KUT C. SO and Rees et al. (1987), have used simulation models to study the buffer allocation problem in pull/just-in-time production systems. So and Pinault ...
The International Journal of Flexible Manufacturing Systems, 1 (1989): 223-237 9 1989 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.
Allocating Buffer Storages in a Flexible Manufacturing System KUT C. SO AT&T Bell Laboratories, PO.B. 900, Princeton, NJ 08540
Abstract. This article reports an approximation scheme to determine buffer capacities required to achieve the
target performance level in a general flexible manufacturing system with multiple products. This work extends our previous study of sequential production lines with a single product. The manufacturing system is operated using a pull mechanism, and the performance level is measured by the average proportion of demands backlogged. Simulation experiments were performed to study the validity of the approximation scheme under various situations.
1. Introduction
Pull production systems have received significant attention in industry with the objective of reducing work-in-process inventory. In a pull system, the succeeding stage demands and withdraws in-process units from the preceding stage only according to the rate and time the succeeding stage consumes the items. Two major advantages of a pull production system are that it helps to reduce and control the work-in-process inventory and that production can respond quickly to changing demands. Ideally, inventory at each stage of a pull production system should be one unit. Practically, a certain amount of interstage inventory is generally required to maintain a steady flow of materials between stages in the production system in order to achieve a certain target performance level. As described in Graves (1987), this inventory consists of pipeline stock, cycle stock, and safety stock. Safety stocks are needed to offset various process variabilities due to processing time variability, machine breakdowns, and demand fluctuations, and could be very substantial. Thus, one basic question facing managers and designers of any pull production system is: In order to achieve a certain target performance level, what is the appropriate buffer capacity between stages (safety stocks) to offset the effects of such variabilities and machine breakdowns? Many researchers have investigated the problem of determining optimal stocking level in the design of production systems. Anderson and Moodie (1969), Kay (1972), Sheskin (1976), Soyster, Schmidt, and Rohrer (1979), Ho, Eyler, and Chien (1979), Yamashina and Okamura (1983), Altiok and Stidham (1983), Sarker (1984), Kubat and Simuta (1985), and others have studied the buffer allocation problem for production lines. Other researchers, including Chang et al. (1983), Salameh and Schmidt (1984), and Graves (1985), have studied the buffer allocation problem for other inventory and production systems. Kimura and Terada (1981), Schonberger (1982), and Monden (1983) provide good discussions on the just-intime (JIT) technique and pull production systems. Various researchers, including Kimura and Terada (1981), Monden (1981), Huang, Rees, and Taylor (1983), Philipoom et al. (1987),
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and Rees et al. (1987), have used simulation models to study the buffer allocation problem in pull/just-in-time production systems. So and Pinault (1988) developed an approximation scheme for determining buffer capacities for production lines operated under a pull mechanism to achieve some target performance level. Bitran and Chang (1987) developed a mathematical programming model to study the kanban system in a deterministic multistage assembly production setting, but their model does not incorporate uncertainties. This article discusses an approximation scheme to determine buffer capacities required to achieve the target performance level in a general flexible manufacturing system with multiple products. This work extends our previous study of sequential production lines with a single product (So and Pinault 1988). The manufacturing system is operated using a pull mechanism, and the performance level is measured by the average proportion of demands backlogged. The scheme can be applied to a rather broad class of flexible manufacturing systems. The system can manufacture multiple types of products. Different types of products can have different deterministic routes through the system. (The scheme can be readily modified to also include some nondeterministic routings such as a test-and-repair operation.) A product can be processed at the same station more than once. The machines in each workstation can process different types of products with negligible switch-over times. The storage buffer at each workstation is finite, and the system is operated using a pull mechanism. The approximation scheme can give managers and system designers a good estimation of the required buffer capacities (thus inventory cost) to achieve the target performance level in the system. Consequently, they can use these estimations easily to analyze and compare the effects of various factors such as machine breakdowns on the total inventory costs. The scheme was recently used by several engineers in a capacitor manufacturing shop to study the effects of processing time variability and machine breakdowns on the performance of the shop. The scheme helped to estimate what buffer capacities would be needed to alleviate these adverse effects to achieve a certain desired performance level. The flexible manufacturing system is modeled as a queueing network with multiple classes of customers and finite buffer capacity. Our approach is to decompose the system into single stations and analyze each station individually. A number of researchers have developed queueing network models to analyze flexible manufacturing systems with finite buffer capacity: see Buzacott and Shanthikumar (1980) and Yao and Buzacott (1986) for examples. Researchers using the method of decomposition to approximate and analyze complex queueing networks include Gelenbe and Pujolle (1976), Sevcik et al. (1977), Chandy and Sauer (1978), Kuehn (1979), Saner and Chandy (1980), Whitt (1983a), and Altiok and Perros (1986). As we discuss later, the pull mechanism of the system naturally justifies the decomposition approach. In each station, different types of items are aggregated into a single stream such that our previous approximation scheme for sequential lines with a single product can be applied to analyze this more general system. A first-come, first-served scheduling rule is used in selecting items to pull and process in each station. Simulation experiments were performed to study the validity of the approximation scheme under various situations. In Section 2, we describe the pull mechanism. In Section 3, we describe the model and notation used in the analysis. In Secction 4, we discuss the approximation scheme. In Section 5, we describe the simulation experiments to validate the approximation scheme. Finally, in Section 6, we summarize our results and discuss some possible areas for future research.
