Optimal buffer allocation strategy for minimizing work-inprocess inventory in unpaced production lines

IIE Transactions (1997) 29, 81±88 Optimal buffer allocation strategy for minimizing work-inprocess inventory in unpaced production lines KUT C. SO Department of Management, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong Received December 1992 and accepted February 1996 Several researchers have previously studied the problem of allocating buffer storage to maximize the throughput of a production line for a given total amount of buffer space. In this paper we study the optimal buffer allocation problem of minimizing the average work-in-process subject to a minimum required throughput and a constraint on the total buffer space. Although these two buffer allocation problems are closely related, our results show, surprisingly, that their optimal buffer allocations have very different patterns. Speci cally, we show that the optimal buffer allocations for the problem considered here generally exhibit a monotonically increasing property where an increasing amount of buffer space is assigned toward the end of the line. This monotonically increasing property generally holds for both the balanced and unbalanced lines. On the basis of our empirical results, we develop a good heuristic for selecting the optimal buffer allocations. 1. Introduction In a tightly coupled production system, disruptions or large variability in operation times at any one station can cause temporary stoppage of the entire system. Such disruptions or variability can be due to machine breakdowns, quality problems or the inherent nature of manual operations. These continual disruptions and variability can result in signi cant loss in the overall capacity of the production system. It is well known that providing buffer space between the stations can reduce the adverse effect of these disruptions and help to recover the loss of capacity in the system. Buffer space can be expensive. There is generally a physical limit to the oor space in the facility. Providing additional buffer space between stations might require some large investment cost to expand the facility. In situations where the items are bulky or must be handled carefully, buffer space between stations should be minimized so that the stations can be placed close to each other to reduce the material handling costs or the transportation times between stations. The optimal use of available buffer space becomes a very important issue. There have been substantial research efforts on the design of buffer storage in production lines. Buzacott and Hani n [1] and Sarker [2] gave two good reviews on earlier research studies, and Buzacott and Shanthikumar [3] provided a comprehensive coverage on existing results in this area. Many of these earlier studies have focused on two-station lines with a nite buffer capacity or longer lines with the same buffer capacity between stations. A main objective of these studies was to investigate the effect of the buffer capacity on the performance of the system. Only a few studies have focused on the effect of unequal storage allocations for compensating for the unequal effects of the variability in the operation times at different stations of the line. Several researchers have recently studied the optimal allocation and placement of buffers to maximize the ef ciency of a production line. Conway et al. [4] conducted simulation experiments to study the buffer allocation problem. They illustrated that the optimal placement of buffer space can signi cantly improve the throughput of a line. Hillier and So [5] and Hillier et al. [6] performed two rather comprehensive studies to characterize the optimal buffer allocation pattern. Extensive numerical results for lines with up to six stations and limited results for lines up to nine stations were provided. These results were then extrapolated to provide some general rules of thumb for optimally allocating buffer space to maximize the throughput for even longer lines. These previous three studies assumed that the lines are balanced. Powell [7] studied the buffer allocation problem for unbalanced lines. Rules of thumb for the optimal sequential placement of buffer space were developed. Glasserman and Yao [8] studied the buffer allocation problems for production lines with more general types of blocking, and provided some results regarding the structure of the optimal allocations. All the above studies focused primarily on maximizing the throughput of a line for a given total amount of 0740-817X # 1997 ``IIE''

