SIMULATION RESULTS ON BUFFER ALLOCATION IN A CONTINUOUS FLOW TRANSFER LINE WITH THREE UNRELIABLE MACHINES

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1 Advances n Producton Engneerng & Management 6 (2011) 1, ISSN Scentfc paper SIMULATION RESULTS ON BUFFER ALLOCATION IN A CONTINUOUS FLOW TRANSFER LINE WITH THREE UNRELIABLE MACHINES Sörensen, K.* & Janssens, G.K.** *Faculty of Appled Economcs, Unversty of Antwerp Prnsstraat 13, B 2000 Antwerp, Belgum **Transportaton Research Insttute (IMOB), Hasselt Unversty Wetenschapspark 5, B-3590 Depenbeek, Belgum E-mal: kenneth.sorensen@ua.ac.be, gerrt.janssens@uhasselt.be Abstract: Engneers desgnng a contnuous flow producton lne consstng of machnes and buffers n seres, have to determne the optmal szes of the ntermedate buffers. When the machnes operatng on the product are unrelable, ths dmensonng decson becomes even more dffcult. Ths paper ams to provde some nsght nto the optmal szes of ntermedate buffers n a contnuous flow transfer lne wth three machnes and two buffers. To ths end, an extensve seres of smulaton experments are performed, and noted the performance (measured as the avalablty,.e., the fracton of tme that the system as a whole s producng output) under a large varety of dfferent nput condtons. The smulaton model for ths purpose s based on Petr-nets and has been extensvely valdated pror to ths research. Usng these smulaton results, we try to fnd rules to determne where to allocate whch amount of buffer space n order to obtan a suffcently hgh avalablty of the system. Dfferent scenaros are smulated: all machnes have equal relablty, one machne has sgnfcantly hgher relablty and one machne has sgnfcantly lower relablty. For each scenaro, a large number of settngs s tested. A careful study of the avalablty of the system as a functon of the szes of the two buffers reveals some nterestng results. We fnd that the optmal buffer szes depend strongly on the (relatve) relablty of the dfferent machnes and that n general the performance of the entre system can be consderably ncreased by correctly choosng the szes of the two buffers. When all machnes are equally relable, both buffers should be equally large. When the frst or the last machne s less relable, the frst respectvely the second buffer should be used to dampen the mpact of ths unrelable machne. Another fndng s that f the mddle machne s consderably less relable than the other two, the buffers have far less effect on the performance of the overall system. The results reported n ths paper can be used by practtoners n the feld of desgn of contnuous flow transfer lne systems to gan nsght nto the optmal dmensonng of buffers n producton systems. Future research n ths area wll consder smulaton models of larger lnear systems or more complex networks of machnes and buffers. Also, the effects of relaxng one or more of the assumptons made n the smulaton model (e.g., on the dstrbuton of up- and down-tmes of the machnes) should be nvestgated. Key Words: Producton Systems, Smulaton, Buffer Allocaton, Unrelable Machnes 1. INTRODUCTION Flow lne producton systems are mostly assocated wth products whch flow naturally. For example: n a petroleum refnng process, the product and the raw materal have a propensty to flow. Producton systems that output dscrete tems such as cars, domestc applances, etc. do not possess ths natural characterstc unless the tems are suffcently small to be treated as a contnuous flow. Wld [19] states that mass producton usng the flow prncple s 15

2 one of the most mportant achevements n manufacturng technology. But even today transfer lnes reman the most effectve soluton for large volume, contnuous producton of lmted varants. In the automotve ndustry they stll are effcent for the producton of cylnder heads, cylnder blocks, crank cases and crank shafts. The problem of desgnng and mprovng flow lne producton systems has receved a great deal of attenton n the lterature. These producton systems consst of a number of stages (arranged n seres) at whch consecutve operatons are performed. The machnes or equpment are ether perfectly relable or subject to falure. Falure at any stage results n the falure of the entre producton system and consequently the overall producton rate s affected. The understandng of the machne