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2. The pull mechanism We first describe the pull mechanism for a sequential production line with a single product. Each stage in the line is modeled as a station with two buffers called the input buffer and output buffer of the station. This model is thus different from the more commonly used model with a buffer between two stations. As is demonstrated later, this model naturally allows us to decompose the line into separate stations. The two buffers associated with a station can actually represent two physical buffers in some systems, or they only represent some constraints prescribed by the control system of the pull mechanism in other systems. In the latter case, the output buffer of a station and the input buffer of its succeeding station actually correspond to the same physical buffer between the two stations. To implement the pull mechanism, we impose a limit on the maximum number of items in each station, or equivalently, we can assume that the total capacity of the input and output buffers of the station is finite. There are two operations in the pull mechanism: withdrawal and production operations. Whenever the total number of items in a station is below its maximum limit, the station will pull one item from the output buffer of its preceding station to its input buffer. This is the withdrawal operation. In actual implementation, such withdrawals can be signaled by using a kanban card system or any other appropriate communication mechanism. If the output buffer of the preceding station is empty, the withdrawal will be backlogged and will need to wait until an item is completed in the preceding station and deposited into its output buffer. Whenever a station is ready to begin to process a new product and its input buffer is nonempty, the station will take an item from its input buffer and start the production operation. Upon the process completion, the item is delivered to the input buffer of the station. Observe that the entire system is controlled by this simple pull mechanism: production at each station is controlled by whether there are items in its input buffer, and material flow between two stations is controlled by the inventory level of items at the two stations. In addition, observe that the production operation in a station does not change the total number of items in the station and therefore would not initiate any withdrawal operation. Therefore, a withdrawal operation in any station can only be initiated when its succeeding station initiates a withdrawal operation and pulls one item from the output buffer of the station. Suppose at some instant the total number of items in a station reaches its maximum limit and its input buffer is empty; the station will remain idle. A withdrawal operation from its succeeding station would result in removing one item from its output buffer, thus initiating a withdrawal operation in the station, pulling one item into its input buffer (if the output buffer of the preceding station is nonempty) and beginning the production operation. In effect, the station only produces at the rate its items are being consumed. This is an important characteristic of a pull system. Furthermore, when a finished product is consumed, the withdrawal operation will propagate down the line immediately and production operation will begin at each station. In other words, each station can react immediately to the consumption of the finished products. This property of the pull mechanism justifies the decomposition of the system into single stations in our analysis. The basic operations of the pull mechanism are the same for the multiple-product case except that two scheduling issues must also be addressed. The first issue relates to the
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production operation. When a station is ready to begin to process a new item, the system needs to decide which type of item should be processed next in the station if there is more than one type of item in its input buffer. In this study, we use the first-come, first-served (FIFO) scheduling scheme for each station of the system. In other words, the item that enters the input buffer of a station first will be processed next in the station. Other scheduling schemes may also be considered and should be a worthy area for future research. For example, a strict priority scheduling scheme may be desirable if different item types have significantly different values or service requirements. The second issue relates to the withdrawal operation. When a withdrawal operation is initiated at a station and there is more than one type of item in the station, which type should be pulled into the input buffer of the station? Since a withdrawal operation at a station always corresponds to a withdrawal operation at its succeeding station, a natural scheme would be to pull the same type of item that is being pulled from the station. This would ensure that the mix of different item types in any station remains constant. Equivalently, we can view that there is a limit on the maxiumum number of items in each station for each type. Alternatively, we can and will model our system as having different input and output buffers for different types of items in the same station although the buffers can physically be used to store any type of item, e.g., floor space. Again, other schemes may also be considered to address this scheduling issue. It is straightforward to extend the above pull mechanism to a general manufacturing system. When a product is being processed at the same station at different stages of the manufacturing process, these different stages of the product are considered as different types of items at that station. A station may pull items from different upstream stations if there is more than one type of item at the station; however, the preceding station for each type at each station is unique. Therefore, once the system decides which type of item to pull in its withdrawal operation, the same pull mechanism still applies.
