Arne Thesen: Optimal allocation of work... /3/98 :5 PM Page OPTIMAL ALLOCATION OF WORK IN A TWO-STEP PRODUCTION PROCESS USING CIRCULATING PALLETS. Arne Thesen Department of Industrial Engineering, University of Wisconsin-Madison 53 University Avenue Madison, WI 5376, U.S.A. Tel: (68) 6-8456 Fax: (68) 6-8454 e-mail: thesen@engr.wisc.edu We present an approach to the optimal allocation of work between stations for a twostep production system using dedicated pallets to transport parts between stations.. It is found that it is always best to assign more than a fair portion of work to the second step. The approach is based on the generation and analytic solution of Markov balance equations. Efficient state-space reduction techniques and even moderately large systems can be analyzed. KEYWORDS: Manufacturing cells, Kanbans, Performance analysis, Optimization. Introduction We consider the problem allocating work to stations in a two-step production process where parts of t different types are first processed in step one on a shared machine, and then in step two on one of t separate, type-specific stations. Such problems frequently occur when the first step in a production process for several related products must be performed at a specialized, capital intensive station, while subsequent processing is performed on more flexible manual stations. While the total amount of work to be performed is fixed, the fraction of this work that is to be performed in step one can be adjusted prior to the production run. Our goal is to find an allocation of work that maximizes the overall production rate. Intuitively, a good decision rule would be to
Arne Thesen: Optimal allocation of work... /3/98 :5 PM Page Common Machine Buffer Buffer Buffer Mean time = t Buffer Figure : Production system with a shared machine, three type-specific stations and three dedicated pallets for each part type. assign S times as much work to the second production step. However, we will show that, for the class of problems studied here, this not the best strategy. A small number of type-specific circulating pallets are used to carry parts between stations (Figure ). Parts are processed at the shared machine in the sequence in which the corresponding pallets appear in the input buffer. Upon completion of processing here, pallets proceed to the appropriate type-specific station. Here the pallet again stay with the part, and, upon completion of processing they return to the input buffer for the shared machine. All processing is in a FCFS sequence, and all processing times are modeled as exponentially distributed random variables with a mean of t for the shared machine, and a common mean of t for all type-specific stations. Identical numbers of
Arne Thesen: Optimal allocation of work... /3/98 :5 PM Page 3 the shared machine. No restrictions were made on the mean processing times for parts of different types. Based on assumptions of exponential service times and enumerable statespaces, the he used semi-markov decision processes (SMDP) (Howard, 97) to identify the next part to be processed upon the termination of each individual system state. The resulting optimal state transition matrix can be used to develop a rule-base for a state dependent sequencing system. Typical applications of this approach can be found in Seidmann and Schweitzer (984), Yih and Thesen (99) and Chen (99). Thesen (998) used a similar approach to evaluate the performance of three simple real-time heuristics for scheduling production on the system studied here. State-space explosion was a common problem for all these applications, and sparse matrix techniques were used to develop number solutions. No analytic results were reported. The use of dedicated pallets is a simple implementation of a kanban system (Sugimori et. al, 977). The performance of kanban systems has been extensively studied. Excellent reviews of recent research can be found in Berkely (99), Price, Gravel and Nsakanda, 994, and Buzacott and Shantikumar (99). Thesen (998) found that the use of dedicated, type-specific pallets could lead to optimal performance when few pallets were used and when the pallets were properly distributed. The problem can also be modeled as a multi-class Jackson network. For example, Buzacott and Shantikumar (993, p388) formulate and solve balance equations to obtain general expressions for the state occupancy rates for such networks. In general, the
