Design of Plant Layouts with Queueing Effects

Transcription

1 Design of Plant Layouts with Queueing Effects Saifallah Benjaafar Department of echanical Engineering University of innesota inneapolis, N July 10, 1997 Abstract In this paper, we present a formulation of the plant layout problem where the objective is to minimize work-in-process. We show that the choice of layout has a direct impact on work-inprocess accumulation, manufacturing lead time, achievable throughput rates, and required material handling capacity. ore importantly, we show that layouts generated using a queueing-based model can be very different from those obtained using conventional layout procedures. In particular, we present a number of surprising and counter-intuitive results. For example, we show that reducing overall distances between departments can increase average work-in-process in the plant. We also show that the relative desirability of a layout can be affected by non-material handling parameters, such as department utilization levels, variability in processing times at departments and variability in product demands.

2 1. Introduction Including queueing effects in design of plant layouts has been notoriously difficult [2, 4, 5, 12]. This difficulty is due to a lack of analytical models which are capable of explicitly capturing the effect of layout configuration on dynamic plant behavior. As a result, most existing plant layout design procedures attempt to simply minimize a static measure of material handling time or cost [6, 11, 15]. This is certainly the case for the widely used quadratic assignment problem (QAP) formulation, where the objective is to minimize the total distances traveled in moving material from one processing department to another [6, 11, 15]. In this paper, we present a reformulation of the quadratic assignment problem, where the objective is to minimize work-in-process (WIP). We show that the choice of layout has a direct impact on work-in-process accumulation, manufacturing lead time, achievable throughput rates, and required material handling capacity. ore importantly, we show that layouts generated using a queueing-based model can be very different from those obtained using the conventional QAP formulation. In particular, we present a number of surprising and counter-intuitive results. For example, we show that reducing overall distances between departments can increase average workin-process in the plant. We also show that the relative desirability of a layout can be affected by non-material handling parameters, such as department utilization levels, variability in processing times at departments and variability in product demands. These results are different from conclusions reached by Fu and Kaku in a recent paper [4], where they argued that the QAP formulation leads to a layout that also minimizes average WIP. In order to obtain average work-in-process due to a particular layout configuration, we model the manufacturing facility as a central server queueing network. Each processing department is modeled as either a single or a multi-server queue with arbitrary distribution of product processing and inter-arrival times. The material handling system operates as a central server in moving material from one department to another. We assume that the material handling system consists of discrete devices (e.g., forklift trucks, human operators, automated guided vehicles, etc.). The distances traveled by the material transporters are determined by the layout configuration, product -2-

3 routings and product demands. In determining the transporter travel time distribution, we account for both empty and full trips by the material transport devices. Detailed description of the queueing model and our assumptions are given in section 4. Because we impose no assumptions regarding the arrival processes of products or their processing times, exact analytical solutions are difficult to obtain. Therefore, we rely on network decomposition and approximation techniques to obtain approximate estimates of average work-inprocess. These approximations have been shown elsewhere to provide fairly reliable estimates of the actual work-in-process for a wide range of parameters [3, 17, 18]. Since our objective in layout design is to obtain a ranked ordering of different layout alternatives, approximations are sufficient, as long as they guarantee accuracy in the ordering of these alternatives. An alternative to analytical approximations is to use computer simulation. However, simulation can be very computing-intensive when hundreds or thousands of layout configurations, as it is often necessary in layout design, must be evaluated. Results from simulation are often difficult to generalize to systems other than those being simulated. This contrasts with analytical models where the mathematical relationships we obtain can be readily used to gain general insights into the fundamental relationship between various parameters. Nevertheless detailed simulation is useful whenever an accurate assessment of an individual layout is required. The organization of the paper is as follow. In section 2, we provide a review of relevant literature. In section 3, we briefly describe the quadratic assignment problem. In section 4, we describe our queueing-based formulation of the layout problem and the corresponding queueing network model. In section 5, we use our model to study the relationship between layout and work-in-process and to compare layouts selected by the QAP to those selected by the queueing model. We also validate our results using simulation. In section 6, we present extensions of our model and conclusions. -3-

