Considering scheduling and preventive maintenance in the flowshop sequencing problem

Computers & Operations Research 34 (2007) 3314–3330www.elsevier.com/locate/cor

Considering scheduling and preventive maintenance in theflowshop sequencing problem

Rubén Ruiz∗, J. Carlos García-Díaz, Concepción MarotoDepartamento de Estadística e Investigación, Operativa Aplicadas y Calidad, Universidad Politécnica de Valencia, Edificio 1-3,

Camino de Vera S/N, 66021, Valencia, Spain

Available online 3 February 2006

Abstract

The aim of this paper is to propose tools in order to implicitly consider different preventive maintenance policies on machinesregarding flowshop problems. These policies are intended to maximize the availability or to keep a minimum level of reliabilityduring the production horizon. It proposes a simple criterion to schedule preventive maintenance operations to the productionsequence. This criterion demonstrates the significance of taking into consideration preventive maintenance together with sequencingand the consequences of not doing so. The optimization criterion considered consists in minimizing the makespan of the sequenceor Cmax. In total, six adaptations of existing heuristic and metaheuristic methods are evaluated for the consideration of preventivemaintenance and they are applied to a set of 7200 instances. The results and experiments carried out indicate that modern Ant Colonyand Genetic Algorithms provide very effective solutions for this problem.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Flowshop; Sequencing; Preventive maintenance

1. Introduction

Production scheduling is one of the most important tasks carried out in manufacturing systems. It is responsible forthe scheduling of jobs in machines and the specification of the sequence and time to be carried out. Some productivesystems present a special configuration that has been widely studied in the literature. This configuration implies anatural ordering of the machines in the shop, in such a way that the jobs go through the same machines in the sameorder. This type of configuration is called flowshop.

Another task closely related to production scheduling in industrial settings is maintenance, understood as the op-erations or techniques that allow to maintain or restore equipment to a specific state and guarantee a given service.Usually, scheduling of maintenance operations and production sequencing are dealt with separately in the literatureand, therefore, also in the industry. Both activities conflict since, as it is known, maintenance operations consumeproduction time whereas delaying maintenance operations until the production sequence allows a period of free timemay increase the probability of machine failure.

∗ Corresponding author. Tel.: +34 96 387 70 07x74946; fax: +34 96 387 74 99.E-mail addresses: [email protected] (R. Ruiz), [email protected] (J. Carlos García-Díaz), [email protected] (C. Maroto).

0305-0548/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.cor.2005.12.007

http://www.elsevier.com/locate/cor

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R. Ruiz et al. / Computers & Operations Research 34 (2007) 3314–3330 3315

The objective of this paper is to provide tools that allow flowshop production scheduling by implicitly taking intoconsideration the necessary maintenance operations to achieve a high level of reliability in machines. Preventivemaintenance (PM) operations are first calculated and then sequenced along with the operations to be performed in theflowshop. This problem has been chosen because it is a very active field of research inside scheduling.

The paper is structured as follows: Section 2 introduces the topic and the basic concepts on preventive maintenance.Section 3 describes the problem under study in this paper: the permutation flowshop. Section 4 deals with the criterion forscheduling preventive maintenance operations to the production sequence. Section 5 experiments with six heuristic andmetaheuristic methods and carries out two experimental computations with 3600 instances each to show the efficiencyof the methods used. Finally, Section 6 presents the conclusions of this paper.

2. Preventive maintenance

In industry, companies are generally aiming at more reliable production systems with higher availability perfor-mance. Reliability and maintainability play a crucial role in ensuring the successful operation of plant processes asthey determine plant availability and thus contribute significantly to process economics and safety. Maintenance andmaintenance policy play a major role in achieving systems’ operational effectiveness at minimum cost. The traditionalapproach to maintenance planning involves selecting optimal policies from known maintenance strategies such asscheduled inspection, preventive maintenance, corrective maintenance, etc.

The analysis and modeling of maintenance operations have aroused the interest of several researchers, such asBorgonovo et al. [1], Brandolese et al. [2], Cassady and Kutanoglu [3], Dekker [4], Marseguerra and Zio [5], McCall[6], Pierskalla and Voelker [7], Valdez-Flores and Feldman [8], van Dijkhuizen and van der Heijden [9] and Zhao[10]. Reliability is a good indicator of the efficiency of a system. This usually decreases when the operation time ofthe machine or its components increases. In order to keep a minimum level of efficiency it is necessary to carry outmaintenance operations during the operational life of the system or machine.

Maintenance is understood as any activity carried out on a system to maintain it or to restore it to a specific state[11]. Maintenance operations can be classified into two large groups: corrective maintenance (CM) and preventivemaintenance (PM). CM corresponds to the actions to carry out when the failure has already taken place. PM is theaction taken on a system while it is still operating, which is carried out in order to keep the system at the desired level ofoperation. PM consists in carrying out the operations in machines and equipment before the failure or the breakdowntakes place, and at fixed time intervals previously established. The objective of PM is to prevent failures before theyhappen and therefore to minimize the probability of failure. The advantage of PM is that the system is always in goodconditions, thus reducing the risk of unexpected failures. The objectives of PM are:

• To increase the reliability of machines and therefore reduce failures during operation, which involves a reduction incosts and an improvement of availability.

• To increase the operational life of the equipment.• To improve production planning and management.• To ensure safety.

PM policies are based on the statistical theory of reliability. The necessary concepts to approach the problem andto achieve the objectives of this paper can be found in the extensive literature on reliability, such as Meeker andEscobar [12], Nelson [13], Rausand and HZyland [14] or Biroloni [11]. Some of these basic concepts are describednext.

