A particle swarm optimization for a fuzzy multi-objective unrelated parallel machines scheduling problem

This paper proposes a novel multi-objective model for an unrelated parallel machine scheduling problem considering inherent uncertainty in processing times and due dates. The problem is characterized by non-zero ready times, sequence and machine-dependent setup times, and secondary resource constraints for jobs. Each job can be processed only if its required machine and secondary resource (if any) are available at the same time. Finding optimal solution for this complex problem in a reasonable time using exact optimization tools is prohibitive. This paper presents an effective multi-objective particle swarm optimization (MOPSO) algorithm to find a good approximation of Pareto frontier where total weighted flow time, total weighted tardiness, and total machine load variation are to be minimized simultaneously. The proposed MOPSO exploits new selection regimes for preserving global as well as personal best solutions. Moreover, a generalized dominance concept in a fuzzy environment is employed to find locally Pareto-optimal frontier. Performance of the proposed MOPSO is compared against a conventional multi-objective particle swarm optimization (CMOPSO) algorithm over a number of randomly generated test problems. Statistical analyses based on the effect of each algorithm on each objective space show that the proposed MOPSO outperforms the CMOPSO in terms of quality, diversity and spacing metrics.     

Minimizing total tardiness and earliness on unrelated parallel machines with controllable processing times

Job scheduling has always been a challenging task in modern manufacturing and the most real life scheduling problems which involves multi-criteria and multi-machine environments. In this research our direction is largely motivated by the adoption of the Just-In-Time (JIT) philosophy in parallel machines system, where processing times of jobs are controllable. The goal of this paper is to minimize total weighted tardiness and earliness besides jobs compressing and expanding costs, depending on the amount of compression/expansion as well as maximum completion time called makespan simultaneously. Jobs due dates are distinct and no inserted idle time is allowed after starting machine processing. Also each machine is capable of processing only some predetermined jobs and operations with probably different speeds. A Mixed Integer Programming (MIP) model is proposed to formulate such a problem and is solved optimally in small size instances. A Parallel Net Benefit Compression-Net Benefit Expansion (PNBC–NBE) heuristic is then presented to acquire the optimal jobs set amount of compression and expansion processing times in a given sequence. To solve medium-to-large size cases, a proposed heuristic, two meta-heuristics and a hybrid technique are also employed. Experimental results demonstrate that our hybrid procedure is a proficient method and could efficiently solve such complicated problems.     

Bottleneck-based heuristics to minimize total tardiness for the flexible flow line with unrelated parallel machines

This paper considers the flexible flow line problem with unrelated parallel machines at each stage and with a bottleneck stage on the line. The objective of the problem is to minimize the total tardiness. Two bottleneck-based heuristics with three machine selection rules are proposed to solve the problem. The heuristics first develop an indicator to identify a bottleneck stage in the flow line, and then separate the flow line into the upstream stages, the bottleneck stage, and the downstream stages. The upstream stages are the stages ahead of the bottleneck stage and the downstream stages are the stages behind the bottleneck stage. A new approach is developed to find the arrival times of the jobs at the bottleneck stage. Using the new approach, the bottleneck-based heuristics develop two decision rules to iteratively schedule the jobs at the bottleneck stage, the upstream stages, and the downstream stages. In order to evaluate the performance of the bottleneck-based heuristics, seven commonly used dispatching rules and a basic tabu search algorithm are investigated for comparison purposes. Seven experimental factors are used to design 128 production scenarios, and ten test problems are generated for each scenario. Computational results show that the bottleneck-based heuristics significantly outperform all the dispatching rules for the test problems. Although the effective performance of the bottleneck-based heuristics is inferior to the basic tabu search algorithm, the bottleneck-based heuristics are much more efficient than the tabu search algorithm. Also, a test of the effect of the experimental factors on the dispatching rules, the bottleneck-based heuristics, and the basic tabu search algorithm is performed, and some interesting insights are discovered.     

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.     