4 2. Literature Review Very little of the existing literature addresses queueing issues in facility layout design. In a recent review of over 150 papers published over the past ten years on plant layout, eller and Gau [14] identified only one paper on the subject. This paper is by Fu and Kaku [5] who to our knowledge, were the first and only ones to explicitly address queueing issues in layout design. Similarly to our study, they used average work-in-process as the layout design criterion. To obtain average work-in-process, they developed a simple queueing network model in which they assumed all arrival processes to be Poisson and all processing times to be exponential, including transportation times. In modeling transportation times, they also ignored empty travel by the material handling devices and accounted only for full trips. These assumptions allowed them to treat the network as a Jackson queueing network - i.e., a network of independent //1 and //n queues - for which a closed-form analytical expression of average work-in-process is available. Using their model, they showed that minimizing average work-in-process is, in fact, equivalent to minimizing average material handling cost, as used in the conventional QAP formulation. The limitations of the Fu and Kaku formulation are in the assumptions used. By assuming that inter-arrival times, processing times, and material handling times are all exponentially distributed, and by not accounting for empty travel times, they failed to capture important dynamics that arise under less restrictive assumptions. These include, for example, effects due to the second moments of transportation and processing times, and variability in product inter-arrival times. As we show in this paper, when the distribution of processing, material handling, and inter-arrival times are appropriately accounted for, the layouts obtained from the queueing model can be very different from those obtained by the QAP formulation. In fact, we show that under certain circumstances, increasing travel distances can reduce average work-in-process. Other related literature include the work of Kouvelis and Kiran [10] who consider a closed queuing network for modeling Flexible anufacturing Systems (FS). Their modeling assumptions are similar to those of Fu and Kaku, except that they assume work-in-process is -4-

5 maintained constant. Therefore, they measure performance by average throughput rather than work-in-process. Johnson and Brandeau [7, 9] and Thonemann and Brandeau [16] have extensively used single stage queueing systems to model discrete material handling devices, such as automated guided vehicles. However, their models do not explicitly capture differences in layout configurations. Several other queueing and simulation models have been proposed for the design and analysis of material handling systems. An excellent review on this subject can be found in [8]. 3. The Quadratic Assignment Problem The quadratic assignment problem (QAP) can be formulated as follows [6, 11, 15]: inimize z = i j k l x ik x il λ ij d kl subject to: K k = 1 x ik = 1 x ik = 1 V i (1) V k (2) x ik = 0, 1 V i, k (3) where x ik = 1 if department i is assigned to location k and x ik = 0 otherwise, d kl is the distance between locations k and l, and λ ij is the amount of material flow (the number of material unit loads) between departments i and j. Constraints 1 and 2 ensure, respectively, that each department is assigned one location and each location is assigned one department. The objective function minimizes material handling cost by minimizing the average distance traveled by an arbitrary unit load of material. If material transport is provided by a discrete material handling device, then the QAP also minimizes the average distance traveled by the device when the device is full - i.e., while carrying a load. Although the above formulation adequately accounts for the cost/time in moving material between departments, it has several important limitations. The objective function is a static -5-

6 measure that does not account for variability in material flows between departments. In systems with discrete material handling devices, the objective function does not capture empty travel by these devices. Also, by minimizing only average distances traveled, information about the higher moments of travel distribution is ignored. This could lead, for example, to selecting a layout with a small mean but a high variance. As we show in section 5, this would result in higher variability in material handling times and cause longer queueing delays. ore generally, by focusing only on average material handling time, the QAP fails to capture congestion effects due to waiting for material handling resources when the number of these resources is finite. Contention for finite resources, coupled with variability, leads to congestion and queueing delays which directly affect overall manufacturing lead times and work-in-process levels in the plant. In the next section, we present a reformulation of the QAP where many of the above limitations are addressed. 4. odel Formulation In order to illustrate the procedure for including queueing effects in layout design, we make the following assumptions (most of these assumptions are made only for illustrative purposes and can be relaxed as discussed in section 6). i) The plant produces N products. Product demands are independently distributed random variables characterized by an average demand D i and a squared coefficient of variation C 2 i for i = 1, 2,, N. The squared coefficient of variation denotes the ratio of the squared mean over the variance. ii) aterial handling is assured by a single discrete material handling device. aterial transfer request are serviced on a first come-first served (FCFS) basis. In the absence of any requests, the material handling transporter remains at the location of its last delivery. iii) The travel time between any pair of locations k and l, t kl, is assumed to be deterministic and is given by t kl = d kl /v, where d kl is the distance between locations k and l and v is the speed of the material handling transporter. -6-

