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Procedia Computer Science 12 (2012) 110 - 115

Complex Adaptive Systems, Publication 2 Cihan H. Dagli, Editor in Chief Conference Organized by Missouri University of Science and Technology

2012- Washington D.C.

A Hybrid EA for Reactive Flexible Job-shop Scheduling

Lin Lina,b*, Mitsuo Gena, Yan Liangb, Katsuhisa Ohnoc

aFuzzy Logic Systems Institute 820-0067, Japan bDalian University of Technology 116620, China cAichi Institute of Technology 470-0392 , Japan

Abstract

In this paper, we consider a reactive flexible job-shop scheduling problem (rFJSP) under uncertainty environment. The most existing reactive scheduling methods are characterized by least commitment strategies such as real-time dispatching that create partial schedules based on local information. In rFJSP, two extensions of these dispatching strategies are to allow the system to select multiple machines assignment, and multiple operation process for each job. So, how to design an effective flexible rescheduling strategy is the key point of this paper. For solving this rFJSP, we propose a hybrid evolutionary algorithm (hEA) with combining genetic algorithm (GA) and particle swarm optimization (PSO). Finally, the experiments verify the effectiveness of proposed hEA, by comparing with different evolutionary approaches for several scale test problems of rFJSP.

Keywords: flexible job shop scheduling problem; reactive scheduling; hybrid evolutionary algorithm

1. Introduction

Flexible job shop scheduling problem (FJSP) is one of the topical research areas in the manufacturing systems. FJSP is expanded from the classical job-shop scheduling problem, which possesses wider availability of machines for all the operations, so it can effectively overcome the uncertainty environments [1]-[4]. Uncertainty is a very important concern in production scheduling since it can cause infeasibilities and production disturbances. Thus scheduling under uncertainty has received a lot of attention in the open literature in recent years. Jackson (1975) made a distinction on the concept of static scheduling and dynamic scheduling [5]. According to the processing characteristics of different scheduling problems, the scheduling problem can be classified as shown in Fig. 1.However, the most of researches considered the rescheduling strategy to determine operation starting times and operation sequence under uncertainty environments. Actually, the flexibility of multiple machines assignment, and flexibility of operation process are very useful to ensure the stability of the manufacturing. In this paper, we are focusing on modelling and optimization of reactive scheduling with flexible job shop type under uncertainty.

* Corresponding author. Tel.: +86-156-4112-3006. E-mail address: lin@dlut.edu.cn

1877-0509 © 2012 Published by Elsevier B.V. Selection and/or peer-review under responsibility of Missouri University of Science and Technology. doi:10.1016/j.procs.2012.09.039

Fig. 1. The classification of scheduling problem

Reactive Scheduling is proposed by Cott and Macchietto (1989)[6], it is one of classical scheduling methods under dynamic environment, and it emphasizes the ability to respond to environmental changes [7]. Recently, Sabuncuoglu and Bayiz (2000) considered a reactive scheduling problem in a stochastic manufacturing environment [8]. They tested several scheduling policies under machine breakdown in a classical job shop system. Chryssolouris and Subramaniam (2001) proposed a genetic algorithm for a reactive job-shop scheduling with considering two performance measures, the mean job tardiness and the mean job cost [9]. Suwa and Sandoh (2007) proposed a when-to-schedule policy in reactive job-shop scheduling with machine breakdowns, which considers timing of schedule revision based on the concept of a control limit policy [10]. Adibia et al. (2010) considered an event driven policy for a reactive job-shop scheduling with random job arrivals and machine breakdowns [11]. They proposed a trained artificial neural network (ANN) updates parameters of variable neighborhood search (VNS) at any rescheduling point. Also, a multi-objective performance measure is applied as objective function that consists of makespan and tardiness. Kianfar et al. (2012) considered a flexible flow shop scheduling problem with non-deterministic and dynamic arrival of jobs and also sequence dependent setup times [12]. The some commonly used dispatching rules and two new methods are incorporated in a simulation model. However, the researches of reactive flexible job shop scheduling are relatively uncommon. The most of the related researches were given in the past five years. Wu (2008) considered a reactive flexible job shop scheduling with the lead time optimization, and proposed a multi-objective immune genetic algorithm for the problem [13]. Fattahi and Fallahi (2010) considered a dynamic scheduling in flexible job-shop [14]. Two objectives are considered to make a balance between efficiency and stability of the schedules. A meta-heuristic algorithm based on the genetic algorithm is developed. Liu et al. (2011) considered a flexible job-shop multi-objective dynamic scheduling problem with the tardiness penalty and manufacturing time [15]. They proposed a genetic algorithm, employed a cycle-driven rescheduling strategies.

In the above researches, the most of researches considered the dynamic rescheduling strategies to adjust the operation starting times and/or operation sequence. Actually, the flexibility of multiple machines assignment, and flexibility of operation process are very useful to ensure the stability of the manufacturing under dynamic environment. In this paper, we focus on the flexibility analysis of FJSP under dynamic environment.

