IIE Transactions (1997) 29, 29±36
Minimizing work-in-process and material handling in the
facilities layout problem
MICHAEL C. FU and BHARAT K. KAKU
College of Business and Management, University of Maryland at College Park, College Park, MD 20742-1815, USA
Received September 1993 and accepted November 1995
We consider the plant layout problem for a job shop environment. This problem is generally treated as the quadratic assignment problem
with the objective of minimizing material handling costs. Here we investigate the relationship between material handling costs and average
work-in-process. Under restrictive assumptions, an open queueing network model can be used to show that the problem of minimizing
work-in-process reduces to the quadratic assignment problem. In this paper, we generalize these results through a simulation model, and
develop a simple secondary measure which allows us to select the layout that minimizes average work-in-process levels from among
solutions that are similar with respect to the objective function for the quadratic assignment problem.
1. Introduction
The facilities layout problem has generally been tackled
by one of two approaches. The ®rst approach seeks a
solution that minimizes the total material handling costs
between all pairs of facilities and is known as the
quadratic assignment problem (QAP) (see, for example,
Kusiak and Heragu, 1987). The second approach is
known as the adjacency requirements formulation and
the objective is to maximize the sum of closeness ratings
(Foulds, 1983). These ratings may be subjective, and
contribute to the objective function only if the two
facilities in question are adjacent in a solution. Both
formulations lead to an NP-hard problem (Burkard,
1984; Foulds, 1983), and heuristic solution methods have
to be applied to most real-size problems. The QAP
model is preferred for most manufacturing and other
operations where the total material handling cost is the
appropriate criterion.
1.1. The QAP formulation
Suppose there are N facilities to be assigned to N
locations. De®ne a ¯ow matrix and a distance (cost)
matrix whose elements are, respectively,
fij = ¯ow from facility i to facility j;
dij = distance between locations i and j (cost of moving one unit of ¯ow from facility i to facility j);
xij ˆ
n
1
0
if facility i is assigned to location j,
otherwise.
0740-817X # 1997 ``IIE''
The QAP is formulated thus:
…QA P†
Min
XX
i;p
subject to
X
fip djq xij xpq
…1†
j;q
xij ˆ 1
8 i;
…2†
xij ˆ 1
8 j;
…3†
j
X
i
xij 2 f0; 1g
8 i; j:
The most successful optimal solution procedures for the
QAP are implicit enumeration algorithms based on the
lower bound proposed by Gilmore (1962) and Lawler
(1963). A good implementation of this bound is given by
Burkard and Derigs (1980). Linearizations of the QAP
have been proposed by, for example, Bazaraa and Sherali
(1980) and Kaufman and Broeckx (1978), but have not
been very successful computationally.
Heuristic procedures for the QAP can be classi®ed as
constructive, improvement, limited enumeration, or heuristic solutions to linearized problems. Hybrid procedures
combining improvement with one of the other three
approaches have been very successful; for example,
Burkard and Bonniger (1983), Bazaraa and Kirca
(1983), and Kaku et al., (1991).
30
1.2. Multi-criteria approaches to the QAP
Attempts at solving the facilities layout problem in a
multi-criteria context have been limited, mainly to work
with objective functions that combine the QAP and the
adjacency requirements objectives. Examples of such an
approach can be found in Rosenblatt (1979), Dutta and
Sahu (1982), and Fortenberry and Cox (1985). More
recently, Kouvelis and Kiran (1990) have incorporated
throughput requirements into the layout design problem
for automated manufacturing systems. They model the
system as a closed queueing network on the basis of the
assumption that the number of pallets and hence the
number of jobs in the system is ®xed. Given some
required throughput, they minimize the sum of transportation costs and the costs of work-in-process (WIP)
inventory as measured by the number of jobs in the
system necessary to maintain that throughput.
We consider queueing effects in a general job shop
where the throughput is ®xed but WIP is not. In this
situation, an open queueing network is the more appropriate model (see, for example, Buzacott and Shanthikumar, 1985), and in previous work (Fu and Kaku, 1995),
we formulated an open queueing network model that
includes layout considerations in the material handling
system with a ®xed capacity (e.g., a ®xed number of
forklifts). Under the analogous assumptions used by
Kouvelis and Kiran (1990) for the closed queueing network model, we showed that the QAP formulation is
equivalent to the goal of minimizing WIP. However,
some of the assumptions on the material handling system
used in the analytical queueing model are not very
realistic. In this paper we explore the link between
material handling and WIP through a more realistic
simulation model that can handle the complexities of a
material handling system. The approximate equivalence
between the QAP and the problem of minimizing WIP
derived from the analytical model allows us to limit the
search space in our simulation experiments. The simulation results indicate that, in general, the layout that
minimizes WIP also minimizes material handling, but
also that there are exceptional situations where a re®nement of the model is necessary. We characterize situations where such an adjustment is necessary, and develop
a secondary measure to incorporate an appropriate adjustment.
