differential evolution: a method for optimization of real … · 2000-08-16 · proj.-nr.: 10247....

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ABSTRACT

A new method for optimization of scheduling problems with nonlinear objective functions and multiple dependentrestrictions is presented. This method is based on an Evolutionary Algorithm but has a special changing operators fora leaded search over the entire solution space. It can be implemented for solving real problems very fast, it requiresonly few control variables, it is robust, easy to use and lends itself very well to parallel computation. The implementa-tion to solve a model representing a real scheduling problem in foundries is presented. This application shows goodresults and the comparison a method based on a stochastic Evolutionary Algortihm, having the reputation for beingvery powerful, shows that the new method converges faster and with more certainty.

1 FIR - Research Institute for Operations Management at Aachen University of Technology (RWTH), Pontdriesch 14/16, 52062 Aachen, Germany,e-mail: [email protected]. Research partially supported by the International Computer Science Institute, Berkeley, California and by the AiF,Proj.-Nr.: 10247.

Differential Evolution:A Method for Optimization of real

Scheduling Problems

Martin Rüttgers1

TR-97-013

March 1997

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PART I INTRODUCTION

Detailed planning as part of production planning processes is not very well supported by computer based methods.One reason for this fact is the complex structure of practical planning problems which are characterized by nonlinearobjective functions and multiple dependent and time varying restrictions. Scheduling problems represent an importantpart of such problems and are of a major importance for the improvement of production process. Many different mo-dels exist for solving these problems but very often they are based on unreal assumptions(DOMSCHKE/SCHOLL/VOSS 1993). So the solution of the models is possible but not of the real problem.

On the other hand, customer requirements force many companies to optimize their production processes to reduceproduction times and costs. For this reason the optimization of scheduling processes in complex environments is animportant field of research. The development of new methods is necessary which lead not only to optimized solutionsbut can also be adapted to real problems. These methods should be robust and easy to use.

The method of Differential Evolution considers these requirements (STORN/PRIECE 1995). Good optimization re-sults can be achieved with short computation times. In this paper the model for a real scheduling problem, the conceptof this method as well as first computation results are presented.

PART II APPLICABILITY OF EXISTING ALGORITHMS

In the field of Operations Research many heuristic methods have been developed to optimize scheduling processes. Anextensive overview of these methods is given in the work of DOMSCHKE/SCHOLL/VOSS (1993, p. 249ff). In thisbook the authors mention the efficiency of some of these methods for solving complex scheduling problems but alsoemphasize the douptfulness of aplying these methods on real problems. This opinion is stressed by NISSEN (1995, p.11ff). He gives two basic reasons for the afore mentioned problem:

• Most methods are based on special models not representing real situations and• most methods are inflexible.They cannot be used while the underlying model changes in time which is very often

necessary to represent real situations.

Because of these two facts, in the last years extensive research has been done in the area of Evolutionary Algorithms.These kinds of algorithms are by their nature very flexible in their application to different models. Many differenttypes of these algorithms have been developed and used for optimization of scheduling processes. All these types arebased on the same idea: A set of solutions is created and the solutions are changed in an iterative procedure by meansof special operators. In each iteration the objective function for each solution is calculated and only the best solutions(by means of the objective function) are taken into the next iteration. The change of the solutions is similar to muta-tion and replication procedures, and the decision for taking a solution to the next iteration similar to a selection proce-dure. These basic ideas show the vicinity to evolution process in nature.

Many different types of Evolutionary Algorithms are described by NISSEN (1995, p.13ff). The main difference be-tween them is based on the way the solutions are represented, the operators that are used for changing the solutionsfrom one iteration step to the next and the objective function (SCHOENBURG/HEINZMANN/FEDDERSEN 1994).

