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Job shop scheduling optimization in real-time production control
395
Applications
Job Shop Scheduling Optimization in RealTime Production Control F.G. Filip, G. Neagu, D.A. Donciulescu Central Institute for Management and lnformatics, Bulet~ard Miciurin 8-10. Bucharest, Romania An optimization algorithm using simulation as an evaluation procedure of the criterion to obtain an optimal schedule of jobs within a workshop is presented. The algorithm, which is recommended for middle-size problems, is embedded in a dedicated minicomputer-based practical system meant for real time production control in a discrete part manufacturing system environment.
Kevwords: Optimization, simulation, production control, hum a n - m a c h i n e interaction, computer-aided decision making, dedicated system.
Florin-Gheorghe FUip was born in Bucharest, Romania, in 1947. He graduated in 1970 in control engineering from the Polytechnic Institute of Bucharest. He received in 1983 the Ph. D. degree for c o n t r i b u t i o n s to hierarchical control of complex systems. Since 1970, dr. Filip has been working at the Central Institute for M a n a g e m e n t and Informatics in Bucharest, where he is the head of the "Real Time Systems" research group. In 1974 Dr. Filip spent half-a-year in Swedish universities as a visiting research fellow and in 1983 he was an invited lecturer on hierarchical, large scale systems at Shenyang Institute of Automation of the Chinese Academy of Sciences. He is the a u t h o r / c o a u t h o r of some forty technical papers and is the coauthor of a book. His current scientific interests include hierarchical optimization and control, C A D / C A M systems and real-time computer control in industry.
North-Holland Computers in Industry 4 (1983) 395-403
1. Introduction Hierarchical multilevel-multilayer management and control systems appear to be very promising design solutions for many practical computer-controlled production systems because large and complicated control problems can conveniently be substituted by several smaller and simpler ones [1,6,7,8,15]. Within the control hierarchy, the real time production control layer acts not only as a passive recorder of data about the plant behaviour and as the supplier of "fresh" information about it, but also as a permanent source of reference for the actual state as compared to the predicted
Dan Alexandru Donciulescu was born in Bachu, Romania, in 1949. He received his MS in Control Engineering in 1972 from the Polytechnic Institute of Bucharest. In 1972, D.A. Donciulescu joined the Central Institute for M a n a g e m e n t and Informatics in Bucharest. He has been working as research engineer and project leader for several development projects of computer-based production control systems. He is an associate assistant professor in computer programming at the Polytechnical Institute of Bucharest. He is the a u t h o r / coauthor of some thirty technical papers. His interests mainly refer to the real time applications of hierarchical systems theory to production control in process industry. Gabriel Neagu was born in Bucharest, Romania, in 1949. In 1973 he received his M.S. in control engineering from the E.E. Institute of Moscow. Since 1974 he has been working at the Control Institute for Management and Informatics in Bucharest as a research engineer and project leader for several real time production control applications and systems in metallurgical and discrete part manufacturing industries. He a u t h o r e d / c o a u t h o r e d some twenty technical papers. His current scientific interests are in the field of real time production control and C A D / C A M Systems.
0166-3615/83/$03.00 '~ 1983 Elsevier Science Publishers B.V. (North-Holland)
396
Applications
trajectory of the system evolution [11]. Thus, the real time system notifies the decision maker about all critical events that have to be taken into account. In order to render the decision making process more efficient and quicker, the system capabilities must be developed towards operative preparation and delivery of decision variants concerning critical events in the controlled system behaviour. According to Kriebel [12], which identifies four levels of relative maturity of information systems, which differently support the four stages of the decision making process (observation, inference, evaluation and choice), the system that gives the recommendation to act is located at the third level, At the operational level, the computer-made decisions suggestions are generally based on models of well structured problems and on optimization procedures within narrow boundaries [13], even though, as Wierzbicki noticed [14] often "the purpose of optimization is not to propose optimal solutions, but rather to generate reasonable alternatives in response to user's requirements, while eliminating clear inferior alternatives". In a discrete part manufacturing system environment, job shop scheduling optimization can be conveniently carried out by using simulation models [4]. Simulation is resorted to because the system concerned is too complex to be described by a simple analytical model and the optimization problem does not have a reasonable straightforward analytical solution [2]. It was observed [5,13] that the optimization procedures may not be as useful as one expects unless they are considered in the context of the entire decision making process. The multistage decision making concept [10] shows its usefulness when coping with uncertainties, preference changes, and decision maker attitudes, experience and motivation, and allows for designing and implementing successful practical computer-aided operative decision making systems where the optimization is but a part. The purpose of this paper is to describe a job shop scheduling optimization algorithm, using simulation as an evaluation procedure of the criterion, embedded in a practical system. The remaining part of this paper is organized as follows. In Section 2, the objectives of the real time production control system are shown. An extension of the optimization algorithm proposed in [4] is given in
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Section 3. An illustrative example is presented in Section 4: and Section 5 describes a real time system at the workshop level, which makes use of the optimization algorithm.
2. System Objectives In a discrete part manufacturing environment, at the workshop level, the foreman is periodically given, by the upper management level, the production program of the next time horizon. The production program indicates the jobs (work orders) to be done and gives the appropriate technological information such as: operation range~ sequence of execution and the time standards. A real time production control system at the workshop level makes the production program implementation effective. Among the major objectives of the system are the following: respect of due dates, increase of the efficiency in the utilization of all manufacturing and manpower resources, diminishing jobs flow time, reduction of the amount of work-in-process and a better coordination with other workshops and support activities, The system is functionally evenl-oriented, where an event is viewed as a discontinuity in the production process. The list of event types is given in Table 1. Job shop scheduling is a central function in production control and its accomplishment depends on the given production program and the actual state of the workshop. In order to obtain a schedule, a simulation procedure may be used to enlist the predictable events, mainly including the optimal sequence of job assignment to various machines, for a scheduling period, AT. An optimi-
"Fable 1 Significant event list No. Significance 1. 2. 3. 4. 5. 6. 7. 8.
New job arrival in the workshop Job entry in a waiting queue of jobs at a machine group Start of an operation Accomplishmentof current operation for a machine Start of a planned maintenance stop Machineset-up after event 5 or 8 Conclusionof job processing in the workshop Unexpectedmachine break down
Computers in Industry
zation algorithm, which extends the results presented in [4], is described in what follows (Section 3).
