Unrelated Parallel Machine Scheduling Research Articles

Unrelated parallel machine scheduling with dedicated machines and common deadline

This paper addresses a scheduling problem in the manufacturing of Polyvinyl Chloride pipes. There are two main attributes of PVC pipes: diameter and color. Each attribute has a corresponding attribute setup time and usually has several different levels. Each extruder produces different PVC pipe products based on the diameters as large, middle and small. The alternatives exist between these extruders, where the large and the middle type extruders can be used to produce the PVC pipes with the other diameters; the small type extruders can be used to produce the PVC pipes with middle diameters but cannot produce those with large diameters. The processing times are longer in all of the alternatives among different types of extruders. The objective is to minimize the total completion time for the unrelated parallel machine problem.Three dedicated machine heuristics are proposed herein for the problem and have been evaluated by comparing with the current scheduling method used in the case plant. The computational results show that the proposed constructive heuristics outperform the current scheduling method with significant improvements and can be used to solve large-size problems in reasonable computational times.

Fuzzy bi-objective formulation for a parallel machine scheduling problem with machine eligibility restrictions and sequence-dependent setup times

In this paper, a fuzzy bi-objective mixed-integer linear programming (FBOMILP) model is presented. FBOMILP encompasses the minimisation workload imbalance and total tardiness simultaneously as a bi-objective formulation for an unrelated parallel machine scheduling problem. To make the proposed model more practical, sequence-dependent setup times, machine eligibility restrictions and release dates are also considered. Moreover, the inherent uncertainty of processing times, release dates, setup times and due dates are taken into account and modelled by fuzzy numbers. In order to solve the model for small-scale problems, a two-stage fuzzy approach is proposed. Nevertheless, since the problem belongs to the class of NP-hard problems, the proposed model is solved by two meta-heuristic algorithms, namely fuzzy multi-objective particle swarm optimisation (FMOPSO) and fuzzy non-dominated sorting genetic algorithm (FNSGA-II) for solving large-scale instances. Subsequently, through setting up various numerical examples, the performances of the two mentioned algorithms are compared. When α = 0.5 (α is a level of risk-taking and when it increases the decision-maker’s risk-taking decreases), FNSGA-II is fairly more effective than FMOPSO and has better performance especially in solving large-sized problems. However, when α rises, it can be stated that FMOPSO moderately becomes more appropriate. Finally, directions for future studies are suggested and conclusion remarks are drawn.

Solving a Time-Indexed Formulation by Preprocessing and Cutting Planes

We consider a time-indexed formulation for the unrelated parallel machine scheduling problem. We show that all polyhedral knowledge known from the single machine problem (in particular, valid inequalities) is applicable to this formulation. We present new facet-inducing valid inequalities and a preprocessing technique involving fixing variables based on reduced costs. We combine both techniques in a basic cutting-plane algorithm and test the performance of the resulting algorithm by running it on randomly generated instances.

A comparison of five hybrid metaheuristic algorithms for unrelated parallel-machine scheduling and inbound trucks sequencing in multi-door cross docking systems

The objective of this paper is to develop five hybrid metaheuristic algorithms, including three hybrid ant colony optimization (hACO) variants, and compare their performances in two related applications: unrelated parallel machine scheduling and inbound truck sequencing in a multi-door cross docking system in consideration of sequence dependent setup, and both zero and nonzero release time. The three hACO variants were modified and adapted from the existing literature and they differ mainly in how a solution is coded and decoded, how a pheromone matrix is represented, and the local search methods employed. The other two hybrids are newly constructed hybrid simulated annealing (hSA) algorithms, which are built based on the authors’ knowledge and experience. The evaluation criteria are computational time and the objective function value, i.e., makespan. Based on the results of computational experiments the simulated annealing-tabu search hybrid turns out to be the best if maximal CPU time is used as the stopping criterion and the 2-stage hACO variant is the best if maximal number of evaluations is the stopping criterion. The contributions of this paper are: (i) being the first to carry out a comparative study of hybrid metaheuristics for the two selected applications, (ii) being the first to consider nonzero truck arrival time in multi-door cross docking operations, (iii) identifying which hACO variant is the best among the three, and (iv) investigating the effect of release time on the makespan.

