Mathematical Problems in Engineering

Volume 2015, Article ID 340769, 8 pages

http://dx.doi.org/10.1155/2015/340769

## A Decomposition-Based Two-Stage Optimization Algorithm for Single Machine Scheduling Problems with Deteriorating Jobs

School of Economics and Management, Nanchang University, Nanchang 330031, China

Received 14 February 2015; Revised 7 May 2015; Accepted 12 May 2015

Academic Editor: Marco Mussetta

Copyright © 2015 Yueyue Liu 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 studies a production scheduling problem with deteriorating jobs, which frequently arises in contemporary manufacturing environments. The objective is to find an optimal sequence of the set of jobs to minimize the total weighted tardiness, which is an indicator of service quality. The problem belongs to the class of NP-hard. When the number of jobs increases, the computational time required by an optimization algorithm to solve the problem will increase exponentially. To tackle large-scale problems efficiently, a two-stage method is presented in this paper. We partition the set of jobs into a few subsets by applying a neural network approach and thereby transform the large-scale problem into a series of small-scale problems. Then, we employ an improved metaheuristic algorithm (called GTS) which combines genetic algorithm with tabu search to find the solution for each subproblem. Finally, we integrate the obtained sequences for each subset of jobs and produce the final complete solution by enumeration. A fair comparison has been made between the two-stage method and the GTS without decomposition, and the experimental results show that the solution quality of the two-stage method is much better than that of GTS for large-scale problems.

#### 1. Introduction

The problem of scheduling jobs on a single machine to minimize total weighted tardiness is extensively studied by many researchers, which also occurs as a subproblem in other scheduling environments such as job shops. The classic scheduling models routinely assume that the job processing times are known and fixed, which however may not be satisfied in many real-world situations. For example, the deterioration effect is known as the fact that a job may need longer processing time if its starting time is postponed. This paper considers the single machine total weighted tardiness problem with proportional deterioration, which is common in steel manufacturing, plastic processing and medical treatments, and so forth [1].

Lawler [2] and Lenstra et al. [3] have shown that the total weighted tardiness problem is NP-hard. Congram et al. [4] proposed an effective local search method for this kind of problem. Bozejko et al. [5] presented a fast local search procedure based on a tabu search approach which employs blocks of jobs and compound moves and applied it to the total weighted tardiness problem.

J. N. D. Gupta and S. K. Gupta [6] and Browne and Yechiali [7] first introduced deterioration into scheduling problems. Recently, there have been growing interests in studying scheduling problems with deteriorating jobs [8]. Bachman and Janiak [9] considered the problems of minimizing maximum lateness under linear deterioration which is NP-hard and presented two heuristic algorithms. Bachman et al. [10] dealt with the single machine scheduling problem with start time dependent job processing times. Cheng et al. [11] introduced a class of machine scheduling problems in which the processing time of a task is dependent on its starting time in a schedule. Hsu and Lin [12] designed a branch-and-bound algorithm for deriving exact solutions according to several properties concerning dominance relations and lower bounds for the single machine problem with deteriorating jobs to minimize the maximum lateness. Wang et al. [13] considered single machine scheduling problems where the processing time is the increasing function of their starting times and the jobs are related by a series-parallel graph. More recent contributions in this line of research can be referred to [14–18], where integrated scheduling problems with deteriorating jobs are studied.

In terms of general single machine scheduling, many different solution approaches have been proposed in the existing literature. Sels and Vanhoucke [19] developed a hybrid dual-population genetic algorithm for the single machine maximum lateness problem which took some specific characteristics into account. Their work has an important implication on the balance between intensification and diversification in the design of search algorithms. Voutsinas and Pappis [20] proposed a branch-and-bound algorithm which uses the suboptimal solution of a heuristic as initial solution for solving the single machine scheduling problem with deteriorating jobs. Jolai et al. [21] focused on the bicriteria scheduling problem of minimizing the number of tardy jobs and maximum earliness for single machine scheduling without allowing idle times. Pakzad-Moghaddam et al. [22] presented a mixed-integer mathematical programming model for a single machine scheduling problem with deteriorating and learning effects. Xu et al. [23] solved the single machine scheduling problem with sequence-dependent setup times and conducted a systematic comparison of hybrid evolutionary algorithms (HEAs), which independently used the six combinations of three crossover operators and two population updating strategies. More recent works that deal with advanced single machine scheduling problems can be found in [24–28].

This paper considers the single machine total weighted tardiness problem with proportional deterioration. Our objective is to minimize the total weighted tardiness of jobs. The problem belongs to the class of NP-hard problems. With the increase of the job number, the time an algorithm consumes to solve the problem increases exponentially. Hence, a two-stage method based on decomposition is presented. First, we perform partition using neural networks to transfer the large-scale problem into smaller-scale subproblems. Then, we find the solution for each subproblem by a hybrid metaheuristic algorithm. Indeed, we employ an improved genetic algorithm, called genetic tabu search (GTS), which uses tabu search (TS) as the mutation operator to increase the performance of local search. Genetic algorithm (GA) is an optimization method based on the principles of genetics and natural selection. However, it is weak in local search and has premature convergence, so we combine it with a tabu search approach to enhance the algorithm. Finally, we combine the obtained sequence for each partition of jobs and give the final solution by means of enumeration.