2.1. Remarks on System Objectives We must emphasize that each scheduling problem is time and space limited because of the hierarchical nature of the control system based on structural and temporal decomposition used for getting real time properties [11]. If no significant disturbance will occur, the schedule will be used during only a validity intert,al, AT,.. After that interval, new intervention information will be provided by the upper level and, consequently, a new more up-dated schedule will be needed. In order to increase the system reliability in cases of failures of data processing or communication equipment, the scheduling horizon is larger than the validity interval [8]. In order to compensate the effect of significant disturbances, it is sometimes necessary to reconstruct the job shop schedule in real time.
3. The Optimization Algorithm The algorithm to be described may be looked upon as a hierarchical, two-level optimization method based on parametric decomposition [7]. At the lower level, an optimal job shop schedule is constructed by using simulation for fixed values of the intervention variables. The intervention vector is modified to minimize a performance measure at the higher level.
3. t. Assumptions The simulation is based on the normal assumptions on the workshop operation [4]: A. There is a set, £,a, of jobs; each job J , , i~.~', enters the workshop at various times, ta,. There are one or more operations d)~, k = 1, R i to be executed from the release ofa~, to its completion. B. The workshop comprises a number, G, of various machine groups, each group, ~j, j = 1, G, containing one or more identical machines (working posts), ~t'~, m = 1, Mp that take over their jobs from a common queue of jobs. C. Each job to be scheduled may or may not have a previously assigned due date, tl~. The first
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category of jobs belongs to the set L-'~, the second one belongs to the set ~2 and tl,, where i~£~az, is chosen oo. D. An operation on one machine must be completed for all items that belong to a job before the next operation on that job can start. E. There is a transit time tt,~ required to transport job J, from one machine group to another before starting the operation 6o,~. F. The processing time values t,~ are available for any operation of each job. G. If a machine J [ , ~, becomes vacant, the highest priority job waiting in its queue is assigned to it. H. Time intervals for machine repairing or verification may be allocated for each machine.
3.2. The Priority Rule In order to indicate which job is to be allocated to a vacant machine, a composite priority rule [9], similar to the one presented in [4] is used. This rule takes into consideration for each job J , the following priority criteria: external priority, Pe.,, Carrol's rule, Pl,i, time in queues, P2.,, remaining processing t i m e / c u r r e n t processing time, p~.,, and the estimated size of the next queue, P4.,. (A review of these criteria is given in the Appendix.) The priority of the ith job is calculated as follows: 4 q=l
where the weights { %, w2, ws, w4 } form the intervention vector in a two-level hierarchical optimization procedure. Notice that each criterion might be used as a screen, defined by threshold and tolerance parameters, in order to eliminate the jobs that are " n o t critical". This is very useful for large problems. However this procedure is optional when small or middle-size problems are concerned and, moreover, is likely to cause optimality losses. Besides, the threshold and tolerance parameters tend to complicate the searching procedure, which will be described in the sequel, by doubling the amount of free variables.
3.3. The Optimal Scheduling The future evolution of the workshop can be simulated as a sequence of significant events over
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398
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J = ~ Q>,(At>,),
the time horizon, AT. The values appropriately chosen of the weight parameters ~%, r/= 1,4, are crucial for the efficiency of the schedule. To establish the optimal values of the weight parameters asks for defining a performance measure. The value of the performance measure indicates, for each simulated sequence, how high are the penalty costs for each job tardiness, Ate.,, or earliness, At2. ~, of the completion time obtained by simulation, t f,, as compared with the due date [4]. In order to reduce job flow time, the time the ith job has waited in various queues, zlt3, , is considered in the penalty function. Since, besides flow time reduction and respect of due dates, a high utilization of production capacities is required, it is reasonable to penalize the time the ruth machine was in the idle state, ~ t 4 .... [9]. Therefore, the performance measure has four components
l'tl ----tL. t2,:- i0,
J~,= ~_~Q-<,(2aG.,). R,
Ate.,= y" tw ~, l,-:] At
"/4= ~ m
The general form of each penalty function, Q~. w h e n ,~ = (fi, y), f i e { l , 2. 3. 4} a n d *~ { ~ , ~ , ~ 1 . . . . . m }, is linearwise and has an obvious economic meaning (Fig. 1). Thus the penalty is calculated by using monotonously increasing cost coefficients, Q .... where v = 1. N(ff~. as follows [7]:
where
J, = ~ Q,.,(at,.;), (tf,-tl,,
04,,,,(~t4,,,)'
=: 1 (;
M= ~M.
J = J1 + J2 + J3 + J4 ,
Ate., = 4[ 0 ,
i ft.ll < l/,. if t,/i~t/~,
iftf,>tl,, if t ~ ~< tl,,
Qt=AQC~.C(~,- ~_~ %.,,(C~,, - ().,, ,). ly--1
C ~,N(I]' / / / / / / / /
/ / / / / /
ctNI~~ I / .'~ / /
C,,o
°%0 '*~,i °<~,z Fig. 1. Penalty cost function.
# I I I
I I o
I
-
t
Computers in Industry
where N(~) is defined by a double inequality
The optimal values of weight coefficients w,v where rl = 1,4, are found out by using a Gausstype linear direct search procedure with each iteration corresponding to a simulation of the evolution of the workshop. Since the values of weight coefficients are likely to be relatively stable, it is advisable that the search be performed only in the "programming phase", when the first job shop schedule is constructed. The possible "correcting" activities, which consist in real time reschedulings within the optimization horizon if necessary, merely consist in simple simulations over even shorter "remaining horizons", contrary to the initial optimization which is carried-out using the "sliding horizon" concept [8].
4. A Numerical Example In order to illustrate the performance and applicability of the algorithm, let us consider a hypothetical workshop consisting of three groups, each group made up of two machines. The characteristics of those fourteen work orders to be scheduled over a horizon of 32 time units are described in Table 2. Some of the jobs, those having the null arrival and transport time for the first operation, are generally remaining from the previous validity interval. Some of them are already on workshop machines. This initial assignment, which is displayed in Table 3, may be looked upon as the initial condition (state) of the problem. Table 4 contains an initial prediction of the future events, including the planned maintenance stops, Different schedules were obtained by using different priority rules (Table 5). The best result, J, was yielded when the composed priority rule with optimal weight parameters was used. The optimal schedule was obtained in 30 evaluations. One possible output of the algorithm, the optimal machine assignment, is displayed in Table 6. 4.1. R e m a r k s on the Numerical E x a m p l e
Simplified linear penalty functions Q~ were used when evaluating the schedule.
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In the first ten cases of simple priority rules, or the equal weight parameters composed rule, the computer time was roughly 30 times shorter than the time employed under the optimal composed priority rule (case 11), since simple simulations and no search were performed.
