Heuristics for hybrid flow shops with controllable processing times and assignable due dates

*Corresponding author. Tel.: #1-765-285-5301; fax: 765-285-8024.E-mail address: [email protected] (J.N.D Gupta).

Computers & Operations Research 29 (2002) 1417}1439

Heuristics for hybrid #ow shops with controllable processingtimes and assignable due dates

Jatinder N.D. Gupta��*, Karin KruK ger�, Volker Lau!�, Frank Werner�,Yuri N. Sotskov�

�Department of Management, Ball State University, Muncie, IN 47306, USA�Otto-von-Guericke-Universita( t, Fakulta( t fu( r Mathematik, PSF 4120, 39016 Magdeburg, Germany

�Institute of Engineering Cybernetics, Surganov St. 6, 220012 Minsk, Belarus

Received 1 July 2000; received in revised form 1 January 2001

Abstract

This paper considers a generalization of the permutation #ow shop problem that combines the schedulingfunction with the planning stage. In this problem, each work center consists of parallel identical machines.Each job has a di!erent release date and consists of ordered operations that have to be processed onmachines from di!erent machine centers in the same order. In addition, the processing times of theoperations on some machines may vary between a minimum and a maximum value depending on the use ofa continuously divisible resource. We consider a nonregular optimization criterion based on due dates whichare not a priori given but can be "xed by a decision-maker. A due date assignment cost is included into theobjective function. For this type of problems, we generalize well-known approaches for the heuristic solutionof classical problems and propose constructive algorithms based on job insertion techniques and iterativealgorithms based on local search. For the latter, we deal with the design of appropriate neighborhoods to "ndbetter quality solution. Computational results for problems with up to 20 jobs and 10 machine centers aregiven.

Scope and purpose

Traditional research to solve multi-stage scheduling problems has focused on regular measures ofperformance based on a single criterion and assumes that several decisions related to due dates andprocessing times have already been made. However, in many industrial scheduling practices, managersdevelop schedules based on multicriteria and have to decide the due dates and processing times as part of thescheduling activities. Further, in practical scheduling situations, there are multiple machines at each stageand the objective function often re#ects the total cost of processing, earliness and tardiness. Such scheduling

0305-0548/02/$ - see front matter � 2002 Elsevier Science Ltd. All rights reserved.PII: S 0 3 0 5 - 0 5 4 8 ( 0 1 ) 0 0 0 4 0 - 5

problems require signi"cantly more e!ort in "nding acceptable solutions and hence have not received muchattention in the literature. For this reason, this paper considers one such hybrid #ow shop schedulingproblem involving nonregular measures of performance, controllable processing times, and assignable duedates. We combine and generalize the existing approaches for the classical #ow shop problem to the problemunder consideration. Computational experiments are used to evaluate the utility of the proposed algorithmsfor the generalized scheduling problems. Brah and Hunsucker (European Journal of Operational Research1991;51:88}99) and Nowicki and Smutnicki (European Journal of Operational Research 1998;106:226}253)describe branch and bound and tabu search algorithms for the approach used in the development of heuristicalgorithms can also be adapted to several other complex practical scheduling problems. � 2002 ElsevierScience Ltd. All rights reserved.

Keywords: Scheduling; Hybrid #ow shop; Controllable processing times; Assignable due dates; Insertion heuristics;Local search

1. Introduction

Scheduling may be de"ned as the allocation of resources over time to process jobs. Theimportance of good scheduling strategies in production environments is obvious. The need torespond to market demands quickly and to run plants e$ciently gives rise to complex schedulingproblems in all but the simplest production environments. However, a large gap exists betweenscheduling theory and practice. MacCarthy and Liu [1] note the failure of classical schedulingtheory to address the total environment within which the scheduling function operates. Schedulingresearch tends to ignore the rich vein of problem environments, methods, and techniques.

One of the classical problems considered in the scheduling literature is the permutation #owshop problem [2], where a set of n jobs is to be processed on a set of m machines. Each job passesthrough the machines in the same order, and on each machine the same job order has to be chosen.Typically the processing time of each operation, which represents the processing of a job ona machine, is given and "xed. The objective is to minimize a regular measure of performance de"nedas a function that is nondecreasing in the completion times of the jobs. Minimization of makespanis the most common optimality criterion considered in scheduling literature. Unfortunately, theabove problem has rather restrictive assumptions to be applied in connection with productionscheduling problems [3].

Inspired by our experience and the description of several practical industrial scheduling situ-ations by Morton and Pentico [4], we consider the following generalization of the permutation#ow shop problem. A set of jobs has to be processed on a set of machine (work) centers, whereeach machine center consists of a set of identical parallel machines, and the non-preemptiveprocessing of a job has to be done on exactly one of the machines of each center. Analogously to theclassical #ow shop problem described above, the routes of the jobs through the machine centers arethe same. The processing times are not necessarily "xed in advance but can be controlleddepending on the use of a continuously divisible resource. Additionally, a release date is given foreach job which is the earliest possible starting time for processing this job. Due dates have to beassigned to each job, and a particular due date assignment cost is a component of the objectivefunction. Thus, the optimality criterion consists of several terms, including earliness and tardiness

1418 J.N.D Gupta et al. / Computers & Operations Research 29 (2002) 1417}1439

penalties. Hence, the problem considered in this paper involves a nonregular measure of perfor-mance implying that the objective function is not necessarily increasing in the completion times ofthe jobs.

Developments in solving the classical scheduling problems have been reviewed by Chen et al.[5]. For optimization criteria other than makespan, branch and bound procedures and heuristicalgorithms for the two-machine #ow shop problems have been developed (see Ref. [5] for details).An overview of scheduling problems with multiple optimization criteria has been given by T'Kindtand Billaut [6]. Several other extensions considered in the literature are rather specializedproblems in contrast to the general ones presented above. For instance, Brah and Hunsucker [7]describe a branch and bound algorithm for the #ow shop problem with machine centers. A frame-work for scheduling problems with controllable processing times has been given in Alidaee andKochenberger [8]. The permutation #ow shop problem with controllable processing times hasbeen considered by Janiak [9] for which a genetic algorithm to minimize makespan has beenproposed by Janiak and Portmann [10]. Moreover, a pseudopolynomial algorithm has beenpresented for determining the optimal resource allocation for a "xed job sequence when minimiz-ing the makespan. A survey of scheduling research involving due date decisions by Cheng andGupta [11] shows that problems with assignable due dates have been considered mainly inconnection with single-stage processing systems and in most cases a common due date has to bedetermined.

This paper generalizes some existing heuristic solution approaches for the classical permutation#ow shop problem to solve the above generalized problem. In particular, we deal with insertionalgorithms to construct a feasible solution. We also discuss iterative algorithms for improvinga known initial solution. In this connection, we mainly focus on the design of appropriateneighborhoods based on the shift neighborhood for classical scheduling problems. The latter twotypes of algorithms have been chosen due to their good performance for the classical problems[12].

The remainder of the paper is organized as follows. After describing the problem in detail inSection 2, we present several constructive algorithms and illustrate them with a numerical examplein Section 3. In Section 4, we then suggest iterative algorithms that are based on the concept of localsearch techniques. We empirically evaluate and discuss the relative performance of the proposedheuristic algorithms in Section 5. Since no exact or heuristic algorithms for the problem underconsideration are available, and the derivation of lower bounds for the objective function value inthis general case would not yield tight bounds for the evaluation of heuristic algorithms, Section5 compares di!erent variants of the proposed algorithms relative to each other. The main aim ofthe computational tests is to "nd those parameters of our algorithms that substantially in#uencethe quality of the results. Finally, Section 6 concludes the paper with a summary of our "ndings andsome fruitful directions for future research in this area.

