Unrelated Parallel Machines Research Articles

Local search based methods for scheduling in the unrelated parallel machines environment

In many real-world situations it is necessary to make timely scheduling decisions. In most cases, metaheuristic algorithms are used to solve various scheduling problems because of their flexibility and their ability to produce satisfactory results in a short time. In recent years, several novel or hybrid metaheuristics have been proposed for scheduling problems. Although such research leads to new insights, it inevitably causes certain problems. First, it becomes unclear which methods perform best, especially if they are not properly compared with existing ones. Second, the proposed methods become increasingly complex, making them more difficult to understand and apply. The goal of this study is to investigate the possibility of defining efficient but simple iterative local search (ILS) methods for the parallel unrelated machines environment with minimisation of the total weighted tardiness. To improve the efficiency of ILS methods, several design decisions, such as the construction of the initial solution and choice of local search operators. The proposed methods have been compared with several metaheuristics, of which they achieve significantly better results. Thus, we conclude that it is not necessary to increase the complexity of metaheuristics to achieve better results. Rather, better results can be obtained with simple but well-designed local search methods.

Improved firefly algorithm with courtship learning for unrelated parallel machine scheduling problem with sequence-dependent setup times

The Unrelated Parallel Machines Scheduling Problem (UPMSP) with sequence-dependent setup times has been widely applied to cloud computing, edge computing and so on. When the setup times are ignored, UPMSP will be a NP problem. Moreover, when considering the sequence related setup times, UPMSP is difficult to solve, and this situation will be more serious in the case of high-dimensional. This work firstly select the maximum completion time as the optimization objective, which establishes a mathematical model of UPMSP with sequence-dependent setup times. In addition, an improved firefly algorithm with courtship learning is proposed. Finally, in order to provide an approximate solution in an acceptable time, the proposed algorithm is applied to solve the UPMSP with sequence-dependent setup times. The experimental results show that the proposed algorithm has competitive performance when dealing with UPMSP with sequence-dependent setup times.

A Two-Population Cooperative Multiobjective Differential Evolution Algorithm for Batching Scheduling Problem

Batch processing machine (BPM) scheduling problem is a NP hard problem for it includes machine allocation, job grouping, and batch scheduling. In this paper, to address the BPM scheduling problem with unrelated parallel machine, a multiobjective algorithm based on multipopulation coevolution is proposed to minimize the total energy consumption and the completion time simultaneously. Firstly, the mixed integer programming model of the problem is established, and four heuristic decoding rules are proposed. Secondly, to improve the diversity and convergence of the algorithm, the population is divided into two populations: each of the populations evolves independently by using different decoding rules, and the two populations will communicate through a common external archive set every certain number of generations. Thirdly, an initialization strategy and a variable neighborhood search algorithm (VNS) are proposed to improve the overall performance of the algorithm. Finally, in order to evaluate the proposed algorithm, a large number of comparative experiments with the state-of-the-art multiobjective algorithms are carried out, and the experimental results proved the effectiveness of the proposed algorithm.

A memetic algorithm for energy-efficient scheduling of integrated production and shipping

ABSTRACT Energy-efficient manufacturing is critical as the industrial sector accounts for a substantial portion of global energy consumption. This research aims to address an energy-efficient scheduling problem of production and shipping for minimizing both makespan and energy consumption. It contributes to an integrated energy-efficient production and shipping system, which is separately studied in most existing research. The production stage allocates jobs onto unrelated parallel machines that can be shut off and adjust their cutting speed to save energy. The shipping stage aims to allocate jobs to vehicles of various sizes with varied unit energy consumption. The problem is modelled as a mixed-integer quadratic program. Considering its complexity, a memetic algorithm (MA) is proposed to incorporate a knowledge-driven local search strategy considering the balance between exploration and exploitation. Two dominance rules are derived from the characteristics of the specific problem and embedded into the proposed MA to enhance its performance. Experimental results demonstrate that the proposed MA outperforms two other population-based algorithms, genetic algorithm and traditional MA, in terms of performance and computing time. This research practically contributes to improving productivity and energy efficiency for the production-shipping supply chain of make-to-order products.

A mathematical formulation and an NSGA-II algorithm for minimizing the makespan and energy cost under time-of-use electricity price in an unrelated parallel machine scheduling.

