Unrelated Parallel Machines Research Articles

Bi-objective unrelated parallel machines scheduling problem with worker allocation and sequence dependent setup times considering machine eligibility and precedence constraints

<p style='text-indent:20px;'>In today's competitive world, scheduling problems are one of the most important and vital issues. In this study, a bi-objective unrelated parallel machine scheduling problem with worker allocation, sequence dependent setup times, precedence constraints, and machine eligibility is presented. The objective functions are to minimize the costs of tardiness and hiring workers. In order to formulate the proposed problem, a mixed-integer quadratic programming model is presented. A strategy called repair is also proposed to implement the precedence constraints. Because the problem is NP-hard, two metaheuristic algorithms, a multi-objective tabu search (MOTS) and a multi-objective simulated annealing (MOSA), are presented to tackle the problem. Furthermore, a hybrid metaheuristic algorithm is also developed. Finally, computational experiments are carried out to evaluate different test problems, and analysis of variance is done to compare the performance of the proposed algorithms. The results show that MOTS is doing better in terms of objective values and mean ideal distance (MID) metric, while the proposed hybrid algorithm outperforms in most cases, considering other employed comparison metrics.</p>

An Enhanced Adaptive Large Neighborhood Search for Unrelated Parallel Machine Scheduling With Sequence Dependent Setup Times

The unrelated parallel machine scheduling problem with sequence dependent setup times (UPMSP-SDST) addressed in this study refers to allocating jobs among a given number of machines and determining their processing sequence on each machine, to minimize the makespan (i.e., the maximum completion time). To deal with large-scale UPMSP-SDST with higher efficiency, this study presents an enhanced adaptive large neighborhood search (EALNS) algorithm with various destroy and repair operators and an efficiency-enhancement mechanism. The efficiency-enhancement mechanism is mainly composed of a simplified calculation method and a hierarchical comparison mechanism, which are applied to improve the implementation process of the greedy-based operators. The simplified calculation method obtains a new makespan by an incremental or decremental transformation, to avoid reluctant calculations. The hierarchical mechanism refines the comparison of different removal or insertion strategies of the operators, thereby arresting high-quality solutions with more metrics. The proposed algorithm is tested on 1640 instances, and numerical results demonstrate the superior performance of the EALNS over the existing methods, especially for large-scale problems. Notably, 834 instances ’ best-known solutions are updated from this study. In addition, deep analysis of the impact of the distribution of setup time on the performance of the algorithm is provided, which further verifies the potential wide applicability of the proposed EALNS.

Unrelated parallel machine scheduling with machine processing cost

In practical scheduling problems, some factors such as depreciation cost, green costs like the amount of energy consumption or carbon emission, other resources consumption, raw material cost, etc., are not explicitly related to the machine processing times. Most of these factors can be generally considered as machine costs. Considering the machine cost as another objective alongside the other classical time-driven decision objectives can be an attractive work in scheduling problems. However, this subject has not been discussed thoroughly in the literature for the case the machines have fixed processing costs. This paper investigates a general unrelated parallel machine scheduling problem with the machine processing cost. In this problem, it is assumed that processing a job on a machine incurs a particular cost in addition to processing time. The considered objectives are the makespan and the total cost, which are minimized simultaneously to obtain Pareto optimal solutions. The efficacy of the mathematical programming approach to solve the considered problem is evaluated rigorously in this paper. In this respect, a multiobjective solution procedure is proposed to generate a set of appropriate Pareto solutions for the decision-maker based on the mathematical programming approach. In this procedure, the ϵ-constraint method is first used to convert the bi-objective optimization problem into single-objective problems by transferring the makespan to the set of constraints. Then, the single-objective problems are solved using the CPLEX software. Moreover, some strategies are also used to reduce the solution time of the problem. At the end of the paper, comprehensive numerical experiments are conducted to evaluate the performance of the proposed multiobjective solution procedure. A vast range of problem sizes is selected for the test problems, up to 50 machines and 500 jobs. Furthermore, some rigorous analyses are performed to significantly restrict the patterns of generating processing time and cost parameters for the problem instances. The experimental results demonstrate the mathematical programming solution approach's efficacy in solving the problem. It is observed that even for large-scale problems, a diverse set of uniformly distributed Pareto solutions can be generated in a reasonable time with the gaps from the optimality less than 0.03 most of the time.

