M-machine Permutation Flowshop Research Articles

A production and distribution framework: Manufacturer dominates

This paper presents a two-stage supply chain problem at the operational level involving a manufacturer and a third-party logistics provider (3PL provider). The integrated problem consists of two subproblems: An m-machine permutation flow shop with inventory considerations for the manufacturer and a routing problem for the 3PL provider. Both agents face tardiness penalty costs in case of late delivery to the customer locations. Assuming imperfect information sharing between the two agents and independence in their own decisions, we investigate the scenario in which the manufacturer dominates the negotiation.In this scenario, the manufacturer imposes the number and composition of vehicles as well as the vehicle departure dates on the 3PL provider. However, lacking the information regarding the actual delivery times he is forced to estimate delivery times at the customer locations in the context of his planning. Following the planning of the manufacturer, the 3PL provider decides on the optimal routing of the vehicles. Both agents aim to minimise their total costs. After formulating the problems as a mixed integer linear program, two metaheuristic algorithms are proposed to solve the scenario. The performance of the heuristics is evaluated on a set of randomly generated instances. The results show that the genetic algorithm is the best performing method.

Permutation flowshop scheduling to obtain the optimal solution/a lower bound with the makespan objective

This paper focuses on developing the optimal solution or a lower bound for N-job, M-machine Permutation Flowshop Scheduling (PFS) problem in a manufacturing system with the objective of minimizing the makespan using Lagrangian Relaxation (LR) technique. Even though LR technique is considered, in general, as a good method to obtain a lower bound, research in this direction with respect to our problem under study appears scarce. We address this gap by developing two MILP based Lagrangian Relaxation models, namely, Lagrangian Relaxation Method 1 (called Proposed Lagrangian Lower Bound Program (PLLBP)) and Alternate Lagrangian Relaxation Method 1 (called ALR) to find the optimal solution or a lower bound on the makespan. Basically, we develop these LR methods to overcome the possible limitation of the general LR procedure involving the sub-gradient approach. Benchmark PFS problem instances are used to evaluate the performance of these methods. It is observed that the PLLBP outperforms the ALR, and it provides better lower bounds than the lower bounds (in most instances) reported in the literature. Even though the PLLBP is superior in terms of solution quality, it has a limitation in that it cannot execute problem instances beyond 500 jobs due to the associated computational effort.

Dominance conditions determination based on machine idle times for the permutation flowshop scheduling problem

In this study, we investigate the lower bound and dominance conditions of the N-job and M-machine permutation flowshop scheduling problem to minimize the makespan based on machine idle times. We have developed and validated three dominance conditions and a lower bound based on a new approach that minimizes the machine idle times in a flowshop to facilitate the efficiency of the branch and bound approach. Most importantly, we have developed a dominance condition that is independent of the preceding and subsequent partial schedules, which has been considered critical but has not previously been developed in flowshop scheduling research. Finally, the efficiency of our developed branch and bound approach based on machine idle times is compared with the efficiencies of some previous branch and bound algorithms from the literature. We show that the dominance conditions that we developed can significantly improve the efficiency of the branch and bound approach.

A filtered beam search method for the m-machine permutation flowshop scheduling problem minimizing the earliness and tardiness penalties and the waiting time of the jobs

This paper addresses the minimization of the absolute deviation of job completion times from a common due date in a flowshop scheduling problem. Besides this main objective, the minimization of the waiting time of the jobs in the production environment, that can be seen as an intermediate inventory cost, is also considered. Initially, a mixed integer programming model for this problem is proposed and, due to its complexity, heuristic approaches are developed. A list-scheduling algorithm for the approached problem is introduced. Moreover, a filtered beam search method that explores specific characteristics of the considered environment is proposed. Numerical experiments show that the presented methods can be successfully applied to this problem.

Flow shop scheduling with jobs arriving at different times

Flow shop scheduling is common in modern lean production systems. In practice, jobs in flow shops can arrive at irregular times. However, scheduling that considers such irregularity has not been adequately investigated in the literature. In this paper, we examine the scheduling of n-jobs in an m-machine permutation flow shop with an unlimited intermediate storage. The jobs in this shop are assumed to have deterministic and known occurrence times. The objective of this scheduling is to reduce the total completion time (Fm|prmu|∑Cj). To improve the quality of solution, we successively put each job in the current best position and reinsert certain jobs based on weight calculations. We then develop and compare the solutions obtained by our simple and constructive heuristic method against the optimal schedules or other simple heuristic solutions with and without jobs arriving at different times. The computational experiments highlight the efficiency, easy implementation, and excellent performance of our proposed heuristic method.

