced consderably. For future works, we recommend thatt someone consder the pckup and drop tmes of parts by the forklft. By relaxng ths smplfcaton assumpton n ths paper, pckup and drop tme affect the transportaton delay tme. Followng works are also suggested for future research: Consderng the capacty of forklft greater than one. The extenson of the model for multple products. Consderng setup tmes. Smulaton study of the system and comparng wth the queung model. 6
. oshref-javad Extenson of the system to nclude transportaton of parts n kanban lnks. Acknowledgements The author expresses hs sncere thanks to Dr. J.T. Black and Dr. K.R. Gue for ther constructve comments, whch have mmensely helped to brng ths paper to the present form. References. Black JT, 007, Desgn rules for mplementng the Toyota Producton System, Internatonal Journal of Producton Research, 45(6), 3639-3664.. Hopp, W.J., and Spearman,.L., 996, Factory Physcs: Foundatons of anufacturng anagement, Irwn, Chcago. 3. Curry, G. L., and Feldman, R.., 009, anufacturng Systems odelng and Analyss, nd Edton, Sprnger Hedelberg Dordrecht London New York. 4. Svasamy, R., and Pukazhenth, N., 009, A Dscrete tme bulk servce queue wth accessble batch: Geo/ NB (L,K) /), OPSEARCH 46(3),3 334. 5. Curry, G.L., and Deuermeyer B.L., 00, Renewal approxmatons of the departure processesof batch systems, IIE Transactons, 34, 95 04. 6. Black, JT., and Hunter, S.L., 003, Lean manufacturng Systems and Cell Desgn, SE: Dearborn, I. 7