82 So buffer space. Although the amount of work-in-process in a line is closely related to (in fact, is bounded by) the total available buffer space, minimizing the total buffer space does not necessarily minimize the amount of workin-process because the buffers can be empty at many time instances. In situations where work-in-process is of much interest, it is useful to determine the optimal strategy of allocating buffer space to minimize workin-process. Very few studies have focused on how buffer allocation can affect the average work-in-process in the system. Smith and Daskalaki [9] studied the buffer allocation in automated assembly systems. They found that capacity allocation should favor downstream buffers in a balanced facility when both throughput and work-inprocess are jointly considered. (See also the discussions in [10].) Very recently, Cheng [11] investigated the throughput and inventory tradeoff in designing production lines. We consider the situations where there is a physical limit such as oor space on the available buffer size in the facility. Within this physical limit, the objective is to determine the strategy of utilizing this space to achieve optimal performance. Speci cally, we assume that the costs of holding work-in-process inventory are high (compared with the physical space) and thus our objective is to minimize the average work-in-process. From Little's law, this objective is also equivalent to minimizing the production lead time of the line, which is the time between a job's entering the line and the job's completing operation at the last station. Therefore our focus here is consistent with the current thrust of reducing manufacturing lead times, where many companies have adopted the strategy for competing on time to stay competitive in the global markets. (See [12] and [13] for two insightful analyses on the trend towards time-based competition.) Speci cally, the objective of our buffer allocation problem is to minimize the average work-in-process subject to a minimum required throughput (to meet the outside demands) and a constraint on the total available buffer space (physical space in the facility). In Section 2 we formulate the model and describe our solution approach. Sections 3 and 4 present the results for a balanced line and an unbalanced line, respectively. On the basis of our results we propose a simple sequential buffer allocation scheme in Section 5 and discuss its performance. Section 6 provides some concluding remarks. 2. Model formulation and solution approach We model the production line by using the classical system of nite queues in series. The system consists of N single-server stations, corresponding to the N workstations of the production line. Every unit receives service from station 1 through station N in a xed sequence. The service times at station j are i.i.d. random variables with mean w j and are independent of other stations. There always is a unit available to begin service at station 1. For j ˆ 2; 3;...;N, the buffer capacity between stations (j 1) and j is a xed non-negative integer q j. Blocking occurs when a unit completes service at a station and the buffer at its downstream station is full; thereby this unit will be held by the server at the station where work was completed and service cannot begin on the next unit by this server. Blocked units are subsequently released to the downstream station as room is cleared in the buffer at the downstream station. A nished item is immediately removed from the system such that station N is never blocked. The two basic performance measures of the system are the throughput and the average work-in-process (WIP). The WIP includes the items at station 1 through station N, including the one item being processed or held at station 1. However, it does not include any raw materials held in front of station 1, as materials are assumed to be always available for processing at station 1. We choose an appropriate timescale such that the expected total processing time for the whole task is equal to N time units. Therefore the maximum throughput of the system isequaltoone,whichcanbeachievedonlywhenthe line is perfectly balanced (w j ˆ 1 for each j) andthereis no variability in the operation times or the buffer capacity q j is in nite for all j. Our buffer allocation problem is to select the optimal buffer allocation that minimizes the average work-inprocess subject to a minimum required throughput r and a constraint on the total buffer space Q. Speci cally, let the vector q =(q 2 ;q 3 ;...;q N ) represent the buffer allocation. Let R(q) andl(q) denote, respectively, the throughput and the average work-in-process in the system with buffer allocation q. Our problem can be formulated as: (P1) Minimize L(q) X N subject to R(q) r; q j Q; q j 0 integer. j ˆ 2 There is a close relationship between problem (P1) and the buffer allocation problem studied by various researchers cited in the preceding section. Their buffer allocation problem is to maximize the throughput subject to a constraint on the total buffer space Q, and can be formulated as: (P2) Maximize R(q) subject to XN q j ˆ Q; q j 0 integer. j ˆ 2 From the monotonicity result that R(q) is increasing in q (see [14]), it follows that the optimal throughput R * (q) for problem (P2) is increasing in the total buffer space Q. Therefore the results obtained for problem (P2) can be directly used to solve the buffer allocation problem of minimizing the total buffer space (and determining the