The rest of the paper is organized as follows. In
Section 2 we report on the simulation results and our
secondary measure, and in Section 3 we present our
conclusions.
2. The simulation experiment
We ®rst describe our procedure for generating test
problems. We begin by generating products, volumes,
Fu and Kaku
and process routes and then derive the ¯ow matrix from
this information, rather than simply randomly generating
the elements of the ¯ow matrix. We then describe the
simulation model that was used to experiment with these
problems, and ®nally discuss the results of the simulations.
2.1. Generating the test problems
We randomly generated 10 problems with 8 facilities
each. Two of these facilities are in ®xed locations, one at
each end of the manufacturing facility. One is considered
to be the raw material storage area and serves as an
`entry' station; and the other is considered to be the
®nished goods storage and shipping area and serves as an
`exit' station. The number of locations is also 8, arranged in a 2-by-4 rectangular grid, and distances are
measured as rectilinear center-to-center.
To provide the ¯ow pattern a structure similar to one
that may be found in practice, we randomly generated
demands and process routes for 10 part types for each
problem, and then translated this information into the
familiar ¯ow matrix of QAPs. The step-by-step procedure for each part type is as follows:
Step 1. Part-type demand in units is 40 times some multiple
between 1 and 100, randomly generated from a
uniform distribution.
Step 2. Part-type batch size is 5, 10, 20, or 40, with the
probability of larger batch sizes increasing as
the demand increases. For example, if part
demand is less than one-quarter of the maximum
possible demand, then one of the four batch
sizes is chosen with equal probability, whereas
if part demand is larger than three-quarters of
the maximum possible demand, then the batch
size is set at 40.
Step 3. The number of departments visited by the part
type is randomly generated from a uniform
distribution between 1 and 5, excluding the ®rst
and last visit, which are ®xed for all part types
as the entry and exit stations, respectively. This
number of departments is then chosen at random
with limited backtracking allowed; a part type
may return to one department that it has already
visited, but only after visiting one or more
departments in between.
Step 4. Parts move through the system in batches, with
the number of batches determined as the demand divided by the batch size. Information on
the process route and number of batches is
translated into ¯ows between speci®c departments for the part type.
This is repeated for all part types with ¯ows between
departments being aggregated into the ¯ow matrix. Other
data that need to be generated are:
Minimizing work-in-process and material handling
31
Table 1. Solutions for the 10 test problems
Problem
Solution
Cost
Deviation %
1
Optimal
Near-opt.
Poor
Optimal
Near-opt.
Poor
Optimal
Near-opt.
Poor
Optimal
Near-opt.
Poor
Optimal
Near-opt.
Poor
Optimal
Near-opt.
Poor
Optimal
Near-opt.
Poor
Optimal
Near-opt.
Poor
Optimal
Near-opt.
Poor
Optimal
Near-opt.
Poor
3958
4062
4966
5094
5264
6756
5578
5672
6880
3186
3192
4296
3620
3712
4752
4144
4252
5612
3354
3474
4498
3932
4288
5044
4790
4796
6194
5396
5754
7304
0.00
2.63
25.47
0.00
3.34
32.63
0.00
1.69
23.34
0.00
0.19
34.84
0.00
2.54
31.27
0.00
2.61
35.42
0.00
3.58
34.11
0.00
9.05
28.28
0.00
0.13
29.31
0.00
6.63
35.36
2
3
4
5
6
7
8
9
10
Assignmentsa
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
3
6
7
7
6
5
4
4
4
2
6
3
6
3
6
7
7
3
5
4
5
3
5
3
3
5
3
5
4
5
6
3
6
3
7
2
3
5
3
3
3
4
7
7
5
6
3
6
6
3
6
4
4
6
2
3
2
4
7
7
7
7
2
2
3
4
6
3
7
5
5
5
4
4
4
2
2
7
4
5
4
7
7
7
4
4
5
6
3
6
2
2
3
6
2
3
7
7
5
4
7
6
2
5
2
3
6
2
2
2
2
6
3
2
5
2
7
7
6
2
5
4
5
4
4
7
5
2
2
7
4
2
3
6
3
5
4
5
3
6
7
2
6
4
7
7
6
3
5
3
4
5
4
5
5
6
2
6
6
6
2
7
5
2
7
4
5
4
7
7
3
5
2
5
6
6
4
2
2
4
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
a
The assignments are stated for departments to locations. For example, the optimal solution to Problem 9 has department 1 in location 1, department 2 in
location 3, department 3 in location 2, and so on.