In the last few years the application of these algorithms to combinatorial optimization problems have shown theirefficiency (RECHENBERG 1973, p. 40ff, SCHWEFEL 1977). The work of MUSIER/EVANS (1989, p. 229ff),CHEN/VEMPATI/ALJABER (1996), LEE/KIM (1995), GRAHAM (et al. 1979), CHENG/SIN (1990) and NEP-PALLI/CHUEN-LUNG/GUPTA (1996, p. 356ff) give many promising results. Without going into details, we can saythat by the application of this algorithm to models representing real problems a new chapter of optimizing theory hasbegan (REEVES 1993). But mainly two facts could be seen as critical for applications to real problems and madefurther research necessary.

• The speed of convergence is slow in many applications so that the necessary time to achieve good solutions be-comes very large and

• many control parameters have to be adjusted to achieve good results which make the real use of the algorithmsquestionable.

From a theoretical point of view the first fact doesn’t lead to any problem because Evolutionary Algorithms lend itselfvery well to parallel computation which allows the reduction of processing times. This is different for real schedulingproblems: The computers being used in most real situations for the optimization don’t support parallel processing.

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Differential Evolution is a new kind of Evolutionary Algorithm which has been developed for optimization over con-tinuous spaces (STORN/PRIECE 1995). The application to this kind of problems has shown that it converges fasterand with more certainty than other methods. By its nature Differential Evolution is a flexible optimization procedurewhich can be used under multiple conditions. The implementation is easy and the method is easy to use. For this rea-sons further research is necessary to make the algorithm applicable to combinatorial optimization problems.

PART III CONCEPT

Differential Evolution represents an iterative solution procedure: A number of individuals are changed by specialchanging-operators and by the means of an objective function only the best individuals survive in the next generation.The main difference to existing Evolutionary Algorithms is the construction of the changing operator. The concept ofthe method is shown in Figure III-1.

First of all, the user configurates the system, i.e. he makes all the adjustments which are necessary to describe theexisting situation. After creating start solutions, the objective function for each individual is calculated and the solu-tion is compared to other solutions by means of the objective function. Only if the value of the objective function of thenew solution is better than this of the other solutions the individuals will be taken into the next generation. If the cri-teria for stopping the algorithm are not fulfilled, the individuals will be changed by means of a mutation and recombi-nation operator and the algorithm jumps back to the calculation of the objective function. At the end, the solution ispresented.

Configuration Start SolutionObjective Function

Change ofSolutions

Test of Solutions

StopPresentation of

Solutions

no

yes

Figure III-1: Concept of method

The mutation operator for changing the individuals is used in each generation and the recombination operator only ifa selected number of iterations no improvement of the best solution can be achieved.

The mutation operators are based on a rule which is not like the basic rules for other mutation operators only stochas-tic. By the concept of Differential Evolution the number of mutations depends on the "distance" between the individu-als. This rule leads to a search over the entire solution space which depends on the shape of the objective function. Ifthe individuals of one generation are spread over the whole solution space, then the individuals of the next generationwill be the same. But if the individuals are concentrated at a special point in solution space, then the individuals of thenext generation stay at this point until a solution somewhere else in solutions space can be found that has a bettervalue of the objecitve function. This leads to a precise search of these parts of solution space promising good results.

Consider an individual being represented by a matrix Aj with

A

a a

a a

j nj

k kl

=

=11 1l

1

1

�

� � �

�

with ,..., .

The columns of the matrix represent the different machines on which the jobs should be scheduled and each line rep-resents a minimum planning time unit ∆t. The variable j counts the individuals of one iteration.

To calculate the distance between the individuals in solution space the Hamilton Distance D(Ai,Ai’) is used. It can becalculated by counting the number of differences in two matrices Ai and Ai’ with i≠i’by

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D A Ai i k l k lalle ak l

( , )' , , ', '

,

= ∑Θ

withΘ Θk l k l kl k l k l k la a, , ', ' '' , , ', ',= = =1 0 if otherwise .