3. The FMS model and notation We consider a flexible manufacturing system with multiple products (see Figure 1). Each type of product has a fixed route, and different types of products may have different routes through the system. A product can be fed back to the same station for processing at different stages of the manufacturing process. However, we eliminate immediate feedback (two consecutive stages of a product requiring processing at the same station) by combining the processing times of any consecutive stages at the same station together, since experience by Kuehn (1979) and Whitt (1983b) has indicated that the approximation by decomposition of queueing networks in general yields better results without immediate feedback. Each station may have multiple (identical) machines. The machines in each station are flexible in the sense that they can process more than one type of product with negligible switchover times. We also assume that transfer times are negligible. We sometimes use the term items to denote the products in each station. Different types of items can refer to different types of products or different stages of the same product. The processing times of the same product at different stations are different, and the processing times of different types of items at the same station can also be different. The processing times of items of the same
227
ALLOCATING BUFFER STORAGES machines
input buffer
:
""..
Infinite supply of raw materials
output buffer
......
%
.
,-m
-
Demands
~
E F-
2
Figure 1. A general flexible manufacturing system with multiple products.
type at each station are i.i.d, random variables with known mean and variance. Machines in each station can break down, and the means and variances of the uptimes and downtimes are known. When a service is interrupted by a machine breakdown, the service will be resumed at the point where it left off as soon as the machine is repaired. All processing times and machine uptimes and downtimes are independent of each other. We assume that the processing batch sizes at different stages of the same product are the same. Thus, without loss of generality, we can assume that the processing batch sizes are one for all products at all stations. Demands for different types of products arrive in single unit (batch) according to different independent stationary renewal processes, and the mean and variance of the interarrival times are known. When a demand arrives and finished products are available, the demand is immediately satisfied. Otherwise, the demand is backlogged. We assume that there is no lost sale due to backlog and all backlogged demands will later be satisfied on a first-come, first-served basis. The system is operated using the pull mechanism described in the preceding section. In particular, we assume that there is a (possibly physical) constraint on the maximum number of products in the system for every stage of every product. Equivalently, we can and will assume that the total capacity of the input and output buffers for each type of item at each station is finite. We also assume that raw materials are always available for each product in front of the input buffer of its first stage. In the event that this might not be true, we can add one additional station in the system and model the first stage of the product as the purchasing process of raw materials for the product. The processing times of this first stage represent the lead times for one unit (batch) of raw materials. However,
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this would imply that just-in-time practices in the purchasing area must be adopted in such a case. The performance measure of the system is the average proportion of demands backlogged for each product, or equivalently, the probability that there is a stockout when there is a demand for that product. It is important to have control over this measure. Not only can backlogged demands result in customer dissatisfaction, and eventually lost sales, but also delay in delivering some products could hold up customers' production operation if such products are used as components in their own manufacturing processes. As an increasing number of manufacturing operations have adopted the just-in-time philosophy, on-time delivery has become very important in those operations. Ideally, the system should have no backlogged demand for all products. However, it is practical only to require the system to achieve a certain target performance level--such as on average 5 percent of demands can be backlogged for a product--because of inevitably existing variability in processing times, demand fluctuations, and machine breakdowns. In our model, different types of products can have different target performance levels. We use the following notation in our model: N = number of stations in the system P = number of products M i = number of machines in station i Bj = number of stages of product j sj~ = processing station for product j at stage k Qjk = maximum number of product j at stage k allowed at station sjk c~j = target performance level (average proportion of demands backlogged) for product j aj = mean interarrival time of demands for product j Oaj = standard deviation of interarrival times of demands for product j dj = 1/aj = demand rate for product i Pjk = mean processing time of product j at stage k %jk = standard deviation of processing times of product j at stage k Yi mean uptime of machines in station i ayi = standard deviation of uptimes of machines in station i xi = mean downtime of machines in station i ffxi standard deviation of downtimes of machines in station i =
:
Our problem is to estimate the required buffer capacities Qjk in order to achieve the target performance level ccj for each product j.
4. The approximation scheme The basic approach in our analysis is first to decompose the system into single stations and aggregate the demand arrival streams of different types of items in each station into a single aggregate demand arrival stream. Then we estimate the total buffer capacity required for this aggregate demand. Finally, we segregate and proportionate the total buffer capacity for each individual type of item.
