A comparative study of heuristicalgorithms to solve maintenance

scheduling problemSyed Asif Raza

Centre for Research on Transportation, Universite de Montreal, Montreal,Canada, and

Umar Mustafa Al-TurkiSystems Engineering Department, King Fahd University of Petroleum and

Minerals, Dhahran, Saudi Arabia

Abstract

Purpose – The purpose of this paper is to compare the effectiveness of two meta-heuristics in solvingthe problem of scheduling maintenance operations and jobs processing on a single machine.

Design/methodology/approach – The two meta-heuristic algorithms, tabu search and simulatedannealing are hybridized using the properties of an optimal schedule identified in the existingliterature to the problem. A lower bound is also suggested utilizing these properties.

Finding – In a numerical experimentation with large size problems, the best-known heuristicalgorithm to the problem is compared with the tabu search and simulated annealing algorithms. Thestudy shows that the meta-heuristic algorithms outperform the heuristic algorithm. In addition, thedeveloped meta-heuristics tend to be more robust against the problem-related parameters than theexisting algorithm.

Research limitations/implications – A future work may consider the possibility of machinefailure along with the preventive maintenance. This relaxes the assumption that the machine cannotfail but it is rather maintained preventively. The multi-criteria scheduling can also be considered as anavenue of future work. The problem can also be considered with stochastic parameters such that theprocessing times of the jobs and the maintenance related parameters are random and follow a knownprobability distribution function.

Practical implications – The usefulness of meta-heuristic algorithms is demonstrated for solving alarge scale NP-hard combinatorial optimization problem. The paper also shows that the utilization ofthe directed search methods such as hybridization could substantially improve the performance of ameta-heuristic.

Originality/value – This research highlights the impact of utilizing the directed search methods tocause hybridization in meta-heuristic and the resulting improvement in their performance forlarge-scale optimization.

Keywords Job sequence loading, Maintenance programmes, Production scheduling

Paper type Research paper

1. IntroductionPreventive maintenance (PM) is considered as an activity that improves a system’sreliability as well as its service life. A sound PM program is essential to improve the

The current issue and full text archive of this journal is available at

www.emeraldinsight.com/1355-2511.htm

The authors would like to thank Professor Shokri Z. Selim, the editor, and anonymous refereesfor their helpful comments. The financial support of King Fahd University is also greatlyacknowledged.

JQME13,4

398

Journal of Quality in MaintenanceEngineeringVol. 13 No. 4, 2007pp. 398-410q Emerald Group Publishing Limited1355-2511DOI 10.1108/13552510710829489

performance for a manufacturing environment. Recently, automation has improved themanufacturing practices; however, it has tied up more investment in themanufacturing facilities. Productivity due to automation has substantially increased,yet a manufacturer can only compete in the market while having reliable automatedmanufacturing facilities. This has resulted in an increased concern for a sound PMprogram in order to ensure high productivity.

The cardinality of PM is realized by both researchers and industry practitioners.There are several studies in order to consider the effect of unreliable systems as well asthe impact of PM. The machine vacation is also addressed in machine schedulingstudies. Schmidt (1988) considered scheduling jobs with due dates on parallel machineshaving availability intervals. Qi et al. (1999) studied scheduling jobs on a singlemachine that required PM. The time elapsed between two maintenance activities couldnot exceed a given value. Lee and Chen (2000) considered the parallel machine versionwith the objective of minimizing the weighted total completion time. Lorigeon et al.(2002) studied two-machine open shops subjected to availability constraint with theobjective of minimizing total completion time. Chen and Powell (2003) considered asituation where jobs were classified into families. No set up was needed whenprocessing jobs of the same family. However, a setup was needed when there was aswitch from one job family to another. Two problems involving identical machineswere considered. In the first problem the total weighted completion time wasminimized and in the second problem the total weighted number of tardy jobs wasminimized. Cassady and Kutanoglu (2003) considered the problem of joint schedulingof jobs and preventive maintenance such that total tardiness was minimized. Aggoune(2004) minimized the make span of a flow shop with availability constraints. Twomaintenance policies were discussed. In the former, maintenance starting times werefixed, whereas in the later, maintenance had to be performed within a given time frame.Chen (2004) considered parallel machines scheduling problem involving job schedulingand resource allocation. The processing times were inversely related to funds allocatedto resources. The objective was to minimize total cost of scheduling and resourceallocation. Adzakpa et al. (2004a) developed heuristics for scheduling jobs on parallelmachines with the objective of minimizing weighted flow time. In another paper,Adzakpa et al. (2004b) considered online scheduling and assignment of maintenance ona single machine with availability constraint, over a given time frame with theobjective of minimizing cost of discharge of maintenance or jobs. Akturk et al. (2003)considered the problem of tool change due to wear on a single computer numericalcontrol machine. The objective was to minimize total completion time. They studiedthe performance of the shortest processing time (SPT) rule for the same problem. Liaoet al. (2005) considered a two parallel machines problem where one machine was notavailable during a fixed and known time period. The objective was to minimize themake span for both non-resumable and resumable cases. Sortrakul et al. (2005)considered an integrated optimization model for production scheduling and preventivemaintenance planning. Mazzini and Armentano (2001) considered single machinescheduling with due dates, ready time and shut down constraint. Tardiness was notallowed; however, earliness was penalized. The fact that shutdowns were known, iswhat made this work different from that of Qi et al. (1999). The work of Qi et al. (1999)was further extended in Raza et al. (2007). The early-tardy minimization was studied asa performance measure. The properties of an optimal schedule were identified. The

