Journal of Applied Mathematics

Volume 2016, Article ID 9598041, 15 pages

http://dx.doi.org/10.1155/2016/9598041

## ILS Heuristics for the Single-Machine Scheduling Problem with Sequence-Dependent Family Setup Times to Minimize Total Tardiness

Department of Computer Science, Universidade Federal de Viçosa, 36570-900 Viçosa, MG, Brazil

Received 19 August 2016; Revised 15 September 2016; Accepted 19 September 2016

Academic Editor: Quanke Pan

Copyright © 2016 Vinícius Vilar Jacob and José Elias C. Arroyo. 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 addresses a single-machine scheduling problem with sequence-dependent family setup times. In this problem the jobs are classified into families according to their similarity characteristics. Setup times are required on each occasion when the machine switches from processing jobs in one family to jobs in another family. The performance measure to be minimized is the total tardiness with respect to the given due dates of the jobs. The problem is classified as -hard in the ordinary sense. Since the computational complexity associated with the mathematical formulation of the problem makes it difficult for optimization solvers to deal with large-sized instances in reasonable solution time, efficient heuristic algorithms are needed to obtain near-optimal solutions. In this work we propose three heuristics based on the Iterated Local Search (ILS) metaheuristic. The first heuristic is a basic ILS, the second uses a dynamic perturbation size, and the third uses a Path Relinking (PR) technique as an intensification strategy. We carry out comprehensive computational and statistical experiments in order to analyze the performance of the proposed heuristics. The computational experiments show that the ILS heuristics outperform a genetic algorithm proposed in the literature. The ILS heuristic with dynamic perturbation size and PR intensification has a superior performance compared to other heuristics.

#### 1. Introduction

Scheduling is a very important decision-making process that occurs in manufacturing systems. Scheduling problems deal with the allocation of resources to jobs over given time periods and its goal is to optimize one or more performance measures. This type of problems has been thoroughly studied since the mid-1950s [1]. Nowadays, scheduling problems are one of the most studied problems because they have great practical and theoretical importance. These problems have many applications in several industries (like chemical, metallurgic, and textile) and most of these problems belong to the class of -hard problems.

The scheduling problem focused on in this paper is stated as follows. There is a set of jobs to be processed on a single machine without interruption or preemption. Jobs are classified into families and are available at time zero. Each family has jobs (), such that . denotes the family of job . The processing time () and due date () of job are previously known. There is a family setup time between jobs and if job is processed immediately after job and . If jobs and belong to the same family (), . The setup times are sequence-dependent; that is, may not be equal to , , .

Setup time is the time required to prepare the necessary resource (machines) to perform a task (operation or job) [2]. Setup operations include obtaining tools, positioning work in process material, return tooling, cleanup, setting the required jigs and fixtures, adjusting tools, and inspecting material [1]. In the problem under study we do not consider the group technology (GT) assumption; that is, a family of jobs is not necessarily processed as a single batch [3]. Therefore a family of jobs could be divided into multiple nonconsecutive batches in an optimal sequence and each of the batches incurs a setup time.

The goal of the problem is to find a production schedule (sequence of jobs) that minimizes the total tardiness with respect to the given due dates of the jobs. The tardiness of a job (defined by ) is defined as the completion time of the job () minus the due date for the job if the job is completed after the due date, and the tardiness is equal to zero if the job is completed before the due date. can be expressed as . If job is processed immediately after job , then if , and if . The total tardiness of jobs (objective function) is computed as

The single-machine total tardiness (SMTT) problem with sequence-dependent family setup times is denoted by following the three-field notation presented by Graham et al. [4] and adapted by Allahverdi et al. [1], where is the single-machine environment, is information of sequence-dependent family or batch setup time, and is the total tardiness objective function.

The total tardiness criterion is very important in manufacture systems because several costs may exist when a job is delivered with tardiness. Among these the following can be quoted: contractual penalties, loss of credibility resulting in a high probability of losing a client for some or for all the possible futures jobs, and damage in the company’s reputation that may distance other clients.

