The Scheduling of Flexible Manufacturing Systems

5. 6. 7. A tool transport system with specified number of

A central storage for tools Tool magaeines for each machine of given capacity

devices.

Z. Doulgeri, R. D. Hibberd, T. M. Husband; Imperial College of Science and Technology Submitted by A. Chisholm (1)

Received on March 6, 1987 - Accepted by the Editorial Committee

End of Part End of End of Pert End of Tool End of Arrival Machining Transport Transport Simulation

ABSTRACT:

The efficient use of capital intensive Flexible ManUfaCturing Systems requires effective scheduling of the system. This paper considers the problems involved in the development on assessment of schedules and proposes an heuristic simulation based approach to the production of schedules. The use of simulation ensures that the schedules are feasible and take account of the constraints of materials handling, tooling and fixtures while the proposed "look ahead" algorithm does not ensure optimal schedules the results indicate that improvementa of 10 - 20% in makespan (lead time) are possible in comparison with the use of traditional dispatching rule based scheduling.

KEY WORDS:

Flexible Kanufacturing Systems, scheduling. simulation. heuristic algorithms.

1. INTRODUCTION

The installation cf Flexible Hanufacturing Systems (FKS) represents a large capital investment. In order to make the most effective uae of this investment, efficient scheduling of the FMS is required. FKS scheduling not only presents all the difficulties associated with job-shop scheduling, but also has a higher degree of complexity due to the new variables involved; these variables being the tools, fixtures and material handling system which need to be scheduled together with parts. As existing job shop scheduling algorithms do not take these variables into account their direct application to the FMS case would result in the development of unrealistic schedulas. To overcome this problem, this paper presents scheduling algorithms which embed an FIB simulation model with can take account of these factors.

1.1 The FMS Simulation model.

The FMS is modelled as a system consisting of:

1. M machines, each capable of performing different kinds of owrations

The three phase structure of the simulation program ia given in Figure 1. The instances vhen the control module is called are shown.

1.2 The structure of the control module.

The main task of the control module is the choice of a part for allocation to an idle machine. This is a scheduling decision. In order for a good scheduling decision to be made, knowledge of the status of the complete system is required. It is realised by creating an image of the system's current state by copying the dynamic file which contains the information on the machines which are idle at that instant of time and on the parts which are available, and waiting for further processing.

A set is created, for each idle machine, consisting of those available parts whose next operation can be performed on that machine. This is called the choice set and it ia worth noting that in general a part may belong to more than one choice set.

This is the firat ,task yf the control module.

I Initialise I 2. A common storage of given capacity which also aerves as

7- 4. A material transport system with a given number of

loading/unloading station Buffer storages for each machine of given capacity

t m s w r t devices I i I [Select next event & advance the clock I

Additional assumptions concerning the operation of the model are :

1. There will be constraints reflecting the inability to process a part on two machines simultaneously or process two parts on the same machine simultaneously.

2. There can be alternative machine tools which can perform the required operations for each processing stage of the parts, and each machine can perform one or more operations.

3. Each part is processed in a specific operational sequence based on its own order of processing stages.

4. Operation durations and transport times are given or drawn randomly from a known distribution.

The production scheduling problem for the PMS model may be expressed as:-

"Determine the schedule of N parts which belong to as specific range of part types with constant production ratios a0 as to minimise makespan."

The model has been structured t facilitate the development of the scheduling The simulation of the systems activities are aeparated from scheduling decisions which are taken in a separate module; the Control Module. This aids the investigation of different control aptegies as only the control module needs to be changed. Control" is called at each decision time. These are the times during the system's operation when a choice between possible scheduling alternatives should be made. Priority is given to part allocation with tooling, fixtures and material handling units acting as constraints on the number of feasible alternatives. Such a priority is justified on the need for high machine tool utilisation given that machines account for a high proportion of the total system's cost. Decision times may therefore also ba called allocation times.

Repeat for each activity

FIG. 1 The structure of the FMS simulation -

A part from each choice set is then selected for allocation to each machine according to a decision rule. Decision rules may vary from simple priority rules to complicated ones, involving weighted sums of attribute values of parta or machines. For example, a rule may use the part's lateness or tardiness. the number of remaining operations for this part or the amount of remaining processing time.

Annals of the CIRP Vol. 36/1/1987 343

2. OFF LINE SCHEDULING BY SIMILATION

A simulation run can be regarded, for scheduling purposes as a heuristic dispatching method for the generation of one non-delay schedule. Non delay schedules assume a part is allocated to a released machine without delay. The construction of the schedule takes place as follows.