ALLOCATINGBUFFER STORAGES
229
4.1 Decomposition of stations Recall that when a demand arrives and a finished product is consumed, a withdrawal operation is initiated by the pull mechanism and the withdrawal operation will propagate down the system immediately. In effect, each station in the route of the product sees the consumption of this product and produces accordingly. Each stage in the process of the product reacts immediately to the demand arrivals for its finished products as if each stage of the product were being consumed at the same instant as the finished product. This justifies the decomposition of the system into single stations for analysis. This natural synchronization of withdrawal and production operations will only be broken at the station when there is withdrawal backlog (its output buffer is empty) for the corresponding items. Thus, if adequate buffer capacity is allowed such that withdrawal backlogs rarely occur, decomposition of stations can still provide a good approximation. We thus decompose the system into single stations and analyze each station individually. In the single-station model, we assume that the items are consumed according to the arrival times of the demands for their corresponding finished products. Furthermore, we assume that there are always items available in front of its front buffer when needed, or equivalently, the output buffer of its preceding station for each type of item is nonempty when pulled in the general system. The latter assumption might not be true in general, and therefore we modify the target performance level ~j slightly to account for this withdrawal backlog (see details in So and Pinault 1988). Nevertheless, c~j's should be kept small in order for the decomposition to provide good approximations. Simulation results by So and Pinault (1988) suggest o9 be less than 0.1, and the approximation works very well in our test cases where ~j's were chosen uniformly between 0.025 and 0.1. Thus, this approximation scheme would be most suitable for systems requiring high performance levels. However, small ~j's do not necessarily imply large buffer capacities. For example, if the workload of a station is low or the machines at the station do not break down, generally small buffer capacities would be sufficient to achieve the desired high performance levels. For some fixed station i (i = 1, 2 . . . . . N), let K denote the number of types of items at station i and g(j) the type of product to which item j corresponds, j = 1, 2 . . . . . K. Also, let pq) and ap~ denote the mean and variance of processing times for item j. We aggregate the arrival streams of demands for products g(1) through g(K) into a single arrival stream approximated by a renewal process using the hybrid procedure proposed by Albin (1984). (Observe that the demand arrivals are not independent ifg(i) and g(j) (with i :~ j) correspond to the same type of product.) Extensive simulation results in Albin (1984) show that the hybrid procedure works very well under various traffic intensities and component processes. Whitt (1984) and Albin and Kai (1986) have studied similar hybrid procedures for approximating departure processes. Albin (1986) studied two approximations for estimating delays for customers from different arrival streams to a queue. Both approximations use the hybrid procedure and obtain good results. In applying the hybrid procedure to aggregate the demand arrivals, we have implicitly imposed the following assumption on the random variables based on their coefficients of variation, where the coefficient of variation of a positive random variable is defined to be the ratio of its standard deviation to its mean: when the coefficient of variation is less than
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K U T C. SO
0.3, a normal distribution is assumed; when the coefficient of variation is between 0.3 and 1, a shifted exponential distribution is assumed; and when the coefficient of variation is greater than 1, a balanced hyperexponential distribution is assumed. These distributions have been well studied in the literature, and they generally fit the actual data well in many production systems (e.g., see Kuehn 1979; Whitt 1982). Another nice property of these distributions is that they are completely characterized by the first two moments, which are generally available or can be readily measured for processing times, interarrival times of demands, and machine uptimes and downtimes. Let a and o~ denote the mean and variance of the interarrival time of this aggregate stream of demands given by the hybrid procedure, respectively. We next approximate the mean and variance of the processing times for the corresponding aggregate items. For 1 < j < K, let~ = dg(j) / F.K= 1 dg(k ) denote the relative frequencies of occurrence of demand arrivals for item j at station i, Consider a random variable Y given by Y = r~(=l 1 {~=k} Yk, where 1 {r is the indicator function of the set {/5 = k}, ~ is a random variable taking on values 1, 2 . . . . . K with probabilities fi, f2 . . . . ,fir, and Yk are i.i.d, random variables with mean/z k and variance o~. Then, K
K
E(Y) = ~ fkE(Yk) and
E(Y 2) =
]~ fkE(Y~) k=l
k=l
Therefore, K
var(Y) = E(Y 2) - E(Y)2 =
K-I
fio~ + k=l
K
~
~
k=l
/=k+l
f~(/~k - kct)2
(1)
Notice that the variance of Y, var(Y), can be accounted for by two terms: the first term on the right side of Equation 1 accounts for the variance of Yk (the individual effect), and the second term accounts for the variance due to the difference in the mean of Yk (the overlapping effect). From the above, we thus approximate the mean and variance of the processing times for the aggregate items, respectively, by K
q =
]~ tiP(k) k=l
K
and
a~ =
K-I
]~ fka~(k)+ C ~ k=l
k=l
K
~
f~(p(k)_ p(/))2
l=k+l
where C represents a scaling factor accounting for the dependence between arrivals of the aggregate demands, since they might not be independent in general. From our empirical results, the simple relation C = aa/a is chosen and appears to work well in many cases where a and a~ are the mean and variance of the interarrival times of the aggregate demands. We remark that the above approximation is exact if all the associated demand arrival streams are independent Poisson processes.