2. Reactive Flexible Job-shop Scheduling

The FJSP is as follows: n jobs {J, j=1, 2, ..., n} are to be scheduled on m machines {Mk, k=1, 2, ..., m}. Each job Jj represents nj ordered operations. The execution of each operation Oj {oij, i=1, 2, ..., I} of job Jj requires one machine k selected from a set of available machines called Ai={aik, k=1, 2, ., m}, and will occupy that machine for pjk time units until the operation is completed. The FJSP problem is to assign operations on machines and to sequence operations assigned on each machine, subject to the constraints that: (1) The operation sequence for each job is prescribed, (2) Each machine can process only one operation at a time. The reactive scheduling deal with the following two types of disturbances: (1) machine breakdown or changes in machine operation that affects the processing times of the operations in these units; (2) order modification or cancellation that change the product

demands and due dates. The purpose of the approaches is thus to update the current production schedule in order to provide an immediate response to the unexpected event. The original schedule is obtained in a deterministic manner and the reactive scheduling corrections are performed either at or right before the execution of scheduled operations. Assumptions of the rFJSP are summarized as follows:

A1. The precedence relationships among the operations of each job are predetermined.

A2. Each operation can be implemented on any machine. In case an operation cannot be processed on a machine,

the processing time of the operation on the machine is set to a very large number. A3. Operation cannot be interrupted. A4. Each machine processes only one operation at a time.

A5. The set-up time for the operations is machine-independent and is included in the processing time. Before introducing the mathematical model, the symbols and notations of the rFJSP model have been defined as

k: index of machines, k=1, 2, ..., m

- Parameters

m: total number of machines

uf total number of operations in job j

Ai={aik}: a set of available machines for operation i

Pr={yr}={(0, 1)}, Vi, i': precedence between operation i and operation i'

//for original schedule at time period t:

N={J/}: set of jobs at time period t

pijkT: processing time of operation Oj on machine k at time period t

djT: due date of job j at time period t

t,/: starting time of operation i of job j at time period t

Bj ={bjik}: machines assignment for each job j at time period t

Cii'T={cii'T}={(0, 1)}, Vi, i': precedence of operation i and operation i' at time period t

//for reactive schedule at time period y:

pijky: processing time of operation oiJ for machine k breakdown or changes at time period y N={J/}: modified set of jobs at time period y d/: modified due date of job j at time period y

- Decision variables

X/={Xjikr}: machines assignment for each job j at time period y tj/: starting time of operation i of job j at time period y In this paper, we consider objective criteria which minimize the total tardiness of jobs. The rFJSP model is given

follows. - Indices

i: index of operations, i=1, 2, ..., I

j: index of jobs, j=1, 2, ..., u

as follows:

xjjke (0,1), Vi,j, k tjf > 0, Vi, j

where Equations (2) and (3) state that the successive operation has to be started after the completion of its precedent operation of the same job, which represents the operation precedence constraints; Equation (4) states that one machine must be selected for each operation.

3. Hybrid Evolutionary Algorithm

Representation for operation sequencing: We combine the random key-based representation for operations sequencing [16]. Fig. 2 presents a scheduling network with 3 jobs and 11 operations. The example of representation is shown in Fig. 3, and we can obtain the same operations sequence into the schedule, (N1- N3-N8- N5- N6-N2-N4-

N7- N9- N10-N11).

locus 1 2 3 4 5 6 7 8 9 10 11

allele 0.10 0.04 0.09 0.05 0.07 0.11 0.03 0.08 0.02 0.01 0.06

Fig. 2. Illustration of scheduling network with 3 job and 11 operations Fig. 3. Illustration of random key-based representation

Representation for machines assignment: After the operation sequence is fixed, the machines assignment can be formulated as a multi-stage problem. For each stage (operation), we decided the state number (which machine should be assigned). Illustration of permutation representation is shown in Fig. 4. As the decoding process, we assign the machine Mu to operation j by using the following equation:

u'=FIX(vi'Ui') (7)

where vt is the value of the i-th gene; U' is the number of available machines for operation i; FIX(x) rounds the elements of x to the nearest integers towards zero.

allele

Fig. 4. Illustration of permutation representation

In this paper, we design 2 kinds of evolutionary operations for rFJSP: exploitation-based evolutionary operation, and exploration-based evolutionary operation [17].

We combine the one-cut point crossover as the exploration-based evolutionary operation. Crossover operates on two chromosomes at a time and generates offspring by combining both chromosomes' features. One-cut point crossover is a simple way to achieve crossover would be to choose a random cut-point and generate the offspring by combining the segment of one parent to the left of the cut-point with the segment of the other parent to the right of the cut-point.

We combine the evolutionary operation of PSO as the exploitation-based evolutionary operation. Different with the operation of PSO, the exploitation-based evolutionary operation replace the chromosome if the generated chromosome is better than the original chromosome. The evolutionary operation process is shown as following steps:

Step 1: Calculate velocities (vf) for each gene i in the chromosome j in generation t.

vfj = + 0,(0. -xfri) + 02(Z1£"1-4r1), vt,j (8)

where m, , ag are parameters of the algorithm, 0, = riag 012 = r2ai, rr and r2 are random numbers between (0, 1). gt is the value of gene i in the best chromosome, and l\ is the value of gene i in the best chromosome in generation t.

