Two-stage flow shop scheduling with dedicated machines

This study is concerned with the manufacturing model that has a common machine at stage one and two parallel dedicated machines at stage two. All jobs need to be processed on the stage-one common machine. After the stage-one processing, the jobs of type 1 (type 2) will route to the first (second) dedicated machine at stage two. We first elaborate several published works on makespan minimisation which are not known to other streams of recent works. While the minimisation of maximum lateness is strongly NP-hard, we develop a linear-time algorithm to solve the case where two sequences of the two job types are given a priori.     

A hybrid VNS-HS algorithm for a supply chain scheduling problem with deteriorating jobs

This paper investigates a coordinated scheduling problem in a two stage supply chain where parallel-batching machine, deteriorating jobs and transportation coordination are considered simultaneously. During the production stage, jobs are processed by suppliers and there exists one parallel-batching machine in each supplier. The actual processing time of a job depends on its starting time and normal processing time. The normal processing time of a batch is equal to the largest normal processing time among all jobs in its batch. During the transportation stage, the jobs are then delivered to the manufacturer. Since suppliers are distributed in different locations, the transportation time between each supplier and the manufacturer is different. Based on some structural properties of the studied problem, an optimal algorithm for minimising makespan on a single supplier is presented. This supply chain scheduling problem is proved to be NP-hard, and a hybrid VNS-HS algorithm combining variable neighbourhood search (VNS) with harmony search (HS) is proposed to find a good solution in reasonable time. Finally, some computational experiments are conducted and the results demonstrate the effectiveness and efficiency of the proposed VNS-HS.     

A method combining rules with genetic algorithm for minimizing makespan on a batch processing machine with preventive maintenance

This paper considers the problem of minimising makespan on a single batch processing machine with flexible periodic preventive maintenance. This problem combines two sub-problems, scheduling on a batch processing machine with jobs’ release dates considered and arranging the preventive maintenance activities on a batch processing machine. The preventive maintenance activities are flexible but the maximum continuous working time of the machine, which is allowed, is determined. A mathematical model for integrating flexible periodic preventive maintenance into batch processing machine problem is proposed, in which the grouping of jobs with incompatible job families, the starting time of batches and the preventive maintenance activities are optimised simultaneously. A method combining rules with the genetic algorithm is proposed to solve this model, in which a batching rule is proposed to group jobs with incompatible job families into batches and a modified genetic algorithm is proposed to schedule batches and arrange preventive maintenance activities. The computational results indicate the method is effective under practical problem sizes. In addition, the influences of jobs’ parameters on the performance of the method are analyzed, such as the number of jobs, the number of job families, jobs’ processing time and jobs’ release time.     

A new complexity proof for the two-stage hybrid flow shop scheduling problem with dedicated machines

This paper considers a two-stage hybrid flow shop scheduling problem with dedicated machines at stage 2. The objective is to minimise the makespan. There is one machine at stage 1 and two machines at stage 2. Each job must be processed on the single machine at stage 1 and, depending upon the job type, the job is processed on either of the two machines at stage 2. We first introduce this special type of the two-stage hybrid flow shop scheduling problem and present some preliminary results. We then present a counter example to the known complexity proof of Riane et al. [Riane, F., Artiba, A. and Elmaghraby, S.E., 2002. Sequencing a hybrid two-stage flowshop with dedicated machines. International Journal of Production Research, 40, 4353–4380.] Finally, we re-establish the complexity of the problem.     

Fast and meta-heuristics for common due-date assignment and scheduling on parallel machines

This study considers common due-date assignment and scheduling on parallel machines. The problem has three decision variables: assigning the common-due-date, allocating jobs to parallel machines, and sequencing the jobs assigned to each machine. The objective is to minimise the sum of due-date assignment, earliness and tardiness penalties. A mathematical programming model is presented, and then two types of heuristics are suggested after characterising the optimal solution properties. The two types of heuristics are: (a) a fast two-stage heuristic with obtaining an initial solution and improvement; and (b) two meta-heuristics, tabu search and simulated annealing, with new neighbourhood generation methods. Computational experiments were conducted on a number of test instances, and the results show that each of the heuristic types outperforms the existing one. In particular, the meta-heuristics suggested in this study are significantly better than the existing genetic algorithm.     