A new individual Anew is generated from an old Aold by putting ( )C D A Amut i i⋅ , ' stochastic changes on this individual.

Cmut>0 is a constant which can be adjusted by the user. The changes can be done in two ways. Either

( )C D A Amut i i⋅ , ' jobs are exchanged stochastically (then we talk about an exchange) or ( )C D A Amut i i⋅ , ' jobs are in-

verted (then we talk about an inversion).

In both cases the mutation procedure can be expressed by

( )A A C D A Anew old mut i i= + ⋅ , ' .

By the inversion a set of ( )C D A Amut i i⋅ , ' jobs is chosen stochastically and the sequence is inverted:

( ) ( )a a a a a a an m n m n m C D A A n m C D A A k lmut i j mut i j

1 1 1 2 1 1, , , , , , , , ,, , , , , , , , ,� �

� ��

�+ + ⋅ + ⋅ +

Inversion

.

The recombination operator on the other hand leads to a new solution by combining to solutions Ai and Ai’ with i ≠ i’.The operator separates the pool of jobs of each individual Ai and Ai’, into two groups X and Y by

a a a a a an m

X

n m n m k l

Y

1 1 1 2 1 2, , , , , ,, , , , , , ,��

��

+ +

.

All jobs from group X remain in the new solution Anew at the old position of solution Ai. The jobs of group Y are put onthis points in the new solution Anew which corresponds to the positions they had in solution Ai’.

PART IV REAL PROBLEM

1 Problem Formulation

The power of the new method is tested in its application to the scheduling problem of core blowers which is character-ized by a nonlinear objective function and many complex, dependent and time varying restrictions. The problem is ofmajor importance for foundries because core blowers are strongly connected to automatic molding plants with highproduction costs. If the cores are not produced early enough, then the molding plants must be stoped. On the otherhand cores grow older and lose quality if they are produced to early. The scheduling problem model for core blowersin foundries can be described by the following premises:

Premise 1: The core blowing process is a four stage process. The stages are core-blowing, core-assembling, core-sleeking and core-drying.

Premise 2: The sequence of process stages is not changeable.Premise 3: On each stage more than two parallel processors exist.Premise 4: Some of the stages can be skipped for special jobs.Premise 5: Not all processors can be reached bei each job.Premise 6: Each processor can processe only one job each time.Premise 7: Ressources of materials are not limited.Premise 8: Special jobs can only be processed on special processors.Premise 9: After each stage storage in a common depot with limited capacity is possible.Premise 10: Production times are processor and job dependent.Premise 11: Productions times are set at the beginning of scheduling process.Premise 12: Set-up times are processor and job dependent.Premise 13: Transportation times between the stages can be neglected.Premise 14: Number of necessary workers depend on the processores.Premise 15: Job-splitting is allowed.Premise 16: Job-lapping is allowed.

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Premise 17: Job-passing is allowed

The optimization of this model should be done on the job floor several times a day. In most cases there are more than15 parallel, heterogeneous processors and more than 200 jobs to be scheduled. In order to avoid interrupt of produc-tion process, the optimization should be done within a couple of minutes. Hardware configuration is based on commonPCs.

2 Module design

The configuration module consists of three major input-parts. In the first part the input of data describing the produc-tion situation is necessary:

• Setup of discirptive terms for each core blower: Set-up time, blowing frequency, necessary staff for blowing, pro-duction costs, set-up costs, machine load,

• capacities for core-blowing, core-assembling, core-sleeking, core-drying,• number of working days, number of shifts per day, shift-duration,• personnel cost,

In the second part, the input of data describing the different jobs is necessary:

• Type of finishing procedure (parallel or successive), rest time after finishing, due date, set-up time, blowing fre-quency, necessary storage capacity, assortment affiliation.