falures s mportant n mprovng the relablty of a transfer lne producton system. Algorthms have been developed to estmate performance characterstcs of flow lnes lke throughput, average buffer contents, and blockng and starvaton probabltes. A frst modellng approach s the open queung network approach, where each server has ts own random processng tme. Whle most approxmaton methods are qute accurate, they are often computatonally complex, due to the number of states ncreasng wth the buffer capactes. In another approach, the goods flow s contnuous and the machnes have producton rates nstead of servce tmes. Ths approach s justfed when cycle tmes are small compared to downtmes and uptmes [5]. In order to mprove the avalablty of the system, two approaches exst. One opton s the utlsaton of stand-by machnes, the other opton s to allocate buffer storage between the producton stages. Stand-by machnes are put nto operaton n case one of the machnes fals. Kubat and Sumta [9] present a dynamc programmng model drected at desgnng a transfer lne wth standby machnes and ntermedate buffers of nfnte capacty. It allocates the places for the stand-by machnes and the buffers n order to maxmze the lne output rate. The use of redundant machnes n an operaton mode called splttng s consdered by Elsayed and Hwang [7]. Each stage n the transfer lne conssts of two machnes wth the same operatve performance, n regular operaton, workng at half producton rates. When ether of the machnes fals, the remanng machne doubles ts producton rate nstantaneously. The other opton to nstall buffer storage between a par of machnes ams to avod the falure of one machne to result n an mmedate falure of the entre flow lne. The decson how to allocate buffers to a producton lne s of great practcal mportance to many ndustral systems lke ol refnng, cannng, beverage producton and many others. The behavour of a system wth multple machnes and buffers s very complex. Industral managers are nterested n performance measures of such a system. Avalablty,.e. the percentage of tme that the lne s producng output, s frequently used as an mportant performance measure. However, closed-form results exst only n very smple cases. In more complex cases ether approxmaton models or smulaton models are used. In the lterature on mult-stage lnes wth fnte ntermedate buffers, De Koster [6dstngushes four classes of models. A frst class deals wth systems n whch the servce tmes are random varables and the products are dscrete. The machnes are not susceptble to falure. A second class assumes determnstc processng tmes, but machnes are unrelable and fal from tme to tme. Products are dscrete. A thrd class deals wth contnuous flow models. Machne speeds are determnstc but machnes may fal. Ths s the class of models ths paper deals wth. Some examples of these models can be found n artcles by varous authors [2, 3, 8, 10 13, 18, 20]. A fourth class deals wth models wth dscrete products, falures of machnes and random processng tmes. Furthermore a dstncton needs to be made nto two types of machnes falures: tmedependent and operaton-dependent falures. In the former type, the falure only depends on the elapsed tme, whch means the machne can fal even f t s not workng on products. The latter type of falure depends on the tme a machne spends workng on a product. In real-lfe manufacturng systems, both types of falures exst. Accordng to Buzacott and Hanfn [2], operaton-dependent falures are more common and represent more than 70 percent. Obtanng analytcal results for a system wth many machnes s consdered to be an mpossble task. Therefore, approxmaton or smulaton models have to be used to 16