Buffer allocation strategy for unpaced production lines 83 corresponding buffer allocation q) subject to a minimum required throughput. Clearly, the amount of work-in-process in the system is bounded above by the total buffer capacity Q plus N items possibly held or processed by the N servers. However, minimizing the total buffer space Q does not necessarily minimize the average work-in-process because a buffer can be empty at some time instant. Therefore, while the problems (P1) and (P2) are closely related, the strategies of allocating buffer space could be very different. Indeed, we will show in the subsequent sections that the optimal buffer allocations for problem (P2) follow a very different pattern from those for (P1). Previous research has established several useful structural properties on the effect of buffer space q on the throughput R(q), including the reversibility (see, for example, [15]), monotonicity and concavity properties (see, for example, [8,14]). These properties have helped reduce the search space for the optimal buffer allocation problem (P2) (see, for example, [6]). Unfortunately, these properties do not hold for the average work-inprocess L(q). To solve problem (P1) for any given r and Q, we basically need to enumerate all feasible buffer allocations q and then select the best buffer allocation. We model the operation times by using some phase-type distributions such that the underlying queueing process can be formulated as a nite state, continuous time Markov chain for any given buffer allocation q. The steady-state probabilities of the resulting Markov chain can be solved by an ef cient Gauss±Seidel procedure (or any other numerical procedure), from which the two performance measures R(q) andl(q) can be computed. However, the number of states of the Markov chain grows rapidly as N and q j increase, and the number of feasible allocations also increases rapidly as N and Q increase. Exact numerical results can be obtained only for small values of N and Q. (There has been much research on analytical approximations for large systems; see, for example, [3] for discussions and references for some of these approximations. However, the degree of accuracy given by these approximations is not good enough for our purpose here.) To reduce further the search space in our study, we limit the capacity of each individual buffer q j to be no more than 9. (Previous results have suggested that the marginal increase in throughput for each additional buffer space at a station is very small when the buffer capacity exceeds 9.) Therefore all results presented in this report should be interpreted in the light of these additional constraints on the individual buffer capacity. However, we believe that the main conclusions of this study (e.g., the monotone increasing property of most optimal buffer allocations) remain valid when these additional constraints are relaxed. Given the computational limitations, our solution approach was to employ exhaustive enumeration for small systems and then to extrapolate the results to larger systems. Thus our goal is to identify any interesting pattern for the optimal buffer allocations by using the results from small systems, and to develop some simple and ef cient heuristic for selecting optimal or nearoptimal buffer allocations for larger systems. We believe that the insights provided in our study can be generalized to much larger systems that are too big for exact calculations. 