Multistage-based genetic algorithm for flexible job-shop ... - CiteSeerX

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The classical Job-shop Scheduling Problem (JSP) concerns determination of a set of jobs on a set of machines so that the makespan is minimized. It is obviously ...
Multistage-based genetic algorithm for flexible job-shop scheduling problem Haipeng Zhang, and Mitsuo Gen Graduate School of Information, Production & Systems, Waseda University Wakamatsu-ku, Kitakyushu 808-0135, JAPAN

Email: [email protected], [email protected]

Abstract Flexible Job-shop Scheduling Problem is expanded from the traditional Job-shop Scheduling Problem, which possesses wider availability of machines for all the operations. Considering the two states of the problem, two definitions (total and partial) of flexibility are offered to separate the different availability information of machines. In this paper, a new multistage operation-based representation is proposed to make the chromosome simpler. By using this approach, all the crossover and mutation methods can be applied to this optimal strategy. The efficiency has been improved after using the new representation, and also the objective values outperform others.

1.

Introduction

The classical Job-shop Scheduling Problem (JSP) concerns determination of a set of jobs on a set of machines so that the makespan is minimized. It is obviously a single criterion combinational optimization and has been proved as a NP-hard problem with several assumptions as follows: each job has a specified processing order through the machines which is fixed and known in advance; processing times are also fixed and known corresponding to the operations for each job; set-up times between operations are either negligible or included in processing times (sequence-independent); each machine is continuously available from time zero; there are no precedence constraints among operations of different jobs; each operation cannot be interrupted; each machine can process at most one operation at a time. The flexible Job-shop Scheduling Problem (f-JSP) extends JSP by assuming that a machine may be capable of performing more than one type of operation (Najid, DauzerePeres & Zaidat, 2002). That means for any given operation, there must exist at least one machine capable of performing it. In this paper two kinds of flexibility are considered to describe the performance of f-JSP (Kacem, Hammadi & Borne, 2002). First total flexibility: in this case all operations are achievable on all the machines available; second, partial flexibility: in this case some operations are only achievable on part of the available machines. Most of the literature on the shop scheduling problem concentrates on JSP case (Gen & Cheng, 1997; Gen & Cheng, 2000; Blazewicz, Domschke & Pesch, 1996). The f-JSP recently captured the interests of many researchers. The first paper that addresses the f-JSP was given by Brucker and Schlie (Brucker & Schlie, 1990), which proposes a polynomial algorithm for solving the f-JSP with two jobs, in which the machines able to perform an operation have the same processing time. For solving the general case with more than two jobs, two types of approaches have been used: hierarchical approaches and integrated - 223 -

approaches. The first was based on the idea of decomposing the original problem in order to reduce its complexity. Brandimarte (Brandimarte, 1993) was the first to use this decomposition for the f-JSP. He solved the assignment problem using some existing dispatching rules and then focused on the resulting job shop subproblems, which are solved using a tabu search heuristic. Mati proposed a greedy heuristic for simultaneously dealing with the assignment and the sequencing subproblems of the flexible job shop model (Mati, Rezg & Xie, 2001). The advantage of Mati’s heuristic is its ability to take into account the assumption of identical machine. Kacem (Kacem, Hammadi & Borne, 2002) came to use GA to solve f-JSP, and he adapted two approaches to solve jointly the assignment and JSP (with total or partial flexibility). The first one is the approach by localization (AL). It makes it possible to solve the problem of resource allocation and build an ideal assignment mode (assignments schemata), the second one is an evolutionary approach controlled by the assignment model, and applying GA to solve the f-JSP. In this paper, we propose a more efficient method called multistage-based GA to solve f-JSP (including total flexibility and partial flexibility) compared with Kacem’s approach. The considered objective is to minimize the makespan, total workloads of the machines and the maximum workloads of machines. This multi-objective optimization will be done by a multistage-based GA which including K stages (the total number of operations for all the jobs), and m state (total number of machines). Computational experiments will be carried out to evaluate the efficiency of our methods with a large set of representative problem instances based on practical data. The rest of the paper is organized as follows: In Section 2, we describe the assumptions of flexible Job-shop Scheduling Problem in detail, and propose the mathematical model of this problem. In Section 3, one heuristic method is applied to solve this problem. Section 4 introduces the GA methods and describes implementations used for this problem. Then, the experimental results are illustrated and analysed in Section 5. Finally, Section 6 provides conclusion and suggestions for further work on this problem.

