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Design of Plant Layouts with Queueing Effects
Saifallah BenjaafarDepartment of Mechanical Engineering
University of MinnesotaMinneapolis, MN 55455
July 10, 1997
AbstractIn 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-in-
process accumulation, manufacturing lead time, achievable throughput rates, and required material
handling capacity. More 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.
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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. More 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 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. 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
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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-in-
process. 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.
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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, Meller 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 M/M/1 and
M/M/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 Manufacturing Systems (FMS). Their modeling
assumptions are similar to those of Fu and Kaku, except that they assume work-in-process is
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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]:
Minimize z = xikxilλ ijdkl ∑l
∑k
∑j
∑i
subject to:
xik = 1∑k = 1
K
V i (1)
xik = 1∑i = 0
M + 1
V k (2)
xik = 0, 1 V i, k (3)
where xik = 1 if department i is assigned to location k and xik = 0 otherwise, dkl 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
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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. More 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. Model 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 Di and a squared coefficient of variation Ci2 for i =
1, 2, …, N . The squared coefficient of variation denotes the ratio of the squared mean over the
variance.
ii) Material handling is assured by a single discrete material handling device. Material 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, tkl, is assumed to be deterministic and is
given by tkl = dkl/v, where dkl is the distance between locations k and l and v is the speed of the
material handling transporter.
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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 i = 0 to M + 1, with the
indices i = 0 and M + 1 denoting, respectively, the loading and unloading departments.
v) The plant consists of M 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, λij, 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 = λ ki∑k = 0
M
= λ ij∑j = 1
M + 1
for i = 1, 2, …m, (4)
λ0 = λM + 1 = Di∑i = 1
N
, and (5)
λ t = λ ij∑j = 1
M + 1
∑i = 0
M
, (6)
where λt is the workload for the material handling transporter.
vi) Processing times at each department are independent and identically distributed with an
expected processing time E(Si) and a squared coefficient of variation Csi2 for i = 0, 1, …, M + 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 = {xij}, where xik = 1 if department i is assigned to location k and xik = 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
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λ0
Loading department Transporter
λM+1
λ t
Unloading department
department 1
Department M
Department 2
λ0λ0
λM
λ2
λ1
Figure 1 Central server queueing network model
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inter-arrival and processing times. Under a given layout, expected WIP at department i can then be
obtained as:
E(WIPi) = ρi
2(Cai2 + Csi
2)gi
2(1 - ρi) + ρi, (7)
where ρi = λ iE(Si) is the average utilization of department i, Cai2 and Csi
2 are, respectively, the
squared coefficients of variation of job inter-arrival and processing times, and
gi ≡ gi(Cai2, Csi
2, ρi) =
exp[-2(1 - ρi)(1 - Cai
2)2
3ρi(Cai2 + Csi
2)] if Cai
2 < 1
1 if Cai2 ≥ 1.
{ (8)
Similarly, expected WIP at the transporter is given by:
E(WIPt) = ρt
2(Cat2 + Cst
2)gt
2(1 - ρt) + ρt, (9)
where ρt = λtE(St) is the average utilization of the transporter, E(St) is the expected travel time per
material transfer request and Cst2 is its squared coefficient of variation. Note that ρt and ρi 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 = {xik}, the first and second moments of transporter
travel time are given by the following:
E(St) = λkrλ ij/λ t2∑
k = 0
M
trij(x)∑j = 1
M + 1
∑i = 0
M
∑r = 1
M + 1
, and
E(St2) = λkrλ ij/λ t
2∑k = 0
M
(trij(x))2∑j = 1
M + 1
∑i = 0
M
∑r = 1
M + 1
,
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where trij(x) = xrkxilxjs(dkl + dls)/v∑s
∑l
∑k
= xrkxildkl/v ∑l
∑k
+ xilxjsdls/v ∑s
∑l
and
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:
tij(x) = xikxjldkl/v∑l = 1
K
∑k = 1
K
. (10)
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 prij of an empty trip
from r to i followed by a full trip from i to j. The probability prij is given by:
prij = pkr∑k = 0
M
pij, (11)
where pij is the probability of a full trip from department i to department j which can be obtained as
pij = λ ij
λ ij∑j = 1
M + 1
∑i = 0
M. (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 trij(x) = tri(x) + tij(x), the first two moments of transporter travel
time per transfer request can now be found as:
E(St) = prijtrij(x)∑j = 1
M + 1
∑i = 0
M
∑r = 1
M + 1
, and (13)
E(St2) = prij(trij(x))2∑
j = 1
M + 1
∑i = 0
M
∑r = 1
M + 1
, (14)
which, upon appropriate substitutions, lead to the desired result. #
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Theorem 2: Given layout x = {xik}, 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:
Ca02 = (Di/ Di∑
i = 1
N
)Ci2∑
i = 1
N
,
Cai2 = πi(ρt
2Cst2 + (1 - ρt
2)Cat2) + 1 - πi , for i = 1, 2, …, M + 1, and
Cat2 =
πiρi2Csi
2∑i = 0
M
+ πi(1 - ρi2)(1 - πi)∑
i = 1
M
+ πi2(1 - ρi
2)ρt2Cst
2∑i = 1
M
+ π0(1 - ρ02)Ca0
2
1 - πi2(1 - ρi
2)( 1 - ρt2)∑
i = 1
M,
where πi = λi/λt for i = 1, 2, …, M + 1.
