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The International Journal of Flexible Manufacturing Systems, 9 (1997): 273298
c 1997 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.
Impact of Routing Flexibility on the Performanceof an FMSA Simulation Study
RAHUL CAPRIHAN
Dayalbagh Educational Institute, Dayalbagh, Agra, India 282 005
SUBHASH WADHWA
Indian Institute of Technology, New Delhi, India 110 016
Abstract. The evolving manufacturing environment is characterized by a drive toward increasing flexibility. One
possible manifestation of flexibility within an FMS is in the form of routing flexibility. Providing this typically is an
expensive proposition, and system designers therefore aim to provide only the required levels commensurate with
a given set of operating conditions. This paper presents a framework based on a Taguchi experimental design for
studying the nature of the impact of varying levels of routing flexibility on the performance of an FMS. Simulation
results indicate that increases in routing flexibility, when made available at the cost of an associated penalty on
operation processing time, is not always beneficial. There is an optimal flexibility level, beyond which system
performance deteriorates, as judged by themakespanmeasure of performance. It is suggested that the proposed
methodology can be used in practice for not only setting priorities on specific design and control factors but also
for highlighting likely factor level combinations that could yield near-optimal shop performance.
Key Words: routing flexibility, dynamic sequencing and dispatching, simulation, Taguchi experimental design
1. Introduction
The evolving manufacturing environment is characterized by a profusion in product variety,
decreasing lead times to delivery, exacting standards of quality, and competitive costs.
Simultaneously, with an increasing trend toward economic globalization, manufacturingsystems must face new challenges to survive and grow in the marketplace. For example,
they must be able to adapt quickly and efficiently to varying market demands that impose
changes in objectives and operating conditions. In an attempt to cope with such multifaceted
problems, new technologies advocate increased automation and flexibility. The challenge
for manufacturing system designers is to select the appropriate type and level of flexibility,
automation, and integration to cope with the market changes in an efficient and effective
manner. It is essential therefore that one appreciate the implications of these issues within
the domain of manufacturing systems scheduling.
For flexible manufacturing systems (FMSs), the definition of flexibility is important.
Browne, Dubois, Rathmill, Sethi, and Stecke (1984) have identified eight different flexi-
bility types in the context of FMSs. They definerouting flexibilityas the ability to handle
breakdowns and to continue producing a given set of part types. We interpret routing
flexibility also to imply the existence of multiple sequencing routes for individual part
types as a means of improving system performance. Browne et al. (1984) refer to this
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IMPACT OF ROUTING FLEXIBILITY ON THE PERFORMANCE OF AN FMS 275
Singh and Mohanty (1991) proposed a fuzzy multiobjective approach to solving the
routing problem in the job shop environment. Grinsted (1995) discussed the benefits of
using alternative routes in a job shop and suggested four decision rules that may be useful.
It was further indicated that the choice of an appropriate rule depends on the manufacturing
objectives. Other references of interest include Wayson (1965), Wilhelm and Shin (1985),
Bobrowski and Mabert (1988), Hutchinson and Pflughoeft (1994), and Caprihan (1995).
2. Motivation and objectives
In this paper, we study the effect of some key design and control parameters on the perfor-
mance of a hypothetical FMS. The two design parameters assumed are routing flexibility
and pallet availability; the two control parameters include a dispatching rule and a sequenc-
ing rule. The performance measure of interest is makespan. Our motivation is to outline amethodology that would help system designers gain quick insights into the relative impor-
tance of design and control factors with respect to defined measures of performance.
The operating conditions within the evolving manufacturing environment are character-
ized by continuous change. Within such an environment, it is useful to ensure that systems
possess appropriate levels of routing flexibility as well as to develop an understanding of
its impact on performance. Further, because system designers and controllers often are
compelled to operate within stringent time constraints when reviewing alternate control or
scheduling decisions, there is little justification, if any, for conducting an exhaustive sim-
ulative search when attempting to find optimal parameter combinations. Not only would
this be computationally prohibitive, it also would be a very time-consuming exercise. There
hence is a need for identifying a procedure that, with reasonable levels of confidence, could
(i) identify important factors that need to be focussed on; (ii) set priorities among the fac-
tors in terms of their relative effects on system performance; and (iii) identify appropriate
parameter combinations for optimizing the assumed measures of performance. This paper
attempts to explore a methodology that fulfills these expectations.
The preceding motivations outline our objectives. More specifically, the objectives of
this paper are as follows:
1. To determine the significance of the impact of the design and control parameters on the
assumed performance measure.
2. To determine the relative impact of the design and control parameters (in terms of their
main factor effects) on the assumed performance measure.
3. To determine appropriate combinations of design and control parameters for optimal
shop performance.
4. To appreciate the impact of increases in routing flexibility on shop performance.
In pursuance of these objectives, we adopt Taguchis experimental design framework
(Phadke, 1989) for conducting the simulation study. Taguchis experimental design proce-
dure provides a convenient framework for establishing both the relative factor effects and
the significance of the assumed factors. Further, it helps identify suitable factor (level) com-
binations for finding near-optimal performance measure estimates. The following sections
address these issues in detail.
