Minimizing the Cost of Lean Production Control Transition

Transcription

1 Proceedings of he 27 Indusrial Engineering Research Conference G. Bayraksan, W. Lin, Y. Son, and R. Wysk, eds. Minimizing he Cos of Lean Producion Conrol Transiion Sean M. Gahagan, Jeffrey W. Herrmann Deparmen of Mechanical Engineering Universiy of Maryland, College Park, MD 2742, USA Absrac Firms ha implemen lean principles commonly adop pull producion conrol echniques (especially kanbans). Changing a manufacuring sysem from push o pull producion conrol, while ulimaely beneficial, can be disrupive. This paper sudies he ransiion process in a single-sage manufacuring sysem. I describes he hree evens ha characerize his ransiion and he associaed coss. Differen combinaions of cos miigaion echniques lead o hree differen scenarios, which require hree differen models. We compare hese models and presen ools o find an opimal ransiion policy. We derive some general lessons abou he condiions ha favor each miigaion echnique. Keywords Lean, Producion Conrol, Opimizaion 1 Inroducion Firms ha implemen lean principles commonly adop pull producion conrol echniques (especially kanbans). The fundamenal elemen of lean producion conrol ransiion is he conversion of a single processing sage from push o pull producion conrol. During his ransiion, he sage experiences a surge of orders as i aemps o build a kanban invenory while i is also processing regular cusomer orders. The surge may overwhelm he capaciy of he saion, resuling in a backlog ha would adversely affec cusomer lead imes. To preven his, we propose wo emporary miigaing echniques adding more resources or deferring some of he cusomer orders. The purpose of his paper is wo-fold; o deermine he bes way o model his ransiion process and, using his model, o explore opimal miigaion policies o reduce he cos of lean ransiion. 1.1 Background The lean lieraure unanimously advocaes he ransiion of push producion conrol o pull where possible [2, 4-7], bu very lile is wrien abou he mechanics of he ransiion process or he behavior of sysems in lean ransiion. Hopp and Spearman [2] discuss he mechanics of push and pull producion conrol, heir role in lean ransiion and even skech ou a lean ransiion scheme. However, hey limi heir analysis o he before and afer seady sae condiions. In fac, o our knowledge, here is no discussion of he ransien effecs of lean producion conrol ransiion in he lieraure. Queueing lieraure hough does discuss he effecs of non-saionary arrival raes on sysem performance. Hall [1] discusses ways o model sysems wih non-saionary arrival raes. He explains ha he size of changes in arrival rae relaive o capaciy dicae which modeling echnique o use. For sysems in which he arrival rae is always much lower han capaciy, seady sae approximaions can be used. In sysems where he arrival rae is much larger han capaciy, a fluid flow approximaion is more appropriae. For sysems where he arrival rae is close o capaciy, he suggess ha only simulaion can accuraely model sysem performance. 1.2 Problem Seup In his work we sudy a single sage of a producion sysem as i ransiions from push producion conrol o pull. We characerize his ransiion in erms of hree evens. The firs even is he arrival of he firs kanban card, which occurs a ime =. We assume ha he number of kanban cards, n k, is predeermined. Furher, we assume ha he arrival rae of he cards, λ k, is consan. The las kanban card arrives a = 1 = (n k - 1) / λ k. We assume 1 >. The sage coninues o receive cusomer orders a a rae of λ a. The oal arrival rae for < < 1 is λ k λ a. The sage has r resources o process he cards and orders. Each resource processes orders a a rae of λ r. The oal processing rae of he sage is rλ r. If λ k λ a > rλ r hen a backlog of orders is accumulaed during he arrival of he cards. The