Study ofheuristic

algorithms

399

properties were later used to construct a single pass heuristic algorithm and a lowerbound estimate to the problem. Two meta-heuristics based on tabu search (TS) andsimulated annealing (SA) algorithms were also proposed. The numericalexperimentation showed that the meta-heuristics could be hybridized using thesome of the properties of an optimal schedule. This resulted in substantial savings incomputational time. The meta-heuristics outperformed the suggested single passheuristic.

The work presented in this paper is motivated by an earlier study reported in Razaet al. (2007). It is shown that the meta-heuristics enables solving large scale schedulingproblems with significant saving in computational time. This can be achieved byembedding the properties of the optimal schedule identified in Qi et al. (1999). Theinterest in the development of the meta-heuristics is stirred by the results reportedaforementioned study, which inferred that the meta-heuristic might outperform theSPT heuristic algorithm. The meta-heuristics can be used to solve large-sized problemswith very good quality, whereas solving the same problems using an implicitenumeration algorithm is intractable (Lyu et al., 1996).

The rest of this paper is organized as follows: in section 2, we define the problem,introduce the notations, and briefly describe some properties of an optimal schedulereported by Qi et al. (1999). In section 3, we develop a lower bound to the problem. Anexisting heuristic, SPT algorithm is briefly presented. Two meta-heuristic algorithms,TS and SA, are also proposed to solve the problem. In section 4, numericalexperimentation with randomly generated test problems is reported. Finally, in section5, the conclusions are drawn from the study along with the future research suggestionsmentioned.

2. The maintenance scheduling problemThe problem is to schedule n jobs, J 1; J 2; . . . ; J n, available at time zero on a singlemachine such that a given performance measure is optimized. The maintenanceduration is t. The machine cannot be continuously operated for a period exceeding T.The processing time pi is deterministically known for a job, Ji and jobs are indexed in anon-decreasing order of their processing times, i.e. p1 # p2 # . . . # pn. No preemptionis allowed, also pn # T . A typical schedule, S, contains a sequence of jobs interruptedby preventive maintenance such that the machine does not run continuously for a timeperiod more than T. The resulting sequence can be viewed as batches of jobs,Bi; ; i ¼ {1; 2; · · ·;L}, separated by preventive maintenance operations M. Thus theschedule, S, is denoted as S ¼ {B1;M ;B2;M ; . . . ;M ;BL}, where M representspreventive maintenance. Each occurrence of M has a cost of time, t, in the performancemeasure. The sum of processing times of jobs in each batch does not exceed T. Let Ci

be the completion time of job Ji in schedule S. Let qi and ni be the summation of theprocessing times of the jobs and the total number of jobs in batch Bi respectively. Thispaper mainly discusses the total completion time minimization as the performancemeasure. Qi et al. (1999) considered minimizing the makespan, Cmax, and identified thatthe Cmax minimization is equivalent to the bin-packing problem (Yao, 1980). Theproblem with total completion time minimization,