7 iv) Products are released to the plant from a loading department and exit the plant through an unloading (or shipping) department. Departments are indexed from to, with the indices and denoting, respectively, the loading and unloading departments. v) The plant consists of processing departments, with each department consisting of a single server (e.g., a machine) with ample storage for work-in-process. Jobs in the queue are processed in first come-first served order. The amount of material flow, λ ij, between a pair of departments i and j is determined from the product routing sequence and the product demand information. The total amount of workload at each department is given by: λ i = k = 0 λ ki λ 0 = λ = λ t = = λ ij for i = 1, 2, m, (4) j = 1 j = 1 N i = 1 where λ t is the workload for the material handling transporter. D i, and (5) λ ij, (6) vi) Processing times at each department are independent and identically distributed with an expected processing time E(S i ) and a squared coefficient of variation C2 si for, 1,, (the processing time distribution is determined from the processing times of the individual products). vii) A layout configuration corresponds to a unique assignment of departments to locations. We use the vector notation x = {x ij }, where x ik = 1 if department i is assigned to location k and x ik = 0 otherwise, to differentiate between different layout configurations. The plant is modeled as an open network of GI/G/1 queues, with the transporter being a central server queue. A graphical depiction of product flow through the network is shown in Figure 1. In order to obtain WIP-related performance, we use network decomposition and approximation techniques (see [3] and [18] for a general review) where the performance of each department, as well as the transporter, is approximated using the first two moments of the associated job -7-

8 λ 1 department 1 λ 2 Department 2 λ 0 λ t Loading department Transporter λ λ +1 Department Unloading department Figure 1 Central server queueing network model -8-

9 inter-arrival and processing times. Under a given layout, expected WIP at department i can then be obtained as: E(WIP i ) = ρ i 2 (C 2 ai + C 2 si )g i + ρ 2(1 - ρ i ) i, (7) where ρ i = λ i E(S i ) is the average utilization of department i, C2 ai and C2 si are, respectively, the squared coefficients of variation of job inter-arrival and processing times, and g i g i (C 2 ai, C 2 si, ρ i ) = Similarly, expected WIP at the transporter is given by: 2 ) 2 exp[ -2(1 - ρ i)(1 - C ai 3ρ i (C 2 ai + C2 si ) ] if C 2 ai < 1 { (8) 1 if C 2 ai 1. E(WIP t ) = ρ t 2 (C 2 at + C 2 st )g t + ρ 2(1 - ρ t ) t, (9) where ρ t = λ t E(S t ) is the average utilization of the transporter, E(S t ) is the expected travel time per material transfer request and C2 st is its squared coefficient of variation. Note that ρ t and ρ i must be less than one for expected work-in-process to be finite. In order to compute expected WIP, the first and second moments of transportation time, as well as the coefficients of variation of inter-arrival times to each department and to the transporter must be known. As we show in the following two theorems, these parameters are directly determined by the layout configurations. Theorem 1: Given a layout configuration x = {x ik }, the first and second moments of transporter travel time are given by the following: 2 E(S t ) = λ kr λ ij /λ t t rij (x), and r = 1 j = 1 k = 0 E(S 2 2 t ) = λ kr λ ij /λ t (t rij (x)) 2, r = 1 j = 1 k = 0-9-

10 s where t rij (x) = x rk x il x js (d kl + d ls )/v = x rk x il d kl /v k l k l + x il x js d ls /v l s corresponds to the travel time, under layout configuration x, from department r to department i and then to department j. Proof: First note that travel time from department i to department j, under layout configuration x, is given by: K K and t ij (x) = x ik x jl d kl /v. (10) k = 1 l = 1 Also note that in responding to a material transfer request, the transporter performs an empty trip from its current location (the location of its last delivery), at some department r, followed by a full trip from the origin of the current request, say department i, to the destination of the transfer request at a specified department j. Then, in order to obtain the first two moments of the transporter travel time, we need to characterize the probability distribution p rij of an empty trip from r to i followed by a full trip from i to j. The probability p rij is given by: p rij = k = 0 p kr p ij, (11) where p ij is the probability of a full trip from department i to department j which can be obtained as p ij = λ ij j = 1 λ ij. (12) Noting that the time to perform an empty trip from department r to department i followed by a full trip to department j is given by t rij (x) = t ri (x) + t ij (x), the first two moments of transporter travel time per transfer request can now be found as: E(S t ) = r = 1 j = 1 p rij t rij (x), and (13) E(S 2 t ) = p rij (t rij (x)) 2, (14) which, upon appropriate substitutions, lead to the desired result. # r = 1 j = 1-10-