Let T a be non-negative random variable representing the lifetime of the production machine. Usually, the reliabilityof a component is given in terms of probability distributions that model the random variable T.

The Weibull distribution model with two parameters, T ≈ W [�, �], is one of the most frequently used modelsbecause of its flexibility to characterize the time to failure of components with non-constant failure rates. The reliabilityfunction of the Weibull model is as follows:

R(t) = exp

[−

(t

�

)�]

, t > 0 (1)

3316 R. Ruiz et al. / Computers & Operations Research 34 (2007) 3314–3330

which depends on two parameters called shape parameter (�) and scale parameter (�). The instantaneous failure rateof this model is

Z(t) = �

��· t�−1, (2)

where it can be seen that the shape parameter � allows to adapt to components with decreasing failure rate when � < 1,with constant failure rate when � = 1 and with increasing failure rate when � > 1. The mean time to failure (MTTF) isthen

MTTF = � · �

[1 + 1

�

], (3)

where �[·] is the tabulated gamma function. Weibull model matches the exponential model when � = 1. When � = 2the Weibull model has a linearly increasing failure rate and is known as the Rayleigh model.

A machine or system is considered repairable if once the failure has taken place, it can be restored to its as-good-as-new original state. The time elapsed between the failure and its return to the as-good-as-new conditions is calleddowntime or repair time. The average time to repair is called mean time to repair or MTTR.

The availability A(t) in time t of a repairable component is the probability that the component is working at time t.If the component is non-repairable, then A(t) = R(t). The limiting availability A is defined as the fraction of time thatthe production system is operational, if observed over an infinite period of time. Traditionally, the optimal preventivemaintenance for a production system has been determined in view of maximizing its limiting availability. The limitingavailability can be calculated as

A = MTTF

MTTF + MTTR. (4)

The so-called model with minimum repair represents an interesting model for the study of reliability of a systemor machine. If repair or substitution of components that have failed returns the complete system to the operationsconditions but the failure rate of the system remains just like before the failure, then the repair is called minimum[15–18]. The minimum repair model usually involves that the failure rate of the system is an increasing function withtime, the repairs do not affect the failure rate of the system, the cost of the repair is lower than the cost of substitutionof the complete system and the failure of the system is immediately detected.

Therefore, several PM policies should be defined, with the aim of determining when it is necessary to carry out PMoperations in the machines according to different criteria. Three possible policies are defined in the following sections.

2.1. Policy I: preventive maintenance at fixed predefined time intervals

This type of policy is widely used in industry. PM operations are planned beforehand, at predefined time intervalswithout considering probabilistic models for the time to failure and making the best use of scheduled stops after weekly,monthly or even annual cyclical production periods. If there are no scheduled stops (high production load), productionis interrupted to carry out PM or is not even performed. In this latter case, production is usually affected when accidentalbreakage of machines takes place due to a lack of PM.

2.2. Policy II: optimum period model for the preventive maintenance maximizing the machines’ availability

In classical maintenance theory, an optimal preventive maintenance interval for an unreliable production system hasbeen determined by maximizing its limiting availability given by (4). Availability combines the effects of a failureand repair, and the effects of downtime caused by the preventive maintenance. A quantitative availability performancecalculation is based on the reliability model of a system and on the failure and repair data of the machines.

According to this policy, PM is performed according to the optimal maintenance period. We assume that the time tofailure follows a Weibull probability distribution, T ≈ W [�, �], with � > 1. tr is the number of time units the repairtakes and tp the number of time units of the PM. The PM action restores the machines to the as-good-as-new state,which allows to model the operation cycle and the PM as a renovation process. It is considered that the repair of thesystem is minimum and therefore the number of failures during each cycle of the renovation process can be modeled as


0.0

0.2

0.4

0.6

0.8

1.0

0 13 17t

Rel

iabi

lity

R(t)

Rs(t)

5 9

Fig. 1. Reliability of a system subject to systematic maintenance vs. a system without maintenance.

a non-homogenous Poisson process (time-dependent failure rate [16]). Let TPM be the interval between two consecutivePM. The objective of this policy is to maximize the availability of the system given by (4).

With these starting hypotheses, and according to Cassady and Kutanoglu [3], the optimal maintenance interval TPMopcan be calculated by means of:

TPMop = � ·[

tp

tr(� − 1)

]1/�

. (5)

In sum, we can state that policy II consists in performing PM whenever a machine is TPMop time units in operation. Forexample, let us suppose that the time to failure follows a Weibull model, T ≈ W [�, �], with � = 1349 and � = 2, andwith tr = 6 h and tp = 1 h. In this case, the optimal PM interval, TPMop, is 550.73 h that, upon applying Eq. (1), involvesapproximately a reliability of 85%.

2.3. Policy III: maintaining a minimum reliability threshold for a given production period t

Suppose a system whose failure rate is increasing with time and therefore it may be affected by failures due to agingor wear. This policy consists in making a systematic PM after a time TPM so as to ensure a minimum reliability of thesystem from time t = 0. We consider that PM restores the machines to the as-good-as-new state. In this case, PM willbe carried out at regular intervals 0, 1TPM, 2TPM, 3TPM, . . . , nT PM that are considered as renovation points.

The reliability function R0(t) can be calculated according to Biroloni [11] and taking into account the followingconsiderations: it is fulfilled that t =nT PM + t∗, where, t∗ < T , and therefore the probability of exceeding time t, R0(t),seen from moment zero, is

R0(t) = R(t) for 0� t < TPM,

R0(t) = [R0(TPM)]nR0(t − nT PM) for nT PM � t < (n + 1)TPM, n�1,

where R0(0) = 1. Overall, it can be considered that R(t − nT PM) = 1, that is, before exceeding the period TPM thesystem does not fail and therefore we can consider that

R0(t) = [R0(TPM)]n. (6)

Fig. 1 shows the reliability level achieved by the system subject to systematic PM R0(t) = RS(t) and the reliability ofthe system, R(t), when it is not subject to the PM policy.