1. The processing time per unit for the part type in a
department is randomly chosen as an integer number of
minutes from the range (1, 10), providing an average
processing time of 5.5 minutes.
2. The setup time for each batch of the part type in a
department is chosen similarly as an integer number of
minutes from the range (R, 10R). Two values were used
for R: 5 and 25, such that the average setup time for the two
cases was 27.5 and 137.5 minutes, respectively.
Finally, the following parameters are adjusted:
1. The speed of the forklifts is set at 40, 45, 50, or 55 feet
per minute, based on the optimal cost of the solution. This
adjustment was made to provide suf®cient forklift capacity
as total ¯ows increased.
2. The number of machines in a department is adjusted so
that utilization in the department does not exceed some
prespeci®ed level.
The ten test problems lead to 12 cases each through
different settings for three parameters. These are the two
values of R mentioned above; three values for the
maximum utilization levels (90, 70 and 50%) in the
departments; and two values for the number of forklifts
(2 or 3).
To test the relationship between the QAP solution cost
and the average WIP in the system we compare the
simulation results for three solutions to each problem.
These are an optimal solution, a `good' solution, and a
poor solution. The optimal solution is found by the
procedure published by Burkard and Derigs (1980).
The good solution is found by a procedure that is a
simpli®ed version of the heuristic developed by Kaku et
al. (1991). We construct nine distinct solutions, improve
them through pairwise interchange, and choose the best
solution with cost greater than the optimal (in other
words, alternate optima are not considered). The poor
32
solution is obtained from the optimal solution by making
pairwise interchanges that worsen the solution, and
choosing the solution whose cost is closest to 30% above
the cost of the optimal. Details on the 10 test problems
and the 3 solutions to each are given in Table 1.
2.2. The simulation model and results
In contrast with the open queueing network model in Fu
and Kaku (1995), the simulation model explicitly takes
into account the queueing effects of a limited number of
forklifts and the possible additional lags due to travel of
the forklift when empty to the next place where it is
needed from its previous drop-off point. A forklift is
assumed to remain at the station at which it unloads until
another request is made. When a request is made, if
more than one forklift is available, then the closest
available one to the requesting station is used. These
characteristics were modeled by using the constructs
available in the SIMAN simulation language.
The simulation model incorporates batches, meaning
the decomposition of service time at a department into a
setup time per batch and an actual per-part processing
time. In the SIMAN implementation, we took the setup
times as exponential and the processing times as deterministic. We also took each batch as the entity in the
system to reduce storage requirements and make the
simulation more ef®cient. We ran each case for six
weeks of production, with a warm-up period of one
week in which no statistics were taken, where a week
comprised ®ve days of three shifts, i.e., 120 hours. Flow
times, department WIP and utilization, and forklift WIP
and utilization were recorded, all in terms of batches,
and statistics based on 40 independent replications were
calculated. These statistics included individual 95% con®dence intervals around the estimated mean, and paired-t
95% con®dence intervals for each pair of differences in
WIP and ¯ow times.
In all 120 instances (12 cases, 10 problems), the poor
solution does decidedly worse than the other two solutions; i.e., in general, better solutions to the QAP problem lead to lower levels of average WIP in the system.
However, in the case of two solutions that are very close
to each other in terms of QAP costs, the WIP results are
not clearly predictable in terms of these costs. These
results seem to be consistent across all 12 cases. Table 2
shows a snapshot summary of a comparison between the
optimal and good solutions based on paired-t comparisons of the mean ¯ow times, which by Little's Law
serves as a surrogate for the total batch WIP. The
notation used in this table indicates the direction of the
comparison and the statistical signi®cance. For example
`' indicates the optimal solution had a lower average
¯ow time and the difference is statistically signi®cant,
whereas `<' indicates the optimal solution had a lower
average ¯ow time, but the difference is not statistically
Fu and Kaku
Table 2. Comparison of Optimal and Near-Optimal Solutions at the
95% level
Case
Problem
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10 11 12
>
>
<
<
<
<
<
<
<
>
<
<
<
>
>
>
<
<
<
<
<
>
<
>
<
<
<
<
>
<
<
>
>
>
<
<
<
<
<
>
<
>
<
>
<
<
<
>
<
<
<
>
<
<
>
>
>
<
<
<
<
<
<
<
<
<
>
>
<
<
>
<
signi®cant.