3 determne varous performance measures. Repeated aggregaton over producton machnes s a useful technque f the output of the aggregated producton machne s close to the pattern of two buffered producton machnes. Aggregaton over producton machnes and an ntermedate buffer has been used already by Buzacott [1], who appled t to a fxed-cycle three stage lne. Aggregaton technques requre the evaluaton of the output parameters for two-stage lnes. Ths has been realzed by assumng exponentally dstrbuted uptmes and downtmes or by showng how more general two-stage lnes can be approxmated by ths type of exponental lnes [4]. The two-machne one-buffer system contnuous flow lne has been studed n an analytcal way by Malathronas et al. [10]. Van Oudheusden and Janssens [17] formulate an approxmatve aggregaton model, n whch exponental uptmes and downtmes are assumed, whch s useful for a lne wth three machnes. Ths model s extended by Sörensen and Janssens [14] to nclude more machnes and more buffers. In Sörensen and Janssens [16], a smulaton model based on Petr nets s developed. The advantages of usng the Petr net formalsm over other smulaton models s that t offers some tools to study the propertes of the system. Moreover, the computatonal requrements are not larger than those of other dscrete-event smulaton models. The Petr net smulaton model has a place for each state of each machne and each buffer. The Petr net transtons descrbe the events that mght occur durng the smulaton and ther effect on the state of the system. By addng a tme aspect to the Petr net model, t becomes a smulaton model that can be used to determne performance measures of an (n-machne n-1 -buffer)- contnuous flow lne wth unrelable machnes. The model ntated n [16] s used to generate the results presented n ths paper. 2. CONTINUOUS FLOW TRANSFER LINES WITH UNRELIABLE MACHINES 2.1 Assumptons Fgure 1: A 3-machne 2-buffer contnuous flow transfer lne. The system studed n ths paper conssts of a machne wth three machnes and two ntermedate buffers n seres. Ths system s depcted n Fgure 1. The followng assumptons are made to model the system: 1. Producton s contnuous. No ndvdual parts can be dentfed. 2. The system s balanced. All machnes operate at the same speed. Producton speed s wthout loss of generalty assumed to be equal to 1. Ths means that every machne outputs 1 unt of product per tme unt. 3. The frst machne cannot be starved. The last machne cannot be blocked. 4. Falure s operaton-dependent. A machne can only break down when t s up, not when t s starved or blocked. 5. Falure tmes and repar tmes are exponentally dstrbuted and statstcally ndependent. The memory-less property of the exponental dstrbuton allows for a smpler smulaton model. When two or more machnes go down, they can be repared smultaneously. In ths paper, we report on extensve smulatons of a model of the above system. 17

4 2.2 Petr net model The Petr net model, used to smulate the performance of a contnuous flow transfer lne system wth three unrelable machnes and two ntermedate buffers, conssts of 16 places and 39 transtons. A markng wth tokens n some places represents a state of the system. For easy reference, the transtons are gven by ther nput and output places. A selecton of transtons s presented n Table Ito obtan a full lst of transtons, we refer to the smulaton model generaton descrbed n Sörensen and Janssens [15] and can be obtaned from the authors. Some addtonal nformaton needs to be kept n varables called token attrbutes. For each machne, a token attrbute s created that contans the amount of tme left untl the next falure (when the machne s down) or the next repar (when t s up). For each buffer, a token attrbute s created that contans the amount of content n the buffer. From these values, t s obvous to decde whch transton fres next. Frng a transton automatcally updates the state of the system. Table I: Selecton of transtons of the Petr net model wth 3 machnes and 2 buffers. Transton No. Event Input places Output places T 1 M 1 down T 2 M 1 down T 3 M 1 down T 4 M 1 down T 5 M 1 down T 38 B 2 full T 39 B 2 full The smulaton logc s as follows: 0. Intalse the smulaton. 1. Identfy all enabled transtons n the Petr net. 2. Calculate the next-event tme for each enabled transton (usng the token attrbutes). 3. Choose the transton wth the earlest event tme. Move the smulaton clock to ths tme. Calculate the progress of tme. 4. Recalculate the token attrbutes. 5. Fre the transton wth the earlest event tme. Go to step Defntons The performance of a producton system s generally measured by ts avalablty, defned as the rato of uptme by total tme. Some papers use other performance measures. For example, De Koster [6] uses the concept throughput, defned as where q(τ) s the actual producton at tme τ. Buzacott and Hanfn [2] use effcency, defned as (1) (2) 18

