Decorous combinatorial lower bounds for row layout problems

A multi-population genetic algorithm to solve multi-objective scheduling problems for parallel machines

• Examines scheduling on m identical parallel machines with a common server. • Considers sequence-dependent setup times in parallel machine scheduling. • Introduces an arc-based MILP formulation for makespan minimization. • Presents a GRASP-enhanced Iterated Greedy algorithm with strong performance. • Reports 21 new best-known solutions across benchmark test instances. The identical parallel machine scheduling problem with a single server and sequence-dependent setup times is a challenging optimization problem with important applications in manufacturing and service industries. In such environments, several machines depend on a common server to perform setup operations before production can begin, which creates strong interdependencies and demands more effective scheduling strategies. This characteristic highlights the practical relevance of the problem. The interaction between machine availability and server operations often becomes a critical bottleneck. This study introduces two complementary approaches. The first is an exact method based on a novel arc-based mixed-integer linear programming (ABF) model, which extends the modeling capability of existing formulations by capturing server-related constraints more effectively. The second is an approximation method built on an Iterated Greedy (IG) algorithm. The IG procedure is improved by two evaluation mechanisms: one model-based evaluation derived from the proposed ABF model, and another employing a greedy randomized adaptive search procedure (GRASP)-based strategy that integrates greedy selection, randomization, and reconstruction to enhance solution quality. Computational experiments are conducted on existing benchmark instances. The results show that the proposed ABF model performs well on small and medium-sized instances compared to existing exact methods, while the IG variants, particularly the proposed GRASP-based version, deliver strong performance against state-of-the-art metaheuristics developed for this problem. In addition, 21 new best-known solutions are reported, further demonstrating the effectiveness of the proposed approaches.

In this paper, we consider a parallel machine scheduling problem with non-renewable resources. Each job consumes several components and must be processed in one stage composed of identical parallel machines. Resources availability operations, jobs assignment and sequencing are considered and optimized simultaneously. In order to find an optimal solution, an exact method is applied to optimize the total completion time. Due to the problem complexity and prohibitive computational time to obtain an exact solution, a metaheuristic approach based genetic algorithm is proposed and several heuristics are adapted to solve it. Moreover, the impact of non-renewable resources procurement methods on production scheduling is analyzed. The system performances are evaluated in terms of measures such as the solution quality and the execution time. The simulation results show that the proposed genetic algorithm gives the same results as the exact method for small instances and performs the best compared to heuristics for medium and large instances.

This paper addresses the problem of makespan minimization on unrelated parallel machines with sequence dependent setup times. The symbiotic organisms search (SOS) algorithm is a new and popular global optimization technique that has received wide acceptance in recent years from researchers in continuous and discrete optimization domains. An improved SOS algorithm is developed to solve the parallel machine scheduling problem. Since the standard SOS algorithm was originally developed to solve continuous optimization problems, a new solution representation and decoding procedure is designed to make the SOS algorithm suitable for the unrelated parallel machine scheduling problem (UPMSP). Similarly, to enhance the solution quality of the SOS algorithm, an iterated local search strategy based on combining variable numbers of insertion and swap moves is incorporated into the SOS algorithm. More so, to further improve the SOS optimization speed and performance, the longest processing time first (LPT) rule is used to design a machine assignment heuristic that assigns processing machines to jobs based on the machine dynamic load-balancing mechanism. Subsequently, the machine assignment scheme is incorporated into SOS algorithms and used to solve the UPMSP. The performances of the proposed methods are evaluated by comparing their solutions with other existing techniques from the literature. A number of statistical tests were also conducted to determine the variations in performance for each of the techniques. The experimental results showed that the SOS with LPT (SOS-LPT) heuristic has the best performance compared to other tested method, which is closely followed by SOS algorithm, indicating that the two proposed algorithms’ solution approaches are reasonable and effective for solving large-scale UPMSPs.

SYNOPTIC ABSTRACTIn this article, we consider a parallel machine scheduling problem that integrates partial maintenance activities due to job-dependent machine deteriorations. In this problem, we are given identical parallel machines, each with an initial maintenance level which is a non-negative integer value to represent its maintenance state, and a set of jobs each with a non-preemptive processing time and a machine deterioration. For each job, its machine deterioration qualifies the decrement of the machine maintenance level after processing it. When each machine processes jobs, one should guarantee a non-negative maintenance level at any time point. Thus, whenever necessary, a maintenance activity must be performed to restore the machine maintenance level. The goal of this problem is to schedule the jobs and the maintenance activities, such that the makespan achieves the minimum. We take the partial maintenance activity pattern, in which each activity can be performed to increase the machine maintenance level to any level not exceeding the maximum. To tackle this problem, we first model it by an integer programming formulation, and then propose an LP-based approximation algorithm with an instance-dependent approximation factor and show the approximability lower bound. Finally, we present a fast heuristic algorithm followed by a numerical example to show how this algorithm works.