PC, is shown to be NP-hard in Qi

et al. (1999). The objective function of the total completion time minimization problemis:

JQME13,4

400

f Sð Þ ¼Xni¼1

Ci

¼Xni¼1

n2 i þ 1ð Þp½i� þXLk¼2

k2 1ð Þnk t

ð1Þ

In equation 1, the first part is the contribution from the processing times of the jobs,and the second part is the contribution from maintenance scheduling in a given jobsequence. Next, we briefly discuss some properties of an optimal schedule. The proofsof the properties can be found in Qi et al. (1999):

. Property 1. In each batch of an optimal schedule, jobs are sequenced according toSPT rule i.e. non-decreasing order of their processing times.

. Property 2. In an optimal schedule, for each batch Bi, the following inequalitymust hold:

T 2 qi , pj; ; J j [ Bk; k ¼ i þ 1; · · ·;L:

. Property 3. In an optimal schedule, n1 $ n2 $ · · · $ nL.

. Property 4. In an optimal schedule, qi=ni # qiþ1=niþ1.

The SPT rule is optimal for total completion time minimization with no maintenanceconstraint. In each batch, the contribution in the total completion time from the jobs inthat batch can be minimized using SPT rule as identified in Property 1. Property 2simply states that a job must be placed in an earlier batch of a schedule given there existsspace for that job in the batch and it does not violate the maintenance constraint. InProperty 3, it is identified that more jobs should be placed in a batch preceding the otherbatch in an optimal schedule. This is an outcome noticed by the observation of Property2. Finally, Property 4 is the extension of Property 1 at the level of batches. According tothis property, the batches are ordered in accordance to the ratios of their sum ofprocessing times of the jobs to the number of jobs in the batch. Qi et al. (1999) proposedan implicit enumeration, Branch and Bound algorithm, and two heuristic algorithms,SPT and fewest batches heuristic (FBH). In a numerical study reported in the work, SPTwas found superior to FBH and it was also very competitive to branch and boundalgorithm. In the next section, we propose a lower bound estimate to the problem as wellas discuss the existing and proposed heuristic algorithms to the problem.

3. Lower bound and heuristic algorithmsA super optimal schedule to the problem is suggested as a lower bound (LB). The LBmay not satisfy the maintenance constraint. A lower bound is schedule SS, constructedsuch that in any batch Bk, qk ¼ T or qk 2 T # pm, where Jm is the last job in the batch.The lower bound is thus given by LB ¼ f ðSSÞ. Next, we briefly discuss heuristicalgorithms tested in this study. This includes discussion on SPT heuristic algorithm,two meta-heuristic algorithms, TS, and SA.

3.1. SPT heuristic algorithmIn SPT heuristic algorithm, the jobs are scheduled in non-decreasing order of theirprocessing times, and inserted maintenance operation is delayed as much as possible

Study ofheuristic

algorithms

401

such that the constraint of maximum continuous operation time (T) is observed. Thealgorithm has the following steps:

(1) Step 1. Let i ¼ 1, j ¼ 1, Bi ¼ Y, S ¼ Y and qi ¼ 0.

(2) Step 2. If qi þ pj # T , put job Jj at the end of batch Bi, qi ¼ qi þ pj. Otherwise,i ¼ i þ 1, and let job Jj be the first job in the new batch Bi, qi ¼ pj.

(3) Step 3. If j ¼ n, stop. Otherwise, j ¼ jþ 1, go to step 2.

(4) Step 4. S ¼ {B1;M ;B2;M ; · · ·;BL}.

3.2. TS algorithmTS was proposed by Glover (1986). It is meta-heuristic that can be superimposed onanother heuristic. TS begins by marching to a local minima. To avoid retracing previoussteps, the method records recent moves in one or more tabu lists. The intent of the list isnot to prevent a previous move from being repeated, but rather to insure it is notreversed. tabu lists are historical in nature and form the TS memory. The role of thememory can change as the algorithm proceeds. Tabu status of a move is overriddenwhen certain criteria (aspiration criteria) are satisfied. More details about this methodcan be found in Glover and Laguna (1997). The properties of an optimal scheduleidentified earlier are used in TS algorithm mainly in the neighborhood generationscheme. This strategy helps in improving the convergence of TS algorithm to a nearoptimal solution with a significant saving in CPU time (Raza et al. 2007). In thisimplementation, the search starts with an arbitrary feasible schedule called the seedsolution. The seed solution is considered as current solution in the search. Severalcandidate solutions (feasible schedules) are generated using a neighborhood generationscheme. The moves of the best candidate solution are checked in the tabu list. If the moveof the best candidate is found in the tabu list and it satisfies the aspiration criterion thenit is also accepted as current solution for the next search iteration; otherwise, this step isrepeated. The search terminates when a stopping criterion is reached. We discuss thedetailed features of the proposed TS algorithm for this problem below:

