Non-identical Parallel Machines Research Articles

Exploring the interplay of costs and flow time in server allocation for flow lines with parallel non-identical machines

ABSTRACT The Server Allocation Problem is a significant challenge in manufacturing systems involving deciding the number and version of machines for each stage. This optimization problem is of utmost importance when there is a trade-off between minimizing investment costs and flow time while achieving target performance, such as a minimum throughput. Manufacturing systems often employ hybrid flow lines, with non-identical machines at each stage. This paper analyzes the trade-off between minimizing costs and flow time when allocating servers in hybrid flow lines by proposing a bi-criteria approach that uses an efficient pattern-based problem representation and a Variable Neighborhood Search solution algorithm. The objectives are to minimize total cost and flow time while ensuring a target throughput. A metric for comparing obtained solutions is proposed, providing insights for practical implementation. A comprehensive numerical analysis is conducted, to validate the proposed approach and demonstrate its applicability in real-world manufacturing settings.

Non-identical parallel machines batch processing problem to minimize the makespan: Models and algorithms

This paper studies a parallel heterogeneous machine batching and scheduling problem in which weighted jobs are first batched, and the batches are then assigned and sequenced on machines of varying capacities. The duration of a batch is the longest time needed to process a job, and the objective is that of minimizing the makespan, or the sum of the batches durations on the machine finishing last. The authors develop polynomial-size mathematical formulations and a variable neighborhood search metaheuristic. Extensive computational results suggest that a flow-based formulation outperforms a compact formulation, despite its larger number of variables. The metaheuristic is capable of producing high-quality solutions within a limited computing time.

A simplified swarm optimization algorithm to minimize makespan on non-identical parallel machines with unequal job release times under non-renewable resource constraints

A simplified swarm optimization algorithm to minimize makespan on non-identical parallel machines with unequal job release times under non-renewable resource constraints

Scheduling parallel extrusion lines

This paper introduces the problem of scheduling jobs on parallel plastic extrusion lines where each line is composed of one or more than one extruder. Although there are some similarities between the introduced problem and the non-identical parallel machines scheduling problems with sequence-dependent setup times, limited additional resources and machine eligibility restrictions, the problem considered in this paper is a generalization of the parallel machine scheduling problem. This is because in parallel machines scheduling each job requires only one machine but in our case some jobs require more than one machine. Thus, our problem reduces to the parallel machine scheduling problem if all jobs require only one machine. This paper describes the problem of scheduling parallel extrusion lines, its industrial context, and develops a mixed-linear formulation to model the problem. This formulation allowed solving instances of up to 15 jobs. In addition, we developed four metaheuristics: a simulated annealing algorithm, a tabu search heuristic, a genetic algorithm, and a greedy randomized adaptive search procedure. These metaheuristics can be used to solve real-life instances of the problem. A numerical experiment shows that the proposed metaheuristics produce excellent solutions. Some of the proposed simulated annealing adaptations and of the tabu search heuristics obtained solutions with less than 2% deviation from the optimum.

Biased random-key genetic algorithm for the job sequencing and tool switching problem with non-identical parallel machines

We address the job sequencing and tool switching problem associated with non-identical parallel machines – a variation of the well-known sequencing and switching problem (SSP) better adapted to reflect the challenges in modern production environments. The NP-hard problem is approached by considering two isolated objective functions: the minimization of the makespan and the minimization of the total flow time. We present two versions of a parallel biased random-key genetic algorithm hybridized with tailored local search procedures that are organized using variable neighborhood descent. The proposed methods are compared with state-of-the-art methods by considering 640 benchmark instances from literature. For both objective functions considered, the proposed methods consistently outperform the compared methods. All known optimal values for both objectives are achieved, and a substantial gap is reported for all instance groups when compared with the best previously published solution values.