11 Theorem 2: Given layout x = {x ik }, the squared coefficients of variation of job inter-arrival and departure times at processing departments and at the transporter can be approximated by the following: C a0 N i = 1 N i = 1 2 = (D i / D i )C i 2, C2 ai = π i (ρ 2 t Cst 2 + (1 - ρ 2 t )Cat 2 ) π i, for i = 1, 2,,, and C2 at = π i ρ 2 i Csi 2 + π i (1 - ρ 2 i )(1 - πi ) + π 2 i (1 - ρ 2 i )ρt 2 C2 st i = 1 i = π i 2 (1 - ρ i 2 )( 1 - ρt 2 ) i = 1 where π i = λ i /λ t for i = 1, 2,,. + π 0 (1 - ρ 2 0)Ca0 2 Proof: We use the following known approximations for characterizing, respectively, the squared coefficients of variation in inter-arrival and departure times at a node i in an open network of GI/G/1 queues [3]: C2 ai = j i λ j p ji λ i (p ji C 2 dj + (1 - p ji )) + λ 0γ i (γ i C 2 a0 + (1 - γ i )), and (15) λ i, C 2 di = ρi 2 Csi 2 + (1 - ρ 2 i )Cai 2, (16) where p ij is the routing probability from node i to node j (nodes include departments and the material handling device), γ i is the fraction of external arrivals that enter the network through node i, and 1/λ 0 and C2 a0 are, respectively, the mean and squared coefficient of variation of the external job inter-arrival times. In our case, γ 0 = 1 and γ for all others since all jobs enter the cell at the loading department, the routing probability from departments through to the material handling transporter is always one, that from the material handling transporter to departments j = 1 through is -11-

12 p tj = λ ij λ ij j = 0 (17) and to the loading department (j = 0) is zero. Parts exit the cell from department (unloading department) so that all the routing probabilities from that department are zero. Substituting these probabilities in the above expression, we obtain C at C a0 N i = 1 N i = 1 2 = (D i / D i 2 )C i 2, (18) 2 2 = (λ i /λ t )C d = π i i C d i, and (19) C2 ai = π i C 2 d π t i for i = 1, 2,,, (20) which, along with equality (10), can be simultaneously solved for the desired results. # The layout design problem can now be formulated as: subject to: K k = 1 x ik = 1 x ik = 1 inimize E(WIP) = E(WIP i ) + E(WIP t ), 2,, (21) k = 1, 2,, K (22) ρ t < 1 (23) x ik = 0, 1, 2,, ; k = 1, 2,, K (24) The above formulation shares the same constraints as the quadratic assignment problem. We require an additional constraint, constraint 23, to ensure that a selected layout is feasible and will not result in infinite work-in-process. The objective function is however different from that of the QAP. In the conventional QAP, the objective function is a positive linear transformation of the -12-

13 expected transporter time. Therefore, a solution that minimizes average travel time between departments is optimal. In the next section, we show that this is not necessarily the case when queuing effects are accounted for and that solutions obtained by the two formulations can be very different. We also note that by virtue of Little's law [13], minimizing epxected WIP is equivalent to minimizing expected product flow time. Therefore, our model can readily be used to optimize lead time performance as well. The quadratic assignment problem is notoriously known for being NP hard. Therefore our model is also NP hard (our objective function is a nonlinear transformation of that of the QAP). Although it is not our intent in this paper to provide a solution algorithm, most of the existing heuristics for the QAP can be readily applied to the our model. For example, an iterative pairwise or multi-step exchange procedure, such as CRAFT [1], can be used to generate a solution. Note that in this case, after each exchange, it is the expected WIP that is calculated and used to evaluate the desirability of the exchange. Other heuristics may be used as well. For a recent review of the quadratic assignment problem and solution procedures, the reader is referred to Pardalos and Wolkowicz [15]. 5. odel Analysis and Insights Examining the expression of expected WIP in the objective function, it is easy to see that it has two sources: the processing departments and the material handling transporter. In both cases, WIP accumulation is determined by (1) the variability in the arrival process, (2) the variability in the processing/transportation times, and (3) the utilization of the departments and the transporter. Because the transporter provides input to all the processing departments, variability in transportation time directly affects the variability in the arrival process to all the departments. In turn, this variability, along with the variability of the department processing times, determines the input variability to the transporter. Because of this close coupling, the variability of any resource affects the work-in-process at all other resources. -13-

