Tabu search methods for a single machine scheduling problem

In this paper we discuss the use of three local search strategies within a tabu search (TS) method for the approximate solution of a single machine scheduling problem. The problem consists of minimizing the sum of the set-up costs and linear delay penalties whenN jobs, arriving at time zero, are to be scheduled for sequential processing on a continuously available machine. Following a review of a previous study of this problem, we first consider a TS method that uses the common approach of making a succession of pairwise job exchanges, or swaps, to move from one trial solution to another. Next, we consider the use of insert moves to define the local neighborhood of each trial solution. These moves consist of transferring a single job from one position to another in the schedule. Finally, we construct a TS method (TS-hybrid) that employs both swap and insert moves. Experiments with benchmark problems of up to 60 jobs illustrate that there is an advantage in using more than one strategy to move from one trial solution to another within a TS method.This work was begun during the author's summer internship at the Advanced Knowledge Systems Group of US West Advanced Technologies.     

A comparison of scheduling algorithms for flexible flow shop problems with unrelated parallel machines,setup times,and dual criteria

This paper considers a flexible flow shop scheduling problem, where at least one production stage is made up of unrelated parallel machines. Moreover, sequence- and machine-dependent setup times are given. The objective is to find a schedule that minimizes a convex sum of makespan and the number of tardy jobs in a static flexible flow shop environment. For this problem, a 0–1 mixed integer program is formulated. The problem is, however, a combinatorial optimization problem which is too difficult to be solved optimally for large problem sizes, and hence heuristics are used to obtain good solutions in a reasonable time. The proposed constructive heuristics for sequencing the jobs start with the generation of the representatives of the operating time for each operation. Then some dispatching rules and flow shop makespan heuristics are developed. To improve the solutions obtained by the constructive algorithms, fast polynomial heuristic improvement algorithms based on shift moves and pairwise interchanges of jobs are applied. In addition, metaheuristics are suggested, namely simulated annealing (SA), tabu search (TS) and genetic algorithms. The basic parameters of each metaheuristic are briefly discussed in this paper. The performance of the heuristics is compared relative to each other on a set of test problems with up to 50 jobs and 20 stages and with an optimal solution for small-size problems. We have found that among the constructive algorithms the insertion-based approach is superior to the others, whereas the proposed SA algorithms are better than TS and genetic algorithms among the iterative metaheuristic algorithms.     

A tabu search heuristic for a sequence-dependent and time-dependent scheduling problem on a single machine

This paper introduces a tabu search heuristic for a production scheduling problem with sequence-dependent and time-dependent setup times on a single machine. The problem consists in scheduling a set of dependent jobs, where the transition between two jobs comprises an unrestricted setup that can be performed at any time, and a restricted setup that must be performed outside of a given time interval which repeats daily in the same position. The setup time between two jobs is thus a function of the completion time of the first job. The tabu search heuristic relies on shift and swap moves, and a surrogate objective function is used to speed-up the neighborhood evaluation. Computational experiments show that the proposed heuristic consistently finds better solutions in less computation time than a recent branch-and-cut algorithm. Furthermore, on instances where the branch-and-cut algorithm cannot find the optimal solution, the heuristic always identifies a better solution.     

Tabu search for scheduling on identical parallel machines to minimize mean tardiness

This paper presents a tabu search approach for scheduling jobs on identical parallel machines with the objective of minimizing the mean tardiness. Initially, we consider a basic tabu search that uses short term memory only. Local search is performed on a neighborhood defined by two types of moves. Insert moves consist of transferring each job from one machine to another and swap moves are those obtained by exchanging each pair of jobs between two machines. Next, we analyze the incorporation of two diversification strategies with the aim of exploring unvisited regions of the solution space. The first strategy uses long term memory to store the frequency of the moves executed throughout the search and the second makes use of influential moves. Computational tests are performed on problems with up to 10 machines and 150 jobs. The heuristic performance is evaluated through a lower bound given by Lagrangean relaxation. A comparison is also made with respect to the best constructive heuristic reported in the literature.     

