Modeling and Simulation of a Serial Production Line with Constant Work-In-Process

Transcription

1 Modelig ad Simulatio of a Serial Productio Lie with Costat Wor-I-Process Mehmet Savsar Iteratioal Sciece Idex, Idustrial ad Maufacturig Egieerig waset.org/publicatio/13084 productio cotrol strategy. Coutless umber of other JIT Abstract This paper presets a model for a ureliable productio lie, which is operated accordig to demad with costat applicatios ad related models ca be see i the literature. Most of the literature deals with the efficiecy of JIT systems wor-i-process (CONWIP). A simulatio model is developed based uder differet operatioal coditios. Either mathematical o the discrete model ad several case problems are aalyzed usig models are developed based o restrictive assumptios or the model. The model is utilized to optimize storage space capacities at itermediate stages ad the umber of abas at the last stage, simulatio models are utilized i the aalysis of JIT systems. which is used to trigger the productio at the first stage. Furthermore, I relatio to the effects of itermediate buffer capacities o a effects of several lie parameters o productio rate are aalyzed usig desig of experimets. push type of serial productio lie ad optimum allocatios of buffers o the lie, several papers have bee published. I particular, papers related to buffer allocatios iclude [12]- Keywords Productio lie simulator, Push-pull system, JIT [22]. system, Costat WIP, Machie failures. I this paper, we developed a discrete mathematical model to aalyze a push-pull system of productio with costat I. INTRODUCTION wor-i-process (CONWIP). Whe a fial product is IMULATION has bee extesively used i modelig ad withdraw from the fiished products ivetory i the last Saalyzig productio cotrol systems. A particular type of stage, a aba is sigaled to the first stage to start the productio cotrol, which has become a commo tred i productio. Fig. 1 illustrates operatio of such a system. idustry, is just i time (JIT) or pull system of productio cotrol. I a JIT system, productio is iitiated accordig to demad for fiished products at each stage to produce what is 1 eeded at the right time ad i the right quatity. Alterative to a purely pull system is the hybrid push-pull system, where the productio at the first stage is scheduled accordig to the demad for the products i the last stage. Withdrawal of 1 M 1 2 M 2.. Demad m M m m+1 fiished products from the last stage triggers the productio at Fig. 1 A Push-pull productio cotrol system the first stage by a iformatio sigalig card, called a aba. Itermediate operatios are performed by a push system. Push-pull systems are commoly used i electroics assembly operatios. Several studies have bee carried out o the implemetatio ad efficiecy of JIT systems. [1]-[6] have aalyzed JIT systems from differet perspectives usig simulatio as well as other meta modelig approaches, icludig eural etwor models. [7]-[10] studied a hybrid push-pull system ad preseted a cotrol algorithm for multi-stage, multi-lie productio systems. [11] compared three pull cotrol policies, amely the aba, base stoc, ad geeralized aba. The effects of abas ad other factors o JIT system performace have bee ivestigated mostly for pull types of Mehmet Savsar is professor ad chairma of the Departmet of Idustrial & Maagemet Systems Egieerig at Kuwait Uiversity, College of Egieerig & Petroleum,, P.O. Box 5969, Safat 13060, Kuwait (phoe: ; fax: ; mehmet@uiv.edu.w or msavsar@yahoo.com). Successive operatios are carried out by completio of each product at each statio (Mi) ad its delivery to the succeedig statio or its buffer store (i), if the statio is busy. It is assumed that the itermediate buffer sizes, which represet maximum wor-i process at each stage, are limited i capacity. Thus, whe the storage of fiished uits i the fial products ivetory reaches a specified maximum capacity, the last statio stops its productio. Similarly, whe a itermediate buffer i is filled up to its maximum capacity, the precedig statio, Mi-1, stops its productio or the completed part stays o the statio util a part is removed from the succeedig buffer. The capacity of buffer i is zi. I productio systems, which operate accordig to demad, equipmet availability is importat sice machie failures sigificatly delivery time of products. While equipmet failures due to wear-outs ca be elimiated, failures due to radom causes could ot be elimiated. Whe optimizig 638