In the third part, the input of data describing the optimization process is necessary:

• Common Data: Type of initial solutions (stochastic, knowledge-based), number of changes in case of knowledge-based generation of initial solutions, smallest planning time, number of individuals per generation

• Mutation Operator: Changing degree (Cmut),• Recombination Operator: Criteria for initialization of recombination, number of recombinations per iteration,

With this input data all necessary information are collected to start the optimization procedure. After configuration ninitial solutions are generated. Depending on the configuration one of the following possibilities for generating thestart solutions can be used:

• Knowledge-based Generation: In this case the user himself creates the first schedule A1, i.e. he puts the jobs onspecial processors and determines the order of jobs on each processor. From this first start solution the other n-1solutions are generated stochastically. The number and the extend of this stochastic changes is given by the con-figuration.

• Stochastic Generation: In this case all n individuals Aj of the initial population are generated statistically, whichmeans that the user has no influence of putting the jobs to special processors and to determine their order. Alsohere the number and extend of the stochastic changes is given by the configuration of the system.

In both cases of generating the initial solutions only those individuals can be generated which do not produce anyconflicts with one of the restrictions. In the objective function the following aspects can be considered: Exceeding ofcapacities, deviation from due dates, production costs, set-up costs and deviation from assortment affiliation. Itsstructure is the following:

F a F a F a F a F a Ftotal Capacity Capacity Date Date Cost Cost Set Set Assort Assort= ⋅ + ⋅ + ⋅ + ⋅ + ⋅ .

The variables acapacity, aDate, aCost, aSet and aAssort are constant and take the rating of the different aspects into accountwhich can be changed by the user.

The variable FCapacity describes all exceedings of capacities from staff, storage-place, assembling, sleeking and drying.All this components where considered in the objective function. By calculating the number of planning units wherethe existing capacities does not satisfy the demand. If K w staff, storage assembling sleeking dryingw with = , , , are

these numbers then the total exceeding of capacity is

( )F C K CCapacity Capacity wall w

Capacity ww= ⋅ ∑ µ µ

mit = const., .

Die variable FDate describes the deviation from due date of all jobs in a schedule by calculating the total time differencebetween due dates and real finishing times:

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( )F C t t C j jDate Date due j real jj

Date= ⋅ − =∑ , , ,µ

µ with = const. , th job.

The variable FCost describes the production costs of the core blowers. If Ki are the production costs for the ith machineand if Θi(t)=1 only if the ith machine is occupied by a job, otherwise Θi(t)=0, then:

( )( )F with const., th machine.Cost = ⋅ ⋅ = =∑C K t C i iCost i ii

CostΘµ

µ,

The variable FSet describes the set-up costs. If Ri are the set-up costs for the ith machine per set-up process and if Zi

counts the number of different set-up processes on the ith machine, then:

( )F C R Z C i iSet Set i i Seti

= ⋅ ⋅ = =∑ µ µ with const., th machine.,

The variable FAssort describes the affiliation of all cores to different assortments by calculation the time differencesbetween the finishing times and due-dates of all cores of one assortment. If Sv are the different assortments, then:

( )F C t t C j jAssort Assort due j real jj Sv

Assort

v

= ⋅ − = =∈∑∑ , , ,

µµ with const., th job.

For changing the individuals the described mutation- and recombination-operator are used. The objective function isformulated in a way that it becomes minimized by the algorithm:

F Mintotal = .

By the means of this a new individual is taken into the next iteration, if the value of the objective function is reducedin comparison to the value of the old individual:

F A F Atotal old total nwu( ) ( )≥ .

The algorithm terminates if one of the following conditions is fulfilled:

• A number of generations is generated,• the reduction of the objective function over the last hundred generations is less than a minimum amount.

The results of an optimization procedure are presented in a Gantt-Diagram which is a graphical presentation of thematrix Ai of the individuals with the smallest value of the objective function. In addition to this, the values of consid-ered staff and machine capacities are shown over the entire planing period. So the user is warned of bottlenecks.