A branch-and-bound algorithm for makespan minimization in differentiation flow shops

This article considers a differentiation flow-shop model, where the jobs are divided into various categories, each of which consists of two stages of operations. All products should be processed first on the single common machine at stage 1. At the second stage, each individual product proceeds to a dedicated machine according to its type. The problem of makespan minimization under the setting with two product types is known to be strongly NP hard. This article considers an arbitrary number of job types by developing a lower bound and two dominance rules, based upon which branch-and-bound algorithms are designed. Computational experiments are carried out to examine the performance of the proposed properties. The statistics show that the proposed properties can substantially reduce the computing efforts required for finding optimal solutions.     

Makespan minimisation for a parallel machine scheduling problem with preemption and job incompatibility

In this paper, an extension of the graph colouring problem is introduced to model a parallel machine scheduling problem with job incompatibility. To get closer to real-world applications, where the number of machines is limited and jobs have different processing times, each vertex of the graph requires multiple colours and the number of vertices with the same colour is bounded. In addition, several objectives related to scheduling are considered: makespan, number of pre-emptions and summation over the jobs’ throughput times. Different solution methods are proposed, namely, two greedy heuristics, two tabu search methods and an adaptive memory algorithm. The latter uses multiple recombination operators, each one being designed for optimising a subset of objectives. The most appropriate operator is selected dynamically at each iteration, depending on its past performance. Experiments show that the proposed algorithm is effective and robust, while providing high-quality solutions on benchmark instances for the graph multi-colouring problem, a simplification of the considered problem.     

带成组加工的二阶段柔性流水作业问题

本文仔细剖析混杂二阶段流水作业问题,其中第一阶段由m台同型机组成,第二阶段由一台批处理机M组成,并以最大完工时间Cmax为极小化目标函数.我们证明了该类问题除一种情况有多项式时间可解外,其余情况为(强)NP-hatd的.文中对所有(强)NP-hard情况均给出了近似算法并作了性能比分析.     

Hybrid flowshop scheduling with family setup time and inconsistent family formation

This research considers a hybrid flowshop scheduling problem where jobs are organised in families according to their machine settings and tools. The family setup time arises when a machine shifts from processing one job family to another. The problem is compounded by the challenges that the formation of job families is different in different stages and only a limited number of jobs can be processed within one setup. This type of problem is common in the production process of standard metal components. This paper aims to propose two approaches to solve this problem. One is a metaheuristic in the form of a genetic algorithm and the other is a heuristic. The proposed approaches are compared and contrasted against the two relevant metaheuristic and heuristic adapted from solving a generalised sequence-dependent setup flowshop problem. Comparative results indicate that the proposed genetic algorithm has better performance on minimising makespan and the heuristic is more effective on reducing family setup time.     

Probabilistic asymptotic analysis of stochastic online scheduling problems

In the stochastic online scheduling environment, jobs with unknown release times and weights arrive over time. Upon arrival, the information on the weight of the job is revealed but the processing requirement remains unknown until the job is finished. In this paper we consider the objective of minimizing the total weighted completion time. With the assumptions that job weights are bounded, machine capacity is adequate, and processing requirements are bounded and identical and independently distributed across the machines and jobs, we show that any nondelay algorithm is asymptotically optimal for the stochastic online single machine problem, flow shop problem, and uniform parallel machine problem. Our simulation studies of these stochastic online scheduling problems show that two generic nondelay algorithms perform very well as long as the number of jobs is larger than 100.     

Two-stage hybrid flow shop scheduling with preventive maintenance using multi-objective tabu search method

This paper investigates an integrated bi-objective optimisation problem with non-resumable jobs for production scheduling and preventive maintenance in a two-stage hybrid flow shop with one machine on the first stage and m identical parallel machines on the second stage. Sequence-dependent set-up times and preventive maintenance (PM) on the first stage machine are considered. The scheduling objectives are to minimise the unavailability of the first stage machine and to minimise the makespan simultaneously. To solve this integrated problem, three decisions have to be made: determine the processing sequence of jobs on the first stage machine, determine whether or not to perform PM activity just after each job, and specify the processing machine of each job on the second stage. Due to the complexity of the problem, a multi-objective tabu search (MOTS) method is adapted with the implementation details. The method generates non-dominated solutions with several parallel tabu lists and Pareto dominance concept. The performance of the method is compared with that of a well-known multi-objective genetic algorithm, in terms of standard multi-objective metrics. Computational results show that the proposed MOTS yields a better approximation.     