2 Iteratioal Sciece Idex, Idustrial ad Maufacturig Egieerig waset.org/publicatio/13084 umbers of abas ad the sizes of i-process buffers, it is ecessary to cosider equipmet reliability i model developmet. I the followig sectio, we preset a discrete mathematical model, which is based o the flow of discrete parts or batches from stage to stage. The model is used to aalyze the behavior of the push-pull system uder various operatioal coditios icludig equipmet availabilities ad radomess i demad. II. MATHEMATICAL MODEL FOR THE PUSH-PULL SYSTEM The basic pricipal of the discrete model is to determie the total time a batch of parts speds i statio i, the time istat at which batch is completed i statio i, ad the time istat at which batch leaves the statio i. Storages 2,., m are called itermediate buffer storages, havig fiite capacity z i, i=2,,m. Iitial iput storage is assumed to have ulimited capacity for the raw material, while the fial output storage has limited capacity for completed batches of products with attached abas. The fial buffer (m+1) is assumed to be the fiished products storage with time betwee part departures beig equal to time betwee demad. The followig otatios are used i the formulatio: i = Time duratio that th batch stays o the i th statio ot cosiderig imposed stoppages due to equipmet failures; i=1,2,..,m. m = Number of statios o the lie. i = Processig time of batch o statio i (this may be a radom variable with certai distributio) i = Repair time of the i th statio required for correctio of a failure durig processig of the th batch. Time to failures ad the repair times are assumed to follow certai distributios, which are geerated ad icorporated ito the model whe the model is solved iteratively. i = Istat of time at which processig of the th batch is completed o the i th statio. i = Istat of time at which th batch departs from the i th statio. 0 = Istat of time at which th batch eters the first statio. i = Istat of time at which i th statio is ready to process the th batch. m+1, = Istat of time at which th batch departs from the fial buffer m+1. = Mea time betwee demad for batches -1 ad from the fial buffer. This time may also be a radom variable with certai distributio. A part stays i a statio for three reasos: (i) The part is beig machied; (ii) The machie has failed durig machiig of the part ad a repair is taig place; (iii) The successive buffer is full ad the part ca ot be trasferred to the ext statio due to a imposed stoppage. The residece time of the th part o the ith statio, i, without cosiderig imposed stoppages is give as follows: i i i (1) The discrete mathematical model of the push-pull system cosists of calculatig part completio times, i, ad part departure times, i, i a iterative fashio. The followig formulatio is developed for i ad i to be used i iterative calculatios. Processig of batch caot be started o statio i util the previous batch, -1 leaves statio i. Therefore the time istat at which i th statio is ready to begi the th batch, deoted by i, is give by i = i,-1. If, i-1, < i, the the th batch must wait i buffer i, sice it has left statio i-1 before statio i is ready to accept it. Therefore, processig of the th batch i the i th statio will start at the istat i. If however, i-1, i, the processig of the th batch i the i th statio ca start immediately at the time istat i-1,. Cosiderig both cases above, oe gets the relatio for the ready time of the th batch to be processed i the ith statio as follows: i = max[ i-1,, i,-1 ] (2) Sice the th batch will stay i statio i for a period of i time uits, its processig will be completed by the time istat i give by: i = max[ i-1,, i,-1 ] + i = i + i (3) where i=2,3,,m. I case of the first statio, a aba must arrive before the batch ca be processed. The arrival of a aba from storage m+1 is modeled as follows: Let =-L 2 where, L 2 =Total umber of batches iitially i statios S 2 ad storages 2,.., m. The, 1 = max[ 1,-1, m+1, ] + 1 (4) Time istat at which th batch is ready to eter the first statio is assumed to be 0 < 1,-1 sice we assumed that there are always batches of parts available i frot of the first statio. However, a aba must arrive from the buffer (m+1) to start the process at statio 1. Now, it remais to determie the time istat at which th batch departs from the i th statio, i. It is foud by cosiderig two cases. Let = z i+1 1 (5) I the first case, i, i1, (6) which idicates that the th batch has bee completed o the i th statio before processig of the (-z i +1) th batch has started o the (i+1) th statio. Sice buffer i+1, which is betwee statio i ad i+1 with capacity z i +1, is full ad statio i has completed the th batch, the th batch may leave the i th statio oly at the istat of time at which the ( z i +1) th batch of the (i+1) th statio has started processig. Therefore, i, i1, (7) I the secod case, i, i1, (8) which idicates that, at the istat i there are free spaces i buffer i +1 ad therefore part ca leave machie i immediately after it is completed; that is, i = i holds uder this case. Cosiderig both cases above, we have the followig relatios for i : i, i, if zi 1 1 (9) max (10) i, i,, i1, 639