2.

Mathematical model

In this paper, the flexible Job-shop Scheduling Problem we are treating is to minimize the makespan, and balance the workload for all machines. Before defining the problem concretely we should add several assumptions to the problem. 1. There is a set of jobs and a set of machines. 2. Each job consists of one fixed sequence of operations. 3. Each machine can process at most one operation at a time. 4. Each machine becomes available to other operations once the operations which are currently assigned to be completed. 5. All machines are available at t = 0. 6. All jobs can be started at t = 0. 7. There are no precedence constraints among operations of different jobs. 8. Any operation cannot be interrupted. 9. Neither release times nor due dates are specified. The f-JSP we considering here is a problem which including n-jobs operated on mmachines. Some symbols and notations have been defined as follows: i: index of jobs, i = 1, 2, … n - 224 -

Ji: the ith job n: total number of jobs k: index of operations, k = 1, 2, … Ki oik: the kth operation of job i (or Ji) Ki: total number of operations in job i (or Ji) j: index of machines, j = 1, 2, … m Mj: the jth machine m: total number of machines pikj: processing time of operation oik on machine j (or Mj) U: a set of machines with the size m Uik: a set of available machines for the operation oik

tikF : completion time of operation oik Wj: workloads (total processing time) of machine Mj The objective function can be described as the following three equations. Eq. (1) gives the first objective makespan and also means to minimize the maximum finishing time considering all the operations. Eq. (2) gives the second objective which is to minimize the maximum of workloads for all machines. Eq. (3) gives the objective total workloads. ⎧ ⎫ (1) min t M = max ⎨ max t ikF ⎬ 1≤i ≤ n ⎩ 1≤ k ≤ K i ⎭ (2) min WM = max {W j }

{ }

1≤ j ≤ m

m

(3) min WT = ∑ W j j =1

(4)

s. t. t + pi,k +1, j ≤ t iF,k +1 F ik

∀i , k , j

(5) t ikF ≥ 0 , ∀i , k Eq. (2) combining with Eq. (3) give a physical meaning to the f-JSP, which referring to reducing total processing time and dispatching the operations averagely for each machine. Considering both of the two equations, our objective is to balancing the workloads of all machines. Eq. (4) and Eq. (5) give two basic processing constrains.

3.

Heuristic method

To demonstrate f-JSP model clearly, we first prepare a simple example. Table 1 gives the data set of an f-JSP including 3 jobs operated on 4 machines. It is obviously a problem with total flexibility because all the machines are available for each operation (Uik=U). There are several traditional heuristic methods that can be used to make a feasible schedule. In this case, we use the SPT (select the operation with the shortest processing time) as selective strategy to find an optimal solution, and the algorithm is based on the procedure in Figure 1. Before selection we first make some initialization: • starting from a table D presenting the processing times possibilities • on the various machines, create a new table D’ whose size is the same one as the table D; • create a table S whose size is the same one as the table D (S is going to represent chosen - 225 -

assignments); • initialize all elements of S to 0 (Sikj=0) • recopy D in D’

Table 1. Data set of a 3-job 4-machine Problem.

J1

J2 J3

o 11 o 12 o 13 o 21 o 22 o 23 o 31 o 32

M1 1 3 3 4 2 9 8 4

M2 3 8 5 1 3 1 6 5

M3 4 2 4 1 9 2 3 8

M4 1 1 7 4 3 2 5 1

procedure: SPT Assignment input: dataset table D output: best schedule S begin for (i=1; i