When the time to failure random variable follows a Weibull model, T ≈ W [�, �], with � > 1 (the failure rateis increasing with time), the period between consecutive PM actions (TPM) can be derived by considering


Eqs. (1) and (6). Let us consider that R0(t) = [R0(TPM)]n and therefore ln[R0(t)] = n ln[R0(TPM)]. On the other hand,

R0(TPM) = exp

[−

(TPM

�

)�]

and therefore

ln[R0(t)] = n ln

{exp

[−

(TPM

�

)�]}

.

and considering that t = nT PM, the expression for the calculation of the time between PM given by the followed isobtained:

TPM =[−�� ln R0(t)

t

]1/(�−1)

. (7)

The following example will help understand this PM policy: let us consider that the time to failure of a machine followsthe distribution T ≈ W [�, �], with � = 1349 and � = 2. The objective is to determine the time between PM so that thereliability after t = 720 h (work horizon) is 95% (R0(720)= 0.95). Considering Eq. (7) we can obtain TPM = 129.65 h,which involves to perform PM approximately every 130 h of operation of the machine. If the machine works 24 h perday, it involves to carry out PM operations every 5.5 days approximately.

Another question is the duration of the preventive maintenance operation, which we call DPM. As we will see inSection 5, we assume that the duration of preventive maintenance actions is uniformly distributed in a wide range ofvalues. This allows for the consideration of short maintenance actions like cleaning, tightening of bolts or lubricationas well as for longer maintenance actions like replacements of parts or thorough inspections.

3. Scheduling: the flowshop problem

One of the most frequent production scheduling problems is the flowshop problem (FSP). In this problem, we find aset N ={1, . . . , n} of n jobs that are processed in a set M={1, . . . , m} of m machines. Each job j, j ∈ N , requires a fixedand known amount of processing time in each machine i, i ∈ M . This amount is represented by pij . In the FSP all n jobsare processed in the m machines in the same order, that is, the jobs follow the same route, starting by machine 1 untilfinishing in machine m. The objective is to find a production sequence of the jobs in the machines so that an establishedcriterion is optimized. The most common optimization criterion is the minimization of the total manufacturing time,called makespan or Cmax. Considering that there are n! possible permutations of jobs and m machines, the total numberof possible sequences is (n!)m. A frequent simplification is to assume that the job permutation is maintained for all themachines, so that “only” n! sequences are considered. In this case, the problem is called permutation flowshop (PFSP)and is denoted by F/prmu/Cmax following Pinedo’s notation [19]. The PFSP is an NP-Hard problem [20].

Additionally, the following assumptions and restrictions are established:

• All the tasks and jobs are independent and available for their process at time 0.• The m machines are continuously available.• Each machine i can only process a job j simultaneously.• Each job j can only be processed in a machine i simultaneously.• The process of a job j in a machine i cannot be interrupted.• The setup and removal times are sequence independent and included in the processing times or can be neglected.• There is an unlimited in-progress product inventory. If a job needs a machine that is occupied, it waits indefinitely

until it is available.

This problem was studied for the first time by Johnson [21] and from that moment a large number of techniquesand algorithms to solve it have appeared, from simple heuristics to complex and modern metaheuristic techniques.Some examples are the CDS heuristic by Campbell et al. [22] and the constructive and improvement heuristics byDannenbring [23]. The NEH heuristic by Nawaz et al. [24] is usually considered as one of the best methods for the


PFSP, especially if it is implemented efficiently. Other more recent algorithms include improvement heuristics like inSuliman [25], modern Genetic Algorithms [26] or Ant Colony Algorithms [27], among others. An updated revision andcomparative study of heuristic and metaheuristic algorithms for PFSP can be seen in Ruiz and Maroto [28].

From the aforementioned assumptions, the second one is especially important from the point of view of PM. Inpractice, not all m machines are continuously available. In addition, sequencing as well as PM are intimately relatedsince the processing of the jobs wears the machines and makes PM necessary to avoid, as far as possible, accidentalbreakage. Furthermore, PM actions affect the production sequence since machines cannot process jobs while PM isbeing carried out. Therefore, it is necessary to consider PM implicitly in production scheduling methods.

4. Scheduling and preventive maintenance

As it has been mentioned, little work has been carried out in which PM operations scheduling and flowshop schedulingare jointly considered. The already mentioned paper of Cassady and Kutanoglu [3] is one of the few cases where anintegrated model for job and PM operations scheduling is shown. However, the authors propose a total enumerationapproach that is tested against very small problems with eight jobs maximum which is not practical. Sloan andShanthikumar [29] also propose a combined method and application to the semiconductor manufacturing sector forthe single machine case. A more complex hybrid flowshop problem is tackled in Allaoui and Artiba [30], but in thiswork, as in the work of Aggoune [31], PM operations are given as problem data and considered as constraints, i.e., notas a result of the machine operation like we consider in this work (it could be said that this is a “Policy I” approach).There is a host of existing work where machine failure is considered and complex formulae as well as simulations areproposed in order to minimize the “expected makespan” measure, some examples are the work of Gourgand et al. [32]for flowshop or Lee and Lin [33] for single machine. While this body of work is very important, it is difficult to applyin practice and what we propose in this paper is a simpler way of considering PM operations scheduling.