Hybrid GRASP Heuristics to Solve an Unrelated Parallel Machine Scheduling Problem with Earliness and Tardiness Penalties

This paper considers an unrelated parallel machine scheduling problem with the objective of minimizing the total earliness and tardiness penalties. Machine and job-sequence dependent setup times and idle times are considered. Since the studied problem is NP-Hard, we test the applicability of algorithms based on Greedy Randomized Adaptive Search Procedure (GRASP) metaheuristic to determine near-optimal solutions. We propose three different heuristics. The first is a simple GRASP heuristic, the second heuristic includes an intensification procedure based on Path Relinking technique, and the third uses an Iterated Local Search (ILS) heuristic in the local search phase of the GRASP algorithm. The results obtained by the heuristics are compared using a set of small, medium and large instances. Comprehensive computational and statistical analyses are carried out in order to compare the performance of the algorithms.

Scheduling of unrelated parallel machines with limited server availability on multiple production locations: a case study in knitted fabrics

This paper studies a complex variation of the parallel machine scheduling (PMS) problem, as encountered at a Belgian producer of knitted fabrics. The aim is to assign jobs to unrelated parallel machines, minimising a weighted combination of job lateness and tardiness. Jobs are assigned specific release, and due dates and changeover times are sequence dependent. Current literature is extended by including geographically dispersed production locations, which influence job due dates and objective function coefficients. Furthermore, the changeover interference due to limited availability of technicians is also studied in this paper. The scheduling problem is solved using a hybrid meta-heuristic, which combines elements from simulated annealing and genetic algorithms. This hybrid meta-heuristic is capable of solving real-scale scheduling problems of up to 750 jobs, 75 machines and 10 production locations within reasonable computation time. This hybrid scheduling procedure is extended with heuristic dispatching rules capable of reducing the impact of changeover interference by 23% on average compared to the random scenario, for the case where a single technician is expected to serve up to 12 machines.

Note on “Unrelated parallel-machine scheduling with rate-modifying activities to minimize the total completion time”

This note shows that the problem studied by Hsu et al. (2011) [2] can be solved in O(nm+3) time even though the rate-modifying rate is larger than 1, where m is the number of machines and n is the number of jobs.

Unrelated parallel-machine scheduling to minimize total weighted completion time

This research deals with the unrelated parallel-machine scheduling to minimize total weighted completion time. There are $$N$$N jobs, each having a weight and requiring a single stage of processing...

Unrelated parallel machine scheduling with setup times and ready times

We consider the problem of scheduling unrelated parallel machines with sequence- and machine-dependent setup times and ready times to minimise total weighted tardiness (TWT). We present a mixed integer programming model that can find optimal solutions for the studied problem. We also propose a heuristic (ATCSR_Rm) and an iterated hybrid metaheuristic (IHM) that can find optimal or nearly optimal solutions for the studied problem within a reasonable time. The proposed IHM begins with effective initial solutions, and then improves the initial solutions iteratively. The IHM integrates the principles of the attraction–repulsion mechanism within electromagnetism-like algorithms with local search. If the search becomes trapped at a local optimum, an elite search procedure is developed to help the search escape. We have compared our proposed IHM with two existing metaheuristics, tabu search (TS) and ant colony optimisation (ACO). Computational results show that the proposed IHM outperforms TS and ACO in terms of TWT for problem instances of all sizes.

Parallel-machine scheduling with controllable processing times and rate-modifying activities to minimise total cost involving total completion time and job compressions

In this paper, we consider unrelated parallel-machine scheduling involving controllable processing times and rate-modifying activities simultaneously. We assume that the actual processing time of a job can be compressed by allocating a greater amount of a common resource to process the job. We further assume that each machine may require a rate-modifying activity during the scheduling horizon. The objective is to determine the optimal job compressions, the optimal positions of the rate-modifying activities and the optimal schedule to minimise a total cost function that depends on the total completion time and total job compressions. If the number of machines is a given constant, we propose an efficient polynomial time algorithm to solve the problem.