5 where Q(t) s what the producton would have been at tme t f the system was perfectly relable. In ths paper, only avalablty s used as a performance measure. The avalablty A of a system wth n machnes and n-1 buffers s defned as the percentage of tme the system s producng output. For a sngle machne, the followng defntons are used: The falure rate λ of a machne s the expected number of falures regstered per tme unt. The mean tme to falure of ths machne s MTTF = 1. The repar rate μ of a machne s the expected number of repars performed per tme unt. The mean tme to repar of ths machne s MTTR = 1. μ The avalablty A of a machne s gven by A =. λ + μ 2.4 Readng the numbers The numbers used n these experments are fcttous. However, when nterpreted correctly, we beleve they can offer a hnt of the sze of the buffers to be nstalled. All buffer szes should be read relatve to the producton speed, whch s assumed to be 1 unt of product per tme unt for each machne. Thus, a buffer wth sze m should be nterpreted as a buffer that can be flled by a machne n m tme unts. In all experments, we fnd that the falure and repar rates do not have a sgnfcant effect on the avalablty of the system, as long as the avalablty of the ndvdual machnes s the same. For example, a system composed of machnes wth µ=4 and λ=1 behaves n terms of avalablty very smlar to the one wth µ=0.04 and λ= SETUP OF THE EXPERIMENTS The followng scenaros are tested: All machnes have equal avalablty (experment 1). One machne has a consderably hgher avalablty than the other two (experment 2). One machne has a consderably lower avalablty than the other two (experment 3). To obtan a relable estmate, each value of avalablty n these experments s computed as the average of 100 ndvdual observatons. Smulaton runs are long enough for any transton effects to have dsappeared (approxmately 20,000 tme unts). For each experment, 5 dfferent total buffer szes (0.1, 0.5, 1, 5, 10 and 100) are dstrbuted among the buffers n 20 equal steps. To obtan an estmate of the mnmal performance of the system, the avalablty of the system s calculated when both buffer szes are set to Experment 1: All machnes have equal avalablty In ths case, the falure and repar rates of the machnes are equal (λ 1 = λ 2 = λ 3 = λ and µ 1 = µ 2 = µ 3 = µ). In fve experment sets, each machne s assumed to be avalable respectvely 50%, 80%, 90%, 95% and 99% of the tme. Assumng exponentally dstrbuted repar tmes wth average duraton of 1 tme unt, ths results n average falure rates of 1, ¼, 1/9, 1/19 and 1/99 tme. 19

6 Table II: Experment 1 (all machnes have equal avalablty): maxmum avalablty of the system for dfferent total buffer szes. A % 25.00% 26.05% 34.29% 34.28% 41.19% 44.67% 49.43% 80% 57.10% 57.91% 65.34% 65.33% 71.30% 74.57% 79.53% 90% 75.01% 75.52% 80.52% 80.51% 84.57% 86.46% 89.57% 95% 86.36% 86.65% 89.60% 89.60% 91.99% 93.01% 94.76% 99% 97.06% 97.13% 97.79% 97.79% 98.35% 98.56% 98.95% Fgure 2: Avalablty of a system wth total buffer sze of 10 and A = 90%. Table IIshows the maxmum avalablty for each tested confguraton. Fgure 2shows some example results from ths smulaton experment. Some general conclusons can be drawn from the results presented above. 1. The more buffer space s nstalled, the hgher the avalablty wll be. 2. The mpact of the buffers s larger when the avalablty of ndvdual machnes s low. For example, when ndvdual machnes are up 50% of the tme, the avalablty of the system can be ncreased by almost 100%. Ths number drops fast wth rsng A. Stated another way, ths fact also means that the mpact of not nstallng ntermedate buffers s much more mportant when ndvdual avalablty s low. The avalablty of a system wthout buffers wth A = 50% (=1,2,3) s equal to 25%, whereas the mnmum avalablty of a system wth A = 99%, s equal to 97%. 3. When the buffers become very large, the theoretcal avalablty almost can be reached. Ths maxmum avalablty s equal to the avalablty of a sngle machne. 4. The mpact of very small buffers on avalablty s very small. Consequently, the dstrbuton of buffer space has a larger mpact on avalablty when the total buffer sze ncreases. For small buffer szes, the dstrbuton s relatvely unmportant. 5. Ideally, the buffer space should be equally dstrbuted among both buffers. 20