At the commencement of the simulation the control module creates a choice set for each run. Using the choaen priority rule a part from each set is allocated to the corresponding machine. The simulation proceeds to the next decision time when an allocation is required. This is repeated until the simulation of the production of the part set has been oomplstsd. The record of the allocation decisions taken constitutes a full non-delay schedule for the system.

Priority dispatching for schedule generation is similar to the dynamic dispatching which is frequently used for dynamic job shop scheduling. ?he inconclusive and sometimes onfli ting evidence about the choice of the best rule F2*3*45 indicates that while the investigation of priority rules may give improved schedules for a given FMS it does not lead to effective algorithms of wide applicability.

2.1 A simulation based DrOCSdure for the generation of all non delay schedules (full enumeration algorithm)

As outlined above at a given stage in the production the record of allocation decisions constitutes the schedule up to that point. The choice sets at the stsge comprise all the candidate parts for extending this (partial) schedule. If all the alternstives ere examined all the feasible non-delay schedules are generated for the FMS and the best may be chosen according to the required performance criterion.

This schedule is not necessarily the optimal schedule, nevertheless t can usually be expected to provide a very good solutionh). Studies of job shop scheduling indicate that the non delay set of schedules is s better basis for heuristic schedule generation than the much larger set of active schedules which is a do 'nant set, ie is guaranteed to contain the optimal schedule&?.

The simulation based algorithm, used in this work, for the generation of all non delay schedules is illustrated below (Pig 2). It u s z s back tracking search, a well tried and

START

D ADVANCE mgB SIDESTEP + BACKTRACK

DECISION NODE

0 TERMINAL NODE

FIG. 2 An example of the full enumeration algorithm - efficient tree search method in operation research problems. The search advances down the branches and sidesteps or backtracks as required. The full algorithm is given in Appendix 1.

Such a full enumeration procedure is to computationally demanding for complex FKs's and in order to reduce computational requirements a heuristic algorithm is suggested.

2.2 A simulation based, heuristic dispatching method using a "looking ahead rule*.

The algorithm belongs to the category of heuristic dispatching. At each decision stage it selects the slternative which gives the best performance estimate. The estimates ere derived by looking ahead using the simulation model with future conflicts overcome using a dispatching rule. Each performance estimate is complete schedules which comprise the partial schedule to the present decision stage,

the alternative choice under consideration and the schedule produced by the model using, as indicated, a dispatching m l e for future decisions.

Having made the decision at a given stags and allocated parts and machines the algorithm proceeds to the next stage where the procedure is repeated.

The algorithm may be visualised in terms of the search tree of non delay schedules. The prediction run for a partial schedule (Pd) is equivalent to a branch visit starting from the node representing Pd down to a terminal node. This is illustrated in Fig 5 . The node with the minimum predicted makespan is encircled at each stage. The full algorithm is given in Appendix 11.

FIG 3 The operahon of the heuristic simulahon algorithm -

3. EVALUATION

?he algorithm has been evaluated on the basis solutions obtained for a number of problems of different sizes and different processing times. The precedence relations were selected at random using NAC library routines. The sizes and the distribution and variences of processing times are given in ?able 1.

Problem size Nos. of Parts N x 6 x 4 10 x 4 20 x 5 20 x 10 Nos-of Machine M

Table 1s Problem sizes investigated

Distribution Uniform (UR) Normal Normal (1.36) (20.10)

Varience High Medium

?able lb Distribution and Variences of Processing Times -

Travel times and other activities present in a real FMS operation were ignored at this stage. Five examples were examined for each combination of problem size and variance of processing times. Each problem was solved using two variations of the algorithm suggested, each created by using two different rules in the extension of the alternative partial schedules st each stage in order to predict this makespan. namesly the FCPS (First come first served) and the SPE (Shortest processing time) rules. Each solution value given by the algorithm was compared with its absolute lower bound and the resulting percentage deviation wee calculated as follows:

S (A) - B1 x loo

n,- S(A)

where S(A) : is the solution value obtained using the algorithm

problem solution B1 : the theoretical lower bound to the overall

B1 is calculated as follows:

M B ~1 - max fmax t tp, mar t tpm 3

p m=l m p=l

344

where tpm is the processing time of part p on machine m.

The smallest problem size was also solved using the f u l l enumeration of non-delay schedules and the best non-delay schedule found. For th i s problem siee the algorithm solution efficiency was also calculated as the percentage ra t io of the best non-delay value to the value found by the algorithm.

