A Tabu Search Algorithm for Parallel Machine Total Tardiness Problem

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<p>Available online at www.sciencedirect.com</p> <p>Computers &amp; Operations Research 31 (2004) 397 414</p> <p>www.elsevier.com/locate/dsw</p> <p>A tabu search algorithm for parallel machine total tardiness problemUmit Bilge , Furkan K rac, Mujde Kurtulan, Pelin Pekgun Department of Industrial Engineering, Bogazici University, Bebek, 80815 Istanbul, Turkey Received 1 April 2001; received in revised form 1 July 2002</p> <p>Abstract In this study, we consider the problem of scheduling a set of independent jobs with sequence dependent setups on a set of uniform parallel machines such that total tardiness is minimized. Jobs have non-identical due dates and arrival times. A tabu search (TS) approach is employed to attack this complex problem. In order to obtain a robust search mechanism, several key components of TS such as candidate list strategies, tabu classications, tabu tenure and intensication/diversication strategies are investigated. Alternative approaches to each of these issues are developed and extensively tested on a set of problems obtained from the literature. The results obtained are considerably better than those reported previously and constitute the best solutions known for the benchmark problems as to date. Scope and purpose Several surveys on parallel machine scheduling with due date related objectives (Oper. Res. 38(1) (1990) 22; EJOR 38 (1989) 156; Oper. Res. 42 (1994) 1025) reveal that the NP-hard nature of the problem renders it a challenging area for many researchers who studied various versions. However, most of these studies make the assumption that jobs are available at the beginning of the scheduling period, which is an important deviation form reality. In this study, as well as distinct due dates and ready times, features such as sequence dependent setup times and di erent processing rates for machines are incorporated into the classical model. These enhancements approach the model to the actual practice at the expense of complicating the problem further. For this complex problem, we present a tabu search (TS) algorithm to minimize total tardiness and provide the best solutions known for a set of benchmark problems. ? 2003 Elsevier Ltd. All rights reserved.Keywords: Scheduling; Parallel machines; Total tardiness problem; Sequence dependent setup times; Tabu search</p> <p>Corresponding author. Tel.: +90-212-263-1500; fax: +90-212-265-1800. E-mail address: bilge@boun.edu.tr (U. Bilge).</p> <p>0305-0548/04/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0305-0548(02)00198-3</p> <p>398</p> <p>U . Bilge et al. / Computers &amp; Operations Research 31 (2004) 397 414</p> <p>1. Introduction The classical parallel machine total tardiness problem (PMTP) can be stated as follows [13]: A set of independent jobs is to be processed on a number of continuously available identical parallel machines. Each machine can process only one job at a time, and each job can be processed on one machine. Each job is ready at the beginning of the scheduling horizon and has a distinct processing time and a distinct due date. The objective is to determine a schedule such that total tardiness is minimized, where tardiness of a job is the amount of time its completion time exceeds its due date. The problem is NP-hard even for a single machine (Du and Leung [4]) and exact methods in which the dimensionality problem is acute are mostly limited to special cases like common due dates and equal processing times (i.e. Root [5], Lawler [6], Elmaghraby and Park [7], Dessouky [8]). A large class of heuristics is based on list scheduling where the jobs are rst prioritised according to some rule and then dispatched in this order to the machine with the earliest nish time. Such heuristics are proposed by Wilkerson and Irwin [9], Dogramaci and Surkis [10], Ho and Chang [11] and Koulamas [3]. Koulamas [12] also developed a decomposition heuristic and a hybrid simulated annealing heuristic, while Bean [13] applied a genetic algorithm (GA) heuristic to the PMTP. In all the studies cited above it is assumed that machines are identical, all jobs are available at time zero and setup times are non-existent. However, in many real-world situations there exist (i) distinct job ready dates, (ii) uniform parallel machines that are capable of processing these jobs at di erent speeds (i.e. new machines versus old machines) and (iii) sequence dependent setups. In this paper, these features are also incorporated into the model so as to dene a problem closer to reality albeit far more complex than the classical one. In this generalized version of parallel machine total tardiness problem (GPMTP), there are n jobs to be processed on m machines of k types. Machines belonging to the same type are identical whereas machines belonging to di erent types are uniform. Each job i has an integer processing time pik on a type k machine, an integer ready time ri , a distinct due date di and a sequence dependent setup k time sji of processing job i