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276 R. CAPRIHAN AND S. WADHWA
Figure 1. Flexible manufacturing system schematic.
3. Flexible manufacturing system description
The hypothetical FMS assumed comprises six flexible machines (this dimension was chosen
as it can be considered to occur most frequently; Shanker and Tzen, 1985), each capable
of processing up to six different part types. Figure 1 is a schematic layout of the assumedsystem. Each part type requires between four and six operations for process completion.
Because the FMS is assumed to possess routing flexibility, alternate machines are available
for processing operations.
We use Chang et al.s (1989) definition of a flexibility indexto vary the degree of routing
flexibility for the assumed FMS. The routing flexibility index (RF) is defined as
RF =
mn
u=1|I(u)|/mn
whereu = index for operations
mn = total number of operations ofn parts onm machines
I(u) = index set of machines that can process operation u
RF, therefore, is a measure of the average number of machines capable of processing
an operation. Chang et al. (1989) point out that, whereas for a conventional job shop
RF = 1, for a shop possessing alternative routing capability, RF 2. Further, the marginal
benefit of increasing RF from 1 to 2 is very large, because this reduces the likelihood of
a bottleneck operation. However, the marginal benefit is expected to decrease rapidly with
further increases in flexibility.
Because routing flexibility is an assumed experimental factor, the RF is varied from 1 to 5.
Tables 1 to 5 depict the relevant part type/processing time details of RF indices between
1 to 5, respectively. It may be noted that, when RF = 1, part type sequences through the
FMS are fixed; that is, no alternate routes are possible. Notice further that an increase in the
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IMPACT OF ROUTING FLEXIBILITY ON THE PERFORMANCE OF AN FMS 277
Table 1. Part type/processing time data (RF = 1).
Alternate machinesPart
type Operation # 1 2 3 4 5 6
A 1 3
2 3
3 2
4 1
5 3
6 5
B 1 4
2 10
3 1
4 8
C 1 2
2 4
3 4
4 6
5 5 6
D 1 5
2 4
3 6
4 8
E 1 12
2 4
3 8
4 5
5 2
F 1 6
2 2
3 5
4 4
5 1 6 4
Note: Cell entries marked with imply the inability of the machine for
processing the specified operation.
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278 R. CAPRIHAN AND S. WADHWA
Table 2. Part type/processing time data (RF = 2).
Alternate machinesPart
type Operation # 1 2 3 4 5 6
A 1 4 3
2 4 3
3 2 3
4 2 1
5 3 4
6 6 5
B 1 4 4
2 10 11
3 1 2
4 8 10
C 1 4 2
2 5 4
3 4 4
4 6 6
5 5 6
D 1 5 5
2 4 4
3 6 6
4 9 8
E 1 13 12
2 4 4
3 8 8
4 6 5
5 2 4
F 1 6 7
2 4 2
3 5 5
4 4 5
5 1 2 6 5 4
Note: Cell entries marked with imply the inability of the machine for
processing the specified operation.
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IMPACT OF ROUTING FLEXIBILITY ON THE PERFORMANCE OF AN FMS 279
Table 3. Part type/processing time data (RF = 3).
Alternate machinesPart
type Operation # 1 2 3 4 5 6
A 1 4 3 4
2 4 4 3
3 2 3 3
4 2 1 2
5 4 3 4
6 6 7 5
B 1 4 5 4
2 10 11 12
3 3 1 2
4 10 8 10
C 1 4 5 2
2 6 5 4
3 4 4 4
4 6 7 6
5 5 6 6
D 1 5 6 5
2 4 5 4
3 7 6 6
4 9 8 9
E 1 13 14 12
2 4 5 4
3 8 8 9
4 6 6 5
5 4 2 4
F 1 6 7 7
2 4 2 4
3 5 5 6
4 4 6 5
5 1 2 26 6 5 4 6
Note: Cell entries marked with imply the inability of the machine for
processing the specified operation.
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280 R. CAPRIHAN AND S. WADHWA
Table 4. Part type/processing time data (RF = 4).
Alternate machinesPart
type Operation # 1 2 3 4 5 6
A 1 4 3 4 5
2 6 4 4 3
3 2 4 3 3
4 2 3 1 2
5 4 3 5 4
6 6 7 5 8
B 1 4 5 5 4
2 10 12 11 12
3 3 1 2 4
4 12 10 8 10
C 1 4 5 6 2
2 6 7 5 4
3 4 4 4 4
4 6 7 7 6
5 5 7 6 6
D 1 5 6 6 5
2 4 5 5 4
3 8 7 6 6
4 9 10 8 9
E 1 14 13 14 12
2 4 5 4 6
3 8 8 10 9
4 6 6 7 5
5 4 2 4 5
F 1 8 6 7 7
2 4 2 4 4
3 5 6 5 6
4 7 4 6 5
5 3 1 2 26 6 5 4 6
Note: Cell entries marked with imply the inability of the machine for
processing the specified operation.
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IMPACT OF ROUTING FLEXIBILITY ON THE PERFORMANCE OF AN FMS 281
Table 5. Part type/processing time data (RF = 5).
