Makespan minimization flowshop with position dependent job processing times—computational complexity and solution algorithms

This paper considers flowshop scheduling problems where job processing times are described by position dependent functions, i.e., dependent on the number of processed jobs, that model learning or aging effects. We prove that the two-machine flowshop problem to minimize the maximum completion time (makespan) is NP-hard if job processing times are described by non-decreasing position dependent functions (aging effect) on at least one machine and strongly NP-hard if job processing times are varying on both machines. Furthermore, we construct fast NEH, tabu search with a fast neighborhood search and simulated annealing algorithms that solve the problem with processing times described by arbitrary position dependent functions that model both learning and aging effects. The efficiency of the proposed methods is numerically analyzed.     

A new approach to the learning effect: Beyond the learning curve restrictions

In this paper, we bring into the scheduling field a new model of the learning effect, where in two ways the existing approach is generalized. First we relax one of the rigorous constraints, and thus in our model each job can provide different experience to the processor. Second we formulate the job processing time as a non-increasing k-stepwise function, that, in general, is not restricted to a certain learning curve, thereby it can accurately fit every possible shape of a learning function. Furthermore, we prove that the problem of makespan minimization with the considered model is polynomially solvable if every job provides the same experience to the processor, and it becomes NP-hard if the experiences are diversified. The most essential result is a pseudopolynomial time algorithm that solves optimally the makespan minimization problem with any function of an experience-based learning model reduced into the form of the k-stepwise function.     

The three-stage assembly flowshop scheduling problem

We consider the three-stage assembly flowshop scheduling problem with the objective of minimizing the makespan. The three-stage assembly problem generalizes both the serial three machine flowshop problem and the two-stage assembly flowshop scheduling problem and is therefore strongly NP-hard. We analyze the worst-case ratio bound for several heuristics for this problem. We also analyze the worst-case absolute bound for a heuristic based on compact vector summation techniques and we point out that, for a large number of jobs, this heuristic becomes asymptotically optimal.Scope and purposeThe three-stage assembly flowshop scheduling problem models situations which arise frequently in manufacturing when various fabrication operations are performed concurrently and then collected and transported into an assembly area for a final assembly operation. The main criterion for this problem is the minimization of the maximum job completion time (makespan). The objective of this paper is to derive algorithms for minimizing the makespan. In doing so, we also demonstrate the reduction of assembly flowshop problems to their embedded serial flowshop problems.     

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.     

两机无等待流水车间调度问题的性质

针对两机无等待流水车间调度问题,提出目标函数最大完工时间最小化的快速算法,并给出算法的复杂度。分析两机无等待流水车间调度问题的排列排序性质,证明了两机无等待流水车间调度问题的可行解只存在于排列排序中,排列排序的最优解一定是两机无等待流水车间调度问题的最优解。最后研究了同时包含普通工件和无等待工件的两机流水车间调度问题的复杂性,为进一步研究两机无等待流水车间调度问题提供了理论依据。     

A new heuristic for scheduling the two-stage flowshop with additional resources

This paper deals with the problem of preemptive scheduling in a two-stage flowshop with parallel unrelated machines at the first stage and a single machine at the second stage. At the first stage, jobs use some additional resources which are available in limited quantities at any time. The resource requirements are of 0–1 type. The objective is the minimization of makespan. The problem is NP-hard. Heuristic algorithms are proposed which solve to optimality the resource constrained scheduling problem at the first stage of the flowshop, and at the same time, minimize the makespan in the flowshop by selecting appropriate jobs for simultaneous processing. Several rules of job selection are considered. The performance of the proposed heuristic algorithms is analyzed by comparing solutions with the lower bound on the optimal makespan. The extensive computational experiment shows that the proposed heuristic algorithms are able to produce near-optimal solutions in short computational time.     

Scheduling a variable maintenance and linear deteriorating jobs on a single machine

We investigate a single machine scheduling problem in which the processing time of a job is a linear function of its starting time and a variable maintenance on the machine must be performed prior to a given deadline. The goals are to minimize the makespan and the total completion time. We prove that both problems are NP-hard. Furthermore, we show that there exists a fully polynomial time approximation scheme for the makespan minimization problem. For the total completion time minimization problem we point out that there exists a fully polynomial time approximation scheme for a special case.     

Solution algorithms for the makespan minimization problem with the general learning model

Most of the papers devoted to scheduling problems with the learning effect concern the Wright’s learning curve. On the other hand, the study about learning has pointed out that the learning curve in practice is very often an S-shaped function, which has not been considered in scheduling. Thus, in this paper, a single processor makespan minimization problem with an S-shaped learning model is investigated. We prove that this problem is strongly NP-hard even if the experience provided by each job is equal to its normal processing time. Therefore, to solve this problem, we prove some eliminating properties that are used to construct a branch and bound algorithm and some fast heuristic methods. Since the proposed algorithms are dedicated for the general case, i.e., where job processing times are arbitrary non-increasing experience dependent functions, their efficiency is verified numerically for the S-shaped model.     

