Permutation flow-shop scheduling problem to optimize a quadratic objective function

A flow-shop scheduling model enables appropriate sequencing for each job and for processing on a set of machines in compliance with identical processing orders. The objective is to achieve a feasible schedule for optimizing a given criterion. Permutation is a special setting of the model in which the processing order of the jobs on the machines is identical for each subsequent step of processing. This article addresses the permutation flow-shop scheduling problem to minimize the criterion of total weighted quadratic completion time. With a probability hypothesis, the asymptotic optimality of the weighted shortest processing time schedule under a consistency condition (WSPT-CC) is proven for sufficiently large-scale problems. However, the worst case performance ratio of the WSPT-CC schedule is the square of the number of machines in certain situations. A discrete differential evolution algorithm, where a new crossover method with multiple-point insertion is used to improve the final outcome, is presented to obtain high-quality solutions for moderate-scale problems. A sequence-independent lower bound is designed for pruning in a branch-and-bound algorithm for small-scale problems. A set of random experiments demonstrates the performance of the lower bound and the effectiveness of the proposed algorithms.     

Scheduling learning dependent jobs in customised assembly lines

The large variety of product models required by customised markets implies lot size reduction. This strongly affects manual-based production activities, since workers need to promptly adapt to the specifications of the next model to be produced. Completion times of manual-based activities tend to be highly variable among workers, and are difficult to estimate. This affects the scheduling of those activities since scheduling precision depends on reliable estimates of job completion times. This paper presents a method that combines learning curves and job scheduling heuristics aimed at minimising the total weighted earliness and tardiness. Workers performance data is collected and modelled using learning curves, enabling a better estimation of the completion time of jobs with different size and complexity. Estimated completion times are then inputted in new scheduling heuristics for unrelated parallel workers, equivalent to machines in this study, created by modifying heuristics available in the literature. Performance of the proposed heuristics is assessed analysing the difference between the optimal schedule objective function value and that obtained using the heuristics, as well as the workload imbalance among workers. Some contributions in this paper are: (i) use of learning curves to estimate completion times of jobs with different sizes and complexities from different teams of workers; and (ii) use of a more complex scheduling objective function, namely the total weighted earliness and tardiness, as opposed to most of the developments in the current scheduling literature. A shoe manufacturing application illustrates the developments in the paper.     

A genetic algorithm based approach for scheduling of jobs containing multiple orders in a three-machine flowshop

This paper investigates a new scheduling problem of multiple orders per job (MOJ) in a three-machine flowshop consisting of an item-processing machine, a lot-processing machine and a batch-processing machine, for a semiconductor manufacturing operation that must form a layer on the wafers. The three-machine flowshop MOJ scheduling problem deals with sequencing customer orders, assigning orders to jobs, and then combining the formed jobs into batches. Genetic algorithms are presented for this scheduling problem to minimise the total weighted tardiness (TWT), in the presence of non-zero order ready times, different order due dates, different order weights and unequal order sizes. Extensive experiments were performed to compare the proposed genetic algorithm (GA)-based approach with the benchmark heuristics presented in previous studies. The experiments led to the conclusions that the GA-based approach is significantly superior over other heuristics in terms of TWT and can find near-optimal solutions in an acceptable amount of computation time.     

Scheduling parallel batch processing machines with arbitrary job sizes and incompatible job families

Motivated by a bottleneck operation in an MLCC (multi-layer ceramic capacitor) production line, we study the scheduling problem of parallel batch processing machines in which a number of jobs can be processed simultaneously in a machine as a batch. Volumes of the jobs are different from each other and each job belongs to the family in which all jobs have the same processing time. In this situation, we analyse three kinds of problems whose performance measures are makespan, total completion time, and total weighted completion time, respectively. Since these problems are known to be NP-hard, we propose a number of heuristics and design genetic algorithms for the problems. Through some computational experiments, we evaluate the performances of the heuristic algorithms proposed, including the genetic algorithms for each of three problems.     

