A fuzzy greedy heuristic for permutation flow-shop scheduling

This paper describes a polynomial-time heuristic for the permutation flow-shop scheduling problem with the makespan criterion. The proposed method consists of two phases: arranging the jobs in priority order and then constructing a sequence. A fuzzy greedy evaluation function is employed to prioritize the jobs for incorporating into the construction phase of the heuristic. Computational experiments using standard benchmark problems indicate an improvement of the new heuristic over the well-known Nawaz, Enscore and Ham (NEH) heuristic. It will be seen that the NEH heuristic is a special case of our more general heuristic.     

A computational study with a new algorithm for the three-machine permutation flow-shop problem with release times

We consider the three-machine permutation flow-shop scheduling problem with release times where the objective is to minimize the maximum completion time. A special solvable case is found for the F2/r_j/C_max problem, which sharpens the boundary between easy and hard cases and can be used to compute a tight lower bound for our problem. Two dominance rules are generalized and applied to generating initial schedules, directing the search strategy and decomposing the problem into smaller ones. The branch and bound algorithm proposed here combines an adaptive branching rule with a fuzzy search strategy to narrow the search tree and lead the search to an optimal solution as early as possible. Our extensive numerical experiments have led to a classification of ‘easy' vs. ‘hard' problems, dependent only on the relative size of the release times. The algorithm has quickly solved approximately 90% of the hardest test problem instances with up to 200 jobs and 100% of the large problems classified as easy.     

A variable neighbourhood search for hybrid flow-shop scheduling problem with rework and set-up times

This paper deals with hybrid flow-shop scheduling problem with rework. In this problem, jobs are inspected at the last stage, and poorly processed jobs were returned and processed again. Thus, a job may visit a stage more than once, and we have a hybrid flow-shop with re-entrant flow. This kind of a shop may occur in many industries, such as final inspection system in automotive manufacturing. The criterion is to minimize the makespan of the system. We developed a 0–1 mixed-integer program of the problem. Since the hybrid flow-shop scheduling problem is NP-hard, an algorithm for finding an optimal solution in polynomial time does not exist. So we generalized some heuristic methods based on several basic dispatching rules and proposed a variable neighbourhood search (VNS) for the problem with sequence-dependent set-up times and unrelated parallel machines. The computational experiments show that VNS provides better solutions than heuristic methods.     

Heuristic algorithms for the minmax regret flow-shop problem with interval processing times

An uncertain version of the permutation flow-shop with unlimited buffers and the makespan as a criterion is considered. The investigated parametric uncertainty is represented by given interval-valued processing times. The maximum regret is used for the evaluation of uncertainty. Consequently, the minmax regret discrete optimization problem is solved. Due to its high complexity, two relaxations are applied to simplify the optimization procedure. First of all, a greedy procedure is used for calculating the criterion’s value, as such calculation is NP-hard problem itself. Moreover, the lower bound is used instead of solving the internal deterministic flow-shop. The constructive heuristic algorithm is applied for the relaxed optimization problem. The algorithm is compared with previously elaborated other heuristic algorithms basing on the evolutionary and the middle interval approaches. The conducted computational experiments showed the advantage of the constructive heuristic algorithm with regards to both the criterion and the time of computations. The Wilcoxon paired-rank statistical test confirmed this conclusion.

     

An approximation algorithm for the m-machine permutation flow shop scheduling problem with controllable processing times

The paper deals with the m-machine permutation flow shop scheduling problem in which job processing times, along with a processing order, are decision variables. It is assumed that the cost of processing a job on each machine is a linear function of its processing time and the overall schedule cost to be minimized is the total processing cost plus maximum completion time cost. A algorithm for the problem with m = 2 is provided; the best approximation algorithm until now has a worst-case performance ratio equal to . An extension to the m-machine (m ≥2) permutation flow shop problem yields an approximation algorithm with a worst-case bound equal to

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, where is the worst-case performance ratio of a procedure used, in the proposed algorithm, for solving the (pure) sequencing problem. Moreover, examples which achieve this bound for = 1 are also presented.     

A penalty-based heuristic algorithm for the permutation flowshop scheduling problem with sequence-dependent set-up times

This paper considers the permutation flowshop scheduling problem with sequence-dependent set-up times and develops a penalty-based heuristic algorithm to find an approximately minimum makespan schedule. The proposed algorithm determines the penalty in time associated with a particular sequence and selects the sequence with the minimum time penalty as the best heuristic solution. Computational results comparing the effectiveness and efficiency of the proposed penalty-based heuristic algorithm with an existing savings index heuristic algorithm are reported and discussed.     

