Unrelated parallel machine scheduling with setup times and a total weighted tardiness objective

This paper presents several search heuristics and their performance in batch scheduling of parallel, unrelated machines. Identical or similar jobs are typically processed in batches in order to decrease setup times and/or processing times. The problem accounts for allotting batched work parts into unrelated parallel machines, where each batch consists of a fixed number of jobs. Some batches may contain different jobs but all jobs within each batch should have an identical processing time and a common due date. Processing time of each job of a batch is determined according to the machine group as well as the batch group to which the job belongs. Major or minor setup times are required between two subsequent batches depending on batch sequence but are independent of machines. The objective of our study is to minimize the total weighted tardiness for the unrelated parallel machine scheduling. Four search heuristics are proposed to address the problem, namely (1) the earliest weighted due date, (2) the shortest weighted processing time, (3) the two-level batch scheduling heuristic, and (4) the simulated annealing method. These proposed local search heuristics are tested through computational experiments with data from dicing operations of a compound semiconductor manufacturing facility.     

A variable neighborhood search for minimizing total weighted tardiness with sequence dependent setup times on a single machine

This paper deals with the single machine scheduling problem to minimize the total weighted tardiness in the presence of sequence dependent setup. Firstly, a mathematical model is given to describe the problem formally. Since the problem is NP-hard, a general variable neighborhood search (GVNS) heuristic is proposed to solve it. Initial solution for the GVNS algorithm is obtained by using a constructive heuristic that is widely used in the literature for the problem. The proposed algorithm is tested on 120 benchmark instances. The results show that 37 out of 120 best known solutions in the literature are improved while 64 instances are solved equally. Next, the GVNS algorithm is applied to single machine scheduling problem with sequence dependent setup times to minimize the total tardiness problem without changing any implementation issues and the parameters of the GVNS algorithm. For this problem, 64 test instances are solved varying from small to large sizes. Among these 64 instances, 35 instances are solved to the optimality, 16 instances' best-known results are improved, and 6 instances are solved equally compared to the best-known results. Hence, it can be concluded that the GVNS algorithm is an effective, efficient and a robust algorithm for minimizing tardiness on a single machine in the presence of setup times.     

Robust scheduling on a single machine to minimize total flow time

In a real-world manufacturing environment featuring a variety of uncertainties, production schedules for manufacturing systems often cannot be executed exactly as they are developed. In these environments, schedule robustness that guarantees the best worst-case performance is a more appropriate criterion in developing schedules, although most existing studies have developed optimal schedules with respect to a deterministic or stochastic scheduling model. This study concerns robust single machine scheduling with uncertain job processing times and sequence-dependent family setup times explicitly represented by interval data. The objective is to obtain robust sequences of job families and jobs within each family that minimize the absolute deviation of total flow time from the optimal solution under the worst-case scenario. We prove that the robust single machine scheduling problem of interest is NP-hard. This problem is reformulated as a robust constrained shortest path problem and solved by a simulated annealing-based algorithmic framework that embeds a generalized label correcting method. The results of numerical experiments demonstrate that the proposed heuristic is effective and efficient for determining robust schedules. In addition, we explore the impact of degree of uncertainty on the performance measures and examine the tradeoff between robustness and optimality.     

Minimizing earliness and tardiness subject to total completion time in an identical parallel machine system

This study addresses the identical parallel machine scheduling problem in which the total earliness and tardiness about a common due date are minimized subject to minimum total flowtime, P∥(E+T)/∑_Ci$P\parallel (E+T)/\sum C_i$. The problem is demonstrated to be transformed into a simplified version of the parallel machine problem with the objective of minimizing makespan subject to minimum total flowtime, P∥_Cmax/∑_Ci$P\parallel C_{\max }/\sum C_i$. Several properties are considered to solve optimally the restricted version of the problem. A streamlined binary integer programming model is developed to solve the P∥_Cmax/∑_Ci$P\parallel C_{\max }/\sum C_i$ problem and the P∥(E+T)/∑_Ci$P\parallel (E+T)/\sum C_i$ problem. Computational experiments indicate that the binary integer programming model is superior to the existing optimization algorithm for the P∥_Cmax/∑_Ci$P\parallel C_{\max }/\sum C_i$ problem in the literature.     