2 final ransiion even is he compleion of processing of he las kanban card a = 2. If λ k λ a << rλ r hen 2 could be as lile as 1 1/λ r. However, if λ k λ a > rλ r and a backlog is creaed, 2 could be as much as n b / (rλ r - λ a ). When he las kanban card has been processed, he sage can be convered o pull producion conrol because cusomer orders can now be saisfied from he now-full downsream buffer. We are ineresed in how he sysem behaves during his ransien phase and how emporary changes o he sysem affec he process. Two ypes of emporary changes are considered in his paper. We say ha r resources may be added o he sysem, increasing he oal processing rae of he sysem o (r r )λ r during he ransiion. We also consider deferring orders a a rae of λ d such ha he oal arrival rae during he ransiion is λ k λ a - λ d. 1.3 Conrol Variables We can explicily idenify he decision variables for managing he ransiion process. Since he number of kanban cards o be inroduced is known a priori, only he rae of heir inroducion λ k is an inpu, where < λ k <. As an alernaive, one could specify he lengh of he ransiion, 1 -, where λ k = n k / ( 1 - ). The miigaion facors are also inpus. The number of addiional resources r is an inpu, where r is an ineger on r <. The deferral rae of cusomer orders λ d is also an inpu, where λ d λ a. 1.4 Transiion Cases From his, we can idenify hree disinc condiions under which ransiion akes place based on he relaionship beween arrival rae and processing rae. This relaionship direcly affecs how much of he ransiion ime occurs before and afer 1. CASE 1: Arrival Rae is Lower han Processing Rae: λ k λ a - λ d << (r r )λ r In his case, he surge of orders does no compleely consume he available capaciy and no backlog is creaed during ransiion. The kanban cards are processed as hey arrive and he ransiion is nearly complee a 1, minimizing 2. CASE 2: Arrival Rae is Higher han Processing Rae: λ k λ a - λ d >> (r r )λ r In his case, he surge of orders compleely consumes he available capaciy and a backlog of cards and orders is creaed during ransiion. Here 1 may be minimized as more, possibly all, of he kanban cards are processed afer he final arrival. CASE 3: Arrival Rae is Equal o Processing Rae: λ k λ a - λ d (r r )λ r In his case, he surge of orders nearly equals he available capaciy. A backlog of orders and cards may be creaed. When arrivals and capaciy are nearly in balance, he formaion of a backlog becomes more dependen on variaion in processing imes. For his condiion, 1 may be a any poin beween and 2. An ineresing feaure of his problem is he fac ha here are decision variables in his idenificaion scheme, meaning ha he naure of he problem iself is a funcion of he inpus. 1.5 Transiion Objecive Holding orders in invenory or backlog, adding resources and deferring orders all have a cos. Our goal is o find he opimal inpu values o minimize oal cos. We define oal cos C o as C o = C d C r C i C b (1) where C d is he cos of orders deferred, C r is he cos of addiional resources, C i is he cos of holding invenory (kanban orders in downsream queue) and C b is he cos of holding backlog (orders in queue during ransiion). We can furher define hese cos componens in erms of he sysem variables we have already defined. C d = λ d ( 2 ) c d (2) C r = r ( 2 ) c r (3) C i = (n k /2) ( 2 ) c i (4) 2 () C b = c b ( Q d ) (5)

3 where c d is he cos per uni of orders deferred, c r is he cos rae per uni of addiional resources, c i is he cos rae per order held in invenory, c b is he cos rae per order held in backlog and Q() is he number of orders held in backlog a ime =. We noe ha, as he deferral rae λ d increases, he deferral cos increases, bu he oher coss decrease due o he smaller backlog and shorer ransiion ime. Similarly, as r increases, he resource cos increases, bu he oher coss decrease due o he smaller backlog and shorer ransiion ime. Increasing he kanban inroducion rae λ k should reduce he ransiion ime unless i is oo large, in which case excessive server increases he backlog. 2 Modeling Approach We consider hree differen echniques o model his process: seady sae approximaion, deerminisic fluid flow approximaion and discree even simulaion. Hall [1] proposed ha hese hree modeling echniques are he bes candidaes for analyzing sysems wih non-saionary arrival raes. Hall s caegorizaion of hese sysems corresponds wih our case definiions discussed above. The following secions describe he models. 2.1 CASE 1 - Seady Sae Model, Arrivals << Capaciy Firs, we consider he case where he surge of orders and kanban cards is much smaller han he capaciy of he resource. Tha is, where λ k λ a - λ d < (r r )λ r. In his case we use a sochasic seady sae (SSS) approximaion. For <, we assume he sysem is in a seady sae. For < < 2 we assume ha he sysem swiches o a second seady sae. For > 2, he sysem revers o a hird seady sae similar o he firs. Since he surge never exceeds capaciy, here is no backlog o deal wih a he end of he ransiion and 1 = 2. We choose o approximae he sysem as a G/G/m server. Hopp and Spearman [2] provide an approximaion for he cycle ime, CT q (G/G/m). Using his, and assuming he coefficiens of variaion for he inerarrival and processing ime are boh 1, we find 2 : 2( r r 1) 1 λk λa ( ) ( ) ( ) λr r r = 1 CTq G / G / m 1/ λr = 1 (6) λ k λa λd λr λr ( r r ) 1 ( ) λr r r We find he backlog by subsiuing he cycle ime approximaion ino Lile s Law: 2 2( r r 1) 1 λk λa λr ( r r ) () d CTq ( G G m)( λa λk λd ) 1 Q = / / = ( λa λk ) λk λa λ d λ (7) r ( r r ) 1 λr ( r r ) Using hese equaions, i was possible o creae a very simple, very fas spreadshee model o evaluae he cos of Case 1 ransiions. 2.2 CASE 2 - Fluid Flow Model, Arrivals >> Capaciy Nex, we consider he case where he surge of orders and kanban cards is much greaer han he capaciy of he resource. Tha is, where λ k λ a - λ d > (r r )λ r. We choose o model his ransiion wih a deerminisic model called a fluid approximaion model. A deerminisic fluid flow (DFF) approximaion model is one in which he flow of arrivals and deparures are modeled as a coninuous variables - flow raes. Fluid approximaion models are easily illusraed. Figure 1 shows a fluid approximaion model of our sysem in ransiion.

