Heuristics for operational fixed job scheduling problems with working and spread time constraints

Operational fixed job scheduling problems select a set of jobs having fixed ready and processing times and schedule the selected jobs on parallel machines so as to maximize the total weight. In this study, we consider working time and spread time constrained versions of the operational fixed job scheduling problems. The working time constraints limit the total processing load on each machine. The spread time constraints limit the time between the start of the first job and the finish of the last job on each machine. For the working time constrained problem, we present a filtered beam search algorithm that evaluates the promising nodes of the branch and bound tree. For the spread time constrained problem we propose a two phase algorithm that defines the promising sets for the first jobs and finds a solution for each promising set. The results of our computational tests reveal that our heuristic algorithms perform very well in terms of both solution quality and time.     

Scheduling jobs with values dependent on their completion times

The paper concerns two scheduling problems with job values and losses of job values (costs) dependent on job completion times. In the first problem, we consider scheduling jobs with stepwise values in parallel processor environment. In the stepwise value, there is given a number of moments at which the job value decreases and between them the job value is constant (thus, the value deteriorates over time). The maximized criterion is the total job value. We prove strong NP-hardness of a single processor case of the problem and construct a pseudo-polynomial time algorithm for a special case with fixed number of unrelated parallel processors and fixed number of common moments of job value changes. Additionally, for uniform and unrelated parallel processors we construct and experimentally test several heuristic algorithms based on the list strategy. The second problem is a single processor one with piecewise linear losses of job values (the loss increases over time). The minimized criterion is the total loss of job value. We prove strong NP-hardness of the problem and existence of a pseudo-polynomial time exact algorithm for its special case. We also construct some heuristic algorithms for this problem and verify experimentally their efficiency.     

Two-machine flow shop scheduling of polyurethane foam production

This paper studies a two-machine flow shop scheduling problem with a supporting precedence relation. The model originates from a real production context of a chemical factory that produces foam-rubber products. We extend the traditional two-machine flow shop by dividing the operations into two categories: supporting tasks and regular jobs. In the model, several different compositions of foam rubber can be mixed at the foam blooming stage, and products are processed at the manufacturing stage. Each job (product) on the second machine cannot start until its supporting tasks (parts) on the first machine are all finished and the second machine is not occupied. The objective is to find a schedule that minimizes the total job completion time. The studied problem is strongly NP-hard. In this paper, we propose a branch-and-bound algorithm incorporating a lower bound and two dominance rules. We also design a simple heuristic and an iterated local search (ILS) algorithm to derive approximate solutions. The performances of the proposed algorithms are examined through computational experiments.     

Minimizing mean absolute deviation of completion time about a common due window subject to maximum tardiness for a single machine

This study deals with the problem of scheduling jobs on a single machine to minimize the mean absolute deviation of the job completion time about a large common due window subject to the maximum tardiness constraint. Using the well-known three-field notation, the problem is identified as MAD/large DueWindow/T_max. The common due window is set to be large enough to allow idle time prior to the beginning of a schedule to investigate the effect of the T_max constraint. Penalties arise if a job is completed outside the due window. A branch and bound algorithm and a heuristic are proposed. Many properties of the solutions and precedence relationships are identified. Our computational results reveal that the branch and bound algorithm is capable of solving problems of up to 50 jobs and the heuristic algorithm yields approximate solutions that are very close to the exact solution.     

A hybrid approach based on the variable neighborhood search and particle swarm optimization for parallel machine scheduling problems—A case study for solar cell industry

This paper studies a solar cell industry scheduling problem which is similar to the traditional hybrid flow shop scheduling (HFS). In a typical HFS with parallel machines problem, the allocation of machine resources for each order should be scheduled in advance and then the optimal multiprocessor task scheduling in each stage could be determined. However, the challenge in solar cell manufacturing is the number of machines can be dynamically adjusted to complete the job within the shortest possible time. Therefore, the paper addresses a multi-stage HFS scheduling problem with characteristics of parallel processing, dedicated machines, sequence-independent setup time, and sequence-dependent setup time. The objective is to schedule the job production sequence, number of sublots, and dynamically allocate sublots to parallel machines such that the makespan time is minimized. The problem is formulated as a mixed integer linear programming (MILP) model. A hybrid approach based on the variable neighborhood search and particle swarm optimization (VNPSO) is developed to obtain the near-optimal solution. Preliminary computational study indicates that the developed VNPSO not only provides good quality solutions within a reasonable amount of time but also outperforms the classic branch and bound method and the current industry heuristic practiced by the case company.     