3. Balanced lines We rst discuss the cases where the production line is perfectly balanced, i.e., w j ˆ 1; 8 j. In each case we enumerated all feasible buffer allocations (with each q j 9) and computed their corresponding R(q) and L(q), from which the optimal buffer allocation for problem (P1) can be determined for any given minimum required throughput r. We summarize the results for the three-station case with exponential operation times in Table 1. Each row in Table 1 gives the optimal allocations for this value of r. Each pair of entries (Q, q * ) corresponds to the optimal allocation q * for a given total available buffer size Q. In particular, the value of Q given in the rst pair in each row gives the minimum total buffer size that is needed to achieve this required throughput r. For instance, when r ˆ 0:6, the minimum required total buffer space is 1, with the corresponding optimal buffer allocation q 2 ; q 3 ˆ 0;1. This buffer allocation remains optimal when the total available buffer size Q is larger than 1. Similarly, when r ˆ 0:65, the minimum required total buffer space is 2, with the corresponding optimal buffer allocation q 2 ; q 3 ˆ 1;1. However, when the total available buffer size Q is equal to or larger than 3, the optimal buffer allocation is given by q 2 ; q 3 ˆ 0;3. Three interesting observations can be made from Table 1. First, the optimal buffer allocations q * have the monotonically increasing property that q 2 q 3. Secondly, minimizing the total buffer size does not necessarily minimize the average WIP. For instance, when r ˆ 0:65, the buffer allocation (1,1) minimizes the total Table 1. Optimal buffer allocations for N ˆ 3 with exponential operation times r Q q * Q q * 0.60 1 (0,1) 0.65 2 (1,1) 3 (0,3) 0.70 3 (1,2) 0.75 5 (2,3) 0.80 8 (3,5) 0.85 13 (5,8)

L(q*) 84 So Table 2. Optimal buffer allocations for N ˆ 3 with Erlang-2 distributed operation times r Q q * Q q * 0.70 2 (0,2) 0.75 2 (1,1) 0.80 4 (1,3) 0.85 6 (3,3) 7 (2,5) 0.90 11 (5,6) 12 (4,8) Table 3. Optimal buffer allocations for N ˆ 3 with Coxian distributed operation times r Q q * Q q * 0.55 2 (0,2) 0.60 3, 4 (1,2) 5 (0,5) 0.65 5 (2,3) 6 (1,5) 0.70 8, 9 (3,5) 10 (2,8) 0.75 12 (6,6) 13 (4,9) required buffer space whereas the buffer allocation (0,3) minimizes the L(q) (assuming the total available buffer space Q 3). In particular, L 0; 3 ˆ2:03 versus L 1; 1 ˆ2:53. Thirdly, the optimal buffer allocations for problem (P2), which generally have a pattern of asbalanced-as-possible allocations with extra buffer spaces assigning at or near the center stations (see [5]), are in general not optimal for problem (P1). We next present the result for N ˆ 3 in which the operation times at different stations still have the same form of distribution, but non-exponential distributions were used. Speci cally, an Erlang-2 distribution (which has a coef cient of variation of 0.71) and a two-stage Coxian distribution with a coef cient of 1.5 were used for each station. The results are summarized in Tables 2 and 3, respectively. The three observations discussed for Table 1 remain valid for both cases. Results for the four-station and ve-station cases with exponential operation times are summarized in Tables 4 and 5. The monotonically increasing property generally holds except for a few cases. The other two observations remain valid. We further analyzed our buffer allocation problem when the total available buffer space is abundant, i.e., Q is large enough that the constraint on the total buffer space can be ignored. In other words, for each given r we consider the case where Q is at least equal to the value given by the last pair of entries in each row in Tables 1±5. In this case we performed a comprehensive set of empirical studies that analyze the optimal buffer allocations for any given value of r. Extensive results for the three-station, four-station and ve-station lines with exponential operation times can be found in [16], and we Table 4. Optimal buffer allocations for N ˆ 4 with exponential operation times r Q q * Q q * Q q * Q q * 0.60 3 (0,1,2) 0.65 4 (1,2,1) 5, 6 (1,1,3) 7, 8 (0,4,3) 9 (0,4,5) 0.70 6 (2,2,2) 7 (1,3,3) 0.75 10,11 (2,4,4) 12 (2,3,7) 0.80 15 (5,5,5) 16 (3,6,7) 0.85 23 (6,8,9) merely summarize the major conclusions here. First, the optimal buffer allocations q * generally have the monotonic property that q j 1 q j. For instance, results from the four-station case show that q 2 q 3 q 4 except for a few cases. Also the optimal allocations in all the exception cases have the nearmonotonic property that q 2 q 3 ˆ q 4 1. Secondly, the optimal buffer allocations for problem (P2), which generally have a pattern of as-balanced-aspossible allocations with extra buffer spaces assigning at or near the center stations (see [5]), are in general not optimal for problem (P1). These as-balanced-as-possible allocations are generally `dominated' by some buffer allocations that exhibit the above monotonic property. To illustrate the second conclusion further, we plot R(q * )againstl(q * ) for the optimal buffer allocation q * in the four-station case in Fig. 1. We refer to this curve as the ef cient frontier for the optimal allocation for R(q*) Fig. 1. Ef cient frontier for the N = 4 case with exponential operation times.