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]:
Cai2 =
λ jpji
λ i
(pjiCdj2 + (1 - pji))∑
j ≠ i
+ λ0γi
λ i
(γiCa02 + (1 - γi)), and (15)
Cdi2 = ρi
2Csi2 + (1 - ρi
2)Cai2, (16)
where pij is the routing probability from node i to node j (nodes include departments and the
material handling device), γi is the fraction of external arrivals that enter the network through node
i, and 1/λ0 and Ca02 are, respectively, the mean and squared coefficient of variation of the external
job inter-arrival times. In our case, γ0= 1 and γi = 0 for all others since all jobs enter the cell at the
loading department, the routing probability from departments i = 0 through M to the material
handling transporter is always one, that from the material handling transporter to departments j = 1
through M + 1 is
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ptj =
λ ij∑i = 0
M + 1
λ ij∑j = 0
M + 1
∑i = 0
M + 1 (17)
and to the loading department (j = 0) is zero. Parts exit the cell from department M + 1 (unloading
department) so that all the routing probabilities from that department are zero. Substituting these
probabilities in the above expression, we obtain
Ca02 = (Di/ Di∑
i = 1
N
)Ci2∑
i = 1
N
, (18)
Cat2 = (λ i/λ t)Cdi
2∑i = 0
M
= πiCdi2∑
i = 0
M
, and (19)
Cai2 = πiCdt
2 + 1 - πi for i = 1, 2, …, M + 1, (20)
which, along with equality (10), can be simultaneously solved for the desired results. #
The layout design problem can now be formulated as:
Minimize E(WIP) = E(WIPi)∑i = 0
M + 1
+ E(WIPt)
subject to:
xik = 1∑k = 1
K
i = 0, 2, …, M + 1 (21)
xik = 1∑i = 0
M + 1
k = 1, 2, …, K (22)
ρt < 1 (23)
xik = 0, 1 i = 0, 2, …, M + 1; 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
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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. Model 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.
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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 (i = 0, 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: D1 = 1.62 parts/hour; E(Si)= 36 min for i = 0, 1, and 2, C12 = 1; and Csi
2 = 1 for i = 0,
1, and 2, and v = 10 ft/min. We consider two layout scenarios, x1 and x2. The distances
between departments are as follows, scenario 1: d01(x1) = d10(x1) = d12(x1) =d20(x1) = d21(x1)
= 100 ft ; and scenario 2: d01(x1) = d10(x1) = d20(x1) = 10 ft, d12(x1) = 190 ft, and d21(x1) =
280 ft. It can be verified that the two scenarios lead to an average travel time of E(St(x1)) =
E(St(x2)) = 17.5 and a transporter utilization ρt(x1) = ρt(x2) = 0.945. The second moments of
transportation time are, however, different: E(St2(x1)) = 325 and E(St
2(x2)) = 644.5. From the
first two moments of travel time, we obtain Cst2(x1) = 0.061224, Cat
2(x1) = 0.98841, Ca02 (x1) = 1,
Ca12 (x1) = Ca2
2 (x1) = 0.580205, Cst2 (x2) = 1.10449, Cat
2(x2) = 1.00129, Ca02 (x2) = 1, and
Ca12 (x2) = Ca2
2 (x2) = 1.046725, from which we can calculate expected WIP as follows:
E(WIP(x1)) = 99.33 and E(WIP(x2)) = 123.76. Since E(WIP(x2)) > E(WIP(x1)), although
ρt(x1) = ρt(x2), 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(x1)) = 102.14 +/- 1.67 and E(WIP(x2)) = 123.12 +/- 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
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M0 M2M1
(a) Product flow sequence
λ0
Transporter
λ2
λ tλ0λ0 λ1
Μ1Μ0
Μ2
(b) Queuing Model
Figure 2 Product flow sequence and the corresponding queueing network model
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are the same with the exception of department processing time, where in this case E(Si) = 36.5 min
for i = 0, 1, and 2. Again, we consider two layout scenarios, x1 and x2; scenario 1: d01(x1) =
d10(x1) = d12(x1) =d20(x1) = d21(x1) = 100 ft ; and scenario 2: d01(x1) = d10(x1) = d20(x1) =
10 ft, d12(x1) = 10 ft, and d21(x1) = 170 ft. Layout x1 results in an average travel time of