2. Problem description

A given set N"�1, 2,2, n� of n jobs is to be processed consecutively on m machine centersM

�,M

�,2,M

�in that order. Each machine center consists either of a single machine (we have

a #ow shop problem in this case) or it consists of a given number of identical parallel machines

J.N.D Gupta et al. / Computers & Operations Research 29 (2002) 1417}1439 1419

(hybrid or #exible #ow shop). For each job i, a nonnegative release date r�

is given which describesthe earliest possible beginning of the processing of this job. The processing of a job i consists ofa sequence of operations O

��,O

��,2,O

��, where O

��represents the processing of job i on a machine

of center M�

during an uninterrupted processing time t��

. For each machine center of a subsetM-�M

�,M

�,2,M

�� and each job i of a certain subset N�-N, the processing time t

��is

assumed to be a linear function of the amount of resource allocated to operation O��

:

t��

"b��

!a��u��

'0,

where parameters a��

'0 and b��

'0 are known before scheduling and u��

is the amount ofresource allocated to operation O

��within the construction of a schedule. The amounts of resource

described by vector u�"(u

��, u

��,2, u

��) are feasible if they satisfy the following constraints:

0)u��

)��

, i"1, 2,2, n,

��M

��

u��

);H,

where ;H is the global amount of resource allocated to the realization of the operations that haveto be processed by a machine center of M, and �

��are given technological constraints on the

maximal amount of the resource allocated to operation O��

, respectively. It is assumed that theinequalities

0);H) ��M

��

��

; 0)��

(b��

/a��

hold. The processing times of the remaining operations are assumed to be equal to given constants.We denote the processing times which can be "xed during the scheduling procedure as controllableprocessing times.

If due dates are to be assigned within the scheduling procedure (i.e. internally by the scheduler aseach job arrives on the basis of job characteristics, shop status information and an estimate of thejob #ow time), this is usually done by one of the following procedures [11]:

(a) d�"r

�#¹I ) ��

��t��

(Total Work Content, TWK);(b) d

�"r

�#¹I ) n

�(Number of Operations, NOP) or

(c) d�"r

�#¹I#��

��t��

(Equal Slack, SLK).

Here ¹I is a parameter known as tightness parameter. The above formulas assign the job a due date,on the basis of the workload of a job in di!erent ways. Small tightness values make it more di$cultfor the jobs to be completed by their due dates. However, larger values of the tightness parameterlead to larger due dates of the jobs but, as we will see later, also to higher &penalties' in the objectivefunction. In this paper, we set the due dates according to the TWK rule which has also been appliedin many other papers (see e.g. [13]). The measure of performance used in this paper contains severalterms based on the assigned due date of each job and can be described as follows:

F"

��

u�E�#

��

v�¹

�#

��

w�C

�#

��

z�d�, (1)

where the earliness of job J�

is de"ned as E�"max�0, d

�!C

��, the tardiness of job J

�is de"ned as

¹�"max�0,C

�!d

��, and C

�denotes the completion time of job i. Moreover, u

�, v

�,w

�and z

�are


the weights associated with job i. The above performance measure is rather general and belongs tothe so-called nonregular criteria, where the objective function is not necessarily increasing in thecompletion times of the jobs. Such due date based criteria are typical for just-in-time production,where jobs that complete early must be held in "nished goods inventory until their due dates, whilejobs that complete after their due dates may cause a customer to shut down operations. Wemention that if u

�*w

�holds for all jobs i, 1)i)n, the third term of the performance function

F could be eliminated by modifying the weights of the earliness and tardiness penalties suitably. Anoverview of such scheduling problems with earliness and tardiness penalties is provided by Bakerand Scudder [14].

Designated as a hybrid yow shop problem with controllable processing times and assignable duedates, the problem considered in this paper is

To xx the controllable processing times by allocating a certain amount of resource, to determine theassignable due date for each job by xxing the tightness value, to assign each job to exactly one of themachines from each machine center and to determine the starting (or completion) times of all jobs onall machine centers where each job is processed in each machine center in the same order such thatthe objective function (1) is minimized.

A solution to the above problem requires us to "x the starting times of the individual operations.However, in order to compute these times, we have to assign the operations to a particular machineof the corresponding center, to "x the processing times of the operations (as they are controllableprocessing times), and to decide where idle times have to be inserted. The objective functionvalue of a schedule can only be computed if the tightness value ¹I has been "xed to compute thedue dates. For simplicity of notations, we use the symbol S for representing a schedule keeping inmind that the designation of a schedule implies that the decisions mentioned above have alreadybeen made.

Several exact optimization and heuristic algorithms are available for classical schedulingproblems with rather restrictive idealizations. However, they cannot immediately be applied to therather general problem under consideration due to several reasons including the following:

� The majority of existing papers on scheduling problems deal with a regular criterion.� The processing times are usually assumed to be given constants.� The machine required to process each operation is prede"ned.� Due dates, even if present, are speci"ed in advance.� A feasible solution is described either by one job sequence for all machines (permutation #ow

shop problem) or a particular job sequence for each machine (#ow or job shop problem).

In order to solve the generalized #ow shop problem considered in this paper, therefore, weextend the existing heuristic algorithms and test their relative e!ectiveness in minimizing thefunction F given by Eq. (1).

3. Constructive algorithms

The constructive algorithms complete a partial sequence either by appending a job (use ofdispatching or priority rules [15]) or inserting a job or operation into a partial solution (see e.g.


[12,16]). In many cases, with a moderate additional computational expense, insertion algorithmsare superior to the priority rules [12,16]. For this reason, we now propose insertion basedconstructive heuristic algorithms and discuss the use and control of various algorithm parameters.The working of the proposed insertion algorithm is illustrated by solving a numerical example.

3.1. Determination of initial processing times

In the insertion algorithm, we start with "xed processing times of all operations including thosewith controllable processing times based on some initial resource allocation. We suggest andcompare the following two resource allocation schemes to control the job processing times:

IR1: allocate the same amount of the resource to all operations with a controllable processing time;and

IR2: allocate the resource to all operations with a controllable processing time such that for alloperations the maximum duration of the processing time is reduced by the same amount(except possibly those where the minimum duration of the processing has already beenreached).

Initially we allocate only a certain percentage of the resource to the operations with a controllableprocessing time which is given by parameter IRA (&initial resource allocation').

3.2. Procedure schedule

Our insertion algorithm operates with a (partial) sequence of jobs that can be transformed intoa complete schedule as follows. If a job has to be processed on a machine center consisting of morethan one machine, the job is processed on the machine that becomes free "rst. Note that thepermutation order is never violated during the construction of the schedule, i.e. for the job sequencep"(p

�, p

�,2, p

�) we have s

�(p

�))s

�(p

�) for i(k, where s

�(p

�) denotes the starting time of job p

�on

the chosen machine of center M�.

For a job that "nishes its last operation before its due date, we shift its last operation to the right(i.e. towards its due date if this improves the objective function value). This may lead to an insertedidle time. Conversely, from the schedule S so constructed, an underlying job sequence p can also bedetermined. We use this procedure when adding a partial sequence p� to an existing schedule S, andwe denote the above construction of the schedule for the jobs of p� as Schedule(S, p+). Thisprocedure neither changes the current processing times nor the current tightness value fordetermining the due dates.

3.3. Procedure construct

The procedure Construct works with a "xed job sequence but may change speci"c controllableprocessing times and the tightness value for determining the due dates if this leads to animprovement of the objective function value. Given a job sequence p"(p

�, p

�,2, p

�), the proced-

ure Construct(S, p) starts with the "rst job p�

and schedules this job using the current processingtimes and determines its completion time C

��. If job p

�is early, we try to increase its completion

time if it will improve the objective function value. It is done by considering only the last operation


of job p�. The current controllable processing time of this operation is either enlarged by releasing

allocated resource (if possible) or it is shifted to the right if this leads to a better objective functionvalue of the current partial schedule (as in procedure Schedule described above). The latter step isperformed in such a way that the completion time of job p

�does not exceed its due date d

��. For

a late job, we try to decrease its completion time as follows. We consider the operations withcontrollable processing times of the last two jobs in the current (partial) sequence. Starting from thelast operation of the last job, we successively allocate additional resource to those operations ofthe job that lead to a decrease in the completion time of this job. This process stops if no furtherresource can be used or the completion time of this job has reached its due date. As long as thecompletion time of the current job is still larger than its due date, we continue with the opera-tions of this job and then with the operations of the currently last job but one (except possiblythe last operation of this job which remains unchanged since it would directly in#uence thecompletion time of the job previously scheduled). Note that the last operation of a job does notchange its completion time even if the processing time of its nonlast operation can be shortened. Inother words, changes in the processing times are only considered if they lead to a decrease in theobjective function value of the current partial solution caused by the completion time of job p

�.