Unrelated parallel-machine scheduling problems with aging effects and deteriorating maintenance activities

This paper explores the problems of unrelated parallel-machine scheduling with aging effects and deteriorating maintenance activities. The jobs are non-resumable. Each machine has at most one maintenance activity which is allowed throughout the planning horizon. Assume that once the maintenance activity is completed, the functions of machine and the properties of jobs e.g. the temperature (hardness) of materials that processed by the machine will be restored to their initial conditions. The length of the maintenance activity is a linear function of its starting time. The objectives of this study are to minimize the total completion time and the total machine load. Three basic types of aging effect model are proposed. We show that all the proposed models can be solved optimally in polynomial time.

A particle swarm optimization for a fuzzy multi-objective unrelated parallel machines scheduling problem

This paper proposes a novel multi-objective model for an unrelated parallel machine scheduling problem considering inherent uncertainty in processing times and due dates. The problem is characterized by non-zero ready times, sequence and machine-dependent setup times, and secondary resource constraints for jobs. Each job can be processed only if its required machine and secondary resource (if any) are available at the same time. Finding optimal solution for this complex problem in a reasonable time using exact optimization tools is prohibitive. This paper presents an effective multi-objective particle swarm optimization (MOPSO) algorithm to find a good approximation of Pareto frontier where total weighted flow time, total weighted tardiness, and total machine load variation are to be minimized simultaneously. The proposed MOPSO exploits new selection regimes for preserving global as well as personal best solutions. Moreover, a generalized dominance concept in a fuzzy environment is employed to find locally Pareto-optimal frontier. Performance of the proposed MOPSO is compared against a conventional multi-objective particle swarm optimization (CMOPSO) algorithm over a number of randomly generated test problems. Statistical analyses based on the effect of each algorithm on each objective space show that the proposed MOPSO outperforms the CMOPSO in terms of quality, diversity and spacing metrics.

A hybrid genetic algorithm for job sequencing and worker allocation in parallel unrelated machines with sequence-dependent setup times

In this paper, the unrelated parallel machine scheduling problem with sequence-dependent setup times and limited human resources is addressed with reference to the makespan minimisation objective. Workers needed for setup operations are supposed to be a critical resource as their number is assumed to be lower than the number of workstations. In addition, each worker is characterised by a specific skill level, which affects setup times. Firstly, a mathematical model able to optimally solve small instances of the problem in hand is illustrated. Then, to deal with large-sized test cases, three different optimisation procedures equipped by different encoding methods are proposed: a permutation encoding-based genetic algorithm (GA), a multi-encoding GA and a hybrid GA that properly moves from a permutation encoding to a multi-encoding once a given threshold on the number of generations is achieved. In particular, three different hybrid GAs featured by different encoding switch thresholds were implemented. An extensive benchmark including both small- and large-sized instances was generated with the aim of both calibrating the genetic parameters and comparing the alternative GAs through distinct ANOVA analyses. Numerical results confirm the effectiveness of the hybrid genetic approach whose encoding switch threshold is fixed to 25 % of the overall generations. Finally, a further analysis concerning the impact of multi-skilled workforce on the performance of both production system and optimisation strategy is presented.

Concentration inequalities for nonlinear matroid intersection

In this work we propose new randomized rounding algorithms for matroid intersection and matroid base polytopes. We prove concentration inequalities for polynomial objective functions and constraints that has numerous applications and can be used in approximation algorithms for Minimum Quadratic Spanning Tree, Unrelated Parallel Machines Scheduling and scheduling with time windows and nonlinear objectives. We also show applications related to Constraint Satisfaction and dense polynomial optimization. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 541-571, 2015

A tabu search algorithm for unrelated parallel machine scheduling with sequence- and machine-dependent setups: minimizing total tardiness

This study considers the problem of scheduling independent jobs on unrelated parallel machines with machine- and sequence-dependent setup times for the objective of minimizing the total tardiness, i.e., Rm│Sijk│∑Tj. Since the parallel machines are unrelated, sequence-dependent setup times must depend on machines. To the best of the authors’ knowledge, the simulated annealing and the iterated greedy algorithms are two existing ones for the new class of scheduling problem with an additional constraint of strict due date constraints for some jobs, i.e., deadlines. In this study, we suggest a tabu search algorithm that incorporates various neighborhood generation methods. A computational experiment was done on the instances generated by the method used in the two previous research articles, and the results show that the tabu search algorithm outperforms the simulated annealing algorithm significantly. In particular, it gave optimal solutions for more than 50 % of small-sized test instances. Also, an additional test was done to compare the performances of the tabu search and the existing iterated greedy algorithms, and the result shows that the tabu search algorithm gives quicker solutions than the iterated greedy algorithm although it gives less quality solutions.