Scheduling distributed flowshops with flexible assembly and set-up time to minimise makespan

The scheduling problems under distributed production or flexible assembly settings have achieved increasing attention in recent years. This paper considers scheduling the integration of these two environments and proposes an original distributed flowshop scheduling problem with flexible assembly and set-up time. Distributed production stage is deployed several homogeneous flowshop factories that process the jobs to be assembled into final products in the flexible assembly stage. The objective is to find a schedule, including a production subschedule for jobs and an assembly subschedule for products, to minimise the makespan. Such a scheduling problem involves four successive decisions: assigning jobs to production factories, sequencing jobs at every factory, designating an assembly machine for each product and sequencing products on each assembly machine. The computational model is first established, and then a constructive heuristic (TPHS) and two hybrid metaheuristics (HVNS and HPSO) are proposed. Numerical experiments have been carried out and results validate the algorithmic feasibility and effectiveness. TPHS can obtain reasonable solutions in a shorter time, while metaheuristics can report better solutions using more yet acceptable time. HPSO is statistically comparable yet less robust compared with HVNS for small-scale instances. For the large-scale case, HPSO outperforms HVNS on both effectiveness and robustness.     

Hierarchical production control for a flow shop with dynamic setup changes and random machine breakdowns

In this paper, we study a manufacturing system consisting of two machines separated by two intermediate buffers, and capable of producing two different products. Each product requires a constant processing time on each of the machines. Each machine requires a constant non-negligible setup change time from one product to the other. The demand rate for each product is considered to be piecewise constant. Each machine undergoes failure and repair. The time-to-failure and time-to-repair are exponentially distributed random variables. The setup change and processing operations are resumable. We model our system as a continuous time, continuous flow process. An optimal control problem is formulated for the system to minimize the total expected discounted cost over an infinite horizon. To determine the optimal control policy structure, a discrete version of the problem is solved numerically using a dynamic programming formulation with a piecewise linear penalty function. A real-time control algorithm is then developed with the objective of maintaining low work-in-process inventory and keeping the production close to the demand. The algorithm uses a hierarchical control structure to generate the loading times for each product on each machine in real time and to respond to random disruptions in the system. The system is simulated using this algorithm to study its performance. The performance of the algorithm is also compared to alternative policies.     

A branch and bound algorithm for scheduling unit size jobs on parallel batching machines to minimize makespan

In this paper, we present a branch and bound algorithm for the parallel batch scheduling of jobs having different processing times, release dates and unit sizes. There are identical machines with a fixed capacity and the number of jobs in a batch cannot exceed the machine capacity. All batched jobs are processed together and the processing time of a batch is given by the greatest processing time of jobs in that batch. We compare our method to a mixed integer program as well as a method from the literature that is capable of optimally solving instances with a single machine. Computational experiments show that our method is much more efficient than the other two methods in terms of solution time for finding the optimal solution.     

Due date assignment in single machine with stochastic processing times

This paper considers two different due date assignment and sequencing problems in single machine where the processing times of jobs are random variables. The first problem is to minimise the maximum due date so that all jobs are stochastically on time. It is shown that sequencing the jobs in decreasing service level (DSL) order optimally solves the problem. The results are then extended for two special cases of flow shop problem. The other problem is to minimise a total cost function which is a linear combination of three penalties: penalty on job earliness, penalty on job tardiness, and penalty associated with long due date assignment. The assignment of a common due date and distinct due dates are investigated for this problem. It is shown that the optimal sequence for the case of common due date is V-shaped.     

