Computers ind. Engng Vol. 16, No. 1, pp. 65-74, 1989 0360-8352/89 $3.00 + 0.00 Printed in Great Britain. All rights reserved Copyright © 1989 Pergamon Press pie

S I M U L A T I O N S T U D I E S O F F L E X I B L E M A N U F A C T U R I N G S Y S T E M S U S I N G S T A T I S T I C A L D E S I G N O F E X P E R I M E N T S

P. K. MISHRA and P. C. PANDEY* Department of Mechanical Engineering, MNR Engineering College, Allahabad, India and

Department of Mechanical Engineering, University of Roorkee, Roorkee 247667, India

(Received for pubfication 2 March 1988)

Abstract--In this paper, a simulation based dynamic scheduling algorithm for the performance evaluation of a Flexible Manufacturing System has been developed. The simulation experiments were conducted for a manufacturing system with six machines, processing six different types of jobs. The system performance was evaluated on the basis of the number of tardy jobs, number of completed jobs, number of waiting and running jobs and average machine utilization. All the simulation experiments were planned as per central composite rotatable design of experiments. It has been observed that the use of designed experiments reduces the computational effort without loss of accuracy.

The system responses were derived in the form of second order polynomials and these have been next utilized to give the optimized values for the system operating parameters. It has been concluded that for the system studied, in order to maximize the number of completed jobs, the mean processing rate and inter-arrival times of the jobs have to be maintained at their highest levels, but for maximizing the number of running jobs and machine utilization, the mean job inter-arrival time must be kept at 0.4.

1. INTRODUCTION

Introduction of highly automated computer controlled machines into manufacturing has been one of the more important and recent developments. All the indications are that such a trend towards automation will continue until near-automated or automated factory becomes a reality. The concept of the flexible manufacturing system (FMS) combines the existing technologies of numerical control manufacturing, automated material handling, and computer hardware and software in order to bring economics of scale of batch work, and is a major step towards unmanned batch production automation.

Despite the obvious success of FMS, its universal adoption by manufacturing industries has not progressed as envisaged. Several reasons can be assigned to this state of affairs. The cost of a FMS (4-12 NC or CNC machine tools) can range from US$1-50 million, may require 1000 m 2 or more of floor space, and need from 1 to 5 years to design and implement. Based soley on capital and other requirements it is easy to understand the reasons for the reluctance of industries to go for FMS in a big way [1].

In view of heavy economic investment in the installation and operation of the flexible manufacturing systems, it is vital that they be operated at the highest possible efficiency. The FMS performance is normally derived from the use of models because experimentation on the actual system is not feasible. Models for FMS however, must consider the interaction effects of the various decision variables subject to technological and other constraints. All such models can be classified under the following two categories:

generative models evaluative models

An excellent review of generative models is given by Buzacott and Yao [2]. These models, however, are useful for systems with relatively few operating parameters and the effects of machine failures, demand uncertainties, etc. are difficult to account for. On the other hand, evaluative models [3-5] are more of a tool to help the decision maker provide an insight into the working of the system but do not lead to optimality. The computer simulation is one of the most widely used evaluative tools for the manufacturing system performance study. The major drawback of this

*Author for correspondence.

65

66 P .K. MISHRA and P. C. PANDEY

D

Loading and unloading area

~ Machine tooL E~] Buffer ~ PeLLet

Fig. 1. FMS shop layout.

methodology however, lies in its high cost of computer time which increases exponentially with the number of decision variables.

In FMS modelling, perturbation analysis (PA) [6] has also been used in a limited way. This has been found to be more efficient computationally but is recommended for systems with few operating variables only.

This paper examines the effect of various operating parameters on the performance of Flexible Manufacturing System through computer simulation. The simulation experiments have been planned using statistically designed experiments and system response equations, in the form of the second degree polynomials derived. The responses chosen are: number of tardy, completed, running and waiting jobs, and average machine utilization. In addition, a sensitivity study and optimization of the operating parameters has been attempted.