PM operations should be performed after each machine has exactly been in operation TPMop time units, if we considerpolicy II (Eq. (5)) or after TPM time units in operation if policy III is applied (Eq. (7)). The problem is that during aPFSP, jobs cannot be interrupted once their processing has started in the machines, therefore it is necessary to establisha criterion to schedule PM operations to the production sequence.

We tested several different criteria, from simple ones to complicated methods. It turned out that a simple criterionyielded very good results in many different situations despite its simplicity. The criterion is as follows: whenever a newoperation is to be processed in each machine, the total processing time accumulated is verified. If this time is higherthan TPMop or TPM, then the process of the next job is postponed and the PM job is carried out first. This schedulingcriterion is conservative since it guarantees that no machine will be working for a longer time than the specified withoutperforming PM.

As we can see, the simplicity of the criterion allows to consider the scheduling of PM operations with ease. As amatter of fact, since no complex formulae are involved, the criterion can be readily implemented in spreadsheets or inexisting methods with just simple codings.

The following example illustrates the operation of the proposed scheduling criterion: let us suppose that we have aflowshop problem with five jobs and three machines, that is, n = 5 and m = 3. The processing times (pij ) are shownin Table 1.

As there are five jobs, we have a total of 5! = 120 possible sequences or solutions for this simple example. We canobtain the optimal solution by enumeration; this solution is {1, 2, 3, 5, 4} with a Cmax of 235 time units. The GanttChart for this solution is shown in Fig. 2.

Table 1Processing times (pij ) for the PFSP example with n = 5 and m = 3

Machines (i) Jobs (j)

1 2 3 4 5

1 3 45 84 24 392 38 51 7 21 163 52 83 19 6 28


Machine 1

Machine 2

Machine 3

makespan: 235

TimeJob 1 Job 2 Job 3 Job 4 Job 5

300250200150100500

Fig. 2. Gantt chart with the optimal solution for the example.

300250200150100500

Machine 1

Machine 2

Machine 3

makespan: 333

Time

PreventiveMaintenance

Job 1 Job 2 Job 3 Job 4 Job 5

Fig. 3. Gantt chart after applying the PM task scheduling criteria to the optimal solution of Fig. 2.

It can be seen that machine 1 is in operation for a total of 195 time units, whereas machines 2 and 3, 133 and 188,respectively.

Let us suppose now that TPM = 120 for all the machines, which means that for each 120 units of operational time,machines have to be stopped to perform PM. For machine 1 we will have to perform at the most � 195

120� = 2 PMoperations, the same applies to machines 2 and 3. Let us suppose also that the durations of these 2 PM operations(DPM) for machine 1 are 31 and 43, for machine 2, 45 and 42 and for machine 3, 8 and 45. Therefore, in the worst casewe will have to perform 6 PM operations in the machines.

Applying the scheduling criterion of these PM operations to the previous optimal solution ({1, 2, 3, 5, 4}), obtainedwithout considering preventive maintenance, yields the Gantt Chart shown in Fig. 3.

As we can see, the first maintenance operation in machine 1 is carried out too soon, because after processing jobs 1and 2, the accumulated total processing time is 3+45=48 and it is not possible to process the third job of the sequence(3), since it has a processing time of 84 units, which would result in an accumulated total processing time of 132 units,which is above TPM (120). In the same way and in this same machine, after carrying out the first PM with duration31, job 3 is processed and after this, once more it is not possible to process the next job (5) since a processing time of84 + 39 = 123 would be accumulated. The same reasoning is applied to all the machines, thus obtaining Cmax of 333units.

It is interesting to calculate the optimal solution of the previous problem but applying the scheduling criterion of thePM operations in all the possible solutions. That is, in the previous case the PM scheduling criterion has been appliedto the optimal solution obtained by ignoring PM. If we calculate the 5! = 120 possible solutions to the PFSP and applyto each one of them the scheduling criterion of PM we obtain the optimal solution shown in Fig. 4.

As it can be seen, the optimal solution in this case is {1, 5, 2, 4, 3}, which is quite different from the initial optimalsolution. In addition, the Cmax is now 252 time units, which implies an increase on the initial optimal solution of only


Job 1 Job 2 Job 3 Job 4 Job 5

300250200150100500

Machine 1

Machine 2

Machine 3

makespan: 252

TimePreventive

Maintenance

Fig. 4. Gantt chart with the optimal solution for the example considering the PM operations.

7.23%. The solution obtained after applying the PM scheduling criterion to the initial optimal solution (Fig. 3, withCmax =333) is 32% worse than the solution obtained if PM is considered implicitly. The explanation is given by the factthat the optimal sequence when PM is considered is not usually the same as if PM is not considered. The opposite case isalso true, in fact, the sequence obtained with PM ({1, 5, 2, 4, 3}) gives a Cmax of 246 if the PM operations are discarded,which is worse than the original Cmax (235). While this result is expected, it is the difference between the makespanvalues which results to be significant. Clearly, implicitly considering the PM operations during the optimization processresults in much better solutions.

The previous example demonstrates the importance of implicitly considering PM operations in the construction ofthe production sequence to optimize Cmax. All the existing constructive heuristics for the PFSP do not consider PMoperations and in many cases the adaptation to consider PM operations is not obvious. For example, the Rapid Access(RA) heuristics by Dannenbring [23] or the CDS method by Campbell et al. [22] are based on the application of thewell-known rule by Johnson [21], therefore the result is only one sequence. Applying the PM scheduling criterion tothis sequence could give very poor results, as it has been seen in the previous example. However, heuristics like NEHby Nawaz et al. [24] and several metaheuristic methods calculate a large number of sequences iteratively, thereforeit is possible to modify these methods to include the application of the PM scheduling criterion (at each stage andcalculation of the Cmax).