Alternate machinesPart
type Operation # 1 2 3 4 5 6
A 1 4 6 3 4 5
2 6 7 4 4 3
3 2 5 4 3 3
4 4 2 4 3 1 2
5 4 3 5 4 6
6 6 7 5 8 8
B 1 4 5 5 4
2 10 12 11 12
3 3 4 1 4 2 4
4 12 10 8 10
C 1 4 5 6 6 2
2 6 7 5 4 7
3 4 5 4 4 4
4 8 6 7 7 6
5 5 7 8 6 6
D 1 5 6 6 5
2 4 5 6 5 4
3 8 7 6 6
4 9 10 10 8 9 10
E 1 14 13 14 12
2 4 5 4 6 7 6
3 8 8 10 9
4 6 6 7 5 8
5 4 2 4 5
F 1 8 8 6 7 10 7
2 4 2 4 4
3 5 6 5 6
4 7 4 6 5
5 3 5 4 1 2 26 6 5 4 6 6
Note: Cell entries marked with imply the inability of the machine for
processing the specified operation.
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282 R. CAPRIHAN AND S. WADHWA
level of routing flexibility is associated with some penalty in terms of increased operation
processing times.
Different shop loading levels are effected by varying the number of available pallets,
used as part-holding devices, from 6 to 14. Each pallet is assumed to accommodate only a
single part. Preliminary simulation runs verified that changing the pallet availability within
this range resulted in the average system utilization varying 60 to 99%.
4. Modeling assumptions
The following assumptions are made with respect to the FMS assumed for the simulation
experiments:
1. Each machine is continuously available for processing; that is, machines never break
down.2. Pre-emption is not allowed; that is, operations that begin processing are completed
without interruption.
3. Machines are never unable to perform a required operation for lack of operator, tool,
or raw material.
4. When the RF 2, the machine selected after a dispatch decision also can be the machine
on which the part has just completed an operation.
5. Each machine can process only one operation at a time.
6. Operation processing times are deterministic.
7. Setup times are sequence independent and included as part of the operation processing
time.
8. Intermachine part transportation times are negligible.
9. Each part type, once started, must be processed to completion; that is, no order cancel-
lations are allowed.
10. Each part can be processed by only one machine at a time; that is, parts are assumed
to be indivisible.
11. Due dates are not specified.
12. No part is rejected due to quality inspection; that is, no rework is allowed.13. Pallet availability is limited.
14. All pallets can be used interchangeably by each part type.
15. All parts are available for processing at the start of the simulation experiment, although
part entry into the shop is dependent on pallet availability.
16. Unlimited buffer locations are assumed before individual machines.
5. Simulation details
Simulation experiments are performed using the SIMAN IV simulation language (Pegden,
Shannon, and Sadowski, 1990), into which user-written C code is linked to capture the
dispatching logic incorporated into the models. Each experiment constitutes a single repli-
cation, which is easily justifiable on account of our having assumed the following: (i) all
parts are assumed available at the start of the simulation run (i.e., part arrivals are not stochas-
tically generated), although part arrivals into the system are dependent on pallet availability;
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IMPACT OF ROUTING FLEXIBILITY ON THE PERFORMANCE OF AN FMS 283
and (ii) prespecified deterministic operation processing times are assumed. A total of 1,000
parts are simulated and the production mix is assumed to be a predetermined constant.
Finally, identical experimental testing conditions for each sequencing and dispatching rule
are ensured using the method of common random numbers (Pegden et al., 1990).
Sequencing of parts from the input buffers of each machine is effected using one of the
following five rules: FCFS, SIO, LIO, LRO, and SRO. Dispatching parts on completion
of an operation is performed using one of the following five rules: WINQ M, NINQ M,
WSPT, NSPT, and XWINQ M. If there happens to be a tie between parts of the same type,
then the FCFS rule is used to break the tie. (See Appendix A for details regarding the
assumed scheduling rules.)
The measure of performance used in the present study is makespan, which is the time
required to completely process all required parts of all types (a total of 1,000 parts is
assumed).The simulation models developed to study the effects of the assumed factors are similar
as far as the application of the sequencing logic is concerned. The essential differences
between the models arise from the manner in which the user-written dispatching logic is
coded. The logic behind each dispatching heuristic (WINQ M, NINQ M, WSPT, NSPT,
and XWINQ M) is a straightforward translation of the respective definitions1.
5.1. Taguchis experimental design framework
The Taguchi experimental design paradigm is based on the technique of matrix experiments
(Phadke, 1989). A matrix experiment consists of a set of experiments where the settings
of the process parameters under study are changed from one experiment to another. The
experimental data generated subsequently is analyzed to determine the effects of various
process parameters. In the statistical literature, matrix experiments are called designed
experiments, and the individual experiments in a matrix experiment are called treatments.
Settings are also referred to aslevelsand parameters asfactors(Phadke, 1989).
Experimental matrices essentially are special orthogonal arrays, which allow the simul-
taneous effect of several process parameters to be studied efficiently. As the name suggests,the columns of an orthogonal array are mutually orthogonal; that is, for any pair of columns
all combinations of factor levels occur and they occur an equal number of times. This, called
thebalancing property, implies orthogonality (Phadke, 1989). The columns of an orthog-
onal array represent the individual factors under study, and the number of rows represent
the number of experiments to be conducted.
