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.     

Application of graph search and genetic algorithms for the single machine scheduling problem with sequence-dependent setup times and quadratic penalty function of completion times

In this paper, we consider the single machine scheduling problem with quadratic penalties and sequence-dependent (QPSD) setup times. QPSD is known to be NP-Hard. Only a few exact approaches, and to the best of our knowledge, no approximate approaches, have been reported in the literature so far. This paper discusses exact and approximate approaches for solving the problem, and presents empirical findings. We make use of a graph search algorithm, Memory-Based Depth-First Branch-and-Bound (MDFBB), and present an algorithm, QPSD_MDFBB that can optimally solve QPSD, and advances the state of the art for finding exact solutions. For finding approximate solutions to large problem instances, we make use of the idea of greedy stochastic search, and present a greedy stochastic algorithm, QPSD_GSA that provides moderately good solutions very rapidly even for large problems. The major contribution of the current paper is to apply QPSD_GSA to generate a subset of the starting solutions for a new genetic algorithm, QPSD_GEN, which is shown to provide near-optimal solutions very quickly. Owing to its polynomial running time, QPSD_GEN can be used for much larger instances than QPSD_MDFBB can handle. Experimental results have been provided to demonstrate the performances of these algorithms.     

Multi-agent based approach for single machine scheduling with sequence-dependent setup times and machine maintenance

Scheduling of single machine in manufacturing systems is especially complex when the order arrivals are dynamic. The complexity of the problem increases by considering the sequence-dependent setup times and machine maintenance in dynamic manufacturing environment. Computational experiments in literature showed that even solving the static single machine scheduling problem without considering regular maintenance activities is NP-hard. Multi-agent systems, a branch of artificial intelligence provide a new alternative way for solving dynamic and complex problems. In this paper a collaborative multi-agent based optimization method is proposed for single machine scheduling problem with sequence-dependent setup times and maintenance constraints. The problem is solved under the condition of both regular and irregular maintenance activities. The solutions of multi-agent based approach are compared with some static single machine scheduling problem sets which are available in the literature. The method is also tested under real-time manufacturing environment where computational time plays a critical role during decision making process.     

Minimizing worst-case regret of makespan on a single machine with uncertain processing and setup times

This research addresses a single machine scheduling problem with uncertain processing times and sequence-dependent setup times represented by intervals. Our objective is to obtain a robust schedule with the minimum absolute deviation from the optimal makespan in the worst-case scenario. The problem is reformulated as a robust traveling salesman problem (RTSP), whereby a property is utilized to efficiently identify worst-case scenarios. A local search-based heuristic that incorporates this property is proposed to solve the RTSP, along with a simulated annealing-based implementation. The effectiveness and efficiency of the proposed heuristic are compared to those of an exact solution method in the literature.     

A beam search heuristic for scheduling a single machine with release dates and sequence dependent setup times to minimize the makespan

This paper considers the problem of scheduling a set of jobs subject to arbitrary release dates and sequence-dependent setup times on a single machine with the objective of minimizing the maximum completion of all the jobs, or makespan. This problem is often found in manufacturing processes such as painting and metalworking. A new mixed integer linear program (MILP) is firstly proposed. Because the problem is known to be NP-hard, a beam search heuristic is developed. Computational experiments are carried out using a well-known set of instances from the literature. Our results show that the proposed heuristic is effective in finding high quality solutions at low computational cost.     

Two-machine robotic cell scheduling problem with sequence-dependent setup times

In this paper, we introduce a new and practical two-machine robotic cell scheduling problem with sequence-dependent setup times (2RCSDST) along with different loading/unloading times for each part. Our objective is to simultaneously determine the sequence of robot moves and the sequence of parts that minimize the total cycle time. The proposed problem is proven to be strongly NP-hard. Using the Gilmore and Gomory (GnG) algorithm, a polynomial-time computable lower bound is provided.     

Design of a genetic algorithm for bi-objective unrelated parallel machines scheduling with sequence-dependent setup times and precedence constraints

This paper presents a novel, two-level mixed-integer programming model of scheduling N jobs on M parallel machines that minimizes bi-objectives, namely the number of tardy jobs and the total completion time of all the jobs. The proposed model considers unrelated parallel machines. The jobs have non-identical due dates and ready times, and there are some precedence relations between them. Furthermore, sequence-dependent setup times, which are included in the proposed model, may be different for each machine depending on their characteristics. Obtaining an optimal solution for this type of complex, large-sized problem in reasonable computational time using traditional approaches or optimization tools is extremely difficult. This paper proposes an efficient genetic algorithm (GA) to solve the bi-objective parallel machine scheduling problem. The performance of the presented model and the proposed GA is verified by a number of numerical experiments. The related results show the effectiveness of the proposed model and GA for small and large-sized problems.     