In many countries, there is an energy pricing policy that varies according to the time-of-use. In this context, it is financially advantageous for the industries to plan their production considering this policy. This article introduces a new bi-objective unrelated parallel machine scheduling problem with sequence-dependent setup times, in which the objectives are to minimize the makespan and the total energy cost. We propose a mixed-integer linear programming formulation based on the weighted sum method to obtain the Pareto front. We also developed an NSGA-II method to address large instances of the problem since the formulation cannot solve it in an acceptable computational time for decision-making. The results showed that the proposed NSGA-II is able to find a good approximation for the Pareto front when compared with the weighted sum method in small instances. Besides, in large instances, NSGA-II outperforms, with 95% confidence level, the MOGA and NSGA-I multi-objective techniques concerning the hypervolume and hierarchical cluster counting metrics. Thus, the proposed algorithm finds non-dominated solutions with good convergence, diversity, uniformity, and amplitude.

Influence of probability distribution initialization methods on the performance of advanced arithmetic optimization algorithm with application to unrelated parallel machine scheduling problem

AbstractThe article investigates the influence of several initialization methods on the performance of the newly proposed advanced arithmetic optimization algorithm, also known as the nAOA. The initialization conditions considered include population sizes, diversity of the population, and the number of iterations. We used 23 different probability distributions with different diversity to test the influence of initialization schemes on the convergence and accuracy of the nAOA. The benchmark classical test functions and the functions defined in the CEC 2020 suite having different properties and modalities were used to compare the possible effects of the initialization methods. The numerical results showed that the nAOA is sensitive to population size and the number of iterations, which must be large for optimal performance. Friedman's and correlation tests were used to gain useful insight into the results we obtained. Findings showed that the performance of the nAOA is not particularly sensitive to the different initialization schemes used for most test functions. The implication is that nAOA is relatively stable and robust. However, the variant of beta distribution, b(3,2), recorded the lowest mean rank for most functions. Hence, it is the best performing initialization distribution scheme for the test problems considered. We can also observe from the experimental results that 47.83% and 100% of the classical and CEC2020 test functions used in this article show significant differences for different initialization methods, respectively. We also investigated the applicability of the nAOA to solve the unrelated parallel machine scheduling problem with sequence‐dependent setup times by using the best‐identified set of initialization methods. The performance of the nAOA is evaluated by comparing its solution to seven other well‐known metaheuristic algorithms, and the results reveal that the nAOA can provide promising results in solving all problem instances of the problem under study using the best initialization schemes.

A robust optimization approach for the unrelated parallel machine scheduling problem

A robust optimization approach for the unrelated parallel machine scheduling problem

A Hybrid Imperialist Competitive Algorithm for the Distributed Unrelated Parallel Machines Scheduling Problem

In this paper, the distributed unrelated parallel machines scheduling problem (DUPMSP) is studied and a hybrid imperialist competitive algorithm (HICA) is proposed to minimize total tardiness. All empires were categorized into three types: the strongest empire, the weakest empire, and other empires; the diversified assimilation was implemented by using different search operator in the different types of empires, and a novel imperialist competition was implemented among all empires except the strongest one. The knowledge-based local search was embedded. Extensive experiments were conducted to compare the HICA with other algorithms from the literature. The computational results demonstrated that new strategies were effective and the HICA is a promising approach to solving the DUPMSP.

Hybrid meta-heuristics for the unrelated parallel machine scheduling problem with setup times

This study focuses on an unrelated parallel machine scheduling problem with machine and job sequence-dependent setup times (UPMST) aiming to minimise the makespan. Recently, adaptive large neighbourhood search with learning automata (LA-ALNS) has been employed to tackle the UPMST with excellent performance. However, its search efficiency is heavily affected by re-visiting the recent solutions, which is known as the short-term cycle. Moreover, it remains challenging to balance the computational cost and quality of the solutions. To address these challenges, we propose a hybrid meta-heuristic method based on LA-ALNS and tabu search (LA-ALNS-TS). In the proposed algorithm, LA-ALNS is used to guide the exploration and TS is used to alleviate the short-term cycle experienced during the search process. In addition, we design two local search operators to achieve a balanced trade-off between the search efficiency and quality of the solutions. A dynamic perturbation scheme is proposed to improve the solution and help the algorithm escape from local optima. Comprehensive experiments were conducted to assess the performance of the algorithm on two exhaustive benchmarks. Experimental results demonstrate the effectiveness and efficiency of the proposed hybrid meta-heuristic method.

