Distribution Scheduling

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A modified genetic algorithm approach for scheduling of perfect maintenance

in distributed production scheduling

S.H. Chung a,, Felix T.S. Chan a, H.K. Chan b

a Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Hong Kongb Norwich Business School, University of East Anglia, Norwich, Norfolk, UK

a r t i c l e i n f o

Article history:

Received 10 March 2008

Received in revised form

14 September 2008

Accepted 3 November 2008Available online 9 January 2009

Keywords:

Distributed scheduling

Multi-factory production

Production scheduling

Perfect maintenance

Genetic algorithms

a b s t r a c t

Distributed Scheduling (DS) problems have attracted attention by researchers in recent years.DS problems in multi-factory production are much more complicated than classical scheduling

problems because they involve not only the scheduling problems in a single factory, but also the

problems in the higher level, which is: how to allocate the jobs to suitable factories. It mainly focuses on

solving two issues simultaneously: (i) allocation of jobs to suitable factories and (ii) determination of

the corresponding production schedules in each factory. Its objective is to maximize system efficiency

by finding an optimal plan for a better collaboration among various processes. However, in many papers,

machine maintenance has usually been ignored during the production scheduling. In reality, every

machine requires maintenance, which will directly influence the machines availability, and

consequently the planned production schedule. The objective of this paper is to propose a modified

genetic algorithm approach to deal with those DS models with maintenance consideration, aiming to

minimize the makespan of the jobs. Its optimization performance has been compared with other

existing approaches to demonstrate its reliability. This paper also tests the influence of the relationship

between the maintenance repairing time and the machine age to the performance of scheduling of

maintenance during DS in the studied models.

& 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Distributed Scheduling (DS) problem in the multi-factory production model is a popular issue to researchers and industrialists in

recent years due to the changes of production environment. Nowadays, markets are frequently shifting. New technologies are

continuously emerging and the number of competitors is globally soaring. To increase the international competitiveness and

responsiveness to the market changes, many companies have to change their production model from traditional single-factory to multi-

factory, by building new factories, merging, factory acquisition, etc. ( Shen and Norrie, 1999; Lau et al., 2006).

In a multi-factory model, each factory is usually capable of manufacturing a variety of product types. Some may be unique in a

particular factory, while some may not. They are subject to different constraints, such as machine technology, labor skills, labor cost,

government policy, tax, local suppliers, transportation facilities, etc. As a result, different factories have different operating costs,

production lead times, customer service levels, efficiency, etc., and these induce the DS problems in multi-factory production

environment (Timpe and Kallrath, 2000).

DS problems can be considered as a set of processes subject to various constraints and are executed in different locations. DS has

gained an increasing importance and is applied in a wide range of areas, from multimedia to industrial control, and extensive efforts have

been invested in solving various open research issues ( Kim et al., 1996; Sandholm, 200 0; Wang and Wu, 2003; Vincent and Stephen,

2004). In general, DS problems in multi-factory production can be divided into two main issues: (i) allocation of jobs to suitable factories

and (ii) determination of the production scheduling in each factory ( Barroso et al., 2002).

Classical production scheduling problems deal with the scheduling of jobs on different machines. Nowadays, high-tech machinery like

that in FMS usually consumes a large amount of capital (Shriskandarajah and Sethi, 1989; Ghosh and Gaimon, 1992; Lee and Vairktarakis,

1998; Shnits et al., 2004; Priore et al., 2006). Although these machines are more advanced and reliable than those in the past, they are still

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Engineering Applications of Artificial Intelligence

0952-1976/$ - see front matter & 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.engappai.2008.11.004

Corresponding author.

E-mail addresses: [email protected] (S.H. Chung), [email protected] (F.T.S. Chan), [email protected] (H.K. Chan).

Engineering Applications of Artificial Intelligence 22 (2009) 10051014
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subject to deterioration because of ageing. Deterioration may result in various consequences, such as lower production rates and lower

product quality. Thus, maintenance is inevitable. However, maintenance will influence the machine availability, meaning that the

operation planned on that machine in the production schedule will be interrupted/delayed. Therefore, optimization of maintenance

planning should be simultaneously planned with the determination of production scheduling.

This paper is divided into the following. Next section gives a literature review. Section 3 describes the studied DS models with the

consideration of maintenance scheduling. Section 4 presents the proposed optimization methodology. Section 5 discusses the

optimization results through some numerical examples. Lastly, the paper will be concluded in Section 6.

2. Literature review

Research in distributed scheduling problem has been conducted by many researchers for many years (Bullinger et al., 1997; Trentesaux

et al., 1998; Gnonia et al., 2003). Timpe and Kallrath (2000) studied the multi-factory models in food and chemical industry. In which,

each factory can produce different types of food according to the needs of the market and is subject to production-capacity constraints.

They proposed a mixed integer programming approach to solve the problem of job allocation. Guinet (2001) studied a multi-factory

model in which the factories are located close to the customers in various countries subject to irregular demand. He proposed a primal-

dual heuristic approach to determine the production plans at the global level and the job shop scheduling in various local workshops.

Jia et al. (2002) proposed a web-based system approach to enable production scheduling in a multi-factory model. They proposed a

GA-based scheduling algorithm to optimize the scheduling problem. Jia et al. (2003) have further modified their GA and compared the

reliability with other classical single-factory scheduling problems and heuristic approaches, and the results obtained were satisfactory.

Leung et al. (2003) studied a multinational lingerie company, which produces a number of product types. The factories are located in

different countries, including China, the Philippines, Thailand, and other Southeast Asian countries. They proposed a goal programming

approach to solve the job allocation problem subject to capacity constraints. In fact, there are many literatures in dealing with distributed

scheduling problems. However, there is a lack of paper studying the maintenance optimization with distributed scheduling.

In literatures, the significance of coordination between maintenance planning and production scheduling has been shown by many

researchers (Benbouzid et al., 2003; Ji et al., 2007). In a repairable system, maintenance can be classified into three types according to the

condition of the machine after the maintenance action ( Pham and Wang, 1996; Cher and Nagarajan, 2008; Yi-Hui, 2007). (i) Perfect

maintenance (PM): after repair, the condition of every component is assumed to be as good as new. The system will have the same

failure rate function as a brand new one (Zhang and Wang, 1996; Zhang and Lam, 1998; Lim and Lee, 2000). (ii) Minimal maintenance

(MM): the condition of the system is assumed to be like just before failure, meaning that the failure rate is the same as the one before

failure (Ascher,1968). This is also named as as bad as old. (iii) Imperfect maintenance (IM): after repair, the components are assumed to

be back to a less-deteriorated condition, in which not all the damage is completely recovered (Nakagawa, 1979; Brown and Proschan,

1983). The failure rate is somewhere between as good as new and as bad as old.

In general, maintenance planning can be classified into planned maintenance and unplanned one (Corder, 1976; Zhang and Wang,

1996; Ascher, 1968; Lim and Lee, 2000; Levitin and Lisnianski, 2000; Gharbi and Kenne, 2000; Wang and Usher, 2005). Planned

maintenance refers to the work performed according to a scheduled plan, such as preventive maintenance. Unplanned maintenance

refers to the work that is performed promptly to avoid serious consequences on the resources and system performance and to keep the

system safe. Rishel and Christy (1996) stated that the relationship between the departments of maintenance and production is conflicting

in nature. The production managers want the machines to operate 24 h per day, while the maintenance managers require the machines to

stop operation for maintenance. For this reason, some factories maintenance policies are emergency maintenance. However, Elsayed and

Dhillon (1979) and Lamber et al. (1971) stated that such policy is very costly compared with other policies. Therefore, a proper design and

integration of maintenance management with production planning and scheduling can increase the effectiveness of management

(Nikolopoulos et al., 2003). If the production schedule obtained from DS does not consider maintenance, the planning determined will be

seriously interrupted because of the mismatch among various processes. Consequently, the system reliability will be damaged and the

purpose of DS will be failed.

Machine maintenance is inevitable. In reality, machine has to be maintained after certain operating time either regularly or irregularly

(Zhou et al., 2004). Every machine usually follows its own distribution form of failure and repair function. Peres and Noyes (2003) carried

out a statistical analysis to identify those functions. They collected the data from 50 pieces of numeric control machines (lathe and

milling). They stated that the failure of multi-component systems, such as FMS, could be characterized by a progressive failure of a certain

number of entities or by a common failure distribution rate for the whole system. They also demonstrated a normalized repair

distribution rate for various types of failure, such as electric, mechanic, computer, and others. Vineyard et al. (1999) studied the failureand repair function of each component in some FMSs, including the mechanical components, electronic, hydraulic, electro-mechanical,

software, etc. They reported that each type of component has different failure distribution and repair distribution rates. They collected

the empirical data and testified it by comparing it with the theoretical one.

Some researchers adopted the machine age to indicate measure and estimate the machine condition (Boukas and Haurie, 1990). It

reflects the expected/estimated inspection time, repairing time, production rate and quality, failure rate, etc. After each time of

maintenance, the machine age has to be adjusted, depending on the type of maintenance carried out. A new set of inspection time,

repairing time, production rate, and product quality will be generated (Tsai et al., 2001).

3. Problem description

The following notations are used to describe the problem studied throughout the paper:

f index for factory, f 1,y, F, where F is the number of factoriesi index for job, i 1,y, I, where I is the number of job.

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j index for operation, j 1,y, Ni, where Ni is the number of operation in job i

h index for machine, h 1,y, Hf, where Hf is the number of machines in factory f

k index for time slot, k 1,y, K, where K is the maximum time horizon

Dif The delivery time required to deliver product from the location of factory f to the location of job i

Tijfh operating lead time of operation j of job i on machine h in factory f

M maximum machine age

Sij starting time of operation j of job i

Eij ending time of operation j of job iCi completion time of job i

wif 1, if job i is allocated to factory f 0, otherwise

dijfhk 1, if operation j of job i occupies time slot k on machine h in factory f

0, otherwise

gijfh 1, if machine h in factory f is maintained after operation j of job i 0, otherwise

The multi-factory model consists of F factories, which are geographically distributed in different locations. Each factory has Hfmachine. Each machine can perform various operations with different operating lead times (Tijfh). Each machine is subject to a maximum

machine age M. Machine age is defined as the cumulated operating time. If the machine age reaches M, maintenance has to be carried out

right after the completion of the operating operation. After maintenance, the machine age will be reset to 0 as shown in Fig. 1. The time

required in maintenance may vary depending on the machine age. In common practice, the relationship between the required

maintenance time and the machine age are usually collected empirically.The problem is to satisfy Ijobs and each job has Ni operations. The traveling time between the factory fand the job i is symbolized by

Dif. The objective of the problem is to minimize the makespan of the jobs. The decision variables are wif, dijfhk, and gijfh. With the solutionwif, dijfhk, and gijfh obtained, the value of Sij, Eij, and Ci can be calculated. The objective function is shown as

Objective Z : MINmaxfCig (3.1)

Completion time (Ci) of job i equals the summation of the completion time of the last operation of the job and the delivering time

between the factory and the job.

The problem is subject to the following constraints:

Precedence constraints:

SijXEij1 i 1; 2; . . . ; I; j 2; 3; . . . ; Ni (3.2)

Processing time constraints:

Eij Sij Xfh

wifTijfh i 1; 2; . . . ; I; j 1; 2; . . . ; Ni (3.3)

Xfhk

dijfhk Xfh

wifTijfh i 1; 2; . . . ; I; j 1; 2; . . . ; Ni (3.4)

Operation constraints:Xfhk

dijfhk 1 i 1; 2; . . . ; I; j 1; 2; . . . ; Ni (3.5)

Processing operation constraints:Xfh

dijfhkp1 i 1; 2; . . . ; I; j 1; 2; . . . ; Ni; k 1; 2; . . . ; K (3.6)

Machine capacity constraints:

Xij

dijfhkp1 k 1; 2; . . . ; K; h 1; 2; . . . ; Hf; f 1; 2; . . . ; F (3.7)

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Time Horizon

Machine Age

M

Maintenance

0

Operating

Idling

Fig. 1. A Sample of modeling of machine age for a machine when reaching maximum machine age.

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Factory constraints:Xf

wif 1 i 1; 2; . . . ; I (3.8)

Refer to the above constraints, constraint (3.2) defines that each operation can only start upon completion of its preceding operation.

Constraint (3.3) defines that once an operation starts, it will carry on without interruption. Constraint (3.4) indicates that the allocated

time slot equals the required operation time. Constraint (3.5) defines that each operation should be carried out on one machine

throughout the horizon. Constraint (3.6) defines that each operation can only be carried out on one machine at each time unit, andconstraint (3.7) defines that each machine can carry out only one operation at a time unit. Constraint (3.8) defines that each job can only

be allocated to one factory.

4. Optimization methodology

The optimization methodology developed in this paper is based on the approach proposed by the authors as published in the paper

(Chan et al., 2006) named Genetic Algorithm with Dominant Gene (GADG). The idea of DGs is to identify and record the best genes in each

chromosome, and the corresponding structure. In that paper, the GADG approach has been compared to various approaches in some

multi-factory production problems to demonstrate its optimization reliability. The encoding of chromosome approach is the same.

However, it is found that in the original GADG approach, the local searching ability is weak. Thus, this paper proposes a modified GADG

approach as discussed in the following sections. The improvement of the optimization performance for the modified approach will be

tested in Section 5.Each chromosome consists ofP

iNi genes. Each gene consists of 6 parameters (FMJOSD), representing the Factory of the job allocated,

the Machine assigned on, the Job and its Operation, the Scheduling of maintenance after the operation, and the Domination of the gene. If

maintenance is scheduled, then the Sparameter will denoted as 1, otherwise 0. Similarly, D parameter will be denoted as 1 if the gene is a

dominant gene, otherwise 0. Fig. 2 shows an example of an encoding and the decoding of a chromosome. In which, 3 jobs (each has

2 operations) is allocated into 2 factories. The scheduling priority of operations is from the highest on the left to the lowest on the right.

Thus, the first gene 111100 has the highest priority to occupy the machine (M1) in factory 1 (F1). Then second gene 213110, indicating

Operations 3 of Job 3, can occupy machine (M1) in factory (F2) earlier than the Operation 2 of Job 2. In addition, the gene indicates that

the machine will be maintained after the operation.

4.1. Basic genetic operations

In this approach, the initial solution pool is randomly generated. However, each chromosome must be valid. The selection operation

applies the roulette wheel selection approach. In order to prevent the loss of the best chromosome during evolutions, elitist strategy isapplied. The best chromosome will be identified and recorded during each evolution. If the best chromosome in the new generation is lost

or becomes weaker after evolution, the recorded one will be inserted back into the mating pool for the next evolution.

The fitness value of a chromosome k is calculated by the makespan of the chromosome k divided by the summation of all the

makespan of all the n chromosome in the solution pool as shown:

fk 1 maxfcig

,Xn

maxfCign

!(4.1)

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FMJOSDFMJOSDFMJOSDFMJOSDFMJOSDFMJOSD111100 213110 121200 212200 222100 223200

M2

M1

M2

M1 J1O1

J1O2

J3O1 Maintenance J2O2

J2O1 J3O2

Factory 1

Factory 2

Time Horizon

Fig. 2. A sample of (a) encoding and (b) decoding of a chromosome.



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4.2. Crossover operation

The main purpose of crossover operation is to import into a chromosome the structure of another chromosome and hopefully which

will increase its strength (fitness value). In the modified approach, the DGs represent the critical structure of a chromosome, meaning this

part creating the most valuable characteristic. During crossover, each pair of parents will form a pair of offspring(s). The whole set of DGs

from one of the parent will be selected to form the initial framework of its offspring. Then it will import the DGs from another parent as

shown in detail below.

The steps for the crossover operation are as shown in the following example.Step 1: offspring O1 imports the DGs from P2 (2nd gene: 211101).

Step 2: O1 imports the DGs from P1 (1st and 8th genes: 123311, 113111).

Step 3: fill in the missing genes (remaining operations of the job) from P1 to O1 (missing genes sequence: 121210, 222110, 111300,

212300, 232200, 113200).

Step 4: adjust the allocation of jobs to other factories according to the imported DG. In the example, since Job 1 is allocated to Factory 1

in P1, the genes with Job 1 have to be adjusted to Factory 2 so that the chromosome becomes valid.

Similar steps will be carried out for O2.

Step 1: offspring O2 imports the DGs from P1 (1st and 8th genes: 123311, 113111).

Step 2: O2 imports the DGs from P2 (2nd gene: 211101).

Step 3: fill in the missing genes from P2 to O2 (missing genes sequence: 222300, 221300, 232100, 212200, 231210, 133200).

Step 4: in this case, Job 3 is allocated to Factory 1 in P2. It is similar to the allocation result in P1. The chromosome is valid without

adjustment.

P1: 123311 121210 111100 222110 111300 212300 232200 113111 113200

P2: 222300 211101 221300 232100 212200 231210 113100 133200 123300Step

1 0 211101 0 0 0 0 0 0 0

2 123311 211101 0 0 0 0 0 113111 0

3 123311 211101 121210 222110 111300 212300 232200 113111 113200

4 123311 211101 221210 222110 211300 212300 232200 113111 113200

O1: 123311 211101 221210 222110 211300 212300 232200 113111 113200

Step

1 123311 0 0 0 0 0 0 113111 0

2 123311 211101 0 0 0 0 0 113111 0

3 123311 211101 222300 221300 232100 212200 231210 113111 133200

4 123311 211101 211100 221300 212201 232100 231210 113111 133200

O2: 123310 211101 211100 v 221300 212201 232100 231210 113110 133200

In the case that if the DGs are conflicted between the two parents either at the same location(s) or job(s), the conflicted DGs in one of

the parents will be ignored. For example, the underlined genes are conflicting in the same location in P1 and P2, and conflicting with the

same job in P3 and P4 as shown in the following chromosomes. In this case, during the formation of O1, the DG (123311) in P1 will be

ignored, while for that of O2, the DG (222301) will be ignored:

P1: 123311 121210 111101 222110 111300 212300 232200 113111 113200

P2: 222301 211100 221300 232100 212200 231210 113100 133200 123300

P3: 123311 121210 111100 222110 111300 212300 232200 113111 113200

P4: 222300 211100 221300 232100 212200 231210 113100 133200 123301

4.3. Mutation operation

Two types of mutation will be applied, and both are one-point mutation rates, which enhance the ability of local searching. In

Mutation 1, a number of genes will be randomly swapped among a chromosome. The pair of the swapping genes must be in the same

factory and on the same machine as shown in the following. This mechanism increases the opportunities of testing different production

priority of the operations and jobs without changing the allocation of jobs.

Before: 123311 121210 111101 222110 111300 212300 232200 113111 113200

After: 121210 123311 111101 222110 111300 212300 232200 113111 113200

In Mutation 2, each time only one of the parameters M, F, or Sundergoes mutation in the selected gene(s). Parameters Jand O will not

mutate here because if they mutate, it will be similar to Mutation 1. In the following example, the fourth gene mutates in M parameter,

changing from M2 to M3.

Before: 123311 121210 111101 222110 111300 212300 232200 113111 113200

After: 123311 121210 111101 232110 111300 212300 232200 113111 113200

However, if the Fparameter mutates, some adjustments have to be taken to change all the related operations of that job to be allocatedto the same factory so that the chromosome becomes valid. For the example shown below, the F parameter in the fourth gene changes

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from F2 to F1. Since the sixth and seventh genes are related to the fourth one because of the job relationship, their Fparameters have to

change to F1 as well.

Before: 123311 121210 111101 222110 111300 212300 232200 113111 113200

After: 123311 121210 111101 122110 111300 112300 132200 113111 113200

4.4. Prevention of premature and local search

To prevent the premature and local search, adaptive strategy is applied. Similarity checking is carried out to measure the saturation of

the chromosomes in the solution pool. During the similarity checking, two genes are regarded as identical when their F, M, J, O, and S

parameters are the same. The D parameter will not be considered because it does not influence the production schedule. For the example

below, the first gene (123311) in P1 indicates that O3 of J3 is allocated to F1 on M2 and will be maintained after the operation. This gene is

identical to the ninth gene (123310) in P2 and the first gene (123311) in P3 because they have the same allocation result, machine

selection result, and maintenance plan for the same job operation. Since the solution pool has only 4 chromosomes, but 3 genes are the

same, the solution pool is saturated. The choice of allocating that job operation to other factory and machine becomes fewer. In this

example, the only choice is to allocate it to F1 on M1. This will lower the ability of the genetic search.

P1: 123311 121210 111101 222110 111300 212300 232200 113111 113200

P2: 222301 211100 221300 232100 212200 231210 113100 133200 123310

P3: 123311 121210 111100 222110 111300 212300 232200 113111 113200

P4: 222300 211100 221300 232100 212200 231210 113100 133200 113301

In order to maintain the diversity of the solution pool, mutation will be carried out on those genes.

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Generate Initial Pool

Crossover/Mutation

Evaluate Fitness Value

Selection operation

Elitist strategies

End

No

Yes

Check Stopping

Fig. 3. The genetic evolution outline proposed in the old approach ( Chan et al., 2006).

Yes

Generate Initial Pool


Check Stopping

Mutation operation Type

Selection operation

Elitist strategies

Crossover operation Mutation operation Type

Stopping until Loop =

(iNi )/10 times

No


Elitist strategies

No

End

Yes

Improve local searching

or

Fig. 4. The modified genetic evolution outline proposed in this paper.



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4.5. Genetic evolutions

The evolution mechanism of the genetic algorithm proposed by Chan et al. (2006) is shown in Fig. 3. In which, the crossover and

mutation operations will perform alternatively during each evolution. However, it is found that such an approach will lower the local

searching ability because the optimization of the production scheduling part has played a less significant role. Thus, in this paper, the

evolution approach is being modified as shown in Fig. 4. In which, the mutation operation Type 1 will run (P

iNi)/10 times, which is 10% of

the total number of operations, during each evolution. Since the function of mutation Type 1 is to change the production priority of the

operations, more production schedule(s) can be tested based on the same set of job allocation result. This can improve the local searchingability.

5. Numerical examples

5.1. Example 1

The objective of this example is to test the optimization performance of the proposed modified GADG approach in classical production

scheduling problem. The problem is found in literatures, which have been studied by Lee and DiCesare (1994), Kumar et al. (2003), and

Chan et al. (2006). The problem consists of 5 jobs and each job has 4 operations. Each operation can be performed on more than one of the

three machines. The objective of the problem is to minimize the makespan of the jobs. The optimization results found by them are shown

in Table 1.

In this paper, the modified GADG approach has been independently run for 50 times on a personal computer with 2.2 GHz to measure

the deviation of the solution obtained. The solution pool is set as 50 and the number of evolution is 5000, which is long enough to obtaina steady solution. The computational time for each trial is about 15s. The average makespan obtained for the 50 trails is 358 unit time,

with the standard deviation of 8.36, and the best result found is 350 unit time. It has 2.78% improvement compared to the original GADG

approach. The detail production schedule for the best result is shown in Fig. 5. This comparison demonstrates that the modified GADG

approach is better.

5.2. Example 2

The optimization performance will also be tested in distributed scheduling problem, which was studied by Chan et al. (2006). The

problem consists of two factories and each has three machines. It has 10 jobs and each has four operations. The best value found in that

paper is 1220 unit time, with the total maintenance time spent equal to 3060 unit time.

Similarly, the modified GADG approach has been independently run for 50 times to measure the deviation of the solution obtained.

The solution pool is set as 50 and the number of evolution is 5000, which is long enough to obtain a steady solution. The computational

time for each trial is about 1 min. The best value obtained is 930 unit time, which has (23.8% improvement compared to the original

approach). The total maintenance time spent is 2350 unit time (180+180+150+140+140+140+140+180+180+140+140+200+160+140+140),which has a 23.2% improvement as shown in the production schedule in Fig. 6. The standard deviation of the solution measured is 12.8.

The results demonstrate that the modified approach is better.

5.3. Example 3

In this example, the objective is to test whether the relationship between the time required for maintenance and the machine age will

influence the results of simultaneously solving maintenance scheduling with DS. For this reason, four different relationship models have

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Fig. 5. The production schedule obtained for Example 1.

Table 1

Optimization results obtained by various approaches for Lee and DiCesares model.

Authors Approach Optimization results

Lee and DiCesare (1994) Petri nets 439

Kumar et al. (2003) Ant colony 420

Chan et al. (2006) GADG 360

Modified GADG 350



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been studied as shown in Fig. 7ad. Fig. 7a shows a linear relationship between the maintenance time required and the machine age. The

maintenance time equals three times of the machine age (i.e., after the machine has being operating for 20 h, then the corresponding

maintenance time required is 60 h). Fig. 7b shows an exponential relationship, which is similar to the one presented by Peres and Noyes(2003) to model the maintenance time for some electric, mechanic, and computer components. Fig. 7c shows a constant relationship,

which is similar to modeling the maintenance time for those maintenance operations in which the whole component will be replaced

with a new one instead of repair, and Fig. 7d shows a 2-level constant relationship. For all the models, the maximum operating limit is

200 units of time. Beyond the limit, the maintenance time required is 720 units of time, which is 20% more than the maximum

maintenance time required for planned maintenance.

Three multi-factory models will be randomly generated. Each model consists of 2 factories and each factory has 3 machines. The

traveling time from the factories to the jobs are randomly generated between 1020 units of time. Each job has 4 operations, and the

operation time is randomly generated between 10150 units of time. Model 1 has 10 jobs, Model 2 has 20 jobs, and Model 3 has 30 jobs.

Each relationship model will be independently used to model the machines.

For the sake of comparison, first of all, maintenance scheduling will not be considered with DS. The maintenance will take place when

the machine age reaches its maximum operating limit (M). In another approach, maintenance scheduling will be considered with DS.

Similarly, the modified GADG approach has been independently run for 50 times. The computational time for each trial is about 10 s,

1 min, and 4 min. For the reason of simplification of comparison, Table 2 summarizes the optimization results, which are the best results

obtained in the 50 runs in each model. In Model 1, based on the same set of problem parameters, the makespan obtained from the secondapproach is 1060 in linear relationship. It goes down to 720 when relationship in machines is changed to the exponential model. It

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Machine AgeMachine Age

1 21 41 61 81 101 121 141 161 181

0

700

600

500

400

300

200

100

Ma

intenanceTime

Ma

intenanceTime

0

700

600

500

400

300

200

100MaintenanceTime

1 14 27 40 53 66 79 92 105

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170

183

196

Machine Age

1 14 27 40 53 66 79 92 105

118

131

144

157

170

183

196

Machine Age

350

300

250

200

150

10050

0

MaintenanceTime

350

300

250

200

150

100

50

0

Fig. 7. Various relationships between maintenance time and machine age for the machines in the Models 1, 2, 3 and 4: (a) linear relationship, (b) exponential relationship,

(c) constant relationship and (d) 2-level constant relationship.

Fig. 6. The production schedule obtained for Example 2.



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increases again to 1030 in constant relationship and reduces to 750 in 2-level constant relationship. Similar patterns can be found in the

other two models. In addition, the makespan obtained in the second approach is better in all the relationships in all the 3 models. This

demonstrates that scheduling of maintenance should be considered simultaneously during DS without being influenced by the

relationship between maintenance time required and the machine age.

6. Conclusion

This paper proposed a modified GADG approach, which is developed based on the approach proposed by Chan et al. (2006). It

improves the local searching ability of the original GADG approach by iteratively solving the jobs allocation problem and scheduling of

production with the maintenance problem. In order to verify the optimization performance of the proposed approach, it has been

compared with other existing approaches, including Petri Nets, Ant Colony, and the original GADG, in example 1, which is a

classical production scheduling problem. The results demonstrate that the optimization reliability of the proposed approach is better. In

example 2, the proposed approach is compared with the original approach proposed by Chan et al. (2006), and the optimization results

indicate that the new approach can obtain a better solution for the problem, studied by Chan et al. (2006), with 23.2% improvement.

Lastly in example 3, this paper tests the influence of the relationship between the maintenance time and the machine age to the

performance of scheduling maintenance with DS. Three DS models have been randomly generated with the application of different

relationship models, including linear, exponential, constant, and 2-level constant, to the machines. It compared the results with the

consideration of maintenance and without maintenance, and the results demonstrate that scheduling of maintenance simultaneously

with DS is better.

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Table 2

Comparison of the performance in different maintenance curves.

Linear Exponential Constant 2-Level

Model 1 No maintenance 1170 830 1220 820

With maintenance 1060 720 1030 750

Model 2 No maintenance 2990 1950 2560 2170With maintenance 2770 1730 2400 1740

Model 3 No maintenance 5130 2990 4240 3750

With maintenance 4380 2860 3730 3480



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