PART V RESULTS AND CONCLUSION

The described algorithm is tested on real problems in two different foundries. The results are compared with an Evo-lutionary Algorithm with a stochastic mutation operator and planning results of a manual planning procedure which isstandard in most foundries. For a small number of jobs the optimum solution of the problem can be calculated andhandled as a lower bound for the problem. The results are shown in Figure V-1

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Differential Evolutionðpð�ðpð�ð�ð Stochastic Evolutionñpñ�ñpñ�ñ�ñ

Optimumòpò�òpò�ò�ò

Manual

Figure V-1: Optimization results (5 machines, 30 jobs)

The results show that the new method converges faster than a normal Evolutionary Algorithm with stochastic basedchanging operator. We found out that the solution of manual planning process can be improved from 37000 to about5000. For the problem, the lower bound can be calculated as 4700. After 17000 iterations which corresponds to acomputation time of about 10 minutes on a pentium processor the method finds its best solution which is close to thelower bound.

In addition to this, the personnel in foundries had no problems in using the algorithm instead of manual planning. Themost important reason for this is the flexibility of the algorithm which makes adaptation to changing situations be-cause of new jobs or machine breakdown very easy.

PART VI REFERENCES

CHEN, C.;VEMPATI, V.S.;ALJABER, N.:

An application of Genetic Algorithms for the flow shop problems.In: European Journal of Operational Research, 80 (1995), S. 389-396.

CHENG, T.C.E.;SIN, C.C.S.:

A state-of-the-art review of parallel-machine scheduling research.In: European Journal of Operational Research, 47 (1990), S. 271-292.

DOMSCHKE, W.;SCHOLL, A.;VOß, S.:

Produktionsplanung.Ablauforganisatorische Aspekte.Springer, Heidelberg 1993.

GRAHAM, R.L.;LAWLER, E.L.;LENSTRA, J.K.;RINNOOY KAN, A.H.G.:

Optimization and approximation in deterministic sequencing and scheduling: Asurvey.In: Annals of Discrete Mathematics, 5 (1979), S. 287-326.

LEE, C.; Parallel Genetic Algorithms for the earlyness-tardiness job scheduling problem withgeneral Penalty Weights.

8

KIM, S.: In: Computers Industrial Engineering, 28 (1995) 2, S. 231-243.

MUSIER, R.F.H.;EVANS, L.B.:

An approximate method for the production scheduling of industrial batch processeswith Units.In: Computers Chemical Engineering, 13 (1989) 1/2, S. 229-238.

NEPPALLI, V.R.;CHEN, C-L.;GUPTA, J.N.D.:

Genetic Algorithms for the two-stage bicriteria flowshop problem.In: European Journal of Operational Research, 95 (1996), S. 356-373.

NISSEN, V.: Evolutionäre Algorithmen.Darstellung, Beispiele, betriebswirtschaftliche Anwendungsmöglichkeiten.Deutscher Universitätsverlag, Wiesbaden 1994.

RECHENBERG, I.: Evolutionsstrategie.Frommann-Holzboog, Stuttgart 1973.

REEVES, C.R.: Modern heuristic techniques for combinatorial problems.John Wiley & Sons, Oxford 1993.

SCHÖNEBURG, E.;HEINZMANN, F.;FEDDERSEN, S.:

Genetische Algorithmen and Evolutionsstrategien.Eine Einführung in Theorie and Praxis der simulierten Evolution.Addisson-Wesley, Bonn, Paris 1994.

SCHWEFEL, H.-P.: Numerische Optimierung von Computer-Modellen mittels der Evolutionsstrategie.Birkhäuser Verlag, Basel, Stuttgart 1977.

STORN;PRIECE:

Differential Evolution - A simple and efficient adaptive scheme for global optimiza-tion over continuos spaces.International Computer Science Institute 1995,Technical ReportTR-95-012.

differential evolution: a method for optimization of real … · 2000-08-16 · proj.-nr.: 10247....

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