7 6. However, the avalablty of the system s qute robust wth regard to the dstrbuton of buffer space. Although the hghest avalablty s reached when the buffer space s equally dstrbuted among the two buffers, devatons from ths deal dstrbuton do not have a large mpact on avalablty, especally when the total buffer space s very large. In ths case, there s more than enough buffer space to allow for a hgh avalablty, even when the dstrbuton of buffer space s not deal. The robustness of avalablty wth regard to the deal buffer space dstrbuton s mportant. A manager wshng to mnmse the total operatng cost may wsh to employ a smaller nventory level of hgh-cost products, as they may represent an mportant workng captal nvestment. Ths robustness s hghest when the total avalable buffer space s ether very small or very large and arses for dfferent reasons n both cases. 3.2 Experment 2: One machne has the hgher avalablty In ths experment, we assume that two out of three machnes are up 80% of the tme. For these machnes, we assume exponentally dstrbuted repar tmes wth average 1 tme unt and exponentally dstrbuted falure tmes wth average 4 tme unts. In three dfferent experment sets, the better machne s placed frst, second and thrd. In three dfferent experments, one of the machnes s up 90%, 95% and 99% of the tme. If we assume exponentally dstrbuted repar tmes wth average of 1 tme unt, falure rates are 1/9, 1/19 and 1/99 respectvely. Dependng on whch of the three machnes s the best, the results are qute dfferent. The maxmum attanable avalablty for each tested confguraton s shown n Table III. Table III: Experment 2 (one machne has hgher avalablty): maxmum avalablty of the system for dfferent total buffer szes. A 1 A 2 A % 80% 80% 62.08% 62.85% 67.43% 70.09% 73.95% 76.61% 79.73% 95% 80% 80% 64.43% 65.08% 67.93% 71.84% 74.91% 77.10% 79.73% 99% 80% 80% 66.27% 66.90% 68.99% 73.21% 75.94% 77.56% 79.77% 80% 90% 80% 62.03% 62.86% 70.22% 70.18% 74.70% 77.19% 79.81% 80% 95% 80% 64.40% 65.15% 72.20% 72.24% 75.71% 77.70% 79.81% 80% 99% 80% 66.23% 66.96% 73.56% 73.60% 76.22% 77.87% 79.81% 80% 80% 90% 62.08% 62.80% 70.05% 67.42% 73.97% 76.60% 79.76% 80% 80% 95% 64.34% 65.12% 71.89% 69.12% 74.94% 77.17% 79.78% 80% 80% 99% 64.40% 65.15% 67.44% 69.47% 75.71% 77.70% 79.81% 21

8 3.2.1 The frst machne has hgher avalablty. Fgure 3: Avalablty of a system wth total buffer sze of 100 and A 1 = 95% (caused by hgher MTTF), A 2 = A 3 = 80%. Fgure 3 shows some detaled example results. The most mportant concluson from ths experment s that avalablty ncreases most f more space s allocated to the second buffer than to the frst. Ths can be understood by reasonng that snce the frst machne has a hgher avalablty than the second the second machne almost never s starved, regardless of the sze of buffer 1. Moreover, the thrd machne has the same avalablty as the second, mplyng that t s starved more often. Therefore, ncreasng the second buffer ncreases the avalablty of the system. Ths effect s notceable n every experment, even n experments wth very small buffer spaces. It s stronger n cases where the frst machne has a very hgh avalablty and the total buffer space s ether very small or very large. In these cases, the optmal avalablty s obtaned when almost all buffer space s allocated to the frst buffer. When the total buffer space s nether very small nor very large, the deal dstrbuton occurs when the greater part of the buffer space s allocated to the frst buffer and a smaller part to the second. Some other conclusons from these experments are: 1. Total avalablty rses wth the amount of buffer space nstalled. 2. Total avalablty rses only lttle wth the qualty of the better machne. 3. The mpact of nstallng buffer space s hgher when the avalablty of the better machne s low. Ths results from the fact that the room for mprovement s lower when ths machne has hgher avalablty. 4. The mprovement caused by the better machne compared to the stuaton where all machnes are equal s hgher when the nstalled buffers are small. Installng buffers thus reduces the effect of havng a better machne. 5. Wth very large buffers, the hghest possble avalablty can almost be reached. Of course, the hghest possble avalablty n ths case s the avalablty of the worst machne, n ths case of machnes 2 and The mpact of very small buffers on avalablty s very small. 22

