rahul caprihan

8/11/2019 Rahul Caprihan

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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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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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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




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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




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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




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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




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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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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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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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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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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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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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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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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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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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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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21/26


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.


22/26


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


23/26


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


24/26


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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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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