14 The conventional QAP model, by focusing only on average travel time, fails to account for the important effect of variability. As we show in the following observations, average travel time can be a poor indicator of expected WIP. Observation 1: Layouts with the same average travel times can have different average WIP. Proof: We use a counter-example to show that layouts with similar average travel time can have very different average WIP levels. Consider a facility with three departments (, 1, and 2). The facility produces a single product, which is manufactured in the fixed sequence shown in Figure 2(a). The corresponding queuing network is shown in Figure 2(b). Other relevant data is as follows: D 1 = 1.62 parts/hour; E(S i )= 36 min for, 1, and 2, C 2 1 = 1; and C2 si = 1 for, 1, and 2, and v = 10 ft/min. We consider two layout scenarios, x 1 and x 2. The distances between departments are as follows, scenario 1: d 01 (x 1 ) = d 10 (x 1 ) = d 12 (x 1 ) =d 20 (x 1 ) = d 21 (x 1 ) = 100 ft ; and scenario 2: d 01 (x 1 ) = d 10 (x 1 ) = d 20 (x 1 ) = 10 ft, d 12 (x 1 ) = 190 ft, and d 21 (x 1 ) = 280 ft. It can be verified that the two scenarios lead to an average travel time of E(S t (x 1 )) = E(S t (x 2 )) = 17.5 and a transporter utilization ρ t (x 1 ) = ρ t (x 2 ) = The second moments of transportation time are, however, different: E(S t 2 (x 1 )) = 325 and E(S t 2 (x 2 )) = From the first two moments of travel time, we obtain C 2 st (x 1 ) = , C 2 at (x 1 ) = , C 2 a0 (x 1 ) = 1, C 2 a1 (x 1 ) = C 2 a2 (x 1 ) = , C 2 st (x 2 ) = , C 2 at (x 2 ) = , C 2 a0 (x 2 ) = 1, and C 2 a1 (x 2 ) = C 2 a2 (x 2 ) = , from which we can calculate expected WIP as follows: E(WIP(x 1 )) = and E(WIP(x 2 )) = Since E(WIP(x 2 )) > E(WIP(x 1 )), although ρ t (x 1 ) = ρ t (x 2 ), our result is proven. # To confirm our analytical result, we simulated a detailed model of the two layout scenarios. For each scenario, we obtained the following 95% confidence interval for the value of expected WIP: E(WIP(x 1 )) = / and E(WIP(x 2 )) = /- 1.79, which certainly support our analytical findings. For the sake of brevity, details of the simulation are omitted. Observation 2: A smaller average travel time does not always lead to a smaller average WIP. Proof: We use again a counter-example to prove that E(WIP) is not necessarily decreasing in average travel time. Consider a facility identical to the one previously described. All parameters -14-

15 0 1 2 (a) Product flow sequence λ 0 λ t λ 1 Μ 0 Transporter λ 2 Μ 1 (b) Queuing odel Μ 2 Figure 2 Product flow sequence and the corresponding queueing network model -15-