We included the PM operation scheduling criterion described to the mentioned NEH heuristic. Recent studies [28]place the NEH method as the best constructive heuristic for the PFSP problem even when compared against morerecent and complex methods. More precisely, in this NEH and in all following methods, the PM scheduling criterionis applied at every calculation of the optimization criterion, i.e., every time the different algorithms calculate the Cmaxfor a given permutation. In this moment the PM scheduling criterion is incorporated and the returned Cmax includesthe PM operations.

Among metaheuristic methods, we have selected two classical and two recent algorithms that can be easily adaptedwith the PM scheduling criterion. Two of the earliest metaheuristics applied to the PFSP are the Tabu search of Widmerand Hertz [34], known as SPIRIT and the Simulated Annealing by Osman and Potts [35] (SA_OP). Therefore, wechoose these two algorithms for the computational evaluation.

On the other hand, we have selected two very recent and advanced methods: the Genetic Algorithm by Ruiz et al.[26] (GARMA) and the Ant Colony Optimization algorithm by Rajendran and Ziegler [27] (PACO). These algorithmsmake use of complex local search operators as well as populations of solutions that might prove useful for the problemconsidered.

For comparison reasons, we include a simple rule that generates random sequences and returns the best sequencegenerated as a result. We will call this latter rule RANDOM and will serve as a worst-case scenario. In total, we takeinto consideration six adapted methods, four of them metaheuristic, one heuristic and the random rule. The choiceof these methods is clear; either they have shown a very good performance in regular flowshop environments or aregeneral enough in the sense that they can be modified so as to consider PM operation scheduling. Therefore, they arethe ideal candidates for testing the PM operation scheduling criterion rather than implementing it in other new methodsnot previously tested in the PFSP problem.


It could be argued that these methods (specially SPIRIT and SA_OP) are not among the best methods availablefor the PFSP. It is known that the critical path-based methods like the Tabu Search TSAB algorithm of Nowicki andSmutnicki [36], the improved version of Grabowski and Wodecki [37] or the Genetic Algorithm with path relinking ofReeves and Yamada [38] are some of the finest (as well as most complex) algorithms available for the PFSP. However,the critical path theorems are not valid under the assumption of PM scheduling and therefore, most of the speed-upsand advantages of these methods are lost. Still, it would seem plausible to rework these algorithms in order to includethe PM scheduling. However, since these algorithms are very intricate, we contacted the corresponding authors of thethree aforementioned papers in order to obtain an executable or the source code of their algorithms. In all three casesthe source code was not provided. Therefore, we have not included these methods in the computational evaluation.

5. Computational evaluation

Section 2 described three policies to determine when to carry out PM operations. Next, we will apply the heuristicand metaheuristic algorithms previously cited with the mentioned policies, with the exception of policy I that actuallyrepresents the common practice where PM depends on the progress of production.

5.1. Policy II

As it was mentioned in Section 2.2, policy II models the PM operations as a renovation process. The interval ofaccomplishment of the PM operations (TPMop) is given by Eq. (5). As it can be seen, the mentioned equation workswith four parameters: �, �, tp and tr.

In order to evaluate the effectiveness of the different heuristic and metaheuristic methods it is necessary to establisha set of experiments that take into consideration a large number of possible productive configurations. As it is known,the number of jobs n and the number of machines m in a PFSP instance clearly determine its difficulty. With theaddition of the PM operations we may find situations where the machines are “new” and require more spaced PMoperations, or “aged” machines where the PM operations are more frequently performed. With all this, the followingexperiments have been designed: first, we have n={20, 50, 100, 200, 500} and m={5, 10, 15, 20}, therefore there are20 configurations of n and m where the processing times (pij ) are distributed as a U [1, 99] as it is common in flowshopscheduling research. For each configuration, � = {2, 3, 4} is defined and the duration of the PM operations (DPM) asU [1, 24], U [1, 49], U [1, 74], U [1, 99], U [1, 124] and U [1, 149]. That is, there are six cases where the average DPMis 25%, 50%, 75%, 100%, 125% or 150% of the average processing times.

In the case of policy II, tp is set at 1 and tr at 8 for all the experiments. � is set according to the number of jobs,therefore � = 1500, 2300, 3200, 4500 and 7200 for n = 20, 50, 100, 200 and 500, respectively. The levels of � arechosen so to make sure that a significant number of PM operations would be carried out in each machine. For example,a small value for � would result in very little or even no PM operations while a very large value would probably impedecarrying out certain processing of jobs on machines without interruptions due to the small amount of time between PMoperations. Finally, for each configuration of n, m, � and DPM there are 10 different problems, which results in a totalset of experiments of 3600 instances.

With all these data it is easy to calculate TPMop with Eq. (5). For example, in the case of n = 100 and � = 3 andaccording to the above � = 3200, tp = 1 and tr = 8, we will obtain a TPMop of 1269.92.

With these instances we test the six algorithms.All the methods considered (with the exception of the NEH procedure)have a stopping criterion. This is set at a fixed elapsed time, which is given by expression n · (m/2) · 25 ms. Thus, forexample, for the instances with n = 500 and m = 20 (the largest), the algorithms iterate for 125 s. All the experimentshave been carried out with a Pentium IV PC with an Intel processor running at 2.8 GHz and with 512 Mbytes of RAMmemory.