Scheduling algorithms to minimise the total family flow time for job shops with job families

This paper considers the job scheduling problem in which jobs are grouped into job families, but they are processed individually. The decision variable is the sequence of the jobs assigned to each machine. This type of job shop scheduling can be found in various production systems, especially in remanufacturing systems with disassembly, reprocessing and reassembly shops. In other words, the reprocessing shop can be regarded as the job shop with job families since it performs the operations required to bring parts or sub-assemblies disassembled back to like-new condition before reassembling them. To minimise the deviations of the job completion times within each job family, we consider the objective of minimising the total family flow time. Here, the family flow time implies the maximum among the completion times of the jobs within a job family. To describe the problem clearly, a mixed integer programming model is suggested and then, due to the complexity of the problem, two types of heuristics are suggested. They are: (a) priority rule based heuristics; and (b) meta-heuristics. Computational experiments were performed on a number of test instances and the results show that some priority rule based heuristics are better than the existing ones. Also, the meta-heuristics improve the priority rule based heuristics significantly.     

Open shop scheduling problem to minimize total weighted completion time

A given number of jobs in an open shop scheduling environment must each be processed for given amounts of time on each of a given set of machines in an arbitrary sequence. This study aims to achieve a schedule that minimizes total weighted completion time. Owing to the strong NP-hardness of the problem, the weighted shortest processing time block (WSPTB) heuristic is presented to obtain approximate solutions for large-scale problems. Performance analysis proves the asymptotic optimality of the WSPTB heuristic in the sense of probability limits. The largest weight block rule is provided to seek optimal schedules in polynomial time for a special case. A hybrid discrete differential evolution algorithm is designed to obtain high-quality solutions for moderate-scale problems. Simulation experiments demonstrate the effectiveness of the proposed algorithms.     

A multi-objective flow shop scheduling with resource-dependent processing times: trade-off between makespan and cost of resources

In traditional flow shop scheduling problems (FSSPs), the processing times are assumed to be pre-known and fixed parameters while in many practical environments, the processing times can be controlled by consumption of extra resources. In this paper, we propose resource-dependent processing times (RDPT) for permutation FSSP in which the processing time of a job depends on the amount of additional resources assigned to that job. To make a trade-off between makespan and required amount of resources, two conflict objective functions are considered: the minimisation of maximum completion time and the minimisation of total cost of resources. In order to solve the problem considered in this paper, a decomposition approach is suggested that strives to deal with the original model via two sub-problems: (i) sequencing problem; and (ii) resource allocation problem. A hybrid discrete differential evolution (HDDE) algorithm with an effective coordinate directions search and a variable neighbourhood search (VNS) are combined to solve two sub-problems. Furthermore, a statistical procedure is employed to adjust the significant parameters of the proposed HDDE and VNS algorithms. This procedure is based on the stepwise regression (SR) technique. The effectiveness of the suggested hybrid algorithm is investigated through a computational study and obtained results show the good performance of our approach with regard to the other algorithms.     

On bilevel machine scheduling problems

Bilevel scheduling problems constitute a hardly studied area of scheduling theory. In this paper, we summarise the basic concepts of bilevel optimisation, and discuss two problem classes for which we establish various complexity and algorithmic results. The first one is the bilevel total weighted completion time problem in which the leader assigns the jobs to parallel machines and the follower sequences the jobs assigned to each machine. Both the leader and the follower aims to minimise the total weighted completion time objective, but with different job weights. When the leader’s weights are arbitrary, the problem is NP-hard. However, when all the jobs are of unit weight for the leader, we provide a heuristic algorithm based on iterative LP-rounding along with computational results, and provide a sufficient condition when the LP-solution is integral. In addition, if the follower weights induce a monotone (increasing or decreasing) processing time order in any optimal solution, the problem becomes polynomially solvable. As a by-product, we characterise a new polynomially solvable special case of the MAX m-CUT problem, and provide a new linear programming formulation for the P||?_j C_j{P||\sum_j C_j} problem. Finally, we present some results on the bilevel order acceptance problem, where the leader decides on the acceptance of orders and the follower sequences the jobs. Each job has a deadline and if a job is accepted, it cannot be late. The leader’s objective is to maximise the total weight of accepted jobs, whereas the follower aims at minimising the total weighted job completion times. For this problem, we generalise some known single-level machine scheduling algorithms.     

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.     