On the n-job, m-machine permutation flow shop scheduling problems with makespan criterion and rework

This paper addresses an n-job, m-machine permutation flow shop scheduling problem (PFSSP) with unlimited intermediate buffers and rework activities. The concept of rework means that processing of a job on a machine may not meet a predefined quality level through its first process. Thus we have a probabilistic cycle of operations for jobs on different machines which is based on two concepts: (1) a failure probability of a job on a machine and, (2) a descent rate that reduces the processing times for rework phase. In this case, the processing times of jobs on machines become random variables with a known probability distribution. The aim of this paper is to examine possible solution approaches for generating the efficient job sequences with the least potential makespan. A wide range of simulation-based approaches are applied to address the proposed problem. These methods contain mathematical formulation, heuristic algorithms, and metaheuristics. The mechanism of the solution approaches is based on firstly using expected processing times to find a job sequence; then evaluating the obtained job sequences by several simulated trials. Using the one-way ANOVA test, these methods have been compared together, and the results show the superiority of metaheuristics, especially simulated annealing, over the other methods

A heuristically directed immune algorithm to minimize makespan and total flow time in permutation flow shops

As an emerging novel evolutionary technique, the immune algorithm has gained a lot of attention and wide applications in various fields of engineering and management. To our knowledge, this paper first considers the application of immune algorithm for the classic permutation flow shop scheduling problem. We present a hybrid heuristic combining immune algorithm and simulated annealing for the n-job, m-machine permutation flow shop scheduling problem to minimize makespan and total flow time. A heuristically directed population-based construction approach in the immune algorithm and a forward/backward shift neighborhood search heuristic in the simulated annealing of the proposed method is utilized to explore the search process for better schedules of jobs as well to speed up the convergence speed of the proposed algorithm. The proposed method is tested with Taillard’s flow shop scheduling benchmark instances for different problems with job sizes varying from 20 to 500. The computational results demonstrate that the proposed heuristic is very competitive with the state-of-the-art procedures in terms of both solution quality and computational times.

The permutation flowshop scheduling problem with exact time lags to minimise the total earliness and tardiness

In this paper, we investigate the problem of n-jobs scheduling in an m-machine permutation flowshop with exact time lags between consecutive operations of each job. The exact time lag is defined as the time elapsed between every couple of successive operations of the same job which is equal to a prescribed value. The aim is to find a feasible schedule that minimises the total tardiness and earliness. We propose three mathematical formulations, which are then solved by running the commercial software CPLEX to provide an optimal solution for small size problems. As the problem is shown to be strongly NP-hard, we propose new improved upper and lower bounds useful for large size problems. We then evaluate their effectiveness through an extensive computational experiment.

The permutation flowshop scheduling problem with exact time lags to minimise the total earliness and tardiness

In this paper, we investigate the problem of n-jobs scheduling in an m-machine permutation flowshop with exact time lags between consecutive operations of each job. The exact time lag is defined as the time elapsed between every couple of successive operations of the same job which is equal to a prescribed value. The aim is to find a feasible schedule that minimises the total tardiness and earliness. We propose three mathematical formulations, which are then solved by running the commercial software CPLEX to provide an optimal solution for small size problems. As the problem is shown to be strongly NP-hard, we propose new improved upper and lower bounds useful for large size problems. We then evaluate their effectiveness through an extensive computational experiment.

Minimization of maximum lateness in an m-machine permutation flow shop with a general exponential learning effect

In this paper, we consider the permutation flow shop scheduling problem with a general exponential learning effect. The objective is to minimize the maximum lateness. A special case that can be solved to optimality by EDD algorithm is provided. To form a hybrid solution framework, several heuristics, a branch-and-bound algorithm and a new Nested-Partition-based solution approach are proposed. Composite bounds and dominance rules are developed to reduce the searching region and to provide guidance on the lower bound. Finally, computational experiments are conducted to evaluate the performance of the algorithms.

Permutation flowshop problems with bi-criterion makespan and total completion time objective and position-weighted learning effects

In this paper, we study the problem of minimizing the weighted sum of makespan and total completion time in a permutation flowshop where the processing times are supposed to vary according to learning effects. The processing time of a job is a function of the sum of the logarithms of the processing times of the jobs already processed and its position in the sequence. We present heuristic algorithms, which are modified from the optimal schedules for the corresponding single machine scheduling problem and analyze their worst-case error bound. We also adopt an existing algorithm as well as a branch-and-bound algorithm for the general m-machine permutation flowshop problem. For evaluation of the performance of the algorithms, computational experiments are performed on randomly generated test problems.

OUTSOURCING DECISIONS IN m-MACHINE PERMUTATION FLOW SHOP SCHEDULING PROBLEMS WITH MACHINE-DEPENDENT PROCESSING TIMES

We consider m-machine permutation flow shop problems with an outsourcing option for a special case where each job's processing time equals the job's processing requirement plus a characteristic value of the machine. The objective is to minimize the sum of the performance measure for in-house jobs (the total completion time or the makespan) and the total outsourcing cost. We prove that two problems are polynomially solvable when the number of machines is fixed.

SOLVING FLOWSHOP SCHEDULING PROBLEMS USING A DISCRETE AFRICAN WILD DOG ALGORITHM

The problem of m-machine permutation flowshop scheduling is considered in this paper. The objective is to minimize the makespan. The flowshop scheduling problem is a typical combinatorial optimization problem and has been proved to be strongly NP-hard. Hence, several heuristics and meta-heuristics were addressed by the researchers. In this paper, a discrete African wild dog algorithm is applied for solving the flowshop scheduling problems. Computational results using benchmark problems show that the proposed algorithm outperforms many other algorithms addressed in the literature.