An imperialist competitive algorithm for a bi-objective parallel machine scheduling problem with load balancing consideration

In this paper, we present a new Imperialist Competitive Algorithm (ICA) to solve a bi-objective unrelated parallel machine scheduling problem where setup times are sequence dependent. The objectives include mean completion time of jobs and mean squares of deviations from machines workload from their averages. The performance of the proposed ICA (PICA) method is examined using some randomly generated data and they are compared with three alternative methods including particle swarm optimization (PSO), original version of imperialist competitive algorithm (OICA) and genetic algorithm (GA) in terms of the objective function values. The preliminary results indicate that the proposed study outperforms other alternative methods. In addition, while OICA performs the worst as alternative solution strategy, PSO and GA seem to perform better.

Chip Attach Scheduling in Semiconductor Assembly

Chip attach is the bottleneck operation in semiconductor assembly. Chip attach scheduling is in nature unrelated parallel machine scheduling considering practical issues, for example, machine-job qualification, sequence-dependant setup times, initial machine status, and engineering time. The major scheduling objective is to minimize the total weighted unsatisfied Target Production Volume in the schedule horizon. To apply Q-learning algorithm, the scheduling problem is converted into reinforcement learning problem by constructing elaborate system state representation, actions, and reward function. We select five heuristics as actions and prove the equivalence of reward function and the scheduling objective function. We also conduct experiments with industrial datasets to compare the Q-learning algorithm, five action heuristics, and Largest Weight First (LWF) heuristics used in industry. Experiment results show that Q-learning is remarkably superior to the six heuristics. Compared with LWF, Q-learning reduces three performance measures, objective function value, unsatisfied Target Production Volume index, and unsatisfied job type index, by considerable amounts of 80.92%, 52.20%, and 31.81%, respectively.

Optimization of production scheduling with time-dependent and machine-dependent electricity cost for industrial energy efficiency

In many industrialized countries, manufacturing industries pay stratified electricity charges depending on the time of day (i.e., peak-load, mid-load, and off-peak-load). In contrast, the emerging smart grid concept may demand that industries pay real-time hourly electricity costs so as to use energy most efficiently. This paper deals with the production and energy efficiency of the unrelated parallel machine scheduling problem. This method allows the decision maker to seek a compromise solution using the weighted sum objective of production scheduling and electricity usage. Reliability models are used to consider the energy cost aspect of the problem. This paper aims to optimize the weighted sum of two criteria: the minimization of the makespan of production and the minimization of time-dependent electricity costs. We suggest a hybrid genetic algorithm with our blank job insertion algorithm and demonstrate its performance in simulation experiments.

Parallel machine scheduling to minimize the makespan with sequence dependent deteriorating effects

A new unrelated parallel machine scheduling problem with deteriorating effect and the objective of makespan minimization is presented in this paper. The deterioration of each machine (and therefore of the job processing times) is a function of the sequence of jobs that have been processed by the machine and not (as considered in the literature) by the time at which each job is assigned to the machine or by the number of jobs already processed by the machine. It is showed that for a single machine the problem can be solved in polynomial time, whereas the problem is NP-hard when the number of machines is greater or equal than two. For the last case, a set of list scheduling algorithms and simulated annealing meta-heuristics are designed and the effectiveness of these approaches is evaluated by solving a large number of benchmark instances.

Unrelated parallel-machine scheduling problems with multiple rate-modifying activities

In this article we investigate scheduling problems with multiple rate-modifying activities on an unrelated parallel-machine setting. We assume that each machine may have multiple rate-modifying activities. The goal is to determine jointly the optimal rate-modifying activity frequencies, the optimal rate-modifying activity positions, and the optimal schedule to minimize the total completion time and the total machine load, respectively. If the number of machines is fixed, we propose two efficient polynomial time algorithms for solving both problems no matter what the processing time of a job scheduled after a rate-modifying activity is greater or less than its processing time scheduled before the rate-modifying activity.