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ALLOCATING BUFFER STORAGES
For each fixed target performance level c~, we then apply the approximation scheme given by So and Pinault (1988) to estimate the total required buffer capacity for the aggregate demands and denote this buffer capacity by Q
4.2 Segregation of items Assuming that the above approximation scheme works well, then on average c~ proportion of the total (aggregate) demand arrivals (withdrawals) will be backlogged. To allocate the required buffer storage for each type of item at station i such that the same performance level c~ can be achieved by each type of item, we proportionate the total buffer capacity Q according to the relative frequencies of occurrences of demand arrivals for each type of item. In particular, we set Qj = ~ Q ] for each j = 1, 2, . . . , K, where [x] represents the smallest integer greater than or equal to x. In our model, different types of products can have different target performance levels. Therefore, we determine Qj for each type of item j at station i by setting a to the corresponding target performance level C~gO.) and repeating the above estimation. In fact, the target performance level crk for each type of product k is slightly modified, as discussed in the preceding subsection, depending on the number of stages of the products to account for withdrawal backlogs. The entire procedure of decomposition of stations and segregation of items to estimate individual buffer capacity for each type of item at station i is then repeated for each station i = 1, 2 . . . . . N to obtain all Qjk.
5. The simulation experiments and results Simulation experiments were performed to study the validity of the approximation scheme of determining buffer capacities to achieve the target performance level in various situations. We studied and compared the following cases: high variability of processing times and interarrival times of demands versus low-variability cases, machine breakdown versus no breakdown cases, and multiple-machine versus single-machine cases. Therefore, there were altogether 2 • 2 • 2 = 8 different sets of experiments, and they are summarized in Table 1. Table 1. Set of simulation experiments Set
Processing times/Demands variability
Machine breakdowns
Number of machines
1 2 3 4 5 6 7 8
Low Low Low Low High High High High
No No Yes Yes No No Yes Yes
Single Multiple Single Multiple Single Multiple Single Multiple
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KUT C. SO
In each set of simulation experiments, 20 systems were studied. For each system, the configuration of the system was generated randomly be selecting the parameters of the system uniformly between their specific lower and upper limits, given in Table 2. For example, the number of stations was always 6, but the number of products was independently and uniformly selected between 2 and 6. The number of stages for each product could be between 3 and 6, and the station for each stage was uniformly selected among station 1 through N (however, no immediate feedback was allowed). The target performance level for each product was independently and uniformly selected between 0.025 and 0.1. Let Si = {(j, k) I product j at stage k is processed at station i}, and the workload of station i, Wi, is defined as
Wi :
~,4
Pjk(Xi "F Yi)
q,k)~Se
ajYiMi
The workload of each station was uniformly selected between 0.85 and 0.98, and the individual workload of each type of item at each station i was also uniformly selected such that the total workload at station i summed up to W/. Notice that the mean processing times Pjk were implicitly determined once the individual workloads were defined. For the high-variability cases, all processing times and interarrival times of demands were assumed to be exponentially distributed. For the low-variability cases, the coefficient of variation of the processing times of each product at each stage and the interarrival times of demands of each products were independently and uniformly selected between 0.1 and 0.6. For the machine breakdown cases, the mean and coefficient of variation of the uptimes and downtimes of the machines at each station were independently and uniformly selected between the limits given in Table 2. For the multiple-machine cases, the number of machines at each station was independently and uniformly selected between 1 and 5. In our simulation experiments, a normal distribution was used if the coefficient of variation of a random variable was less than 0.3 and a shifted exponential distribution was used if the coefficient of variation was between 0.3 and 1. This assumption is consistent with that used in our analysis. After the configuration of a system was generated, we used our approximation scheme to allocate the total buffer capacity for each product at each stage Qjk. Then, the system
Table 2. Lower and upper limits of the parameters Parameters
Lower Limit
Upper Limit
N P Bj ozj aj
6 2 3 0.025 0.1 50.0 1.0 5.0 0.3 0.85
6 6 6 O.1 1.0 100.0 1.0 10.0 1.0 0.98
Yi %jy, xi
ax,/Xi Wi
ALLOCATING BUFFER STORAGES
233
was simulated for 200,000 total demand arrivals for all products using the pull mechanism described in Section 2. The total amount of demand arrivals that were backlogged and the total amount of arrivals demands for each product were recorded. The simulation estimate of performance level was then computed and compared with the target performance level for each product. For each set of simulation experiments, let aj, k denote the target performance level for product j in system k and %",k the corresponding simulation estimate of performance level. D e f i n e "~j,k = ~ Thus, if the simulation estimate of performance level agrees with the target performance level for product j in system k, yj,k = 1. Let Pk denote the number of products in system k. Let y and -2 s represent the mean and variance of the "Yj,k, respectively; they are given by 20 =
k=l
P~ and
j=l
s-2 =
20
e~
k=t
j=l
Ptot~l
Ptot~l
where Ptotat = ~2k~ Pk. The quantities y and s were computed for each set of simulation experiments and are given in Table 3. The total number of products for which 3'j,k lies between the specific ranges is also given in Table 3.