3 Iteratioal Sciece Idex, Idustrial ad Maufacturig Egieerig waset.org/publicatio/13084 if > z i+1 +1; i =1, 2, 3,, m-1, where = z i+1 1 ad, max (11) m, m,, m1, Where, m+1, = Departure time if the th batch from the fial buffer m+1. Departure time of a fiished product from the fial buffer (m+1) depeds o two coditios ad calculated as follows: I the first case, m, m+1,-1 +. I this case, m+1, = m+1,-1 +. (12) I the secod case, m, > m+1,-1 +. I this case, m+1, = m,. (13) Combiig both cases above, the followig geeral relatio is obtaied for m+1, : m+1, = max[ m,, m+1,-1 + ] (14) I real-world situatios, itermediate buffers may cotai batches of parts that are ofte left from a previous shift or day. Therefore, it is importat to start the iteratios with some iitial coditios as follows: Let: l i =Number of batches iitially i buffer i ad statio S i m ad, 1 L Total umber of batches o statios l i i S,..,S m ad storages,.., m+1. Whe iteratios are started, oe ca assume that, at the iitial time istat t=0, parts 1,2,, L i+1 are already processed o statio S i, sice these batches are iitially i the lie right after statio S i. Therefore, give the iitial values of l 1,.,l m+1, the iitial values of i ad i for =1,2,.,L i+1 are expressed by: i = i = 0; i = 1, 2,.., m. (15) I order to carry out iterative computatios, a simulatio procedure is developed ad implemeted o the computer to determie several productio lie performace measures, which iclude average umber of batches completed by the lie durig a specified period, average umber of batches completed by each statio durig the same time, percetage of time for which each statio is up ad dow due to imposed or iheret stoppages. III. COMPUTATIONS OF THE MODEL The discrete model is coded ito a simulatio program ad implemeted o the computer to calculate system performace measures. I additio to the variables described for the discrete model, the simulatio allows several distributios, icludig: expoetial, uiform, Weibull, ormal, log ormal, Erlag, gamma, beta ad costat values to be specified for failure ad repair times of the equipmet i each statio. Iterative simulatio model basically calculates the time istat at which each part eters a statio, duratio of its stay, ad the time it leaves the statio. This is cotiued util, for example oe shift, which is the specified simulatio time T sim, is completed. I order to obtai reliable results, several simulatio rus have to be obtaied ad the average performace measures should be calculated. The results of each iterative simulatio are utilized with statistical tests to determie if the specified coditios are met to stop the umber of simulatio iteratios (i.e., shifts). If the coditios are ot met, simulatio iteratios are cotiued with further rus. For each simulatio realizatio, calculatios of i, i, i, ad m+1, are performed iteratively with the cosideratio give to equipmet failures ad repairs as the parts flow through the system. Reliable results caot always be obtaied from a sigle simulatio realizatio. Therefore, additioal rus have to be performed ad the results tested statistically util the error i the lie productio rate is less tha a value with a probability, both of which are predefied. This is accomplished by comparig the average productio output rate from simulatio ( Q ) to the expected value (Q ) usig the cofidece iterval calculatio give below. Here, N N N i (16) Q ad T sim i1 where N i =productio output obtaied from simulatio ru i ad is the umber of simulatio rus. Z Pr1 2 V( Q) Q Q Q Z 1 2 V( Q) 1 Q (17) The aim is to have a estimated output rate, Q, as close to the actual mea output rate Q as possible. To achieve this, Z V ( Q ) 2 Q is miimized by obtaiig more rus. As this value gets closer to 0, Q Q with probability 1. A value is etered by the user; the simulatio program calculates Z V ( Q ) 2 Q after each iteratio; compares this quatity with ad termiates the program if it is less tha. If it is ot less tha after a maximum umber of rus, the program is still termiated to avoid excessive computatio. The iterative simulatio model allows oe to determie various parameters ad depedet variables with sigificat effects o productivity ad other performace measures. Estimatio idices are obtaied for such variables as the total, iheret, ad imposed time losses due to failures ad stoppages for each statio as follows: Q 60 / is the omial productivity of statio i, where i i i is the cycle time for statio i; Qr 60N i / T i sim is the relative productivity of statio i; K loss 1 Q r / Q i i is the total loss factor of statio i; K ih 1 tri /( tri t fi ) is iheret loss factor of statio i; ad K imp K loss K ih is the imposed loss factor for statio i, i = 1, 2,.., m. The terms t fi ad tri are mea times to failure ad to repairs, of statio i respectively. After determiig these loss factors, they are compared for all statios. The statio with the highest total loss factor is the 640