A matheuristic approach for solving a simultaneous lot sizing and scheduling problem with client prioritization in tire industry

This paper introduces an integrated lot sizing and scheduling problem inspired from a real-world application in off-the-road tire industry. This problem considers the assignment of different items on parallel machines with complex eligibility constraints within a finite planning horizon. It also considers a large panel of specific constraints such as: backordering, a limited number of setups, upstream resources saturation and customers prioritization. A novel mixed integer formulation is proposed with the objective of optimizing different normalized criteria related to the inventory and service level performance. Based on this mathematical formulation, a problem-based matheuristic method that solves the lot sizing and assignment problems separately is proposed to solve the industrial case. A computational study and sensitivity analysis are carried out based on real-world data with up to 170 products, 70 unrelated parallel machines and 42 periods. The obtained results show the effectiveness of the proposed approach on improving the company’s solution. Indeed, the two most important KPIs for the management have been optimized of respectively 32% for the backorders and 13% for the overstock. Moreover, the computational time have been reduced significantly.

Hybrid tabu search algorithm for unrelated parallel machine scheduling in semiconductor fabs with setup times, job release, and expired times

This research is motivated by a scheduling problem arising in the ion implantation process of wafer fabrication. The ion implementation scheduling problem is modeled as an unrelated parallel machine scheduling (UPMS) problem with sequence-dependent setup times that are subject to job release time and expiration time of allowing a job to be processed on a specific machine, defined as: R|rj,eij,STsd|Cmax. The objective is first to maximize the number of processed jobs, then minimize the maximum completion time (makespan), and finally minimize the maximum completion times of the non-bottleneck machines. A mixed-integer programming (MIP) model is proposed as a solution approach and adopts a hybrid tabu search (TS) algorithm to acquire approximate feasible solutions. The MIP model has two phases and attempts to achieve the first two objectives. The hybrid TS algorithm has three phases and attempts to achieve all three objectives. In a real setting, computational results demonstrate that the maximum number of processed jobs can be acquired within a short time utilizing the hybrid TS algorithm (average 8 s). By comparing the two approaches, the TS outperforms the MIP model regarding solution quality and computational time for the second objective, minimizing the makespan. Furthermore, the third phase of the hybrid TS algorithm shows the effectiveness further to enhance the utilization of the ion implantation equipment.

Scheduling in a flexible flow shop with unrelated parallel machines and machine-dependent process stages: Trade-off between Makespan and production costs

This paper aims to tackle bi-objective scheduling problem in a flexible flow shop containing unrelated parallel machines in the first stage. Due to the different technology levels of the parallel machines, their process speeds and production costs vary to each other. Therefore, minimizing the maximum completion time (Makespan) and the total production cost are considered as two objective functions. In addition, setup times are considered sequence-dependent and this system considers machine-dependent process steps and the process steps of an order depend on the assigned machine in the first stage. First, the problem is described and formulated as a bi-objective mathematical model. Since the problem is known to be strongly NP-hard, an approximate solution method is introduced based on the Non-dominated Sorting Genetic Algorithm (NSGA-II) to provide proper solutions for decision makers. The performance of the proposed solution method is investigated in comparison to another powerful multi-objective algorithm (SPEA 2) by solving different test problems. The computational results using various metrics such as Error Ratio (ER) and Generational Distance (GD) show the effectiveness of the proposed method in terms of optimality. The other indices such as Spacing (S), Diversification (D), and Mean Ideal Distance (MID) emphasize the superiority of the proposed algorithm compared to the rival algorithm in solving medium and large instances. In addition, supplementary analysis provided proper trade-off between two objectives for managers to select the best solution based on their preferences.

Optimising Makespan In a Hybrid Flowshop with Unrelated Parallel Machines, Shared Resource, Sequence Dependent Setup Times, and Machine Eligibility Using Meta and Hyper Heuristic Approaches

Optimising Makespan In a Hybrid Flowshop with Unrelated Parallel Machines, Shared Resource, Sequence Dependent Setup Times, and Machine Eligibility Using Meta and Hyper Heuristic Approaches

Developing a Matheuristic for the Integrated Planning of a Cold Rolling Steel Plant

Steel-producing firms are confronted with the challenge of determining plans that coordinate production processes across a set of processing stages. While several approaches for integrated planning of hot rolling mills have been proposed, the cold rolling process is still poorly investigated. Therefore, we present a model formulation to simultaneously form campaigns of coils in cold rolling steel plants. The model considers multiple processing stages with unrelated parallel machines, sequence-dependent setup operations and order-specific due dates. To solve the model, we propose a matheuristic based on Fix-and-Relax and Fix-and-Optimize that utilizes a production line-based decomposition strategy. The results of a numerical study are presented that replicates the realistic production environment of a real cold rolling plant and a planning horizon of up to 10 days. The proposed matheuristic consistently outperforms a commercial solver in terms of solution quality and time.