(1) Seed solution. A seed solution is any random sequence of jobs that satisfies thepreventive maintenance requirement.

(2) Neighborhood. A neighborhood solution S0 is obtained by swapping tworandomly selected jobs. The policy adopted has an equal chance in the processof a random generation of a neighborhood. In this policy, we swap two jobsbetween any two distinct batches in an existing schedule. If any of the swapsresult in an infeasible schedule, i.e. maintenance constraint is not met, then weuse following procedure to achieve feasibility on the same job sequence:. Step 1. Remove the maintenance, i.e. M, from the infeasible neighbor

schedule, S0, and restore the job sequence.. Step 2. In the job sequence schedule maintenance using SPT heuristic

algorithm.. Step 3. Rearrange jobs in each batch in SPT order.. Step 4. Re-index the job batches such the following rule is satisfied:

q1 þ t

n1#

q2 þ t

n2# · · · #

qL þ t

nL:

JQME13,4

402

(3) Candidate list size. It is a list containing a subset of neighborhood movesexamined. A candidate list size of 20 is selected for each of the iteration afterusing the conclusions from a series of tests performed in Raza (2002).

(4) Tabu restriction. In our implementation, attributes of a schedule are jobsswapped in a schedule that are recorded in the tabu list. The tabu list can store amaximum of seven moves and is updated using first in first out (FIFO) strategy.

(5) Aspiration criterion. It is satisfied when the best neighbor solution of the currentiteration is found to be better than the previously visited best solution.

(6) Stopping criterion. The algorithm is stopped after 10,000 iterations of noimprovement.

3.3. SA algorithmSA was proposed by Kirkpatrick et al. (1983). SA follows an analogy from annealing ofmetal. During the search process SA not only accepts better solutions (downhill move),but also accepts bad solutions (uphill move) with some probability. This feature of SAenables the search to escape a local minimum. The SA algorithm requires a seedsolution, metropolis criterion, cooling schedule, acceptance probability function, andstopping criterion. The SA algorithm also makes use of properties of an optimalschedule aforementioned in this paper and hence the search for a near optimal is betterdirected. The algorithm starts with a seed solution (feasible schedule) at a hightemperature such that the most feasible neighborhood solutions of the seed solution areaccepted. At a particular temperature the metropolis loop is executed for a fixedMarkov chain length in order to achieve the quasi equilibrium state at thattemperature. At each loop, the neighbor solution is accepted if it outperforms itsgenerator solution; however, a poor solution is also accepted but the acceptance followsa probabilistic acceptance function. A temperature decrement rule is applied once quasiequilibrium state is reached at a particular temperature. The metropolis loop uses theneighborhood generation scheme same as suggested in TS algorithm. The coolingschedule and acceptance probability functions in particular to this SA algorithmimplementation are described as follows:

(1) Cooling schedule. The main parameters of a cooling schedule are: initialtemperature; temperature decrement rule; and final temperature at which theannealing process is stopped:. Initial temperature. We use this method for estimating Y0 proposed by White

(1984). In this approach, the system is considered hot enough if Yo .. s,where s is the standard deviation of the cost function at initial temperatureY0. The following equation uses the stated criterion:

c ¼ 23

lnR

Y 0 ¼ cs

s is determined based on 100 randomly generated neighbors of an arbitraryseed solution and R is the percentage of accepted solutions which is over 90percent for this implementation.

Study ofheuristic

algorithms

403

. Temperature decrement rule. In most SA approaches, geometric temperaturedecrement rule is employed. If the temperature at iteration k is Yk then thetemperature at iteration kþ 1 is given by:

Ykþ1 ¼ aYk

where 0:8 # a # 0:99 in most SA applications (Sait and Youssef, 1999). Inour implementation, a ¼ 0:99.