Dominance rules for the parallel machine total weighted tardiness scheduling problem with release dates

We address the parallel machine total weighted tardiness scheduling problem with release dates. We describe dominance rules and filtering methods for this problem. Most of them are adaptations of dominance rules based on solution methods for the single-machine problem. We show how it is possible to deduce whether or not certain jobs can be processed by a particular machine in a particular context and we describe techniques that use this information to improve the dominance rules. On the basis of these techniques we describe an enumeration procedure and we provide experimental results to determine the effectiveness of the dominance rules.     

Single CNC machine scheduling with controllable processing times to minimize total weighted tardiness

Advanced manufacturing technologies, such as CNC machines, require significant investments, but also offer new capabilities to the manufacturers. One of the important capabilities of a CNC machine is the controllable processing times. By using this capability, the due date requirements of customers can be satisfied much more effectively. Processing times of the jobs on a CNC machine can be easily controlled via machining conditions such that they can be increased or decreased at the expense of tooling cost. Since scheduling decisions are very sensitive to the processing times, we solve the process planning and scheduling problems simultaneously. In this study, we consider the problem of scheduling a set of jobs on a single CNC machine to minimize the sum of total weighted tardiness, tooling and machining costs. We formulated the joint problem, which is NP-hard since the total weighted tardiness problem (with fixed processing times) is strongly NP-hard alone, as a nonlinear mixed integer program. We proposed a DP-based heuristic to solve the problem for a given sequence and designed a local search algorithm that uses it as a base heuristic.     

An O(n 2 ) algorithm for scheduling equal-length preemptive jobs on a single machine to minimize total tardiness

In this paper, we study the problem of scheduling n equal-length preemptive jobs on a single machine to minimize total tardiness, subject to release dates. The complexity status of this problem has remained open to date. We provide an O(n²) time algorithm to solve the problem.     

A hybrid genetic algorithm for the single machine scheduling problem with sequence-dependent setup times

This paper presents a hybrid approach based on the integration between a genetic algorithm (GA) and concepts from constraint programming, multi-objective evolutionary algorithms and ant colony optimization for solving a scheduling problem. The main contributions are the integration of these concepts in a GA crossover operator. The proposed methodology is applied to a single machine scheduling problem with sequence-dependent setup times for the objective of minimizing the total tardiness. A sensitivity analysis of the hybrid approach is carried out to compare the performance of the GA and the hybrid genetic algorithm (HGA) approaches on different benchmarks from the literature. The numerical experiments demonstrate the HGA efficiency and effectiveness which generates solutions that approach those of the known reference sets and improves several lower bounds.     

A discrete differential evolution algorithm for the single machine total weighted tardiness problem with sequence dependent setup times

This paper is concerned with solving the single machine total weighted tardiness problem with sequence dependent setup times by a discrete differential evolution algorithm developed by the authors recently. Its performance is enhanced by employing different population initialization schemes based on some constructive heuristics such as the well-known NEH and the greedy randomized adaptive search procedure (GRASP) as well as some priority rules such as the earliest weighted due date (EWDD) and the apparent tardiness cost with setups (ATCS). Additional performance enhancement is further achieved by the inclusion of a referenced local search (RLS) in the algorithm together with the use of destruction and construction (DC) procedure when obtaining the mutant population. Furthermore, to facilitate the greedy job insertion into a partial solution which will be employed in the NEH, GRASP, DC heuristics as well as in the RLS local search, some newly designed speed-up methods are presented for the insertion move for the first time in the literature. They are novel contributions of this paper to the single machine tardiness related scheduling problems with sequence dependent setup times. To evaluate its performance, the discrete differential evolution algorithm is tested on a set of benchmark instances from the literature. Through the analyses of experimental results, its highly effective performance with substantial margins both in solution quality and CPU time is shown against the best performing algorithms from the literature, in particular, against the very recent newly designed particle swarm and ant colony optimization algorithms of Anghinolfi and Paolucci [A new discrete particle swarm optimization approach for the single machine total weighted tardiness scheduling problem with sequence dependent setup times. European Journal of Operational Research 2007; doi:10.1016/j.ejor.2007.10.044] and Anghinolfi and Paolucci [A new ant colony optimization approach for the single machine total weighted tardiness scheduling problem. http://www.discovery.dist.unige.it/papers/Anghinolfi_Paolucci_ACO.pdf, respectively. Ultimately, 51 out of 120 overall aggregated best known solutions so far in the literature are further improved while other 50 instances are solved equally.     

