Scheduling Policies, Batch Sizes, and Manufacturing Lead Times

Transcription

1 To appear in IIE Transactions Scheduling Policies, Batch Sizes, and Manufacturing Lead Times Saifallah Benjaafar and Mehdi Sheikhzadeh Department of Mechanical Engineering University of Minnesota Minneapolis, Minnesota 55455, USA Abstract: In this paper, we examine the impact of scheduling policies on batch sizing decisions in a multi-item production system. We also investigate the joint effect of scheduling policies and batch sizing decisions on production lead times. In particular, we compare the performance of a first come-first-served (FCFS) policy with a group scheduling (GS) policy and study the effect of both on the optimal batch size. We show that a group scheduling policy can lead to significant performance gains, as measured by reduced lead times and higher production rates, relative to the FCFS policy and characterize conditions under which these gains are realized. We also study the impact of the GS policy on other system operating parameters. In particular, we find that using a group scheduling policy eliminates the need for batching, preserves system capacity despite the presence of setups, and accommodates higher product mix variety. These results are shown to be very different from those obtained for the FCFS policy and to have important implications for the management and control of multi-item production facilities. The author's research is supported by the National Science Foundation under grant No. DMII

2 1 Introduction In this paper, we examine the impact of scheduling policies on batch sizing decisions in a multi-item production system. We also investigate the joint effect of scheduling policies and batch sizing decisions on production lead times. In particular, we compare the performance of a first come-first-served (FCFS) policy with a group scheduling (GS) policy and study the effect of both on the optimal batch size. We show that a group scheduling policy can lead to significant performance gains, as measured by reduced lead times and higher production rates, relative to the FCFS policy and characterize conditions under which these gains are realized. We also study the impact of the GS policy on other system operating parameters. In particular, we find that using a group scheduling policy eliminates the need for batching, preserves system capacity despite the presence of setups, and accommodates higher product mix variety. These results are shown to be very different from those obtained for the FCFS policy and to have important implications for the management and control of multi-item production facilities. Despite its importance, the relationship between batch sizes and dynamic performance measures of production systems, such as flow time, work-in-process inventory, and production rates, has received little rigorous treatment in the literature. Although a number of analytical models have been proposed for the general performance evaluation of manufacturing systems [2] [3] [10], none of these models deals explicitly with the relationships between batch sizes and performance. In the queueing literature, a significant body of work exists on queues with bulk arrivals and bulk service times [8]. However, exact results exist only for simple models. None of these models account for setup times between batches and/or the possibility of alternating priority scheduling. In the production and manufacturing literature, Karmarkar [4] was the first to examine, using a queuing model of a single machine, the impact of batch sizes and setup times on levels of WIP and lead times. This model is extended in [5] and [6] to the multiitem/multi-machine case where a procedure for obtaining optimal batch sizes is also -2-

3 described. Using a variation on the single machine model in [4], Kekre [7] studies the effect of increasing product mix variety on the size of optimal batches. His findings indicate that increased variety tends to increase the optimal batch size. A queueing model similar to the one described in [4] is independently proposed by Zipkin [12] and used to model the aggregate behavior of a batch production facility. Benjaafar [1] evaluates the relationship between batching and machine sharing decisions in multi-product manufacturing systems. He shows that in the presence of setup times greater machine sharing between multiple part types generally induces larger batch sizes. All of these studies assume a FCFS policy in scheduling arriving batches. That is, no attempt is made to economize on setups by giving priority to batches of part types for which the production facility is already setup. Similarly, no attempt is made to systematically group batches based on their setup requirement and to produce them in an alternating priority. In fact, Kekre [7] argued that using a setup avoiding, or a look-ahead, policy leads to little additional benefits, as measured by average waiting time, relative to the FCFS policy. Using an approximation model, he concluded that "efforts to reduce queue times by processing all parts available from the queue for which the cell is already setup and saving setup time results in little gain" [7, p. 330]. In this paper, we show that this conclusion is incorrect and that indeed a setup avoiding, or group scheduling, policy can lead to significant gains. More importantly, we show that such a policy has very important and surprising implications for batch sizing decisions that are very different from those of the FCFS policy. The policy has also several counter-intuitive implications for planning and managing multi-product manufacturing facilities regarding system capacity, part mix variety, and setup requirements. 2 The First Come-First Served Policy Processing parts in batches is preferable to the processing of parts in lots of size one when setup times are significant. By batching parts that have similar manufacturing -3-