. Final temperature. In this implementation, the lowest allowable temperatureis set to 10-3.

(2) Markov chain length. The Markov chain length describes the number of timesthe Metropolis loop is executed at a given temperature to attainquasi-equilibrium (Eglese, 1990). In this study, the Markov chain length is 20.

(3) Acceptance probability function. In the present SA algorithm, we use thestatistical acceptance probability function (Sait and Youssef, 1999; Lyu et al.,1996). At a given temperature Yi, the acceptance probability function pa of asolution (schedule) S0 is given as:

pa ¼1 If f S0

� �, f Sð Þ

exp D=Yi

� �Otherwise

8<: ð2Þ

where D ¼ f ðS0Þ2 f ðSÞ ¼ f(S0) 2 f(S).

(4) Stopping criterion. SA stops if 10,000 iterations result in no improvementobserved. This value is selected based on some pre-experimental work with thealgorithm.

4. Numerical experimentationFollowing the numerical experimentation scheme suggested in Qi et al. (1999), weextend the analysis to large size problems. The processing times of jobs are randomlygenerated between 1 to 30 using uniform distribution. The selected parameters formaximum allowed time for continuous machine operation, T are 50, 60, 70, and 80respectively. The maintenance time, t, also has four distinct values, 10, 20, 30, and 40.The numerical experimentation reported uses job sizes, n ¼ 25, 30, 35, and 40. For eachcombination of T and t, ten problems are solved having the same job size. Theeffectiveness of each algorithm is calibrated using three distinct performancemeasures. These three performance measures are, deviation from LB, improvementover SPT heuristic algorithm, and the CPU time needed for convergence. The percentrelative deviation from LB is measured with:

Cost of heuristic algorithm 2 Cost of LB

Cost of LB£ 100:

The percentage relative improvement achieved by a meta-heuristic, TS or SAalgorithm is measured with:

Cost of meta-heuristic algorithm 2 Cost of SPT heuristic

Cost of SPT heuristic£ 100:

JQME13,4

404

The algorithms studied in this research are coded using FORTRAN 90. The numericalsimulation study is carried out on a stand alone 950 MHZ, Intel Processor with 256 MBRAM. Next, we conduct a comparison of the heuristic algorithms and a sensitivityanalysis for some related parameters as follows.

4.1. Effect of maximum allowable continuous operation timeIn Figure 1, the effect of maximum allowable continuous operation time (T) is studied.The performance measure considered is the deviation from LB. It is observed that withan increase in T the performance of SPT heuristic algorithm, as well as the LB,improves. This finding is further established in Figure 2, where the improvement of theTS and SA algorithm over SPT heuristic algorithm is reported. From the figure, we cannotice that the improvement TS and SA algorithm achieve over SPT heuristicalgorithm improves with a decrease in T. As identified earlier in this paper, the SPTheuristic algorithm is very competitive to two other algorithms, a branch and boundalgorithm and a FBH algorithm, as reported in Qi et al. (1999). Both the TS and SAalgorithms are still able to outperform SPT heuristic algorithm. The relative

Figure 2.Effect of maximum

allowed operation time (T)on error of SPT heuristic

Figure 1.Effect of maximum

allowed operation time (T)on performance

Study ofheuristic

algorithms

405

improvement at T ¼ 80 is about 0.60 percent, which increases up to 1.30 percent atT ¼ 50. The performance of TS and SA are nearly comparable; however, in somestudies SA supersedes TS algorithm.

4.2. Effect of maintenance timeSimilar to the study made for T, an analysis is also carried out to study the impact ofmaintenance time (t). From Figure 3, we can infer that the performance of SPT heuristicalgorithm, as well as the LB, deteriorates with the increase in maintenance time. Thisobservation is opposite to the finding from a sensitivity analysis with T. Theimprovement of TS and SA algorithms over SPT heuristic algorithm increases with anincrease in t. The relative improvement TS and SA algorithms are able to achieve overSPT heuristic algorithm is over 0.40 percent at t ¼ 10. The relative improvementincreases to up to 1.40 percent at t ¼ 40. Similar to a sensitivity analysis using T, wealso conclude that SA algorithm is marginally superior to TS algorithm.