Table 3. Summary of simulation results Set of experiments
P,o,,,t 3'j.k < 0.4 0.4 < 3'j.k < 0.8 < "Yj.k< 1.2 - %k < 1.6 -< "Y~.k< 2.0 < %k < 2.4 _< "gj.~ < 2.8 < "l'j,k< %.~ > 3.2
0.8 1.2 1.6 2.0 2.4 2.8 3.2
1
2
3
4
5
6
7
8
0.35 0.32 78 47 25 4 2 0 0 0 0 0
0.41 0.53 72 45 14 8 3 0 1 0 0 1
1.25 1.01 79 16 18 13 7 10 1 8 1 5
1.34 1.00 67 13 12 9 l0 6 6 4 5 2
0.82 0.58 91 24 24 22 13 4 1 2 1 0
0.93 0.63 83 17 27 14 11 6 6 2 0 0
1.03 0.86 83 25 14 14 12 5 7 2 2 2
1.18 0.86 86 14 14 22 15 10 4 3 1 3
6. Conclusions and future research This article studies an approximation scheme to determine buffer capacities required to achieve the target performance level in a general flexible manufacturing system with multiple products. This work extends our previous study of sequential production lines with a single product. The manufacturing system is operated using a pull mechanism, and the performance level is measured by the average proportion of demands backlogged. The analysis is based on the decomposition of the system into single stations and aggregation of different types of items into a single aggregate item type such that our previous approximation
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KUT C. SO
scheme for sequential lines with a single product can be applied to this more general system. A first-come, first-served scheduling rule is used in selecting items to pull and process in each station. Simulation experiments were performed to study the validity of the approximation scheme under various situations. The simulation results are summarized in Table 3. The approximation scheme generally gives very good estimates on the required buffer capacities. In most cases, the simulated amount of demand backlogged is less than twice of the target amount ('rj.k _ 3.2 in our simulation experiments. The simulation results show that the approximation scheme performs better for the highvariability cases (simulation sets 5 through 8). This is expected, as most of our analyses are based on results under the exponential assumption and are then scaled appropriately according to the degree of variation based on empirical results. Therefore, we expect that the approximation scheme would give better estimations for the exponential cases than for the normal or shifted exponential cases. In particular, it appears that the scheme has overestimated the required buffer capacities in simulation sets 1 and 2, the no machine breakdown and low-variability cases. However, the required buffer capacities are generally very low in such cases. In fact, the approximation scheme has already allocated very small buffer capacities (less than 5) in many of those cases. Thus, an additional one buffer space can easily lead to a large improvement in performance and any overestimation by the scheme can be easily adjusted through a more detailed simulation analysis in such cases. The approximation scheme appears to give slightly better results for the no-machinebreakdown cases than for the breakdown cases, for example, by comparing the simulation results in set 5 versus set 7 or set 6 versus set 8. However, the difference appears to be very small and not significant. The results in sets 3 and 4 are much better than the corresponding sets 1 and 2 in terms of 7. This probably arises from the fact that the additional required buffer capacities due to machine breakdowns are in general greater than those due to processing time or demand variations. Consequently, the slight overestimation by the scheme in sets 1 and 2 due to processing time and demand variations becomes insignificant in the corresponding machine breakdown cases. There is no significant difference in the performance of the approximation scheme between the single- and multiple-machine cases in our simulation results. The approximation scheme for determining buffer capacities could be useful in two major applications. First, it can be used to determine the required physical buffer capacities in the design of a manufacturing system. Alternatively, the estimation given by the scheme can be used as an initial estimate of the required buffer capacities and further simulation studies can be employed in fine-tuning this estimate in searching for minimal buffer capacities. A good initial estimate can greatly reduce the amount of simulation efforts. The approximation scheme can also be used as a tool to study the effects of various parameters of the manufacturing system on the required buffer capacities, which in turn would directly affect the inventory cost of the system. To illustrate this application, consider the following examples. Suppose there are four stations and two products in the system. Each product has three stages, and product 1 requires processing at stations 1, 2, and 4 in sequence, while product 2 requires processing at stations 1, 3, and 4 in sequence. Suppose