An exact decomposition method for unrelated parallel machine scheduling with order acceptance and setup times

In this paper, an unrelated parallel machine scheduling problem is studied, where order acceptance, sequence and machine-dependent setup times and the maximum available times of machines are additionally considered. The objective is to maximize the profit, which is the difference between the total revenues of accepted jobs and the cost associated with makespan. A mixed integer programming (MIP) model is formulated. To tackle this problem efficiently, an exact decomposition method, which is a two-layer logic-based Benders decomposition (LBBD) based (denoted by TL-LBBD) method, is developed, where an inner LBBD is embedded into an outer LBBD. Specifically, in the outer LBBD method, the master problem is used to determine the acceptance of jobs, whereas the subproblem examines the schedule given the accepted jobs from the master problem. The subproblem could be further decomposed into an assignment master problem and a sequencing subproblem by the inner LBBD method as well. Extensive computational experiments are conducted and the results show that the developed TL-LBBD method produce better quality solutions in significantly less computation time than solving the MIP model directly and the classic LBBD method. Moreover, the maximum scales of the problem instances that could be solved to optimality by the developed TL-LBBD method within 30 min are also evaluated.

Order acceptance and scheduling on unrelated parallel machines

This paper considers an order acceptance and scheduling problem on unrelated parallel machines. The objective is to select and schedule a subset of accepted orders that maximizes the total profit, which is defined as the sum of revenues of all the accepted orders minus their weighted tardiness penalties. We formulate the problem as a mixed-integer linear programming model. To solve this strongly NP-hard problem, we also propose a heuristic algorithm based on the Lagrangian relaxation technique and some greedy search operations.

An Improved Line-Up Competition Algorithm for Unrelated Parallel Machine Scheduling with Setup Times

It is well known that with the development of economic globalization and increasing competition in the market, enterprises are facing a huge challenge in the unrelated parallel machine scheduling problem with setup time (UPMST). Determining the processing order of all jobs and assigning machines to production scheduling has become more complex and has research implications. Moreover, a reasonable production scheduling scheme can not only complete the production plan efficiently but also contribute to reducing carbon emissions. In this paper, a mathematical model with the goal of the shortest completion time is studied for the UPMST problem. An improved line-up competition algorithm (ILCA) is proposed to solve this model, and the search accuracy and rate of the algorithm are improved by the newly proposed heuristic workpiece allocation rules and variation strategies. From the perspective of evaluation purposes, the effectiveness and stability of the method are significantly superior to other methods, and it is competitive in solving the UPMST problem.

Unrelated parallel machine energy-efficient scheduling considering sequence-dependent setup times and time-of-use electricity tariffs

Given that about half of the produced energy in the world is consumed in industries, there has been an increasing concern about optimizing energy consumption in manufacturing sectors. As one of the most effective ways, proper production scheduling to reduce energy consumption is of crucial importance among researchers and manufacturers. This paper addresses an unrelated parallel machine energy-efficient scheduling problem with sequence-dependent setup times by considering different energy consumption tariffs. The setup times are studied in two modes: disjointed from/jointed to processing time. For each one of these problems, two mixed-integer linear programming models have been formulated. The presented models for the problem with setup time disjointed from processing time can solve up to 16 machines and 45 jobs. In contrast, this capability is changed to 20 machines and 40 jobs for processing time jointed to the setup time problem. Furthermore, a fix and relax heuristic algorithm is presented for large-size instances, which can solve instances of up to 20 machines and 100 jobs for each of the two considered problems.