4 Iteratioal Sciece Idex, Idustrial ad Maufacturig Egieerig waset.org/publicatio/13084 chose for improvemet. If K imp>k ih (i), stoppages are maily due to blocig ad starvatio; therefore it is ecessary to icrease the capacity of buffers immediately precedig ad succeedig it. If K ih (i)>k imp (i), stoppages are maily caused by iheret failures, that is breadows; therefore, the reliability of statio i, should be icreased or its mea repair time should be decreased i order to gai improvemet i total lie productivity. After the suggested chages are made, iterative simulatio is repeated to see the effects of the proposed chages i the desig. IV. STORAGE SPACE ALLOCATION Optimum storage space or buffer allocatio problem has bee studied by several researchers with respect to allocatio of a fixed amout of total buffer space o a serial productio lie. The problem ca be stated as follows: Give a total amout of acceptable buffer space of Z uits, allocate this total space to idividual buffers S 2,,S m+1, the quatities z 2, z 3,..,z m+1 respectively such that the total productio output rate of the lie, Q(z), is maximized. The problem is stated as follows: Choose z 2, z 3,..,z m+1 so as to Maximize Q(z) m Subject to: 1 zi Z (18) i2 z i 0 ad iteger (i=2,3,..m+1) This problem has bee discussed by [7] for productio lies with expoetial processig times i all statios. The optimizatio model is a liearly costraied iteger oliear programmig problem that is difficult to solve due to the fact that Q(z) has to be evaluated by either cotiuous time Marov chais or by some other stochastic processes approximatio. [7] evaluated Q(z) for the serial lie usig Marov chais approach ad idicated that the umber of states are too large ad exceeds well over 20,000 equatios to be solved to obtai the value of Q(z) for a give buffer size combiatio. Eve if it was practical to solve the problem, it would ot be still applicable to the cases with equipmet failures ad o-expoetial process times. The buffer allocatio model is applied to the push-pull system i this paper. However, we obtai the solutio for Q(z) usig the iterative solutio procedure preseted above. This procedure is ot restricted to expoetial process times ad all reliable equipmet, sice it is based o simulatio. A fixed umber of buffers are specified ad the iterative computatios are performed to determie optimum combiatios by evaluatig all buffer combiatios. The optimum correspods to the maximum productio output rate. For small size problems, such as lies with up to 8 statios ad up to 10 buffer capacities, computatioal time is i the order of miutes, depedig o the accuracy required. However, for larger problems, such as more tha m=10 statios ad more tha Z=15 buffer capacities, computatioal time is relatively large sice umber of possibilities evaluated is large. If a small accuracy with 5% error is acceptable, large problems ca also be solved i a reasoable time. V. SIMULATED CASE PROBLEMS The model is illustrated by several case problems. Table I ad Table II are the iput data ad the output results obtaied for a 5-statio lie with all statios available 85% of the time. Processig times, failure distributios, their parameters, repair distributios ad their parameters are show i Table I. Statio TABLE I INPUT DATA FOR PRODUCTION LINE SIMULATION Process No. of Failure Repair Distrib. Time Failures Distrib Expo(85) Normal Expo(85) Normal Expo(85) Normal Expo(85) Normal Expo(85) Normal The outputs for 2000 time uits of simulatio with =0.05, =0.005, ad maximum iteratios=100 are show i Table II. The results iclude relative productio rate of each statio ad various loss factors due to equipmet failures as discussed i sectio III. The output also icludes suggestios for lie improvemet. TABLE II UNITS Average Lie Output (Parts/Time Uit.)= Stadard Dev. of Lie Output Rate= Optimum Buffer Allocatios for Buffers m=2 to 6 are as Follows: Statio Relative Imposed Loss Factors Iheret Loss Factor Suggestios: Statio No. 1 has the Maximum Total Loss Factor. Dow Time is Maily Imposed. Icrease the Capacity of Storage Adjacet to This Statio. Also Icrease Reliability ad Productivity of Adjacet Statios ad Try Simulatio Agai. Error<Epsilo Is Reached at Iteratio = 100 Maximum Iteratio Is Reached At Iteratio = 100 Total Computatio Time = Secods I the secod case problem, a push-pull productio lie with 5 serial statios is cosidered as before. All statios are assumed to be reliable, except oe statio which was placed at 641

5 Iteratioal Sciece Idex, Idustrial ad Maufacturig Egieerig waset.org/publicatio/13084 the begiig (B), i the middle (M), or at the ed (E) of the lie to see the effects of ureliable statio at differet segmets of the lie. Failure ad repairs for the ureliable statio are as i Table I. Stadard deviatio of the repair times was tae as 15% of the mea. Sice availability is A=MTBF/(MTBF+MTTR), selected parameters represeted 85% equipmet availability for the particular statio, whose effect o the lie was ivestigated. I other words, we wated to see how the buffers would be allocated if the ureliable statio was at the begiig, at the middle or at the ed of the lie. The system was aalyzed by the simulatio over a period of 2000 time uits. All itermediate buffer combiatios, which add up to less tha or equal to total buffer capacity Z (Z=0,1,23,5,10) are evaluated i order to determie lie productivity, as show i Fig. 2, ad optimum buffer combiatios (z 2, z 3, z 4, z 5 ), which resulted i maximum productio rate, as show i Table III, for three differet locatios of the ureliable equipmet at the begiig (B), i the middle (M), ad at the ed (E) of the lie. The correspodig productio output rates are give as the percetage of omial rate, which would be 100 if the lie