Step 2: Generate a new chromosome j' with calculating each gene i in generation t by following equation:

123456789 10 11

0.56 0.36 0.78 0.23 0.34 0.70 0.50 0.41 0.85 0.15 0.32

4, = 4 + vfj, Vi, j (9)

Step 3: Evaluate the fitness fitj' of the chromosome j'; if the fitness fitj' is better than the fitness fitj of chromosome j, replace the chromosome j by vj = [vfj, ],.

Selection operation is the evolutionary operation of GA, provides the driving force in a GA. With too much force, a genetic search will be slower than necessary. Typically, a lower selection pressure is indicated at the start of a genetic search in favor of a wide exploration of the search space, while a higher selection pressure is recommended at the end to narrow the search space. The selection directs the genetic search toward promising regions in the search space.

Selection Policy: Parental selection is Fitness Proportionate ("Roulette Wheel"). Selection of an individual from the "clutch" to enter the population can use the same mechanism or be deterministic, i.e. the best offspring is always picked.

Clutchsize Policy: i.e. the number of offspring cloned in a single iteration of the algorithm. This will affect the balance between exploitation and exploration in the early stages of the run before the population has converged.

In this paper, we combine proposed exploitation-based evolutionary operation as the deletion policy, combine roulette wheel selection as fitness proportionate, and combine the clutchsize policy for scheduling problems, where the probability of clutchsize ps = clutch_size /population_size.

4. Experimental Comparisons and Discussions

In order to test the effectiveness and performance of the proposed hybrid evolutionary algorithm (hEA), different EAs are used to evaluate the same test set of rFJSP problems. The compared algorithms are a priority-based GA (priGA) proposed by Zhang and Gen (2006) [18], a random key-based PSO (rkPSO) proposed by Guo et al. (2009) [19]. The evolutionary parameter's initial settings are taken as shown in Table 1. The results are compared in terms of solution optimization and convergence rate. All of tests are coded under the service-oriented evolutionary computation architecture (SoECA) software developed by Hao and Lin [20]. All experiments are conducted 30 runs on a machine running on Intel Xeon 2.00GHz CPU and 4 GB of memory.

The 3 representative FJSP instances (represented by problem nxm) were selected for simulation. The original schedules of FJSP are generated by SPT (Shortest Processing Time) dispatching rule, where select the operation with the shortest processing time [16]. There are 2 types of disturbances: (1) machine breakdown or changes in machine operation that affects the processing times of the operation in these units; (2) due dates modification or cancellation. The parameters of disturbances are represented as random variables depend on the distribution functions. Table 2 summarizes the experimental results. The result clearly indicates that all of results by hEA are better than each of the other EA approaches.

Table 1. Parameters and strategies of prGA, rkPSO, proposed hEA Table 2. Performance Comparisons with Different EA

Approaches by 3 rFJSP Tests with 30 runs

priGA rkPSO hEA problem disturbances times priGA algorithms rkPSO hEA

Iteration 5000 5000 5000 8x8 1 1.45 1.37 1.15

Population size Representation 200 priority-based 200 randomkey-based 200 random key-based machine breakdown 10x10 2 1 2 1.56 1.21 2.72 2.05 0.96 2.86 1.12 0.27 2.18

intager state real number state real number state 15x10 1 2 1.61 3.45 1.37 1.49 1.11 0.02

Selection tournament(2) roulette wheel 8x8 1 1.53 0.60 0.43

Clutchsize Evol. Operations Shiftt (pS =0.65) Pm =0.65, pc =0.2, ps =0.3 probability of clutch size 20% One cut-point (p c =0.65) due date modification 10x10 2 1 2 2.53 2.08 2.60 1.97 1.25 1.88 0.48 0.71 1.59

Swapping (p W =0.20) w =1.4, a ,=2.0, a 2=2.0 Pm =0.65, pc =0.2, ps =0.3 15x10 1 2 1.72 2.92 1.48 1.13 0.05 1.71

Mutation (p M =0.20) w =1.4, a 1=2.0, a 2=2.0

5. Conclusion

In this paper, we formulated a reactive flexible job-shop scheduling problem under uncertainty environment. In

rFJSP, two extensions of these dispatching strategies were to allow the system to select multiple machines assignment, and multiple operation process for each job. For solving this rFJSP, we proposed a hybrid evolutionary algorithm (hEA) with combining genetic algorithm and particle swarm optimization. Finally, the experiments verified the effectiveness of proposed hEA, by comparing with different evolutionary approaches for several scale test problems of rFJSP.

Acknowledgements

This work is partly supported by the Japan Society of Promotion of Science (JSPS), Grant-in-Aid for Scientific Research (C) (No.245102190001), and also is partly supported by the Fundamental Research Funds (Software+X) of Dalian University of Technology (No.DUT12JR05, No.DUT12JR12).

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