This paper addresses the problem of scheduling n jobs on a single machine and on m identical parallel machines to minimize the completion time variance of jobs. This problem of scheduling jobs on parallel machines is motivated by a case study in an automobile ancillary unit. First, a heuristic to solve the single-machine scheduling problem is proposed. The parallel-machine scheduling problem is solved in two phases: job-allocation phase and job-sequencing phase. Two heuristics are proposed in the job-allocation phase, whereas in the job-scheduling phase, the single-machine scheduling approach is used. In this paper, both versions of parallel-machine scheduling problem (restricted and unrestricted) are considered. A good upper bound is obtained using a genetic algorithm, to evaluate the performance of the proposed heuristics for the parallel-machine scheduling problem. An extensive computation evaluation of the proposed heuristics is presented for both single-machine scheduling problem and the parallel-machine scheduling problem (especially considering the case study), along with the comparison of performances with the existing heuristics in the literature.

This research deals with the problem of scheduling N jobs on M unrelated parallel machines. Each job has a due date and requires a single operation. A setup that includes detaching one die and attaching another from the appropriate die type is incurred if the type of the job scheduled is different from the last job on that machine. Due to the mechanical structure of machines and the fitness of dies to each, the processing time for a job depends on the machine on which the job is processed, and some jobs are restricted to be processed on certain machines. Furthermore, the required detaching and attaching times depend on both the die type and the machine. This type of problems may be encountered, for example, in plastics forming industry where unrelated parallel machines are used to process different components and setups for auxiliary equipment (e.g., dies) are necessary. This type of scheduling problems is also frequently encountered in injection molding departments where many different parallel machines are also used to produce different components and for which setups are required for attaching or detaching molds. In general, the dies (or molds) are quite expensive (tens of thousands dollars each) and thus the number of each type of dies available is limited. Therefore, dies should be considered as secondary resources, the fact of which distinguishes this research from many past studies in unrelated parallel-machine scheduling in which secondary resources are not restricted. This type of problems is NP-hard (So, 1990). When dealing with a large instance encountered in industry, in the worst case, it may not be able to obtain an optimal solution in a reasonable time. In this research heuristics based on guided search, record-to-record travel, and tabu lists from the tabu search (TS) are presented to minimize the maximum completion time (i.e., makespan or Cmax) and maximum tardiness (i.e., Tmax), respectively, to promote schedule performance. Computational characteristics of the proposed heuristics are evaluated through extensive experiments. The rest of this research is organized in six sections. Previously related studies on parallel machine scheduling are reviewed in Section 2. The record-to-record travel and tabu lists are briefly described in Section 3. The proposed heuristic to minimize makespan and the computational results are reported in Section 4. The proposed heuristic to minimize maximum tardiness and the computational results are reported in Section 5. Conclusions and suggestions for future research are discussed in Section 6. O pe n A cc es s D at ab as e w w w .ite ch on lin e. co m

We consider the parallel-machine flexible-resource scheduling (PMFRS) problem in which a set of jobs must be scheduled over a set of parallel machines, where the processing time of each job is a function of the amount of allocated resource. Resource flexibility provides the capability to dynamically reassign a renewable resource across machines to break processing bottlenecks and improve system performance as measured by schedule makespan. The PMFRS problem has many important applications, including production scheduling of manufacturing cells where a cross-trained work force can be dynamically reallocated among cells. The problem is also NP-hard, motivating the development of effective heuristics that approximately determine the allocation of resource to jobs, the sequence of jobs on each machine, and the associated job start times that minimize system makespan. This paper explores heuristics for the PMFRS problem, and in particular the application of tabu-search methodology to this problem setting. We review an existing heuristic (SBH), define two tabu-search heuristics, and discuss extensive computational experience with the procedures. The computational results indicate that the heuristics are effective in obtaining approximate solutions to the PMFRS problem. In particular, the approach that uses tabu-search methodology in tandem with SBH consistently yields high-quality solutions with modest computational effort.