L(q*) Buffer allocation strategy for unpaced production lines 85 Table 5. Optimal buffer allocations for N ˆ 5 with exponential operation times r Q q * Q q * Q q * Q q * Q q * Q q * Q q * 0.50 1 (0,0,0,1) 0.55 2 (0,1,1,0) 3 (0,1,0,2) 4 (0,0,2,2) 5 (0,0,2,3) 0.60 4 (1,1,1,1) 5 (0,2,1,2) 6 (0,1,2,3) 0.65 7 (1,2,2,2) 8 (1,1,3,3) 9,10 (1,1,3,4) 11 (0,4,3,4) 12 (0,4,3,5) 13 (0,4,3,6) 14 (0,4,3,7) 0.70 10 (2,2,3,3) 11 (1,3,4,3) 12 (1,3,3,5) 0.75 14 (3,4,4,3) 15 (3,4,4,4) 16,17 (2,4,5,5) 18,19 (2,4,4,8) 20 (2,3,8,7) 21 (2,3,8,8) 22 (2,3,8,9) 0.80 22 (4,5,7,6) 23 (4,5,6,8) 24 (3,7,7,7) 25 (3,6,8,8) 26 (3,6,8,9) problem (P1). This ef cient frontier gives the minimum average work-in-process to achieve a required throughput, or alternatively the maximum achievable throughput for a given average work-in-process level. We indicate in Fig. 1 the optimal buffer allocations corresponding to problem (P2) for the four-station case with exponential operation times to show how they relate to this ef cient frontier. Observe that most of the optimal allocations for problem (P2) are away from the ef cient frontier and are not among the optimal solutions for problem (P1). We remark that although the difference between the ef cient frontier and the optimal solutions for (P2) is small, the potential improvement can be gained by simply rearranging the buffer allocations without any additional investment to the original system. U3 U5 U4 U6 4. Unbalanced lines We now extend our results to the case where the line is unbalanced. Previous studies (see, for example, [17]) have suggested that the mean and variance of the operation times have a much more signi cant impact on the performance of a line than the higher moments. A recent study by Powell [7] further suggests that an imbalance in means of the operation times has a much greater impact on allocating buffer space than an imbalance in standard deviations. Therefore to reduce the degrees of freedom in the unbalanced cases we consider here only the case where the operation times at different stations have the same distribution but with different means w j.inparticular we assume that the operation times are exponentially distributed. We consider the three-station and fourstation cases. Furthermore, we focus our discussions on the cases where Q is large such that we can relax the constraint on the total available buffer space. For computational convenience, however, we limit q j 9 for all j. Let w =(w 1 ;w 2 ;...;w N ). For N ˆ 3, we assumed that the mean operation times at the three stations were given by 0.7, 1.0, and 1.3 and considered all six possible assignments of these three mean operation times to the three stations: U1: w = (1.3, 1.0, 0.7) U2: w = (1.3, 0.7, 1.0) U3: w = (0.7, 1.3, 1.0) U4: w = (1.0, 1.3, 0.7) U2 U1 R(q*) Fig. 2. Ef cient frontiers for the N = 3 unbalanced cases: U1±U6. Table 6. Optimal buffer allocations for unbalanced lines with N ˆ 3 r Case U1 U2 U3 U4 U5 U6 0.50 (0, 0) (0, 0) (0, 0) (0, 0) (0, 0) (0, 0) 0.55 (0, 1) (0, 0) (0, 1) (0, 1) (0, 1) (0, 0) 0.60 (1, 0) (0, 1) (0, 2) (1, 1) (0, 1) (0, 1) 0.65 (2, 1) (1, 1) (1, 2) (2, 1) (0, 3) (1, 1) 0.70 (3, 3) (2, 2) (1, 6) (3, 3) (0, 6) (1, 3) 0.75 (7, 9) (4, 5) (4, 9) (7, 7) (2, 8) (2, 7)

L(q*) 86 So U5: w = (0.7, 1.0, 1.3) U6: w = (1.0, 0.7, 1.3) Therefore U1 and U2 represent the cases where the bottleneck is at the rst station, U3 and U4 have the bottleneck at the second station, and U5 and U6 have the bottleneck at the last station. In each case we computed R(q) and L(q) for each possible buffer allocation q = (q 2 ; q 3 ), from which the optimal buffer allocation q * for any required throughput r can be determined. Optimal buffer allocations for selected values of r in each of the six cases are given in Table 6. (Additional results can be found in [16].) The main conclusion is that the optimal buffer allocations continue to exhibit the monotonically increasing property q 2 q 3, except for three cases (U1 with r = 0.60, U1 with r = 0.65, and U4 with r = 0.65). In Fig. 2 we plot the ef cient frontiers for all six cases. Observe that Cases U1 and U2 (where the bottleneck is the rst station) provide