5. DICOTR-D A Real Time Shopfioor Level
System
at the
One must be aware of those many uncertainties the simulation model used to obtain job shop scheduling contains: the assumption that all the machines of a group have the same performance, or the standard processing or transit times can be exactly observed etc. Fortunately the hidden danger of the blind trust of the human in the "'technologically respectable" output of the computer is not as great as one may expect. Moreover the human is sometimes tempted to turn down the computer suggestions because they may seem too mechanicist, and, instead, simply ask for some assistance of his decisions which may primarily take into account the output of the computerized algorithms. For example the human decision maker would better alter some standard processing or transport times in accordance with the skills of his people he is aware of. He may also prefer to modify some event time instants or even the order in the future event list, or to generate and introduce his own event sequence in the future event file. 5. I. General Features
The forthcoming solution for designing computer-aided production control systems is characterized by the following general features [5]: 1. It includes models and fast optimization procedures as well as the possibility for the decision maker to easily modify model and criterion parameters or even alter the recommendations of the computer in accordance with his experience and judgement. 2, The machine is able to evaluate the human decisions in short time. 3. The system must be generated by the user himself in order to provide the human with greater job satisfaction and to prevent the possible phenomenon of "polarization of EDP users" discussed in [3].
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Table 2 J o b s Characteristics Job i
External priority P~,
Number of operations
Arrival time ta;
Due time t/,
Operation number k
Group number 1
Transport time I t)
Processing lime ~
l 2 3 1 2 3 1 2 3 1 2 3 2 3 2 3 1 2 3 l 2 3 1 2 3 l 2 3 2 3 2 3 2 3 2 3 3
0 ~ 2 t i 2 I
7 7 I1 5 6 13 8
l
5
10
l 2 3 1 2 3 1 2 3 1 2 3 1 2 l 2 1 2 3 1 2 3 l 2 3 1 2 3 l 2 1 2 l 2 l 2 1
0 2 4 3 3 ~ 2 2 1 2 2 0 3 2 0 3 2 t} 0 0 0 i 0 2 0 I 1 ! 0
15 5 12 q 4 9 5 7 3 4 9 7 10 7 4 8 7 2 5 1 2 4 2 5 i 5 2 3 8
12
1
3
0
9
Ri 1
3
3
1
2
2
3
1
3
1
3
1
26
4
4
3
1
22
5
1
2
2
6
5
2
2
20
7
4
3
4
22
8
1
3
4
9
2
3
4
-
lO
3
3
0
20
ll
4
2
0
10
12
1
2
0
12
13
1
2
0
14
1
2
1
15 16
2 I
1 1
0 0
Table 3 I n i t i a l C o n d i t i o n of the W o r k s h o p Machine Job
M? 0
M2~ 10
M12 11
5.2. The Main Functions of the System The optimization of job shop scheduling is but one function of the real time production control system. Other functions of the system are the
Mz2 12
M13 16
M~~ 15
following: -- Work-in-progress control through an accurate event reporting. - Q u a l i t y data collection from the quality checking centre.
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F.G. Filip et al. / Job Shop Scheduling
Table 4 Initial Future Event File Event number
Time instant
Job
Group
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 1 1 1 2 2 4 4 4 2 2 2 1 1 8 9 4 6 10 14
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0 0 0 0
1 1 1 1 1 1 1 1 1 1 2 2 2 2 3 3 1 1 2 2
Machine
Event Type
*
l 1 1
-
1
1 1 1 1 1 4 4 4 3 1 4 4 5 6 5 6
-
2 1 2 1 2 1 l 1 1 1
* The symbol - may equally take the values 1 or 2.
Real time jobs resequencing provided that some internal or interface disturbances appear and their effect is beyond a tolerance value.
Table 6 Machines Assignment
-
Table 5 P e r f o r m a n c e of the o p t i m i z a t i o n a l g o r i t h m in c o m p a r i s o n with o t h e r methods No 1 2 3 4 5 6
7 8 9 10
11
Priority criteria Carroll's rule Carroll's rule m u l t i p l i e d by external priority Total waiting time Total waiting time m u l t i p l i e d by external priority R e m a i n i n g processing t i m e / c u r r e n t processing time R e m a i n i n g processing t i m e / c u r r e n t processing time m u l t i p l i e d by external priority Size of the next queue Size of the next queue multiplied by external priority First In First Out C o m p o s e d priority rule with equal weight parameters ( w = 1 ) C o m p o s e d priority rule with o p t i m a l weight p a r a m e t e r s
J/,~ 1.293 1.293 1.469 1.316 1.472
1.296 2.360 2.360 2.375
1.274 1
M]
M~
M~
M~
1
0
2 3 4 5 6 7 8 9 10 11 12 13 14 16 17 18 19 20 21 22 24 26 27 28 30 32
1 1 1 7 7 7 1
10 10 2 2 2 2 2 2 4 4
11 11 10 10 10 10 10 5 5 5
12 12 13 14 14 6 6 6 6 6
1
4
-
2
1 1 3 3 3 3 3 3 3 0 0 0 0 0 0 0
4 4 9 9 9 8 8 8 8 8 8 0 0 0 0 0
7 7 4 4 4 4 4 4 4 4 3 3 3
2 2 2 2 5 1 1 1 1 1 1 9 9 9 9 9
Machine
Mit
M3
Job Time 16 16 16 16 16 16 16 16 16 12 12 12 12 12 13 13 13 13 2 2 2 2 2 2 2 2 14
15 15 15 15 15 15 15 15 1l 11 11 11 10 6 6 6 6 6 6 7 7 7 7 7 7 1 1
401
402
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- Changes implementation to in-progress work orders consisting for example in lot splitting (in case of disturbances in materials or tool handling), modification of technological routes (in case of machine breakdown), modification of the standard times for certain operation (because of off-specification materials or changes in technological routing), order cancellation, new orders (in case of reworks or rejects). - Reporting about the current state of the work orders, machines and group queues as well as predicted evolution of the workshop. - A t t e n d a n c e data collection via specialized terminals. Interfacing the higher computerized management and control level. It is obvious that the above-mentioned functions belong to a standard (prototype) system. Depending on the application concerned, at system analysis phase, the designer and the user can cooperate in choosing the appropriate set.