4 requirements, the frequency of setups is reduced, which in turn reduces part lead times and provides greater production capacity. Excessive batching can, however, result in performance deterioration. Increasing batch sizes increases the batch processing times which, in turn, increase the batch lead time (or flow time) through the manufacturing facility. The relationship between batch sizes and lead times, under a FCFS batch scheduling policy, can be illustrated, as suggested by Kekre [7] and Benjaafar [1], by modeling a machine as a multi-class single server queue. Parts arrive to the machine in batches of size Q where they wait for service. When the machine finishes its current batch, it is set up for the next batch, if different from the current one, and starts processing parts from that batch. The product mix consists of K part types each with an average demand D i. The demand is assumed to be evenly distributed among part types so that D i = D/K where D is the total average production demand. This assumption is made in order to be able to examine the impact of part variety on batch sizing decisions and on lead time. The processing time is denoted by X and is assumed to be generally distributed with a mean of 1/µ and variance of σ 2. This assumption allows for different part types to have different processing times. A batch incurs a setup time of τ when the previous batch on the machine is of a different type. Otherwise, no setup is required. The average arrival rate of batches to the machine is given by λ = D/Q, with batch inter-arrival times assumed to be exponentially distributed (the robustness of this assumption is discussed in [4]). Assuming an unlimited buffer size and first-come first-served sequencing of batches at the machine, average part flow time can be calculated, based on that of a M/G/1 queue, as F FCFS = E(S) + λe(s 2 ) 2(1 - λe(s)), (1) where E(S), and E(S 2 ) are respectively the first and second moment of batch processing and setup time with E(S) = Q/µ + τ(k - 1)/K and E(S 2 ) = E(S) 2 + Qσ 2 + τ 2 (K - 1)/K

5 given by In order to ensure system stability, there is a minimum feasible batch size that is Q min (FCFS) = Dτ(K - 1) K(1 - ρ), (2) where ρ = D/µ and represents actual machine utilization (i.e. excluding time spent on setups). Equivalently, for a fixed batch size, there is a maximum feasible setup time, τ max, Q K(1 - ρ) τ max (FCFS) = D(K - 1), (3) and a maximum feasible level of part variety, K max, K max (FCFS) = Dτ Dτ - Q(1 - ρ). (4) The value of Q min is an increasing function of setup time τ and part variety K. The value of τ max is, on the other hand, a decreasing function of K and an increasing function Q. The value of K max is an increasing function of Q and a decreasing function of τ. These bounds mean that, in order to ensure production feasibility, part mix variety and setup requirements must be limited while a minimum batch size must always be maintained. Since a fraction of system capacity is spent on setups, the maximum feasible production rate, or maximum throughput, is given by Q TH max (FCFS) = Q τ(k - 1) + µ K (5) which is an increasing function of Q and a decreasing function of K and τ. In the limit case, as Q, TH max µ, with µ being the machine's maximum throughput in the absence of any setups. It is easy to see that increasing Q increases the batch processing time but decreases the frequency of setups and thus increases system capacity. Depending on the length of setup times and the level of machine utilization, larger batches may initially result in a net reduction of part flow times. However, as Q is further increased, the advantages of batching are eventually eroded and flow time starts to increase with increases in Q. Noting -5-