was all reliable, balaced with costat processig times. A importat observatio related to buffer size allocatio ca be see i Table III. If the lie has a ureliable statio at the start or at the ed, the buffer capacity available should be located immediately after the last statio, except i the case of 5 ad 10 buffer sizes, i which case oe uit is allocated to buffer 4, after statio 3 i the ceter of the lie. A similar observatio is see i the case if the ceter statio is ureliable. I this case if there is oe buffer space to be allocated o the lie, it is preferred to be at buffer 3, immediately after statio 2. Lie Productivity (%) B M E Total Buffer Capacity (Z) Fig. 2 Lie productivity as a fuctio of Z Additioal buffer sizes are mostly allocated to the ed of the lie after statio 5, except i the case of 2 ad 10 buffer sizes, i which case oe space is allocated after the ceter statio. The mai reaso that the buffer spaces are mostly allocated to the ed of the lie could be due to the fact that the lie is operated as a push-pull system ad therefore the first statio ca ot start processig a part uless a part or batch is withdraw from the last statio. Buffer spaces after the last statio helps icreasig part availability durig demad. The TABLE III BUFFER CAPACITY DISTRIBUTION TO STATIONS Begiig Middle Ed Z z 2 z 3 z 4 z 5 z 6 z 2 z 3 z 4 z 5 z 6 z 2 z 3 z 4 z 5 z same three cases were evaluated for a purely push type of productio lie, where the last statio does ot have a limit o outputtig its product ad the first statio ca start without waitig for a part withdrawal from the last statio. The results, which are ot show here, are almost opposite of what is obtaied for the push-pull system ad the buffer allocatio is preferred to be immediately after the first statio ear the start of the lie i all cases. VI. EFFECTS OF LINE CONFIGURATIONS,MAINTENANCE POLICIES AND LINE PARAMETERS ON LINE PERFORMANCE I order to see effects of various productio related parameters ad factors o lie performace measure, such as the productio rate, several experimets were set up ad results were obtaied. I particular, the followig productio lie factors were tae ito cosideratio: 1. Productio lie legth (3, 5,ad 9 statios); 2. Buffer capacities betwee statios (0, 2, 4, 6, 8); 3. Process time variability measured by its coefficiet of variatio (CV pt =0, 0.2, 0.5, 0.7); 4. Demad iterval variability (CV dm =0, 0.2, 0.5, 0.7); 5. Type of maiteace applied (Desig out maiteace resultig i full reliability [REL], reliability cetered maiteace [CM-PM], corrective maiteace [CM]) Process time at each statio was assumed to be ormally distributed with mea of 3.0 time uits ad varied accordig to the coefficiet of variatio (CVp selected. Similarly, time iterval betwee the demads for withdrawal of products from the fiished products storage was assumed to be ormally distributed with mea of 3.0 time uits ad also varied accordig to the coefficiet of variatio (CV dm ) selected. The productio lies are simulated over 2400 time uits. 10 rus are carried out for each combiatio ad average values are recorded. Figures 3-5 illustrates the productio output rate as a fuctio of various lie cofiguratio ad factors metioed above. CM-i, CM-PM-i, ad REL-i represet two levels of maiteace ad full reliability case for each statio i. I order to compare effects of corrective maiteace (CM) oly to the CM with prevetive maiteace (PM), reliability cetered maiteace (RMC) cocept was icorporated ito the model. Uder RMC, equipmet is subjected to PM just before a failure is expected. Mea time betwee failures (MTBF) must be determied i advace. I this case, it is assumed that failures due to wear outs are elimiated ad oly 642

6 Iteratioal Sciece Idex, Idustrial ad Maufacturig Egieerig waset.org/publicatio/13084 radom failures remai. This idea ca be implemeted aalytically if time betwee failures are uiformly distributed. This cocept has bee explaied i detail by Savsar[23]. Followig is a mathematical procedure to separate radom failures from the wear-out failures. This separatio is eeded i order to be able to see the effects of maiteace o the productivity ad availability of a lie whe simulatig the system. Let f( = Probability distributio fuctio (pdf) of time betwee failures. F( = Cumulative probability distributio fuctio (cdf) of time betwee failures. R( = Reliability fuctio (Probability that the equipmet survives by time. h( = Hazard rate (or istataeous failure rate). Hazard rate h( cosists of two compoets, the first due to radom failures ad the secod due to wear-out failures as: h( = h 1 ( + h 2 ( (19) h 1 ( = Hazard rate due to radom failures. h 2 ( = Hazard rate due to wear-out failures. Sice the equipmet failures are either due to chace causes or wear-outs, reliability of the equipmet, which is the probability that equipmet survives by time t, ca be expressed as follows: R( = R 1 ( R 2 ( (20) where, R 1 ( = Reliability due to chace causes (or radom failures) ad R 2 ( = Reliability due to wear-outs. Sice the hazard rate due to radom failures is idepedet of time ad therefore