2. SYSTEM MODELLING

The flexible manufacturing system (Fig. 1) considered in this paper, has been assumed to process p~ (i = 1, 2 , . . . P) kinds of parts. Each part of type i requires K i numbers of operations before completion. The processing of the parts is completed on a number of identical general purpose machine tools M m ( m = 1, 2 , . . . M ) each provided with a buffer of capacity B,, (m = 1, 2 . . . . M). As soon as the parts are received at the loading/unloading (L/U) station they are loaded into a number of pallets H~ (h = I, 2 . . . . P) and thereafter despatched for processing. The part along with the pallet is stored in the empty buffer space of the concerned machine, otherwise returned back to the L/U station.

The system described in Fig. I operates under the following constraints: 1. The jobs do not recycle. 2. One machine processes one part at a time. 3. All the buffers are full to start with (at t = 0). 4. Operations once started cannot be interrupted before completion. 5. Machine waiting due to non-availability of tools, jigs, fixtures etc., is negligible. 6. The processing times are inclusive of the setup times and are independent of the sequence

followed. 7. All parts are processed as per pre-determined sequence. 8. The velocity of pallet travel between the work stations is constant. 9. The inter-arrival and processing times of the parts vary randomly.

Flexible manufacturing systems 67

Update cLock.,]

NO

I Define data I

_I - t Generate arrivals ]

Generate new sequence of machines

I CO,curate Es~ I=

YES ~ ~NO .

Assign priority I index, buffer, pallet I

~ YES

1 SeLect job of I highest priority

n ~ YES - ~No

I Schedule Job I

:[ Scan system status I

1 I Hat° job I

Fig. 2. Schematic flow chart for generation of schedules.

3. S C H E D U L I N G M E T H O D O L O G Y

In order to develop an efficient scheduling methodology it is assumed that the arrival of the parts to the loading and unloading station is as per the Poisson law, whereas the intertravel time of the pallets, interfailure time of the machines, repair time and machining times follow an exponential distribution [7, 8]. Total number of operations on the jobs vary randomly. The assignment of due date, to a new arrival, has been based on the following consideration.

Kanet et al., [9] and others [10, 11] have suggested that an index based on the total work content of the job is a good basis for setting the due date in jobshops. Conway et al., [13] on the other hand, are of the view that the due date in a job shop be equal to the total work content of the

68 P.K. MISHRA and P. C. PANDEY

job multiplied by a factor of 4 to 8. In view of this, the due date for a new arrival has been assumed to be uniformally distributed between four and six times the total work content of the job.

Based on earliest start times of the parts (EST) the algorithm (Fig. 2) generates a list of workpieces competing for each machine. Depending upon the buffer capacity and pallet avail- ability, priority is assigned to the first operation in the list as per equation (1).

P/, = (DD~- rain (EST(P~)), (i = 1, 2 . . . . P)/WKR~ (1)

Where,

EST(P~) EST(Pi)

Fm HT(mi 1, m)

DD~ For - I

ARTi WKRi

= MAX (Fro, Fo, . + HT(mi t, mi) for partially finished jobs = MAX (F,,, ART~ + HT(m~_ i, mi), for new arrivals. = finish time of the previously assigned job to machine m, = transportation time of the parts between two consecutive workstations, = due date for ith part, = finish time of (Oi i)th operation on ith part, = arrival time of part i at the loading and unloading station, = work remaining on ith part.

It can be noticed that the use of the scheduling and rescheduling procedure enables an effective utilization of the processing facilities. The algorithm also provides revised schedules in case of significant operational changes e.g. machine breakdowns and repairs.

4. SIMULATION METHODOLOGY

Based on the next event time flow mechanism pilot simulation, runs were made assuming the system to be initially "empty and idle". However, allowance must be made for the system to reach the steady state [24]. From preliminary runs it was found that after 20 simulations (each sample of size 10), the system virtually acquires steady state and hence, the statistics of the initial 20 runs have been ignored in all the experiments [15].