Two-phase branch and bound algorithm for robotic cells rescheduling considering limited disturbance

This paper addresses a robotic cell rescheduling problem and focuses on trade-off between the total completion time of all jobs and the disturbance of a reschedule. We first define and measure the disturbance of a reschedule as the deviation of completion time of the jobs already scheduled between the reschedule and the initial schedule. To guarantee the steady performance of the system, we consider a special case that the processing sequence of the jobs already scheduled cannot be changed. The addressed rescheduling problem is transformed into a series of deterministic local scheduling problems with the objective of minimizing the total completion time of all jobs provided that the disturbance is within a given limit. A two-phase branch and bound algorithm is developed to efficiently solve the local scheduling problems. To improve the efficiency of the search procedure, a dynamic enumeration mechanism is applied to eliminate redundant constraints. Furthermore, two search strategies are proposed to direct the search procedure toward finding an optimal solution and a near-optimal solution. Finally, computational results demonstrate the efficiency of our algorithm.     

A branch and bound enhanced genetic algorithm for scheduling a flowline manufacturing cell with sequence dependent family setup times

This paper addresses the permutation flowline manufacturing cell with sequence dependent family setup times problem with the objective to minimize the makespan criterion. We develop a cooperative approach including a genetic algorithm and a branch and bound procedure. The latter is probabilistically integrated in the genetic algorithm in order to enhance the current solution. Moreover, the application of the branch and bound algorithm is based upon the decomposition of the problem into subproblems. The performance of the proposed method is tested by numerical experiments on a large number of representative problems.     

An Enhanced Dynamic Slope Scaling Procedure with Tabu Scheme for Fixed Charge Network Flow Problems

A heuristic algorithm for solving large scale fixed charge network flow problems (FCNFP) based on the dynamic slope scaling procedure (DSSP) and tabu search strategies is presented. The proposed heuristic integrates the DSSP with short-term memory intensification and long-term memory diversification mechanisms in the tabu scheme to improve the performance of the pure DSSP. With the feature of dynamically evolving memory, the enhanced DSSP evaluates the solutions in the search history and iteratively adjusts the linear factors in the linear approximation of the FCNFP to produce promising search neighborhoods for good quality solutions. The comprehensive numerical experiments on various test problems ranging from sparse to dense network structures are reported. The overall comparison of the pure DSSP, the enhanced DSSP, and branch and bound (B&B by cutting-edge MIP optimizer in CPLEX) is shown in terms of solution quality and CPU time. The results show that the enhanced DSSP approach has a higher solution quality than the pure DSSP for larger scale problems. Research has been partially supported by NSF and CRDF grants.     

Deteriorating job scheduling to minimize the number of late jobs with setup times

Deteriorating jobs scheduling problems have been widely studied recently. However, research on scheduling problems with deteriorating jobs has rarely considered explicit setup times. With the current emphasis on customer service and meeting the promised delivery dates, we consider a single-machine scheduling problem to minimize the number of late jobs with deteriorating jobs and setup times in this paper. We derive some dominance properties, a lower bound, and an initial upper bound by using a heuristic algorithm to speed up the search process of the branch-and-bound algorithm. Computational experiments show that the algorithm can solve instances up to 1000 jobs in a reasonable amount of time.     

Single machine batch scheduling problem with fuzzy batch size

In a batch scheduling problem, jobs are grouped (group is called batch) and scheduled in batches, and a setup time is incurred when starting a new batch. Processing times are assumed to be identical for all jobs. Setup times are assumed to be identical for all batches. Though all batch sizes cannot exceed a common upper bound, the upper bound is flexible and satisfaction degree with respect to the upper limit to be maximized is given. Also the other two objectives, i.e., the maximum completion time and the flow-time are to be minimized. Usually there exists no solution optimizing three objectives at a time. Therefore we define non-dominated solutions consisting of batch size, batch number and allocation of jobs to batches. First we propose an efficient algorithm for a sub-problem with fixed upper limit of batch size, fixed batch number based on a Lagrange relaxation procedure. Based on the properties of non-dominated solutions clarified in this paper, we propose an efficient algorithm to find some non-dominated solutions. Finally we summarize the results in this paper and discuss further research problems.     

A branch and price algorithm to minimize makespan on a single batch processing machine with non-identical job sizes

In this paper, we consider the scheduling problem on a single batch processing machine with non-identical job sizes; in which the machine has a limited capacity and can process a group of jobs simultaneously as a batch. The processing time of a batch is the longest processing time of all jobs in the batch. The objective is to minimize the makespan. We formulate the problem using Dantzig–Wolfe decomposition as a set partitioning problem. Based on the set partitioning formulation, we present a tight lower bound using column generation method. A heuristic algorithm is also developed to generate the basic solution in the column generation method. A branch and price algorithm which combines the column generation technique with branch and bound method is then presented to obtain the optimal solution of the problem. The efficiency of the proposed branch and price algorithm is ultimately compared to the branch and bound algorithm from the literature, based on the generated sample problems.     