costat, we let h 1 (=. Thus, the reliability of the equipmet due to radom failures with costat hazard rate would be as follows: R 1 ( = e -t (21) h( = + h 2 ( (22) It is ow that h( =f(/r( = f(/[1-f(] = + h 2 ( (23) h 2 ( = h( - h 1 ( = f(/[1-f(] - (24) f f ( 1 F( f ( h2 ( R ( [ ][ ] [1 F( )] t t t 1 F( e e e 2( 2 t F 1 F( e 1 R ( 1 t e 2 ( 2 t R( t e df2 ( f 2 ( dt (25) R 2 ( = R(/R 1 ( = [1-F(]/ e -t (26) h 2 ( = f 2 (/R 2 ( (27) These derivatios show that, total time betwee failures, f( ca be separated ito two distributios, time betwee failures due to radom causes [f 1 (] ad time betwee failures due to wear-outs [f 2 (]. Sice the failures due to radom causes could ot be elimiated, we must cocetrate o the failures due to wear-outs i order to elimiate them by appropriate maiteace policies. By the procedure described above, it is possible to separate the two types of failures ad develop the best maiteace policy to elimiate the wear-out failures. This separatio is aalytically possible for uiform distributio. However, it is ot possible aalytically for other distributios. It is assumed that whe a prevetive maiteace policy is implemeted, failures due to wear-outs are elimiated ad oly failures due to radom causes remai. These radom failures are assumed to follow expoetial distributio with costat hazard rate sice they are completely radom with uow causes ad the memoryless property of expoetial is applicable. For uiformly distributed time betwee failures, t, i the iterval 0<t<, probability distributio fuctio of time betwee failures without itroductio of PM is give by: f ( 1 /. (28) If we let =1/, the, reliability is give as 1-t ad the total failure rate is give as: h(=f(/r(=/(1-. (29) Let us assume that hazard rate due to radom failures is a costat give by h 1 (=, the the hazard rate due to wear-out failures could be determied by: h 2 (=h(-h 1 (=/(1--= 2 t/(1- (30) The correspodig time to failure probability desity fuctios for each type of failure rate is: ( t ) f1 ( e 0 t (31) 2 ( t ) f 2 ( t e, 0 t (32) The reliability fuctio for each compoet would be is as follows: ( R1 ( e 0 t (33) t R2 ( (1 e, 0 t (34) R( R1 ( R2 ( (35) Whe the prevetive maiteace (PM) is itroduced, failures due to wearouts are elimiated ad thus the machiery fails oly due to radom causes, which are expoetially distributed as give by f 1 (. Samplig for the time to failures i simulatios is thus based o expoetial distributio with mea ad a costat failure rate of =1/. I case of CM without PM, i additio to the radom failures, wear-out failures are also preset ad thus the time betwee equipmet failures is uiformly distributed betwee 0 ad as give by f(. The justificatio behid this assumptio is that uiform distributio implies a icreasig failure rate with two compoets, amely, failure rate due to radom failures ad failure rate due to wearout failures as give by h 1 ( ad h 2 ( respectively. Iitially whe t = 0, failures are due to radom effect with a costat rate =1/. As the equipmet operates, wearout failures come ito play ad thus the total failure rate h( icreases with time t. Samplig for the time betwee failures i simulatio is based o a uiform distributio with mea /2 ad a icreasig rate, h(. I the simulatio experimets cosidered, time to failure is assumed uiformly distributed betwee 0 ad 200 time uits with a mea of 100 time uits for all statios for the case of CM oly. I the case of PM, wearouts are elimiated ad time to failure exteds; it becomes 643

7 Iteratioal Sciece Idex, Idustrial ad Maufacturig Egieerig waset.org/publicatio/13084 expoetially distributed with a mea of 200 time uits. Time to repair was assumed ormally distributed with mea of 15 time uits ad stadard deviatio of 3 time uits. Fig. 3 illustrates simulatio results for the case whe CV=0.0 for process time (CVp ad the demad (CV dm ). As it is see i fig. 3, 800 uits (2400/3.0) are produced o ay legth of lie if the lie is fully reliable ad there is o other source of variability. However, if the lie is uder failures with CM oly, productio rate is sigificatly reduced whe lie legth is icreased. Whe PM is itroduced i additio to CM, productio rate is betwee the CM ad REL cases. It ca be see from figure 3 that betwee the cases of ureliable lies, the lowest productio rate is for a 9-statio lie with CM oly, while the highest rate is for a 3-statio lie with CM ad PM together. Fig. 4 shows the results for CVpt=0.0 ad CV dm =2.1; fig. 5 shows the results for CVpt=2.1 ad CV dm =0.0; fig. 6 shows the results for CVpt=2.1 ad CV dm =2.1. As it ca be see from figs. 3-6, as the process time ad demad variability icrease, productio rate decreases. It is also clear from figs. 5 ad 6 that as the process time becomes variable, the productio rate ca o loger reach to the maximum level of 800 uits eve for the reliable lie. I all cases however, as the lie legth icreases ad the buffer capacities decrease, productio rate decrease. Also, i all cases CM oly results i lower productio rate. CM-3 CM-9 CM-PM-5 REL-3 CM-5 CM-PM-3 CM-PM-9 REL Buffers Fig. 3 Lie productio rate uder differet factors (CV pt =CV dm =0.0) CM-3 CM-9 CM-PM-5 REL Buffers CM-5 CM-PM-3 CM-PM-9 REL-5 Fig. 4 Lie productio rate uder differet factors (CV pt =0.0; CV dm =2.1) VII. EXPERIMENTAL DESIGN I order to see sigificace of the effects of sigificat factors o lie productio rate, a geeral factorial desig was set up with five factors each at three levels. Thus, lie legths of 3, 5, ad 7; buffer capacities of 0, 2, ad 6; process time CV of 0, 0.2, ad 0.7; demad CV of 0, 0.2, ad 0.7; ad maiteace policies of REL, CM, ad CM-PM cases were CM-3 CM-5 CM-9 CM-PM-3 CM-PM-5 CM-PM-9 REL-3 REL-5 REL Buffers Fig. 5 Lie productio rate uder differet factors (CV pt =2.1; CV dm =0.0) 644