A review and classification of heuristics for permutation flow-shop scheduling with makespan objective

Makespan minimization in permutation flow-shop scheduling is an operations research topic that has been intensively addressed during the last 40 years. Since the problem is known to be NP-hard for more than two machines, most of the research effort has been devoted to the development of heuristic procedures in order to provide good approximate solutions to the problem. However, little attention has been devoted to establish a common framework for these heuristics so that they can be effectively combined or extended. In this paper, we review and classify the main contributions regarding this topic and discuss future research issues.     

Complexity of a scheduling problem with controllable processing times

We consider the problem of scheduling a set of independent jobs on a single machine so as to minimize the total weighted completion time, subject to the constraint that the total compression cost is less than or equal to a fixed amount. The complexity of this problem is mentioned as an open problem. In this note we show that the problem is NP-hard.     

A two-machine flow shop scheduling problem with controllable job processing times

The paper deals with a two-machine flow shop scheduling problem in which both the sequence of jobs and their processing times are decision variables. It is assumed that the cost of performing a job is a linear function of its processing time, and the schedule cost to be minimized is the total processing cost plus maximum completion time cost. In is shown that the decision form of this problem is NP-complete, even when the processing times on one machine only are controllable and all the processing cost units are identical. Two heuristic methods for solving the problem are proposed and their worst-case analysis is presented.     

Single-machine scheduling with convex resource dependent processing times and deteriorating jobs

In this study, we consider scheduling problems with convex resource dependent processing times and deteriorating jobs, in which the processing time of a job is a function of its starting time and its convex resource allocation. The objective is to find the optimal sequence of jobs and the optimal convex resource allocation separately. This paper focus on the single-machine problems with objectives of minimizing a cost function containing makespan, total completion time, total absolute differences in completion times and total resource cost, and a cost function containing makespan, total waiting time, total absolute differences in waiting times and total resource cost. It shows that the problems remain polynomially solvable under the proposed model.     

A three-machine permutation flow-shop problem with minimum makespan on the second machine

In this paper, we consider a three-machine permutation flow-shop scheduling problem where the criterion is to minimize the total completion time without idle times subject to the minimum makespan on the second machine. This problem is NP-hard while each of the objective functions alone can be optimized in polynomial time. We develop a branch-and-bound algorithm with effective branching rules and dominance properties which help to reduce the search space. By our computational experiments, the branch-and-bound algorithm is comparable to a recent mixed integer programming solver and, for some kinds of problem instances, the branch-and-bound algorithm outperforms the solver. On the other hand, the computational result would indicate that the hierarchical scheduling problems are essentially hard in both theoretical and computational points of view.     

A fast tabu search algorithm for the permutation flow-shop problem

A fast and easily implementable approximation algorithm for the problem of finding a minimum makespan in the permutation flow shop is presented. The algorithm is based on a tabu search technique with a specific neighborhood definition which employs a block of jobs notion. Computational experiments (up to 500 jobs and 20 machines) show its excellent numerical properties.     

Single machine scheduling with total tardiness criterion and convex controllable processing times

We consider a single machine scheduling problem with total tardiness criteria and controllable job-processing times specified by a convex resource consumption function. The objective is to have the total tardiness limited into a given range, and minimize the total resource consumption. A polynomial time algorithm of O(n ²) is presented for the special case where jobs have a common due date.     

Single machine scheduling problem with controllable processing times and resource dependent release times

We consider two single machine scheduling problems with resource dependent release times and processing times, in which the release times and processing times are linearly decreasing functions of the amount of resources consumed. The objective is to minimize the total cost of makespan and resource consumption function that is composed of release time reduction and processing time reduction. In the first problem, the cost of reducing a unit release time for each job is common. We show that the problem can be solved in polynomial time. The second problem assumes different reduction costs of job release times. We show that the problem can be reduced polynomially from the partition problem and thus, is NP-complete.     