4.3. Effect of problem sizeThe impact of an increase in the problem size, i.e. n, is also explored. With an increasein the problem size, the improvement TS and SA algorithm are able to achieve alsoincreases. This finding is reported in Figure 4. At n ¼ 20, the average reduction in totalcompletion time achieved by TS and SA is 25 and 35 units respectively. With n ¼ 30,this improves to 91 and 98 for TS and SA algorithms respectively. Finally for n ¼ 40,these values are 99 and 102 for TS and SA algorithm (Figure 5).

A comparative study of the CPU time taken by TS and SA algorithm is presented inFigure 6. We infer from the plot that the increase in the CPU time needed forconvergence of TS and SA algorithm unlike an implicit enumeration algorithmincreases non-exponentially. Both TS and SA algorithms are able to produce solutionsof good quality; however, as the problem size increases there is a marginal increase inthe CPU time requirement.

Figure 3.Effect of maintenance timeon performance

JQME13,4

406

Figure 4.Effect of maintenance

time (t)

Figure 5.Effect of problem size on

performance ofmeta-heuristic

Figure 6.Effect of problem sizeincrease on CPU time

Study ofheuristic

algorithms

407

5. Conclusion and future research suggestionsIn this paper, we studied the problem of scheduling jobs and maintenanceoperations on a single machine such that total completion time is minimized. Twometa-heuristics, TS, and SA are proposed to solve the problem. The properties ofan optimal schedule identified in the literature are embedded into TS and SAalgorithms in order to increase their efficiency. The strategy is shown to be veryeffective in reaching a near optimal solution with a reduced computational time.Three criteria are used to measure the performance of the existing and proposedheuristic algorithms that include deviation from a lower bound, improvement overSPT heuristic algorithm, and the CPU time taken. In a study based on numericalexperimentation, we conclude that:

. The lower bound and SPT heuristic algorithm are sensitive to maintenancerelated parameters. These parameters include, maximum time allowed forcontinuous operation (T) and the duration of maintenance (t).

. With an increase in the maintenance time (t), the performance of SPT heuristicalgorithm, as well as the lower bound, deteriorates and the TS and SAalgorithms are also able to improve over SPT heuristic algorithm with anincrease in the maintenance time for a given T. This is, because with an increasein t, the maintenance operations’ contribution gets substantial. Therefore therecould be some other sequencing minimizing the total completion time better thanthe SPT sequencing.

. The impact of maximum allowable continuous operation time (T) is opposite tothat of maintenance time. For a given maintenance time, a decrease in Tdeteriorates the performance of SPT heuristic algorithm as well as the lowerbound. The aforementioned rationale mentioned for performance improvementof TS and SA algorithms also implies under this situation.

. In this study, it is shown that embedding the characteristics of an optimalschedule improves the performance of meta-heuristics and it also helps in moredirected search which helps reduce the computational substantially.

There are several directions in which this research can be extended. An extendedanalysis would consider the possibility of machine failure along with the preventivemaintenance. This relaxes the assumption that the machine cannot fail but it israther maintained preventively. The multi-criteria scheduling can also beconsidered as an avenue of future work. The problem can also be consideredwith stochastic parameters such that the processing times of the jobs and themaintenance related parameters are random and follow a known probabilitydistribution function.

References

Adzakpa, K.P., Adjallah, K.H. and Lee, J. (2004a), “A new effective heuristic for the intelligentmanagement of the preventive maintenance tasks of the distributed systems”, AdvancedEngineering Informatics, Vol. 17 Nos 3-4, pp. 151-63.

Adzakpa, K.P., Adjallah, K.H. and Yalaoui, F. (2004b), “Scheduling with tool changes to minimizetotal completion time: basic results and SPT performance”, Journal of IntelligentManufacturing, Vol. 15, pp. 131-40.

JQME13,4

408

Aggoune, R. (2004), “Minimizing the makespan for the flow shop scheduling problem withavailability constraints”, European Journal of Operational Research, Vol. 153 No. 3,pp. 534-43.