The purpose of conducting an orthogonal experiments is twofold:
1. To determine the factor combinations that will optimize a defined objective function
(i.e., to determine the optimal level for each factor).2. To establish the relative significance of individual factors in terms of their main effects
on the objective function.
Taguchi suggests using a summary statistic, , called thesignal-to-noise(S/N)ratio, as
the objective function for matrix experiments. Phadke (1989) discusses the rationale for
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284 R. CAPRIHAN AND S. WADHWA
using as the objective function. Taguchi classifies objective functions into one of three
categories: the smaller-the-better type, the larger-the-better type; and the nominal-the-best
type. S/N ratios are measured in decibels (dB).
An important goal in conducting a matrix experiment is to determine optimum factor
levels. The optimum level for a factor is that which results in the highest value of in the
experimental region. The effect of a factor level (also called the main effect) is defined as
the deviation it causes from the overall mean2. The process of estimating the main effects
of each factor is called analysis of means.
Taguchi makes a fundamental assumption in the method suggested for determining the
optimal factor combination (based on the optimum level for each factor) for a defined objec-
tive function. He assumes that the variation ofas a function of the factor levels is additive
in nature; that is, cross-product terms involving two or more factors are not allowed. The
assumption of additivity essentially implies the absence of significant interaction effectsbetween factors. Taguchi suggests that a verification experiment (with factors at their op-
timum levels) be run to validate the additivity assumption. After running a verification
experiment, Phadke (1989) points out
if the predicted and observed are close to each other, then we may conclude that the
additive model is adequate for describing the dependence ofon the various parameters.
...On the contrary, if the observation is drastically different from the prediction, then we
say the additive model is inadequate. ...This is evidence of a strong interaction among
the parameters.
In fact, Taguchi considers the ability to detect the presence of interactions to be the
primary reason for using orthogonal arrays to conduct matrix experiments.
5.2. Standard orthogonal arrays
Taguchi has tabulated 18 basic orthogonal arrays, called standard orthogonal arrays. To
illustrate the notational scheme used for standard orthogonal arrays, consider as an example
the L 25(56)array, which has 25 rows with six 5-level factors. For brevity, we henceforth
refer to the L25(56)array simply as the L 25 array. The number of rows of an orthogonal
array represents the number of experiments to be conducted. To be a viable choice, the
number of rows must be at least equal the degrees of freedom required for the problem.
The number of columns of an array represents the maximum number of factors that can be
studied using that array. Further, to use a standard orthogonal array directly, we must be
able to match the number of levels of the factors with the number of levels of the columns
in the array. Importantly, orthogonality of a matrix experiment is not lost by keeping oneor more columns of an array empty.
The real benefit in using matrix experiments is the economy they afford in terms of
the number of experiments to be conducted. In the present study, because we need to
experiment with four factors, each at five levels, a full factorial experiment would have
required 54 = 625 experiments. In contrast, having found theL25 orthogonal array to be
suitable for our purposes, only 25 experiments need to be conducted.
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IMPACT OF ROUTING FLEXIBILITY ON THE PERFORMANCE OF AN FMS 285
6. Matrix experiment details
To study the impact of the assumed factors within the FMS considered, standard orthogonal
array experiments are performed. As mentioned above, Taguchis standard L 25 orthogo-
nal array (see Table 6) is found suitable for experimentation purposes. This enables the
simultaneous consideration of six factors at five levels. In the present case only four factors
are considered, so the first four columns of the L 25 orthogonal array are used, with the
fifth and sixth columns being excluded for experimentation purposes without affecting the
orthogonality of the matrix. The respective factors along with their assumed levels are
Factor Levels
Routing flexibility (1, 2, 3, 4, 5)
Number of pallets (6, 8, 10, 12, 14)Dispatching rule (WINQ M, NINQ M, WSPT, NSPT, XWINQ M)
Sequencing rule (FCFS, SIO, LIO, LRO, SRO)
Table 6. StandardL 25(56)orthogonal array.
1 1 1 1 1 1
1 2 2 2 2 2
1 3 3 3 3 31 4 4 4 4 4
1 5 5 5 5 5
2 1 2 3 4 5
2 2 3 4 5 1
2 3 4 5 1 2
2 4 5 1 2 3
2 5 1 2 3 4
3 1 3 5 2 4
3 2 4 1 3 5
3 3 5 2 4 1
3 4 1 3 5 2
3 5 2 4 1 3
4 1 4 2 5 3
4 2 5 3 1 4
4 3 1 4 2 5
4 4 2 5 3 1
4 5 3 1 4 2
5 1 5 4 3 2
5 2 1 5 4 3
5 3 2 1 5 4
5 4 3 2 1 5
5 5 4 3 2 1
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286 R. CAPRIHAN AND S. WADHWA
Table 7. Factor-level details used in the matrix experiment.