The production scheduling problem employing non-identical parallel machines with due dates considering carbon emissions and multiple types of energy sources

This paper addresses the production scheduling problem with non-identical parallel machines in a high-mix, low-volume production environment with due dates, considering carbon emissions savings and diverse types of energy sources for machine operation. To this end, we present a bi-objective mixed-integer programming model that minimizes both the total production-related cost and the total carbon emissions in the production process to determine the optimal production strategy. For each machine, we consider various types of energy sources with different electricity generation costs and different amounts of carbon emissions. The proposed model is validated with an application to a manufacturer in Ulsan, the industrial capital of the Republic of Korea. Our results showcase the potential use of non-identical parallel machines to minimize the total cost and carbon emissions in green manufacturing. Furthermore, the results identify the trade-off between different energy sources and the total cost under different carbon emissions limits. Generally, as the carbon emissions limit increases, our proposed model tends to replace natural gas with coal to minimize the total cost. Also, we perform sensitivity analyses with respect to the energy price of different types of energy sources combined with nuclear and renewable energy sources. We find that coal provides more stability than natural gas in terms of the total cost, particularly when there are fluctuations in the price for coal and natural gas. Additionally, we determine the optimal combination of energy sources for various carbon emissions limits, aiming to minimize the total cost while simultaneously satisfying all production-related and environmental constraints.

Hybridizing local searching with genetic algorithms for the job sequencing and tool switching problem with non-identical parallel machines

The job sequencing and tool switching problem (SSP) is an old and highly studied problem. However, the SSP with non-identical parallel machines (SSP-NPM) is a new generalization of this problem. Since the SSP-NPM has been very recently introduced, the number of solution methods for this problem is low. Additionally, exact solution methods are mostly insufficient to find a good solution within a reasonable computational time since this problem is NP-hard. Therefore, heuristic approaches are more appropriate, especially when solving large-size problems. Thus, this study presents a new hybrid algorithm that is a combination of a genetic algorithm (GA) and a local search (LS) technique. Both of these methods are greatly preferred by authors for solving job scheduling problems. Thereby, this study combines their strengths to solve the SSP-NPM. Two versions of the proposed method are tested with benchmark instances. These versions differ in terms of their tool optimization policies: the first uses the well-known “keep tool needed soonest” (KTNS) policy, while the second uses the “randomly remove tool” (RRT) policy. Computational experiments showed that combining some of the features of GA and LS methods for the SSP-NPM contributed greatly to the quality of the solution. Two versions of the proposed hybrid method were compared to a mixed-integer programming model and several schemes of an iterated local search algorithm. Our computational results show that both versions of the proposed hybrid method were significantly better than the other methods, and were able to find the solutions within a reasonable computational time.

Minimizing total completion time on non-identical parallel batch machines with arbitrary release times using ant colony optimization

This paper considers the problem of scheduling a group of jobs with arbitrary release times, non-identical sizes, and different processing times on non-identical parallel batch processing machines to minimize the total completion time. A mixed-integer programming (MIP) model is firstly constructed in this paper to solve this problem. Then since the studied problem is strongly NP-hard, a modified elite ant system algorithm with the local search (MEASL) is also proposed to solve it, which is compared with several meta-heuristic algorithms and the commercial optimization solver (Gurobi) through extensive simulation experiments. Finally, the experimental results verify the effectiveness of the proposed algorithm.

Credibility based chance constrained programming for parallel machine scheduling under linear deterioration and learning effects with considering setup times dependent on past sequences

The industry has expressed significant concern regarding the issue of parallel machines and the influence of learning and deterioration. This research investigates non-identical parallel machine scheduling, taking into account the simultaneous consideration of learning effects, deterioration, and past-sequence-dependent setup times. Due to the existence of uncertain parameters in real-world scenarios, the processing times and due dates are assumed to be triangular fuzzy numbers. A fuzzy nonlinear mathematical model with two objective functions is presented and solved using the fuzzy Chance Constraint Programming approach. The two objectives are the summation of earliness and tardiness, as well as makespan. To achieve an efficient near-optimal Pareto front for the problem, a hybrid NSGA-II and VNS multi-objective meta-heuristic is proposed and the results are discussed. Finally, the augmented ε-constraint method is utilized to address issues with small dimensions. The computational analysis demonstrates the effectiveness of this proposed algorithm in tackling problems, especially those with substantial dimensions.

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Purpose This research deals with a scheduling problem that minimizes weighted sum of earliness and tardiness penalties in a single setup-operator and non-identical parallel machine system. Methods We first present a mixed integer linear programming formulation for the problem, and then, show the optimal solutions for relatively small problems can be easily found with the model and a commercial optimization software. However, since the problem is NP-hard and the size of a real problem can be large, we propose three heuristic algorithms including genetic algorithm to solve the practical big-size problems in a reasonable computational time. Results Through the computational experiments we found the heuristic algorithms show very good performances for the practical big-size problems. Conclusion The heuristic procedures proposed in this study can easily be used in the real situations.