A branch-and-bound algorithm to minimize total weighted completion time on identical parallel machines with job release dates

In this paper, we consider an identical parallel machine scheduling problem with release dates. The objective is to minimize the total weighted completion time. This problem is known to be strongly NP-hard. We propose some dominance properties and two lower bounds. We also present an efficient heuristic. A branch-and-bound algorithm, in which the heuristic, the lower bounds and the dominance properties are incorporated, is proposed and tested on a large set of randomly generated instances.     

Rescheduling of parallel machines with stochastic processing and setup times

This paper tackles rescheduling for the unrelated parallel machine problem with sequence dependent setup times and different rates of breakdowns or urgent jobs arrivals. The jobs’ processing and setup times are stochastic for better depiction of the real world. A new repair rule which will be referred to as Minimum Weighted C_max Difference (MWCD) is developed and compared to existing algorithms using simulation.     

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.     

求解异构并行机调度问题的混合多目标教-学优化算法

以异构并行机调度问题为研究对象,考虑了一类以优化总加权完工时间和加权延误总和的调度问题。首先,基于问题描述构建了该问题的混合整数规划模型。其次,提出了混合多目标教-学优化算法。在算法设计中,结合问题的特点设计序列编码方法,并采用分解技术来实现多目标调度问题的求解。此外,该算法通过融合多种交叉算子来定义个体进化过程,并通过与变邻域搜索算法的混合来提升其优化效果。最后,给出了仿真实验与分析,测试结果验证了多目标教-学优化算法求解该调度问题的优越性。     

A quadratic time algorithm to maximize the number of just-in-time jobs on identical parallel machines

In this paper, we study a scheduling problem on identical parallel machines, in which a processing time and a due date are given for each job, and the objective is to maximize the number of just-in-time jobs. A job is called just-in-time if it is completed exactly on its due date. We discuss the known results, show that a recently published greedy algorithm for this problem is incorrect, and present a new quadratic time algorithm which solves the problem.     

Hybrid genetic algorithms with dispatching rules for unrelated parallel machine scheduling with setup time and production availability

This article considers the unrelated parallel machine scheduling problem with sequence- and machine-dependent setup times and machine-dependent processing times. Furthermore, the machine has a production availability constraint to each job. The objective of this problem is to determine the allocation policy of jobs and the scheduling policy of machines to minimize the total completion time. To solve the problem, a mathematical model for the optimal solution is derived, and hybrid genetic algorithms with three dispatching rules are proposed for large-sized problems. To assess the performance of the algorithms, computational experiments are conducted and evaluated using several randomly generated examples.     

A branch and bound algorithm for minimizing makespan on a single machine with unequal release times under learning effect and deteriorating jobs

We present a single-machine problem with the unequal release times under learning effect and deteriorating jobs when the objective is minimizing the makespan. In this study, we introduced a scheduling model with unequal release times in which both job deterioration and learning exist simultaneously. By the effects of learning and deterioration, we mean that the processing time of a job is defined by increasing function of its execution start time and position in the sequence. A branch-and-bound algorithm incorporating with several dominance properties and lower bounds is developed to derive the optimal solution. A heuristic algorithm is proposed to obtain a near-optimal solution. The computational experiments show that the branch-and-bound algorithm can solve instances up to 30 jobs, and the average error percentage of the proposed heuristic is less than 0.16%.     

Some unrelated parallel machine scheduling problems with past-sequence-dependent setup time and learning effects

In this paper, we study an unrelated parallel machine scheduling problem with setup time and learning effects simultaneously. The setup time is proportional to the length of the already processed jobs. That is, the setup time of each job is past-sequence-dependent. The objectives are to minimize the total absolute deviation of job completion times and the total load on all machines, respectively. We show that the proposed problem is polynomially solvable. We also discuss two special cases of the problem and show that they can be optimally solved by lower order algorithms.     

Soft computing for scheduling with batch setup times and earliness-tardiness penalties on parallel machines

A model for scheduling grouped jobs on identical parallel machines is addressed in this paper. The model assumes that a set-up time is incurred when a machine changes from processing one type of component to a different type of component, and the objective is to minimize the total earliness-tardiness penalties. In this paper, the algorithm of soft computing, which is a fuzzy logic embedded Genetic Algorithm is developed to solve the problem. The efficiency of this approach is tested on several groups of random problems and shows that the soft computing algorithm has potential for practical applications in larger scale production systems.     