A memetic algorithm for the re-entrant permutation flowshop scheduling problem to minimize the makespan

A common assumption in the classical permutation flowshop scheduling model is that each job is processed on each machine at most once. However, this assumption does not hold for a re-entrant flowshop in which a job may be operated by one or more machines many times. Given that the re-entrant permutation flowshop scheduling problem to minimize the makespan is very complex, we adopt the CPLEX solver and develop a memetic algorithm (MA) to tackle the problem. We conduct computational experiments to test the effectiveness of the proposed algorithm and compare it with two existing heuristics. The results show that CPLEX can solve mid-size problem instances in a reasonable computing time, and the proposed MA is effective in treating the problem and outperforms the two existing heuristics.     

A discrete differential evolution algorithm for the permutation flowshop scheduling problem

Very recently, Pan et al. [Proceedings of the 9th Annual Conference on Genetic and Evolutionary Computation, GECCO07, pp. 126–33] presented a new and novel discrete differential evolution algorithm for the permutation flowshop scheduling problem with the makespan criterion. On the other hand, the iterated greedy algorithm is proposed by [Ruiz, R., & Stützle, T. (2007). A simple and effective iterated greedy algorithm for the permutation flowshop scheduling problem. European Journal of Operational Research, 177(3), 2033–49] for the permutation flowshop scheduling problem with the makespan criterion. However, both algorithms are not applied to the permutation flowshop scheduling problem with the total flowtime criterion. Based on their excellent performance with the makespan criterion, we extend both algorithms in this paper to the total flowtime objective. Furthermore, we propose a new and novel referenced local search procedure hybridized with both algorithms to further improve the solution quality. The referenced local search exploits the space based on reference positions taken from a reference solution in the hope of finding better positions for jobs when performing insertion operation. Computational results show that both algorithms with the referenced local search are either better or highly competitive to all the existing approaches in the literature for both objectives of makespan and total flowtime. Especially for the total flowtime criterion, their performance is superior to the particle swarm optimization algorithms proposed by [Tasgetiren, M. F., Liang, Y. -C., Sevkli, M., Gencyilmaz, G. (2007). Particle swarm optimization algorithm for makespan and total flowtime minimization in permutation flowshop sequencing problem. European Journal of Operational Research, 177(3), 1930–47] and [Jarboui, B., Ibrahim, S., Siarry, P., Rebai, A. (2007). A combinatorial particle swarm optimisation for solving permutation flowshop problems. Computers & Industrial Engineering, doi:10.1016/j.cie.2007.09.006]. Ultimately, for Taillard’s benchmark suite, four best known solutions for the makespan criterion as well as 40 out of the 90 best known solutions for the total flowtime criterion are further improved by either one of the algorithms presented in this paper.     

Reduction of permutation flowshop problems to single machine problems using machine dominance relations

The Permutation Flowshop Scheduling Problem with Makespan objective (PFSP-M) is known to be NP-hard for more than two machines, and literally hundreds of works in the last decades have proposed exact and approximate algorithms to solve it. These works—of computational/experimental nature—show that the PFSP-M is also empirically hard, in the sense that optimal or quasi-optimal sequences statistically represent a very small fraction of the space of feasible solutions, and that there are big differences among the corresponding makespan values. In the vast majority of these works, it has been assumed that (a) processing times are not job- and/or machine-correlated, and (b) all machines are initially available. However, some works have found that the problem turns to be almost trivial (i.e. almost every sequence yields an optimal or quasi-optimal solution) if one of these assumptions is dropped. To the best of our knowledge, no theoretical or experimental explanation has been proposed by this rather peculiar fact.Our hypothesis is that, under certain conditions of machine availability, or correlated processing times, the performance of a given sequence in a flowshop is largely determined by only one stage, thus effectively transforming the flowshop layout into a single machine. Since the single machine scheduling problem with makespan objective is a trivial problem where all feasible sequences are optimal, it would follow that, under these conditions, the equivalent PFSP-M is almost trivial. To address this working hypothesis from a general perspective, we investigate some conditions that allow reducing a permutation flowshop scheduling problem to a single machine scheduling problem, focusing on the two most common objectives in the literature, namely makespan and flowtime. Our work is a combination of theoretical and computational analysis, therefore several properties are derived to prove the conditions for an exact (theoretical) equivalence, together with an extensive computational evaluation to establish an empirical equivalence.     

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.     

Makespan distribution of permutation flowshop schedules

The makespan distribution of permutation flowshop schedules has been a topic of debate for almost fifty years. Many researchers have confirmed or doubted the famous claim that the makespan distribution of permutation flowshop schedules is asymptotically normal if the number of jobs is sufficiently large. This paper theoretically and empirically investigates the makespan distribution of permutation flowshop schedules and shows that the normality claim is not valid for the job-dominated and machine-dominated flowshops. Errors in the proof of normality of the makespan distribution of permutation flowshop schedules are pointed out. It is shown that the makespan distribution of a permutation flowshop scheduling problem depends on the number of jobs as well as the number of machines.     