Criteria selection and analysis for single machine dynamic on-line scheduling with multiple objectives and sequence-dependent setups

Fertile opportunities exist for research involving dynamic and stochastic scheduling with multiple conflicting objectives and sequence-dependent setups as little has been reported in the literature to date. This research focuses on understanding and identifying the criteria that could be combined into a single rule using the linear weighted aggregation approach to consider the contradicting needs of cycle time and delivery accuracy. Eight dispatching criteria are compared and evaluated using discrete event simulation. In most studies, the basic concept is to combine different dispatching criterion that performs the best in each objective into a single rule but this may be insufficient. Simulation results show that it is necessary to take into account not only the criterion’s strength in optimizing a performance objective but also the degree of trade-off it has on the other conflicting performance objectives of interest. A correlation analysis of the objectives used is also presented.     

A two-stage heuristic for single machine capacitated lot-sizing and scheduling with sequence-dependent setup costs

This paper considers a single machine capacitated lot-sizing and scheduling problem. The problem is to determine the lot sizes and the sequence of lots while satisfying the demand requirements and the machine capacity in each period of a planning horizon. In particular, we consider sequence-dependent setup costs that depend on the type of the lot just completed and on the lot to be processed. The setup state preservation, i.e., the setup state at the end of a period is carried over to the next period, is also considered. The objective is to minimize the sum of setup and inventory holding costs over the planning horizon. Due to the complexity of the problem, we suggest a two-stage heuristic in which an initial solution is obtained and then it is improved using a backward and forward improvement method that incorporates various priority rules to select the items to be moved. Computational tests were done on randomly generated test instances and the results show that the two-stage heuristic outperforms the best existing algorithm significantly. Also, the heuristics with better priority rule combinations were used to solve case instances and much improvement is reported over the conventional method as well as the best existing algorithm.     

Hybrid flexible flowshops with sequence-dependent setup times and machine availability constraints

One of the common assumptions in the field of scheduling is that machines are always available in the planning horizon. This may not be true in realistic problems since machines might be busy processing some jobs left from previous production horizon, breakdowns or preventive maintenance activities. Another common assumption is the consideration of setup times as a part of processing times, while in some industries, such as printed circuit board and automobile manufacturing, not only setups are an important factor but also setup magnitude of a job depends on its immediately preceding job on the same machine, known as sequence-dependent setup times. In this paper, we consider hybrid flexible flowshops with sequence-dependent setup times and machine availability constraints caused by preventive maintenance. The optimization criterion is the minimization of makespan. Since this problem is NP-hard in the strong sense, we propose three heuristics, based on SPT, LPT and Johnson rule and two metaheuristics based on genetic algorithm and simulated annealing. Computational experiments are performed to evaluate the efficiencies of the algorithms.     

A hybrid Lagrangian-simulated annealing-based heuristic for the parallel-machine capacitated lot-sizing and scheduling problem with sequence-dependent setup times

This paper examines the parallel-machine capacitated lot-sizing and scheduling problem with sequence-dependent setup times, time windows, machine eligibility and preference constraints. Such problems are quite common in the semiconductor manufacturing industry. In particular, this paper pays special attention to the chipset production in the semiconductor Assembly and Test Manufacturing (ATM) factory and constructs a Mixed Integer Programming (MIP) model for the problem. The primal problem is decomposed into a lot-sizing subproblem and a set of single-machine scheduling subproblems by Lagrangian decomposition. A Lagrangian-based heuristic algorithm, which incorporates the simulated annealing algorithm aimed at searching for a better solution during the feasibility construction stage, is proposed. Computational experiments show that the proposed hybrid algorithm outperforms other heuristic algorithms and meets the practical requirement for the tested ATM factory.     

A branch-and-cut algorithm for a production scheduling problem with sequence-dependent and time-dependent setup times

This paper introduces and compares three different formulations of a production scheduling problem with sequence-dependent and time-dependent setup times on a single machine. The setup is divided into two parts: one that can be performed at any time and another one that is restricted to be performed outside of a given time interval. As a result, the setup time between two jobs is a function of the completion time of the first job. The problem can be formulated as a time-dependent traveling salesman problem, where the travel time between two nodes is a function of the departure time from the first node. We show that the resulting formulation can be strengthened to provide better linear programming relaxation lower bounds. We also introduce several families of valid inequalities which are used within a branch-and-cut algorithm. Computational experiments show that this algorithm can solve some instances with up to 50 jobs within reasonable computing times.     