6 that the expression of average flow time is a convex function of Q, the batch size that minimizes flow time can be obtained by the first order condition of optimality as Q * (FCFS) = ae + a2 e 2 + ad(be - cd), (6) ad where a, b, c, d, and e are all positive parameters with a = K(2µ - D), b = 2µτ(K - 1)(µ - D) + KDµ 2 σ 2, c = Dµ 2 τ 2 (K - 1)(K - 2)/K, d = 2µK(µ - D), and e = 2Dτµ 2 (K - 1). The value of Q*(FCFS) can be shown to be monotonically increasing with part variety, K, and setup time, τ, and is, thus, generally greater than one. For a given batch size, it should be noted that the average time spent on setups per batch is given by T setup (FCFS) = τ(k - 1)/K (7) which means that only K - 1 out of every K batches incur a setup. In other words, with a probability of 1/K, two consecutive batches are of the same type. This, in turn, means that the length of a production run per setup is variable and may be greater than a single released batch. Thus, in optimizing the batch size under a FCFS scheduling policy, we are not optimizing the length of each production run but only the size in which parts are released to the system. 3 The Group Scheduling Policy In practice, there is often an attempt to minimize the frequency of setups by dynamically reordering batches so that part types for which the machine is currently setup are processed first. This means that the machine would process all the batches of a particular type, that are currently in queue, before switching to another part type. In this section we examine the effect of using such a batch sequencing rule on system performance, on the optimal batch size, and on the setting of other system operating parameters. Specifically, we consider a sequencing rule where once the machine is setup (setups are assumed to be non-zero) for a particular part type, it continues processing batches from that type until all batches are exhausted. It is subsequently setup for the next -6-

7 part type. The machine is assumed to switch from one part type to the next in a cyclic order. This sequencing rule is known in the queuing literature as a cyclic and exhaustive alternating priority policy and is usually studied in the context of polling systems [9]. In the manufacturing literature, such rules are often referred to as family or group scheduling (GS) rules. Proposition 1: Q * (GS) = 1, Q min (GS) = 1 and τ max (GS)=, K max =. Proof: The expression of average part flow time for the GS scheduling policy is given by [9] F GS = E(X) + λe(x 2 ) + τ(k - λe(x)), (8) 2(1 - λe(x)) where E(X) and E(X 2 ) are respectively the first and second moment of batch processing time, excluding setup time, with E(X) = Q/µ and E(X 2 ) = E(X) 2 + Qσ 2. Expression (8) can also be rewritten as F GS = a d Q + f d, (9) where a and d are as defined earlier and f = K 2 [τσµ(µ - D/K) + Dµ 2 σ 2 /K]. Since a, d and f are all positive parameters independent of Q, average part flow time is a linearly increasing function of Q and, thus, the optimal batch size, Q *, is equal to one. The stability condition for this policy is simply λe(x) < 1, or equivalently D/µ < 1. Consequently, the minimum feasible batch size, Q min, is one, and both the maximum feasible setup time, τ max, and the maximum feasible product variety, K max, are unbounded. These results lead to the following corollary: Corollary 1: For τ > 0, TH max (GS) > TH max (FCFS) and TH max (GS) is independent of τ, where TH max ( ) is the maximum feasible throughput rate. Proof: It follows from the stability conditions for the two policies that the maximum feasible throughput rates (i.e., system capacity) are, respectively, given by -7-

8 Q TH max (FCFS) = Q τ(k - 1) + µ K (10) and TH max (GS) = µ. (11) Noting that TH max (FCFS) is a strictly decreasing function of τ, it is easy to show that TH max (GS) > TH max (FCFS). Proposition 1 and corollary 1 have important implications for system operation and management. They go counter to long held beliefs regarding the inevitability of batching in the presence of setups. In fact, these results not only do they show that batching is not necessary when a setup avoiding policy is in place but that it is not even optimal. Furthermore, the GS policy improves system capacity by maintaining a maximum feasible production rate that is unaffected by increases in setup times. In practice, this capability is important for systems where sustaining high production volumes is desirable and/or where parts with highly different setup requirements are simultaneously produced. The GS policy also preserves system capacity despite increases in part variety. This capability is important for manufacturing systems with a highly diversified product portfolio or those that compete based on customized products, among others. Note that with a FCFS policy, system capacity quickly deteriorates with increases in setup time or part variety. The impact of the FCFS and GS policies on Q *, Q min,τ max, and K max is graphically depicted in Figures 1, 2, and 3. The above somewhat counterintuitive results can, in part, be explained by the following proposition: Proposition 2: Let T setup (GS) be the average time spent on setup per batch, then for fixed D, µ and Q, T setup (GS) is a constant independent of both product variety, K, and setup time, τ. The value of T setup is given by T setup (GS) = Q(1/D - 1/µ). -8-