Results are shown i n tables 2, 3 and 4.

3.1 Solution time

-- 20 x 5 61.1 842.70 100.8

x) x 10 82.23 650.23 210.51

In general t h e solution deviation values are small i n problems with a larger variety of parts in relation to the number of machines (eg 20 x 5)-

This is not unexpected as there are always l ikely to be schedulable jobs i n the choice s e t and idleness may be avoided. Where the reverse is true the derivation of an effect ive schedule is more d i f f icu l t and idleness may occur.

In almost a l l cases the use of FCFS for the look ahead simulation run gives a bet ter performance than the SF? rule. ?he variance values are a lso s ignif icant with the solution efficiency decreasing as the variance decreases.

Problem average number of CPU i n sac. s ize decision stages nodes

0.428

best ND 1845.53 45.940

18.86

Tabla 2. Average Number of Decision Stages and Nodes, and CPU time.

The CW time increases rapidly with the number of machines but only l inear ly with the number of parts. Nevertheless the time appears acceptable even f o r large problems. In present terms a 10 machine system is f a i r l y large. The comparison of the f u l l enumeration of the 6 x 4 problem with the heuris t ic solution i l l u s t r a t e s that even with small problems the f u l l procedure is too computationally demanding for pract ical use.

3.2 Efficiency of the Schedules Developed

problem UR [1.36] N[20,10] N [20,51 s ize FCFS SPT FCFS SF'T FCFS SF'T

6 x 4 98.57 100 97.62 96.78 95.30 94.28

Table 3a Average efficiency and heuris t ic algorithm compared with best AD schedule

problem UN [1,36] N[20,10]

6 I 4 92.85 93.29 91.45 90.18 91.71

problem UR 11,361 N[ 20,101 N b . 5 1

6 I 4 23.16 23.16 18.71 23.61 22.28 24.86

Table 4 Worst case values of solution efficiency and solution deviation.

3.3 Comparison of the Heuristic Algorithm and Dispatchiq Rules -

The solution values obtained by the algorithm were also compared to those obtained using the simple dispatching rule based Simulation with SFT used as the dispatching rule.

problem size UR [1,36] ~[20,10] B [20,5]

~ 6 x 4 8.01 1 10.95

10 x 4 14 * 95 13 * 56 10.92

20 x 5 13-29 13 * 59

problem UR [1,361 NC 20.10 1

6 x 4 12.67 9.84 10.4 15.8 16.61

10 x 4 6.9 5.18 5.23 7.33

20 x 10 14.00 12.67 15.47

Table 5 Average percentage improvement achieved.

In a l l cases the heuris t ic look ahead algorithm gives s ignif icant improvements over the simple use of a dispatching rule despite the use of the SPT rule i n the 'look ahead' procedure.

120 x 5 0.0 1.15 0.56 2.99 1.49 3.03 I 20 x 10 12.63 13.66 11.01 13.10 14.39 16.10

Fable 3 b Percentage deviation of heuris t ic algorithm aolution compared w i t h the minimum possible makespan ( the lower bound)

The comparison of the full and heuris t ic resul ts for the 6 x 4 problem (Table 3a) shows that heuris t ic achieves a high level of solution efficiency with the minimum deviation being only 5.7% from the best A.D. schedule. If we compare these resul ts with the same ones as analysed i n Table 3b, it can be seen that the comparison with the lower bound makespan undervalues the heuris t ic algorithm. The worst resu l t s (N(M,15) dis t r ibut ion and SF'T role) suggests that deviations from the lower bound is 16.6% whereas 3a shows that it is only 5.7%.

4. CASE STUDY

The effects of the suggested simulation based heuris t ic algorithm has been investigated through a case study. The FlIS configuration i n t h e case study is a simplified version of one found in the manufacturing industry. It consists of f ive machines two of them are similar and can perform different kinds of machining operations, buffer storages i n f ront of each machine of capacity one, two robots which serve the system by transporting par t s between different working s ta t ions and a central storage which serves as a loading and unloading s ta t ion. There are three different types of parts which are to be produced i n the ra t io of 2:l:l i n periodic sets of eight parts. Naohining and transport times are given and each machine has a tool magazine which comprises a l l tools necessary to perform the required operations. Results are shown i n table 6 which compares the effectiveness of the schedule developed using the algorithm with those developed using common diepatching rules.