A population-based variable neighborhood search for the single machine total weighted tardiness problem

This paper investigates the single machine total weighted tardiness problem, in which a set of independent jobs with distinct processing times, weights, and due dates are to be scheduled on a single machine to minimize the sum of weighted tardiness of all jobs. This problem is known to be strongly NP-hard, and thus provides a challenging area for metaheuristics. A population-based variable neighborhood search (PVNS) algorithm is developed to solve it. This algorithm differs from the basic variable neighborhood search (VNS). First, the PVNS consists of a number of iterations of the basic VNS, and in each iteration a population of solutions is used to simultaneously generate multiple trial solutions in a neighborhood so as to improve the search diversification. Second, the PVNS adopts a combination of path-relinking, variable depth search and tabu search to act as the local search procedure so as to improve the search intensification. Computational experiments show that the proposed PVNS algorithm can obtain the optimal or best known solutions within a reasonable computation time for all standard benchmark problem instances from the literature.     

Parallel machine scheduling with splitting jobs by a hybrid differential evolution algorithm

The problem of parallel machine scheduling for minimizing the makespan is an open scheduling problem with extensive practical relevance. It has been proved to be non-deterministic polynomial hard. Considering a job’s batch size greater than one in the real manufacturing environment, this paper investigates into the parallel machine scheduling with splitting jobs. Differential evolution is employed as a solution approach due to its distinctive feature, and a new crossover method and a new mutation method are brought forward in the global search procedure, according to the job splitting constraint. A specific local search method is further designed to gain a better performance, based on the analytical result from the single product problem. Numerical experiments on the performance of the proposed hybrid DE on parallel machine scheduling problems with splitting jobs covering identical and unrelated machine kinds and a realistic problem are performed, and the results indicate that the algorithm is feasible and efficient.     

A heuristic for job shop scheduling to minimize total weighted tardiness

This paper considers the job shop scheduling problem to minimize the total weighted tardiness with job-specific due dates and delay penalties, and a heuristic algorithm based on the tree search procedure is developed for solving the problem. A certain job shop scheduling to minimize the maximum tardiness subject to fixed sub-schedules is solved at each node of the search tree, and the successor nodes are generated, where the sub-schedules of the operations are fixed. Thus, a schedule is obtained at each node, and the sub-optimum solution is determined among the obtained schedules. Computational results on some 10 jobs and 10 machines problems and 15 jobs and 15 machines problems show that the proposed algorithm can find the sub-optimum solutions with a little computation time.     

Parallel machine earliness/tardiness scheduling problem under the effects of position based learning and linear/nonlinear deterioration

This paper considers a parallel machine earliness/tardiness (ET) scheduling problem with different penalties under the effects of position based learning and linear and nonlinear deterioration. The problem has common due-date for all jobs, and effects of learning and deterioration are considered simultaneously. By the effects of learning we mean that the job processing time decreases along the sequence of partly similar jobs, and by the effects of deterioration we mean slowing performance or time increases along the sequence of jobs. This study shows that optimal solution for ET scheduling problem under effects of learning and deterioration is V-shape schedule under certain agreeable conditions. Furthermore, we design a mathematical model for the problem under study and algorithm and lower bound procedure to solve larger test problems. The algorithm can solve problems of 1000 jobs and four machines within 3 s on average. The performance of the algorithm is evaluated using results of the mathematical model.     

Minimizing total tardiness on a single machine with controllable processing times

The concept of time-cost trade-off is commonly considered in PERT/CPM, but it is seldom considered in the scheduling area. Such concept implies that the processing times of jobs are controllable by increasing or decreasing the available resources, such as manpower and equipment. In this paper, we focus on the single machine total tardiness problem with controllable processing times. First, a mixed-integer programming (MIP) model is formulated to find the optimal solution. Then, we propose both a linear programming model and a net benefit of compression (NBC) algorithm to obtain a set of optimal amounts of compression for a given sequence. To solve medium- to large-size problem instances, we develop a heuristic based on the NBC algorithm. To verify the proposed heuristic, the MIP model is used as a comparison for small-size problem instances, whereas for medium- to large-size instances the variable neighborhood search, a useful local search method, is employed. Computational results show that the proposed heuristic has a good performance.     