9 3.2.2 The thrd machne has hgher avalablty. Fgure 4: Avalablty of a system wth total buffer sze of 1 and A 3 = 95%, A 1 = A 2 = 80%. Some example results are shown n Fgure 4. When the thrd machne s better than the other two, the avalablty s hghest when the frst buffer s larger than the second. Because the thrd machne has a sgnfcantly hgher avalablty, the second machne almost never s blocked, regardless of the sze of the second buffer. The avalablty of the entre system can therefore be ncreased most by nstallng a large frst buffer and a smaller second buffer. The conclusons 1 to 6 n the prevous secton are equally vald n ths case The second machne has hgher avalablty. When the second machne s sgnfcantly better than the other two, the relaton between the avalablty of the entre system and buffer allocaton s sgnfcantly less strong than n the case where all machnes have equal avalablty. An example of ths behavour can be seen n Fgure 5. For example, the maxmum ncrease of avalablty n Fgure 5 s less than 3%. An explanaton can be found n the fact that, snce machne 2 s sgnfcantly better, buffer 1 s close to empty most of the tme and buffer 2 s close to full most of the tme, regardless of the respectve szes of these buffers. Ths mples that machne 1 almost never s blocked and that machne 3 almost never s starved. When ths happens, the buffers become useless and lose ther role as buffers. For some cases, ths happens regardless of the allocaton of the buffer space. In ths stuaton, t s best to nstall as lttle buffer space as possble. It occurs when the total buffer volume s ether very low or very hgh. As a general concluson, t can be stated that when the second machne s better than the other two machnes, avalablty rses slowly wth a rse n total buffer volume. 23

10 Fgure 5: Avalablty of a system wth total buffer sze of 5 and A 2 = 90%, A 1 = A 3 = 80%. 3.3 Experment 3: One machne has lower avalablty In ths experment, we assume that two of the three machnes have 95% avalablty. For these machnes, we assume exponentally dstrbuted repar tmes wth average 1 tme unt and exponentally dstrbuted falure tmes wth average 19 tme unts. Ths corresponds to an average avalablty of 95%. In three dfferent experment sets, the worst machne s placed n frst, second or thrd place. We assume avalablty of 50%, 80% and 90% for ths machne. If we assume exponentally dstrbuted repar tmes wth average of 1 tme unt, falure rates are 1, 1/4 and 1/9 respectvely. Results from ths experment are shown n Table IV. Dependng on whch of the three machnes s the best, the results are qute dfferent. However as can be seen n Table IV the maxmum avalablty s roughly the same n all scenaros. Table IV: Experment 3 (one machne has lower avalablty): maxmum avalablty of the system for dfferent total buffer szes A 1 A 2 A % 95% 95% 47.49% 47.84% 48.72% 49.64% 50.06% 50.07% 50.28% 80% 95% 95% 73.76% 74.29% 75.68% 78.62% 79.71% 80.04% 80.25% 90% 95% 95% 82.22% 82.63% 83.86% 87.82% 88.58% 89.63% 90.22% 95% 50% 95% 47.52% 47.71% 48.28% 49.64% 49.94% 50.08% 50.20% 95% 80% 95% 73.81% 74.12% 75.09% 78.69% 79.19% 79.88% 80.18% 95% 90% 95% 82.17% 82.58% 83.52% 87.78% 88.16% 89.38% 90.17% 95% 95% 50% 47.54% 47.87% 48.71% 50.01% 50.06% 50.07% 50.05% 95% 95% 80% 73.78% 74.28% 75.62% 79.93% 79.68% 80.02% 80.06% 95% 95% 90% 82.23% 82.65% 83.84% 87.60% 88.60% 89.62% 90.04% Some general conclusons can be drawn from these data: 1. If the avalablty of the worst machne s low, the avalablty of the system s close to the maxmum attanable avalablty. Of course, the maxmum avalablty s equal to the avalablty of the worst machne. Ths can be called the weakest-lnk effect. The worst machne keeps the performance of the system low, regardless of the qualty of the better machnes. 24