Scheduling a batch processing machine with non-identical job sizes: a clustering perspective

Batch processing machines that process a group of jobs simultaneously are often encountered in semiconductor manufacturing and metal heat treatment. This paper considered the problem of scheduling a batch processing machine from a clustering perspective. We first demonstrated that minimising makespan on a single batching machine with non-identical job sizes can be regarded as a special clustering problem, providing a novel insight into scheduling with batching. The definition of WRB (waste ratio of batch) was then presented, and the objective function of minimising makespan was transformed into minimising weighted WRB so as to define the distance measure between batches in a more understandable way. The equivalence of the two objective functions was also proved. In addition, a clustering algorithm CACB (constrained agglomerative clustering of batches) was proposed based on the definition of WRB. To test the effectiveness of the proposed algorithm, the results obtained from CACB were compared with those from the previous methods, including BFLPT (best-fit longest processing time) heuristic and GA (genetic algorithm). CACB outperforms BFLPT and GA especially for large-scale problems.     

A two‐phase approach for single machine scheduling problems: Minimizing the total absolute deviation

Abstract

This paper presents a heuristic for solving a single machine scheduling problem with the objective of minimizing the total absolute deviation. The job to be scheduled on the machine has a processing time, p_i , and a preferred due date, d_i . The total absolute deviation is defined as the sum of the earliness or tardiness of each job on a schedule 5. This problem is proved to be NP‐complete by Garey et al. [8]. As a result, we developed a two‐phase procedure to provide a near‐optimal solution to this problem. The two‐phase procedure includes the following steps: First, a greedy heuristic is applied to the set of jobs, N, to generate a “good” initial sequence. According to this initial sequence, we run Garey's local optimization algorithm to provide an initial schedule. Then, a pairwise switching algorithm is adopted to further reduce the total deviation of the schedule. The effectiveness of the two‐phase procedure is empirically evaluated and has been found to indicate that the solutions obtained from this heuristic procedure are often better than other heuristic approaches.     

A branch-and-bound algorithm for a two-stage hybrid flowshop scheduling problem minimizing total tardiness

A two-stage hybrid flowshop-scheduling problem is considered with the objective of minimizing total tardiness of jobs. In the hybrid flowshop, there is one machine at the first stage and multiple identical parallel machines at the second stage. Dominance properties and lower bounds are developed for the problem and a branch-and-bound algorithm is suggested using them. Results of computational experiments show that the suggested algorithm can find optimal solutions for problems with up to 15 jobs in a reasonable amount of central processing unit time.     

Minimising total tardiness for the identical parallel machine scheduling problem with splitting jobs and sequence-dependent setup times

This paper focuses on an identical parallel machine scheduling problem with minimising total tardiness of jobs. There are two major issues involved in this scheduling problem; (1) jobs which can be split into multiple sub-jobs for being processed on parallel machines independently and (2) sequence-dependent setup times between the jobs with different part types. We present a novel mathematical model with meta-heuristic approaches to solve the problem. We propose two encoding schemes for meta-heuristic solutions and three decoding methods for obtaining a schedule from the meta-heuristic solutions. Six different simulated annealing algorithms and genetic algorithms, respectively, are developed with six combinations of two encoding schemes and three decoding methods. Computational experiments are performed to find the best combination from those encoding schemes and decoding methods. Our findings show that the suggested algorithm provides not only better solution quality, but also less computation time required than the commercial optimisation solvers.     

Parallel machine scheduling with step-deteriorating jobs and setup times by a hybrid discrete cuckoo search algorithm

This article considers the parallel machine scheduling problem with step-deteriorating jobs and sequence-dependent setup times. The objective is to minimize the total tardiness by determining the allocation and sequence of jobs on identical parallel machines. In this problem, the processing time of each job is a step function dependent upon its starting time. An individual extended time is penalized when the starting time of a job is later than a specific deterioration date. The possibility of deterioration of a job makes the parallel machine scheduling problem more challenging than ordinary ones. A mixed integer programming model for the optimal solution is derived. Due to its NP-hard nature, a hybrid discrete cuckoo search algorithm is proposed to solve this problem. In order to generate a good initial swarm, a modified Biskup–Hermann–Gupta (BHG) heuristic called MBHG is incorporated into the population initialization. Several discrete operators are proposed in the random walk of Lévy flights and the crossover search. Moreover, a local search procedure based on variable neighbourhood descent is integrated into the algorithm as a hybrid strategy in order to improve the quality of elite solutions. Computational experiments are executed on two sets of randomly generated test instances. The results show that the proposed hybrid algorithm can yield better solutions in comparison with the commercial solver CPLEX^® with a one hour time limit, the discrete cuckoo search algorithm and the existing variable neighbourhood search algorithm.