Factor name (label) Factor level Factor-level details (name or value)
Routing flexibility (RF) 1 1
2 2
3 3
4 4
5 5
Dispatching rule (DR) 1 WINQ M
2 NINQ M
3 WSPT
4 NSPT
5 XWINQ M
Sequencing rule (SR) 1 FCFS
2 SIO
3 LIO
4 LRO
5 SRO
Number of pallets (NP) 1 6
2 83 10
4 12
5 14
The levels for each factor used in the matrix experiment are shown in Table 7. Table 8
shows the resulting matrix experiment table with the factor level details.
Because makespan is the assumed performance measure for our study, it can be suitably
modified into the corresponding S/N ratio for incorporation into the matrix experiment. Itmay be noted here that, although the real benefit in using S/N ratios is for situations where
multiple replications are performed (Roy, 1990), we resort to their use to remain consistent
with their conventional usage in the Taguchi method of experimentation. Accordingly, the
makespan performance measure, classified in the smaller-the-better category, is modified to
i = 10 log10 (makespan)2
for incorporation into the orthogonal array experiment.
7. Matrix experiment results
The results obtained from the matrix experiment are detailed in Table 9. The data analy-
sis procedure using the Taguchi experimental framework involves the analysis of means
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IMPACT OF ROUTING FLEXIBILITY ON THE PERFORMANCE OF AN FMS 287
Table 8. Factor-level details.
Experiment # RF DR SR NP
1 1 WINQ M FCFS 6
2 1 NINQ M SIO 8
3 1 WSPT LIO 10
4 1 NSPT LRO 12
5 1 XWINQ M SRO 14
6 2 WINQ M SIO 10
7 2 NINQ M LIO 12
8 2 WSPT LRO 14
9 2 NSPT SRO 6
10 2 XWINQ M FCFS 811 3 WINQ M LIO 14
12 3 NINQ M LRO 6
13 3 WSPT SRO 8
14 3 NSPT FCFS 10
15 3 XWINQ M SIO 12
16 4 WINQ M LRO 8
17 4 NINQ M SRO 10
18 4 WSPT FCFS 12
19 4 NSPT SIO 14
20 4 XWINQ M LIO 6
21 5 WINQ M SRO 12
22 5 NINQ M FCFS 14
23 5 WSPT SIO 6
24 5 NSPT LIO 8
25 5 XWINQ M LRO 10
(ANOM) and analysis of variance (ANOVA). ANOM helps identify the optimal factorcombinations, whereas ANOVA establishes the relative significance of factors in terms of
their contribution to the objective function. Using the simulation results data summarized
in Table 9, the ANOM and ANOVA is presented next.
7.1. Analysis of means
The main factor effects, calculated using the formulas given in Phadke (1989), are summa-
rized in Table 10. The notational convention adopted for analysis ism j k = main factor effect for thekth level of factor j ,
3
i = observed S/N ratio for thei th orthogonal experiment4,
m = overall mean value of = [n
i =1i ]/n,
wheren = number of experiments performed (i.e., 25).
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288 R. CAPRIHAN AND S. WADHWA
Table 9. Matrix experiment simulation results.
Experiment # Observed makespan (mins) Observed makespan (i ) (dB)
1 6152 75.78
2 5470 74.76
3 5382 74.62
4 5100 74.15
5 5019 74.01
6 4695 73.43
7 4566 73.19
8 4398 72.87
9 5680 75.09
10 4839 73.70
11 4463 72.99
12 6174 75.81
13 4785 73.60
14 4472 73.01
15 4465 73.00
16 5370 74.60
17 4835 73.69
18 4618 73.29
19 4347 72.76
20 5428 74.69
21 4952 73.90
22 4974 73.93
23 5591 74.95
24 4789 73.60
25 4959 73.91
Based on the analysis of means, the optimum levels for each factor resulting from the
matrix experiment is shown italicized in the Makespan column of Table 10. It may be
noted that the main effects values are measured in decibels because they refer to S/N ratios.
Accordingly, the predicted factor level combination that should optimize (i.e., minimize)
the makespan is RF2, DR4, SR2, NP5, which easily is interpreted to mean the routing
flexibility = 2, the dispatching rule is NSPT, the sequencing rule is SIO, and the number of
pallets = 14. Interestingly, the predicted best settings do not correspond to any of the rows
in the matrix experiment.
Figure 2 plots the main effects of each factor level. The optimal level for each factor easily
is identified as the level that results in the highest value of in the factor-level range. Note
that the prediction of the optimum factor level combination is conditioned by the variation
of as a function of the factor level, satisfying the additivity assumption. To justify the
validity of this assumption, we need to carry out a verification experiment with optimum
factor-level settings. The results of the verification experiment are reported in Section 8.
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IMPACT OF ROUTING FLEXIBILITY ON THE PERFORMANCE OF AN FMS 289
Table 10. Factor main effects for matrix experiment simulation results.