9 Figure 1 The effect of setup time on average flow time in the FCFS and GS policies (K = 2, µ = 1, D = 0.6, Q = 11) Figure 2 The effect of batch size on average flow time in the FCFS and GS policies (K = 2, µ = 1, D = 0.8, τ = 1.3) -9-

10 Figure 3 The effect of part variety on average flow time in the FCFS and GS policies (Q = 10, µ = 1, D = 0.8, τ = 2.63) -10-

11 Proof: Let us first define average cycle time, T c, as the average time interval between two successive setups of the machine for the same part type. The value of T c can be calculated as T c = Kτ + N c Q/µ, (12) where N c is the average number of batches produced during a cycle time and is itself given by N c = λt c = DT c /Q. (13) The average time spent on setup per batch can now be obtained as T setup (GS) = (K/N c )τ, (14) where the ratio K/N c corresponds to the frequency with which setups are performed during a cycle time. Substituting for the values of N c and T c, the above expression can be rewritten as T setup (GS) = Q(1/D - 1/µ). (15) The fact that T setup (GS) is independent of both τ and K is itself surprising and counter-intuitive. It can however be explained by the fact that as either τ or K increases, the machine would spend more time processing batches after each setup, making these setups increasingly less frequent. We should note that this is not the case for the FCFS policy where the average time spent on setups per batch is given by T setup (FCFS) = (K - 1)τ/K (16) and is indeed increasing in both τ and K. If we divide T setup by the batch size Q to obtain the average time spent on setup per part, we find that in the case of the GS policy the value of this average to be independent of Q, which partly explains why a batch size of one is desirable; on the other hand, for the FCFS policy the average time spent on setup per part is decreasing in Q which similarly justifies the need for larger batch sizes in this case. As it is the case with the FCFS policy, the length of each production run under the GS policy is also variable. That is, the number of parts produced after each setup may be -11-

12 greater than the size, Q, in which parts are released to the machine. In fact, the average number of batches produced following each setup is given by (see Appendix) N Q = Dτ Q(1 - D (17) ), µ which means that the corresponding number of parts produced per setup is N P = Dτ (1 - D (18) ). µ This number can be viewed as the realized average production batch size. The value of N P is an increasing function of setup and machine utilization and is generally greater than one. Thus, although the optimal release batch size is one, parts could be produced in larger quantities. The GS policy will, in effect, result in a dynamic batch sizing policy that responds to changes in system loading and setups. It is interesting to note that N p is independent of Q and K so that regardless of the size of the release batch and the level of part variety, the same average number of parts will be produced per setup. This is not the case in the FCFS policy, where the realized batch size is determined by both Q and K and is independent of system loading conditions (under the FCFS policy, the probability that a given batch of size Q would incur a setup is (K- 1)/K, independently of machine utilization). An additional advantage to the GS policy is, thus, relieving production managers from determining the batch size in which parts should be released and in letting the actual production batch size be determined dynamically as a function of the system's loading conditions. In view of proposition 1 and Corollary 1, it is easy to find instances where the GS policy performs significantly better than the FCFS policy. For example, this will be the case when Q is sufficiently small, setup time is long, or product variety is high. In fact, when Q is below Q min (FCFS), τ is greater than τ max (FCFS) or K is larger than -12-

13 K max (FCFS), the FCFS policy leads to an infinitely large flow time (see Figures 1, 2 and 3). More generally, we have the following result: Proposition 3: F FCFS F GS if and only if (1) ρ ρ 0 and (2) Q Q 0, where and ρ 0 = [(K 2 - K + 2) - (K - 1)(K 3 - K 2 + 4)]/2, Q 0 = D(K - 1)[τ(2(1 - ρ) + K(K - 1)) + KDσ 2 )]/[ρ 2 - ρ(k 2 - K + 2) + (K 2 - K + 2)]. Proof: Let F ASY (FCFS) denote the asymptote of average flow time as a function of Q for the FCFS policy (a lower bound for F FCFS ). We first show that F ASY (FCFS) F GS if and only if ρ ρ 0 Noting that the expression of average flow time for the FCFS policy can be written as F FCFS = aq 2 + bq - c, (17) dq - e its asymptote can be obtained as F ASY (FCFS) = d a Q + bd + ae. (18) d 2 Therefore, F ASY (FCFS) F GS is equivalent to or, bd + ae d 2 f d (19) d(b - f) + ae 0 (20) which reduces to ρ 2 - ρ(k 2 - K + 2) + (K 2-2K + 2) 0. (21) The above inequality can be rewritten as (ρ - ρ 0 )(ρ - ρ 1 ) 0, with ρ 0 = [(K 2 - K + 2) - (K - 1)(K 3 - K 2 + 4)]/2 and ρ 1 = [(K 2 - K + 2) + (K - 1)(K 3 - K 2 + 4)]/2. Noting that ρ 1 1, the above inequality is equivalent to -13-