Once the Cmax of each algorithm has been obtained for the 3600 instances, we calculate the best solution obtainedfor each instance by any of the six algorithms, which we call Minsol. With this, we transform the results to calculatethe relative percentage deviation with respect to this best solution with the following expression:

relative percentage deviation (RPD) = Algsol − Minsol

Minsol· 100, (8)


Table 2Average relative percentage deviation (RPD) for the six algorithms grouped by n and m for the PM policy II

Instance RANDOM NEH SA_OP SPIRIT GARMA PACO

20 × 5 5.77 2.43 0.87 3.16 0.15 0.6520 × 10 9.36 2.25 1.38 2.85 0.26 0.2120 × 15 5.61 0.59 0.50 2.41 0.08 0.0520 × 20 6.15 2.51 0.90 5.02 0.12 0.7950 × 5 7.47 2.18 1.05 2.42 0.18 0.2850 × 10 8.69 1.76 1.14 2.46 0.16 0.1050 × 15 5.03 1.46 0.64 4.98 0.07 0.2650 × 20 7.75 4.19 1.66 3.18 0.17 0.93100 × 5 6.83 1.89 0.97 1.83 0.22 0.08100 × 10 6.72 1.03 0.77 4.81 0.08 0.12100 × 15 5.74 4.27 1.69 3.01 0.18 1.12100 × 20 8.72 4.32 1.79 2.42 0.29 0.48200 × 5 5.43 1.40 0.61 2.37 0.09 0.14200 × 10 6.09 4.03 1.58 5.28 0.11 1.29200 × 15 7.14 4.19 1.25 2.29 0.22 0.64200 × 20 7.64 3.65 1.14 2.05 0.14 1.67500 × 5 4.84 2.54 0.75 5.01 0.06 0.71500 × 10 7.58 6.01 2.02 2.98 0.36 1.43500 × 15 6.48 3.32 1.01 1.38 0.22 0.88500 × 20 6.26 2.63 0.85 4.80 0.03 1.21

Average 6.77 2.83 1.13 3.24 0.16 0.65

where Algsol is theCmax obtained for a given algorithm and instance. This RPD measure is a common and straightforwardway of comparing algorithms since an RPD of 5% for a given algorithm means that this algorithm is 5% over the bestobtained solution on average. Obviously, lower values of RPD are preferred.

The results of the experiments, averaged for each one of the n and m configurations (180 data per average) are shownin Table 2.

As it could be expected, the RANDOM rule provides the worst average results. The NEH heuristic obtains muchbetter results, which even exceed those obtained by SPIRIT tabu search. This is a good result since the total time of NEHand SPIRIT to evaluate the 3600 instances has been 2.81 and 27.19 h, respectively. SA_OP obtains a better result thanSPIRIT. From the experiment, the two best algorithms are the GARMA Genetic Algorithm and the PACO Ant ColonyAlgorithm. In almost all the occasions, the best solution for the 3600 instances was obtained by PACO or GARMA,although it seems that GARMA is better than PACO.

In order to verify the statistical validity of the results shown in Table 2 and to ascertain which is the best algorithm,we have performed a multifactor ANOVA where the response variable is log(RPD + 0.1) and the factors are n, m, �,DPM and type of algorithm. The transformation of the response variable was necessary in order to ensure compliancewith ANOVA’s three important hypotheses; normality, homogeneity of variance and independence of the residuals. Themeans plot and least significative differences (LSD) intervals for the type of algorithm factor is shown in Fig. 5.

Recall that overlapping LSD intervals for two given means depict that no statistically significant differences areobserved for the means and thus can be considered equal. As it can be seen, there are statistically significant differencesbetween the six algorithms considered, GARMA being the best algorithm for the PM policy II. However, as shown inTable 2, the average differences between PACO and GARMA are not large and it can be stated that both algorithmsare able to find very good solutions.

5.2. Policy III

The objective of policy III is to maintain a minimum level of reliability for a production period t. The interval of thePM operations (TPM) is given by Eq. (7). In this case we have four parameters: �, �, R0(t) and t. Since both policies IIand III are not comparable, it is necessary to define a new set of instances to evaluate the effectiveness of the differentadapted methods. The same configurations of n, m, �, � and DPM as in the case of policy II are considered, and therefore


Algorithm

log

(RP

D+

0.1)

-1.9

-0.9

0.1

1.1

2.1

SPIRIT

SA_OP

PACONEH

GARMARANDOM

Fig. 5. Means plot and LSD intervals (at the 95% confidence level) for the type of algorithm factor in the PM policy II.

Table 3Average relative percentage deviation (RPD) for the six algorithms grouped by n and m for the PM policy III

Instance RANDOM NEH SA_OP SPIRIT GARMA PACO

20 × 5 8.93 8.57 6.02 7.92 5.71 0.1920 × 10 8.64 7.41 3.74 6.14 2.57 0.5620 × 15 7.80 6.70 3.31 5.26 1.82 1.0620 × 20 6.86 5.72 2.81 4.52 1.43 0.8850 × 5 15.10 11.91 11.18 12.27 10.83 0.0550 × 10 11.65 6.67 3.77 5.11 2.54 0.2650 × 15 11.43 6.72 3.40 4.21 1.39 0.5350 × 20 10.52 5.77 3.10 3.67 1.07 0.55100 × 5 15.54 12.13 11.85 12.62 11.71 0.03100 × 10 10.42 4.36 2.71 3.94 2.09 0.16100 × 15 11.23 5.15 2.84 3.82 1.76 0.29100 × 20 10.84 5.06 2.85 3.29 1.32 0.35200 × 5 11.90 8.73 8.88 10.71 8.48 0.04200 × 10 8.23 2.12 1.85 5.34 1.09 0.12200 × 15 9.49 2.92 1.87 6.01 0.89 0.25200 × 20 9.67 3.38 2.06 6.05 1.01 0.40500 × 5 3.20 0.59 1.08 3.97 0.58 0.01500 × 10 5.77 0.38 0.93 7.49 0.24 0.01500 × 15 7.09 0.74 1.12 9.22 0.33 0.10500 × 20 7.35 1.17 1.32 9.50 0.53 0.09