The just-in-time scheduling problem in a flow-shop scheduling system

We study the problem of maximizing the weighted number of just-in-time (JIT) jobs in a flow-shop scheduling system under four different scenarios. The first scenario is where the flow-shop includes only two machines and all the jobs have the same gain for being completed JIT. For this scenario, we provide an O(n³) time optimization algorithm which is faster than the best known algorithm in the literature. The second scenario is where the job processing times are machine-independent. For this scenario, the scheduling system is commonly referred to as a proportionate flow-shop. We show that in this case, the problem of maximizing the weighted number of JIT jobs is NP-hard in the ordinary sense for any arbitrary number of machines. Moreover, we provide a fully polynomial time approximation scheme (FPTAS) for its solution and a polynomial time algorithm to solve the special case for which all the jobs have the same gain for being completed JIT. The third scenario is where a set of identical jobs is to be produced for different customers. For this scenario, we provide an O(n³) time optimization algorithm which is independent of the number of machines. We also show that the time complexity can be reduced to O(n log n) if all the jobs have the same gain for being completed JIT. In the last scenario, we study the JIT scheduling problem on m machines with a no-wait restriction and provide an O(mn²) time optimization algorithm.     

Schedule execution for two-machine flow-shop with interval processing times

This paper addresses the issue of how to best execute the schedule in a two-phase scheduling decision framework by considering a two-machine flow-shop scheduling problem in which each uncertain processing time of a job on a machine may take any value between a lower and upper bound. The scheduling objective is to minimize the makespan. There are two phases in the scheduling process: the off-line phase (the schedule planning phase) and the on-line phase (the schedule execution phase). The information of the lower and upper bound for each uncertain processing time is available at the beginning of the off-line phase while the local information on the realization (the actual value) of each uncertain processing time is available once the corresponding operation (of a job on a machine) is completed. In the off-line phase, a scheduler prepares a minimal set of dominant schedules, which is derived based on a set of sufficient conditions for schedule domination that we develop in this paper. This set of dominant schedules enables a scheduler to quickly make an on-line scheduling decision whenever additional local information on realization of an uncertain processing time is available. This set of dominant schedules can also optimally cover all feasible realizations of the uncertain processing times in the sense that for any feasible realizations of the uncertain processing times there exists at least one schedule in this dominant set which is optimal. Our approach enables a scheduler to best execute a schedule and may end up with executing the schedule optimally in many instances according to our extensive computational experiments which are based on randomly generated data up to 1000 jobs. The algorithm for testing the set of sufficient conditions of schedule domination is not only theoretically appealing (i.e., polynomial in the number of jobs) but also empirically fast, as our extensive computational experiments indicate.     

A novel particle swarm optimization algorithm for permutation flow-shop scheduling to minimize makespan

It is well known that the flow-shop scheduling problem (FSSP) is a branch of production scheduling and is NP-hard. Now, many different approaches have been applied for permutation flow-shop scheduling to minimize makespan, but current algorithms even for moderate size problems cannot be solved to guarantee optimality. Some literatures searching PSO for continuous optimization problems are reported, but papers searching PSO for discrete scheduling problems are few. In this paper, according to the discrete characteristic of FSSP, a novel particle swarm optimization (NPSO) algorithm is presented and successfully applied to permutation flow-shop scheduling to minimize makespan. Computation experiments of seven representative instances (Taillard) based on practical data were made, and comparing the NPSO with standard GA, we obtain that the NPSO is clearly more efficacious than standard GA for FSSP to minimize makespan.     

The dominance digraph as a solution to the two-machine flow-shop problem with interval processing times

The flow-shop minimum-length scheduling problem with n jobs processed on two machines is addressed where processing times are uncertain: lower and upper bounds for the random processing time are given before scheduling, but its probability distribution between these bounds is unknown. For such a problem, there often does not exist a dominant schedule that remains optimal for all possible realizations of the job processing times, and we look for a minimal set of schedules that is dominant. Such a minimal dominant set of schedules may be represented by a dominance digraph. We investigate useful properties of such a digraph.     

A survey of scheduling with controllable processing times

In classical deterministic scheduling problems, the job processing times are assumed to be constant parameters. In many practical cases, however, processing times are controllable by allocating a resource (that may be continuous or discrete) to the job operations. In such cases, each processing time is a decision variable to be determined by the scheduler, who can take advantage of this flexibility to improve system performance. Since scheduling problems with controllable processing times are very interesting both from the practical and theoretical point of view, they have received a lot of attention from researchers over the last 25 years. This paper aims to give a unified framework for scheduling with controllable processing times by providing an up-to-date survey of the results in the field.