immediately after job j on a type k machine. For a given processing order of the jobs, the earliest completion time Ci and the tardiness Ti can be computed for each job, where tardiness is dened as Ti = max{0; Ci di }. The objective is to nd the processing order of the jobs that minimizes the sum of the tardiness of all the jobs. This paper presents a tabu search (TS) approach to the GPMTP dened above. Tabu search (Glover and Laguna [14], Reeves [15]) is a meta-heuristic that guides a local heuristic search procedure to explore the solution space beyond local optimality. TS allows intelligent problem solving by the incorporation of adaptive memory and responsive exploration. Key elements of the search path are selectively remembered and strategic choices are made to guide the search out of local optima and into diverse regions of the solution space. The adaptive memory usage is a clever compromise between the rigid memory structure of exact techniques like Branch &amp; Bound and the memoryless heuristics like local search procedures. A large number of successful applications of TS for scheduling problems can be found in literature. Laguna et al. [16] study the single machine scheduling problem with the objective of minimizing the sum of setup costs and delay penalties and propose a TS algorithm that uses a hybrid neighbourhood. James and Buchanan [17] develop enhanced TS strategies for the single machine early/tardy scheduling problem. Hubscher and Glover [18] apply a candidate list strategy and introduce an in uential diversication to parallel machine scheduling to minimize the makespan. Nowicki and Smutnicki</p> <p>U . Bilge et al. / Computers &amp; Operations Research 31 (2004) 397 414</p> <p>399</p> <p>[19] present a TS to minimize makespan in a ow shop with parallel machines, and employ a neighbourhood based on blocks of operations on a critical path. A similar block approach is used by Liaw [20] for makespan minimization in an open shop. Park and Kim [21] compare simulated annealing and TS for a parallel machine scheduling problem where jobs have equal due dates and equal ready times for minimizing holding costs. When jobs are allowed to have distinct arrival times as well as due dates, di erent processing rates on machines and sequence dependent setup times, the literature becomes really sparse. There are only two studies reported on this more general problem to our knowledge and both of them deal with minimizing the total earliness-tardiness costs: Serifoglu and Ulusoy [22] present a GA with a new crossover operator, while Balakrishnan et al. [23] report a compact mathematical model to solve small sized (up to 10 jobs) problems. The TS algorithm proposed here is tested using the problem set given in Serifoglu and Ulusoy [22] and the results are compared to their results for the case where the weight of the earliness penalty is zero (In this case their problem also reduces to total tardiness problem). The next section describes the key aspects of the TS approach used. Section 3 compares several alternative approaches leading towards a robust TS algorithm tailored to solve the problem at hand, and evaluates the performance of this algorithm through numerical experimentation. The paper concludes with discussion of results and further studies in Section 4. 2. Description of the tabu search approach This section outlines the totally deterministic TS algorithm tailored to the GPMTP by discussing several of the key concepts such as solution encoding, initial solutions, tabu classications, candidate list structures, tabu tenure and intensication/diversication strategies. 2.1. Solution representation Considering a schedule with nj jobs on each machine j, where n = m nj , the solution is j=1 represented as m partial schedules as shown in Fig. 1. The objective value of a solution represented in this manner is obtained by going through the sequence of jobs over each machine and summing up the tardiness of each job, calculated by taking into account the distinct ready times, due dates, di erent processing times on di erent types of machines, and sequence dependent setup times. 2.2. Initial solutions In the presence of distinct ready times, sequence dependent set up times and uniform machines, list scheduling heuristics using priority rules such as shortest processing time or minimum slack time do not produce good results. Due to the same reason, more sophisticated list scheduling heuristics developed for the classical PMTP [3,9,11] cannot be readily used, either. Earliest Due Date (EDD) based list scheduling, however, performs agreeably well and is used to generate starting solutions for TS. Here, jobs are ordered with respect to their EDD and then scheduled on the machine that will complete them earliest.</p> <p>400</p> <p>U . Bilge et al. / Computers &amp; Operations Research 31 (2004) 397 414</p> <p>Machine 1</p> <p>1</p> <p>2</p> <p>3</p> <p>..</p> <p>..</p> <p>..</p> <p>..</p> <p>..</p> <p>..</p> <p>n1</p> <p>Machine 2 : : : Machine j : : : Machine m</p> <p>1</p> <p>2</p> <p>3</p> <p>..</p> <p>..</p> <p>..</p> <p>..</p> <p>..