A comparison of different mathematical models for the job sequencing and tool switching problem with non-identical parallel machines

This paper addresses the generalisation of the NP-hard job sequencing and tool switching problem with non-identical parallel machines and sequence-dependent setup times where a set of jobs is to be scheduled on unrelated parallel machines with machine-dependent processing and tool switching times. Three different mathematical models for two different objectives are presented and applied to newly generated test instances. The instances are compared and analysed using a commercial solver and an iterated local search heuristic. Overall, it is shown that the solution quality obtained by the mathematical models depends on the size of the problem instance as well as the tool requirements. The precedence-based formulation is superior in general to the position-based and time-index-based formulation for dense problem instances while the position-based formulation works well for sparse problems. With an increasing problem size, the metaheuristic requires significantly less time to find near-optimal solutions than the mathematical models.

An Exact Decomposition Method for Unrelated Parallel Machine Scheduling with Order Acceptance and Setup Times

An Exact Decomposition Method for Unrelated Parallel Machine Scheduling with Order Acceptance and Setup Times

Investigating total earliness and tardiness costs through unrelated parallel machine scheduling in uncertain job shop environment using robust optimisation and design of experiment

In real world production systems, uncertain events such as random machine breakdown and processing time can occur anytime. These events lead to disruption of normal activities and consequently invalidate the initial schedule. Considering uncertainty in the scheduling process enables organisations to resume their activities effectively after uncertain events occur. The focus of this paper is proactive scheduling approach with an objective of minimising the total cost (lateness/earliness penalty and tooling cost). Robust optimisation is used to solve the scheduling problem considering processing time, setup times and tooling cost as uncertain parameters. Numerous scenarios are solved using data from local job shop. Multiple performance measurement criteria are evaluated to assess the significance of results obtained using robust and deterministic models. Design of experiment (DOE) has been implemented to evaluate the effects of different factors on the total cost and computational times.

Mathematical Models for Minimizing Total Tardiness on Parallel Additive Manufacturing Machines

In this research we tackle the scheduling problem in additive manufacturing for unrelated parallel machines. Both the nesting and scheduling aspects are considered. Parts have several alternative build orientations. The goal is to minimize the total tardiness of parts. We propose a mixed-integer linear programming model which considers the nesting subproblem as a 2D bin-packing problem, as well as a model which simplifies the nesting subproblem to a 1D bin-packing problem. The computational efficiency and properties of the proposed models are investigated by numerical experiments. Results show that the total tardiness optimization significantly increases the complexity of the problem, only the simple instances are solved optimally, whereas the makespan variant is able to solve all testing instances. Using the 1D bin-packing simplification allows for solving more instances to optimality, but with a risk of obtaining nesting-infeasibility. We also observed the compromise between the total tardiness and makespan objectives, which originates from the dilemma of “packing more parts to benefit from the common machine setup/recoating time” or “packing less parts to maintain the flexibility for handling distributed duedates”.

A mixed integer formulation and an efficient metaheuristic for the unrelated parallel machine scheduling problem: Total tardiness minimization

In this paper, the unrelated parallel machine scheduling problem with the objective of minimizing the total tardiness is addressed. For such a problem, a mixed-integer linear programming (MILP) formulation, that considers assignment and positional variables, is presented. In addition, an iterated local search (ILS) algorithm that produces high-quality solutions in reasonable times is proposed for higher size instances. The ILS robustness was determined by comparing its performance with the results provided by the MILP. The instances used in this paper were constructed under a new approach which results in tighter due dates than the previous generation method for this problem. The proposed MILP formulation was able to solve instances of up to 150 jobs and 20 machines. Regarding the ILS, it yielded high-quality solutions in a reasonable time, solving instances of a size up to 400 jobs and 20 machines. Experimental results confirm that both approaches are efficient and promising. • A new mixed integer programming formulation for the Unrelated Parallel Machine Scheduling Problem. • The formulation uses positional variables, is linear and does not use the Big-M technique. • The formulation solves instances up to 150 jobs and 20 machines to optimality. • An efficient iterated local search metaheuristic is presented. • A potential issue in the commonly used instance-generation procedure is discussed and a new set of instances is proposed.

Scheduling unrelated parallel machines with a common server and sequence dependent setup times

In this article we consider the problem of scheduling unrelated parallel machines with sequence dependent setup times and a common server subject to unavailability periods. The problem is inspired by the industrial context of a manufacturer that produces mechanical parts for hydraulic and electrical sector. The originality resides in the fact that the common server, used for setup of all machines have fixed unavailability periods, rarely considered in the literature. We have formulated the problem by a Mixed Integer Linear Program (MILP). We executed different tests of the MILP, to study the influence of each parameter on the resolution time. The preliminary results show that, while the algorithm presented takes exponential time to solve, it can be used to give effective schedules for small size problems and can be used as a basis for heuristic approaches.