Optimisation of flow-shop scheduling with batch processor and limited buffer

This paper deals with a flow-shop scheduling problem with limited intermediate buffer. Jobs are grouped in incompatible job families. Each job has to be processed by a batch processor followed by a discrete processor in the same order. The batch processor can process several jobs simultaneously so that all jobs of the same batch start and complete together. We assume that the capacity of batch processor is bounded. The batch processing time is identical for batches of the same family. A batch which has completed processing on the batch processor may block the processor until there is a free unit in the buffer. The objective is to determine a batching and scheduling for all jobs so as to minimise mean completion time. A lower bound and two heuristics algorithm are developed. Moreover, a two-stage method embedded with a Differential Evolution (DE) algorithm is also developed. DE is one of the latest evolutionary computation algorithms, which implements mutation, crossover, and selection operators to improve the candidate solutions iteratively. Three variants of DE are first compared with a continuous Genetic Algorithm employing the random key representation. Then, one variant of the DE with the best convergence speed is selected. Numerical experiments are conducted to evaluate the performances of the selected two-stage meta-heuristic and two heuristics.     

Scheduling hybrid flowshops to minimize maximum tardiness or maximum completion time

The problem of scheduling independent jobs on several serial workshops consisting of identical parallel machines is studied. Each job is processed by one machine in each workshop. This workshop environment is called a hybrid Flowshop. Each job has its own due-date and the objective is to minimize maximum tardiness or maximum completion time. Given that the problem is NP-hard, a set of list algorithms is developed to solve it. To evaluate the quality of these heuristics, lower bounds on the optimal solution have been derived and compared to the value of the heuristics on 1920 problems. Our results indicate that a heuristic based on Nawaz et al. (1983) method outperformed the other approaches.     

A multi-point simulated annealing heuristic for solving multiple objective unrelated parallel machine scheduling problems

This study considers the problem of job scheduling on unrelated parallel machines. A multi-objective multi-point simulated annealing (MOMSA) algorithm was proposed for solving this problem by simultaneously minimising makespan, total weighted completion time and total weighted tardiness. To assess the performance of the proposed heuristic and compare it with that of several benchmark heuristics, the obtained sets of non-dominated solutions were assessed using four multi-objective performance indicators. The computational results demonstrated that the proposed heuristic markedly outperformed the benchmark heuristics in terms of the four performance indicators. The proposed MOMSA algorithm can provide a new benchmark for future research related to the unrelated parallel machine scheduling problem addressed in this study.     

Scheduling about an unrestricted common due windowwith arbitrary earliness/tardiness penalty rates

We consider the NP-hard problem of scheduling jobs on a single machine about an unrestricted due window to minimize total weighted earliness and tardiness cost. Each job has an earliness penalty rate and a tardiness penalty rate that are allowed to be arbitrary. Earliness or tardiness cost is assessed when a job completes outside the due window, which may be an instant in time or a time increment defining acceptable job completion. In this paper we present properties that characterize the structure of an optimal schedule, present a lower bound, propose a two-step branch and bound algorithm, and report results from a computational experiment. We find that optimal solutions can be quickly obtained for medium-sized problem instances.     

Scheduling a batch processing machine with non-identical job sizes

The problem of scheduling batch processors is important in some industries and, at a more fundamental level, captures an element of complexity common to many practical scheduling problems. We describe a branch and bound procedure applicable to a batch processor model with arbitrary job processing times, job weights and job sizes. The scheduling objective is to minimize total weighted completion time. We find that the procedure returns optimal solutions to problems of up to 25 jobs in reasonable CPU time, and can be adapted for use as a heuristic for larger problems.     

Single machine batch scheduling with sequential job processing

The problem of scheduling n jobs on a single machine in batches to minimize some regular cost functions is studied. Jobs within each batch are processed sequentially so that the processing time of a batch is equal to the sum of the processing times of the jobs contained in it. Jobs in the same batch are completed at the same time when the last job of the batch has finished its processing. A constant set-up time precedes the processing of each batch. The number of jobs in each batch is bounded by some value b. If b < n, then the problem is called bounded. Otherwise, it is unbounded. For both the bounded and unbounded problems, dynamic programming algorithms are presented for minimizing the maximum lateness, the number of late jobs, the total tardiness, the total weighted completion time, and the total weighted tardiness when all due dates are equal, which are polynomial if there is a fixed number of distinct due dates or processing times. More efficient algorithms are derived for some special cases of both the bounded and unbounded problems in which all due dates and/or processing times are equal. Several special cases of the bounded problem are shown to be NP-hard. Thus, a comprehensive classification of the computational complexities of the special cases is provided.     