In test problems 3, 4, and 9, the QAP costs of the good
solutions are very close to the optimal costs ± with a
difference of less than 2% ± and the good solutions may
be termed near-optimal solutions. For these three problems the near-optimal solution sometimes has lower
WIP than the optimal solution. A breakdown of the total
WIP into its two components, department WIP and
forklift WIP, shows further that the department WIP is
practically identical; the difference lies in the forklift
WIP. To understand this result, a detailed comparison of
the placement of departments in the two solutions is
required. For this purpose we make a distinction between
the four central locations and the four corner locations in
the 2-by-4 grid because the former have shorter average
distances to other locations than the latter. The optimal
and near-optimal solutions for test problem 9 are shown
in Fig. 1 to facilitate this discussion. Comparing the
optimal and near-optimal solutions, we see that departments 3, 6, and 7 remain in central locations whereas
departments 1, 4, and 8 remain in corner locations. The
only difference is in departments 5 and 2, which exchange positions with respect to central and corner
locations. Department 5, which is in a corner location
in the optimal solution, has a larger ¯ow through it (856
batches) than department 2 (456 batches). Thus the
forklift is more likely to be at a corner location in the
optimal solution at any given point in time and will
Figure 1 Optimal and near-optimal solutions for test problem 9.
Minimizing work-in-process and material handling
therefore take longer to respond to the next service call,
contributing to higher forklift WIP. However, this is not
a critical factor when there is a fairly clear QAP cost
difference between the optimal and the good solution,
because the in-transit forklift WIP dominates the WIP
waiting for a forklift. If the two solutions are very close
to each other, the in-transit WIP for both is nearly the
same, and the waiting WIP becomes a factor.
A similar effect can be observed in Problems 3 and 4;
however, this type of visual analysis is not reliable for
all situations, e.g., changes in locations for more than
two departments, or larger problems with more classes of
locations than the two in our example problems. To
overcome this limitation, we have devised an easily
calculated measure to capture the effect of layout on
the time spent waiting for material handling service in
the form of the expected travel time for an empty forklift
to arrive at the station demanding service.
The issue of empty travel time for a material handling
system has been studied in the context of the design of
an automated guided vehicle system (AGVS) by Maxwell and Muckstadt (1982), who used a model that
minimized the empty travel time of vehicles in the
AGVS moving along ®xed unidirectional paths. Adding
in the known loading, unloading and loaded travel times,
it is possible to calculate the minimum number of
vehicles required in the AGVS. The timing of service
requests is not considered, so this model provides a
lower bound on the total vehicle service time. The
results from this model are used in a heuristic dispatching procedure to determine sequences for the different
routes, assuming a constant rate of output and consumption of components. Finally, a deterministic simulation
permits the authors to determine blocking effects and
WIP storage requirements at the stations.
The job shop environment we study is quite different:
there are multiple routes that can be traveled in either
direction, blocking is not typically a concern, and demand for forklift service is not likely to be at some
constant rate. Further, our measure of empty forklift
travel requires a simple one-step calculation. We de®ne
the following additional terms:
fi!
fi
f
=
=
=
=
P
total ¯ow from department i = P j fij ;
total ¯ow into department iP= j fji ;P
total ¯ow in the system = i fi! ˆ i fi ;
average velocity of a forklift.
At any given moment, station i demands forklift
service with probability fi! =f and the forklift is at station
j with probability fj =f . The time for the empty forklift
to travel from station j to station i is given by dij =. We
assume that the exit station (numbered N) never demands
service, and that a forklift never leaves empty from the
entry station (numbered 1).
Thus an estimate for the expected travel time for an
33
empty forklift is
NX
ÿ1X
N
Nÿ1 X
N
dij fi! fj
1 X
dij fi! fj ;
ˆ 2
f
f
f
iˆ1 jˆ2
iˆ1jˆ2
so our secondary measure is given by
NX
ÿ1X
N d f !f
ij i j
f2
iˆ1 jˆ2
:
Our basic algorithm to minimize WIP is as follows:
Step 1. Find the best QAP solutions. If one is signi®cantly
better than all the others, choose it.
Step 2. If near-optimal solutions exist, check the secondary measure to discard solutions whose measures are considerably higher than others.