5.3. The System Architecture DICOTR-D which belongs to the DICOTR systems family [5], has been developed as a dedicated complex: its hardware and software components have been chosen or specially developed in order to efficiently implement the above-defined functions. The system implements, according to user option at system generation time, two kinds of functional regimes: a) transactional regime, when the operator has the initiative in activating various system functions such as event reporting or workshop state interogation, and b) real time regime, when the system indicates the next action to be taken by the operator in case of normal evolution of the system, notifies on deviations from predicted evolution or suggests suitable decisions to be made. The system structure is shown in Fig. 2. The data organisation has been set up so as to meet the requirements of the above-described functions and to enable interfaces with other components of the overall management and control structure. This gives a functional autonomy to the system within the scheduling validity period [7]. The system data base contains several categories of files such as: WOD (work orders descrip-
Fig. 2. The DI('OTR-D Systet~l~
tion), (7US (description of the current state of job. machines and queues) and IDA (internal data about the future event list, including the initial future event list). The main program modules are MON (System monitor), MOD (mode selection), INT (interface to higher management level for receiving the production program and preparing the reports needed by higher levels), OPT (optimal solution of weight parameters), SIM (creating the future event file by using simulation), REP (local reporting), EMO (event monitoring: collecting data about various events in the workshop), QUA (quality data collection), OMO (modifications made by the operator to the optimization problem formulation such as: standard times modifications, initial future event list alterations etc.), ATT (attendance date collection), COR (corrections made by the operator in the job shop schedule and evaluation of the human decisions). The hardware structure of the system is made up of the Romanian minicomputer INDEPEND E N T 100, which is similar to the PDP 11/34 machine, CRT terminals and special attendance reporting terminals. The operating system is RSX llM.
6. Conclusions
The two-level optimization algorithm using simulation to evaluate the performance criterion is quite beneficial in solving middle-size problems of job shop scheduling. For cases of job resequencing
Computers in lndustrv
under time pressure, a simple simulation can be used. A system has been presented which helps the computer-aided decision concept be implemented by a real time supply of information necessary for evaluating the current state of the workshop and by suggesting the decision to be made in case of deviation from a specified trajectory. The system is flexible enough and allows the user to choose and implement, under computer guidance, the version that suits the user's skill and confidence in computer. Efforts are currently being made to extend the family of scheduling algorithms based on various assumptions to be chosen by the user at system generation time. It is hoped that the system (which is being tested in an actual application) will stimulate the user to continuously improve his or her skill and knowledge.
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[11] M. Guran, F.G. Filip, A.D. Donciulescu, M. Muratcea, L. Orh}anu, N. Predoiu and T. R~.dulescu, "Multilevel approach to real time management and control systems", Ec. Computation and Ec. Cybern. Studies and Research, vol. XV, no. 1, 23-33 (1981). [12] C.H. Kriebel, "The Future MIS", Interactive Oriented Data Case Systems (Edited by W.C. House), Petrocelli/ Charter, New York (1977). [13] R.H. Sprague, H.Y. Watson, "MIS Concepts", Interactive Decision Oriented Data Base Systems (Edited by W.C. House), Petrocelli/Charter. New York (1977). [14] A.P. Wierzbicki, "Muhiobjective trajectory optimization and model semiregularization", Prep. of IFAC Congress, Kyoto, Japan, paper 49.2 (I981). [15] T.J. Williams, "Systems engineering of hierarchical computer control of large industrial steel mill complexes". Handbook of Large Scale System Engineering Applications (Edited by M.G. Singh and A. Titli), North Holland, Amsterdam, 372-399 (1979).
Appendix A, Priori~, Criteria 1. Carroll's rule
References [1] O. Bjorke, "Partly unmanned machining", Advanced Manufacturing Technology (Edited by P.L. Blake), North Holland, Amsterdam, 271-284 (1980). [2] G.D. Brewer, Politicians, Beaurocrats and the Consultant (Basic Book, New York. 1973). [3] U. Briefs, "Re-thinking industrial work: computer effects on technical white-collar workers", Computers in Industry, vol.2, no 1, 76-81 (1981). [4] J.C. Emery, "Jobshop scheduling by means of simulation and an optimum-seeking search", Proc. of the Third Conf. on Applications of Simulation, Los Angeles, USA, 363 37l (1969). [5] D.A. Donciulescu and F.G. Filip, "DISPATCHER - A decision supporting system based on hierarchical approach", Proc. of the Fifth International Conf. on Control Systems and Computer Sci., Bucharest, Romania, 189-194 (1983). [6] G. Doumeingts and G.D. Breuil, "A method for structuring production planning and control systems", Production Management Systems (Edited by D. Falster and A. Rolstadhs), North Holland, Amsterdam, 85-105 (1981). [7] F.G. Filip, "Contributions to Hierarchical Control of Complex Systems", Ph. D. Thesis, Politechnical Institute of Bucharest, Romania (1981). (In Romanian.) [81 F.G. Filip and D.A. Donciulescu, " O n an on line dynamic coordination method in process industry", Automatica, vol.19, no.3, 317-320 (1983). [9] F.G. Filip, D.A. Donciulescu, M. Muratcea, G. Neagu and L. Or~.,~anu, "Modeling, multilevel optimization and simulation in computer aided production control", Proc. of International AMSE Conf. Modelling and Simulation. Paris-Sud, vol. 2, 54-60 (1982). [10] W. Findeisen, "Decentralized and hierarchical control under consistency or disagreement of interests", Automatic& vol. 18, no. 6, 647-664 (1982).
P,., ~ < / 4
(1)
t v -s,
whenv,>s,>O
Ui
c,= t ;
whenst~< 0
(2)
when s, >1 vi
<"
tw~,
ta nroi
k
~, k
(3)
R,-k
s,~tl,-,+
Z (,:=t,/)=4 /=k+l
,
(4)
where: t,~ = the current processing (set-up and run) time for the @,~ operation of theo,~,job; t~;* = the amount of time thea~, job had to wait for previous operations: R, = the total number of the operations for the J , job: nro, = the remaining number of operations for the,:/,, job, except the current one: tl, = the due date for the,t,, job; t = the current time: t/= the processing time for dJ/ operation of the J , job: t t / = the transport time from the machine group of d)/ i operation to the group of C/ operation.
2. Time in queue P2.i & tw,k - l + tq,~
(5)
where: tq,a = the time already spent by tbeo~, job in the queue for the 0 k operation.