Factor-level Main effect
main effects Applicable formula value makespan
mRF1 (1 + 2 + 3 + 4 + 5)/5 74.664
mRF2 (6 + 7 + 8 + 9 + 10)/5 73.656
mRF3 (11 + 12 + 13 + 14 + 15)/5 73.682
mRF4 (16 + 17 + 18 + 19 + 20)/5 73.806
mRF5 (21 + 22 + 23 + 24 + 25)/5 74.058
mDR1 (1 + 6 + 11 + 16 + 21)/5 74.140
mDR2 (2 + 7 + 12 + 17 + 22)/5 74.276
mDR3 (3 + 8 + 13 + 18 + 23)/5 73.866
mDR4 (4 + 9 + 14 + 19 + 24)/5 73.722
mDR5 (5 + 10 + 15 + 20 + 25)/5 73.862
mSR1 (1 + 10 + 14 + 18 + 22)/5 73.942
mSR2 (2 + 6 + 15 + 19 + 23)/5 73.780
mSR3 (3 + 7 + 11 + 20 + 24)/5 73.818
mSR4 (4 + 8 + 12 + 16 + 25)/5 74.268
mSR5 (5 + 9 + 13 + 17 + 21)/5 74.058
mNP1 (1 + 9 + 12 + 20 + 23)/5 75.264
mNP2 (2 + 10 + 13 + 16 + 24)/5 74.052
mNP3 (3 + 6 + 14 + 17 + 25)/5 73.732
mNP4 (4 + 7 + 15 + 18 + 21)/5 73.506
mNP5 (5 + 8 + 11 + 19 + 22)/5 73.312
The ANOM plots shown in figure 2 reveal the relative magnitude of effects by factors
on the makespan: The number of pallets is seen to affect the makespan the most, followed
by the routing flexibility. The effect of both of the control rules is seen to be relatively lesspronounced. However, a better feel for the relative effects is obtained by conducting the
analysis of variance. ANOVA also is needed for estimating the error variance for the factor
effects and the variance of the prediction error (Phadke, 1989), which provide the necessary
input for justifying the additivity assumption.
7.2. Analysis of variance
The formulas used in conducting the ANOVA are detailed in Appendix B. Table 11 showsthe resulting ANOVA tableau. From the ANOVA tableau, the error variance (2e), defined as
Error variance = (SSE/degrees of freedom for error)
is calculated to be (2e)makespan = 0.170(dB)2. It may be noted that the error variance is
calculated using the method of pooling (Phadke, 1989).
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290 R. CAPRIHAN AND S. WADHWA
Table 11. ANOVA using the simulated results estimated as S/N ratios.
Sum of Mean
Factor DOFa squares square F
Number of pallets 4 11.930 2.983 17.533
Routing flexibility 4 3.489 0.872 5.127
Dispatching rule 4 1.032b 0.258
Sequencing rule 4 0.782b 0.196
Error 8 0.907b 0.113
Total 24 18.141
(Error) (16) (2.722) (0.170)
aDOF is degrees of freedom.
bIndicates the sum of squares added together to estimate the pooled errorsum of squares, indicated by parentheses. TheFratio is calculated using
the pooled error mean square.
Figure 2. Analysis of means plot of factor main effects (performance measure is makespan; optimal factor level
is indicated on the graph).
Phadke (1989) suggests using theFratio resulting from the ANOVA only to establish the
relative magnitude of the effect of each factor on the objective function and to estimate
the error variance. However, probability statements regarding the significance of indivi-
dual factors are not made. From the ANOVA tableau, the relative effects of the factors
the number of pallets and the routing flexibility are seen to be important, followed by the
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IMPACT OF ROUTING FLEXIBILITY ON THE PERFORMANCE OF AN FMS 291
Table 12. ANOVA using the original simulated results.
Sum of Mean
Factor DOFa squares square Fb
Number of pallets 4 4313074 1078268 18.080
Routing flexibility 4 1154163 288540 4.838
Dispatching rule 4 378559c 94639
Sequencing rule 4 269754c 67438
Error 8 305888c 38236
Total 24 6421440
(Error) (16) (954201) (59637)
aDegrees of freedom.
bThe critical Fratio (at = 0.5; i.e., F0.5,4,16) = 3.01.cIndicates the sum of squares added together to estimate the pooled error
sum of squares, indicated by parentheses. The F ratio is calculated using
the pooled error mean square.
factors the dispatching rule and the sequencing rule, in that order. This is in agreement with
the ANOM results.
To highlight the statistical significance of the impact of individual factors on the make-
span, in Table 12 we present the ANOVA using the original simulated results (i.e., without
converting to S/N ratios). The resulting F ratios (again calculated using the method of
pooling) are seen to be critical for the factors the number of pallets ( F = 18.080) and the
routing flexibility (F = 4.838).
8. Testing for additivity
To validate the assumption of additivity, a verification experiment needs to be conducted
with the optimal factor settings (Phadke, 1989). The result of the verification experiment
then is compared with a predicted optimal value, resulting in a prediction error. If the
prediction error happens to fall within a two-standard-deviation confidence limit of thevariance of prediction error, the additivity assumption can be assumed justified (Phadke,
1989). Validation of the additivity assumption essentially implies the absence of significant
interaction effects between factors.
8.1. Verification experiment
A verification experiment was performed with the optimal factor combination (RF2, DR4,
SR2, NP5). The observed optimal makespan was 4,339 mins; that is,obs.opt = 72.74 dB.
