Discrete Dynamics in Nature and Society

Volume 2015, Article ID 320140, 8 pages

http://dx.doi.org/10.1155/2015/320140

## A Local Search Algorithm for the Flow Shop Scheduling Problem with Release Dates

^{1}Software College, Northeastern University, Shenyang 110819, China^{2}School of Economics & Management, Shenyang University of Chemical Technology, Shenyang 110142, China

Received 25 August 2014; Accepted 4 November 2014

Academic Editor: Baoqiang Fan

Copyright © 2015 Tao Ren et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper discusses the flow shop scheduling problem to minimize the makespan with release dates. By resequencing the jobs, a modified heuristic algorithm is obtained for handling large-sized problems. Moreover, based on some properties, a local search scheme is provided to improve the heuristic to gain high-quality solution for moderate-sized problems. A sequence-independent lower bound is presented to evaluate the performance of the algorithms. A series of simulation results demonstrate the effectiveness of the proposed algorithms.

#### 1. Introduction

In a flow shop scheduling model, each job must be processed on a set of machines in identical order. The goal is to determine the job sequence to optimize a certain predetermined objective function. At any given time, each machine can process at most one job, and each job can be handled by at most one machine. Meanwhile, each job cannot be preempted by the other jobs. Flow shop scheduling problems widely exist in industrial production and mechanical manufacturing. For example, in a steel-making process, molten steel is casted into semifinished slabs by a conticaster; after being heated by the heat furnace, the slabs are rolled into products in rolling mill. Obviously, it is a typical flow shop production model. As most of the problems are strongly NP-hard, it is impossible to obtain the global optimum solution in polynomial time. So the study of flow shop scheduling algorithms is very important for reducing running time and boosting productivity.

Since the first scheduling rule was presented by Johnson [1] for the two-machine flow shop problem with objective of makespan (i.e., the maximum completion time) minimization, many works have been conducted on this research area. A comprehensive survey of flow shop makespan problems by 2010 can be found in Potts and Strusevich [2] or Bai and Ren [3]. The up-to-date research works are mentioned as follows. A. Rudek and R. Rudek [4] proved the ordinary NP-hardness for two-machine flow shop makespan problem when job processing times are described by nondecreasing position dependent functions (aging effect) on at least one machine and indicated the strong NP-hardness if job processing times are varying on both machines. Aydilek and Allahverdi [5] presented a polynomial time heuristic algorithm for the two-machine flow shop makespan problem with release dates. For the minimizing makespan in an -machine flow shop with learning considerations problem, Chung and Tong [6] proposed a dominance theorem and a lower bound to accelerate the branch-and-bound algorithm for seeking the optimal solution. For the criterion of makespan in flow shop model, a high-performing constructive heuristic with an effective tie-breaking strategy was proposed by Ying and Lin [7] to improve the quality of solutions. Similarly, Gupta et al. [8] proposed an alternative heuristic algorithm that is compared with the benchmark, Palmer’s, CDS, and NEH algorithms, to solve -job and -machine flow shop scheduling problem with minimizing makespan. For the result of the job-related criterion, Bai [9] presented the asymptotic optimality of the shortest processing time-based algorithms for the flow shop problem to optimize total quadratic completion time with release dates. Bai and Zhang [10] extended the results to a general objective, total -power completion time ().

In this paper, the flow shop scheduling problem for the minimization of makespan with release dates is addressed. Contrary to the static setting in which the jobs are simultaneously available, jobs arrive over time according to their release dates, which more closely approaches practical scheduling environments. Lenstra et al. [11] proved that the two-machine flow shop makespan problem with release dates is strongly NP-hard. It implies that the optimal solution of this problem cannot be found in polynomial time; heuristic algorithm may be more effective to obtain an approximate solution for large-sized problems. Therefore, a new modified GS algorithm (MGS) based on the algorithm of Gonzalez and Sahni [12] is presented for slow shop minimizing makespan with release dates. Then an improved scheme is provided to promote performance of the MGS algorithm. Moreover, a sequence-independent lower bound of the problem is presented. Computational experiments reveal the performances of the MGS algorithm, improved scheme, and lower bound in different size problems.

The remainder of this paper is organized as follows. The problem is formulated in Section 2, and the MGS algorithm and improved scheme are provided in Sections 3 and 4, respectively. The new lower bound and computational results are given in Section 5. This paper closes with the conclusion in Section 6.

#### 2. Problem Statement

In a flow shop problem, a set of jobs has to be processed on different machines in the same order. Job , , is processed on machines , , with a nonnegative processing time and a release date , which is the earliest time when the job is permitted to process. Each machine can process at most one job and each job can be handled by at most one machine at any given time. The machine processes the jobs in a first come, first served manner. The permutation schedule is considered in this paper, and the intermediate storage between successive machines is unlimited. The completion time of job , , on machine , , is denoted by . The goal is to determine a job sequence that minimizes the makespan, that is, .

#### 3. The Modified GS Algorithm

Gonzalez and Sahni [12] presented the GS algorithm to solve the flow shop makespan problem. Based on its idea, a new heuristic named modified GS (MGS) algorithm is presented to deal with the flow shop makespan problem with release dates. A formal expression of the MGS algorithm is presented as follows.

##### 3.1. The MGS Algorithm

*Step 1. *Divide the machines into groups.

*Step 2. *For each machine group , , whenever machine becomes idle or new jobs arrive, process the available jobs by Johnson’s rule (i.e., first schedule the jobs with in order of nondecreasing and then schedule the remaining jobs in order of nonincreasing , where denotes the processing time of job in group on machine , ); if no job is available, go to Step 3.

*Step 3. *Wait until a job arrives and go to Step 2. If all the jobs have been scheduled, go to Step 4.

*Step 4. *Terminate the program and calculate the objective values of the schedules. Select the minimum one as the final solution, .

The flowchart of the algorithm is shown in Figure 1. An example is proposed to show the execution of the MGS algorithm.