3. Remaining processing time/current processing time P3.i ~ tr,~/t~
(6)
R,
tr, k&
E
j:k+l
(t/+tt/)=t, k
(7)
4, Size of next queue
P4,i~ l/l~ +1
(8)
Applications
Job Shop Scheduling Optimization in RealTime Production Control F.G. Filip, G. Neagu, D.A. Donciulescu Central Institute for Management and lnformatics, Bulet~ard Miciurin 8-10. Bucharest, Romania An optimization algorithm using simulation as an evaluation procedure of the criterion to obtain an optimal schedule of jobs within a workshop is presented. The algorithm, which is recommended for middle-size problems, is embedded in a dedicated minicomputer-based practical system meant for real time production control in a discrete part manufacturing system environment.
Kevwords: Optimization, simulation, production control, hum a n - m a c h i n e interaction, computer-aided decision making, dedicated system.
Florin-Gheorghe FUip was born in Bucharest, Romania, in 1947. He graduated in 1970 in control engineering from the Polytechnic Institute of Bucharest. He received in 1983 the Ph. D. degree for c o n t r i b u t i o n s to hierarchical control of complex systems. Since 1970, dr. Filip has been working at the Central Institute for M a n a g e m e n t and Informatics in Bucharest, where he is the head of the "Real Time Systems" research group. In 1974 Dr. Filip spent half-a-year in Swedish universities as a visiting research fellow and in 1983 he was an invited lecturer on hierarchical, large scale systems at Shenyang Institute of Automation of the Chinese Academy of Sciences. He is the a u t h o r / c o a u t h o r of some forty technical papers and is the coauthor of a book. His current scientific interests include hierarchical optimization and control, C A D / C A M systems and real-time computer control in industry.
North-Holland Computers in Industry 4 (1983) 395-403
1. Introduction Hierarchical multilevel-multilayer management and control systems appear to be very promising design solutions for many practical computer-controlled production systems because large and complicated control problems can conveniently be substituted by several smaller and simpler ones [1,6,7,8,15]. Within the control hierarchy, the real time production control layer acts not only as a passive recorder of data about the plant behaviour and as the supplier of "fresh" information about it, but also as a permanent source of reference for the actual state as compared to the predicted
Dan Alexandru Donciulescu was born in Bachu, Romania, in 1949. He received his MS in Control Engineering in 1972 from the Polytechnic Institute of Bucharest. In 1972, D.A. Donciulescu joined the Central Institute for M a n a g e m e n t and Informatics in Bucharest. He has been working as research engineer and project leader for several development projects of computer-based production control systems. He is an associate assistant professor in computer programming at the Polytechnical Institute of Bucharest. He is the a u t h o r / coauthor of some thirty technical papers. His interests mainly refer to the real time applications of hierarchical systems theory to production control in process industry. Gabriel Neagu was born in Bucharest, Romania, in 1949. In 1973 he received his M.S. in control engineering from the E.E. Institute of Moscow. Since 1974 he has been working at the Control Institute for Management and Informatics in Bucharest as a research engineer and project leader for several real time production control applications and systems in metallurgical and discrete part manufacturing industries. He a u t h o r e d / c o a u t h o r e d some twenty technical papers. His current scientific interests are in the field of real time production control and C A D / C A M Systems.
0166-3615/83/$03.00 '~ 1983 Elsevier Science Publishers B.V. (North-Holland)
396
Applications
trajectory of the system evolution [11]. Thus, the real time system notifies the decision maker about all critical events that have to be taken into account. In order to render the decision making process more efficient and quicker, the system capabilities must be developed towards operative preparation and delivery of decision variants concerning critical events in the controlled system behaviour. According to Kriebel [12], which identifies four levels of relative maturity of information systems, which differently support the four stages of the decision making process (observation, inference, evaluation and choice), the system that gives the recommendation to act is located at the third level, At the operational level, the computer-made decisions suggestions are generally based on models of well structured problems and on optimization procedures within narrow boundaries [13], even though, as Wierzbicki noticed [14] often "the purpose of optimization is not to propose optimal solutions, but rather to generate reasonable alternatives in response to user's requirements, while eliminating clear inferior alternatives". In a discrete part manufacturing system environment, job shop scheduling optimization can be conveniently carried out by using simulation models [4]. Simulation is resorted to because the system concerned is too complex to be described by a simple analytical model and the optimization problem does not have a reasonable straightforward analytical solution [2]. It was observed [5,13] that the optimization procedures may not be as useful as one expects unless they are considered in the context of the entire decision making process. The multistage decision making concept [10] shows its usefulness when coping with uncertainties, preference changes, and decision maker attitudes, experience and motivation, and allows for designing and implementing successful practical computer-aided operative decision making systems where the optimization is but a part. The purpose of this paper is to describe a job shop scheduling optimization algorithm, using simulation as an evaluation procedure of the criterion, embedded in a practical system. The remaining part of this paper is organized as follows. In Section 2, the objectives of the real time production control system are shown. An extension of the optimization algorithm proposed in [4] is given in
('omputers in Dulustm
Section 3. An illustrative example is presented in Section 4: and Section 5 describes a real time system at the workshop level, which makes use of the optimization algorithm.
2. System Objectives In a discrete part manufacturing environment, at the workshop level, the foreman is periodically given, by the upper management level, the production program of the next time horizon. The production program indicates the jobs (work orders) to be done and gives the appropriate technological information such as: operation range~ sequence of execution and the time standards. A real time production control system at the workshop level makes the production program implementation effective. Among the major objectives of the system are the following: respect of due dates, increase of the efficiency in the utilization of all manufacturing and manpower resources, diminishing jobs flow time, reduction of the amount of work-in-process and a better coordination with other workshops and support activities, The system is functionally evenl-oriented, where an event is viewed as a discontinuity in the production process. The list of event types is given in Table 1. Job shop scheduling is a central function in production control and its accomplishment depends on the given production program and the actual state of the workshop. In order to obtain a schedule, a simulation procedure may be used to enlist the predictable events, mainly including the optimal sequence of job assignment to various machines, for a scheduling period, AT. An optimi-
"Fable 1 Significant event list No. Significance 1. 2. 3. 4. 5. 6. 7. 8.
New job arrival in the workshop Job entry in a waiting queue of jobs at a machine group Start of an operation Accomplishmentof current operation for a machine Start of a planned maintenance stop Machineset-up after event 5 or 8 Conclusionof job processing in the workshop Unexpectedmachine break down
Computers in Industry
zation algorithm, which extends the results presented in [4], is described in what follows (Section 3).