Step 3. Pick one of the ®nalists, possibly based on other
additional considerations.
We ran another set of simulations to test the usefulness of
this secondary measure. Ten new problems were generated,
each having the characteristic that there is at least one nearoptimal QAP solution within 0.5% of the optimal. This time
alternate optimal solutions were accepted, provided that they
were not simply mirror images of the optimal. (A mirror
image is obtained by rotating any solution around its horizontal or vertical axis.) For the ratio of setup to processing
times we set R ˆ 10; maximum machine or assembly workstation utilization at 70%; and the number of forklifts at 2.
These problems and their solutions are presented in Table 3,
with asterisks indicating the solution with the lowest secondary measure. We see that two of the problems had
alternate optima and three had two additional solutions
within 0.5% of the optimal.
The results of the simulation are shown in Table 4.
Asterisks in the forklift WIP column indicate the lowest
value (though not always statistically signi®cant). Comparison with Table 3 indicates a perfect correspondence
between the lowest secondary measure and the lowest
forklift WIP; i.e., the secondary measure consistently
picks the solution with the lower WIP. Looking more
closely at Table 3, we observe that there are three
problems for which the secondary measure for the alternate optimal or near-optimal solution is better (smaller)
than that for the optimal solution: problems 3, 5, and 8.
For problem 3, the alternate optimal solution has a
secondary measure that is only 0.07% better than that
for the optimal solution, whereas for problem 8 the nearoptimal solution is 0.08% worse in terms of the QAP
cost but 0.42% better in terms of the secondary measure.
For both these problems, the forklift WIP is smaller for
the alternate/near-optimal solution, although the difference is not statistically signi®cant and there is no
statistically signi®cant difference between the mean ¯ow
times. However for problem 5, the secondary measure is
34
Fu and Kaku
Table 3. Solutions for the second set of 10 test problems
Problem
Solution
Cost
Deviation (%)
Sec. measure
1
Optimal
Near-opt.
Optimal
Near-opt.
Optimal
Alt.-opt.
Near-opt.
Optimal
Near-opt.
Optimal
Near-opt.
Optimal
Near-opt.
Optimal
Alt.-opt.
Near-opt.
Optimal
Near-opt.
Optimal
Near-opt.
Optimal
Near-opt.
Near-opt.
5362
5386
5224
5244
4135
4135
4153
2612
2622
4078
4094
3682
3684
2450
2450
2458
4811
4815
4643
4647
4770
4776
4778
0.00
0.45
0.00
0.38
0.00
0.00
0.44
0.00
0.38
0.00
0.39
0.00
0.05
0.00
0.00
0.33
0.00
0.08
0.00
0.09
0.00
0.13
0.17
*1.712845
1.731464
*1.919127
1.957579
1.771174
*1.769924
1.780842
*1.790738
1.823147
1.923231
*1.794920
*1.819641
1.827726
*1.838522
1.882022
1.842749
1.720258
*1.712955
*1.966118
1.967324
*2.046220
2.049782
2.119964
2
3
4
5
6
7
8
9
10
6.67% smaller for the near-optimal solution, and the
corresponding forklift WIP is lower at a statistically
signi®cant level. The mean ¯ow time is also lower for
the near-optimal solution, although not by enough to be
declared statistically signi®cant.
2.3. The effect of transfer batches
We next investigated the effect of transfer batches on
our results (we thank one of the anonymous referees for
suggesting this line of investigation). In the usual sense,
a transfer batch is the quantity transported between
departments. It has been shown that using transfer
batches smaller than the production batch size leads to
reduced WIP. For this set of experiments, we set the size
of the transfer batch to be half the size of the production
batch, in effect doubling the number of batches that have
to be transported. To keep forklift utilization at approximately the same level as in the previous experiments, we
doubled the number of forklifts in the material handling
system. Many policies on selecting the next waiting
transfer batch to be processed at a department are
possible. In our study, we assumed a ®rst-come, ®rstserved queue discipline. Of course, setups would not be
incurred if transfer batches of the same class followed
each other on the same machine. We ran Case 5 from the
®rst set of problems (see Table 2) under this new
Assignments
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
6
2
4
3
2
3
5
4
7
2
6
7
7
2
4
2
6
2
5
5
5
6
4
2
6
7
6
3
2
2
3
3
6
5
5
5
3
2
3
5
6
3
3
7
7
7
5
4
2
7
5
5
3
6
6
5
2
2
2
5
6
7
7
7
6
7
3
3
3
4
3
5
2
7
7
7
2
2
4
4
6
6
7
3
5
4
4
7
6
2
2
2
3
5
3
5
4
4
4
5
5
3
3
4
3
4
5
4
3
3
2
2
4
4
6
7
7
6
4
6
6
6
7
4
7
7
3
4
6
7
6
2
5
4
4
6
5
5
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
scenario. The results remained essentially the same, with
the optimal QAP solution doing better at minimizing
WIP for the same problems as before.