Next, if for the resulting schedule S� inequality C��

!d��

*B holds, i.e. the completion timeexceeds the due date of this job by a given speci"c bound B, we check whether the change of thetightness parameter ¹I� :"¹I for determining the due dates may improve the objective functionvalue of the current partial schedule. To this end, we successively consider tightness values of theform ¹I�"¹I�#�¹I ) k, where k takes consecutive positive integers 1, 2 and so on. These valuesare considered as long as they improve the objective function value of schedule S�. Then weconsider the second job p

�in sequence p and continue until all jobs of p have been scheduled. In

each step, the procedure of increasing value ¹I is applied, if any of the jobs p�of p considered so far

satis"es the inequality C��

!d��

*B. In our tests, we applied B"2.

3.4. Insertion orders and positions

The insertion order of the jobs has been shown to have a substantial in#uence on the quality ofthe results [12]. In our tests, we compare the following four insertion orders:

O1: insert the jobs according to nondecreasing due dates (initially determined by the startingvalue of parameter ¹I);

O2: insert the jobs according to nondecreasing release dates;O3: insert the jobs according to nonincreasing sums of the weights (i.e. u

�#v

�#w

�#z

�) and

O4: insert the jobs in random order.

For the problem considered in this paper, insertion orders O1}O3 have been chosen since therelease and due dates and the job weights are expected to signi"cantly in#uence the quality of thesolution. For minimizing the makespan, all sequence positions are often candidates for insertinga given job. However, minimizing a nonregular measure of performance like the earliness andtardiness penalties, all sequence positions may not be suitable candidates to insert a given job. Forinstance, a job with a large due date is not expected to be a candidate for the "rst positions in thesequence. Therefore, we apply a restricted insertion algorithm, where the notation Ins(k) means thatonly the last k positions in the current sequence are considered for inserting a job. The variant


where all positions are considered for inserting the current job is denoted as Ins(all). If only the lastposition is considered for inserting a job, the application of order O1 corresponds to the applicationof the earliest due date rule, where in each step a job is appended at the end of the current partialsequence (variant Append).

3.5. The insertion algorithm

Among the partial sequences considered in each step when inserting a job, we choose theone with the best objective function value. Then procedure Construct is only applied to thelast job in the current partial sequence, where we use a steplength of �¹I"0.1 for changingparameter ¹I.

The insertion procedure works in a greedy manner. First the jobs are assigned to the machinesin a &best possible' way without violating the permutation sequence. After having chosen apartial sequence, changes in the processing times and the tightness parameter are acceptedonly if they improve the objective function value. Algorithm Ins(k) can be summarized asfollows.Algorithm Ins(k)1. determine the initial resource allocation for all operations with controllable

processing times and the resulting operation durations;2. de"ne the insertion order IO :"(i

�, i�,2, i

�) as an ordered job sequence;

3. l :"0; p :"�; "x the initial tightness value ¹I;While IOO� DoBegin

4. select the "rst job i of the ordered sequence IO and remove i from IO;5. FH :"R; kH :"min�k, l#1�;

For j :"1 To kH DoBegin

6. u :"l!kH#j#1;7. insert job i at position u in sequence p and let

p�"(p�,2, p

��, i, p

,2, p

) be the resulting sequence,

S� be the current schedule of the jobs p�, p

�,2, p

��and p�"(i, p

,2, p

);

8. determine S� by applying procedure Schedule(S �, p�);9. If F(S�)(FH Then

Begin FH :"F(S�); jH :"j End9. End;

10. l:"l#1;11. insert in sequence p job i at position jH and denote the resulting

permutation by p"(pH, p) and let SH be the current schedule of pH;

12. apply procedure Construct(SH,p) and let S� be the resulting schedule

and ¹I again be the tightness value obtained;End;

13. S� together with value ¹I is the generated heuristic solution.


Table 1The data for the example

Job r�

t��

t��

a��

t��

u�

v�

w�

z�

1 4.4 9.2 7.9 4.4 3.1 7.2 9.9 0.8 0.52 15.6 4.7 4.0 3.9 8.0 6.8 2.7 0.6 0.73 6.2 6.5 6.5 } 8.6 7.3 1.9 0.2 0.34 14.3 8.1 6.6 3.3 9.9 9.5 2.9 0.5 0.55 11.5 3.0 2.6 4.5 9.1 2.9 2.1 0.3 0.9

3.6. A numerical illustration

We consider a small example to illustrate the constructive insertion algorithm. For the sake ofsimplicity, since the processing times are assumed to be real, we operate here only with real datahaving one decimal place (except the amount of resource used).

Let n"5 and m"2. Moreover, assume that machine center M�

consists of two machines andthat jobs 1, 2, 4 and 5 have a controllable processing time on M

�. Let the global amount of resource

be ;"0.52, and we consider the minimization of function (1). The remaining data are given inTable 1, where for the operations with controllable processing times the maximal processing timet��

, the minimum processing time t��

and the slope a��

are given. We apply Algorithm Ins(2), usebound B"2, a steplength �¹I"0.1 for changing value ¹I, and we start with the initial resourceallocation IRA"0.5. This leads to the processing times t

��"8.9, t

��"4.5, t

��"7.9, t

�"2.7.

The amount of resource used is ;�"0.247 (notice that ;�O0.5; since, due to our simpli"cationabove, processing times are only considered with one decimal place).

Starting with the initial value ¹I"1.4, we get the due dates d�"21.2,d

�"33.0,

d"27.3, d

�"39.3 and d

"28.1. We start with job 1 and then insert job 3. Considering the

partial sequence p�"(3, 1) we get C"21.3 and shift job 3 on the machine of M

�towards its due

date which yields a contribution of job 3 to the objective function value of 13.6. We now obtainC

�"24.7 and thus F(S�)"78.6 (here and in the following, we denote by S� always the current

schedule of sequence p�). When considering the partial sequence p�"(1, 3), we "rst get C�"16.4.

Since C�(d

�, we shift job 1 towards its due date which reduces the contribution of job 1 from 58.3

to 27.6. Sequencing now job 3 yields F(S�)"43.6. Consequently, we select p�"(1, 3) and applyprocedure Construct. This decreases the processing time of job 1 on M

�to 7.9 which leads to

C"27.4 and F(S�)"41.5. The tightness value ¹I"1.4 is not enlarged since no completion time

deviates from the due date by more than bound B"2.Now we insert job 5 at positions 2 and 3 which yields F(S)"80.8 for sequence p"(1, 5, 3)

(after shifting job 2 towards its due date) and F(S�)"81.2 for sequence p�"(1, 3, 5). Thus p�"(1,5, 3) is chosen. The application of procedure Construct reduces C

to 30.0 and thus we get

F(S�)"80.6. The tightness value ¹I remains unchanged since for ¹I"1.5, we get F(S�)"91.8.Next, we insert job 2 at positions 3 and 4. For sequence p"(1, 5, 2, 3), we obtain "rst C

�"29.2,

shift job 2 on machine M�

towards its due date, and then we get C"36.7 which yields

F(S)"137.6. Considering sequence p�"(1, 5, 3, 2), we get F(S�)"133.7. Thus, we select p�. The


application of procedure Construct does not decrease the processing time t��

since job 2 hasa waiting time and thus the completion time would not be reduced by allocating additionalresource to job 2 (notice that job 3 has no controllable processing time). Also, the increase of thetightness value by 0.1 is not accepted since for ¹I"1.5, we obtain F(S)"144.7.