解不同交货期并行机调度问题的并行遗传算法

为有效地解决不同交货期窗口下的非等同并行多机提前/拖后调度问题,设计了一种分段编码的混合遗传算法。此编码方式能反映工件的分配序列,并利用调度优先级规则和最好适应值规则相结合的启发式算法对其顺序进行了调整,加快了收敛速度。同时为了更好地适应调度实时性和解大规模此类问题的需要,基于遗传算法自然并行性特点的基础上,实现了主从式控制网络模式下并行混合遗传算法。计算结果表明,此算法是有效的,优于遗传算法,有着较高的并行性,并能适用于大规模不同交货期窗口下非等同并行多机提前/拖后调度问题。     

An improved scatter search algorithm for the single machine total weighted tardiness scheduling problem with sequence-dependent setup times

In this paper, a scatter search algorithm with improved component modules is proposed to solve the single machine total weighted tardiness problem with sequence-dependent setup times. For diversification generation module, both random strategy based heuristics and construction heuristic are adopted to generate the diversified population. For improvement module, variable neighborhood search based local searches are embedded into the algorithm to improve the trial solutions and the combined solutions. For reference set update module, the number of edges by which the two solutions differ from each other is counted to measure the diversification value between two solutions. We also propose a new strategy in which the length of the reference set could be adjusted adaptively to balance the computing time and solving ability. In addition, a discrete differential evolution operator is proposed with another two operators constitute the combination module to generate the new trial solutions with the solutions in the subsets. The proposed algorithm is tested on the 120 benchmark instances from the literature. Computational results indicate that the average relative percentage deviations of the improved algorithm from the ACO_AP, DPSO, DDE and GVNS are −5.16%, −3.33%, −1.81% and −0.08%, respectively. Comparing with the state-of-the-art and exact algorithms, the proposed algorithm can obtain 78 optimal solutions out of 120 instances within a reasonable computational time.     

Parallel machine scheduling problems using memetic algorithms

In this paper, we investigate how to apply the hybrid genetic algorithms (the memetic algorithms) to solve the parallel machine scheduling problem. There are two essential issues to be dealt with for all kinds of parallel machine scheduling problems: job partition among machines and job sequence within each machine. The basic idea of the proposed method is that (a) use the genetic algorithms to evolve the job partition and then (b) apply a local optimizer to adjust the job permutation to push each chromosome climb to his local optima. Preliminary computational experiments demonstrate that the hybrid genetic algorithm outperforms the genetic algorithms and the conventional heuristics.     

The single-machine total weighted tardiness scheduling problem with position-based learning effects

This study addresses the problem of minimizing the total weighted tardiness on a single-machine with a position-based learning effect. Several dominance properties are established to develop branch and bound algorithm and a lower bound is provided to derive the optimal solution. In addition, three heuristic procedures are developed for near-optimal solutions. Computational results are also presented to evaluate the performance of the proposed algorithms.     

An exact algorithm for the single-machine total weighted tardiness problem with sequence-dependent setup times

This study proposes an exact algorithm for the single-machine total weighted tardiness problem with sequence-dependent setup times. The algorithm is an extension of the authors' previous algorithm for the single-machine scheduling problem without setup times, which is based on the SSDP (Successive Sublimation Dynamic Programming) method. In the first stage of the algorithm, the conjugate subgradient algorithm or the column generation algorithm is applied to a Lagrangian relaxation of the original problem to adjust multipliers. Then, in the second stage, constraints are successively added to the relaxation until the gap between lower and upper bounds becomes zero. The relaxation is solved by dynamic programming and unnecessary dynamic programming states are eliminated to suppress the increase of computation time and memory space. In this study a branching scheme is integrated into the algorithm to manage to solve hard instances. The proposed algorithm is applied to benchmark instances in the literature and almost all of them are optimally solved.     

A genetic local search algorithm for minimizing total weighted tardiness in the job-shop scheduling problem

This paper considers the job-shop problem with release dates and due dates, with the objective of minimizing the total weighted tardiness. A genetic algorithm is combined with an iterated local search that uses a longest path approach on a disjunctive graph model. A design of experiments approach is employed to calibrate the parameters and operators of the algorithm. Previous studies on genetic algorithms for the job-shop problem point out that these algorithms are highly depended on the way the chromosomes are decoded. In this paper, we show that the efficiency of genetic algorithms does no longer depend on the schedule builder when an iterated local search is used. Computational experiments carried out on instances of the literature show the efficiency of the proposed algorithm.