A fuzzy-mixed-integer goal programming model for a parallel-machine scheduling problem with sequence-dependent setup times and release dates

This paper presents a new mixed-integer goal programming (MIGP) model for a parallel-machine scheduling problem with sequence-dependent setup times and release dates. Two objectives are considered in the model to minimize the total weighted flow time and the total weighted tardiness simultaneously. Due to the complexity of the above model and uncertainty involved in real-world scheduling problems, it is sometimes unrealistic or even impossible to acquire exact input data. Hence, we consider the parallel-machine scheduling problem with sequence-dependent set-up times under the hypothesis of fuzzy processing time's knowledge and two fuzzy objectives as the MIGP model. In addition, a quite effective and applicable methodology for solving the above fuzzy model are presented. At the end, the effectiveness of the proposed model and the denoted methodology is demonstrated through some test problems.     

带准备时间的单机调度问题的混合进化算法研究

提出了一种混合进化算法（HEA）用于求解具有序列相关依赖且带准备时间的单机调度问题, 其优化目标为最小化总延迟。该混合进化算法由局部搜索和进化算法框架混合而成。HEA具有一些新的特点, 例如在局部搜索中采用了一种新提出的基于块移动的邻域结构, 这种邻域结构合理地限制了搜索空间, 提高了算法的搜索效率; 在HEA中采用了一种新的组合算子——块顺序交叉算符（BOX）来产生新的子代工作序列。用本算法对当前国际文献中公开的两组共64个算例进行了测试, HEA改进了9个算例在当前文献中的最优解, 表明了所提出的HEA算法的优越性。与之前的国际文献中最好的四个启发式算法进行了详细比较, 表明了HEA算法的优势。     

Artificial chromosomes with genetic algorithm 2 (ACGA2) for single machine scheduling problems with sequence-dependent setup times

Artificial chromosomes with genetic algorithm (ACGA) is one of the latest versions of the estimation of distribution algorithms (EDAs). This algorithm has already been applied successfully to solve different kinds of scheduling problems. However, due to the fact that its probabilistic model does not consider variable interactions, ACGA may not perform well in some scheduling problems, particularly if sequence-dependent setup times are considered. This is due to the fact that the previous job will influence the processing time of the next job. Simply capturing ordinal information from the parental distribution is not sufficient for a probabilistic model. As a result, this paper proposes a bi-variate probabilistic model to add into the ACGA. This new algorithm is called the ACGA2 and is used to solve single machine scheduling problems with sequence-dependent setup times in a common due-date environment. A theoretical analysis is given in this paper. Some heuristics and local search algorithm variable neighborhood search (VNS) are also employed in the ACGA2. The results indicate that the average error ratio of this ACGA2 is half the error ratio of the ACGA. In addition, when ACGA2 is applied in combination with other heuristic methods and VNS, the hybrid algorithm achieves optimal solution quality in comparison with other algorithms in the literature. Thus, the proposed algorithms are effective for solving the scheduling problems.     

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.     

并行机成组调度问题的启发式算法

研究了优化目标为总拖后/提前时间最小化的并行机成组调度问题,提出了一种三阶段启发式近似求解算法。首先把并行机问题看成单机问题,以最小化总拖后时间为优化目标排列工件的加工次序;然后将工件按第一阶段所求得的次序指派到最先空闲的并行的机器上;最后采用改进的GTW算法对各机器上的工件调度插入适当的空闲时间。计算表明该算法能够在很短的时间内给出大规模调度问题的近似最优解。     

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.     

An evolutionary clustering search for the no-wait flow shop problem with sequence dependent setup times

This paper examines the m machine no-wait flow shop problem with setup times of a job separated from its processing time. The performance measure considered is the makespan. The hybrid metaheuristic Evolutionary Cluster Search (ECS_NSL) proposed in Nagano et al. (2012) is extended to solve the scheduling problem. The ECS_NSL performance is evaluated and the results are compared with the best method reported in the literature. Experimental tests show superiority of the ECS_NSL regarding the solution quality.     

An iterated local search for the multi-objective permutation flowshop scheduling problem with sequence-dependent setup times

Due to its simplicity yet powerful search ability, iterated local search (ILS) has been widely used to tackle a variety of single-objective combinatorial optimization problems. However, applying ILS to solve multi-objective combinatorial optimization problems is scanty. In this paper we design a multi-objective ILS (MOILS) to solve the multi-objective permutation flowshop scheduling problem with sequence-dependent setup times to minimize the makespan and total weighted tardiness of all jobs. In the MOILS, we design a Pareto-based variable depth search in the multi-objective local search phase. The search depth is dynamically adjusted during the search process of the MOILS to strike a balance between exploration and exploitation. We incorporate an external archive into the MOILS to store the non-dominated solutions and provide initial search points for the MOILS to escape from local optima traps. We compare the MOILS with several multi-objective evolutionary algorithms (MOEAs) shown to be effective for treating the multi-objective permutation flowshop scheduling problem in the literature. The computational results show that the proposed MOILS outperforms the MOEAs.