Scheduling two jobs with fixed and nonfixed routes

The shop-scheduling problem with two jobs andm machines is considered under the condition that the machine order is fixed in advance for the first job and nonfixed for the second job. The problems of makespan and mean flow time minimization are proved to be NP-hard if operation preemption is forbidden. In the case of preemption allowance for any given regular criterion theO(n _*) algorithm is proposed. Here,n _* is the maximum number of operations per job.     

Worst-case and numerical analysis of heuristic algorithms for flowshop scheduling problems with a time-dependent learning effect

In this paper, we investigate a time-dependent learning effect in a flowshop scheduling problem. We assume that the time-dependent learning effect of a job was a function of the total normal processing time of jobs scheduled before the job. The following objective functions are explored: the makespan, the total flowtime, the sum of weighted completion times, the sum of the kth power of completion times, and the maximum lateness. Some heuristic algorithms with worst-case analysis for the objective functions are given. Moreover, a polynomial algorithm is proposed for the special case with identical processing time on each machine and that with an increasing series of dominating machines, respectively. Finally, the computational results to evaluate the performance of the heuristics are provided.     

Single-machine scheduling jobs with exponential learning functions

In a manufacturing system workers are involved in doing the same job or activity repeatedly. Hence, the workers start learning more about the job or activity. Because of the learning, the time to complete the job or activity starts decreasing, which is known as “learning effect”. In this paper, an exponential sum-of-actual-processing-time based learning effect is introduced into single-machine scheduling. By the exponential sum-of-actual-processing-time based learning effect, we mean that the processing time of a job is defined by an exponential function of the sum-of-the-actual-processing-time of the already processed jobs. Under the proposed learning model, we show that under a sufficient condition, the makespan minimization problem, the sum of the θth (θ > 0) power of completion times minimization problem, and some special cases of the total weighted completion time minimization problem and the maximum lateness minimization problem remain polynomially solvable.     

Some scheduling problems with deteriorating jobs and learning effects

Although scheduling with deteriorating jobs and learning effect has been widely investigated, scheduling research has seldom considered the two phenomena simultaneously. However, job deterioration and learning co-exist in many realistic scheduling situations. In this paper, we introduce a new scheduling model in which both job deterioration and learning exist simultaneously. The actual processing time of a job depends not only on the processing times of the jobs already processed but also on its scheduled position. For the single-machine case, we derive polynomial-time optimal solutions for the problems to minimize makespan and total completion time. In addition, we show that the problems to minimize total weighted completion time and maximum lateness are polynomially solvable under certain agreeable conditions. For the case of an m-machine permutation flowshop, we present polynomial-time optimal solutions for some special cases of the problems to minimize makespan and total completion time.     

Algorithms for a realistic variant of flowshop scheduling

This paper deals with a realistic variant of flowshop scheduling, namely the hybrid flexible flowshop. A hybrid flowshop mixes the characteristics of regular flowshops and parallel machine problems by considering stages with parallel machines instead of having one single machine per stage. We also investigate the flexible version where stage skipping might occur, i.e., not all stages must be visited by all jobs. Lastly, we also consider job sequence dependent setup times per stage. The optimization criterion considered is makespan minimization. While many approaches for hybrid flowshops have been proposed, hybrid flexible flowshops have been rarely studied. The situation is even worse with the addition of sequence dependent setups. In this study, we propose two advanced algorithms that specifically deal with the flexible and setup characteristics of this problem. The first algorithm is a dynamic dispatching rule heuristic, and the second is an iterated local search metaheuristic. The proposed algorithms are evaluated by comparison against seven other high performing existing algorithms. The statistically sound results support the idea that the proposed algorithms are very competitive for the studied problem.     

Minimizing makespan on a three-machine flowshop batch scheduling problem with transportation using genetic algorithm

This paper studies the minimization of makespan in a three-machine flowshop scheduling problem in which a batch processing machine is located between two single processing machines on first and third stages. In this study also transportation capacity and transportation among machines times are explicitly considered.We establish a mixed integer programming model and propose a heuristic algorithm based on the basic idea of Johnson's algorithm. Since the problem under study is NP-hard, a genetic algorithm is also proposed to minimize makespan. The effectiveness of our solution procedures is evaluated through computational experiments. The results obtained from the computational study have shown that the genetic algorithm is a viable and effective approach that is capable to produce consistently good results.     

A two-stage hybrid flowshop scheduling problem in machine breakdown condition

This paper considers a two-stage hybrid flowshop scheduling problem in machine breakdown condition. By machine breakdown condition we mean that the machine may not always be available during the scheduling period. Machine failure may occur with a known probability after completing a job. Probability of machine failure depends on the previous processed job. The problem to be studied has one machine at the first stage and M parallel identical machines at the second stage. The objective is to find the optimal job combinations and the optimal job schedule such that the makespan is minimized. The proposed problem is compatible with a large scope of real world situations. To solve the problem, first, we introduce one optimal approach for job precedence when there is one machine in both stages and then provide a heuristic algorithm when there are M machines in stage two. To examine the performance of the heuristic, some experiments used are provided as well.