Finally, we insert job 4 at positions 4 and 5. If we consider p�"(1, 5, 3, 4, 2), we getF(S�)"191.8. For sequence p "(1, 5, 3, 2, 4) we get F(S )"119.9. Thus, we select the lattersequence p . The application of procedure Construct cannot reduce the completion time C

�since

decreasing the processing time t��

by 0.1 would require an amount 0.030 of the resource anddecreasing t

��by 0.1 would require an amount of 0.027 but only the amount 0.024 is still available

and, increasing the tightness value by 0.1, we obtain for ¹I"1.5 the function value F(S )"192.7.Therefore, schedule S together with ¹I"1.4 is the heuristic solution obtained.

4. Iterative algorithms

Iterative algorithms based on local search apply some neighborhood structure to improve thecurrent starting solution iteratively. To avoid stagnation in a local optimum, often a metaheuristic(e.g. simulated annealing, threshold accepting or tabu search) is included to guide the search toglobally better solutions. For scheduling problems, where a complete schedule is described by oneor several job sequences, typical neighborhoods allow small changes in one or a few job sequences.For instance, if one job sequence describes a schedule, two adjacent jobs can be interchanged (APIneighborhood), or a job from an arbitrary position may be shifted to another position (shift orreinsert neighborhood), or two arbitrary jobs in the sequence can be interchanged (pairwiseinterchange or swap neighborhood). Whereas in the "rst neighborhood a solution has only O(n)neighbors, the application of the latter two neighborhoods usually produces better results sincea solution has O(n�) neighbors and thus early stagnation in a local optimum is more often avoided.In this section, we "rst discuss some basic moves for generating a neighbor and then describe thecomposite neighborhoods used in our iterative algorithms.

4.1. Basic neighborhoods

For the problem under consideration, more general neighborhood de"nitions are required sincewe have to consider changes in the use of the resource (and thus in the resulting processing times),in the assigned due dates (by modifying the ¹I value), in the distribution of the jobs to the machinesof a work center, or in the inserted idle times between the operations since the optimizationcriterion is nonregular. In this paper, we only focus on generalizations of the shift neighborhood.Similar investigations can be carried out with generalizations of the pairwise interchange neighbor-hood. In the following, we suggest and investigate three types of moves that perform changes in thecurrent solution in di!erent ways. In all moves, initially the value of ¹I is changed since leaving theinitial value ¹I unchanged produces bad solutions. First, we introduce two types of moves thatmake changes in the job sequence which may lead to changes in other features of the solution.

The "rst type of a move that we consider is a change in the tightness value together with a shift ofa job within the current sequence. Applying the procedure Schedule described before, we candetermine a schedule (which is uniquely determined except ties in the earliest starting times of themachines of each center). Let S be the current schedule and ¹I be the current tightness value.


� Move I*Shift: First we determine the new tightness value ¹I� by choosing randomly a valuefrom the interval [¹I!�

�, ¹I#�

�]. A probability parameter prdec gives the probability that

the tightness value is decreased, i.e. a value ¹I�3[¹I!��, ¹I] is chosen. Having ¹I� deter-

mined, we select in the permutation p"(p�, p

�,2, p

�) corresponding to the current schedule

S an arbitrary job p�

with C��

Od��

and shift it to position j, jOi. If the selected job is early(C

��(d

��), it will be shifted to the right if this job is late (C

��'d

��) it will be shifted to the left.

Let p� be the resulting job sequence, S be the current schedule of the "rst r!1 jobs in p� andp�"(p�

�, p�

��,2, p�

�), where r"min�i, j� is the smallest position with di!erent jobs in se-

quences p and p�. Then the jobs of p� are rescheduled by applying procedure Schedule(S, p�). Sucha move changes the job sequence and possibly the distribution of the jobs to the machines ofa machine center (this is possible only for the jobs from p�). However, when rescheduling the jobsof p� neither the current processing times nor the current tightness value are changed.

For the problem considered, one may expect that shifts of a job by a large number of positionsoften do not generate a neighbor with a good objective function value. This leads to the suggestionto restrict the maximal number of positions by which a job may be shifted. The variant Shift(k)allows shifts of a job by at most k positions in the current job sequence.

The following move is also based on a shift of a job, however, contrary to move I, the tightnessparameter may be changed not only at the beginning, but also during the &rescheduling' of the jobsafter performing the shift. For instance, if the ¹I value was initially decreased too much, this movewill correct (at least partially) the ¹I value during the reconstruction of the solution. Also someprocessing times could possibly be changed by procedure Construct.

� Move II*Shift/reconstruction: The "rst steps are the same as in move I, i.e. we determine the newtightness value ¹I� and in the permutation p"(p

�, p

�,2, p

�), we perform a left shift of a late job

or a right shift of an early job. As in move I, we can restrict the maximal number of positions bywhich a job may be shifted. Let p� be the resulting permutation, r be the smallest position withdi!erent jobs in sequences p and p�, S be the partial schedule for the "rst r!1 jobs in p� andp�"(p�

�, p�

��,2, p�

�).

After shifting the selected job, we apply procedure Construct(S, p�) to complete the partialschedule S. In our experiments, we tested several steplengths �¹I. Schedule S together with the"nal value of ¹I� is the generated neighbor in move II. To avoid such an &in#exible' increase ofvalue ¹I�, we also consider a re"ned version, where from the last accepted value ¹I�, anothervalue, obtained as a random number from the interval (¹I�!�¹I/2, ¹I�#�¹I/2), is con-sidered and the better one is taken for the further steps. The latter procedure is also applied in thecase k"0 (no increase of parameter ¹I has been accepted) or if C

��!d

��(B holds for job

p�

currently considered (an increase of parameter ¹I is not tested). We denote this re"nement as&random-TI'.

The following move changes the tightness value and, as a consequence, possibly the processingtimes of some operations. The job sequence underlying the current schedule will not be changed.

� Move III*Tightness/reallocation: Having ¹I� determined, we calculate the new due dates of thejobs. If no late job has a larger contribution to the objective function value (i.e. the sum of all four


terms resulting from this job) than in the starting solution S with the old ¹I value, the neighborgenerated by move III has been obtained. Otherwise we randomly select a late job J

�with

a larger contribution to the objective function value than in S. Next, we try to reduce thecontribution of this job to the objective function value by allocating additional resource to anoperation of this or the previous job in the current sequence in a similar way as it has beendescribed for procedure Construct(S,p�). If the available resource amount is exceeded by sucha change, we must remove allocated resource from randomly selected operations as long as;H isnot exceeded which yields schedule S�. If F(S�)(F(S), we continue with S� (together with ¹I�),otherwise we continue with S (together with ¹I). Then we select another late job with a largercontribution to the objective function value than in schedule S and repeat the above procedure.The step of selecting a late job is performed at most three times when generating move III. Theschedule SH together with the "nal value ¹I� is the neighbor generated by move III.

4.2. Composite neighborhoods and local search

Recent investigations [13] demonstrate the superiority of composite neighborhoods, whereseveral means of generating a neighbor are considered and the application of the individual meansis controlled by probability parameters. Such composite neighborhoods allow a higher #exibility inthe generation of neighbors, and the use of probabilities permits us to generate certain promisingtypes of neighbors more often without ignoring alternative types of neighbor generation. Using themoves de"ned above, we consider the following two types of composite neighborhoods for theproblem under consideration, where the set of neighbors N(S) of a feasible schedule S is de"ned asfollows:

� neighborhood A: N(S)"�S� � S� is obtained from S by a move I or a move III�;� neighborhood B: N(S)"�S� � S� is obtained from S by a move II or a move III�.