Arne Thesen: Optimal allocation of work... /3/98 :5 PM Page 4 optimal allocation of work. The success of our approach is due to an efficient state-space reduction technique and the existence of a closed form solutions to the balance equations.. Performance Analysis The production rate for the system described above can be estimated from the utilization of the shared station as: Where: R ts = ρ / t R ts ρ t = Production rate for a system with t part-types and p pallets per part-type. = Utilization of the shared machine. = Mean processing time at the shared machine (step one). In order to estimate the utilization of the shared machine, we first build a statetransition model of the production process. Then we develop a suitable set of balance equations, and from that we develop a closed form expression for the production rate for a system with a given configuration. This equation is then optimized to find the allocation of processing times that yields the optimal production rate. We will exploit the assumption of equal processing times for all part-types at each step to develop a model that differs from conventional state-space models in that significantly fewer states are used. This feature significantly reduces the problem of state-space explosion that is usually encountered for most models of this type. State-space development
Arne Thesen: Optimal allocation of work... /3/98 :5 PM Page 5 Shared machine Type machine Type machine 3 4 5 6 7 8 9 3 4 3 5 6 7 8 Figure : Conventional state transition diagram for two-station/two-pallet system. State zero is the only state where the shared machine is idle pallets, including the sequence of pallets in different buffers. Observe that state is the only state where the shared station is idle. Also observe that state 5 describes the state where a part of type (a square) is in the shared machine, three pallets are waiting in this machines buffer, and no parts are processed at any of the stations. The successor-state to state 5 is state 9. Here a part of type is processed at station and a part of type (a circle) is processed at the shared machine. The successor-state to state 9 is either state 4 or state 7, depending on which part finishes first. The state-space enumeration approach described above has the drawback that the state-space will grow exponentially with the number of pallets and stations. This is because each premutation of pallets in the shared machine buffer calls for a different state. However an alternative state-space model can be developed where information
Arne Thesen: Optimal allocation of work... /3/98 :5 PM Page 6 probability, all physical states with identical populations of pallets at the different stations. The resulting model has significantly fewer states, however, the new state occupancy rates are identical to the ones observed for the original model. On the other hand, the branching behavior is quite different. This is because each new state now represent all states with a given pallet distribution, and a state will have different successor states depending on which part-type triggers the end-of-service event. Hence successor states will be selected with probabilities proportional to the relative population of different pallets in the shared machine (each pallet at the station is equally likely to be served). Figure 3 shows the reduced state-space for the two-parts-two pallets system. Shared machine Step - machine with largest buffer population Step - machine with smallest buffer population 3 4 Figure 3. Reduced statespace for system with two part-types and two palets for each type. Each digit describer the pallet population at the indicated station. Balance equations Assuming that all processing times are exponentially distributed, it can be shown
Arne Thesen: Optimal allocation of work... /3/98 :5 PM Page 7 state, a set of balance equations can now be formulated and solved for the probability that the system is in any one of its different states. Although the statespace is potentially large, the utilization of the shared machine, and hence the overall production rate can be computed from information about the one state where this machine is idle. Symbolic solution of the balance equations yields the following simple expression for the occopancy probability for state zero (the only state where the shared machine is idle): π ( T, P ) = PS i = t c i t i where: π (T,P) = Probability that the system with P pallets and T part types is in state zero. = Probability that the shared machine is idle. c i = Scenario dependent coefficients, see below. t i = Mean processing time at step i. For the two-type, two-pallet system illustrated in Figures and 3, the probability of being in state zero is: π (,) = 3 4 t t t t + + 4 + 6 + 6 t Representative expressions for π for a few other systems configurations are given in Appendix. For other configurations, the values of c i can easily be computed recursively for systems with any number of part-type and one pallet for each type, as well as for systems with two part-types and any number of stations. At this time we are not aware of t t t