Makespan minimization for scheduling unrelated parallel machines with setup times

This study considers the problem of scheduling jobs on unrelated parallel machines with machine-dependent and job sequence-dependent setup times. In this study, a restricted simulated annealing (RSA) algorithm which incorporates a restricted search strategy is presented to minimize the makespan. The proposed RSA algorithm can effective reduce the search effort required to find the best neighborhood solution by eliminating ineffective job moves. The effectiveness and efficiency of the proposed RSA algorithm is compared with the basic simulated annealing and existing meta-heuristics on a benchmark problem dataset used in earlier studies. Computational results indicate that the proposed RSA algorithm compares well with the state-of-the-art meta-heuristic for small-sized problems, and significantly outperforms basic simulated annealing algorithm and existing algorithms for large-sized problems.     

Scheduling a batch processing machine with incompatible job families

The problem of scheduling batch processors is important in some industries and, at a more fundamental level, captures an element of complexity common to many practical scheduling problems. We describe a branch and bound procedure applicable to a batch processor model with incompatible job families. Jobs in a given family have identical job processing times, arbitrary job weights, and arbitrary job sizes. Batches are limited to jobs from the same family. The scheduling objective is to minimize total weighted completion time. We find that the procedure returns optimal solutions to problems of up to about 25 jobs in reasonable CPU time, and can be adapted for use as a heuristic for larger problems.     

The two-machine flowshop no-wait scheduling problem with a single server to minimize the total completion time

This work studies the scheduling problem where a set of jobs are available for processing in a no-wait and separate setup two-machine flow shop system with a single server. The no-wait constraint requires that the operations of a job have to be processed continuously without waiting between two machines. The setup time is incurred and attended by a single sever which can perform one setup at a time. The performance measure considered is the total completion time. The problem is strongly NP-hard. Optimal solutions for several restricted cases and some properties for general case are proposed. Both the heuristic and the branch and bound algorithms are established to tackle the problem. Computational experiments indicate that the heuristic and the branch and bound algorithm are superior to the existing ones in term of solution quality and number of branching nodes, respectively.     

A tabu search algorithm for parallel machine total tardiness problem

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 classifications, tabu tenure and intensification/diversification 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 purposeSeveral 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 different 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.     

Minimizing total tardiness in a two-machine re-entrant flowshop with sequence-dependent setup times

We consider a two-machine re-entrant flowshop scheduling problem in which all jobs must be processed twice on each machine and there are sequence-dependent setup times on the second machine. For the problem with the objective of minimizing total tardiness, we develop dominance properties and a lower bound by extending those for a two-machine re-entrant flowshop problem (without sequence-dependent setup times) as well as heuristic algorithms, and present a branch and bound algorithm in which these dominance properties, lower bound, and heuristics are used. For evaluation of the performance of the branch and bound algorithm and heuristics, computational experiments are performed on randomly generated instances, and results are reported.     

Solution method for the location planning problem of logistics park with variable capacity

In this paper we study a logistics park location planning problem in which the capacity of the logistics park is determined by the sectors used to establish it in an open site. Since the size of each sector is not necessarily the same in every potential site, the capacity of the logistics park is thus variable, which makes this problem different from the traditional location problems in which the capacity of each facility is fixed. The task of this problem is to determine the location of the logistics parks, the sectors to be used to establish the logistics park in each open site, and the allocation of customers to the established logistics parks so as to minimize the total costs for establishing the logistics parks and supplying the demands of customers. The size mode is introduced to deal with the nonlinear establishment cost function and consequently this problem is formulated as an integer linear programming (ILP) model. Since CPLEX can only solve the ILP model with small-size problems, a tabu search (TS) hybrid with filter and fan (F&F) is presented to obtain near optimal solutions. In the hybrid algorithm, the TS is used to improve the solution by changing the allocation of customers to open sites while the F&F is used to further improve the solution by adjusting the status of sites (i.e., open or closed). In addition, an elite solution pool is constructed to store good solutions found in the searching history. Whenever the hybrid algorithm is trapped in local minima, a new start solution will be generated from the elite pool so as to improve the search diversity. To evaluate the performance of the proposed hybrid TS method, the column generation (CG) method with an acceleration strategy is developed to provide tight lower bounds. Computational results showed that the proposed hybrid algorithm can obtain optimal solutions for most of the small size problems and satisfactory near-optimal solutions with comparison to lower bounds for large size problems.     