2.1. Remarks on System Objectives We must emphasize that each scheduling problem is time and space limited because of the hierarchical nature of the control system based on structural and temporal decomposition used for getting real time properties [11]. If no significant disturbance will occur, the schedule will be used during only a validity intert,al, AT,.. After that interval, new intervention information will be provided by the upper level and, consequently, a new more up-dated schedule will be needed. In order to increase the system reliability in cases of failures of data processing or communication equipment, the scheduling horizon is larger than the validity interval [8]. In order to compensate the effect of significant disturbances, it is sometimes necessary to reconstruct the job shop schedule in real time.
3. The Optimization Algorithm The algorithm to be described may be looked upon as a hierarchical, two-level optimization method based on parametric decomposition [7]. At the lower level, an optimal job shop schedule is constructed by using simulation for fixed values of the intervention variables. The intervention vector is modified to minimize a performance measure at the higher level.
3. t. Assumptions The simulation is based on the normal assumptions on the workshop operation [4]: A. There is a set, £,a, of jobs; each job J , , i~.~', enters the workshop at various times, ta,. There are one or more operations d)~, k = 1, R i to be executed from the release ofa~, to its completion. B. The workshop comprises a number, G, of various machine groups, each group, ~j, j = 1, G, containing one or more identical machines (working posts), ~t'~, m = 1, Mp that take over their jobs from a common queue of jobs. C. Each job to be scheduled may or may not have a previously assigned due date, tl~. The first
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397
category of jobs belongs to the set L-'~, the second one belongs to the set ~2 and tl,, where i~£~az, is chosen oo. D. An operation on one machine must be completed for all items that belong to a job before the next operation on that job can start. E. There is a transit time tt,~ required to transport job J, from one machine group to another before starting the operation 6o,~. F. The processing time values t,~ are available for any operation of each job. G. If a machine J [ , ~, becomes vacant, the highest priority job waiting in its queue is assigned to it. H. Time intervals for machine repairing or verification may be allocated for each machine.
3.2. The Priority Rule In order to indicate which job is to be allocated to a vacant machine, a composite priority rule [9], similar to the one presented in [4] is used. This rule takes into consideration for each job J , the following priority criteria: external priority, Pe.,, Carrol's rule, Pl,i, time in queues, P2.,, remaining processing t i m e / c u r r e n t processing time, p~.,, and the estimated size of the next queue, P4.,. (A review of these criteria is given in the Appendix.) The priority of the ith job is calculated as follows: 4 q=l
where the weights { %, w2, ws, w4 } form the intervention vector in a two-level hierarchical optimization procedure. Notice that each criterion might be used as a screen, defined by threshold and tolerance parameters, in order to eliminate the jobs that are " n o t critical". This is very useful for large problems. However this procedure is optional when small or middle-size problems are concerned and, moreover, is likely to cause optimality losses. Besides, the threshold and tolerance parameters tend to complicate the searching procedure, which will be described in the sequel, by doubling the amount of free variables.
3.3. The Optimal Scheduling The future evolution of the workshop can be simulated as a sequence of significant events over
Applications
398
('omputers m lndustrv
J = ~ Q>,(At>,),
the time horizon, AT. The values appropriately chosen of the weight parameters ~%, r/= 1,4, are crucial for the efficiency of the schedule. To establish the optimal values of the weight parameters asks for defining a performance measure. The value of the performance measure indicates, for each simulated sequence, how high are the penalty costs for each job tardiness, Ate.,, or earliness, At2. ~, of the completion time obtained by simulation, t f,, as compared with the due date [4]. In order to reduce job flow time, the time the ith job has waited in various queues, zlt3, , is considered in the penalty function. Since, besides flow time reduction and respect of due dates, a high utilization of production capacities is required, it is reasonable to penalize the time the ruth machine was in the idle state, ~ t 4 .... [9]. Therefore, the performance measure has four components
l'tl ----tL. t2,:- i0,
J~,= ~_~Q-<,(2aG.,). R,
Ate.,= y" tw ~, l,-:] At
"/4= ~ m
The general form of each penalty function, Q~. w h e n ,~ = (fi, y), f i e { l , 2. 3. 4} a n d *~ { ~ , ~ , ~ 1 . . . . . m }, is linearwise and has an obvious economic meaning (Fig. 1). Thus the penalty is calculated by using monotonously increasing cost coefficients, Q .... where v = 1. N(ff~. as follows [7]:
where
J, = ~ Q,.,(at,.;), (tf,-tl,,
04,,,,(~t4,,,)'
=: 1 (;
M= ~M.
J = J1 + J2 + J3 + J4 ,
Ate., = 4[ 0 ,
i ft.ll < l/,. if t,/i~t/~,
iftf,>tl,, if t ~ ~< tl,,
Qt=AQC~.C(~,- ~_~ %.,,(C~,, - ().,, ,). ly--1
C ~,N(I]' / / / / / / / /
/ / / / / /
ctNI~~ I / .'~ / /
C,,o
°%0 '*~,i °<~,z Fig. 1. Penalty cost function.
# I I I
I I o
I
-
t
Computers in Industry
where N(~) is defined by a double inequality
The optimal values of weight coefficients w,v where rl = 1,4, are found out by using a Gausstype linear direct search procedure with each iteration corresponding to a simulation of the evolution of the workshop. Since the values of weight coefficients are likely to be relatively stable, it is advisable that the search be performed only in the "programming phase", when the first job shop schedule is constructed. The possible "correcting" activities, which consist in real time reschedulings within the optimization horizon if necessary, merely consist in simple simulations over even shorter "remaining horizons", contrary to the initial optimization which is carried-out using the "sliding horizon" concept [8].
4. A Numerical Example In order to illustrate the performance and applicability of the algorithm, let us consider a hypothetical workshop consisting of three groups, each group made up of two machines. The characteristics of those fourteen work orders to be scheduled over a horizon of 32 time units are described in Table 2. Some of the jobs, those having the null arrival and transport time for the first operation, are generally remaining from the previous validity interval. Some of them are already on workshop machines. This initial assignment, which is displayed in Table 3, may be looked upon as the initial condition (state) of the problem. Table 4 contains an initial prediction of the future events, including the planned maintenance stops, Different schedules were obtained by using different priority rules (Table 5). The best result, J, was yielded when the composed priority rule with optimal weight parameters was used. The optimal schedule was obtained in 30 evaluations. One possible output of the algorithm, the optimal machine assignment, is displayed in Table 6. 4.1. R e m a r k s on the Numerical E x a m p l e
Simplified linear penalty functions Q~ were used when evaluating the schedule.
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399
In the first ten cases of simple priority rules, or the equal weight parameters composed rule, the computer time was roughly 30 times shorter than the time employed under the optimal composed priority rule (case 11), since simple simulations and no search were performed.