4.1. Performance indices

The following performance indices have been employed:

Number of tardy jobs (NT) Number of completed jobs (NC) Number of running jobs (NR) Number of waiting jobs (NW) Average machine utilization (U) 4.1.1. Number of tardy jobs (NT). The lateness of job i, is measured by Li,

Li = (Cp~ - DD~) if Cpi > DD,, and L, = O, if Cp, <~ DO,

where Cp~ = completion time of job i. Tardiness can be obtained as T, = max (0, L~) and the number of tardy jobs is equal to:

P

NT = ~ 6 (T,) i = l

where

6 ( x ) = l if x > 0

6(x)-- 0, otherwise.

4.1.2. Number of completed jobs (NC). It is equal to:

P

NC = Y, ~(T,) i = 1

Flex ib l e m a n u f a c t u r i n g s y s t e m s 69

x, :]

z=

11

T - f ( x )

Z = = constant

Fig. 3. C y b e r n e t i c b l a c k b o x m o d e l o f F M S .

where

6(x) = l i fx~<0

6(x) = 0, otherwise.

4.1.3. Number of running jobs (NR). This refers to the number of jobs which are partially complete at the end of the planning horizon.

4.1.4. Number of waiting jobs (NW). This refers to number of jobs which are queuing up for their first operation at the end of the simulation run.

4.1.5. Average machine utilization (U). If T= is the time interval during which the machine M,, was actually used and S is the planning horizon, then U is given by:

U =Tm x 100/S.

All the simulation experiments were planned as per statistical design of experiments.

5. S T A T I S T I C A L D E S I G N O F S I M U L A T I O N E X P E R I M E N T S

For this purpose the system described in Fig. 1 can be modelled as a cybernetic black box (Fig. 3). In Fig. 3, X~'s are the factors whose effects on the system performance have to be investigated, Zm's are the factors held constant during the investigation and Yu's are the measured values of the responses. The factors X~ have been selected as part processing time, pallet travel time and the job interarrival time. Whereas the levels of X~ are given in Table 1 and chosen responses (I",) are:

number of tardy jobs, number of completed jobs; number of running jobs, number of waiting jobs; and average machine utilization.

The simulation experiment to study the factorial effects of the FMS operating parameters, was planned in accordance with the statistical technique of experimental design [16, 17]. With a well

Table 1. Scheme of simulation experimentation using central composite rotatable design

S. No. x I x 2 x 3 S. No. x I x 2 x 3

1. --1 --1 --1 11. 0 - - I .682 0 2. 1 --1 --1 12. 0 1.682 0 3. -- 1 I - - I 13. 0 0 --1.682 4. 1 1 - 1 14. 0 0 1.682 5. --1 - - I 1 15. 0 0 0 6. I - - I I 16. 0 0 0 7. -- I I 1 17. 0 0 0 8. 1 1 I 18. 0 0 0 9. -- 1.682 0 0 19. 0 0 0

10. 1.682 0 0 20. 0 0 0

70 P . K . MIsHRA a n d P. C. PANDEY

Table 2. Range of parameters selected for experimentation

Levels in coded form

Variables 1.682 t 0 - I - 1,682

(x~) Mean processing time 2.0 1.645 1.125 0.605 0,25 (x2) Mean pallet travel time 2,0 1.645 1.125 0.605 0,25 (x~) Mean arrival time 2.0 1,676 1.20 0.724 0.40

designed experiment, it should be possible to determine accurately, with much reduced effort, the effect of change in one or more variables on the process output and interaction effect of the factors, if any. If all the investigated factors are quantitative in nature, then it should be possible to approximate the response Yu by a polynomial equation (2).

k k k

L = bo + E b,x, + E + E b ;xj (2) i=1 i=1 i < j

where x~(i = 1, 2 . . . . k ) are the coded levels of k quantitative variables and b0, b;, etc. are the least square estimates of the regression coefficients.