Unrelated Parallel Machine Scheduling with Job and Machine Acceptance and Renewable Resource Allocation

In this paper, an unrelated parallel machine scheduling problem with job (product) and machine acceptance and renewable resource constraints was considered. The main idea of this research was to establish a production facility without (or with minimum) investment in machinery, equipment, and location. This problem can be applied to many real problems. The objective was to maximize the net profit; that is, the total revenue minus the total cost, including fixed costs of jobs, job transportation costs, renting costs of machines, renting cost of resources, and transportation costs of resources. A mixed-integer linear programming (MILP) model and several heuristics (greedy, GRASP, and simulated annealing) are presented to solve the problem.

A hyper-heuristic for distributed parallel machine scheduling with machine-dependent processing and sequence-dependent setup times

Today, because the market is scattered around the world, manufacturing activities are not limited to a single location and have spread globally. As a result, the discussion of scheduling the factory has changed from a classic single to a network scheduling as a need in the real world. In this regard, this study considers the scheduling of multiple factories by taking into account the job transportation time between factories. The main problem here is that each job would be assigned to which factory and machine. In this research, unrelated parallel machines are considered in which the processing time of jobs depends on the machine and setup time. To minimize the makespan, first, a mixed-integer linear model was proposed in which two types of modeling have been combined. Then, a hyper-heuristic algorithm (HHA) was designed to solve the problem in a reasonable time by choosing the best method among four low-level heuristic methods that are precisely designed according to the properties of the problem. Finally, the efficiency of the proposed algorithm has been compared with the imperialist competitive algorithm (ICA) by conducting experiments. The results show that the proposed algorithm performs very well compared to the ICA and, in more than 75% of the test problems, the proposed algorithm was superior. Also, based on the analysis, in comparing the proposed algorithm with the ICA, it can be concluded that there is a significant difference between the results, and in all cases, the HHA was remarkably better. Considering the challenges and rapid changes of today’s market that traditional centralized production planning does not have enough flexibility to respond to them, the results of this research are expected to be useful and attractive for planners in this field.

Unrelated parallel machines scheduling with dependent setup times in textile industry

The problem tackled in this paper is an unrelated parallel machines scheduling problem with machine and sequence dependent setup times, machine eligibility and different resource types constraints. This subject is inspired from a textile industry partner case. The different characteristics of this problem have never been combined as it is depicted in this paper. Two resolution methods are exposed. A new mathematical programming is proposed to have an exact resolution approach of this complex problem. An improved genetic algorithm (GA) with dedicated operators is also presented in order to solve the industrial case in a reasonable computational time.

A parallel hybrid PSO-GA algorithm for the flexible flow-shop scheduling with transportation

In this paper, a Mixed-Integer Linear Programming (MILP) model to simultaneously schedule jobs and transporters in a flexible flow shop system is suggested. Wherein multiple jobs, finite transporters, and stages with parallel unrelated machines are considered. In addition to the mentioned technicalities, the jobs are able to omit one or more stages, and may not be executable by all the machines, and similarly, transportable by all the transporters. To the best of our knowledge, no study in the literature has featured efficacy of the parallel computing in simultaneous scheduling of jobs and transporters in the flexible flow shop system which remarkably shortens run time if the solution approaches are designed accordingly. To this end, we employ Gurobi solver, Parallel Genetic Algorithm (PGA), Parallel Particle Swarm Optimization (PPSO) and hybrid Parallel PSO-GA Algorithm (PPSOGA) to deal with the problem instances. Furthermore, a parallel version of Ant Colony Optimization (ACO) algorithm adapted from the state-of-the-art literature is developed to verify the performance of our suggested solution methods. Using 60 problem instances generated via uniform distribution, the suggested solution approaches are compared against one another. After assessing the results of the computational experiments, it is deduced that PPSOGA algorithm outperforms PGA, PPSO, Parallel Ant Colony Optimization (PACO) and Gurobi solver in terms of the quality of the solutions. The efficiency and run time of the suggested approaches are then assessed through two prominent statistical tests (i.e., Wald and Analysis of Variance (ANOVA)). Eventually, it comes to spotlight that PPSOGA algorithm is computationally rewarding and dependable.