</p> <p>..</p> <p>..</p> <p>n2</p> <p>1</p> <p>2</p> <p>3</p> <p>..</p> <p>..</p> <p>..</p> <p>..</p> <p>nj</p> <p>1</p> <p>2</p> <p>3</p> <p>..</p> <p>..</p> <p>..</p> <p>nm</p> <p>Fig. 1. Solution representation.</p> <p>2.3. Neighbourhood generation Insert moves and pairwise exchanges (swaps) are two of the frequently used move types in permutation problems. An insert move identies two particular jobs and places the rst job in the location that directly precedes the location of the second job. A swap move, on the other hand, places each job in the location previously occupied by the other, and can be considered as a move that combines two insert moves. In the parallel machine scheduling problem the new locations may be on di erent machines as well as on the same machine. Swap moves involving jobs on di erent machines do not cause a change in the number of jobs on machines. The neighbourhood used in this study has a hybrid structure. It consists of the complete insert neighbourhood where all the intra-machine and inter-machine insert moves are considered, with the addition of a partial swap neighbourhood which consists of inter-machine swap moves only. In other words, only those swaps that involve two jobs each on di erent machines are considered. Hence, the neighbourhood includes also the moves that create di erent sequences without changing the number of jobs on machines. 2.4. Candidate list strategies For situations where the neighbourhood of a solution is large or its elements are expensive to evaluate, candidate list strategies are essential to restrict the number of solutions examined on a given iteration [16]. The purpose of these rules is to screen the neighbourhood so as to concentrate on promising moves at each iteration. When the aggressive nature of TS in selecting the next solution is considered, rules for generating and evaluating good candidates become critical for the e ciency of</p> <p>U . Bilge et al. / Computers &amp; Operations Research 31 (2004) 397 414</p> <p>401</p> <p>the search process. Since jobs have distinct ready times, di erent processing times on di erent types of machines and sequence dependent setup times, calculation of total tardiness for a given move is a tedious task. Although this is implemented in an e cient way by rst determining the a ected jobs and updating the tardiness values for only those jobs, move value calculation is still time consuming. Therefore, a good candidate list strategy, which saves time, is critical for the e ciency of the TS algorithm. In this study three candidate list strategies, which are described below, are tested. Since the swap neighbourhood is already small, these strategies are applied only for insert moves. 2.4.1. The maximum tardy vs. maximum early approach In this approach, only the jobs on the machine with the highest contribution to total tardiness are chosen as candidates for inserting on the machine with the highest contribution to total earliness, as explained below: Calculate the contribution of each machine j to total tardiness as iIj Ti , where Ij is the set of jobs scheduled on machine j. Select the machine with highest contribution to total tardiness, and call it machine k T . Calculate the contribution of each machine j to total earliness as iIj Ei , where Ei is the earliness of job i given by Ei = max{0; di Ci }. Select the machine with highest contribution to total earliness, and call it machine k E . Consider every job on machine k T for an insertion on machine k E . Since this approach has been shown to be quite fast and decreases the size of the neighbourhood considerably, it is called the High Candidate List Strategy. 2.4.2. The maximum tardy approach In this approach, the machine with the highest contribution to total tardiness is specied as explained above. Then, only the jobs on this machine are considered for an insert operation on any other machine. Since the neighbourhood screening introduced is less, this approach is slower and it is called the Low Candidate List Strategy. 2.4.3. The ready time closeness approach In this approach, job i is allowed to be inserted after job j only if the completion time of job j in the current schedule is within a range of the ready time of job i, i.e. |ri Cj | 6 a threshold value. Thus, the amount of time job i spends waiting (when ri Cj ) or the inserted idleness on the machine (when Cj ri ) is bounded. The threshold value is a measure of closeness of the two jobs. Since the aim of candidate list strategy is to save time, the threshold value has to be simple and easy to compute. Moreover, it should not be too restrictive, and it should be dependent on data specic to the problem instance. For these reasons, instead of trying to develop sophisticated bounds that apply under di erent cases, the threshold value is chosen to be the sum of the maximum processing time and the maximum setup time of all the jobs. This strategy is called the Closeness Candidate List Strategy.</p> <p>402</p> <p>U . Bilge et al. / Computers &amp; Operations Research 31 (2004) 397 414</p> <p>2.5. Tabu classication In...</p>