For the second order central composite rotatable experimental design, effects of each of the variables xi on the yield Yu has been studied, through simulation, at five different levels [16, 17].

The levels of the variables Xl, x2, x3 are given in Table 2, whereas their coded values are derived by the use of transforming equations (3-5).

x I = 2.163 + 1.92(s)

x2 = 2.163 + 1.92(pt)

x3 = 2.523 + 2 .102(a) .

(3)

(4)

(5)

A plan of the simulation experiment is given in Table 1 and each treatment combination was replicated twice to achieve an acceptable degree of precision.

From the simulated data second order polynomial response equations have been derived and the results given in Table 4. The response equations have next been employed to yield the optimum

Table 3. Analysis of variance

First order Second order Lack of Grand Performance Source term term fit Exp. error total indices d.f. 3 6 5 5 19 F-ratio

No. of tardy jobs ss 1478.645 364.783 72.285 16.615 1932.226 ms 492.881 50.79 14.457 3.323 4.377* d.f. 3 6 5 5 19

No. of completed jobs ss 2714.496 313.658 77.441 16,413 3122.009 ms 904,832 52,276 15.488 3.282 4.718*

d.f. 3 6 5 5 19 No. of running jobs ss 229.673 66.743 6.850 6.013 309.280

ms 76.591 11.124 1.370 1.202 1.139" d.f, 3 6 5 5 19

No. o f waiting jobs ss 2295.468 170.899 74.990 17.315 2558.672 ms 765,156 28.483 14,998 3.463 4.331" d,f. 3 6 5 5 19

Average machine utilization ss 1057.349 188.027 12.976 7.473 ms 619. I 16 31.338 2,595 1,494 1.736*

ss = sum of squares; d.f. = degree of freedom; ms = mean square; F-ratio = Fisher F0.95(f5) = 5.1. *Adequate; F < F095~5,51.

Ratio (observed ratio of Ms/Exp. error).

Table 4. Computed values of the regression coefficients

Performance indices b o b~ b 2 b3 b,~ b22 b33 b12 bl3 b23

No. of tardy jobs 55.61 - 3 8 . 3 0 18.81 - 8 . 3 0 11.70 - 8 . 6 7 - 4 . 1 2 - 6 . 8 3 0.80 12.32 No. of completed jobs 31.75 - 8 . 4 5 - 4 1 . 4 6 41.68 - 5 . 5 6 7.69 - 3 . 8 4 15.47 - 0 . 3 5 - 10.65 No. of waiting jobs 20.63 17.86 18.46 ~38.99 - 0 . 8 4 1.57 13.52 0.32 -2 .071 - 7 . 7 2 No. o f running jobs - 2 . 1 9 14.61 4.51 4.19 - 4 . 4 3 0.26 - 4 . 3 7 - 4 . 5 2 6.56 1.20 Average machine utilization (%) 58.07 46.61 - 16.46 --23.29 - 11.75 2.60 3.43 - 3.09 1.05 3.48

Flexible manufacturing systems

Table 5. Optimized values of the FMS operating parameters

71

Objective function

Mean Mean Mean processing interarrival pallet travel

time time time Remarks

No. of tardy jobs 2.0 2.0 0.25 Minimize No. of completed jobs 0.25 2.0 0.25 Maximize No. of waiting jobs 0.25 1.49 0.25 Minimize No. of running jobs 2.0 0.40 1.9 Minimize Mean machine utilization 2.0 0.4 0.25 Maximize

o

E

s o - (o )

5 0 --

4 0 - -

30 L

20

10

6o - ( b ]

5 0 -

.~ 4 0 -

"- 30 o

E ~ 20

10

0

0 St Mean processing tlrne

x Pr~Mean paLLet t ravel time

o, Mean InterarrivoL time

0% - - - - Design of experiments \

Q ~ ~ ~ Conventional one variabLe of o time

..--.- ,,~... x-- .--...~..x ~ --.....