8 Iteratioal Sciece Idex, Idustrial ad Maufacturig Egieerig waset.org/publicatio/13084 CM-3 CM-5 CM-9 CM-PM-3 CM-PM-5 CM-PM-9 REL-3 REL-5 REL Buffers Fig. 6 Lie productio rate uder differet factors (CV pt =2.1; CV dm =2.1) cosidered. The ANOVA results show i Table IV idicate that followig factors are sigificat: A: Lie legth: B: Buffers; D: Demad CV: E: Maiteace policy; ad three iteractios AE, BD, ad DE % of the variatio is explaied by these sigificat terms. It is iterestig that process time variatio was ot a sigificat factor for this model. I the geeral factorial desig model, the factors are cosidered as qualitative ad therefore the model is hierarchical. The productio rate is give as fuctio of sigificat factors ad their iteractios. Fig. 7, the ormal probability plot for the residuals shows that the ormality assumptio is valid. TABLE IV ANOVA FOR SELECTED FACTORIAL MODEL RESPONSE:PRODUCTION RATE Sum of Mea F Source Squares DF Square Value Prob. > F Model 4.3E E < sigificat A 4.2E E < B 7.9E E < D 8.4E E < E 1.94E E < AE < BD 1.13E < DE < Residual 1.07E Corrected Tot. 4.4E+6 Total DF: 242 The Model F-value of implies the model is sigificat. There is oly a 0.01% chace that a "Model F- Value" this large could occur due to oise. Values of "Prob > F" less tha 0.05 idicate model terms are sigificat. I this case A, B, D, E, AE, BD, DE are sigificat model terms. Values greater tha 0.1 idicate the model terms are ot sigificat. Other ANOVA related statistics are as follows: Std. Dev.=21.93; R-Squared=0.9756; Mea=591.67; Adj R- Squared=0.9734; C.V.=3.71; Pred R-Squared=0.9707; PRESS=1.279E+5; Adeq Precisio= The "Pred R-Squared" of is i reasoable agreemet with the "Adj R-Squared" of "Adeq Precisio" measures the sigal to oise ratio. A ratio greater tha 4 is desirable. The ratio of idicates a adequate sigal. Fial equatio, which relates the productio rate to the coded values of the sigificat factors, is give as follows: Productio Rate= *A[1]+7.34*A[2]-73.22*B[1] +7.29*B[2]+59.28*D[1]+20.67*D[2]-95.47*E[1]-24.07*E[2] *A[1]E[1]+0.56*A[2]E[1]+12.91*A[1]E[2]+2.19* A[2]E[2]+37.60*B[1]D[1]-11.48*B[2]D[1]-3.48*B[1]D[2] * B[2]D[2]-20.44*D[1]E[1]-6.75*D[2]E[1]-5.98* D[1]E[2]-1.61*D[2]E[2] DESIGN-EXPERT Plot ProductioRate Normal %Probability Normal Plot of Residuals Studetized Residuals Fig. 7 Normal probability plot of residuals VII. CONCLUDING REMARKS This paper has preseted a iterative mathematical model ad a computer simulatio procedure for a multi-stage productio flow lie operated accordig to demad at the last statio, while usig a push system at the itermediate statios. Based o the discrete mathematical model, simulatio process icorporates a three-stage procedure which allows the user to eter a set of data describig the system uder study, simulate the system iteratively util selected statistical criteria are satisfied, obtai the output, ad apply specific recommedatios for productivity improvemet util satisfied productio output is achieved. The simulatio model is very useful i estimatig productio lie productivity for realistic systems. It allows the lie desiger or maagers to evaluate effects of storage capacity ad repair/maiteace policies o productivity of a system. The model was utilized to see the optimum allocatios of storage uit capacities alog the lie if the equipmet were subject to radom failures. If all the equipmet had similar 645

9 Iteratioal Sciece Idex, Idustrial ad Maufacturig Egieerig waset.org/publicatio/13084 failure rates, it was observed that the optimum allocatio of buffer storages followed a bowl shape, meaig that more buffer spaces were allocated to the ceter statios. If oly oe statio was subject to failures, most of the buffers were allocated to the fial storage to achieve maximum productio output irrespective to the locatio of the ureliable statio beig either at the start, at the middle, or at the ed of the lie. As a future study, the suggested iterative model ca be icorporated ito iteractive computer software to be effectively utilized by egieers ad maagers. Simulatio model was utilized to ivestigate the effects of lie cofiguratios, maiteace policies, buffer capacities, process time variability, ad demad variability o productio rate of the lie. A factorial desig was set up to ivestigate the sigificat factors that affect the productio rate. It was foud that lie legth, buffer capacities, maiteace policies, ad