Arne Thesen: Optimal allocation of work... /3/98 :5 PM Page 8 3. Optimization Since state zero is the only state where the shared machine is idle, the production rate of the system can be computed as ( ρ )/t. Without a loss of generality, we can scale the mean processing times such that that t + t =. In this case, the production rate for a system with S stations and P pallets per station can be shown to be: P, S ( t, where a is shorthand for t / t. t ) = PS i = c α i i ( + ) α R [Equation ] 4. 3.5 3..5..5..5. Four pallets per part-type One pallet per part-type...4.6.8. Mean processing time at the type-specific stations (t ) Figure 4: Production rates for three station system using between and 4 pallets. The behavior of Equation is illustrated in Figure 4 for production systems with three type-specific stations and between one and four pallets for each part type. It is seen that the production rate is quite sensitive to the value of t. Using a symbolic software package such as MathLab, the optimal value of t can now be found. Optimal values for t and the resulting production rates for system configurations with two to four pert-types
Arne Thesen: Optimal allocation of work... /3/98 :5 PM Page 9 3.47 3.56 4.58.7..6 4.63 3.67 4.67.3..7 3. 4. 5..33.5. 4. Discussion Intuitively we had expected that the best allocation of work to be one where work was allocated in proportion to the number of processors available for each step. Hence we had expected to see t = t * 3 for a system processing three part types. This is not what happened. For the case where only one pallet of each type was used, it was always best to assign no work to the shared processor. This finding may be explained by the observation that the interval between arrivals at the type-specific station may be quite lengthy when only a single pallet of each type is used. In order to investigate the efficiency of the proposed allocation of work, we compared the resulting production rates for a number of different scenarios with the production rates that were observed when the intuitive assignment t =S*t was used. The results are shown in Table 4 where the percent improvement in performance is used as a performance measure. Table : Improvement in production rates when compared to the t =S*t policy Pallets Stations 3 4.% 4.7% 6.%.9% 4.8% 5.6% 3.%.9% 3.5% 4.3%.%.8%
Arne Thesen: Optimal allocation of work... /3/98 :5 PM Page number of part-types increase. This may cause type-specific machines to starve unless the step one processing times are reduced in the way suggested by the present algorithm. In summary, it is always best to assign more than a fair portion of work to the second step, when this is done, the effect of the starvation caused by congestion at the first state is minimized. The success of this approach followed from our ability to develop analytic solutions to a set of balance equations. While we have successfully solved such equations numerically for systems with more than 5, states, we have not been able to develop analytic solutions for cases with more than about 5 states. Fortunately, we were able to employ a state-space reduction technique that reduced the number of equations by several orders of magnitude. For example, a conventional state transition model of the 3- station/3-pallet model has 5,48 states while our model has states. The technique is based on the fact that the only state of interest is the state where the shared machine is empty. Since identical processing rates were assumed for all parts at the shared machine (/t ) and at the step two machines (/t ), all states with similar part populations exhibited identical behavior, and, they could be merged into a small number of super states. Of course, this limits our findings to the class of problems where all parts have similar processing times at each of the two steps. However, the general trends observed here should be similar for other scenarios, and numeric techniques may be used in these cases. We also explored the use of other common heuristic rules to control the system