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APPENDIX 2

Algorithm 2: A simulation based, heu r i s t i c dispatching method using a looking ahead ru le .

Let d = 1

P o - ! ) the nu l l p a r t i a l schedule

a - 0 the a l loca t ion t i m e

nd Y[ the set of machines associated with a

cdm : the choice set of machine mEMd

Performance Measure 1 flow time machine

u t i l i s a t i o n

Makeapan Mean

I I Algorithm 250 90.8 53

Dispatching Rule

FCPS 283 94.3 46

L W 211 94.6 47

LPT 323 90-8 41

SPT 211 93.3 47

Table 6 Makespan f o r schedules derived us ing ( i ) t he algorithms and ( i i ) var ious d i s w t c h i n g rules.

It is clear that the suggested algorithm y i e l d s the minimum values of the makespan and mean flow time and the maximum mean machine u t i l i s a t i o n . The improvement achieved i n terms of the makespan values compared with well known dispatching rules is i n t h s mugs of 10 t o 22%.

5. CONCLUSION

This paper presents a new heur i s t i c scheduling algorithm f o r FXS. The method produces a good f eas ib l e schedule based on a simulated model of the system, and it can therefore be used as a powerful t oo l t o cont ro l FHS operation i n a p rac t i ca l sense. It is found t h a t the algorithm is e f fec t ive i n reducing the makespan and mean flow time of production while achieving an increased u t i l i s a t i o n f o r the machine tools.

APPENDIX 1

Algorithm 1: A simulation based, backtracking search of non-delay schedules f o r FMS

Let

while

d > O

do -

d = 1

Po - 0 - 0 the a l loca t ion time

M d ' M

cam :

the n u l l partial schedule

the set of machines assoc ia ted with a

It comprises a l l p a r t s whose f i r s t operation can he performed on m

Kd CdtX- -. XCdm - ~~ - Pd (Pa l , * * + Pam) wMd

- pd .. (Plr P2...9 Pd)

- if Pd is a so lu t ion then while

Kd # record it

do

advance

- Simulate the system u n t i l the next a l loca t ion time a

- the cont ro l module generates Ma and Cdm

backtrack d - d - 1 Kd = Kd- Pd

while

Kd # $

DO

c rea t e a l t e rna t ive partial schedule for s t age d

- choose pdmC Cdm me M,j

- Pd - (Pd' 9 . * ?Pdm)

- Kd Kd - Pd

- pd = (PI. P2.*-9 Pd)

- i f Pd is a so lu t ion then

save makespan (Pd)

e l se

p r e d i c t i ts makespan with a look ahead simulation rug

d' = d

- Simulate the system u n t i l the next a l loca t ion time a*

d' = d' + 1

Md' : the set of machines associated with a *

Cdim: the choice set of machine m, m E & .

Do choose pd'E Cd" W mEMd*

Pd' = (Pd",. * ePdsm)

pd' * (PI s.-s pdepd')

If Pd' is a f u l l schedule then

save makespan (Pd') and ex i t : endif

2) Find the minimum makespan (Pd') and save Pd a s the partial so lu t ion with the bes t predicted performance so far.

If Pd is a so lu t ion STOP

Simulate the system u n t i l t he next a l loca t ion time a.

d i d + ] .

The cont ro l module generates:

Md : the set of machines assoc ia ted with a

Cdm : the choice set of machine me Md

Go t o s t e p 3 .

RFFERENCES

(1) Doulgeri, Z., Hihberd, R.D., 1985, A simulation t o Develop and Assess Control Algorithms f o r FMS, Proceedings of 1st Conference on Simulation i n Flanufacturing, S t r a t fo rd . IPS, 185-189.

(2) Day, J.B., Hottenstein, M.P., 1970, Review of Sequencing

( 3 ) Conway. R . W . , Maxwell, W.L., Miller, L.W., (eds.) . 1967,

(4) Panwalkar, S.S., Iskaader, w.. 1977, A survey of Scheduling Rules, Operations Research, Volume 25. Number 1, January-February.

(5) Baker, K.R., 1974, In t roduct ion t o Sequencing and

(6) Jeremiah, B . , Lschandrani, A.. Schrage, L., 1964. Heur is t ic Rules Towards Optimal Scheduling, Dept. Indus t r i a l Eng. Research Report, Cornell University.

Research, Xaval Research Logis t ics Quarterly, 17:11-79.

Theory of Scheduling. Addison Wesley, London.

Scheduling, Wiley, New York.

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