Since the SMTT problem without sequence-dependent setup times is a binary -hard problem [5], it follows that the problem considered in this paper is at least -hard in the ordinary sense [6].

Scheduling problems that consider explicitly setup times are of great importance in manufacturing systems. Extensive literature reviews for these problems, considering different shop environments, were presented by [1, 2]. The SMTT problem with sequence-dependent setup times (without considering the grouping of jobs into families), denoted by , is one of the most researched topics in the scheduling literature. For this problem there are many studies on metaheuristics such as genetic algorithms [7–10], Simulated Annealing [11], Ant Colony Optimization [12, 13], Greedy Randomized Adaptive Search Procedure [14, 15], Iterated Local Search [16], and iterated greedy [17]. There are also some studies on exact algorithms for [18–21].

Some studies were realized on single-machine scheduling problems with family consideration and sequence-independent family setup times (). To better understand , let us suppose that the jobs , , and belong to different families. If job (or ) is processed immediately before job , then (or ), where is the considered sequence-independent setup time. Note that the setup time does not depend on the family (or ) previously processed; it only depends on family . Kacem [22] addressed the problem and proposed a set of methods to obtain lower bounds for the optimal total tardiness. Schaller [23] developed Branch and Bound and heuristic algorithms for the problem. Exact methods were used by Pan and Su [24] and Baker and Magazine [25] to solve the problem, where is the maximum lateness. To minimize the total earliness and tardiness cost (), exact and heuristic methods were developed by Schaller and Gupta [26]. Gupta and Chantaravarapan [27] considered the problem where jobs in each family are processed together; that is, the group technology assumption is considered. These authors provided a mixed-integer linear programming (MILP) model capable of solving small-sized problems.

Single-machine scheduling problems involving sequence-dependent family setup times have been much less studied [2]. van der Veen et al. [28] addressed the problem, where is the maximum completion time (makespan). These authors proposed a polynomial time algorithm for this problem. For the problem , where is the total completion time, Karabati and Akkan [29] proposed a Branch and Bound algorithm. Jin et al. [30] addressed the problem and developed a Simulated Annealing heuristic. For the same problem, Jin et al. [31, 32] proposed dominance relations and Tabu Search heuristics. Sels and Vanhoucke [33] considered the problem where each job has a release time (the earliest time at which the job can star its processing). They developed a genetic algorithm with local search. Recently, Herr and Goel [34] addressed a SMTT problem with sequence-dependent family setup times and resource constraints; that is, each job requires a certain amount of resource that is supplied through upstream processes. Schedules must be generated in such a way that the total resource demand does not exceed the resource supply up to any point in time. This problem can be denoted as , where is the quantity of a resource required by the machine to process the job . The authors proposed a MILP model and a heuristic algorithm to solve the problem.

To the best of our knowledge, there is only a paper in the literature discussing the problem. Chantaravarapan et al. [6] propose a MILP model and a hybrid genetic algorithm (HGA) for this problem. The effectiveness of the HGA was evaluated by instances with up to 60 jobs. The experiments presented in [6] showed that HGA performs better than other heuristics.

Motivated by the computational complexity and the practical relevance of the problem, in this work we test the applicability of the metaheuristic Iterated Local Search (ILS) to find good quality solutions for realistic size problem instances. We use ILS because it is a simple and generally applicable heuristic that has proved to be very effective for solving a wide range of difficult combinatorial optimization problems, especially vehicle routing problems [35–39] and scheduling problems [40–45]. Furthermore, ILS has very few control parameters and it does not require specific knowledge of the problem as in sophisticated heuristic algorithms.

The traditional ILS consists of an iterative process that combines a perturbation phase and a local search phase [46, 47]. During the perturbation phase, the current solution is modified in a probabilistic fashion similar to the mutation operator used in genetic algorithms. In the local search phase, the perturbed solution is improved leading to a local minimum solution. An acceptance criterion is used to decide whether the search continues from the local minimum solution or from the one that served as the starting point of the most recent perturbation phase.