Tuning stochastic local search algorithms for tackling large instances is difficult due to the large amount of CPU-time that testing algorithm configurations requires on such large instances. We define an experimental protocol that allows tuning an algorithm on small tuning instances and extrapolating from the obtained configurations a parameter setting that is suited for tackling large instances. The key element of our experimental protocol is that both the algorithm parameters that need to be scaled to large instances and the stopping time that is employed for the tuning instances are treated as free parameters. The scaling law of parameter values, and the computation time limits on the small instances are then derived through the minimization of a loss function. As a proof of concept, we tune an iterated local search algorithm and a robust tabu search algorithm for the quadratic assignment problem.

A vibration damping optimization algorithm for a parallel machines scheduling problem with sequence-independent family setup times

The flexible open shop scheduling problem is the combination of two classical open shop and a parallel machine scheduling problems. Due to the computational complexity of solving the open shop scheduling problem, so far, limited research has considered the assumption of flexibility in this problem. In this study, a bi-objective flexible open shop scheduling with independent setup time is considered. The considered objective functions are to minimize the maximum completion time of jobs and total tardiness. Using the first objective function, an attempt is made to balance the amount of workloads on the machines so that the completion time of the last job is minimized, while using the second one, from the customers' point of view, an attempt is made to reduce the amount of tardiness in delivery of jobs to the customers. Here, firstly, a mixed-integer nonlinear programming model is proposed. Then, to deal with the multi-objective problem, the weighted Lp-metric method is used. Since, this problem is a non-deterministic polynomial-time hard problem (NP-hard), a scatter search algorithm is developed to obtain near-optimal solutions in a reasonable time. In this regard, for the first time, a novel solution representation method is also presented. To ensure the efficiency of the proposed algorithm, the results were compared with NSGA-II using several instances. After analyzing the results using four evaluation metrics, it is observed that the scatter search algorithm performs better than the NSGA-II.

In this paper, a parallel machine scheduling problem was considered, where the processing time of a job is a simple linear function of its starting time. The objective is to minimize makespan. A fully polynomial time approximation scheme for the problem of scheduling n deteriorating jobs on two identical machines was worked out. Furthermore, the result was generalized to the case of a fixed number of machines.

This paper introduces the parallel machine scheduling problems with work related musculoskeletal disorder (WMSD) risk considerations. By the WMSD consideration, we mean that job processing times are an increasing function of occupational repetitive technical actions (OCRA) risk factor. The OCRA index was recruited for risk assessment of occupational repetitive technical actions. WMSD risks were modeled with cumulative mean value of OCRA index. Due to NP-Hard structure of parallel machine scheduling problems with WMSD considerations and learning rate, the solution cannot be found always. However, problem can be solved by transforming assignment problem. In spite of the fact that the computational effort remains O (n 4 ), problem is solved within more efficient time. In this study, a model that includes learning effect and WMSD risk was proposed. WMSD risk was considered as cumulative mean of risk value throughout shift. In order to the balance between productivity and WMSD risk, jobs’ foreseeable cycle time (FCT) value was changed. It is aimed to decrease mean of risk along schedule without increasing total basic process time. The value of sacrifice from the FCT was compensating from job which have optimal, acceptable, borderline or slight risk level. Process time and risk values were recalculated by using new FCT value. Thus, balanced process times and risk values obtained. Proposed model was solved with Lingo software and sequence of jobs was obtained. Total flow time and mean of risk were compared for balanced and none balanced schedules. Total flow time and mean of risk belong to balanced schedule is smaller than none balanced schedule. It was shown that handled problem is solvable at the polynomial time and total flow time can be improved by bringing balance between WMSD risks and productivity.

Improved antlion optimization algorithm via tournament selection and its application to parallel machine scheduling

Identical parallel machine scheduling problem for minimizing the makespan is a very important production scheduling problem which has been proven to be NP-hard. The problem further compounds with additional constraints. Genetic algorithms (GA) have shown great advantages in solving the combinatorial optimization problem in view of its characteristic that has high efficiency and that is fit for practical application. In this chapter we present a spreadsheet based GA approach for minimizing the makespan for scheduling of a set of tasks for identical parallel machines and worker assignment to machines. The results obtained from the proposed approach are compared with two sets of benchmark problems consisting of 100 problems each. It has been demonstrated that the performance of proposed approach is superior to the results that have been obtained earlier. The proposed approach produces optimal solution for almost 95% of the problems demonstrating the effectiveness of the proposed approach. An empirical analysis of GA parameters has also been carried out to see the effect on the performance of the proposed algorithm.