GRASP algorithms for the unrelated parallel machines scheduling problem with additional resources during processing and setups

ABSTRACT This paper addresses an unrelated parallel machines scheduling problem with the need of additional resources during the processing of the jobs, as well as during the setups that machines need between the processing of any two jobs. This problem is highly complex, and therefore in this paper we propose several constructive heuristics to solve it. To improve the performance of these heuristics, we propose several variations, including randomisation with different probability distributions and a local search phase, having this way GRASP algorithms. The results of extensive experiments over randomly generated instances show several findings on the different parameters that characterise our constructive algorithms. In particular, we highlight the fact that non-uniform probability distributions might be advisable for choosing elements of a restricted candidate list in GRASP algorithms.

Preemptive and non-preemptive scheduling on two unrelated parallel machines

Preemptive and non-preemptive scheduling on two unrelated parallel machines

Unrelated parallel machine scheduling with eligibility constraints and delivery times to minimize total weighted tardiness

This paper addresses an unrelated parallel machine scheduling problem with job-machine-dependent delivery times and eligibility constraints motivated by a distributed manufacturing environment. The objective is total weighted tardiness minimization. We present a mixed integer linear programming (MILP) formulation and conduct an analysis of the scheduling problem to derive precedence properties that can be integrated in local search procedures to increase computational efficiency. We implement a variable neighborhood search (VNS) algorithm for the problem at hand and examine the effect of integrating the properties on the performance. The tests show that the theoretical findings can reduce computational effort significantly. Furthermore, we propose another heuristic approach based on the Apparent Tardiness Cost rule and a memetic biased random-key genetic algorithm. In experiments, we compare the MILP and (meta-)heuristics on a large set of randomly generated problem instances. The VNS procedure outperforms the other algorithms.

Energy-efficient unrelated parallel machine scheduling with general position-based deterioration

ABSTRACT This paper investigates an energy-efficient scheduling problem on unrelated parallel machines considering general position-based deterioration which arises from the labour-intensive textile industry. The actual processing time of a job is not only associated with the job and the machine but also with its position in the processing sequence. The objective is to minimise the total energy consumption with a bounded makespan. To address this problem, we first establish a mixed-integer linear programming (MILP) model. Afterwards, the initial model is improved by deriving lower and upper bounds on the makespan, and an upper bound on the number of jobs processed on each machine. We also develop an iterative heuristic embedded with a variable neighbourhood search procedure (IHVNS). The algorithm obtains initial solutions iteratively by solving assignment problems and then repairs and improves them with the VNS procedure. Computational results demonstrate that the improved model is up to 230 times faster than the original one. Moreover, the proposed heuristic yields excellent solutions with average gaps of less than 0.73% for large-scale instances. Especially, the results reveal that the IHVNS algorithm is more suitable than MILP models for solving large-scale problems with tight makespan restrictions.

A Pareto-based discrete particle swarm optimization for parallel casting workshop scheduling problem with fuzzy processing time

The parallel casting workshop scheduling problem is essentially a type of unrelated parallel machine scheduling (UPMS) problem with fuzzy order processing time. In this paper, a tri-objective parallel casting workshop optimization model is investigated to minimize the order advance/delay penalty, makespan, and workload imbalance, and a formula of the solution scale with the number of orders and workshops is derived. In line with the multi-objective integer programming model, a new Pareto-based discrete particle swarm optimization (PDPSO) is proposed. In this algorithm, the particles are encoded with integers. A novel two-phase heuristic is developed for integer-encoded particle updates. The first phase determines the order assignment through a weighting vote, and the second phase sorts the orders according to their dominance in each workshop. Every particle’s update is simultaneously guided by the best compromise solution selected from the personal and global Pareto solution. Meanwhile, the Pareto simulated annealing strategy is introduced in PDPSO to enhance its exploration ability. Multiple simulations under different scales are tested to validate the effectiveness of our proposed model and algorithm. The results show that the proposed PDPSO outperforms the other two multi-objective algorithms and has large advantages in finding the Pareto frontier (PF) with an acceptable solution number and solving speed, especially for small and medium-sized instances.