the best throughput and work-in-process tradeoff, whereas Cases U5 and U6 (where the bottleneck is the last station) provide the worst throughput and work-in-process tradeoff. We now consider the four-station case. For N ˆ 4, we assumed that the mean operation times at the four stations were given by 0.7, 1.0, 1.0, and 1.3, and considered the following four cases: U7: w = (1.3, 1.0, 1.0, 0.7), U8: w = (1.0, 1.3, 0.7, 1.0), U9: w = (1.0, 0.7, 1.3, 1.0), U10: w = (0.7, 1.0, 1.0, 1.3). Therefore the bottleneck station is the rst, second, third, and last station in cases U7, U8, U9, and U10, respectively. Again, in each case we computed R(q) and L(q) for each possible buffer allocation q =(q 2,q 3,q 4 ), from which the optimal buffer allocation q * for any required throughput r can be determined. The optimal buffer allocations for some selected values of r in each of the four cases are given in Table 7. In Fig. 3 we plot the ef cient frontiers for all four cases. The main conclusions from the results for the fourstation cases parallel those of the three-station cases. In particular, q 2 q 3 q 4 in most cases, and cases where the bottleneck is the rst station (case U7) provide the best throughput and work-in-process tradeoff. Overall, the empirical results for the unbalanced case also show that the optimal buffer allocations generally possess the monotonically increasing property q j 1 q j. (For the cases where the buffer capacity is decreasing, q j 1 exceeds q j by no more than 1.) This implies that the optimal strategy for allocating buffer size for problem (P1) generally is different from that for problem (P2) in which the bottleneck station should generally be allocated the buffer space to maximize the throughput. For instance, for case U8 with r ˆ 0:55, the optimal buffer allocation for problem (P1), q *ˆ 0;1;1, gives R(q * ) = 0.55480 and L(q * ) = 2.30. With a total buffer space Q ˆ 2, the allocation (1,0,1), which provides a Table 7. Optimal buffer allocations for unbalanced lines with N ˆ 4 r buffer space at the second (bottleneck) station, maximizes the throughput with R 1; 0; 1 ˆ0:56816 whereas L 1; 0; 1 ˆ3:02. Our results also suggest that the slower operations should be assigned to the beginning of the line to provide the best performance (in terms of the throughput and work-in-process tradeoff). An intuitive explanation is that the bottleneck at the front of a line limits the ow of materials into the line. Once the materials nish processing at the front (bottleneck) station, they can be processed at downstream stations with minimal blocking, thereby minimizing the amount of work-in-process in the line. R(q*) Case U7 U8 U9 U10 0.45 (0, 0, 0) (0, 0, 0) (0, 0, 0) (0, 0, 0) 0.50 (0, 0, 1) (0, 0, 0) (0, 0, 0) (0, 0, 1) 0.55 (0, 2, 4) (0, 1, 1) (0, 0, 2) (0, 0, 2) 0.60 (1, 1, 2) (1, 1, 1) (0, 1, 3) (0, 0, 6) 0.65 (2, 2, 5) (2, 1, 3) (1, 1, 5) (0, 2, 3) 0.70 (3, 5, 9) (3, 3, 5) (1, 3, 7) (0, 4, 6) 0.75 (8, 7, 9) (7, 8, 8) (3, 7, 9) (2, 6, 9) U10 Fig. 3. Ef cient frontiers for the N = 3 unbalanced cases: U7±U10. U8 U9 U7

Buffer allocation strategy for unpaced production lines 87 5. A sequential buffer allocation scheme To determine the optimal buffer allocation for a given required throughput r, one in principle needs to evaluate all feasible buffer allocations and then select the best allocation. This becomes computationally intractable for lines with large N or Q, as the number of possible allocations is in the order of Q N 1. Therefore it is useful to develop some simple and ef cient heuristic that can give optimal or near-optimal allocations. Based on our empirical results, we propose the following buffer allocation scheme. Basically, our proposed scheme selects the buffer allocation by assigning a minimum amount of buffer space to each station sequentially, starting from q 2 to q N. For all practical purposes let us impose a maximum limit Q j on each individual buffer capacity, i.e., q j Q j. The scheme works as follows: Step 1. Choose the smallest q 2 such that R q 2 ; Q 3 ; Q 4 ;...;Q N r. (Assume that R Q 2 ; Q 3 ; Q 4 ;...;Q N r, otherwise the problem is infeasible.) Step 2. Once q 