14 ρ ρ 0. (22) Next, we show that F FCFS F GS implies ρ ρ 0 and Q Q 0. Using the fact that if F FCFS F GS then F ASY F GS, it follows that if F FCFS F GS then ρ ρ 0. Now, noting that F FCFS F GS is equivalent to aq 2 + bq - c dq - e a d Q + f d, and using the fact that dq - e 0 and d(b - f) + ae 0 (see (20)), then the above ineqality can be rewritten as or Q cd - ef d(b - f) + ae, (23) where Q Q 0, Q 0 = D(K - 1)[τ(2(1 - ρ) + K(K - 1)) + KDσ 2 )]/[ρ 2 - ρ(k 2 - K + 2) + (K 2 - K + 2)]. Hence, we have shown F FCFS F GS ρ ρ 0 and Q Q 0. To show, the reverse, we note that when ρ ρ 0 and Q 0 Q, we have aq 2 + bq - c dq - e d aq + d f, which is equivalent to F FCFS F GS. The value of ρ 0 is an increasing function of K so that the range of utilizations over which the GS policy is more desirable increases as K decreases. The value of Q 0 is similarly an increasing function of K which in this case means that as K increases the range of batch sizes over which the GS policy is more desirable increases. The value of Q 0 is also an increasing function of setup time and processing time variance so that with increases in either τ or σ 2 the range of batch sizes that makes the GS policy superior increases. We should note that condition (2) could have been equivalently expressed in terms of either a critical setup time parameter or a critical part variety parameter. Numerical comparisons of average flow time between the GS and the FCFS policies are provided in Table 1. The GS policy can be seen to yield significant gains in -14-

15 performance over a wide range of operating conditions. It is particularly important to note that the difference in flow time at the optimal batch size for both the GS and the FCFS policies can be substantial. As anticipated from proposition 3, the difference in performance between the two policies tends however to diminish with increases in batch size. Using simulation, similar behavior was observed for the flow time variance. A small sample from the simulation is listed in Table 2. As mentioned in the introduction, Kekre [7] argued that a group scheduling policy (a look-ahead policy in Kekre's terminology) would result in little reduction in part flow times. Clearly, the results of this paper show that this is not necessarily true. Kekre bases his argument on a group scheduling policy similar to the one described here, except that the switching between part types follows a first come-first served discipline with setups occurring only if there is at least one part already in the queue. Such a policy is evidently superior to the strict cyclic GS policy described here and, thus, should result in at least equal benefits. Using simulation, this result was verified and two policies were, indeed, found to have similar characteristics and to provide relatively equal performance. A sample of the simulation results is included in Table 3 - the notation GS/FCFS and GS/CYC is used to differentiate between the policy described by Kekre and the strict cyclic policy discussed in this paper. As expected, the dynamic GS/FCFS performs slightly better than the cyclical GS/CYC policy, especially for larger batch sizes and higher number of part types. Similar results were also obtained recently by Wemmerlöv and Vakharia [11] in a simulation study. Because of the state-based nature in which changeover between batch types are made under a first come-first served discipline, analytical results are, however, difficult to obtain. 4. Conclusion -15-