Single-machine batch delivery scheduling and common due-date assignment with a rate-modifying activity

We consider batch delivery scheduling on a single machine, where a common due-date is assigned to all the jobs and a rate-modifying activity on the machine may be scheduled, which can change the processing rate of the machine. Thus the actual processing time of a job is variable depending on whether it is processed before or after the rate-modifying activity. The objective is to determine the optimal job sequence, the optimal partition of the job sequence into batches, the optimal assigned common due-date, and the optimal location of the rate-modifying activity simultaneously to minimize the total cost of earliness, job holding, weighted number of tardy jobs, due-date assignment, and batch delivery. We derive some structural properties of the problem, based on which we design polynomial-time algorithms to solve some special cases of the problem.     

Applied column generation-based approach to solve supply chain scheduling problems

More and more enterprises have chosen to adopt a made-to-order business model in order to satisfy diverse and rapidly changing customer demand. In such a business model, enterprises are devoted to reducing inventory levels in order to upgrade the competitiveness of the products. However, reductions in inventory levels and short lead times force the operation between production and distribution to cooperate closely, thus increasing the practicability of integrating the production and distribution stages. The complexity of supply chain scheduling problems (integrated production and distribution scheduling) is known to be NP-hard. To address the issues above, an efficient algorithm to solve the supply chain scheduling problem is needed. This paper studies a supply chain scheduling problem in which the production stage is modelled by an identical parallel machine scheduling problem and the distribution stage is modelled by a capacitated vehicle routing problem. Given a set of customer orders (jobs), the problem is to find a supply chain schedule such that the weighted summation of total job weighted completion time and total job delivering cost are minimised. The studied problem was first formulated as an integer programme and then solved by using column generation techniques in conjunction with a branch-and-bound approach to optimality. The results of the computational experiments indicate that the proposed approach can solve the test problems to optimality. Moreover, the average gap between the optimal solutions and the lower bounds is no more than 1.32% for these test problems.     

Single machine group scheduling with family setups to minimize total tardiness

This paper considers the single machine scheduling problem with independent family (group) setup times where jobs in each family are processed together. A sequence-independent setup is required to process a job from a different family. The objective is to minimize total tardiness. A mixed-integer linear programming model capable of solving small-sized problems is described. In view of the NP-hard nature of the problem, two-phase heuristics including simulated annealing algorithms are proposed to find optimal or near-optimal schedules. Empirical results show that the proposed heuristic algorithms are quite effective in minimizing total tardiness for a single machine group scheduling problem with family setup times.     

Batch scheduling problem for a machinery factory with fixed-position layout

We consider a batch scheduling problem for a two-stage flow shop with fixed-position layout. In the first stage, a fixed number of jobs are assembled on a batch machine with a family batch setup time and a common processing time. In the second stage, the assembled jobs are individually performed for system integration on a discrete machine. The finished job is immediately packed and shipped if the payment has been made; otherwise, it is moved to a temporary storage area, incurring additional removal time. This study develops a mixed integer programming (MIP) to solve the problem of minimising the total completion time and proposes two heuristics for large-size problems. Computational results show that the proposed methods can be applied to resolve real-world problems similar to those in this study.     

New scheduling algorithms and digital tool for dynamic permutation flowshop with newly arrived order

The permutation flowshop scheduling problem has been widely studied under static environment by assuming machines and jobs are available at the time of zero. However, in reality, new orders arrive at production systems randomly, which leads to sheer complexity in scheduling due to the dynamic changes given various constraints of resources. Previous studies simply attach new orders directly after the existing schedule. Recent study shows mixing jobs of old and new orders could result in better scheduling solutions. But the heuristic algorithms are still lacking to implement the job mixing policy. To address this problem, a novel scheduling strategy is herein proposed by integrating match-up strategy and real-time strategy (MR) in order to make use of the remaining time before the old order due date. Based on the new MR strategy, eleven new heuristics are introduced with ten existing and one new priority rules. Computational results illustrate the effectiveness of the new heuristics. A digital tool is developed for ease of application of these heuristics, and it is validated by case studies.