The frequency of performing a speci"c move in one of the above neighborhoods is controlled bya parameter prob which gives the probability of applying move III in the above neighborhoodsA and B. Thus, a (pseudo)random number x from the interval (0,1) is generated and if x(prob,then we generate a neighbor by move III in neighborhoods A and B, respectively. We emphasizethat, in contrast to classical scheduling problems, there are in"nitely many neighbors of the currentschedule.

5. Computational results

We now report the results of empirical tests to evaluate the relative e!ectiveness of the proposedalgorithms in minimizing objective function (1). As stated earlier, no exact optimization or heuristicalgorithms are available for the problem considered here. Further, any lower bounds on theoptimal function value generalized from the known bounds for the classical scheduling problemsare expected to be very poor. Therefore, we compare the proposed algorithms relative to eachother. In these comparisons, our main objective is to "nd the preferred parameter settings and thefeatures of the algorithms that have a strong in#uence on the solution quality. We hope that these


investigations and experiments will stimulate further research for general scheduling problemsfound in practical production scheduling environments.

All algorithms have been implemented in FORTRAN 90 and run on a Pentium PC (133 MHz).We "rst describe the generation of the test problems. Then we present some computational resultsabout the relative performance of various constructive and iterative algorithms.

5.1. Test problems

A large number of test problems were generated and used in the empirical experiments. In thissection, we present the results for only a few representative problem classes. The results for theother problem classes not reported here were not substantially di!erent than the results presentedbelow.

The "xed processing times are uniformly distributed real numbers from the interval (1,100).Controllable processing times are only possible when the corresponding machine center consists ofonly one machine and is a &bottleneck' in the entire production process. In the case of constantprocessing times, the number of parallel identical machines per center is a random integer from theset �1, 2, 3�. As in [10], we consider problems with

R1: ;H"0.5;� andR2: ;H"0.9;�,

where ;�"��

, summed over all operations with controllable processing times and ��

areuniformly distributed real numbers from the interval (1, 4). The release dates are randomlygenerated values depending on the number of jobs. More precisely, for the "rst job we set r

�"0,

and the release dates of the remaining jobs are determined by r�"20nx, where x is a random real

number from the interval (0,1).Although we performed experiments for a large number of problems with very di!erent

combinations of n and m, we present the results only for four representative problem classes(combinations of n and m) of medium size. For m"5 ("ve machine centers), we present the resultsfor problems with n"10, 15 and 20 jobs. Further, to observe the in#uence of a larger number ofmachine centers, we present the results for problems with n"10 jobs and m"10 machine centers.

For each of the four combinations of n and m mentioned above, we generated 20 instances. Foreach instance, we generated two di!erent weight sets �u

�, v

�,w

�, z

��, i"1, 2,2, n, for objective

function (1). We denote these weight sets as=1 and=2, respectively). We used both variants R1and R2 for the available amount of the resource. Thus, we present the results for 80 instances ofeach problem class.

5.2. Comparative evaluation of constructive algorithms

We now discuss the results obtained with the insertion algorithm. In particular, we hierarchicallyconsider the following four questions:

� Which insertion order of the jobs should be chosen?� How should the resource be initially allocated to the operations with controllable processing

times?


Table 2The number of best values for di!erent insertion orders for algorithm Ins(2)

n�m O1 O2 O3 O4

10�5 544 106 123 2715�5 578 170 51 120�5 594 180 26 010�10 490 248 31 31Total 2206 704 231 59

� How should the value of k (i.e. number of positions considered for insertion) be chosen?� How should the initial tightness value ¹I be chosen?

5.2.1. Ewectiveness of insertion ordersThe di!erences between the individual insertion orders are expected to be larger when k,

representing the last k positions considered for insertion, is small. Therefore, we investigated thee!ectiveness of the four insertion orders O1}O4 using the insertion algorithm variant Ins(2) usingthe initial value ¹I"1. The amount of resource used is IRA );H where IRA3�0, 0.25, 0.5, 0.75, 1�.

Each of the 80 problem instances was solved using each of the four insertion orders O1}O4 byapplying variant IR1 and IR2 to each instance and each of the "ve initial values of parameter IRA.Table 2 shows the number of times each insertion order gave the best results. In approximately70% of the cases, insertion order O1 produced the best objective function value. While substan-tially worse than order O1, the insertion order O2 was the second best order. This is probably dueto the large in#uence of the due date on the objective function value even though release and duedates are correlated. Insertion orders O3 and O4 performed poorly since the insertion of the currentjob only in the last position does not generate a good approximate solution. Due to its superiority,in the following experiments, we only apply insertion order O1.

5.2.2. Initial resource allocationAmong the two initial resource allocation variants IR1 and IR2, we observed that variant IR1 is

slightly superior. For IRA*0.25, variant IR1 found a better solution in approximately 60% of allcases. However, the di!erences in the objective function values were not substantial. For thisreason, we applied only variant IR1 in our subsequent experiments as the results with resourceavailability R2 will be quite similar.

We applied the "ve variants of the insertion algorithm, Ins(2)}Ins(6) to solve each probleminstance in each problem class. For these 800 runs, we calculated the number of times a speci"cvariant of parameter IRA provided the best and the worst value. Since the results do notsigni"cantly depend on the problem class and the value of k in Ins(k), we aggregated the abovevalues over all 800 runs. We found that variant IRA"1 most often produced the best objectivefunction value (369) followed by variant IRA"0.5 (150). On the other hand, the variant IRA"0.5provided the smallest number (68) of worst values followed by the variant IRA"0.25 (84),IRA"0.75 (90), and IRA"1 (136). In most cases, the variant IRA"0 performed poorly.


Table 3Improvements of the average objective function values over variant Append

n�m IRA weights Ins(2) Ins(3) Ins(4) Ins(5) Ins(6) Ins(all)

10�5 0.5 =1 7.82 19.25 18.14 19.91 19.91 19.910.5 =2 8.72 13.88 15.21 18.11 18.90 19.491 =1 9.67 16.39 19.54 20.99 21.06 21.061 =2 5.88 14.66 16.73 19.08 19.87 20.41

15�5 0.5 =1 6.99 10.64 14.20 13.19 18.30 19.500.5 =2 8.17 11.62 17.42 17.28 20.97 21.281 =1 7.75 13.74 15.99 15.87 21.25 21.671 =2 9.55 11.04 16.89 16.94 18.72 20.47

20�5 0.5 =1 8.04 9.90 12.51 16.30 17.50 20.450.5 =2 5.76 11.16 15.00 15.50 18.14 24.251 =1 8.39 11.64 14.22 15.33 17.59 20.991 =2 4.05 12.38 15.65 18.16 20.37 25.28

10�10 0.5 =1 13.36 13.26 15.74 12.91 14.92 15.420.5 =2 9.68 12.71 15.48 17.25 18.35 18.351 =1 12.12 12.55 15.47 14.05 15.57 15.431 =2 10.04 12.71 15.47 17.29 17.91 18.27

Therefore, the variant IRA"0.5 is a good compromise. Due to the small computational times, thevariant IRA"1 can be used in an alternative run. In our subsequent empirical tests, we appliedboth variants IRA"0.5 and IRA"1.

5.2.3. Ewect of the number of insertion positionsWe tested the insertion variants Ins(k), 2)k)n, and variant Ins(all) by applying insertion order

O1 for problems with an amount IR1 of resource against the priority rule Append with the initialtightness value ¹I"1. Since we did not observe signi"cant di!erences in the results for di!erentresource allocations in the tests so far, we used only resource allocation R1. For each problem classconsidered, we found the average objective function value for the 20 problem instances. Using thisaverage value for each individual variant, we calculated the percentage improvement over variantAppend. The results for variants Ins(2)}Ins(6) and Ins(all) are given in Table 3.