A Bottleneck-Assignment Based Branch-and-Bound Algorithm to Minimize the Makespan in an m-Machine Permutation Flowshop

This work aims at the minimization of makespan in m-machine permutation flowshops and presents an efficient branch-and-bound algorithm to solve this problem optimally. A job-based lower bound, integrated with a machine-based lower bound, is proposed which in turn is deployed in solving a bottleneck assignment problem to obtain a tight lower bound on the makespan. The algorithm is evaluated by solving many randomly generated problems of different problem sizes and the results of an extensive computational investigation are presented. In addition, the proposed branch-and-bound algorithm is compared with an existing bounding scheme.

Bi-criteria minimization for the permutation flowshop scheduling problem with machine-based learning effects

In traditional scheduling problems, the processing time for the given job is assumed to be a constant regardless of whether the job is scheduled earlier or later. However, the phenomenon named “learning effect” has extensively been studied recently, in which job processing times decline as workers gain more experience. This paper discusses a bi-criteria scheduling problem in an m-machine permutation flowshop environment with varied learning effects on different machines. The objective of this paper is to minimize the weighted sum of the total completion time and the makespan. A dominance criterion and a lower bound are proposed to accelerate the branch-and-bound algorithm for deriving the optimal solution. In addition, the near-optimal solutions are derived by adapting two well-known heuristic algorithms. The computational experiments reveal that the proposed branch-and-bound algorithm can effectively deal with problems with up to 16 jobs, and the proposed heuristic algorithms can yield accurate near-optimal solutions.

An Improved Immune Algorithm for Manufacturing Scheduling

This paper addresses an n-job, m-machine permutation flow shop scheduling problem to minimize makespan criterion. We present a modification of the best-known immune algorithm of Engin and Döyen (2004) [A new approach to solve hybrid flowshop scheduling problems by artificial immune system. Future Generations Computer Systems 2004;20:1083-1095] for this problem. To evaluate the solution quality and efficiency of the proposed method, the experiments are carried out in two phases on a set of benchmark problems and analyzed. We show, through computational experimentation, that this modification considerably improves its performance without affecting its time complexity. The proposed method also has been compared with the currently best simulated annealing from the literature.

Parallelization of a Branch and Bound Algorithm on Multicore Systems

The general m-machine permutation flowshop problem with the total flow-time objective is known to be NP-hard for m ≥ 2. The only practical method for finding optimal solutions has been branch-and-bound algorithms. In this paper, we present an improved sequential algorithm which is based on a strict alternation of Generation and Exploration execution modes as well as Depth-First/Best-First hybrid strategies. The experimental results show that the proposed scheme exhibits improved performance compared with the algorithm in [1]. More importantly, our method can be easily extended and implemented with lightweight threads to speed up the execution times. Good speedups can be obtained on shared-memory multicore systems.

A Genetic Algorithm to Minimize the Total Tardiness for M-Machine Permutation Flowshop Problems

The m-machine, n-job, permutation flowshop problem with the total tardiness objective is a common scheduling problem, known to be NP-hard. Branch and bound, the usual approach to finding an optimal solution, experiences difficulty when n exceeds 20. Here, we develop a genetic algorithm, GA, which can handle problems with larger n. We also undertake a numerical study comparing GA with an optimal branch and bound algorithm, and various heuristic algorithms including the well known NEH algorithm and a local search heuristic LH. Extensive computational experiments indicate that LH is an effective heuristic and GA can produce noticeable improvements over LH.

A note on machine scheduling with sum-of-logarithm-processing-time-based and position-based learning effects

Recently, Biskup [2] classifies the learning effect models in scheduling environments into two types: position-based and sum-of-processing-time-based. In this paper, we study scheduling problem with sum-of-logarithm-processing-time-based and position-based learning effects. We show that the single machine scheduling problems to minimize the makespan and the total completion time can both be solved by the smallest (normal) processing time first (SPT) rule. We also show that the problems to minimize the maximum lateness, the total weighted completion times and the total tardiness have polynomial-time solutions under agreeable WSPT rule and agreeable EDD rule. In addition, we show that m-machine permutation flowshop problems are still polynomially solvable under the proposed learning model.

A discrete particle swarm optimization algorithm with self-adaptive diversity control for the permutation flowshop problem with blocking

This paper proposes a discrete particle swarm optimization (DPSO) algorithm for the m-machine permutation flowshop scheduling problem with blocking to minimize the makespan, which has a strong industrial background, e.g., many production processes of chemicals and pharmaceuticals in chemical industry can be reduced to this problem. To prevent the DPSO from premature convergence, a self-adaptive diversity control strategy is adopted to diversify the population when necessary by adding a random perturbation to the velocity of each particle according to a probability controlled by the diversity of the current population. In addition, a stochastic variable neighborhood search is used as the local search to improve the search intensification. Computational results using benchmark problems show that the proposed DPSO algorithm outperforms previous algorithms proposed in the literature and that it can obtain 111 new best known upper bounds for the 120 benchmark problems.