11 2. The fact whether the worst machne s frst, second or thrd does not have a large mpact on the maxmum avalablty of the system. Total avalablty s roughly the same n all three cases. 3. When the avalablty of the worst machne s rather large, the room for mprovement grows. In these cases, the weakest-lnk effect s much less promnent and the provson of buffer space becomes more mportant The frst machne has lower avalablty In ths case, there s a slght preference for a larger frst buffer. However, the relatonshp between buffer allocaton and avalablty s weak The thrd machne has lower avalablty When the thrd machne s worse than the other two machnes, slghtly more buffer space should be allocated to the second buffer. Agan, the relatonshp between buffer allocaton and avalablty s weak The second machne has lower avalablty The deal buffer allocaton n ths case s when both buffers are of equal sze. Ths s llustrated n Fgure 6. Even wth relatvely small buffers, the avalablty s close to optmal at all tmes, especally when the avalablty of the second machne s low. Therefore, devatons from the deal buffer allocaton only cause small decreases n avalablty. Ths can be understood by reasonng that most of the tme buffer 1 s full and buffer 2 s empty, resultng from the fact that machne 1 s better than machne 2 and machne 2 s worse than machne 3. The buffers lose much of ther functon n ths case. Fgure 6: Avalablty of a system wth total buffer sze of 5 and A 2 = 50%, A 1 = A 3 = 95%. 4. CONCLUSIONS In ths paper, extensve smulaton results on the allocaton of buffer space n a contnuous flow transfer lne wth unrelable machnes are dscussed. Smulaton results are collected from a Petr net smulaton model of a lne wth three machnes and two buffers. It s shown 25

12 how these results can be used to fnd some generally applcable rules of thumb for the allocaton of buffer space n a contnuous flow-lne. REFERENCES [1] Buzacott, J.A. (1967). Automatc transfer lnes wth buffer stocks. Internatonal Journal of Producton Research, 5(3): [2] Buzacott, J.A. and Hanfn, L.E. (1978). Model of automatc transfer lnes wth nventory banks: a revew and comparson. AIIE Transactons, 10(2): [3] Collard, P. and Proth, J.M.. Effet des stocks tampons dans une fabrcaton en lgne. JORBEL (Belgan Journal of Operatons Research, Statstcs and Computer Scence), 24(2):3 27, 1984 [4] De Koster, M.B.M. (1987). Estmaton of lne effcency by aggregaton. Internatonal Journal of Producton Research, 25(4): [5] De Koster, M.B.M. (1988). An mproved algorthm to approxmate the behavor of flow lnes. Internatonal Journal of Producton Research, 26(4): [6] De Koster, M.B.M. (1989). Capacty orented analyss and desgn of producton systems. In Lecture Notes n Economcs and Mathematcal Systems, volume 323, 245 pp., Sprnger- Verlag, Berln [7] Elsayed, E.A. and Hwang, C.C. (1986). Analyss of two-stage manufacturng systems wth buffer storage and redundant machnes. Internatonal Journal of Producton Research, 24(1): [8] Fox, R.A. and Zerbe, D.R. (1974). Some practcal system avalablty calculatons. AIIE Transactons, 6(3): [9] Kubat, P. and Sumta, U. (1985). Buffers and backup machnes n automatc transfer lnes. Internatonal Journal of Producton Research, 23(6): [10] Malathronas, J. P.; Perkns, J.D. and Smth, R.L. (1983). The avalablty of a system of two unrelable machnes connected by an ntermedate storage tank. AIIE Transactons, 15(3): [11] Mtra, D. (1988). Stochastc theory of a flud model of producers and consumers coupled by a buffer. Advances n Appled Probablty, 20: [12] Murphy, R.A. (1975). The effect of surge on system avalablty. AIIE Transactons, 7(4): , [13] Murphy, R.A. (1978). Estmatng the output of a seres producton system. AIIE Transactons, 10(2): [14] Sörensen, K. and Janssens, G.K.. Buffer allocaton and requred avalablty n a transfer lne wth unrelable machnes. Internatonal Journal of Producton Economcs, 74: [15] Sörensen, K. and Janssens, G.K. (2004). Automatc smulaton model generaton of a contnuous flow transfer lne wth unrelable machnes. Qualty and Relablty Engneerng Internatonal, 20(4): [16] Sörensen, K. and Janssens, G.K. (2004). A Petr net model of a contnuous flow transfer lne wth unrelable machnes. European Journal of Operatonal Research, 152(1) : [17] Van Oudheusden, D.L. and Janssens, G.K. (1994). Avalablty of three-machne two-buffer systems. JORBEL (Belgan Journal of Operatons Research, Statstcs and Computer Scence), 34(2): [18] Wjngaard, J. (1979). The effect of nterstage buffer storage on the output of two unrelable producton unts n seres, wth dfferent producton rates. AIIE Transactons, 11(1): [19] Wld, R. (1972). Mass-Producton Management: the Desgn and Operaton of Producton Flow- Lne Systems. J. Wley & Sons, London [20] Yeralan, S.; Franck, W.E. and Quasem, M.A. (1986). A contnuous materals flow producton lne wth staton breakdown. European Journal of Operatonal Research, 27: