J. Magalhães-Mendes

A two-step optimization approach for job shop scheduling problem using a genetic algorithm

Jorge Magalhães Mendes

Working Paper 06.2013

Departamento de Engenharia Civil CIDEM Instituto Superior de Engenharia do Porto Instituto Politécnico do Porto Rua Dr. António Bernardino de Almeida, 431 4200-072 Porto PORTUGAL e-mail: [email protected]

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J. Magalhães-Mendes

A two-step optimization approach for job shop scheduling problem using a genetic algorithm Abstract: This paper presents a two-step optimization approach to solve the complex scheduling problem in a job shop environment. This problem is also known as the Job Shop Scheduling Problem (JSSP). The JSSP is a difficult problem in combinatorial optimization for which extensive investigation has been devoted to the development of efficient algorithms. The proposed approach is based on a genetic algorithm. Genetic algorithms are an optimization methodology based on a direct analogy to Darwinian natural selection and mutations in biological reproduction. The chromosome representation of the problem is based on random keys. The schedules are constructed using a schedule generation scheme in which the priorities and delay times of the operations are defined by the genetic algorithm and obtaining parameterized active schedules. After a schedule is obtained a local search heuristic using Monte Carlo method is applied to improve the solution. The approach is tested on a set of standard instances taken from the literature and compared with other approaches. The computation results validate the effectiveness of the proposed approach. Keywords: Scheduling; Discrete Optimization; Genetic Algorithm; Job Shop; heuristics.

1. Introduction and background The JSSP consists of a set of different machines (e.g., lathes, milling machines, drills, etc.) that perform operations on jobs. Each job consists of a sequence of operations, each of which uses one of the machines for a fixed duration. Once started, the operation cannot be interrupted. Each machine can process at most one operation at a time. A schedule is an assignment of operations to time intervals on the machines. The problem is to find a schedule of minimal time to complete all jobs (French 1982). Scheduling operations problems arise in diverse areas such as flexible manufacturing, production planning and scheduling, logistics, supply chain problem,

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J. Magalhães-Mendes etc. A common feature of many of these problems is that no efficient solution algorithms are known that solve each instance to optimality in a time bounded polynomial in the size of the problem. Discrete optimization can help to overcome these difficulties (Dorndorf and Pesch 1995). To make the right decision, the manager must have reliable and complete information and adequate scheduling models. Single and small scale productions are characterized by many work orders. So, the application of new tools based on discrete optimization not only in production process but also in determination of priority of technological operations will be appropriate (Mendes 2010). The JSSP is considered as a particularly hard combinatorial optimization problem (Lawler et al. 1982). Over the last few decades, a good number of algorithms have been developed for solving JSSP. However, no single algorithm is capable of solving all kinds of JSSPs optimally (or near optimally) within a reasonable time limit. Thus, there is scope to analyze the difficulties of JSSPs as well as to design algorithms that may be able to solve most of the standard problems (Hasan et. al. 2008). In early stages, Giffler and Thompson (1990), Carlier and Pinson (1989), Pinson (1990), Brucker et al. (1994), Williamson et al. (1997) have been successful in solving small instances, including the notorious 10×10 instance of Fisher and Thompson proposed in 1963 and only solved twenty years later. More recently, many approximate methods have been developed to solve the JSSP, such as simulated annealing (SA) (Lourenço 1995), tabu search (TS) (Nowicki and Smutnicki (1996, 2005), Pezzela and Merelli (2006), Zhang et al. (2008)), genetic algorithms (GA) (Aarts et al. (1994), Croce et al. (1995), Dorndorf and Pesch (1995), Wang and Zheng (2001), Gonçalves et al. (2005), Essafi et al. (2008), Hasan et al. (2008), Mendes (2012), Qing-doa-er-ji and Wang (2012)), particle swarm optimization

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J. Magalhães-Mendes Magalhães (PSO) (Sha and Hsu (2006)), (2006)), greedy randomized adaptive search procedure (GRASP), Binato et al. (2002),, Aiex et al. (2003)) and others heuristics (Rego Rego and Duarte (2009), Pardalos and Shylo (2006), Pardalos et al. (2010)). The rest of this paper is organized as follows: The problem definition is described in Sect. 2. and the types of schedules in Sect. 3. The approach proposed is described in Sect. 4. The computational results are presented and discussed in Sect. 5. Finally, in Sect. 6 we present some concluding remarks.

2. Problem definition efinition The JSSP may be described as follows: n jobs are to be scheduled on m machines. Each job i represents ni ordered operations. The execution of each operation j of job i (noted as oij) requires one machine m selected from a set of machines for a fixed duration, see Figure 1. Each machine can process at most one operation at a time and once an operation initiates ates processing on a given machine it must complete processing on that machine without interruption. The operations of a given job have to be processed in a given order. The problem consists in finding a schedule of the operations on the machines, taking into nto account the precedence constraints, which minimizes the makespan (Cmax), that is, the finish time of the last operation completed in the schedule.

Figure 1. An example of a Job Shop System.

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J. Magalhães-Mendes Let J = {0, 1, …, N, N+1} denote the set of operations to be scheduled and M = {1,..., m} the set of machines. The operations 0 and N+1 are dummy, have no duration and represent the initial and final operations. The operations are interrelated by two kinds of constraints: •

First, the precedence constraints, which force each operation j to be scheduled after all predecessor operations, Pj, are completed;

•

Second, operation j can only be scheduled if the machine it requires is idle. Further, let dj denote the (fixed) duration (processing time) of operation j.

Let A(t) be the set of operations being processed at time t, and let rj,m = 1 if operation j requires machine m to be processed and rj,m = 0 otherwise. Let Fj represent the finish time of operation j. A schedule can be represented by a vector of finish times (F1, …, Fm, ... , Fn+1). Additionally, the standard job-shop scheduling problem makes the following assumptions (Hasan et. al. 2008): •

Each job consists of a finite number of operations;

•

The processing time for each operation in a particular machine is defined;

•

There is a pre-defined sequence of operations that has to be maintained to complete each job;

•

Delivery times of the products are undefined;

•

There is no setup cost or tardy cost;

•

A machine can process only one job at a time;

•

Each job visits each machine only once;

•

No machine can deal with more than one type of task;

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J. Magalhães-Mendes •

The system cannot be interrupted until each operation of each job is finished;

•

No machine can halt a job and start another job before finishing the previous one;

•

Each and every machine has full efficiency.

3. Types of schedules This section presents a classification of schedules for project scheduling problems. Classifying schedules is the basic work to be done before attacking scheduling problems (Kolisch 1995). Schedules can be classified into one of the following three types of schedules: •

Semi-active schedules. These are feasible schedules obtained by sequencing activities as early as possible. In a semi-active schedule the start time of a particular activity is constrained by the processing of a different activity on the same resource or by the processing of the directly preceding activity on a different resource;

•

Active schedules. These are feasible schedules in which no activity could be started earlier without delaying some other activity or breaking a precedence constraint. Active schedules are also semi-active schedules. An optimal schedule is always active;

•

Non-delay schedules. These are feasible schedules in which no resource is kept idle at a time when it could begin processing some activity. Non-delay schedules are active and hence are also semi-active.

The set of active schedules is usually very large and contains many schedules with poor quality. To reduce the solution space was used the concept of parameterized active schedules (Gonçalves et al. 2005). 6

J. Magalhães-Mendes Magalhães 4. A two-step step approach The approach presented in this paper is based on a genetic algorithm to perform its optimization process. Figure 2 shows the architecture of approach. The approach has two main steps: •

Step1: Combines ombines a genetic algorithm with a schedule generation scheme (SGS). This SGS generates parameterized active schedules. This step allows to obtain a schedule for each chromosome;

•

Step2: This step makes use of a local search procedure that hat attempts to improve the solution obtained previously.

Figure 2. Architecture of the approach.

The Genetic Algorithms (GAs) follows the principles of The Origin of Species proposed by Charles Darwin (1859). One fundamental advantaged of GAs from traditional methods is described by Goldberg (1989): in many optimization methods, we move gingerly from a single solution in the decision space to the next using some transition rule to determine the next solution.

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J. Magalhães-Mendes First of all, an initial population of potential solutions (individual) is generated randomly. A selection procedure based on a fitness function enables to choose the individual candidate for reproduction. The reproduction consists in recombining two individuals by the crossover operator, possibly followed by a mutation of the offspring. Therefore, from the initial population a new generation is obtained. From this new generation, a second new generation is produced by the same process and so on. The stop criterion is normally based on the number of generations. The GA based-approach uses a random key alphabet U (0, 1) and an evolutionary strategy identical to the one proposed by Goldberg (1989).

4.1 Details of the GA A real-coded GA is adopted in this article. Compared with the binary-code GA, the realcoded GA has several distinct advantages, which can be summarized as follows (Y.-Z. Luo et al. 2006): •

It is more convenient for the real-coded GA to denote large scale numbers and search in large scope, and thus the computation complexity is amended and the computation efficiency is improved;

•

The solution precision of the real-coded GA is much higher than that of the binary-coded GA;

•

As the design variables are coded by floating numbers in classical optimization algorithms, the real-coded GA is more convenient for combination with classical optimization algorithms.

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J. Magalhães-Mendes In this paper a chromosome represents a solution to the problem and is encoded as a vector of random keys (random numbers). Each solution chromosome is made of 2n genes where n is the number of operations (excluding 0 and n+1): Chromosome = (genel , .., genen , gene n+1 , ... , gene 2n )

The priority decoding expression uses the following expression: PRIORITY j = gene j

j = 1,..., n

(1)

The delay time used by each activity is given by the following expression: Delay time = gene jm +1 × 1.5 × MaxDur

(2)

where MaxDur is the maximum duration of all activities. The factor 1.5 is obtained after some experimental tuning. A maximum delay time equal to zero is equivalent to restricting the solution space to non-delay schedules and a maximum delay time equal to infinity is equivalent to allowing active schedules. To reduce the solution space was used the value given by expression (2), see Gonçalves et al. (2005).

4.2 Construction of a schedule Schedule generation schemes (SGS) are the core of most heuristic solution procedures for scheduling operations. This heuristic makes use of the priorities and the delay times defined by the genetic algorithm and constructs parameterized active schedules. In this work was used the SGS described in Gonçalves et al. (2005) and Mendes et al. (2009).

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J. Magalhães-Mendes 4.3 Local search Local search algorithms move from solution to solution in the space of candidate solutions (the search space) until an optimal solution or a stopping criterion is found. In this work is applied the two exchange local search, based on the work of Nowicki and Smutnicki (1996). The local search procedure begins by identifying the critical path in the solution obtained by the schedule generation procedure. Any operation on the critical path is called a critical operation. It is possible to decompose the critical path into a number of blocks where a block is a maximal sequence of adjacent critical operations that require the same machine. The critical path thus gives the neighborhood of moves. After the critical path will be known, the Monte Carlo Method selects the block where we make the swapping operations. In fact, this process selects the most critical resource to make the swapping between two consecutive operations.

4.4 Initial population The initial population is generated randomly. The quality of this population is poor and one way to improve it is to incorporate some chromosomes generated by priority rules. In this paper are selected the priority rules GRPW (greatest rank positional weight) and SPT (shortest processing time) to improve some chromosomes of the initial population.

4.5 Evolutionary strategy There are many variations of genetic algorithms obtained by altering the reproduction, crossover, and mutation operators. Reproduction is a process in which individual (chromosome) is copied according to their fitness values (makespan). Reproduction is accomplished by first copying some of the best individuals from one generation to the 10

J. Magalhães-Mendes Magalhães next, in whatt is called an elitist strategy. In this paper the fitness proportionate selection, also known as roulette-wheel roulette selection, is the genetic operator for selecting potentially useful solutions for reproduction. The characteristic of the roulette wheel selection is stochastic sampling. The fitness value is used to associate a probability of selection with each individual chromosome. If fi is the fitness of individual i in the population, its probability of being selected is,

pi =

fi

,

N

∑

i = 1,..., n

(3)

fi

i =1

A roulette wheel model is established to represent the survival probabilities for all the individuals in the population. Then the roulette roulette wheel is rotated for several times (Goldberg 1989). The Figure 3 shows this evolutionary strategy with the operators selection, recombination (or crossover) and mutation (Mendes 2010).

Figure 3. Evolutionary strategy.

After selecting, crossover may proceed in two steps. First, members of the newly selected (reproduced) chromosomes in the mating pool are mated at random. Second,

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J. Magalhães-Mendes each pair of chromosomes undergoes crossover as follows: an integer position k along the chromosome is selected uniformly at random between 1 and the chromosome length l. Two new chromosomes are created swapping all the genes between k+1 and l, see Mendes (2012). The mutation operator preserves diversification in the search. This operator is applied to each offspring in the population with a predetermined probability. We assume that the probability of the mutation in this paper is 0.001.

4.5 GA configuration Though there is no straightforward way to configure the parameters of a genetic algorithm, we obtained good results with values: population size of 5 × number of activities in the problem; mutation probability of 0.001; top (best) 1% from the previous population chromosomes are copied to the next generation; stopping criterion of 10000 generations.

5. Computational results This section presents computational results with the implementation proposed in this paper: •

GA-MC-SPc (Genetic Algorithm with Local Search using Monte Carlo and Single Point crossover). To evaluate the performance of proposed approach, we considered the following

classes of problems, taken from the literature: •

ABZ5, ABZ6, ABZ7, ABZ8 and ABZ9 proposed by Adams et al. (1988);

•

FT6, FT 10 and FT20 originally proposed by Fisher and Thompson (1963);

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J. Magalhães-Mendes •

LA1-LA40 proposed by Lawrence (1984);

•

ORB1-ORB10, proposed by Applegate and Cook (1991). Tables 1, 2, 3 and 4 summarize the experimental results. It lists number of jobs,

number of operations, instance, best known solution (BKS) and the algorithm GA-MCSPc. The last row of each table shows the value of the average relative deviation (ARD). The ARD is calculated in the following way: NIS

RE = ∑

Cmax i − BKSi

i =1

ARD =

(4)

BKSi

RE NIS

(5)

where NIS is number of instances solved.

Table 1. Experimental results for instances ABZ5-ABZ9. Jobs Operations Instance BKS /job 10 10 ABZ5 1234 10 10 ABZ6 943 15 20 ABZ7 656 15 20 ABZ8 645 15 20 ABZ9 661 % ARD

GA-MCSPc 1234 943 688 704 715 4,44%

Table 2. Experimental results for instances FT06, FT10 and FT20. Jobs Operations Instance BKS /job 6 10 20

6 10 5

FT06 55 FT10 930 FT20 1165 % ARD

GA-MC-SPc 55 930 1165 0,00%

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J. Magalhães-Mendes

Table 3. Experimental results for instances LA01-LA40. Jobs

Operations /job

Instance

BKS

GA-MC-SPc

10 10 10 10 10 15 15 15 15 15 20 20 20 20 20 10 10 10 10 10 15 15 15 15 15 20 20 20 20 20 30 30 30 30 30 15 15 15 15 15

5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 15 15 15 15 15

LA01 LA02 LA03 LA04 LA05 LA06 LA07 LA08 LA09 LA10 LA11 LA12 LA13 LA14 LA15 LA16 LA17 LA18 LA19 LA20 LA21 LA22 LA23 LA24 LA25 LA26 LA27 LA28 LA29 LA30 LA31 LA32 LA33 LA34 LA35 LA36 LA37 LA38 LA39 LA40

666 655 597 590 593 926 890 863 951 958 1222 1039 1150 1292 1207 945 784 848 842 902 1046 927 1032 935 977 1218 1235 1216 1157 1355 1784 1850 1719 1721 1888 1268 1397 1196 1233 1222

666 655 597 590 593 926 890 863 951 958 1222 1039 1150 1292 1207 945 784 848 842 902 1049 927 1032 939 977 1218 1251 1225 1166 1355 1784 1850 1719 1721 1888 1278 1408 1205 1244 1228

% ARD

0,18%

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J. Magalhães-Mendes

Table 4. Experimental results for instances ORB1-ORB10. here Jobs 10 10 10 10 10 10 10 10 10 10

Operations Instance /job 10 ORB1 10 ORB2 10 ORB3 10 ORB4 10 ORB5 10 ORB6 10 ORB7 10 ORB8 10 ORB9 10 ORB10 % ARD

BKS 1059 888 1005 1005 887 1010 397 899 934 944

GA-MCSPc 1059 888 1005 1005 887 1010 397 899 934 944 0,00%

The computational results shows that the algorithm GA-MC-SPc in Table 1 is able to find 2 of the best solutions in 5, Table 2 has found all the best solution, Table 3 shows that is able to find the best know solution for 30 instances, i.e., about 75% of the instances and in Table 4 has found all the best solutions. In order to verify the performance of the proposed approach, GA-MC-SPc is compared with some algorithms reported in the literature in recent years, see Table 5. The performance of GA-MC-SPc was evaluated on a standard set of 50 benchmark instances belonging to two classical sets known LA from Lawrence (1984) and ORB from Applegate and Cook (1991). The approach GA-MC-SPc obtained all the optimal solutions for the instances ORB and has the same results than the approaches Nowicki and Smutnicki (2005), Zhang et al. (2008) and Pardalos et al. (2010). For LA instances the approach GA-MCSPc is positioned in the top ten best results of the state-of-art (ranked in fifth place with the approach proposed by Qing-doa-er-ji and Wang (2012)). The computational time dispended is in the range [45, 11876] seconds.

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J. Magalhães-Mendes Table 5. Comparison of the ARD for the different number of problems authors considered. NIS

40

Instances

LA01-LA40

Authors

Algorithm

ARD(%)

Pardalos et al. (2010)

GES

0.00

Sha and Hsu (2006)

HPSO

0.02

i-TSAB

0.05

Nowicki and Smutnicki (2005) Pezzela and Merelli (2006)

-

0.11

GA-MC-SPc

0.18

HGA

0.18

F&F-PRD

0.29

PSO

0.37

P. Active

0.41

-

0.43

GR-SA-RA

0.97

Gonçalves et al. (2005)

Active

1.10

Gonçalves et al. (2005)

Non-Delay

1.20

Aarts et al. (1994)

GLS2

1.75

Binato et al. (2002)

-

1.87

Aarts et al. (1994)

GLS1

2.05

SBI

3.67

GA-MC-SPc i-TSAB

0.00

This paper Qing-doa-er-ji and Wang (2012) Rego and Duarte (2009) Sha and Hsu (2006) Gonçalves et al. (2005) Aiex (2003) Hasan et al. (2008)

Adams et al. (1988) 10

This paper ORB1-ORB10 Nowicki and Smutnicki (2005) Zhang et al. (2008) Pardalos et al. (2010)

TS

0.00 0.00

GES

0.00

This computational experience has been performed on a computer with an Intel Core 2 Duo CPU T7250 @2.00 GHz. The algorithm proposed in this work has been coded in VBA under Microsoft Windows NT.

6. Conclusions and further research A GA two-step approach to solving the job shop scheduling problem has been proposed. This approach combines a genetic algorithm with a schedule generation scheme to obtain schedules. After a schedule is obtained, a local search heuristic is applied to improve the solution. The chromosome representation of the problem is based on random keys. The schedules are constructed using a priority rule in which the

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J. Magalhães-Mendes priorities are defined by the genetic algorithm. Schedules are constructed using a procedure that generates parameterized active schedules. The experimental results show that the proposed approach has a good performance when compared with others proposed algorithms tested using 50 instances taking from the literature. Further work could be conducted to explore the possibility of genetically correct the chromosomes supplied by the genetic algorithm to reflect the solutions obtained by the local search heuristic.

Acknowledgements This work has been partially supported by the CIDEM (Centre for Research & Development in Mechanical Engineering). CIDEM is a unit of FCT – Portuguese Foundation for the Science and Technology.

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