16 are the same with the exception of department processing time, where in this case E(S i ) = 36.5 min for, 1, and 2. Again, we consider two layout scenarios, x 1 and x 2 ; scenario 1: d 01 (x 1 ) = d 10 (x 1 ) = d 12 (x 1 ) =d 20 (x 1 ) = d 21 (x 1 ) = 100 ft ; and scenario 2: d 01 (x 1 ) = d 10 (x 1 ) = d 20 (x 1 ) = 10 ft, d 12 (x 1 ) = 10 ft, and d 21 (x 1 ) = 170 ft. Layout x 1 results in an average travel time of E(S t (x 1 )) = 17.5 and layout x 2 in E(S t (x 2 )) = The resulting utilization of the transporters is ρ t (x 1 ) = and ρ t (x 2 ) = Thus, we have E(S t (x 1 )) > E(S t (x 2 )) and ρ t (x 1 ) > ρ t (x 2 ). The second moments of transportation time can be calculated as E(S t 2 x 1 ) = 325 and E(S 2 t x 2 ) = , from which we obtain C 2 st (x 1 ) = , C 2 at (x 1 ) = , C 2 a0 (x 1 ) = 1, C 2 a1 (x 1 ) = C 2 a2 (x 1 ) = , C 2 st (x 2 ) = , C 2 at (x 2 ) = , C 2 a0 (x 2 ) = 1, and C 2 a1 (x 2 ) = C 2 a2 (x 2 ) = (note that C 2 st (x 2 ) > C 2 st (x 1 )). We can now calculate expected WIP as follows: E(WIP(x 1 )) = and E(WIP(x 2 )) = Since E(WIP(x 2 )) > E(WIP(x 1 )), although ρ t (x 2 ) < ρ t (x 1 ), our result is proven. Again, we confirmed or result using simulation. The 95% confidence intervals for expected WIP are: E(WIP(x 1 )) = / and E(WIP(x 2 )) = 205 +/- 5.72, which certainly support our conclusion. # The above observations highlight the important effect that variability in travel times between departments can have on the desirability of a layout. In both observations, layouts with the smaller travel time variance were superior, even when their average travel times were higher. In fact, in the example of observation 2, the average travel time in layout x 2 was less than half that of x 1. These results clearly show that minimizing average travel time (or transporter utilization) is not always desirable. In fact, reductions in average travel time if they come at the expense of increasing travel time variance should at times be avoided. Note that the increase in travel time, due to the higher travel time variance, does not only affect WIP accumulation at the transporter, but also the level of WIP at the processing departments. The greater variability in travel times translates into greater variability in the arrival process to the department which, in turn, leads to longer queues at these departments. These results point to the need to explicitly account for travel time variance when selecting a layout. A layout that exhibits a small variance may, indeed, be more desirable than one with a smaller travel time average. -16-

17 Travel distances are not, however, the only factor that affects the relative desirability of a layout. Non material handling factors such as department utilization levels, variability in department processing times, and variability in demand levels could determine whether one layout configuration is more desirable than another. Observation 3: The relative desirability of a layout can be affected by non-material handling factors. Proof: We use a series of examples to show that varying either utilization levels or processing time and demand variability can affect the relative desirability of a layout. We consider the same example described in observation 2 and the same two layout scenarios x 1 and x 2. In Table 1, we show the effect of varying department processing time on the performance of x 1 and x 2, and in Table 2, we show the effect of varying processing and demand variability (for the sake of brevity, only WIP values are reported). It is easy to see that the same layout can be superior under one set of parameters and inferior under another. For example, x 2 has a smaller average WIP than x 1 when average processing time is 32 min and a much larger average WIP than x 1 when average processing time is 37 min. Similarly, x 1 has a smaller average WIP than x 2 when C2 a0 = 1 and C2 s.5 and a larger average WIP when C2 a0 = C2 si = 2. # The above results show that the relative desirability of a layout is highly dependent on many operating parameters. For example, a layout that is effective in an environment where department utilization is high may not be appropriate if departments were lightly loaded. Similarly, a layout that is effective when processing/arrival times are highly variable may not be appropriate if processing/arrival times were deterministic. Generally speaking, layouts that reduce travel time variability are more desirable when department utilization is high or variability in either inter-arrival or processing times is low. When either department utilization is low or processing/demand variability is high, minimizing average travel times becomes more important. However, definite guidelines are difficult to identify because of the many interacting parameters. Therefore, an evaluation of the queueing model will often be required to generate an accurate layout ranking. -17-

18 Table 1 The effect of utilization on the relative desirability of layouts (C2 a0 = 1, C2 si = 1, D = 1.62 parts/hr) E(S i ) E(WIP(x 1 )) E(WIP(x 2 )) 32 min min min min min min 2,588 3,088 Table 2 The effect of variability on the relative desirability of layouts (E(S i ) = 35 min, D = 1.62 parts/hour) Variability coefficients E(WIP(x 1 )) E(WIP(x 2 )) (C2 a0 = 1, C2 s.5) (C2 a0 = 1, C2 si = 1) (C2 a0 = 1, C2 si = 2) (C2 a0 = 2, C2 si = 2)