Average 9.58 5.31 3.83 6.55 2.87 0.30

a set of 3600 instances is also obtained. The aim is a 95% reliability after the production period t, therefore R0(t)=0.95.In order to calculate TPM it is still necessary to determine period t, which can be easily obtained from the processingtimes (pij ) of the instances. Since these are distributed as a U [1.99], then, t � n · 50. Thus, we have t = 1000, 2500,5000, 10 000 and 25 000 for n = 20, 50, 100, 200 and 500, respectively. For example, in the case of n = 500 and � = 4we have � = 7200 and t = 25 000, therefore TPM = 1766.65.

With this new set of instances we carry out the evaluation of the six algorithms considered as in policy II. The resultsare shown in Table 3.

Noticeably, the mean deviations are higher than in the case of policy II. The explanation is given by the fact thatwhen setting R0(t) = 0.95, the reliability maintained in the machines is very high and the PM operations are carried


-1.8

-0.8

0.2

1.2

2.2

Algorithm

log

(RP

D+

0.1)

GARMANEH PACO

SA_OPSPIRIT

RANDOM

Fig. 6. Means plot and LSD intervals (at the 95% confidence level) for the type of algorithm factor in the PM policy III.

β

log

(RP

D+

0.1)

-2

-1

0

1

2

3

2 43

GARMA

PACO

RANDOM

Algorithm

NEH

SA_OP

SPIRIT

Fig. 7. Means plot and LSD intervals (at the 95% confidence level) for the interaction between the factors type of algorithm and � in the PMpolicy III.

out much more often than in policy II. From this, we conclude that the problems are much more difficult for almostall the algorithms as the frequency of the PM operations increases. With these results, we now have that the algorithmPACO is the most effective of the comparison. Additionally, PACO is in this case more effective than GARMA by aquite considerable difference. The relationships between the other algorithms that were already obtained in the case ofpolicy II are maintained for this policy III.

Another ANOVA has been carried out to verify the statistical validity of the results. The LSD plot for the type ofalgorithm factor is shown in Fig. 6.

Once again, there are statistically significant differences between the six algorithms considered, PACO being thebest algorithm for the PM policy III.

TheANOVA analysis, which also controls � and DPM as factors, allows to study the effects on the different algorithmswhen the durations of the PM operations or � are increased. For example, the interaction between � and the type ofalgorithm is shown in Fig. 7.

It is interesting to see that the algorithm GARMA is almost as good as PACO when �=2 whereas for �=3 or 4 PACOis clearly better. There is not a clear trend or indication that increasing � has a negative effect over the performance ofthe algorithms.


GARMA

PACO

RANDOMAlgorithm

NEH

SA_OP

SPIRIT

DPM

log

(RP

D+

0.1)

-2

-1

0

1

2

3

25 50 75 100 125 150

Fig. 8. Means plot and LSD intervals (at the 95% confidence level) for the interaction between the factors type of algorithm and DPM in the PMpolicy III.

Now, the interation between DPM and the type of algorithm is shown in Fig. 8.As we can see, there is almost no effect on the different algorithms as regards the distribution of the PM durations.

For some algorithms, longer PM operations (on average) result in slightly more difficult problems while in others thereis no clear trend.

The last two figures show an interesting result. The algorithms adapted show a robust performance in a wide rangeof situations covered by the different values of � and DPM.

5.3. Evaluating the impact of considering scheduling and preventive maintenance

Once both policies have been evaluated, the result is that for policy II the algorithms GARMA and PACO are the mosteffective with a slight advantage to the former. In the case of policy III (more difficult problems), PACO provides muchbetter results than the rest. Section 4 described that implicitly considering preventive maintenance operations duringthe construction of the sequence could provide a better result than simply sequencing these operations afterwards. Insum, it is interesting to carry out the same analysis for the sets of experiments generated.

Thus, we test PACO algorithm (with the same stopping criterion considered previously) with the sets of experiments(policies II and III) without considering the PM operations, that is, as a standard permutation flowshop. Then, weapply the scheduling criterion of the PM operations to the solution obtained to achieve a sequence where preventivemaintenance has been considered “a posteriori”. This solution is compared with that obtained by the PACO algorithmin Tables 2 and 3. Obviously, the expected results is that implicitly considering PM operations in the scheduling shouldprovide better results than if considered “a posteriori”. However, with this experiment we aim at quantifying thisdifference. The result is shown in Table 4.

The table shows three columns per policy. Column “After” indicates the average relative percentage deviation resultingfrom adding the PM operations after having obtained the sequence resulting from the PACO algorithm. Column “During”contains the average relative percentage deviation obtained by algorithm PACO where the PM operations are consideredimplicitly at every iteration (Tables 2 and 3). Finally, the column labeled � shows the difference between the first two.Columns “After” and “During” are expressed as the average relative percentage deviation with respect to the solutionobtained by the PACO algorithm without considering the PM operations (as in a standard flowshop).

The first relevant result is that implicitly considering PM operations always provides better results than if these areapplied “a posteriori”. In the case of policy II, the PM operations are scarce and very distanced, therefore the sequencesare an average of only 4.59% longer with maintenance than without it (case “After”). Even so, considering maintenanceimplicitly in the optimization process results in sequences of an average 1.45%, which is shorter than those obtainedif maintenance is considered “a posteriori” (case “During”).