E(St(x1)) = 17.5 and layout x2 in E(St(x2)) = 8.25. The resulting utilization of the transporters
is ρt(x1) = 0.945 and ρt(x2) = 0.4455. Thus, we have E(St(x1)) > E(St(x2)) and ρt(x1) >
ρt(x2). The second moments of transportation time can be calculated as E(St2|x1) = 325 and
E(St2|x2) = 198.25, from which we obtain Cst
2(x1) = 0.061224, Cat2(x1) = 0.993961, Ca0
2 (x1) = 1,
Ca12 (x1) = Ca2
2 (x1) = 0.580502, Cst2(x2) = 1.912764, Cat
2(x2) = 1.001311, Ca02 (x2) = 1, and
Ca12 (x2) = Ca2
2 (x2) = 1.091104 (note that Cst2 (x2) > Cst
2 (x1)). We can now calculate expected WIP
as follows: E(WIP(x1)) = 185.195 and E(WIP(x2)) = 210.966. Since E(WIP(x2)) >
E(WIP(x1)), although ρt(x2) < ρt(x1), our result is proven. Again, we confirmed or result using
simulation. The 95% confidence intervals for expected WIP are: E(WIP(x1)) = 182.14 +/- 6.58
and E(WIP(x2)) = 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 x2 was less than half that of x1 .
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.
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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 x1 and x2. In Table 1, we
show the effect of varying department processing time on the performance of x1 and x2, 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, x2 has a smaller average WIP than x1
when average processing time is 32 min and a much larger average WIP than x1 when average
processing time is 37 min. Similarly, x1 has a smaller average WIP than x2 when Ca02 = 1 and
Csi2 = 0.5 and a larger average WIP when Ca0
2 = Csi2 = 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.
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Table 1 The effect of utilization on the relative desirability of layouts(Ca0
2 = 1, Csi2 = 1, D = 1.62 parts/hr)
E(Si) E(WIP(x1)) E(WIP(x2))
32 min 25.76 20.55
33 min 30.55 26.18
34 min 38.44 35.51
35 min 53.99 54.02
36 min 99.33 108.20
37 min 2,588 3,088
Table 2 The effect of variability on the relative desirability of layouts(E(Si) = 35 min, D = 1.62 parts/hour)
Variabilitycoefficients
E(WIP(x1)) E(WIP(x2))
(Ca02 = 1, Csi
2 = 0.5) 38.75 47.24
(Ca02 = 1, Csi
2 = 1) 53.99 54.02
(Ca02 = 1, Csi
2 = 2) 86.41 84.47
(Ca02 = 2, Csi
2 = 2) 95.02 92.96
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A special instance when minimizing average travel time also minimizes average work-in-
process is when Cai2 = Csi
2 = Cat2 = Cst
2 = 1 for all i = 0, 1, …, M + 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, ρt, also minimizes average work-in-process.
Since ρt is a linear transformation of average travel time, minimizing average travel time minimizes
ρt.
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 ρt < 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, λmax, is given by
λmax = 1/ prij( xrkxilxjs(dkl + dls)∑s
∑l
∑k
)/v∑j = 1
M + 1
∑i = 0
M
∑r = 1
M + 1
, (25)
which is clearly a function of the transportation distances and the distribution of transportation
times. By limiting λmax , the maximum achievable output rate from the system is limited. This
means that the choice of layout could directly affect the available production capacity. Maximizing
throughput by maximizing λmax 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, nmin, for a given material handling workload, λt:
nmin = λ t prij( xrkxilxjs(dkl + dls)∑s
∑l
∑k
)/v∑j = 1
M + 1
∑i = 0
M
∑r = 1
M + 1
. (26)
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.
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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. More 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.
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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. DMII-9309631, the U.S. Department of Transportation under grant No.
USDOT/DTRS93-G-0017, and the University of Minnesota Graduate School.
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