2 is determined, choose the smallest q 3 such that R q 2 ; q 3 ; Q 4 ;...;Q N r. Continue in this fashion to select q 4 ;...;q N 1 sequentially. Step 3. Set q N to be the best q, q ˆ 0; 1; 2;...;Q N,such that L q 2 ;...;q N 1, q) is minimized subject to R q 2 ;...;q N 1, q r. (Our empirical results suggest that L sometimes decreases as q N increases. Therefore, to enhance the performance, the scheme exhausts all possible q N instead of just picking the smallest one.) Clearly, P the above scheme will evaluate no more than N j ˆ 2 Q j 1 buffer allocations. When Q j is large, one can approximate R q 2 ; q 3 ;...;q j 1, Q j, q j 1 ;...;q N by the minimum of R q 2 ; q 3 ;...;q j 1 and R q j 1 ;...;q N. Also, our results in the previous sections have indicated that the optimal buffer allocations generally possess the monotonically increasing property q j 1 q j,exceptinafew cases where q j 1 ˆ q j 1. To reduce the computations further, one can modify the scheme by choosing only the smallest (nonnegative) q j q j 1 1 in Step 2 (and Step 3 accordingly) in the basic scheme. We applied the above scheme (with the modi cations described above) to the cases given in Tables 1±7, and compared the buffer allocations given by the scheme (denoted by q 0 ) with the optimal allocations (assuming that Q is large enough that q * giveninthelastentryin each row of Tables 1±5 is optimal). The sequential buffer allocation scheme gives the optimal buffer allocations in all 29 cases for the balanced lines and in 55 out of the total 64 cases for the unbalanced lines. Table 8 gives the values of q 0, L(q 0 ), q *, L(q * )andl(q 0 ) L(q * )in each of the nine non-optimal cases. In ve out of these Table 8. Non-optimal cases given by the sequential buffer allocation scheme Case r q 0 L(q 0 ) q * L(q * ) L(q 0 ) L(q * ) U2 0.70 (1, 6) 2.71 (2, 2) 2.69 0.02 U6 0.65 (0, 3) 3.18 (1, 1) 2.98 0.20 0.70 (0, 6) 4.76 (1, 3) 4.25 0.51 U9 0.65 (0, 3, 3) 4.80 (1, 1, 5) 4.53 0.27 0.70 (0, 6, 7) 6.89 (1, 3, 7) 6.29 0.60 0.75 (2, 9, 9) 12.80 (3, 7, 9) 12.11 0.69 U10 0.65 (0, 1, 6) 5.23 (0, 2, 3) 5.12 0.11 0.70 (0, 3, 8) 7.77 (0, 4, 6) 7.74 0.03 0.75 (1, 8, 9) 15.17 (2, 6, 9) 14.99 0.18 nine cases, q 0 represents the second-best allocation. 6. Concluding remarks In this paper we studied the buffer allocation problem with the objective of minimizing the average work-inprocess subject to a minimum required throughput and a constraint on the total buffer space. Both the balanced and unbalanced lines were considered. Our results show that the optimal strategy of allocating buffer size for this problem exhibits a rather interesting pattern that is different from the buffer allocation problem of maximizing the throughput subject to a constraint on the total buffer space. Speci cally, monotonically increasing allocations, whereby an increasing amount of buffer space is assigned toward the end of the line, are shown to be optimal for our problem in most cases. Furthermore, our empirical results suggest that when the line is unbalanced, the slowest operations should be assigned to the beginning of the line to provide the best throughput and average work-in-process tradeoff. (We remark that these conclusions are consistent with those observed by Smith and Daskalaki [9] and Cheng [11].) On the basis of our results, we develop a good heuristic for selecting the optimal buffer allocations. Our results suggest that it is generally a good strategy to assign minimal amount of buffer space to the beginning of the line. When a line is balanced, this effectively implies that congestion is created at the upstream stations. This will then limit the material ow into the downstream stations. Once the materials have been processed by the upstream stations, they can then ow through the downstream stations without much blocking. Therefore, although more buffer spaces are being allocated to the downstream stations, most of them could be empty and the amount of work-in-process is minimized. When a line is unbalanced, it remains valid that minimal buffer space should generally be assigned to the beginning of the line, regardless of the location of the bottleneck station.