5. DICOTR-D A Real Time Shopfioor Level
System
at the
One must be aware of those many uncertainties the simulation model used to obtain job shop scheduling contains: the assumption that all the machines of a group have the same performance, or the standard processing or transit times can be exactly observed etc. Fortunately the hidden danger of the blind trust of the human in the "'technologically respectable" output of the computer is not as great as one may expect. Moreover the human is sometimes tempted to turn down the computer suggestions because they may seem too mechanicist, and, instead, simply ask for some assistance of his decisions which may primarily take into account the output of the computerized algorithms. For example the human decision maker would better alter some standard processing or transport times in accordance with the skills of his people he is aware of. He may also prefer to modify some event time instants or even the order in the future event list, or to generate and introduce his own event sequence in the future event file. 5. I. General Features
The forthcoming solution for designing computer-aided production control systems is characterized by the following general features [5]: 1. It includes models and fast optimization procedures as well as the possibility for the decision maker to easily modify model and criterion parameters or even alter the recommendations of the computer in accordance with his experience and judgement. 2, The machine is able to evaluate the human decisions in short time. 3. The system must be generated by the user himself in order to provide the human with greater job satisfaction and to prevent the possible phenomenon of "polarization of EDP users" discussed in [3].
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Applications
Computers in lndustr~
Table 2 J o b s Characteristics Job i
External priority P~,
Number of operations
Arrival time ta;
Due time t/,
Operation number k
Group number 1
Transport time I t)
Processing lime ~
l 2 3 1 2 3 1 2 3 1 2 3 2 3 2 3 1 2 3 l 2 3 1 2 3 l 2 3 2 3 2 3 2 3 2 3 3
0 ~ 2 t i 2 I
7 7 I1 5 6 13 8
l
5
10
l 2 3 1 2 3 1 2 3 1 2 3 1 2 l 2 1 2 3 1 2 3 l 2 3 1 2 3 l 2 1 2 l 2 l 2 1
0 2 4 3 3 ~ 2 2 1 2 2 0 3 2 0 3 2 t} 0 0 0 i 0 2 0 I 1 ! 0
15 5 12 q 4 9 5 7 3 4 9 7 10 7 4 8 7 2 5 1 2 4 2 5 i 5 2 3 8
12
1
3
0
9
Ri 1
3
3
1
2
2
3
1
3
1
3
1
26
4
4
3
1
22
5
1
2
2
6
5
2
2
20
7
4
3
4
22
8
1
3
4
9
2
3
4
-
lO
3
3
0
20
ll
4
2
0
10
12
1
2
0
12
13
1
2
0
14
1
2
1
15 16
2 I
1 1
0 0
Table 3 I n i t i a l C o n d i t i o n of the W o r k s h o p Machine Job
M? 0
M2~ 10
M12 11
5.2. The Main Functions of the System The optimization of job shop scheduling is but one function of the real time production control system. Other functions of the system are the
Mz2 12
M13 16
M~~ 15
following: -- Work-in-progress control through an accurate event reporting. - Q u a l i t y data collection from the quality checking centre.
Computers in lndustry
F.G. Filip et al. / Job Shop Scheduling
Table 4 Initial Future Event File Event number
Time instant
Job
Group
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 1 1 1 2 2 4 4 4 2 2 2 1 1 8 9 4 6 10 14
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0 0 0 0
1 1 1 1 1 1 1 1 1 1 2 2 2 2 3 3 1 1 2 2
Machine
Event Type
*
l 1 1
-
1
1 1 1 1 1 4 4 4 3 1 4 4 5 6 5 6
-
2 1 2 1 2 1 l 1 1 1
* The symbol - may equally take the values 1 or 2.
Real time jobs resequencing provided that some internal or interface disturbances appear and their effect is beyond a tolerance value.
Table 6 Machines Assignment
-
Table 5 P e r f o r m a n c e of the o p t i m i z a t i o n a l g o r i t h m in c o m p a r i s o n with o t h e r methods No 1 2 3 4 5 6
7 8 9 10
11
Priority criteria Carroll's rule Carroll's rule m u l t i p l i e d by external priority Total waiting time Total waiting time m u l t i p l i e d by external priority R e m a i n i n g processing t i m e / c u r r e n t processing time R e m a i n i n g processing t i m e / c u r r e n t processing time m u l t i p l i e d by external priority Size of the next queue Size of the next queue multiplied by external priority First In First Out C o m p o s e d priority rule with equal weight parameters ( w = 1 ) C o m p o s e d priority rule with o p t i m a l weight p a r a m e t e r s
J/,~ 1.293 1.293 1.469 1.316 1.472
1.296 2.360 2.360 2.375
1.274 1
M]
M~
M~
M~
1
0
2 3 4 5 6 7 8 9 10 11 12 13 14 16 17 18 19 20 21 22 24 26 27 28 30 32
1 1 1 7 7 7 1
10 10 2 2 2 2 2 2 4 4
11 11 10 10 10 10 10 5 5 5
12 12 13 14 14 6 6 6 6 6
1
4
-
2
1 1 3 3 3 3 3 3 3 0 0 0 0 0 0 0
4 4 9 9 9 8 8 8 8 8 8 0 0 0 0 0
7 7 4 4 4 4 4 4 4 4 3 3 3
2 2 2 2 5 1 1 1 1 1 1 9 9 9 9 9
Machine
Mit
M3
Job Time 16 16 16 16 16 16 16 16 16 12 12 12 12 12 13 13 13 13 2 2 2 2 2 2 2 2 14
15 15 15 15 15 15 15 15 1l 11 11 11 10 6 6 6 6 6 6 7 7 7 7 7 7 1 1
401
402
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- Changes implementation to in-progress work orders consisting for example in lot splitting (in case of disturbances in materials or tool handling), modification of technological routes (in case of machine breakdown), modification of the standard times for certain operation (because of off-specification materials or changes in technological routing), order cancellation, new orders (in case of reworks or rejects). - Reporting about the current state of the work orders, machines and group queues as well as predicted evolution of the workshop. - A t t e n d a n c e data collection via specialized terminals. Interfacing the higher computerized management and control level. It is obvious that the above-mentioned functions belong to a standard (prototype) system. Depending on the application concerned, at system analysis phase, the designer and the user can cooperate in choosing the appropriate set.