For most of the problems considered here, the restriction to the last six positions for inserting a jobis su$cient. In particular, for small problems, the results with Ins(6) are rather good in comparisonwith Ins(all). Only for weight set=2 and problems with n"20, variant Ins(all) found percentageimprovements that are 5}6% larger compared to Ins(6). For the latter problems with IRA"0.5,variant Ins(8) obtained percentage improvements of 24.51% (even slightly better than Ins(all)!).Further, for these problems with IRA"1, variant Ins(8) obtained 20.99% and Ins(9) 24.98%improvement over variant Append. For problems with n'20, the number of positions consideredfor insertion slightly increased with n. However, for any n, it is not necessary to consider allpositions for inserting a job into the job sequence (since a job with a large due date is usually nota candidate for the "rst position).


Table 4Percentage improvements of the average value with Algorithm Ins(6) and the initial value ¹I over the variant with initialvalue ¹I"1

n"10, m"5TI 1.4 1.6 1.8 2.0 2.2 2.4 Best1 Best2=1 9.60 15.58 18.66 21.98 20.87 19.30 24.37 24.74=2 11.72 18.62 20.75 25.64 25.56 25.93 27.17 28.83

n"15, m"5TI 2.1 2.4 2.7 3.0 3.3 3.6 Best1 Best2=1 16.21 26.27 30.70 28.15 27.35 22.66 33.04 32.83=2 14.99 21.01 25.49 27.82 24.06 19.33 30.75 30.91

n"20, m"5TI 2.8 3.2 3.6 4.0 4.4 4.8 Best1 Best2=1 21.59 25.75 22.97 22.45 19.64 15.12 28.98 26.02=2 19.98 25.68 25.19 24.49 22.25 18.60 30.90 28.45

n"10, m"10TI 1.4 1.6 1.8 2.0 2.2 2.4 Best1 Best2=1 14.69 21.92 25.89 23.39 16.47 16.50 29.01 27.70=2 15.98 25.77 28.91 26.22 21.81 16.04 31.31 30.30

5.2.4. Appropriate initial tightnessWe experimented with initial values of ¹I between 1 and 5. In Table 4, we give the percentage

improvements of the average values of the individual variants over the run with the initial value¹I"1 for other initial values of ¹I that depend on the number of jobs where IRA"0.5. Theresults for runs with IRA"1 were similar and hence are not reported here. We observed that anappropriate initial value of ¹I linearly increases with the number of jobs. Since the function valueshave a large range, we also give the results when taking the minimum of three runs. Best1 refers tothe minimum value of the three runs with ¹I3�0.16n, 0.18n, 0.20n� and Best2 refers to theminimum value of the three runs with ¹I3�0.18n, 0.20n, 0.22n�.

As reported in Table 4, an appropriate initial value of parameter ¹I has substantial in#uence onthe quality of the solution. For all problem classes considered, as compared to the initial value¹I"1, the use of an appropriate initial value of ¹I improves the average objective function valueby 25}30%. Due to the large range of the function values, we recommend applying three di!erentinitial values which, on average, increases the percentage improvements by about 3% over the bestindividual initial value (see Best1 and Best2). The variant Best1 is, on average, superior to variantBest2. Comparing the results for n"10 and both values of m, we observed that the initial value of¹I should be slightly smaller with a larger number of machine centers. It can be seen that di!erentweight sets =1 and =2 do not signi"cantly in#uence the results within one problem class.Moreover, we observed that, when starting with an appropriate initial value of ¹I, the value of thisparameter increased only moderately (starting with ¹I"0.20n, the value of ¹I never increased bymore than 0.6). Using ¹I"0.20n, Algorithm Ins(6), on average, improved the objective functionvalue of Algorithm Append with ¹I"1 as initial value by about 40%. Therefore, we recommendthe individual runs with ¹I"0.18n or 0.20n.


5.3. Comparative evaluation of iterative algorithms

In connection with the iterative algorithms we simultaneously investigated the followingquestions:

� How should the initial change of the tightness value be controlled (use of parameters prdec,��

and ��)?

� How should the generation of a neighbor in the composite neighborhoods be controlled(parameter prob)?

� How should the re"ned reconstruction of a schedule in neighborhood B be controlled (use ofsteplength �¹I, procedure random-TI)? Does the re"ned construction of a schedule in neighbor-hood B improve the results in comparison with neighborhood A?

� Should one restrict the length of a shift, i.e. the maximal number of positions by which a job maybe shifted within the sequence?

First, we tested the iterative algorithms based on neighborhood A. For all problems we generated2000 feasible solutions. The starting solution has been determined by algorithm Ins(6) with aninitial tightness value ¹I"0.2n. Concerning the individual parameters, we considered the follow-ing variants:

(a) prob3�0.0, 0.1, 0.2,2, 0.9, 1.0�;(b) prdec3�0.0, 0.1, 0.2,2, 0.9, 1.0�;(c) (�

�; �

�)3�(0.1; 0.1),(0.2; 0.1),(0.2; 0.4),(0.3; 0.3),(0.4; 0.2)� and

(d) Shift(1), Shift(3), Shift(6), Shift(9) and Shift.

We considered all combinations obtained by selecting one parameter value given in (a)}(d)above. In Table 5, we give the percentage improvements of the average objective function value ofeach problem class and weight set over the starting solution for the following representativealgorithm variants:

1: prob"0.9, prdec"0.7, (��, �

�)"(0.3, 0.3), Shift(3);

2: prob"0.9, prdec"0.7, (��, �

�)"(0.3, 0.3), Shift(6);

3: prob"0.9, prdec"1.0, (��, �

�)"(0.3, 0.3), Shift(6);

4: prob"0.9, prdec"0.7, (��, �

�)"(0.3, 0.3), Shift;

5: prob"0.5, prdec"0.7, (��, �

�)"(0.3, 0.3), Shift(6);

6: prob"0.9, prdec"0.7, (��, �

�)"(0.4, 0.2), Shift(3);

7: prob"0.9, prdec"0.7, (��, �

�)"(0.4, 0.2), Shift(6);

8: prob"0.9, prdec"0.7, (��, �

�)"(0.4, 0.2), Shift(1);

9: prob"0.7, prdec"0.7, (��, �

�)"(0.4, 0.2), Shift(6);

10: prob"0.9, prdec"0.7, (��, �

�)"(0.2, 0.1), Shift(6);

11: prob"0.9, prdec"0.7, (��, �

�)"(0.1, 0.1), Shift(6);

BEST: best improvement obtained by some tested variant.Our tests with neighborhood A can be summarized as follows. We observed that move I should

be applied considerably more often than move III. Parameter prob should be preferably between0.7 and 1.0. Based on our tests, we favor the value prob"0.9. Concerning the values �

�and �

�, we


Table 5Results for the iterative algorithms with neighborhood A

Weights Algorithm variant BEST

1 2 3 4 5 6 7 8 9 10 11

n"10, m"5=1 7.62 7.95 6.23 7.49 7.76 7.98 8.32 6.80 7.83 7.81 6.81 8.45=2 7.86 7.46 5.58 7.82 6.97 7.12 7.54 6.70 7.21 7.60 7.35 7.86

n"15, m"5=1 10.62 11.56 10.94 11.34 11.51 11.52 11.62 10.28 12.35 11.69 10.04 12.40=2 9.92 9.59 8.92 10.80 9.53 9.85 9.43 8.12 9.86 9.56 8.50 10.88

n"20, m"5=1 16.63 16.14 13.99 15.37 13.92 15.36 15.80 13.42 15.80 14.71 11.15 16.64=2 15.02 16.96 13.74 15.41 12.85 14.33 16.53 13.49 16.32 14.42 12.80 16.96

n"10, m"10=1 14.18 14.24 13.04 13.93 13.57 13.97 14.55 12.98 14.34 13.66 11.91 14.82=2 13.17 12.74 11.73 12.62 12.68 12.66 12.71 11.56 13.22 13.18 10.29 13.58

Ave. 11.88 12.08 10.52 11.85 11.10 11.59 12.06 10.43 12.12 11.58 9.86 12.70

found that small values perform poorly (see column 11 in Table 5). The best values were found for��"0.4 and �

�"0.2 as well as �

�"�

�"0.3, where the former variant is slightly superior.