Unrelated parallel machine scheduling with multiple time windows: An application to earth observation satellite scheduling

In this paper, we investigate a variant of the unrelated parallel machine scheduling problem with multiple time windows that is widely employed in the scheduling of earth observation satellites. Two mixed-integer linear programming formulations are proposed based on the concept of non-immediate precedence and the deep analysis of the relationship of time windows. To facilitate the problem solving, we develop a bidirectional rolling horizon preprocessing algorithm to reduce the problem size. Based on detailed overlapping analysis, more inequalities are obtained to tighten the formulations. Furthermore, the dominance among derived inequalities is analyzed, and only nondominated inequalities are appended to limit the size of the problem. The proposed formulation and algorithm can optimally solve instances of up to 800 jobs and 20 machines within 20 min. Moreover, to verify the practicality of our method, we also conducted numerous experiments on realistic instances of satellite scheduling. The experimental results prove that our algorithm is effective for obtaining optimal solutions and can be applied in practice.

Heuristic and metaheuristic methods for the parallel unrelated machines scheduling problem: a survey

Heuristic and metaheuristic methods for the parallel unrelated machines scheduling problem: a survey

Bi-objective unrelated parallel machine joint scheduling of jobs and preventive maintenance with a dynamic speed-scaling technique

Nowadays, industries contribute to a substantial consumption of energy. This consumption leads to significant greenhouse gas emissions, energy dependence and ever-increasing costs for manufacturers. Controlling consumption and improving energy efficiency in industry are therefore major challenges in terms of environmental issues. In this context, this article examines an energy-efficient bi-objective unrelated parallel machine joint scheduling of jobs and preventive maintenance activities problem to minimize both makespan and total energy consumption. The parallel machines are speed-scaling. To solve the problem, we propose a bi-objective mixed integer linear programming (MILP) model. Since the problem involves assigning jobs to machines and selecting an appropriate processing speed level for each job, we characterize each individual by two vectors: a job-machine assignment vector and a speed vector. To the best of our knowledge, no papers considering both production scheduling and Preventive Maintenance periods with minimizing the bi-objective makespan and the total of energy consumption. The gap between theory and practice prevents the satisfying performance of the schedules in industry. The purpose of this paper is to establish a joint model for integrating the production and PM scheduling to optimize the bi-objective of the total of energy consumption (TEC) and makespan (Cmax) simultaneously; and to make the theory applied in practice more efficaciously. The performance of the proposed mixed binary integer programming model is evaluated based on the exact method of Branch and Bound integrated on the CPLEX solver plus analyses using MATLAB software. A study of the results proved the performance of the model developed.

Unrelated parallel machine scheduling problem with special controllable processing times and setups

The controllable processing times (CPTs) have many practical applications, enabling the length of the job processing to vary within an interval flexibly with additional costs. In this paper, we study an unrelated parallel machine scheduling problem with machine- and sequence-dependent setup times and a special case of CPTs without extra costs. The objective is to maximize the difference between the sum of realized state of processing of all jobs and makespan. To solve the problem, a mixed integer programming (MIP) model is formulated first and then a logic-based Benders decomposition (LBBD) method is developed, in which the master problem is for job assignments and the subproblem is furthered decomposed into a sequencing problem for minimum total setup times on each machine and a processing time determination problem. Several LBBD-based heuristics are also employed by imposing different optimality gaps for the master problem. The performance of the MIP formulation, the exact LBBD method and the LBBD-based heuristics is compared through extensive computational experiments. The results demonstrate that the exact LBBD method with the preprocessing procedure which generates initial feasible solutions and cuts is effective and the proposed LBBD-based heuristics reduce the computation time significantly at the expense of slightly solution quality reduction.