Table 4Average relative percentage deviation (RPD) for the PACO algorithm considering the preventive maintenance operations after and during theoptimization for policies II and III

Instance Policy II Policy III

After During � After During �

20 × 5 6.45 2.40 4.05 24.03 6.41 17.6220 × 10 7.05 1.94 5.11 20.54 6.58 13.9620 × 15 7.02 2.13 4.89 18.94 6.59 12.3520 × 20 6.57 2.14 4.44 17.05 6.28 10.7750 × 5 4.56 1.82 2.74 30.11 6.27 23.8450 × 10 5.70 1.37 4.33 23.71 8.04 15.6750 × 15 6.30 1.61 4.69 23.35 8.69 14.6650 × 20 6.53 2.13 4.40 23.19 9.11 14.08100 × 5 3.34 1.42 1.93 93.86 63.26 30.60100 × 10 4.07 1.18 2.89 87.91 69.62 18.28100 × 15 5.17 1.25 3.92 83.59 65.69 17.90100 × 20 6.08 1.46 4.62 77.97 61.62 16.35200 × 5 2.47 1.07 1.40 28.19 7.87 20.32200 × 10 2.96 1.07 1.89 22.87 9.15 13.72200 × 15 4.22 1.12 3.10 24.63 10.48 14.15200 × 20 4.90 1.41 3.49 25.60 11.28 14.32500 × 5 1.55 0.73 0.82 20.07 8.38 11.69500 × 10 1.73 0.75 0.98 20.86 9.05 11.81500 × 15 2.14 0.83 1.31 22.61 10.01 12.60500 × 20 3.03 1.23 1.81 23.67 10.83 12.84

Average 4.59 1.45 3.14 35.64 19.76 15.88

The situation is completely different for policy III. In this case, the resulting average increase from considering thepreventive maintenance “a posteriori” is 35.64% while if this is considered implicitly the increase is reduced to 19.76%.That is, implicitly considering preventive maintenance operations in the optimization process achieves an improvementof 15.88%. Additionally, this improvement is very significant for some groups of problems, for example, for the caseof n = 100 and m = 5, the improvement is more than 30%.

As mentioned, this result was expected but one should consider that the proposed rule for sequencing PM operationsis very simple and can be easily added to existing methods and even in this case, the results obtained are very good.

5.4. Practical application

All aforementioned methods, including the PM operations scheduling criterion, have been implemented in a pro-duction scheduling software and are now being tested in realistic settings (see [39] for more details). This software isknown as ProdPlanner and is aimed at the ceramic tile manufacturing sector where complex maintenance and schedul-ing problems are found. Fig. 9 shows an screenshot of the software where preventive maintenance along with tasks(ceramic tile lots) are being scheduled.

Obtaining real results is difficult mainly due to the reluctance of plant managers to change PM plans. Some facilitiesin ceramic tile manufacturing are very expensive (like for example the kilns) and the plant managers prefer to adhere toold practices that consist in stopping the production facilities at predefined intervals and to carry out forced preventivemaintenance actions like changing the kiln rollers. Many times the workers find that the kiln rollers are still practicablebut are forced to change them since the next scheduled stop might be too far away in time and not changing them mightbe far too risky. It has to be mentioned that kiln rollers are very expensive. Clearly, collecting and storing reliability dataregarding this and other equipment and subsequently applying some of the reliability models that we have shown herewould clearly benefit the ceramic tile sector as well as other manufacturing companies. Additionally, all this process iscarried out with independence of the workload at the factories and many times a scheduled stop is highly inconvenientif there exists a backlog in production orders.


Fig. 9. Screenshot of scheduling software considering scheduling with PM operations.

6. Conclusions and future research

In this paper, we have proposed tools, in the form of adaptations of heuristic and metaheuristic methods to implicitlyconsider preventive maintenance operations in sequencing problems. These problems, specifically permutation flow-shops, have been solved up to now by assuming that the different machines in the shop are continuously available,which obviously is not true in practice.

Two preventive maintenance policies have been proposed, one based on the maximization of the availability ofthe machines, and other that aims at maintaining a minimum level of reliability after the production period. Then,we have described a new criterion to schedule preventive maintenance operations to the production sequence. Thiscriterion demonstrates the importance of implicitly considering preventive maintenance operations in the productionsequence.

Six adaptations of heuristic and metaheuristic methods that include the mentioned criterion have been scheduled. Themethods chosen for adaptation have shown very good performance in regular flowshops and thus are ideal candidatesfor testing the PM operation scheduling criterion. Extensive computational evaluations with two sets of problems anda total of 7200 instances have given as a result that modern Ant Colony Algorithms and Genetic Algorithms providevery good solutions. All this demonstrates the feasibility of considering preventive maintenance operations togetherwith the sequencing of production in flowshop problems.

We are currently advancing in the practical applications and extending the work shown here to more complexproduction environments like hybrid flexible flowshops with sequence dependent setup times. Another interesting lineof research is to devise multiobjective metaheuristics where both policies II and III can be jointly considered wherethe aim is to obtain feasible production schedules with high performing production criteria (Cmax or due date-basedobjectives like weighted tardiness) and high reliability and availability as well.

All the codes used in this paper, as well as the sets of instances, are available upon request to the authors.

Acknowledgments

The authors are partly funded by the Spanish Department of Education and Science under the research projectref. DPI2004-06366-C03-01 financed by FEDER funds. The authors also thank the two anonymous referees for theirinsightful comments on an earlier version of this paper.


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Considering scheduling and preventive maintenance in the flowshop sequencing problem

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