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 genetic algorithm application using fuzzy processing times in non-identical parallel machine scheduling problem

There are many scheduling problems which are NP-hard in the literature. Several heuristics and dispatching rules are proposed to solve such hard combinatorial optimization problems. Genetic algorithms (GA) have shown great advantages in solving the combinatorial optimization problems in view of its characteristic that has high efficiency and that is fit for practical application [1]. Two different scale numerical examples demonstrate the genetic algorithm proposed is efficient and fit for larger scale identical parallel machine scheduling problem for minimizing the makespan. But, even though it is a common problem in the industry, only a small number of studies deal with non-identical parallel machines. In this article, a kind of genetic algorithm based on machine code for minimizing the processing times in non-identical machine scheduling problem is presented. Also triangular fuzzy processing times are used in order to adapt the GA to non-identical parallel machine scheduling problem in the paper. Fuzzy systems are excellent tools for representing heuristic, commonsense rules. That is why we try to use fuzzy systems in this study.     

基于NSGA2算法的并行机多目标调度问题研究

针对并行机多目标调度问题,以完工时间和总延迟时间最小为目标函数建立了数学模型,从而将具有解决复杂组合优化问题的非劣排序遗传算法NSGA2应用于求解多目标并行机调度问题。文中详细描述了用NSGA2算法求解并行机调度问题的步骤,并通过Matlab仿真,表明YhqNSGA2算法求解多目标并行机调度问题的可行性和有效性。     

A comparison of hybrid genetic algorithm and hybrid particle swarm optimization to minimize makespan for assembly job shop

Very often, studies of job shop scheduling problem (JSSP) ignore assembly relationship and lot splitting. If an assembly stage is appended to JSSP for the final product, the problem then becomes assembly job shop scheduling problem (AJSSP). To allow lot splitting, lot streaming (LS) technique is examined in which jobs may be split into a number of smaller sub-jobs for parallel processing on different stages such that the system performance may be improved. In this study, the system objective is defined as the makespan minimization. In order to investigate the impact of LS on the system objective under different real-life operating conditions, part sharing ratio (PSR) and system congestion index (SCI) are considered. PSR is used to differentiate product-specific components from general-purpose, common components, and SCI for creating different starting conditions of the shop floor. Both PSR and CSI are useful as part sharing (also known as component commonality) is a common practice for manufacturing with assembly operations and system loading is a significant factor in influencing the shop floor performance. Since the complexity of AJSSP is NP-hard, a hybrid genetic algorithm (HGA) and a hybrid particle swarm optimization (HPSO) are proposed and developed to solve AJSSP in consideration of LS technique. Computational results show that for all test problems under various system conditions, HGA can significantly outperform HPSO. Also, equal-sized lot splitting is found to be the most beneficial LS strategy especially for medium-to-large problem size.     

A hybrid artificial immune algorithm for a realistic variant of job shops to minimize the total completion time

This paper investigates an extended problem of job shop scheduling to minimize the total completion time. With aim of actualization of the scheduling problems, many researchers have recently considered realistic assumptions in their problems. Two of the most applied assumptions are to consider sequence-dependent setup times and machine availability constraints (MACs). In this paper, we deal with a specific case of MACs caused by preventive maintenance (PM) operations. Contrary to the previous papers considering fixed or/and conservative policies, we consider flexible PM operations, in which PM operations may be postponed or expedited as required. A simple technique is employed to schedule production jobs along with the flexible MACs caused by PM. To solve the given problem, we present a novel meta-heuristic method based on the artificial immune algorithm (AIA) incorporating some advanced features. For further enhancement, the proposed AIA is hybridized with a simple and fast simulated annealing (SA). To evaluate the proposed algorithms, we compare our proposed AIA with three well-known algorithms taken from the literature. Finally, we find that the proposed AIA outperforms other algorithms.