ALLOCATINGBUFFER STORAGES
235
that Pij = 0.9 for all i, j, a~ = a2 = 1.0, M~ = M4 = 2, M2 = M3 = 1, Yl = 150.0, X 1 : 10.0, x2 = x3 = x4 = 0, cq = 0.05, c~2 = 0.1, and that all positive random variables are assumed to be exponentially distributed. Applying the approximation scheme gives the following buffer capacities: Qll = 71, Q12 = 36, Q13 = 19
and
Q21 = 54, Q22 = 29,
023
=
15
Since the two products are essentially identical in terms of their demands and processing times requirements, this example compares and gives estimates on different required buffer capacities (inventory cost) for different performance levels (a~ = 0.05 versus a2 = 0.1). We have simulated this system for 500,000 total demand arrivals, and the simulation estimates of performance level turned out to be 0.03 (cq') for product 1 and 0.12 (c~2') for product 2. Consider also the same example except that now y~ = 1,500.0 and x~ = 100.0, and the approximation scheme gives Qll = 311, Ql2 = 36, Q13 = 1 9
and
Q21 =217, Q22 = 2 9 , Q23 =15
We have also simulated this system for 500,000 total demand arrivals, and the simulation estimates of performance level again turned out to be 0.03 (a() for product 1 and 0.12 (c~2') for product 2. Thus, this example illustrates the effect of machine breakdowns on the required buffer capacities. Notice that the average proportion of time when a machine is up in station 1 is the same (= 15/16) as the previous example, but the mean downtime (consequently the uptime) of the machines in station 1 is longer in this example. Consequently, the required buffer capacities in station 1 are much higher in this example, since larger amount of inventory is necessary to feed the downstream stations during the longer down period of the machines in station 1. This implies that it is necessary to keep the down period of the machines short (ideally to zero) in any pull manufacturing system in order to reduce inventory cost. The effects of other parameters of the system can be studied similarly. Our model and approximation scheme can be easily extended to the case where some of the items in the system may require some assembly operations. Also, our model uses a first-come, first-served scheduling rule in pulling and producing items. Other more sophisticated scheduling rules can also be considered to improve the performance for each product. For example, if the cost of the work-in-process of one type of product is significantly higher than the others, a strict priority scheduling rule might help to reduce the total inventory costs, since in general smaller buffer capacity is required for items with higher priority (more expensive) than for items with lower priority (less expensive) under such scheduling rule. Further studies on developing similar approximation schemes under different scheduling rules would be useful. Finally, the cases where the switch-over times are nonzero should also be studied.
Acknowledgments The constructive criticism and suggestions for improvement from two anonymous reviewers and an associate editor are greatly appreciated.
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KUT C. SO
References Albin, Susan L., "Approximating a Point Process by a Renewal Process. II: Superposition Arrival Processes to Queues," Operations Research, Vol. 32, No. 5, pp. 1133-1162 (September 1984). Albin, Susan L., "Delays for Customers from Different Arrival Streams to a Queue" Management Science, Vol. 32, No. 3, pp. 329-340 (March 1986). Albin, Susan L., and Kai, Sheng-Roan, '~_pproximation for the Departure Process of a Queue in a Network," Naval Research Logistics Quarterly, Vol. 33, No. 1, pp. 129-143 (February 1986). Altiok, Tayfur, and Perros, Harry G., "Open Networks of Queues with Blocking: Split and Merge Configurations," liE Transactions, Vol. 18, No. 3, pp. 251-261 (September 1986). Altiok, Tayfur, and Stidham, Shaler, Jr., "The Allocation of Interstage Buffer Capacities in Production Lines," liE Transactions, Vol. 15, No. 4, pp. 292-299 (December 1983). Anderson, D.R., and Moodie, C.L., "Optimal Buffer Storage Capacity in Production Line Systems" International Journal of Production Research, Vol. 7, No. 3, pp. 233-240 (1969). Bitran, Gabriel R., and Chang Li, "A Mathematical Programming Approach to a Deterministic Kanban System," Management Science, Vol. 33, No. 4, pp. 427-441 (April 1987). Buzacott, John A., and Shanthikumar, J. George, "Models for Understanding Flexible Manufacturing Systems," AllE Transactions, Vol. 12, No. 4, pp. 339-350 (December 1980). Chandy, K. Mani, and Sauer, Charles H., "Approximate Methods for Analyzing Queueing Network Models of Computer Systems," ACM Computing Surveys, Vol. 10, No. 3, pp. 281-317 (September 1978). Chang, C. Alec; Brown, Larry G.; and Johnson, L. Ray, "Analysis of Buffering Techniques in MRP Systems with a Matrix Loading Model" liE Transactions, Vol. 15, No. 4, pp. 