3. Conclusions
A simple analytical queueing network model (Fu and
Kaku, 1995) showed that under certain conditions in a
job shop environment, the QAP with an objective of
minimizing material handling costs also serves to minimize average work-in-process. This result serves as a
starting point in this paper in an investigation of the
relationship between material handling cost and WIP. A
simulation study allowed us to relax the assumptions of
the analytical model as well as vary the levels of some
important parameters. The result was found to hold
under more general conditions, within the bounds of
these parameters. In particular, department WIP seems
to be fairly insensitive to layout considerations, so the
layout dependence enters primarily through the material
handling system. This result appears to hold even when
transfer batches are smaller than the production batches.
We emphasize that this result is obtained for a job shop
environment and that in an automated manufacturing
system where buffer space may be limited, blocking
could occur and the result may not hold. As explained
Minimizing work-in-process and material handling
35
Table 4. Output statistics for case 13
Problem
Department
Flow time
WIP
1
2
3
4
5
6
7
8
9
10
600
601
499
501
593
594
593
699
699
552
546
510
512
544
545
545
560
560
474
476
533
534
538
4
4
4
4
4
4
4
7
7
4
4
3
3
5
5
4
4
4
6
6
5
5
5
32.5
32.5
27.2
27.2
24.7
24.8
24.7
19.1
19.0
25.8
25.8
21.7
21.7
17.3
17.3
17.3
28.9
28.9
21.0
21.1
28.5
28.6
28.5
Forklift
Utilization
0.4
0.4
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.2
0.2
0.2
0.3
0.4
0.3
0.3
0.3
0.3
0.3
in the previous section, for situations where two or more
QAP solutions are very close, we need a means to
discriminate between candidate solutions in terms of
WIP. We have developed a simple secondary measure
for this purpose, and simulation results indicate that this
combination of two measures works well in selecting the
layout that minimizes average WIP levels. A possible
extension to this work would be the development of a
procedure to minimize a weighted sum of the material
handling costs and WIP levels.
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0.65
0.65
0.65
0.65
0.60
0.60
0.60
0.62
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0.01
0.01
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0.00
0.00
0.01
0.01
0.01
0.01
0.01
0.01
0.01
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Biographies
Michael C. Fu is an Associate Professor of Management Science and
Statistics in the College of Business and Management at the University
of Maryland at College Park, with a joint appointment in the Institute for
Systems Research. He holds a Ph.D. and M.S. in applied mathematics
from Harvard University, bachelor's and master's degrees in electrical
engineering and computer science from MIT, and a bachelor's degree in
mathematics from MIT. His research involves stochastic modeling ±
including discrete-event simulation, queueing theory and inventory
control ± and its application to manufacturing problems. He has served
Fu and Kaku
as a consultant for Westinghouse Electric Corporation and the Johns
Hopkins University Applied Physics Laboratory on manufacturing and
transportation problems. He is the co-author with Jian-Qiang Hu of the
forthcoming book Conditional Monte Carlo: Gradient Estimation and
Optimization Applications. He is a member of IEEE and INFORMS,
serves as an Associate Editor for the journals IIE Transactions and
INFORMS Journal on Computing, and served on the Program Committee for the Spring 1996 INFORMS National Meeting in Washington,
D.C.
Bharat K. Kaku is an Assistant Professor of Management Science and
Statistics in the College of Business and Management at the University
of Maryland at College Park. He holds a B.E. degree in mechanical
engineering from Bhopal University, an M.B.A. from the University of
Delhi, a master's degree in Industrial Administration from Carnegie
Mellon University, and a Ph.D. in Operations Management also from
Carnegie Mellon University. His research interests are in the areas of
facilities layout, group technology, and ¯exible manufacturing systems.
He has worked with the Engineering Research Center at the University
of Maryland to design facilities layouts and to provide guidance on
manufacturing-related issues for small companies in Maryland. Dr
Kaku is a member of INFORMS and POMS and serves on the Editorial
Review Board of Production and Operations Management.