Akturk, M.S., Ghosh, J.B. and Gunes, E.D. (2003), “Scheduling with tool changes to minimize totalcompletion time: a study of heuristics and their performance”, Naval Research Logistics,Vol. 50, pp. 15-30.

Cassady, C.R. and Kutanoglu, E. (2003), “Minimizing job tardiness using integratedpreventive maintenance planning and production scheduling”, IIE Transactions,Vol. 35, pp. 503-13.

Chen, Z.L. (2004), “Simultaneous job scheduling and resource allocation on parallel machines”,Annals of Operations Research, Vol. 129 Nos 1-4, pp. 135-53.

Chen, Z.L. and Powell, W.B. (2003), “Exact algorithms for scheduling multiple families of jobs onparallel machines”, Naval Research Logistics, Vol. 50, pp. 823-40.

Eglese, R.W. (1990), “Simulated annealing: a tool for operational research”, European Journal ofOperational Research, Vol. 46 No. 3, pp. 271-81.

Glover, F. (1986), “Future paths for integer programming and links to artificial intelligence”,Computers and Operations Research, Vol. 13 No. 5, pp. 533-49.

Glover, F. and Laguna, M. (1997), Tabu Search, Kluwer Academic Publishers, New York,NY.

Kirkpatrick, S., Gelatt, C.D. and Vecchi, M.P. (1983), “Optimization by simulated annealing”,Management Science, Vol. 22 No. 1, pp. 671-80.

Lee, C.Y. and Chen, Z. (2000), “Scheduling jobs and maintenance activities on parallel machine”,Naval Research Logistics, Vol. 47 No. 2, pp. 145-65.

Liao, C.J., Shyur, D.L. and Lin, C.H. (2005), “Makespan minimization for two parallel machineswith an availability constraint”, European Journal of Operational Research, Vol. 160 No. 2,pp. 445-56.

Lorigeon, T., Billaut, J.C. and Bouquard, J.L. (2002), “A dynamic programming algorithm forscheduling jobs in a two-machine open shop with an availability constraint”, Journal of theOperational Research Society, Vol. 53 No. 11, pp. 1239-46.

Lyu, J., Gunasekaran, A. and Ding, J.H. (1996), “Simulated annealing algorithm for solving thesingle machine early/tardy problem”, International Journal of Systems Science, Vol. 27No. 7, pp. 605-10.

Mazzini, R. and Armentano, V.A. (2001), “A heuristic for single machine scheduling withearly and tardy costs”, European Journal of Operational Research, Vol. 128 No. 10,pp. 129-46.

Qi, X., Chen, T. and Tu, F. (1999), “Scheduling the maintenance on a single machine”, Journal ofthe Operational Research Society, Vol. 50 No. 10, pp. 1071-8.

Raza, S.A. (2002), “Simultaneous maintenance and jobs scheduling on single machine undervarious maintenance policies”, Master’s thesis, King Fahd University of Petroleum andMinerals, Dahran.

Raza, S.A., Al-Turki, U.M. and Selim, S.Z. (2007), “Early tardy minimization for joint schedulingof jobs and maintenance operations on a single machine”, International Journal ofOperations Research, pp. 32-41.

Sait, S.M. and Youssef, H. (1999), Iterative Computer Algorithms with Applications inEngineering, IEEE Computer Society, Washington, DC.

Schmidt, G. (1988), “Scheduling independent tasks with deadlines on semi-identical processors”,Journal of Operational Research Society, Vol. 39, pp. 271-7.

Study ofheuristic

algorithms

409

Sortrakul, N., Nachtmann, H. and Cassady, C.R. (2005), “Genetic algorithms for integratedpreventive maintenance planning and production scheduling for a single machine”,Computers in Industry, Vol. 56 No. 2, pp. 161-8.

White, S.R. (1984), “Concept of scale in simulated annealing”, Proceedings of the InternationalConference on Computer Design, IEEE, Piscataway, NJ, pp. 646-51.

Yao, A.C. (1980), “New algorithms in bin packing”, Journal of the ACM, Vol. 27 No. 2,pp. 207-27.

Corresponding authorSyed Asif Raza can be contacted at: asif_s@encs.concordia.ca

JQME13,4

410

To purchase reprints of this article please e-mail: reprints@emeraldinsight.comOr visit our web site for further details: www.emeraldinsight.com/reprints