88 So Therefore this optimal strategy of allocating buffer space can be viewed as a mechanism for controlling the in ow of materials into the line. The buffering strategy then further helps to regulate the material ow in downstream stations to reduce work-in-process. To illustrate this, consider the situation where jobs arrive according to a Poisson process with rate (say, equal to the required throughput r) and the buffer capacity between stations is in nite (thus no buffer control at each station). In this case, station 1 can begin processing only when there are jobs waiting in front of the line. Assuming that processing times are exponential, we can `decompose' the system into NM/M/1 queues, each with the same Poisson arrival rate and exponential service rate 1=w j. Then the average work-in-process in the system (not counting the jobs waiting in front of station 1) is given by L 0 ˆ w 1 XN w 1 j ˆ 2 j : For example, consider the balanced line w j ˆ 1 with N ˆ 4 and ˆ 0:7. For this case, L 0 ˆ 7:70, compared with L(q * ) = 4.96 for r ˆ 0:7 (where q * = (1,3,3) given in Table 4). Similarly, consider the unbalanced line case U8 with ˆ 0:7. For this case, L 0 ˆ 14:11, compared with L(q * ) = 5.90 for r ˆ 0:7 (where q * = (3,3,5) given in Table 7). Therefore the optimal ( nite) buffer allocations help to regulate material ow in the line to provide a substantial work-in-process reduction from the case where the in ow of materials is controlled (by the actual job arrivals) but there is no buffer control in the line. Our mathematical model and results are based on the two assumptions that there are always materials available for processing at the beginning of the line and that the last station can never be blocked. It would be interesting to investigate models where either one of these two assumptions is relaxed. Speci cally, one can consider the situation where there is external Poisson job arrival stream and station 1 can only start processing when there are jobs waiting (make-to-order environment). In contrast, one can consider the situation where there is a nite buffer after the last station where the nished items are consumed by an external Poisson demand stream such that the last station can also be blocked (make-tostock environment). Unfortunately, in either case, the resulting Markov chain has an in nite number of states and there exists no product form solution for the steadystate probability distribution. Simulation experiments seem to be the only means of studying these two environments, and have not been addressed in this paper. References [1] Buzacott, J.A. and Hani n, L.E. Models of automatic transfer lines with inventory banks ± a review and comparison. IIE Transactions, 10, 197±207 (1978). [2] Sarker, B.R. Some comparative and design aspects of series production systems. 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Annals of the Institute of Statistical Mathematics, 27, 201±212 (1975). [16] So, K.C. Optimal buffer allocation strategy for minimizing the average work-in-process inventory subject to a minimum production rate in unpaced production lines. Working Paper Series, no. DS92014, Graduate School of Management, University of California, Irvine (1992). [17] Muth, E.J. Numerical methods applicable to a production line with stochastic servers. Algorithmic Methods in Probability, Neuts, M.F. (ed) (TIMS Studies in Management Science, vol. 7), pp. 143±159 (1977). Biography Kut C. (Rick) So is Chair of Management and Head of the Department of Management at the Hong Kong Polytechnic University. His main research interests involve effective production and inventory control, design of production and service systems, and marketing/production interface. Professor So's research papers have appeared in numerous academic journals including Management Science, Operations Research, Advances in Applied Probability, IIE Transactions, International Journal of Production Research, Naval Research Logistics, European Journal of Operational Research. He currently serves as an Associate Editor of IIE Transactions.