305-312 (December 1983). Gelenbe, E., and Pujolle, G., "The Behaviour of a Single Queue in a General Queueing Network," Acta Informatica, Vol. 7, No. 2, pp. 123-136 (1976). Graves, Stephen C., "A Multi-Echelon Inventory Model for a Repairable Item with One-for-One Replenishment," Management Science, Vol. 31, No. 10, pp. 1247-1256 (October 1985). Graves, Stephen C., "Safety Stocks in Manufacturing Systems," Working paper, Sloan School of Management, Massachusetts Institute of Technology (January 1987). Ho, Y. C.; Eyler, M. A.; and Chien, T. T., "A Gradient Technique for General Buffer Storage Design in a Production Line," International Journal of Production Research, Vol. 17, No. 6, pp. 557-580 (November 1979). Huang, Philip Y.; Rees, Loren P.; and Taylor, Bernard W., III, "A Simulation Analysis of the Japanese Just-inTime Technique (with Kanbans) for a Multiline, Multistage Production System," Decision Science, Vol. 14, pp. 326-344 (1983). Kay, E., "Buffer Stocks in Automatic Transfer Lines," International Journal of Production Research, Vol. 10, No. 2, pp. 155-165 (April 1972). Kimura, Osamu, and Terada, Hirosuke, "Design and Analysis of Pull System, a Method of Multi-Stage Production Control," International Journal of Production Research, Vol. 19, No. 3, pp. 241-253 (May 1981). Kubat, Peter, and Sumita, Ushio, "Buffers and Backup Machines in Automatic Transfer Lines" International Journal of Production Research, Vol. 23, No. 6, pp. 1259-1270 (November 1985). Kuehn, Paul J., "Approximate Analysis of General Queueing Networks by Decomposition," IEEE Transactions on Communications, Vol. 27, No. 1, pp. 113-126 (January 1979). Monden, Yasuhiro, "How Toyota Shortened Supply Lot Production Time, Waiting Time, and Conveyance Time,' Industrial Engineering, Vol. 13, No. 9, pp. 22-29 (September 1981). Monden, Yasuhiro,/b3~ataProduction System: Practical Approach to Production Management, Industrial Engineering and Management Press, Atlanta, GA (1983). Philipoom, Patrick R.; Rees, Loren P.; Taylor, Bernard W., III, and Huang, Philip Y., ' ~ n Investigation of the Factors Influencing the Number of Kanbans Required in the Implementation of the JIT Technique with Kanbans" International Journal of Production Research, Vol. 25, No. 3, pp. 457-472 (March 1987). Rees, Loren P.; Philipoom, Patrick R. ; Taylor, Bernard W., III; and Huang, Philip Y., "Dynamically Adjusting the Kanbans in a Just-in-Time Production System Using Estimated Values of Leadtimes," liE Transactions, Vol. 19, No. 2, pp. 199-207 (June 1987). Salameh, M.K., and Schmidt, J.W., "Safety Stock Analysis in a Multi-Level Inventory System," liE Transactions, Vol. 16, No. 4, pp. 348-354 (December 1984).
ALLOCATING BUFFER STORAGES
237
Sarker, Bhaba H., "Some Comparative and Design Aspects of Series Production Systems," liE Transactions, Vol. 16, No. 3, pp. 229-239 (September 1984). Sauer, Charles H., and Chandy, K. Mani, "Approximate Solution of Queueing Models," Computer, Vol. 13, No. 4, pp. 25-32 (April 1980). Schonberger, Richard J., Japanese Manufacturing Techniques: Nine Hidden Lessons in Simplici~, Free Press, New York, NY (1982). Sevcik, K.C. ; Levy, A.I. ; Tripathi, S.K.; and Zahorjan, J.L., "Improved Approximations of Aggregated Queueing Network Subsystems," in Computer Performance, K. Mani Chandy and Martin Reiser (Ed.), Elsevier NorthHolland, Inc., New York, NY, pp. 1-22 (1977). Sheskin, Theodore J., "Allocation of Interstage Storage Along an Automatic Production Line," AIIE Transactions, Vol. 8, No. 1, pp. 146-152 (March 1976). So, Kut C., and Pinault, Steven C., "Allocating Buffer Storages in a Pull System," International Journal of Production Research, Vol. 26, No. 12, pp. 1959-1980 (December 1988). Soyster, Allen L.; Schmidt, J. W.; and Rohrer, M.W., "Allocation of Buffer Capacities for a Class of Fixed Cycle Production Lines," AIIE Transactions, Vol. 11, No. 2, pp. 140-146 (June 1979). Whitt, Ward, "Approximating a Point Process by a Renewal Process I: Two Basic Methods," Operations Research, Vol. 30, No. 1, pp. 125-147 (January 1982). Whitt, Ward, "Performance of the Queueing Network Analyzer," Bell System Technical Journal, Vol. 62, No. 9, pp. 2817-2843 (November 1983b). Whitt, Ward, "The Queueing Network Analyzer," Bell System Technical Journal, Vol. 62, No. 9, pp. 2779-2815 (November 1983a). Whitt, Ward, ''Approximations for Departure Processes and Queues in Series;' Naval Research Logistics Quarterly, Vol. 31, No. 4, pp. 499-521 (December 1984). Yamashina, H., and Okamura, K., ''Analysis of In-Process Buffers for Multi-Stage Transfer Line Systems," International Journal of Production Research, Vol. 21, No. 2, pp. 183-195 (March 1983). Yao, David D., and Buzacott, John A., "Models of Flexible Manufacturing Systems with Limited Local Buffers," International Journal of Production Research, Vol. 24, No. 1, pp. 107-118 (January 1986).