The following equation was then used to predict the optimum performance measure value
(Phadke, 1989):
pre.opt = m + (mNP5 m) + (mRF2 m)
= 73.973 + (73.312 + 73.973) + (73.656 + 73.973)
= 72.995 dB
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IMPACT OF ROUTING FLEXIBILITY ON THE PERFORMANCE OF AN FMS 293
pallets and routing flexibility and only then on the control rules for improving shop per-
formance. The result is noteworthy, because shop controllers instead would have typically
been tempted to experiment with alternative control rules without regard to the possible
benefits of trying other controllable factors. The use of the Taguchi experimental design
procedure provides an expedient platform for quickly focusing in on the parameters that
need to be given priority.
For the experimental frame considered, the impact of increasing both the number of
pallets and the routing flexibility on makespan is not as straightforward as it may appear to
be at first. One generally assumes that an increase in the number of available pallets would
result in higher utilization levels for the machines in the system. It thereby is hoped that the
makespan would be reduced. Similarly, one is tempted to believe that increasing levels of
routing flexibility would ensure smaller waiting times for parts in machine queues because
of the possibility of the parts being directed toward less-congested queues. This intuitivelywould imply a reduction in makespan, because the waiting time for all parts most likely
would fall. We caution that such intuitive predictions for flexible systems often can lead to
erroneous results, as highlighted next.
Attention is drawn to the operating conditions specified in an earlier section. Notice that
the assumed processing time data detailed in Tables 1 through 5 reflects that increases in
routing flexibility are associated with a corresponding penalty. The penalty takes the form
of an increase in the operation processing time for every additional machine made available
(i.e., with increasing values of routing flexibility) for processing a given part type operation.
Whereas it is obvious that an increase in routing flexibility without an associated penaltywould not have an adverse affect on shop performance, the case investigated poses a more
challenging problem. In our opinion, the case with a penalty is representative of more
practical situations and consequently warrants research attention.
The ANOM plot of figure 2 justifies our intuition with regards to the impact of increasing
the number of pallets on makespan; that is, the latter improves with additional pallets being
made available. However, as can be seen from the plot, the marginal improvement decreases
as we increase the number of pallets.
Further, the ANOM plot of figure 2 provides two useful insights with regard to routing
flexibility: an increase in the RF from 1 to 2 improves system performance appreciably even
at the cost of an associated penalty, but continuing the increase in RF does not guarantee
an improvement in performance in terms of makespan reduction.
It also is interesting to report that about 300 random experiments (from the possible 54 =
625 experiments) were performed for comparison purposes with the makespan estimate
resulting from the optimal factor setting of the Taguchi experiment using the same data set.
Only two of the random experiments performed found better makespan values5. However,
not only were these very close to Taguchis optimal value but were also well within the
two-standard-deviation confidence limits of the variance of prediction error. Importantly,the two cases differed only in terms of the levels for the factor routing flexibility. Another
interesting observation made from the random experiments conducted was that the highest
flexibility level; that is, RF = 5 was not the optimum. This clearly indicates that increasing
the routing flexibility beyond a certain level sometimes can be counterproductive. (This
observation, however, is domain specific, being applicable for the assumed experimental
control structure, and therefore is not to be interpreted in a generic sense.) The ANOM
plots of figure 2 indeed had predicted a similar behavioral pattern.
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294 R. CAPRIHAN AND S. WADHWA
In our opinion, therefore, the following two-step methodology may be adopted by system
controllers for improving shop performance:
1. Identify intuitively sound alternatives for both design and control parameters (factors)
and perform a Taguchi experiment with suitably selected factor levels. This will enablesetting priorities among the identified factors.
2. Perform detailed simulation experiments on a full factorial basis for the smaller subset
of factors with significant effects on the performance measure to thereby identify more
reliable factor level combinations for performance improvement.
10. Conclusions
In this paper, we proposed a methodology based on the Taguchi experimental design pro-
cedure that can be used by system designers to gain quick insights into the behavior of
assumed design and control parameters within FMS environments. We suggest that the role
of increasing routing flexibility (when made available at the cost of an associated penalty on
operation processing time) should not be taken for granted as a direction for performance
improvement. In this regard, as is demonstrated through the use of the proposed method-
ology, for the hypothetical system and conditions assumed in this study, there is an optimal
flexibility level, beyond which the performance is seen to deteriorate.
The proposed methodology can further help system designers and controllers not only insetting priorities to focus on the assumed design and control factors but also in highlighting
likely factor-level combinations that would result in near-optimal shop performance.
Appendix A
The following notation will be followed in defining the scheduling rules (both sequencing
and dispatching rules) used for the simulation study detailed in this paper.