Moreover, the probability parameter prdec should be within the interval [0.6, 0.9]. In particular,the value prdec"0.7 usually performed well. The use of �

�"0.4 and �

�"0.2 as well as

prdec"0.8 and 0.9 produced good results. This shows that the ¹I value should be decreased moreoften in order to reduce the objective function value. However, the variant prdec"1.0 performspoorly (see column 3 of Table 5). Thus, decreasing only value ¹I in the neighbor generation cannotbe recommended. Concerning the length of a shift, variant Shift(6) is recommended. In particular,the variant Shift(1) clearly performs worse than Shift(6) (see column 8 of Table 5) whereas Shift(3)and Shift in general perform slightly worse. For problems with n'20, we also found that the lengthof a shift should be appropriately restricted.

On the basis of the foregoing computational results, we recommend the use of variantprob"0.9, prdec"0.7,�

�"0.4,�

�"0.2 and Shift(6) (see column 7 of Table 5). The best variant

produced an average improvement that is only 0.11% higher than the recommended variant. Suchsmall di!erences could be mainly caused by the randomized nature of the algorithms. However,slight changes in the values of one or a few parameters lead to a similar solution quality. Also, forthe variant recommended above, most of the percentage improvements were found within the "rst1,000 generated solutions (e.g. 11.51% for the variant in column 7 of Table 5) whereas for mostother variants the intermediate results after the "rst 1,000 generated solutions are often worse. Thetotal computational time for a set of 20 instances ranged from about 7 s for n"10, m"5 up toabout 19 s for n"20, m"5.

With the above settings for parameters ��

and ��, we experimented with neighborhood B. In

contrast to neighborhood A, neighborhood B uses the procedure Construct as it allows a &moreprecise' determination of parameter ¹I (which was very important in the constructive variant) and


Table 6Results for the iterative algorithms with neighborhood B

Weights Algorithm variant BEST

1 2 3 4 5 6 7 8

n"10, m"5=1 8.17 7.98 8.25 8.26 8.08 7.98 8.41 7.92 8.74=2 7.84 7.87 7.46 8.05 7.84 8.05 7.72 7.90 8.14

n"15, m"5=1 11.95 12.23 11.59 12.08 12.17 12.20 11.32 12.40 12.86=2 10.89 10.53 10.91 11.82 11.08 11.52 11.59 11.36 12.43

n"20, m"5=1 14.35 15.39 16.54 16.55 16.44 16.41 15.35 15.55 16.55=2 14.78 15.86 16.01 17.05 16.09 16.49 16.83 14.71 17.11

n"10, m"10=1 15.09 15.05 14.68 15.02 14.75 15.39 15.35 14.52 15.51=2 13.77 13.80 13.35 13.51 13.23 13.51 13.69 13.58 14.23

Ave. 12.11 12.34 12.35 12.79 12.46 12.69 12.53 12.24 13.20

additional changes in the controllable processing times. Based on the test results with neighbor-hood A, we restricted the range of the parameters as follows:

(a) prob3�0.8, 0.9, 1.0�;(b) prdec3�0.6, 0.7, 0.8�;(c) Shift(6), Shift;(d) �¹I3�0.025, 0.05, 0.1� and(e) application of random-TI or no application of random-TI.

Table 6 shows the percentage improvements over the starting solution for the following variantsof iterative algorithms:

1: prob"0.8, prdec"0.8, Shift(6), �¹I"0.025, no random-TI;2: prob"0.8, prdec"0.8, Shift(6), �¹I"0.05, no random-TI;3: prob"0.9, prdec"0.7, Shift(6), �¹I"0.05, no random-TI;4: prob"1.0, prdec"0.7, Shift(6), �¹I"0.05, no random-TI;5: prob"1.0, prdec"0.7, Shift, �¹I"0.05, no random-TI;6: prob"1.0, prdec"0.7, Shift(6), �¹I"0.1; no random-TI;7: prob"1.0, prdec"0.7, Shift(6), �¹I"0.05, random-TI;8: prob"0.8, prdec"0.8, Shift(6), �¹I"0.05, random-TI;BEST: best improvement obtained by some tested variant.

The total computational time for 20 instances of a problem class ranged from 15 s forn"10,m"5 up to 45 s for n"20,m"5 when using neighborhood B. As for neighborhood A, werecommend prdec"0.7 and Shift(6). Concerning parameter prob, we obtained best results forprob"1.0, i.e. contrary to neighborhood A, the additional consideration of move III did not


improve the results. This is probably due to the re"ned reconstruction of a schedule from theunderlying sequence when applying move II (which also allows changes in the controllableprocessing times similarly as move III). Although the di!erences in the percentage improvementsbetween prob"0.9 and prob"1.0 are small, nevertheless it can be observed that more recordvalues for neighborhood A have been obtained with prob"0.9 and more record values forneighborhood B with prob"1.0.

Concerning parameter �¹I, we did not observe big di!erences for the individual variants. Thevariant �¹I"0.025 worked poorly for the larger problems (n"20, see column 1 of Table 6), andvariant �¹I"0.05 is slightly superior to �¹I"0.1. However, the additional use of random-TI didnot improve the results (see columns 7 and 8 of Table 6). It indicates that the randomized structureof the algorithms is su$cient. Thus, we recommend the variant given in column 4 of Table 6. Thepercentage improvements are about 0.7% higher than those for the corresponding variantsoperating with neighborhood A. We also found that the tested randomized algorithms are ratherrobust with respect to moderate changes in some parameters. Moreover, about 80}85% of the overallpercentage improvements over the function value of the starting solution was obtained from the"rst 200 generated solutions. This observation and the small computational times recommend therepeated application of several shorter variants of the algorithms suggested in this paper, whereboth neighborhoods A and B can be used.

Finally, we mention that we also experimented with simulated annealing variants which didnot improve the results. Our explanation for this behavior is the importance of the continuoustightness parameter which leads to in"nitely many neighbors in the case of iterative improvement.

6. Conclusions

This paper considered a hybrid #ow shop scheduling problem with release dates, controllableprocessing times, and multiple identical parallel machines at various machine centers. The optimal-ity criterion is the sum of the earliness and tardiness penalties, weighted completion time of jobs,and the costs of due date assignments. Since the problem is NP-hard in the strong sense, wedeveloped and tested several heuristics to solve this problem. These heuristics are generalizations ofthose algorithms that have successfully been applied to classical scheduling problems. In this paper,we mainly concentrated on the determination of appropriate control mechanisms for theseheuristics. In particular, we derived constructive algorithms based on insertion techniques andlocal search algorithms based on shift neighborhoods. From our empirical experiments, we candraw the following conclusions and recommendations:

� For the problem considered, the di!erences between the variants of our insertion algorithm aresigni"cant (and larger than those with insertion algorithms for the classical scheduling prob-lems). This con"rms the importance of appropriate control mechanisms for the schedulingproblems like the one considered in this paper. For the due date based optimization criterion, allsequence positions need not be considered as candidate positions for a job to be inserted. In mostcases, for problems with 20 or fewer jobs, the insertion of the jobs according to the EDD rule inthe last six positions is su$cient. The consideration of more than n/2 insertion positions is notnecessary.


� The initial tightness value for computing the job due dates strongly in#uences the solutionquality of the insertion algorithm. We found that a good initial value should linearly depend onthe number of jobs.

� In comparison with Algorithm Append applied with the initial value ¹I"1 of the tightnessparameter (which basically corresponds to an application of a generalized EDD rule as priorityrule), the use of algorithm Ins(6) with an appropriate initial tightness value yields, on average,percentage improvements of about 40%. This shows that an insertion based algorithm is a moreappropriate constructive heuristic than priority rule based heuristics.