4 Proceedings of he 27 Indusrial Engineering Research Conference G. Bayraksan, W. Lin, Y. Son, and R. Wysk, eds. Trigger for Transiion Modificaions Change Variables Aler Resource 1 True Order Arrivals Order Deferral Processing Discard Fals e Kanban Card Arrivals Figure 2: Simulaion Model in Arena Figure 1: Deerminisic fluid flow (DFF) approximaion model In he figure, he blue line represens he cumulaive flow of order and card arrivals during ransiion. The red line shows he flow of compleed orders from he sysem. The slopes of hese lines are equivalen o he flow raes of arrivals and deparures from he sysem. Evaluaion of his model is sraighforward geomery. We address he ransiion in wo phases; < 1 and 1 < 2. In he former, he backlog is building, while in he laer i is being consumed. We firs solve o find he backlog, Q( 1 ). Q ( ) 1 ( n 1) k = a k d r r λk ( λ λ λ λ ( r )) (8) We can hen use his resul o find 2 : ( n 1) k ( λa λk λr ( r r )) λk 2 = 1 (9) λ λ λ a d r ( r r ) Using hese equaions, we buil a second spreadshee model o calculae he cos of Case 2 ransiions. 2.3 CASE 3 - Simulaion Model, Arrivals Capaciy Finally, we address he case where he arrival rae is approximaely equal o capaciy, or where λ k λ a - λ d (r r )λ r. In his condiion, Hall [1] recommends he use of simulaion o model he sysem. Simulaion is a very powerful, bu compuaionally expensive modeling echnique. We buil a simulaion model of our sysem using Arena [3]. Figure 2 shows our simple sysem as modeled. The simulaion model iself is fairly sraighforward, bu collecing good performance daa from a poenially shor, ransien period requires careful seup of he replicaion parameers. Our model uses a warm-up period of 1, imes he order processing ime, which i repeas for each replicaion. I mainains a coun of how many kanban cards are in he sysem and i sops he replicaion when he las card exis. We use 1 replicaions. To evaluae performance we used he defaul repors which provide saisics on number of orders in backlog and replicaion lengh. 3 Comparison In order o compare he models, we used hem o esimae he performance of a sysem over a range of ransiion arrival raes cenered abou he capaciy of he sysem. For his comparison, he sysem was configured as follows:

5 Table 1: Comparison Seup Inpu Variables Inpu Value Mean order inerarrival ime, iniial.13 (75 orders per uni ime) Mean processing ime per order.1 (1 orders per uni ime per resource) Number of resources, iniial 1 Number of kanban cards o be inroduced 24 Number of resources, during ransiion 2 Wih hese inpus fixed, we varied he kanban inroducion rae λ k o manipulae he raio of ransiion load o capaciy, λ k λ a - λ d / (r r )λ r, from.8 o 1.2. We looked a he average number of orders in backlog indicaed by each model. Figure 3 shows a plo of each model oupu. 3 SSS Average Number of Orders in Backlog DFF SIM Arrival Rae / Processing Rae Figure 3: Average Number of Orders in Backlog versus Arrival Rae / Capaciy We expeced he simulaion model o mos closely approximae he behavior of he sysem, which we prediced would be a smooh, monoonically increasing curve as he arrival rae slowly overcame he processing capaciy of he sysem. As expeced, he SSS approximaion followed he simulaion resuls iniially, bu increased asympoically o infiniy as he arrivals approached capaciy from he lef. The DFF repored backlogs well below he simulaion model bu caugh up o and followed closely wih i a higher arrival raes. If we assume ha he simulaion model is he closes approximaion o he behavior of he sysem hen he comparison is jus as Hall prediced wih he SSS approximaion mos useful when he arrival rae is well below capaciy (Case 1), he DFF approximaion mos useful a raes well above capaciy (Case 2) and he simulaion model bes a raes near capaciy (Case 3). One marked difference beween he models hough is no refleced in he oupu - he processing ime. The resuls of he SSS and DFF approximaions were available from heir respecive spreadshee models in a spli second. The simulaion model ook nearly 2 minues o process each daa poin. The flexibiliy of he simulaion model has a high compuaional price ha affecs is usefulness for opimizaion of he ransiion. 4 Opimizaion To ease he compuaional burden of an exclusively simulaion-based opimizaion of he ransiion parameers, we propose a muli-model opimizaion scheme ha uilizes all hree models. Since he SSS and DFF models require so much less compuing power han he simulaion model, we can use hem o reduce he size of he domain space before using he simulaion model. For a given scenario, we sugges finding he minimum cos ransiion parameers using he SSS and DFF models, consraining he arrival rae o capaciy raio o, say <.8 and >1.2, respecively. Spreadshee applicaions like Excel have powerful solver uiliies ha can be used o find he opimal parameers very quickly. Then we can use he simulaion model, which also comes packaged wih an opimizaion engine, o explore he remaining domain space for superior soluions. The opimal soluion can hen be idenified as he lowes resul from he hree separae opimizaions. I should be noed ha hese arrival rae o capaciy raio hresholds are

6 sricly guesses, based largely on he resuls of he comparison case sudy above. Proper hreshold selecion for such a scheme is a subjec for fuure sudy. 5 General Lessons The objecive of our modeling is o undersand ransiion cos. Using he comparison case sudy above, we colleced cos daa for each rial wih he cos raes all se o 1. Figure 4, demonsraes he effec of each cos componen on overall ransiion cos wih respec o arrival rae / capaciy. I also shows how ransiion ime is affeced Transiion Cos 25 2 Backlog Invenory Transiion Time 15 Resources Overall 1 1 Time Arrrival Rae / Processing Rae Figure 4: Transiion Cos and Time versus Arrival Rae / Capaciy From his, we can see ha in an opimum ransiion, he arrival rae is exacly balanced wih he processing rae of he sysem. If he rae is oo low, he processing resources are underuilized. Too high, and he backlog cos sars o cach up wih invenory cos. We see here ha he effec of backlog and resource coss are small compared o he conribuion of invenory, bu real-world cos raes could be quie differen and drasically change hese relaionships. In general, balancing capaciy wih arrival rae is he answer o a low cos lean ransiion. 6 Conclusion Conversion from push producion conrol o pull is an imporan, bu poorly undersood par of lean manufacuring. In order o undersand he cos of ransiion, we developed a cos model for ransiion and described hree disinc ypes of ransiion. We developed hree models, a sochasic seady sae approximaion, a deerminisic fluid flow approximaion and a simulaion model, ha all approximae he behavior of a single sage undergoing lean producion conrol ransiion. We illusraed he differences in he models by applying hem o a es case and we proposed a muli-model opimizaion approach ha leverages he srenghs of each model o quickly find he opimal ransiion parameers. We used our case sudy o demonsrae ha an opimal ransiion balances he arrival rae wih capaciy. In he fuure we will expand our models o include muliple sages undergoing lean ransiion and endeavor o beer undersand how o opimize he many more decision variables such a model would presen. References 1. Hall, R., 1991, Queueing Mehods for Services and Manufacuring, Prenice Hall, New Jersey. 2. Hopp, W. J., and M. L. Spearman, 1996, Facory Physics, Irwin/McGraw-Hill, Boson, Massachuses. 3. Kelon, W., R. Sadowski, and D. Surrock, 24, Simulaion wih Arena, 3rd ediion, McGraw-Hill, Boson, Massachuses. 4. Liker, J., 24, The Toyoa Way, McGraw-Hill, New York. 5. Shingo, S., 1989, A Sudy of he Toyoa Producion Sysem from an Indusrial Engineering Viewpoin, Produciviy Press, Cambridge, Massachuses. 6. Slack, N. (Ed.) The Blackwell Encyclopedia Dicionary of Operaions Managemen, Blackwell Publishers Ld., Oxford, Unied Kingdom. 7. Womack, J., Jones, D. and Roos, D., The Machine Tha Changed he World: The Sory of Lean Producion, 1991, Harper Perenial, New York.