I I I I I I I I 0 . 2 5 0 . 5 0 0 . 7 5 1 .00 1.25 1 .50 1.75 2 . 0 0

$ ==

p, a D

/

x . ~ / x "

I I I I I I I I

- 20 f (¢) ~.~ to

I I i I i I I 0 0 . 2 5 O. 50 0 . 7 5 1.00 1.25 1 .50 1 .75 2 . 0 0

$

P, = O P

Fig. 4(a--c)

72 P.K. MISHRA and P. C. PANDEY

o

f,

E

4 0

30

20

10

0

80

70

60

50

40

( d )

d~. x I

x ~

I I I I I I I I

(e)

20 I I 1 I I I l l 1 I 0 0.25 0.50 0.7"5 1.00 1.25 1.50 1.','5 2.00

$ =

,9, ,, Q

Fig . 4 ( d a n d e)

Fig. 4. (a) Effect of interarrival times of jobs, mean processing time and mean pallet travel time on number of tardy jobs. (b) Effect of interarrival times of jobs, mean processing times and mean pallet travel times on number of completed jobs. (c) Effect of interarrival times of jobs, mean processing times and mean pallet travel times on number of running jobs. (d) Effect of interarrival times of jobs, mean processing times and mean pallet travel times on number of waiting jobs. (e) Effect ofinterarrival times of jobs, mean

processing times and mean pallet travel times on mean machine utilization.

values of the system operating parameters. Optimization was achieved through the use of the flexibility tolerance method [18], and the optimized results are given in Table 5.

To check the validity of the postulated model [equation (2)] a complete analysis of variance (ANOVA) was performed and a sample calculation given in Table 3. The various response equations have been plotted graphically (Figs 4a-e) and discussed below.

6. RESULTS AND DISCUSSION

Figures 4a-e have been obtained by plotting the regression equations derived as per the procedure outlined above. The simulation results, obtained by varying one variable at a time are also shown in these figures. It can be observed that the conventional and experimentally designed simulation results are identical whereas designed experiments require less computational effort.

Figure 4a shows that the mean number of tardy jobs is virtually unaffected by the mean pallet travel time except for values below 1.0, whereas increasing the processing time and interarrival time leads to a reduction in the number of tardy jobs. The rate of reduction however, diminishes at higher values of P, and a. The mean number of running jobs, given in Fig. 4c, is virtually unaffected by P, and a but increases gradually with s. Figure 4d shows that the number of waiting jobs decreases rapidly with the interarrival time, but for interarrival times in excess of 1.25, this effect

Flexible manufacturing systems

Table 6. Significance test of individual variables (F-test)

73

No. o f No. o f No. of No. o f Average l)¢gt~e of tardy completed running waiting machine

Variable Source freedom jobs jobs jobs jobs utilization

Mean processing time (s) x I 1 112.3" 5.5* 188.8" 75.8* 345.2* Mean pallet travel t ime (pt) x 2 1 13.2" 133.7" 7.4* 67.8* 104.9" Mean arrival t ime (a) x 3 1 3.1 118.1" 17.2" 101.9" 88.5* Square terms

s ~ x 2 1 12.5" 3.1 19.3" 0.1 42.4* p,: x! 1 6.9* 5.9* 0.1 0.3 2.1 a 2 x~ 1 1.1 1.0 13.1" 14.4" 2.5

Interaction spt x l , x 2 1 2.3 13.3" 11.1" 0.01 1.6 sa x t , x 3 1 0.03 0.01 13.6" 0.2 0.1 pta xz, x 3 I 6.4* 5.3* 0.7 3.1 1.7

*Fo.gsl, lo ~ = 4.96. Significant: F > F0.9~ll:0 ).

is very small, whereas with increasing mean pallet travel time and procesing time, the mean number of waiting jobs increases as shown in Fig. 4d.

From Fig. 4e it can be seen that the effect of processing time on machine utilization is dominant upto a equal to 1.5, whereas increasing interarrival and pallet travel lead to a continuous decrease in machine utilization.