Two-stage flowshop earliness and tardiness machine scheduling involving a common due window

This paper studies a non-preemptive two-stage flowshop scheduling problem to minimize the earliness and tardiness under the environment of a common due window. The window size and the window location are considered to be given parameters. The just-in-time problem exists naturally and has many practical applications. The problem is shown to be NP-complete in the strong sense. We develop a branch and bound algorithm and a heuristic to solve the problem. We conduct the computational experiments to test the performances of the algorithms. A strong lower bound is derived for the branch and bound algorithm that can efficiently solve 15 jobs problem for about 5 minutes. The heuristic is shown to be efficient and effective, which can solve the problem of 150 jobs for about 20 seconds and provide near-optimal solution. We justify that the heuristic is an excellent solution approach for large problem instances. We also show that four special cases are either polynomial solvable or NP-complete in the ordinary sense.     

Two-machine flowshop scheduling with a truncated learning function to minimize the makespan

Scheduling with learning effects has continued to attract the attention of scheduling researchers. However, the majority of the research on this topic has been focused on the single-machine setting. Moreover, under the commonly adopted learning model in scheduling, the actual processing time of a job drops to zero precipitously as the number of jobs increases, which is at odds with reality. To address these issues, we study a two-machine flowshop scheduling problem with a truncated learning function in which the actual processing time of a job is a function of the job's position in a schedule and the learning truncation parameter. The objective is to minimize the makespan. We propose a branch-and-bound and three crossover-based genetic algorithms (GAs) to find the optimal and approximate solutions, respectively, for the problem. We perform extensive computational experiments to evaluate the performance of all the proposed algorithms under different experimental conditions. The results show that the GAs perform quite well in terms of both efficiency and solution quality.     

Scheduling deteriorating jobs with CON/SLK due date assignment on a single machine

This paper considers the problem of scheduling deteriorating jobs and due date assignment on a single machine. The actual processing time of a job is a linear increasing function of its starting time. The problem is to determine the optimal due dates and the processing sequence simultaneously to minimize costs for earliness, due date assignment and weighted number of tardy jobs. We present polynomial-time algorithms to solve the problem in the case of two popular due date assignment methods: CON and SLK.     

Hybrid particle swarm optimization with mutation for optimizing industrial product lines: An application to a mixed solution space considering both discrete and continuous design variables

This article presents an artificial intelligence-based solution to the problem of product line optimization. More specifically, we apply a new hybrid particle swarm optimization (PSO) approach to design an optimal industrial product line. PSO is a biologically-inspired optimization framework derived from natural intelligence that exploits simple analogues of collective behavior found in nature, such as bird flocking and fish schooling. All existing product line optimization algorithms in the literature have been so far applied to consumer markets and product attributes that range across some discrete values. Our hybrid PSO algorithm searches for an optimal product line in a large design space which consists of both discrete and continuous design variables. The incorporation of a mutation operator to the standard PSO algorithm significantly improves its performance and enables our mechanism to outperform the state of the art Genetic Algorithm in a simulated study with artificial datasets pertaining to industrial cranes. The proposed approach deals with the problem of handling variables that can take any value from a continuous range and utilizes design variables associated with both product attributes and value-added services. The application of the proposed artificial intelligence framework yields important implications for strategic customer relationship and production management in business-to-business markets.     