16 The objective of this paper was to show that a group scheduling policy can lead to substantial improvements in performance relative to a first-come first-served policy. In addition to its impact on flow time, such a policy is found to have important and surprising implications for other system operating parameters. In particular, we found that using a group scheduling rule eliminates the need for batching, preserves system capacity despite the presence of setups, and accommodates higher product mix variety. The capability to produce a large variety of items in small batch sizes without significant setup time penalties and in short lead times has become in recent years increasingly important to the design and operation of most manufacturing systems. When setup times cannot be significantly educed, a group scheduling policy, similar to the one proposed here, certainly offers an attractive alternative for system designers and managers. Since the proposed group scheduling is not necessarily optimal, further performance improvements, through additional refinement of the policy, may be possible. Future research should thus focus on characterizing the optimal scheduling policy and examining its effect on batch sizing decisions. -16-

17 Table 1 Flow time comparisons between the FCFS and the GS policies K Q F FCFS F GS

18 Table 2 Flow time variance comparisons between the FCFS and the GS policies K Q V FCFS V GS

19 Table 3 Flow time comparisons between the GS/FCFS and the GS/CYC policies K Q F GS/FCFS F GS/CYC

20 Appendix Proposition 4: The average number of parts produced per setup is given by N p = Dτ/(1 - D/µ). Proof: Noting that and we get T c = Kτ + N c Q/µ, N c = λt c = DT c /Q, N c = D Kτ Q(1 - D µ ). Since N c is the total average number of batches produced per cycle, the average number of batches of each part type produced per cycle is N Q = D τ Q(1 - D µ ) and the corresponding avearge number of parts is N P = D τ (1 - D µ ). References -20-

21 [1] Benjaafar, S., "Modeling and Analysis of Machine Sharing in Automated Manufacturing Systems," The European Journal of Operational Research, 91, 1, 56-73, [2] Bitran, G. R. and S. Dasu, "A Review of Open Queueing Network Models of Manufacturing Systems," Queueing Systems, 12, , [3] Buzacott, J. A. and J. G. Shanthikumar, Stochastic Modeling of Manufacturing Systems, Prentice Hall, New Jersey, [4] Karmarkar, U. S., "Lot Sizes, Lead Times and In-Process Inventories," Management Science, 33, 3, , [5] Karmarkar, U. S. and S. Kekre, "Lot Sizing in Multi-Item Multi-Machine Job Shops" IIE Transactions, 17, 3, , [6] Karmarkar, U. S., Kekre, S., Kekre and S. Freeman, "Lot Sizing and Lead-time Performance in a Manufacturing Cell," Interfaces, 15, 2, 1-9, [7] Kekre, S., "Performance of a Manufacturing Cell with increased Product Mix," IIE Transactions, 19, 3, , [8] Kleinrock, L., Queuing Systems, Vol. I, John Wiley, New York, New York, [9] Takagi, H., "Queueing Analysis of Polling Models," ACM Computing Surveys, 20, 1, 5-28, [10] Viswanadham, N. and Y. Narahari, Performance Modeling of Automated Manufacturing Systems, Prentice Hall, Englewood Cliffs, New Jersey, [11]Wemmerlöv, U. and A. J. Vakharia, "On the Impact of Family Scheduling Procedures," IIE Transactiona, 25, 4, , [12] Zipkin, P. H., "Models for Design and Control of Stochastic, Multi-Item Batch Production Systems," Operations Research, 34, 1, ,

22 Biographical Sketch Saifallah Benjaafar is an Assistant Professor in the Department of Mechanical Engineering at the University of Minnesota. He holds Ph.D. and M.S. degrees from the School of Industrial Engineering at Purdue University and a B.S. degree from the Department of Electrical and Computer Engineering at the University of Texas, Austin. His research interests are in design, modeling and control of manufacturing systems. He is the author of over 30 technical papers in this area. He has served as area editor for the 1994 IERC Conference Proceedings and as a guest editor for a special issue of the International Journal of Flexible Manufacturing Systems on manufacturing flexibility. He is also a member of the editorial board of the International Journal of Industrial Engineering. Mehdi Sheikhzadeh is a doctoral student in the Department of Mechanical Engineering at the University of Minnesota. He holds a M.S. degree in Industrial Engineering from the University of Minnesota and a B.S. degree from Tehran University. His research interests are in aplying queueing models to multiple product manufacturing systems. -22-