To test the significance of the linear, square and product terms in equation (2), significance testing was carried out and the results are given in Table 6. It can be noticed that the effect of linear and square terms of the mean interarrival time on tardy jobs is virtually insignificant whereas, the square and product terms have very little effect on the number of waiting jobs and machine utilization.

The optimized values of the FMS operating parameters are given in Table 5. It can be noted that, in order to minimize the number of tardy jobs, the mean processing time and interarrival times both have to be maintained at their respective highest levels within the range selected. The mean pallet travel time must be fixed at the minimum level. Maximization of the number of completed jobs however, requires the mean processing time to be as small as possible. Therefore, the conditions that favour maximization of the number of completed jobs are also valid for the minimization of the inprocess inventory. When the number of running jobs as well as mean machine utilization are both to be maximized, the mean interarrival time must be maintained at around 0.40 (Table 5).

It should, however, be noted that it is not possible to optimize the system performance as a whole to satisfy all the objective functions simultaneously. This however could be achieved through goal programming and work is currently in progress in this direction.

7. C O N C L U S I O N S

The FMS simulation experiments when based on the statistical design of experiments and performed conventionally yield identical results. However, the computational effort needed for the former is reduced appreciably with an added advantage that the response equations thus derived are capable of yielding the individual effects of linear, square and product terms. The response equations could also be used for the development of control systems and the optimization of the FMS performance.

It can also be concluded that, for the systems studied the minimization of the number of tardy jobs would require the mean processing time and the mean interarrival time of the jobs to be maintained at 2.0, whereas the mean pallet travel time be set at 0.25. The optimized values of the mean processing time, mean interarrival time, pallet travel time, etc. depend upon the particular criterion selected for the system performance optimization.

The results obtained from this study, conducted on a hypothetical FMS, are expected to apply to real situations as well because the range of the operating parameters and the performance criterion selected as also valid in practice to a certain extent.

74 P.K. MISrtRA and P. C. PANDEY

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Computer Application Production & Engineering, (Edited by E. A. Warman). North Holland, Amsterdam (1983). 5. K. Iwata, Y. Murotsu, F. Oba and K. Yasuda. Production scheduling of flexible manufacturing systems, CIRP Annals

31, 319 (1982). 6. R. Suri and W. Dille. On line optimization of flexible manufacturing systems using perturbation analysis. 1st

ORSATIMS Conf. on Flexible Manufacturing Systems, Michigan (1984). 7. D. D. Yao and J. A. Buzacott. Modelling the performance of flexible manufacturing systems. Int. J. Prod. Res. 20623,

5,945 (1985). 8. J. A. Buzacott and J. A. Shanthikumar. Models for understanding flexible manufacturing systems. AIlE Trans. 12, 339

(1980). 9. J. J. Kanet and J. C. Hayya. Priority dispatching with operation due dates in a job shop. J. Oper. Mgmt 2, 155 (1982).

10. J. J. Kanet. On anomalies in dynamic ratio type scheduling rules: A clarifying analysis. Mgmt Sci. 28, 1337 (1982). 11. K. R. Baker. Sequencing rules and due date assignments in a job shop. Mgmt Sci. 30, 1093 (1984). 12. R. W. Conway, W. L. Maxwell and L. W. Miller. Theory of Scheduling. Addison Wesley, London (1967). 13. R. W. Conway. Some tactical problems in digital simulation. Mgmt Sci. 10, 47 (1963). 14. P. K. Mishra, P. C. Pandey and C. K. Singh. Simulation studies of flexible manufacturing systems. 2nd Int. Conf. on

Simulation in Manufacturing, Chicago, U.S.A. (1986). 15. W. G. Cochran and G. M. Cox. Experimental Designs. Asia Publishing House, New Delhi (1962). 16. K. C. Peng. The Design and Analysis of Scientific Experiments. Addison-Wesley, London (1967). 17. D. M. Himmelblau. Applied Nonlinear Programming. McGraw Hill, New York (1972).