� Concerning iterative improvement algorithms, we obtained slightly higher percentage improve-ments with neighborhood B than with neighborhood A. Due to the re"ned reconstruction of theschedule after performing a shift operation in the job sequence, use of move II was su$cient withneighborhood B. However, in order to enable su$cient changes in the resource allocation tooperations with controllable processing times, the application of move III with a small probabil-ity is recommended with neighborhood A.

� The inclusion of simulated annealing led to worse results than with iterative improvement. Thisbehavior is probably due to the di!erent structure of the set of feasible solutions in comparisonwith classical permutation problems (the problem considered here includes several continuouscomponents describing a schedule).

� Although the performance of the proposed iterative algorithms depend on a number ofparameters, they are rather robust in the sense that moderate changes in some parameters do notlead to signi"cantly worse results. Since no single iterative improvement algorithm based onneighborhood B required more than 3 s of CPU time per instance, the application of severalvariants of the algorithms suggested can be recommended.

� Changing a particular term in the objective function is not expected to substantially in#uence thedesign of the algorithms presented in this paper. Therefore, the proposed algorithms can beapplied to problems with other complex objective functions similar to the one considered here.Due to the robustness of the suggested heuristics, several other generalizations of the problemmay be included as well. For example, we might be able to include the in#uence of setup timesrequired before an operation can start its processing in a machine center.

We conclude the paper with some fruitful directions for future research. Firstly, the neighbor-hoods for the problem considered here can be re"ned further. For instance, the distribution of jobsto the machines of a center can be re"ned, and alternatives in the insertion of idle times fornonregular criteria can be considered. Secondly, alternative composite neighborhoods, includinga composition of shift and pairwise interchange neighborhoods should be considered. Since&shorter' shifts were found to be preferable, one can also use probabilities to perform shorter shifts(or pairwise interchanges) with larger probability and &longer' shifts with smaller probability.Thirdly, for the insertion algorithm, one can look for improved procedures within the scheduleconstruction, or one can drop our assumption of considering only permutation schedules where thesame job sequence is followed on each machine center. Fourthly, one can also include otherconstraints that are important in connection with applications in production scheduling but whichhave not been considered in such a general scheduling environment. In particular, the inclusionof such constraints that require a more general neighborhood structure are of interest. In contrastto the latter problems, additional consideration of setup or transportation times between the


operations can be treated by the iterative algorithms developed for the classical schedulingproblems, where a neighbor is generated by appropriate change in one or several job sequences.

Acknowledgements

This work was supported by Deutsche Forschungsgemeinschaft, Project ScheMA, and by INTAS,Projects 93-257 and 96-0820. The authors express deep appreciation to two anonymous reviewers fortheir constructive comments and suggestions that improved the presentation of this paper.

References

[1] MacCarthy BL, Liu J. Addressing the gap in scheduling research: a review of optimization and heuristic methods inproduction scheduling. International Journal of Production Research 1993;31:59}79.

[2] Ignall E, Schrage L. Application of the branch and bound technique to some #ow-shop scheduling problems.Operations Research 1965;13:400}12.

[3] Gupta JND. A review of #owshop scheduling research. in: Ritzman LP, Krajewski LJ, Berry WL, Goodman SM,Hardy ST, Vitt LD, editors. Disaggregation problems in manufacturing and service organizations. The Hague,Netherlands: Martin Nijho! 1979, p. 363}88.

[4] Morton TE, Pentico DW. Heuristic scheduling systems. Wiley: New York, 1993.[5] Chen B, Potts CN, Woeginger GJ. A review of machine scheduling: complexity, algorithms and applications In: Du

D-Z, Pardalos PM, editors. Handbook of combinatorial optimization. Dordrecht, Netherlands: Kluwer, 1998.p. 21}169.

[6] T'Kindt V, Billaut J-C. Multicriteria scheduling problems: a survey. RAIRO Recherche Operationnelle 2001, toappear.

[7] Brah SA, Hunsucker JL. Branch and bound algorithm for a #owshop with multiple processors. European Journalof Operational Research 1991;51:88}99.

[8] Alidaee B, Kochenberger G. A framework for machine scheduling problems with controllable processing times.Journal of Production and Operations Management 1996;5(4):391}405.

[9] Janiak A. General #ow-shop scheduling with resource constraints. International Journal of Production Research1988;26:1089}103.

[10] Janiak A, Portmann MC. Genetic algorithm for the permutation #ow-shop scheduling problem with linear modelsof operations. Annals of Operations Research 1998;83:95}114.

[11] Cheng TCE, Gupta MC. Survey of scheduling research involving due date decisions. European Journal ofOperational Research 1989;38:156}66.

[12] Sotskov YN, Tautenhahn T, Werner F. On the application of insertion techniques for job shop problems with setuptimes. RAIRO Recherche Operationnelle 1999;33:209}45.

[13] Cheng TCE. Analytical determination of optimal TWK due dates in a job shop. International Journal of SystemSciences 1985;16:777}87.

[14] Baker KR, Scudder GD. Sequencing with earliness and tardiness penalties: a review. Operations Research1989;38:22}36.

[15] Panwalkar SS, Iskander W. A survey of scheduling rules. Operations Research 1977;25:45}61.[16] Nawaz M, Enscore EE, Ham I. A heuristic algorithm for the m-machine, n-job #ow shop sequencing problem.

OMEGA 1983;11:91}5.

Jatinder N. D. Gupta, is currently Professor of Management, Information and Communication Sciences, and Industryand Technology at the Ball State University, Muncie, Indiana, USA. He holds a PhD in Industrial Engineering (with


specialization in Production Management and Information Systems) from Texas Tech University. Coauthor of a text-book in Operations Research, Dr. Gupta serves on the editorial boards of several national and international journals. Hehas published research and technical papers in Annals of Operations Research, Computers and Operations Research,European Journal of Operational Research, IIE Transactions, International Journal of Information Management, Journal ofManagement Information Systems, Mathematics of Operations Research, Naval Research Logistics, Operations Research. Hiscurrent research interests include information technology, scheduling, planning and control, organizational learning ande!ectiveness, systems education, and knowledge management.Karin KruK ger studied mathematics at the Technical University of Magdeburg (Germany) from 1973 to 1977. She

received her Ph.D. degree from the same university in 1982. Afterwards she continued work in the "eld of practicalscheduling problems.Volker Lau4 is currently a Ph.D. candidate at the Otto-von-Guericke-University of Magdeburg (Germany) dealing

with scheduling problems with nonregular optimization criteria. Earlier, he studied physics at the University ofKaiserslautern (Germany) from 1989 to 1997.Yuri N. Sotskov is currently the head of the Laboratory of Mathematical Cybernetics in the Institute of Engineering

Cybernetics of the National Academy of Sciences of Belarus and an invited Professor at the University of Technology ofTroyes (France). He received M.Sc. degree from the Belarusian State University (Minsk), Ph.D. degree from the Instituteof Mathematics of the National Academy of Sciences of Belarus, D.Sc. degree from the Institute of Cybernetics of theNational Academy of Sciences of Ukraine (Kiev), and the Professor certi"cate from the Russian Academy of Sciences(Moscow) in 1994. He is interested in research areas related to operations research, scheduling theory, discreteoptimization and graph theory. He has published three books and 80 papers in English and Russian.Frank Werner is currently Professor of Mathematics and Operations Research at the Otto-von-Guericke-University

Magdeburg in Germany. He received his Ph.D. in Mathematics from the University of Magdeburg. His research interestsare discrete optimization and scheduling, particularly the development of exact and heuristic algorithms for schedulingproblems. He has published about 70 papers in this "eld in such journals as the Annals of Operations Research,International Journal of Production Research, Computers and Operations Research, and Mathematical and ComputerModelling.


Documents

Heuristics for hybrid flow shops with controllable processing times and assignable due dates