This paper is organized as follows. In Section 2, the basic definitions and notations of the problem are presented. Section 3 describes the detailed algorithms in the two-stage method, including the neural network, the genetic algorithm, and tabu search. Computational results are shown in Section 4. Section 5 presents the conclusion.

#### 2. Problem Description

We consider the problem of scheduling a set of jobs on a single machine with the following assumptions: (i)All jobs are ready at time zero. (ii)The machine can handle only one job at a time. (iii)There are no precedence relations between jobs and preemption is not allowed. (iv)Setup times are not explicitly considered. (v)The basic processing times, the latest starting times, and the deterioration factor are known.

The single machine total weighted tardiness problem addressed in this paper is affected by the deterioration effect, which is a phenomenon commonly observed in many cases. For example, in the process of steel-making, as the temperature of an ingot drops below a specified level, it must be heated again to the temperature required for rolling.

The problem can be described as follows. There is a set of jobs (index , ) that have to be scheduled on a single machine. Each job is characterized by a basic processing time , a latest starting time , a weight , and a due date . For a given sequence of jobs, the (earliest) completion time , tardiness , and cost of job can be computed. The objective is to find a job sequence in which the given set of jobs are scheduled such that the total weighted tardiness with respect to the given due dates is minimized.

Besides, the actual processing time of each job is a nondecreasing function of its starting time. When the starting time of job () is earlier than or equal to the latest starting time which is given, the actual processing time equals the basic processing time . Otherwise, is a function of as defined below:where is the deterioration factor.

The notations that will be used throughout this paper are summarized in Notation section.

#### 3. A Two-Stage Solution Method Based on Decomposition

##### 3.1. Methodology Overview

To tackle complex optimization problems efficiently, the utilization of problem-specific properties is highly important in the design of optimization algorithms. As for the problem studied in this paper, since each job has several features including a basic processing time , a latest starting time , a weight , a due date , and a deterioration factor , the distributions of these data tend to have certain characteristics which can be extracted and utilized as special information to guide the search behavior of optimization algorithms. To improve the time efficiency of the optimization process, we transform the original large-scale scheduling problem into several small-scale scheduling problems based on the features of each job using a back-propagation (BP) neural network.

Firstly, we get the optimal solutions of some small-scale problems for training the neural network. After establishing the neural network which can roughly predict the position of each job, we divide the original set of jobs into several subsets, each corresponding to a scheduling subproblem. For example, if we are to have 4 subproblems, then we could use the neural network to predict which jobs will belong to the first 1/4 in the final schedule (and thus will be placed into the first subset), which will belong to the next 1/4 (and thus will be placed into the second subset), and so forth. Then, we use the proposed GTS to obtain the solution for each subproblem and combine them to yield the final solution.

##### 3.2. The First Stage

The BP neural network is a typical artificial neural network that can approximate complex nonlinear mapping functions. It has been widely adopted in application areas like classification, fitting, and compression. A typical BP neural network consists of three layers including an input layer, a hidden layer, and an output layer. The number of nodes in the input layer depends on the dimension of the input vector. In this paper, each job has five features including a basic processing time , a latest starting time , a weight , a due date and a deterioration factor . Generally, a single hidden layer can realize the arbitrary nonlinear mapping by increasing the number of neuron nodes appropriately. The number of nodes in the hidden layer, which can adjust the accuracy of the neural network, affects the training results and the training time. With an increase of the node number, the training results improve while the training time increases correspondingly. The number of neurons in the output layer is decided according to the practical problem, and, in this study, it is equal to the number of subproblems we are planning to divide into. The Quasi-Newton method, which is a fast optimization technique based on Taylor series expansion, is applied to train the neural network.

The network is trained based on sample data with known output derived from small-scale problem instances. The data consist of the information of each job with the above features and the sequential index of the subset it belongs to. The data is grouped into two parts, the training data for training the neural network and the test data for testing the correctness of the classification.

##### 3.3. The Second Stage

Although GA can be directly applied to complex combinatorial optimization problems, each generation of the algorithm must maintain a large population size. With the expansion of the problem size, the computational time needed will increase dramatically. Besides, GA usually converges prematurely, which is mainly caused by a lack of diversity in the population. In addition, the mutation operator is inadequate for a systematic local search. Compared with GA, TS has faster convergence rate. However, the search performance of TS greatly depends on the initial solution.

Population-based GA and single-trajectory TS have complementary characteristics. GA explores well the search space while TS intensifies the search in promising regions. According to the strengths and weaknesses of these two algorithms, we apply TS to replace the mutation operator in GA.

We present the general framework of the GTS in Algorithm 1. GTS starts from an initial random population (line 3). Then, the crossover operator is employed to generate new offspring solutions (line 6). Besides, the mutation operator is implemented with TS to enhance the local search performance (line 7). Subsequently, the population updating rule decides whether such a mutated solution should be inserted into the population and which existing individual should be replaced (line 9).