Besides the standard ILS heuristic, this paper presents other contributions: the ILS is enriched by two special features. The first feature consists in using a variable perturbation size to escape from local optimal solutions. This feature is inspired from the Variable Neighborhood Search (VNS) heuristic that systematically changes the neighborhood within the search [48]. The perturbation size at the beginning is fixed at and incremented by 1, if the solution is not improved until a value (the maximum perturbation size). If a solution improves in any perturbation size, it is again fixed at . The second feature consists in using a Path Relinking (PR) technique [49] to intensify the search of good solutions. PR generates new solutions by exploring paths that connect elite solutions. The performance of our ILS heuristic with the different features is carefully analyzed.

The remainder of the paper is organized as follows. In Section 2, we present the mathematic formulation for the problem under study. In Section 3, we describe all the phases of the proposed ILS heuristics. In Section 4, we show the design of instances and the calibration of our heuristics and report the computational results. Finally, in Section 5, we conclude this paper and give future directions.

#### 2. Problem Model

In this section we present a mixed-integer linear programming (MILP) model of the problem. This mathematical model is originally from [6]. The resulting MILP is used to assess the performance of the developed heuristics for small-size problem instances. The goal of the model is to determine the optimal sequence of jobs to be processed on the single machine. A sequence is a permutation of jobs , where is the th job of the processing sequence (i.e., is the job processed in the position ).

The following parameters are used in the model (input data): = number of families, = number of jobs in family (), = total number of jobs (), = due date of the th job in family (, ), = processing time of the th job in family (, ), = sequence-dependent setup time of preceding family and following family (), = setup time of family before the first position ().

The following decision variables are used within the model: = completion time of job at position . = tardiness of job at position .

The resulting MILP model is as follows:

The objective function (the total tardiness) to be minimized is defined in (2). Constraint set (3) assures that one position of the sequence can contain only one job. Constraint set (4) guarantees that each job should be processed only once. Constraint sets (5) and (6) calculate the completion time of job in the first position of the sequence (). Constraint set (5) controls the setup time of the first position, resulting in the completion time of the first position in (6). Constraint sets (7) and (8) calculate the completion times from the second position to the last position of the sequence. For any two consecutive jobs, constraint set (7) checks whether or not the preceding job and the following job are from the same family. If so, there is no setup time between them (). Otherwise, setup time exists (). Constraint (9) computes the tardiness value () of jobs at each position . Constraints (10) represent nonnegativity conditions of the decision variables and , while constraints (11) and (12) ensure that and are binary variables.

#### 3. Proposed ILS Heuristics

To solve the problem, in this section, we propose three heuristics based on the Iterated Local Search (ILS) metaheuristic. ILS is a simple and generally applicable metaheuristic that iteratively applies a* perturbation* procedure as a diversification mechanism and a* local search* (LS) as an improvement heuristic. At each iteration, a new initial solution is generated by randomly performing an appropriate modification, called perturbation, to a good locally optimal solution previously found (current solution). Instead of generating a new initial solution from scratch, the perturbation mechanism generates a promising initial solution by retaining part of the structure that made the current solution a good solution. The perturbed solution is improved by the LS heuristic obtaining a new solution . The solution is accepted as the new current solution under some conditions defined by the* acceptance criteria*. A detailed explanation of the ILS metaheuristic can be found in [47].

The first proposed heuristic, called ILS_BASIC, is a standard implementation of ILS. The pseudocode of ILS_BASIC is described in Algorithm 1. To implement the basic ILS algorithm, four procedures are specified: (i) CONSTRUCTION, where an initial solution is constructed; (ii) LOCAL_SEARCH, which improves the solution initially obtained and the perturbed solution; (iii) PERTURBATION, where a new starter point is generated through a perturbation of the current solution; (iv) ACCEPTANCE_CRITERION, which determines from which solution the search should continue. The best solution found over all iterations is returned by the ILS algorithm.