Arne Thesen: Optimal allocation of work... /3/98 :5 PM Page Two extensions of our model should be mentioned. The first used limited buffer sizes, and employed one more pallet than could be accommodated in a station s buffer. The performance deteriorated. This was because blocking frequently occurred at the shared machine. The performance was in fact almost identical to the one observed for several other systems using efficient rules without blocking avoidance. The second modification routed pallets after processing was completed. The performance improved. We observe that this performance was almost identical to the one seen for other scheduling strategies with look-ahead capabilities. We conclude that an important determinant in the performance of a system is the information provided to the scheduling system. 5. Conclusion We have studied the performance of a class of kanban driven two-step production systems making parts of several different types. The allocation of work between the two steps can be adjusted prior to the production run. All parts are first processed on a shared machine in step one, and then on separate type-specific for step two. Processing times were assumed to be exponentially distributed with a mean of t for step one and a mean of t for step two for all part types. We found that the production rate of a system could in all cases be improved by allocating disproportional more work to the second step in the process. This effect was most significant for systems configurations with many part-
Arne Thesen: Optimal allocation of work... /3/98 :5 PM Page blocking. We also studied two extensions on the system where limited buffer sizes caused blocking to be induced. The first extension ignored any information about blocking, the second extension incorporated information about buffer availability times. The resulting performance was similar to the performance of other systems using the same amount of information. 6. References Berkely, B. J. 99, A Review of the Kanban Production Control Research Literature, Production and Operations Management,, 4, 393-4. Borst, S.C. and O.J. Boxma, O.J., 997, Polling Models with and without Switchover Times, Operations Research, Vol 45, 536,545. Buzacott, J.A. and G. Shantikumar, 99, A General Approach for Coordinating Production in Multiple-Cell Manufacturing Systems, Production and Operations Management,,, 34-44. Buzacott and Shantikumar 993, Stochastic Models of Manufacturing Systems, Prentice- Hall. Chen, J.C., 99, State Dependent Scheduling for Manufacturing Systems, Ph.D. Dissertation. Department of Industrial Engineering, University of Wisconsin-Madison. Chen, J.T. and A. Thesen, 998, Performance Evaluation of Manufacturing Cells using Rotation Rules, Proceedings of the 3 rd International Conference on Computers in Industrial Engineering, Chicago, IL Harrison, J.M. and L.M. Wein, 99, Scheduling Network of Queues: Heavy Traffic Analysis of a Two-Station Closed Network. Operations Research 38, 5-64. Howard, R.A., 97, Dynamic Probabilistic Systems. Volume II: Semi-Markov and Decision Processes, John Wiley & Sons, New York, New York. Nelson, B., 996, Stochastic Modeling, Analysis & Simulation, McGraw-Hill, New York, New York. Price, W, M. Gravel and A.L. Nsakanda, 994, A Review of Optimization Models of Kanban-Based Production Systems, European Journal of Operational Research, Vol. 75, 994.
Arne Thesen: Optimal allocation of work... /3/98 :5 PM Page 3 Seidmann, A. and A. Tennenbaum, 994, Throughput Maximization on Flexible Manufacturing Systems, IIE Transactions, Vol. 6, No., 9-. Seidmann, S. and R. Weber, 993, A Survey of Markov Decision Models for Control of Networks of Queues, Queueing Systems, 3, 9-34. Sugimori, Y., K. Kusunoki, K. Cho, and S. Uchikawa (977), Toyota Production System and Kanban System Materialization of Just-In-Time and Respect-for-Human System, International Journal of Production Research, 5, 6, 553-564. Thesen, Arne (998). Some simple but efficient push and pull heuristics for production sequencing for certain flexible manufacturing systems. International Journal of Production Research, to appear. Yao, D.D. and J.A. Buzacott, 984, Modeling a Class of State-Dependent Routing in FMSs, Annals of OR, 3, 53-68. Yih, Y. and A. Thesen, 99, Semi-Markov Decision Models for Real-Time Scheduling, International Journal for Production Research, Vol. 9, No., 33-346. Appendix Pallets States π 3 /(+α+α ) 6 /(+α+4α +6α 3 +6α 4 ) 3 /(+α+4α +8α 3 +4α 4 +α 5 +α 6 ) 4 5 /(+α+4α +8α 3 +6α 4 +3α 5 +5α 6 +7α 7 +7α 8 ) Three pallet types Pallets States π 4 /(+3α+6α +6α 3) /(+3α+9α +4α 3 +54α 4 +9α 5 +9α 6 3 /(+3α+9α +6α 3 +78α 4 +α 5 +5α 6 +5α 7 +68α 8 +68α 9 ) 4 35 /(+3α+9α +7α 3 +8α 4 +4α 5 +69α 6 +,89α 7 +4,83α 8 +,3α 9 +,5α +34,65α +34,65α ) Four pallet types Pallets Reduce π d States /(+4α+α +4α 3 +4α 4) 5 /(+4α+6α +6α 3 +4α 4 +6α 5 +44α 6 +5α 7 +5α 8