5.3. The System Architecture DICOTR-D which belongs to the DICOTR systems family [5], has been developed as a dedicated complex: its hardware and software components have been chosen or specially developed in order to efficiently implement the above-defined functions. The system implements, according to user option at system generation time, two kinds of functional regimes: a) transactional regime, when the operator has the initiative in activating various system functions such as event reporting or workshop state interogation, and b) real time regime, when the system indicates the next action to be taken by the operator in case of normal evolution of the system, notifies on deviations from predicted evolution or suggests suitable decisions to be made. The system structure is shown in Fig. 2. The data organisation has been set up so as to meet the requirements of the above-described functions and to enable interfaces with other components of the overall management and control structure. This gives a functional autonomy to the system within the scheduling validity period [7]. The system data base contains several categories of files such as: WOD (work orders descrip-
Fig. 2. The DI('OTR-D Systet~l~
tion), (7US (description of the current state of job. machines and queues) and IDA (internal data about the future event list, including the initial future event list). The main program modules are MON (System monitor), MOD (mode selection), INT (interface to higher management level for receiving the production program and preparing the reports needed by higher levels), OPT (optimal solution of weight parameters), SIM (creating the future event file by using simulation), REP (local reporting), EMO (event monitoring: collecting data about various events in the workshop), QUA (quality data collection), OMO (modifications made by the operator to the optimization problem formulation such as: standard times modifications, initial future event list alterations etc.), ATT (attendance date collection), COR (corrections made by the operator in the job shop schedule and evaluation of the human decisions). The hardware structure of the system is made up of the Romanian minicomputer INDEPEND E N T 100, which is similar to the PDP 11/34 machine, CRT terminals and special attendance reporting terminals. The operating system is RSX llM.
6. Conclusions
The two-level optimization algorithm using simulation to evaluate the performance criterion is quite beneficial in solving middle-size problems of job shop scheduling. For cases of job resequencing
Computers in lndustrv
under time pressure, a simple simulation can be used. A system has been presented which helps the computer-aided decision concept be implemented by a real time supply of information necessary for evaluating the current state of the workshop and by suggesting the decision to be made in case of deviation from a specified trajectory. The system is flexible enough and allows the user to choose and implement, under computer guidance, the version that suits the user's skill and confidence in computer. Efforts are currently being made to extend the family of scheduling algorithms based on various assumptions to be chosen by the user at system generation time. It is hoped that the system (which is being tested in an actual application) will stimulate the user to continuously improve his or her skill and knowledge.
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[11] M. Guran, F.G. Filip, A.D. Donciulescu, M. Muratcea, L. Orh}anu, N. Predoiu and T. R~.dulescu, "Multilevel approach to real time management and control systems", Ec. Computation and Ec. Cybern. Studies and Research, vol. XV, no. 1, 23-33 (1981). [12] C.H. Kriebel, "The Future MIS", Interactive Oriented Data Case Systems (Edited by W.C. House), Petrocelli/ Charter, New York (1977). [13] R.H. Sprague, H.Y. Watson, "MIS Concepts", Interactive Decision Oriented Data Base Systems (Edited by W.C. House), Petrocelli/Charter. New York (1977). [14] A.P. Wierzbicki, "Muhiobjective trajectory optimization and model semiregularization", Prep. of IFAC Congress, Kyoto, Japan, paper 49.2 (I981). [15] T.J. Williams, "Systems engineering of hierarchical computer control of large industrial steel mill complexes". Handbook of Large Scale System Engineering Applications (Edited by M.G. Singh and A. Titli), North Holland, Amsterdam, 372-399 (1979).
Appendix A, Priori~, Criteria 1. Carroll's rule
References [1] O. Bjorke, "Partly unmanned machining", Advanced Manufacturing Technology (Edited by P.L. Blake), North Holland, Amsterdam, 271-284 (1980). [2] G.D. Brewer, Politicians, Beaurocrats and the Consultant (Basic Book, New York. 1973). [3] U. Briefs, "Re-thinking industrial work: computer effects on technical white-collar workers", Computers in Industry, vol.2, no 1, 76-81 (1981). [4] J.C. Emery, "Jobshop scheduling by means of simulation and an optimum-seeking search", Proc. of the Third Conf. on Applications of Simulation, Los Angeles, USA, 363 37l (1969). [5] D.A. Donciulescu and F.G. Filip, "DISPATCHER - A decision supporting system based on hierarchical approach", Proc. of the Fifth International Conf. on Control Systems and Computer Sci., Bucharest, Romania, 189-194 (1983). [6] G. Doumeingts and G.D. Breuil, "A method for structuring production planning and control systems", Production Management Systems (Edited by D. Falster and A. Rolstadhs), North Holland, Amsterdam, 85-105 (1981). [7] F.G. Filip, "Contributions to Hierarchical Control of Complex Systems", Ph. D. Thesis, Politechnical Institute of Bucharest, Romania (1981). (In Romanian.) [81 F.G. Filip and D.A. Donciulescu, " O n an on line dynamic coordination method in process industry", Automatica, vol.19, no.3, 317-320 (1983). [9] F.G. Filip, D.A. Donciulescu, M. Muratcea, G. Neagu and L. Or~.,~anu, "Modeling, multilevel optimization and simulation in computer aided production control", Proc. of International AMSE Conf. Modelling and Simulation. Paris-Sud, vol. 2, 54-60 (1982). [10] W. Findeisen, "Decentralized and hierarchical control under consistency or disagreement of interests", Automatic& vol. 18, no. 6, 647-664 (1982).
P,., ~ < / 4
(1)
t v -s,
whenv,>s,>O
Ui
c,= t ;
whenst~< 0
(2)
when s, >1 vi
<"
tw~,
ta nroi
k
~, k
(3)
R,-k
s,~tl,-,+
Z (,:=t,/)=4 /=k+l
,
(4)
where: t,~ = the current processing (set-up and run) time for the @,~ operation of theo,~,job; t~;* = the amount of time thea~, job had to wait for previous operations: R, = the total number of the operations for the J , job: nro, = the remaining number of operations for the,:/,, job, except the current one: tl, = the due date for the,t,, job; t = the current time: t/= the processing time for dJ/ operation of the J , job: t t / = the transport time from the machine group of d)/ i operation to the group of C/ operation.
2. Time in queue P2.i & tw,k - l + tq,~
(5)
where: tq,a = the time already spent by tbeo~, job in the queue for the 0 k operation.
3. Remaining processing time/current processing time P3.i ~ tr,~/t~
(6)
R,
tr, k&
E
j:k+l
(t/+tt/)=t, k
(7)
4, Size of next queue
P4,i~ l/l~ +1
(8)