Multi-machine Scheduling Problem Research Articles

A cutting plane approach for the multi-machine precedence-constrained scheduling problem

A cutting-plane approach is developed for the problem of optimally scheduling jobs with arbitrary precedence constraints on unrelated parallel machines to minimize weighted completion time. While the single machine version of this problem has attracted much research efforts, enabling solving problems with up to 100 jobs, not much has been done on the multiple machines case. A novel mixed-integer programming model is presented for the problem with multiple machines. For this model, many classes of valid inequalities that cut off fractional linear programming solutions are developed. This leads to an increase of the linear programming lower bound from 89.3 to 94.6% of the corresponding optimal solution, and a substantial reduction in the computational time of an optimal branch-and-bound algorithm for this problem. This enables us to report optimal solutions for problem instances with up to 25 jobs and 5 machines, which is more than twice the size of problems for which optimal solutions have been reported in the literature thus far. For a special case of the problem—that of minimizing makespan—application of our model helps solve 18 of 27 previously unsolved problem instances to optimality.

Pareto-based Multi-criteria Evolutionary Algorithm for a Parallel Machines Scheduling Problem with Sequence-dependent Setup Times

This paper addresses an unrelated multi-machine scheduling problem with sequence-dependent setup time, release date and processing set restriction to minimize the sum of weighted earliness/tardiness penalties and the sum of completion times, which is known to be NP-hard. A Mixed Integer Programming (MIP) model is proposed to formulate the considered multi-criteria problem. Also, to solve the model for real-sized applications, a Pareto-based algorithm, namely controlled elitism non-dominated sorting genetic algorithm (CENSGA), is proposed. To validate its performance, the algorithm is examined under six performance metric measures, and compared with a Pareto-based algorithm, namely NSGA-II. The results are statistically evaluated by the Mann–Whitney test and t-test methods. From the obtained results based on the t-test, the proposed CENSGA significantly outperforms the NSGA-II in four out of six terms. Additionally, the statistical results from Mann–Whitney test show that the performance of the proposed CENSGA is better than the NSGA- II in two out of six terms. Finally, the experimental results indicate the effectiveness of the proposed algorithm for different problems.

Multi-machine scheduling with interval constrained position-dependent processing times

This paper investigates multi-machine scheduling problems with interval constrained actual processing times. The actual processing time of each job is assumed to be restricted in a given interval otherwise the extra earliness or tardiness time should be used to patch up the flaw of job. The objectives are to find the optimal job sequence to minimize the total load of machines, the number of exceeding-interval jobs and the makespan of job schedule, respectively. This paper shows that both of the total load minimization problem and the exceeding job number minimization problem are polynomially solvable. For the makespan minimization problem, this paper proves that it is NP-hard, and proposes a fully polynomial time approximation scheme (FPTAS) for the case with two parallel machines.

Multi-machine scheduling with general position-based deterioration to minimize total load revisited

In this paper, we investigate multi-machine scheduling problems with general position-dependent effects where the actual processing time functions of jobs are not only associated with the job's position in the sequence but also associated with the machine number and the job number. All the processing time functions of jobs are not restricted to any specific form. The objective is to minimize the total load on machines. Analytical results show that all the considered multi-machine scheduling problems can be optimally solved in polynomial time.

Using heuristic algorithms to solve the scheduling problems with job-dependent and machine-dependent learning effects

The multi-machine scheduling problems with job-dependent and machine-dependent learning effects are proposed in this paper. Since it is almost impossible to obtain the analytic results for this complicated multi-machine scheduling problems with learning effects, four heuristic algorithms are used to solve this newly proposed model, where the variants of well-known genetic algorithm (GA), simulated annealing (SA), ant colony optimization (ACO) and particle swarm optimization (PSO) are coded in the commercial software MATLAB. The objective is to minimize the makespan of this new model. For this kind of scheduling problem, the numerical experiments show that the GA and SA outperform ACO and PSO.

A Multi-Machine Scheduling Problem with Global Non Idling Constraint

We deal here with a multi-machine scheduling problem with non idling constraints, i.e constraints which forbids the interruption of the use of the processors. We reformulate the problem, while using specific pyramidal profile functions, next get a min-max feasibility criterion and finally derive exact polynomial algorithms for both the feasibility problem and the makespan optimization problem.

Multi-machine scheduling problem with setup times

In this paper we consider a multi-machine scheduling problem with setup times, which is determined in the literature as the flexible job shop problem. It belongs to the strongly NPcomplete complexity class.We propose an algorithm based on the tabu search method. The new elimination criteria were used in the construction process of blocks of the critical path

Evaluation of Feedback among Multiple Scheduler Profiles in Fuzzy Genetic Scheduling

This paper extends the earlier studies conducted on multiple scheduler profile in fuzzy genetic scheduling. Multiple schedulers can set up individual fuzzy membership bounds which results in different evaluation of multi-objective problem of single machine scheduling. A new software application enables feedback among schedulers by applying seeding of individual scheduler's population by best chromosomes from other scheduler's population. Few experiments are performed on the aforementioned software application to evaluate the performance of the multi objective single machine scheduling problem by varying the level and frequency of feedback. More improvement is observed as the frequency of the feedback is increased but no significant improvement is observed when the level is increased.