Maximizing production rate and workload smoothing in assembly lines using particle swarm optimization

Particle swarm optimization (PSO) one of the latest developed population heuristics has rarely been applied in production and operations management (POM) optimization problems. A possible reason for this absence is that, PSO was introduced as global optimizer over continuous spaces, while a large set of POM problems are of combinatorial nature with discrete decision variables. PSO evolves floating-point vectors (called particles) and thus, its application to POM problems whose solutions are usually presented by permutations of integers is not straightforward. This paper presents a novel method based on PSO for the simple assembly line balancing problem (SALBP), a well-known NP-hard POM problem. Two criteria are simultaneously considered for optimization: to maximize the production rate of the line (equivalently to minimize the cycle time), and to maximize the workload smoothing (i.e. to distribute the workload evenly as possible to the workstations of the assembly line). Emphasis is given on seeking a set of diverse Pareto optimal solutions for the bi-criteria SALBP. Extensive experiments carried out on multiple test-beds problems taken from the open literature are reported and discussed. Comparisons between the proposed PSO algorithm and two existing multi-objective population heuristics show a quite promising higher performance for the proposed approach.     

A note on “minimizing a weighted quadratic function of job lateness in the single machine system”

A set of jobs is to be scheduled on a single machine where an idle time is allowed to be inserted before the processing of the first job begins. The objective is to find an optimal sequence that minimizes the weighted sum of a quadratic function of job lateness. Sen et al. [Sen, T., Dileepan, P., Lind, M.R., 1995. Minimizing a weighted quadratic function of job lateness in the single machine system. International Journal of Production Economics 42(3), 237–243] presented a branch-and-bound algorithm for the problem; however, as shown in this note, their algorithm does not work because the branching rule for adjacent jobs is not applicable to non-adjacent jobs.     

A bicriteria approach to maximize the weighted number of just-in-time jobs and to minimize the total resource consumption cost in a two-machine flow-shop scheduling system

We analyze a two-machine flow-shop scheduling problem in which the job processing times are controllable by the allocation of resources to the job operations and the resources can be used in discrete quantities. We provide a bicriteria analysis of the problem where the first criterion is to maximize the weighted number of just-in-time jobs and the second criterion is to minimize the total resource consumption cost. We prove that although the problem is known to be NP-hard even for constant processing times, a pseudo-polynomial time algorithm for its solution exists. In addition, we show how the pseudo-polynomial time algorithm can be converted into a two-dimensional fully polynomial approximation scheme for finding an approximate Pareto solution.     

A note on constructive heuristics for the flowshop problem with blocking

This paper analyzes the minimization of the makespan criterion for the flowshop problem with blocking. In this environment, there are no buffers between successive machines, and therefore intermediate queues of jobs waiting in the system for their next operations are not allowed. As the problem is NP-hard, a constructive heuristic that explores specific characteristics of the problem is developed. The small computational effort of such strategy, which is valuable in practical applications, is one of the reasons that motivated this study. The performance of a combination of the proposed method with existing ones is examined through a comparative study. The new methods outperform the NEH algorithm, currently the best constructive heuristic for this problem, in problems with up to 500 jobs and 20 machines.     

Unrelated parallel machine scheduling with setup consideration and a total weighted completion time objective

This paper addresses the problem of scheduling a set of independent jobs on unrelated parallel machines with job sequence dependent setup times so as to minimize a weighted mean completion time. The study of the problem stemmed from a real service industry problem. This problem is at least NP-hard in the ordinary sense, even when there are only two identical machines with no setups. Seven heuristic algorithms are proposed and tested by simulation. The results and analysis of quite extensive computational experiments are reported and discussed. The findings through the computational results are presented. Whether this problem is strongly NP-hard is left as an open question.     

A hybrid PSO algorithm for a multi-objective assembly line balancing problem with flexible operation times,sequence-dependent setup times and learning effect

This paper addresses multi-objective (MO) optimization of a single-model assembly line balancing problem (ALBP) where the operation times of tasks are unknown variables and the only known information is the lower and upper bounds for operation time of each task. Three objectives are simultaneously considered as follows: (1) minimizing the cycle time, (2) minimizing the total equipment cost, and (3) minimizing the smoothness index. In order to reflect the real industrial settings adequately, it is assumed that the task time is dependent on worker(s) (or machine(s)) learning for the same or similar activity and sequence-dependent setup time exists between tasks. Finding an optimal solution for this complicated problem especially for large-sized problems in reasonable computational time is cumbersome. Therefore, we propose a new solution method based on the combination of particle swarm optimization (PSO) algorithm with variable neighborhood search (VNS) to solve the problem. The performance of the proposed hybrid algorithm is examined over several test problems in terms of solution quality and running time. Comparison with an existing multi-objective evolutionary computation method in the literature shows the superior efficiency of our proposed PSO/VNS algorithm.     