19 A special instance when minimizing average travel time also minimizes average work-inprocess is when C2 ai = C2 si = C 2 at = C2 st = 1 for all, 1,,. This is the case when the processing times, inter-arrival times, and travel times are all exponentially distributed. Average work-in-process can then be shown to be a function of only the utilization levels. Therefore, a layout that minimizes the utilization of the transporter, ρ t, also minimizes average work-in-process. Since ρ t is a linear transformation of average travel time, minimizing average travel time minimizes ρ t. Finally, we should note that in addition to affecting work-in-process and lead time, the choice of layout determines production capacity. The stability condition ρ t < 1 puts a limit on the maximum feasible number of unit transfer loads per unit time that can be moved by the transporter. This limit, λ max, is given by s λ max = 1/ p rij ( x rk x il x js (d kl + d ls ))/v, (25) r = 1 j = 1 k which is clearly a function of the transportation distances and the distribution of transportation times. By limiting λ max, the maximum achievable output rate from the system is limited. This means that the choice of layout could directly affect the available production capacity. aximizing throughput by maximizing λ max could be used as an alternative layout design criterion. In this case, layouts would be chosen so that the available material handling capacity is maximized. Such a design criterion could be appropriate in high volume/make-to-stock environments where minimizing lead time or work-in-process is not critical. The stability condition can also be used to determine the minimum required number of transporters, n min, for a given material handling workload, λ t : l s n min = λ t p rij ( x rk x il x js (d kl + d ls ))/v. (26) r = 1 j = 1 k Similarly, we could obtain the minimum required transporter speed, for a fixed number of transporters, or the minimum required transfer batch size. Note that in determining these feasibility requirements, we account for both full and empty travel by the material handling device. l -19-

20 6. Discussion and Conclusion In this paper, we presented a formulation of the plant layout problem where the objective is to minimize work-in-process. We used the model to explore the relationship between layout configuration and operational performance. We showed that the conventional criterion of selecting layouts based on average material handling distances can be a poor indicator of queueing effects. In a series of counter-intuitive observations, we showed that the conventional QAP formulation can lead to the selection of very different layouts from those obtained using the queueing-based model. In particular, our analysis highlighted the important effect that variance in material handling times plays in determining layout desirability. This effect is ignored in conventional layout procedures. We also showed that non-material handling factors, such as processing time variability or process utilization, can directly affect layout performance. Certain simplifying assumptions we have made in our current queueing model are easy to relax. For example, the model can be extended to allow for multiple transporters and multiple processing units at each department. Approximations for GI/G/n queues would simply have to be used (see [3, 17]) for details). Similarly, the assumptions of deterministic distances between pairs of departments and deterministic transporter speed can be eliminated. Accounting for variability in these two parameters can be easily accommodated by appropriately modifying the second moment of transportation time. Other assumptions are, however, more difficult to relax. This includes, for example, the modeling of control policies other than FCFS for either routing transporters or for sequencing products. In these cases, simulation may be the only practical approach. Our queueing model is based on known approximations of GI/G/1 queues. While these approximations are fairly robust, more customized approximations could be constructed for specific applications [3]. However, we should note that the usefulness of the queueing model is not as much in its accuracy, as it is in its ability to capture key effects in the relationship between layout and operational performance and in its ability to lead to consistent rankings of different layout alternatives. ore importantly, it is a tool that can be used at the design stage to rapidly evaluate a large number of alternatives, a task that may be difficult to achieve using simulation. -20-

21 The queueing model also offers an opportunity to design simultaneously the layout and the material handling system (e.g., determining the number of transporters, transporter speed, travel paths, etc.) and to examine the effect of both on expected WIP. The ability to evaluate layout and material handling concurrently is indeed absent from most existing layout procedures. Several avenues for future research are possible. Better analytical approximations should be developed to take advantage of the special structure of the layout problem. Either analytical or simulation models that account for different routing and dispatching policies of the material handling system should be constructed. These models should then be used to study the effect of different policies on layout performance. Furthermore, it would be useful to use the queueing model to evaluate and compare the performance of different classical layout configurations, such as product, process, and cellular layouts, under varying conditions. This may lead to identifying new configurations that are more effective in achieving short lead times and small WIP levels. In previous sections, we have argued that variance in travel distances is as important as the mean of these distances. Therefore, identifying configurations that minimize both mean and variance is important. Examples of such layouts, as shown in Figure 3, could include a Star layout, where departments are equi-distant from each other, or a Hub-and-Spoke layout, where each hub consists of several equi-distant departments and is serviced by a dedicated transporter. Acknowledgement: The author's research is supported by the National Science Foundation under grant No. DII , the U.S. Department of Transportation under grant No. USDOT/DTRS93-G-0017, and the University of innesota Graduate School. -21-

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