Some Research on Non-Identical Parallel Machine Scheduling Problem - An Example to Plane De-Icing Operation

Under the context of deicing in the airport, and in order to solve a kind of Non-identical Parallel Multi-machine Schedule Problem for minimizing the number of passengers delayed, an ant colony algorithm was proposed. Considering the characteristic of scheduling model and its constraints, an improved pheromone update strategy was developed by deploying different parameters. Simulation was made for proving the effectiveness of the algorithm, and the sensitivity analysis was conducted. Besides, the improved ant colony algorithm was compared further with FIFO algorithm used in realistic backgrounds. In all, the result indicated that the improving ant colony algorithm was valid, and it can fit for large-scale non-identical parallel Multi-machine scheduling problem.

Estimations of an absolute error and the scheme of the approached solution problems of the scheduling theory

For single and multi-machine scheduling problems with the criterion of minimization maximum lateness the metrics ρ has been used for the first time. A theorem of estimating the absolute error has been proved. The idea of the offered approach consists in construction by an initial instance of a problem of other instance for which it is possible to find the optimum or approximated solution, with the minimal distance up to an initial instance in entered metric.

Efficient algorithms for machine scheduling problems with earliness and tardiness penalties

In this paper, we study the multi-machine scheduling problem with earliness and tardiness penalties and sequence dependent setup times. This problem can be decomposed into two subproblems—sequencing and timetabling. Sequencing focuses on assigning each job to a fixed machine and determine the job sequence on each machine. We call such assignment a semi-schedule. Timetabling focuses on finding an executable schedule from the semi-schedule via idle-time insertion. Sequencing is strongly NP-hard in general. Although timetabling is polynomial-time solvable, it can become a computational bottleneck if the procedure is executed many times within a larger framework. This paper makes two contributions. We first propose a quantum improvement to the computational efficiency of the timetabling algorithm. We then apply it within a squeaky wheel optimization framework to solve the sequencing and overall problem. Finally, we demonstrate the strength of our proposed algorithms by experiments.

Multi-machine scheduling with deteriorating jobs and scheduled maintenance

In this paper, we investigate a multi-machine scheduling problem in which job processing times are increasing functions of their starting times and machines are not always available. Job processing times are assumed to follow simple linear deteriorations. Moreover, each machine is assumed to have a maintenance period which is known in advance. Both the resumable and non-resumable cases are discussed with the objective of minimizing the makespan. A lower bound and a heuristic algorithm are derived for each case. Numerical results are also provided to evaluate the efficiency of the proposed procedures.

Scheduling with common due date, earliness and tardiness penalties for multimachine problems: A survey

The just-in-time production philosophy has led to a growing interest in schedulingproblems considering both earliness and tardiness penalties. Most publications considering these so-called nonregular objective functions are devoted to single machine problems. In the case of multimachine problems, there are some papers dealing with parallel machine problems. However, for multi-operation scheduling problems such as shop scheduling problems, investigations have begun only recently. In this paper, inspired by recent own work (partly yet unpublished), we survey some results on multimachine scheduling problems with a given common due date, where the focus is on possible approaches for shop scheduling problems.

A note on the SPT heuristic for solving scheduling problems with generalized due dates

Two NP-hard scheduling problems on parallel identical machines with generalized due dates are studied. We focus on two objectives: (i) minimizing maximum tardiness and (ii) minimizing total tardiness. In both cases, we introduce a shortest processing time first (SPT)-based heuristic and simple lower bounds on the optimal cost. Our numerical study indicates that the SPT heuristic performs extremely well in all settings. Scope and purpose In traditional scheduling literature, due dates are job dependent, i.e., a specific job has to be completed by a given due date. When generalized due dates are considered, a certain number of jobs must be completed by each specified due date. The identity of the jobs is immaterial. In this note, we address two hard multi-machine scheduling problems with generalized due dates. We introduce a simple and efficient heuristic, which is shown numerically to produce very close to optimal schedules.