demad variability had sigificat effects o productio rate. REFERENCES [1] Chu, C. ad Shih, W. Simulatio Studies i JIT Productio, Iteratioal Joural of Productio Research, 30 (11), 1992, pp [2] Fuuawa, T. ad Hog S.C., The Determiatio of Optimal Number of Kabas i a Just-I-Time Productio System, Computers Idustrial Egieerig, 24 (4), 1993, pp [3] Savsar, M. ad Aljawii, A., Simulatio Aalysis of Just-I-Time Productio Systems, Iteratioal Joural of Productio Ecoomics, 42, 1995, pp [4] Savsar, M., Effects of Kaba Withdrawal Policies ad Other Factors o the Performace of JIT Systems: A Simulatio Study, It. Joural of Prod. Res., 34 (10), 1996, pp [5] Savsar, M., Simulatio Aalysis of a Push-Pull System for a Electroic Assembly Lie, It. Joural of Prod. Ecoomics, 51, 1997, pp [6] Savsar, M. ad Choueii, H. M., A Neural Networ Procedure for Kaba Allocatio i JIT Productio Cotrol Systems, It. Joural of Prod. Research, 38 (14), 2000, pp [7] Olhager, J. ad Ostlug, B., A Itegrated Push-Pull Maufacturig Strategy Europea Joural of Operatioal Research, 45, 1990, pp [8] Hodgso, T.J. ad Wag, D., Optimal Hybrid Push/Pull Cotrol Strategies for Parallel Multistage System: Part II, Iteratioal Joural of Productio Research, 29 (7), 1991, pp [9] Wag, H. ad Xu, C., Hybrid Push/Pull Productio Cotrol Strategy Simulatio ad its Applicatios, Productio Plaig ad Cotrol, 8, 1997, pp [10] Beamo, B.M. ad Bermud, J.M., A hybrid push-pull Ccotrol algorithm for multi-stage, multi-lie productio systems, Productio Plaig & Cotrol,11(4), 2000, pp [11] Duri, C., Frei, Y., ad Dimascolo, M., Compariso amog three pull cotrol policies: Kaba, Base Stoc ad Geeralized Kaba, Aals of Operatios Research, 93(1), 2000, pp [12] Hillier, F.S. ad So, K. C., The Effect of the Coefficiet of Variatio of Operatio Times o the Allocatio of Storage Space i Productio Lie System, IIE Trasactios, (23), 1991, pp [13] Hillier, F.S., So, K. C., ad Bolig, R. W., Notes: Toward Characterizig the Optimal Allocatio of Storage Space i Productio Lie Systems with Variable Processig Times, Maagemet Sci. 39(1), 1993, pp [14] Papadopoulos, H. T. ad Heavey, C., Queuig Theory i Maufacturig Systems Aalysis ad Desig: A Classificatio of Models for Productio ad Trasfer Lies, Europea Joural of Operatioal Research, (92), 1996, pp [15] Papadopoulos, H. T., ad Vouros, G. A., A Model Maagemet System (MMS) for the Desig ad Operatio of Productio Lies, It. Joural of Productio Research, 35(8), 1996, [16] Powel, S. G. ad Pye, D. F., Allocatio of buffers to serial productio lies with bottleecs IIE Trasactios, 28, 1996, pp [17] Vouros, G. A. ad Papadopoulos, H.T., Buffer Allocatio i Ureliable Productio Lies Usig a Kowledge Based System, Computers & Operatios Research, 25(12), 1996, pp [18] Vouros, G. A., Vidalis, M. I., ad Papadopoulos, H. T., A Heuristic Algorithm for Buffer Allocatio i Ureliable Productio Lies, Iteratioal Joural of Quatitative Methods, 6(1), 2000, pp [19] Spiellis, D.D. ad Papadopoulos, C.T., Stochastic Algorithms for Buffer Allocatio i Reliable Productio Lies, Mathematical Problems i Egieerig, 5, 2000a, pp [20] Spiellis, D.D. ad Papadopoulos, C.T., A Simulated Aealig Approach for Buffer Allocatio i Reliable Productio Lies, Aals of Operatios Research, 93, 2000b, pp [21] Gershwi, S.B. ad Schor, J.E., Efficiet algorithms for buffer space allocatio, Aals of Operatios Research, 93, 2000, pp [22] Savsar, M. ad Youssef, A. S., A Itegrated Simulatio-Neural Networ Meta Model Applicatio i Desigig Productio Flow Lies, WSEAS Trasactios o Electroics, 2 (1), 2004, pp [23] Savsar, M. Effects of Maiteace Policies o the Productivity of Flexible Maufacturig Cells, OMEGA, Vol. 34, 2006, pp Mehmet Savsar is a professor ad chairma of the Idustrial & Maagemet Systems Egieerig Departmet at Kuwait Uiversity. Prof. Savsar received his B.Sc. degree from Blac Sea Techical Uiversity i Turey, his M.Sc. ad PhD. Degrees from the Pesylvaia State Uiversity, PA, USA i the area of Idustrial Egieerig. He has bee with Kuwait Uiversity sice Prior to joiig Kuwait Uiversity, he has wored as a faculty member at Aatolia Uiversity i Turey ad at Kig Saud Uiversity i Riyadh, Saudi Arabia ad as a researcher at Pesylvaia State Uiversity, USA. Prof. Savsar has taught a variety of courses i the areas of Productio Plaig ad Ivetory Cotrol, JIT Productio Cotrol, Quality Cotrol, Maiteace ad Reliability, Operatios Research, Stochastic Processes, Computer Simulatio, Plat Layout ad Facilities Plaig, Egieerig Cost Aalysis, ad Maufacturig Systems. His research iterests iclude: Modelig ad Aalysis of Productio Systems, Plat Layout, Reliability ad Maiteace Maagemet, Flexible Maufacturig Systems, Quality Cotrol, ad Just-I-Time (JIT) Productio Cotrol. He has over 100 publicatios i refereed iteratioal jourals ad iteratioal coferece proceedigs. He has completed several research projects. He serves i editorial boards of several iteratioal jourals ad is a referee to several jourals, icludig EJOR, IJAMT, Simulatio, It. J. of Systems Sciece, It. J. of Productio Ecoomics, It. Joural of Prod. Research. msavsar@yahoo.com 646