Scheduling a single batch processing machine with arbitrary job sizes and incompatible job families

Motivated by a bottleneck operation in a multi-layer ceramic capacitor production line, we study a scheduling problem of batch processing machine in which a number of jobs are processed simultaneously as a batch. The performance measures considered include makespan, total completion time, and total weighted completion time. We first present a new simple integer programming formulation for the problem, and using this formulation, one can easily find optimal solutions for small problems. However, since the problem is NP-hard and the size of a real problem is very large, we propose a number of heuristic algorithms and design a hybrid genetic algorithm to solve practical big-size problems in a reasonable computational time. To verify performance of the algorithms, we compare them with lower bounds for the problems. From the results of these computational experiments the heuristic algorithms including the genetic algorithm show different performances for the three problems.     

Flexible job shop scheduling with due window—a two-pheromone ant colony approach

Recently, the companies reduce the manufacturing costs and increase capacity efficiency in the competitive environment. Therefore, to balance workstation loading, the hybrid production system is necessary, so that, the flexible job shop system is the most common production system, and there are parallel machines in each workstation. In this study, the due window and the sequential dependent setup time of jobs are considered. To satisfy the customers’ requirement, and reduce the cost of the storage costs at the same time, the sum of the earliness and tardiness costs is the objective. In this study, to improve the traditional ant colony system, we developed the two pheromone ant colony optimization (2PH-ACO) to approach the flexible job shop scheduling problem. Computational results indicate that 2PH-ACO performs better than ACO in terms of sum of earliness and tardiness time.     

Hybrid flow shop scheduling with parallel batching

In this research the problem of parallel batch scheduling in a hybrid flow shop environment with minimizing C_max is studied. In parallel batching it is assumed that machines in some stages are able to process a number of operations simultaneously. Since the problem is NP-hard, three heuristic algorithms are developed to give near optimal solutions. Since this problem has not been studied previously, therefore, a lower bound is developed for evaluating the performance of the proposed heuristics. Several test problems have been solved using these heuristics and results compared. To further enhance the solution quality, a three dimensional genetic algorithm (3DGA) is also developed. Several test problems have been solved using 3DGA and the results indicate its superiority to the other heuristics.     

A note: Multi-machine scheduling with general position-based deterioration to minimize total load

The majority of the papers dealing with scheduling deteriorating jobs ignores general deterioration forms, and considers only special cases. Moreover, most of these papers consider deterioration, most of these papers consider deterioration based on job starting times, and only a few study position-based deterioration. Finally, very few researchers focus on the measure of total load, which becomes important in a setting of deterioration on multi-machines. In this note, we study general, non-decreasing, job-dependent and position-dependent deterioration function. The machine setting is parallel identical machines, and the objective function is total load. We introduce a polynomial time solution for this problem.     

基于PSO的信息熵数据融合非均匀分簇路由算法

针对无线传感器网络分簇算法中能量分布不均衡导致的"热区"和簇头负载过重问题,提出了一种基于PSO算法优化簇头选举的非均匀分簇算法。在候选簇头选举和竞争半径计算过程中综合考虑节点动态能量、节点密度和节点距基站距离,将网络进行非均匀分簇,并引入PSO算法进行最终簇头选举。根据节点能量、节点密度和距基站距离确定簇间单跳多跳结合的路由规则,选取代价函数小的节点作为下一跳节点。基于节点信息熵确定融合阈值,进行簇内数据融合剔除冗余数据。仿真结果表明,改进算法的数据传输量比EEUC算法和UCRA算法分别提高了20%和10%,提升了数据的融合效率,有效延长了网络生命周期,簇头能量消耗得到均衡,减少了网络能量消耗,网络的整体性能显著优于其他对比算法。