T = Scheduling horizon (total available production time)t = Time at which the decision is to be made
n = Number of parts in the shop
i = Part index
j = Operation index
j (t) = Imminent operation of parti ; that is, all operations 1 j < j (t)are completed
pi,j = The processing time for the j th operation of thei th part
Ri,j = The time at which thei th part becomes ready for its j th operation
ROi (t) = Remaining number of operations on thei th part
Ni,j (t) = The set of parts in the queue corresponding to the j th operation of thei th part at
timet
Wi,j (t) = The total work content of the queue; that is, the sum of the imminent operation
times of the Ni,j (t)parts in that queue
Ki (t) = The priority of parti at timet
Mi,j +1 = The set of machines capable of processing the (j + 1)th operation of thei th part
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IMPACT OF ROUTING FLEXIBILITY ON THE PERFORMANCE OF AN FMS 295
Ni,j +1,m(t) = The set of parts in the m th machine queue, mMi,j +1, corresponding to the
j + 1th operation of thei th part at timet
Wi,j +1,m(t) = The total work content of themth machine queue,mMi,j +1, that is, the sum
of the imminent operation times of the Ni,j +1,m(t)parts in that queue
A.1. Sequencing rules used
SIO = Select the part with the shortest imminent operation (SIO) time; that is, select the
minimumKi (t), where Ki(t) = pi,j (t)LIO = Select the part with the longest imminent operation (LIO) time; that is, select the
maximumKi (t), where Ki(t) = pi,j (t)
SRO = Select the part with the smallest number of remaining operations (SRO); that is,select the minimum Ki (t), where Ki(t) = ROi (t)
LRO = Select the part with the largest number of remaining operations (LRO); that is,
select the maximum Ki (t), where Ki(t) = ROi (t)
FCFS = Select the part according to the rule of first come, first served (FCFS); that is, select
the minimum Ki (t), where Ki (t) = Ri,j foriNi,j (t)
A.2. Dispatching rules used
NINQ M = Select that machine to process the next operation which has the shortest
queue; that is, select the minimum Ki (t), where Ki (t) = |Ni,j +1,m(t)|
formMi,j +1WINQ M = Select that machine to process the next operation which has the least
work; that is, select the minimum Ki (t), where Ki (t) = Wi,j +1,m(t)for
mMi,j +1WSPT = Select that machine to process the next operation which has the least
work; that is, select the minimum Ki (t), where Ki (t) = Wi,j +1,m(t)formMi,j +1. In the event of a tie, select the machine that will process the
operation in the (work) shortest processing time (WSPT). If a tie still
persists, make a random choice from among the tied machines
NSPT = Select that machine to process the next operation which has the small-
est number in queue; that is, select the minimum Ki (t), where Ki (t) =
|Ni,j +1,m(t)| for mMi,j +1. In the event of a tie, select the machine
that will process the operation in the (number) shortest processing time
(NSPT). If a tie still persists, make a random choice from among the tied
machines
XWINQ M = Select that machine to process the next operation which has the least
expected work; that is, select the minimum Ki (t), where Ki (t) =
Wi,j +1,m(t)+ formMi,j +1. Wi,j +1,m(t)
+ includes the processing times
of operations already assigned to the machine in question but yet to arrive
at the machines input buffer6
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296 R. CAPRIHAN AND S. WADHWA
Appendix B: Formulas for analysis of variance
We use the following formulas in conducting the ANOVA (Phadke, 1989).
Total sum of squares (SST)= Sum of the sums of squares due to various factors (SSB)
+ Sum of squares due to error (SSE)
where
SST = Grand total sum of squares (GTSS)
Sum of squares due to the mean (SSM)
Now,
GTSS =
25i =1
2i = (75.78)2 + (75.76)2 + + (73.91)2
= 136819.016(dB)2
Further,
SSM = n m2 = 25 (73.973)2
= 136800.875(dB)2
Therefore,
SST = GTSS SSM = 18.141(dB)2
Also,
SSB =
cj =1
lj
ljk=1(m
j k m)
2wherec = number of factors;lj = number of levels for factor j .
This essentially can be broken up into
SSB = SSB1 + SSB2 + SSB3 + + SSBC
In our case,
SSB = SSBRF + SSBDR + SSBSR + SSBNP
Now,
SSBRF = 5 [(74.664 + 73.973)2 + (73.656 + 73.973)2
+ +(74.058 + 73.973)2]
= 3.489(dB)2
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IMPACT OF ROUTING FLEXIBILITY ON THE PERFORMANCE OF AN FMS 297
Similarly, the other components are calculated to be
SSBDR = 1.033(dB)2
SSBSR = 0.782(dB)
2
SSBNP = 11.930(dB)2
Therefore,
SSB = 3.489 + 1.033 + 0.782 + 11.930
= 17.234(dB)2
Finally,
SSE = SST SSB = 18.141 17.234
= 0.907(dB)2
Notes
1. To link user-written C routines with SIMAN IV simulation models, a separate SIMAN IV executable file needs
to be created. Consequently, every dispatching heuristic results in a unique SIMAN IV executable file. This
entailed the creation of five executable files for the present study.
2. The overall mean value of for the experimental region is defined as
m =
ni =1
i
n
wheren = number of experiments performed and i = experiment number.
3. The term j is assigned the following labels: RF = routing flexibility, DR = dispatching rule, SR = sequencing
rule, NP = number of pallets.
4. For the performance measure makespan,iis calculated as
i = 10 log
10(makespan)2.
5. Case 1 (RF: 3; DR: NSPT; SR: SIO; NP: 14) and Case 2 (RF: 4; DR: NSPT; SR: SIO; NP: 14). The observed
makespan for Case 1 was 4,269 mins and for Case 2 it was 4,334 mins.
6. Although intermachine part transportation times have been neglected, such situations can arise when dispatch
decisions are effected at identical event times.
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