Minimizing makespan subject to minimum flowtime on two identical parallel machines

We consider the problem of scheduling jobs on two parallel identical machines where an optimal schedule is defined as one that gives the smallest makespan (the completion time of the last job) among the set of schedules with optimal total flowtime (the sum of the completion times of all jobs). We propose an algorithm to determine optimal schedules for the problem, and describe a modified multifit algorithm to find an approximate solution to the problem in polynomial computational time. Results of a computational study to compare the performance of the proposed algorithms with a known heuristic shows that the proposed heuristic and optimization algorithms are quite effective and efficient in solving the problem. Scope and purpose Multiple objective optimization problems are quite common in practice. However, while solving scheduling problems, optimization algorithms often consider only a single objective function. Consideration of multiple objectives makes even the simplest multi-machine scheduling problems NP-hard. Therefore, enumerative optimization techniques and heuristic solution procedures are required to solve multi-objective scheduling problems. This paper illustrates the development of an optimization algorithm and polynomially bounded heuristic solution procedures for the scheduling jobs on two identical parallel machines to hierarchically minimize the makespan subject to the optimality of the total flowtime.

Minimizing the sum of earliness/tardiness in multi-machine scheduling: a mixed integer programming approach

We developed a mixed integer programming formulation for minimizing job earliness and tardiness (ET) in a multi-machine scheduling problem. Our formulation allows setup times to depend on the job-to-job sequence, processing times to depend on the job–machine combination, due dates and cost penalties to vary for each job, and machine characteristics to be non-uniform. The model can easily provide the optimal solution to problems involving about nine jobs and three machines. Thus, the formulation shown here can provide the optimal solutions necessary for the development and validation of industrial-scale, multi-machine ET heuristics, which to date, have only been tested against other heuristics.

Minimizing the sum of job earliness and tardiness in a multimachine system

We provide a heuristic algorithm for minimizing job earliness and tardiness cost in a multimachine scheduling problem. Our formulation explicitly separates machine setup times from processing times, assumes that setup times may depend on the job-to-job sequence, and assumes that processing times may depend on the job-machine combination. The new algorithm can easily provide approximate solutions to problems involving about 10 machines and 100 jobs when using a personal computer. The accuracy of the solutions was measured using a large variety of problem types taken from the literature. Relative accuracy was measured by comparing the new algorithm's results with those from other heuristics, and absolute accuracy was measured by comparing with results obtained using integer programming methods. The new algorithm was significantly more accurate than the other heuristics, and its average deviation was only about 10% when compared to the optimal solutions. In many cases the new algorithm yielded the optimal solution.

Heuristics for multimachine minmax scheduling problems with general earliness and tardiness costs

We consider the problem of scheduling N jobs on M parallel machines so as to minimize the maximum earliness or tardiness cost incurred for each of the jobs. Earliness and tardiness costs are given by general (but job-independent) functions of the amount of time a job is completed prior to or after a common due date. We show that in problems with a nonrestrictive due date, the problem decomposes into two parts. Each of the M longest jobs is assigned to a different machine, and all other jobs are assigned to the machines so as to minimize their makespan. With these assignments, the individual scheduling problems for each of the machines are simple to solve. We demonstrate that several simple heuristics of low complexity, based on this characterization, are asymptotically optimal under mild probabilistic conditions. We develop attractive worst-case bounds for them. We also develop a simple closed-form lower bound for the minimum cost value. The bound is asymptotically accurate under the same probabilistic conditions. In the case where the due date is restrictive, the problem is more complex only in the sense that the set of initial jobs on the machines is not easily characterized. However, we extend our heuristics and lower bounds to this general case as well. Numerical studies exhibit that these heuristics perform excellently even for small- or moderate-size problems both in the restrictive and nonrestrictive due-date case. © 1997 John Wiley & Sons, Inc.

Heuristics for multimachine minmax scheduling problems with general earliness and tardiness costs

Heuristics for multimachine minmax scheduling problems with general earliness and tardiness costs

Heuristics for Multimachine Scheduling Problems with Earliness and Tardiness Costs

We consider multimachine scheduling problems with earliness and tardiness costs. We first, analyze problems in which the cost of a job is given by a general nondecreasing, convex function F of the absolute deviation of its completion time from a (common) unrestrictive due-date, and the objective is to minimize the sum of the costs incurred for all N jobs. (A special case to which considerable attention is given is the completion time variance problem.) We derive an easily computable lower bound for the minimum cost value and a simple “Alternating Schedule” heuristic, both of which are computable in O(N log N) time. Under mild technical conditions with respect to F, we show that the worst case optimality (accuracy) gap of the heuristic (lower bound) is bounded by a constant as well as by a simple function of a single measure of the dispersion among the processing times. We also show that the heuristic (bound) is asymptotically optimal (accurate) and characterize the convergence rate as O(N−2) under very general conditions with respect to the function F. In addition, we report on a numerical study showing that the average gap is less than 1% even for problems with 30 jobs, and that it falls below 0.1% for problems with 90 or more jobs. This study also establishes that the empirical gap is almost perfectly proportional with N−2, as verified by a regression analysis. Finally, we generalize the heuristic to settings with a possibly restrictive due date and general asymmetric, and possibly nonconvex, earliness and tardiness cost functions and demonstrate its excellent performance via a second numerical study.