A hybrid Electromagnetism-like Mechanism/tabu search procedure for the single machine scheduling problem with a maximum lateness objective

This paper presents a hybrid meta-heuristic search procedure to solve the well-known single machine scheduling problem to minimize the maximum lateness over all jobs, where precedence relations may exist between some of the jobs. The hybridization consists of a well-designed balance between the principles borrowed from an Electromagnetism-like Mechanism algorithm and the characteristics used in a tabu search procedure. The Electromagnetism-like Mechanism (EM) algorithm follows a search pattern based on the theory of physics to simulate attraction and repulsion of solutions in order to move towards more promising solutions. The well-known tabu search enhances the performance of a local search method by using memory structures by prohibiting visited solutions during a certain time of the search process. The hybridization of both algorithms results in an important trade-off between intensification and diversification strategies. These strategies will be discussed in detail. To that purpose, a new set of data instances is used to compare different elements of the hybrid search procedure and to validate the performance of the algorithm.     

片上网络映射问题的改进禁忌搜索算法

为求解通信时延受约束的低能耗片上网络(NoC)映射问题,提出一种改进禁忌搜索算法.该算法由局部搜索和精英重组2个步骤经过多次迭代完成,局部搜索采用简化的robust tabu search (RoTS),精英重组步骤选用COHX交叉操作.实验结果表明:文中算法与RoTS相比具有优化性能好、搜索空间小的优点,映射结果比分支限界法平均节能16.1%,适于求解大规模NoC映射问题.     

Using tabu search with ranking candidate list to solve production planning problems with setups

This study considers production planning problems involving multiple products, multiple resources, multiple periods, setup times, and setup costs. It can be formulated as a mixed integer program (MIP). Solving a realistic MIP production planning problem is NP-hard; therefore, we use tabu search methods to solve such a difficult problem. Furthermore, we improve tabu search by a new candidate list strategy, which sorts the neighbor solutions using post-optimization information provided by the final tableau of the linear programming simplex algorithm. A neighbor solution with higher priority in the ranking sequence has a higher probability of being the best neighbor solution of a current solution. According to our experiments, the proposed candidate list strategy tabu search produces a good solution faster than the traditional simple tabu search. This study also suggests that if the evaluation of the entire neighborhood space in a tabu search algorithm takes too much computation and if an efficient and effective heuristic to rank the neighbor solutions can be developed, the speed of tabu search algorithm could be significantly increased by using the proposed candidate list strategy.     

A robust hybrid approach based on particle swarm optimization and genetic algorithm to minimize the total machine load on unrelated parallel machines

This paper dealt with an unrelated parallel machines scheduling problem with past-sequence-dependent setup times, release dates, deteriorating jobs and learning effects, in which the actual processing time of a job on each machine is given as a function of its starting time, release time and position on the corresponding machine. In addition, the setup time of a job on each machine is proportional to the actual processing times of the already processed jobs on the corresponding machine, i.e., the setup times are past-sequence-dependent (p-s-d). The objective is to determine jointly the jobs assigned to each machine and the order of jobs such that the total machine load is minimized. Since the problem is NP-hard, optimal solution for the instances of realistic size cannot be obtained within a reasonable amount of computational time using exact solution approaches. Hence, an efficient method based on the hybrid particle swarm optimization (PSO) and genetic algorithm (GA), denoted by HPSOGA, is proposed to solve the given problem. In view of the fact that efficiency of the meta-heuristic algorithms is significantly depends on the appropriate design of parameters, the Taguchi method is employed to calibrate and select the optimal levels of parameters. The performance of the proposed method is appraised by comparing its results with GA and PSO with and without local search through computational experiments. The computational results for small sized problems show that the mentioned algorithms are fully effective and viable to generate optimal/near optimal solutions, but when the size of the problem is increased, the HPSOGA obtains better results in comparison with other algorithms.