Solving multi-objective Modified Distributed Parallel Machine and Assembly Scheduling Problem (MDPMASP) with eligibility constraints using metaheuristics

ABSTRACT We present a new generalization and metaheuristics for the distributed parallel machine and assembly scheduling problem (DPMASP), namely MDPMASP with eligibility constraints. There is a set of unrelated factories or production lines. Each has a series of non-identical parallel machines with varying processing speeds. Then, it was disposed of as a single assembly machine in a series. Each product is assembled from a set of components (jobs). Each item necessitates multiple unidentical jobs. The objectives are to minimize mean flow time and the number of tardy jobs. We suggest four basic heuristics and three metaheuristics to tackle the problem. The Taguchi approach is used to discuss and calibrate various metaheuristic parameters. Algorithms are compared using the four performance measures. The computational results show that the proposed algorithms can solve moderate-sized problems efficiently and near-optimal solutions in most cases. Moreover, based on the four performance measures, MGA is the best method, followed by MSA, MPSO, SH2, SH4, SH1 and SH3.

Optimizing multiple qualifications of products on non-identical parallel machines

In some manufacturing contexts, such as semiconductor manufacturing, machines must be qualified, or eligible, to process a product, and machines cannot be qualified for all products. This paper investigates the problem of optimizing a given number of new qualifications of products to machines to maximize a flexibility measure that evaluates the balance of the qualification configuration of a work center in terms of utilization rate of machines on a set of non-identical parallel machines. Motivated by empirical observations, new solution approaches, notably inspired by heuristics for discrete location problems and based on the analysis of dual variables, are proposed and compared on industrial data from a semiconductor manufacturing facility and on randomly instances. The use of dual variables leads to heuristics that are effective both in terms of solution quality and computational time. The best proposed approach is currently used in the decision support system of a semiconductor manufacturing facility.

Non-identical parallel machines batch processing problem with release dates, due dates and variable maintenance activity to minimize total tardiness

Combination of job scheduling and maintenance activity has been widely investigated in the literature. We consider a non-identical parallel machines batch processing (BP) problem with release dates, due dates and variable maintenance activity to minimize total tardiness. An original mixed integer linear programming (MILP) model is formulated to provide an optimal solution. As the problem under investigation is known to be strongly NP-hard, two meta-heuristic approaches based on Simulated Annealing (SA) and Variable Neighborhood Search (VNS) are developed. A constructive heuristic method (H) is proposed to generate initial feasible solutions for the SA and VNS. In order to evaluate the results of the proposed solution approaches, a set of instances were randomly generated. Moreover, we compare the performance of our proposed approaches against four meta-heuristic algorithms adopted from the literature. The obtained results indicate that the proposed solution methods have a competitive behaviour and they outperform the other meta-heuristics in most instances. Although in all cases, H + SA is the most performing algorithm compared to the others.

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.

Tabu Search Algorithm Based on Lower Bound and Exact Algorithm Solutions for Minimizing the Makespan in Non-Identical Parallel Machines Scheduling

Recently, several heuristics have been interested in scheduling problems, especially those that are difficult to solve via traditional methods, and these are called NP-hard problems. As a result, many methods have been proposed to solve the difficult scheduling problems; among those, effective methods are the tabu search algorithm (TS), which is characterized by its high ability to adapt to problems of the large size scale and ease of implementation and gives solution closest to the optimum, but even though those difficult problems are common in many industries, there are only a few numbers of previous studies interested in the scheduling of jobs on unrelated parallel machines. In this paper, a developed TS algorithm based on lower bound (LB) and exact algorithm (EA) solutions is proposed with the objective of minimizing the total completion time (makespan) of jobs on nonidentical parallel machines. The given solution via EA was suggested to enhance and assess the solution obtained from TS. Moreover, the LB algorithm was developed to evaluate the quality of the solution that is supposed to be obtained by the developed TS algorithm and, in addition, to reduce the period for searching for the optimal solution. Two numerical examples from previous studies from the literature have been solved using the developed TS algorithm. Findings show that the developed TS algorithm proved its superiority and speed in giving it the best solution compared to those solutions previously obtained from the literature.

An effective hybrid meta-heuristic for flexible flow shop scheduling with limited buffers and step-deteriorating jobs

This paper addresses a flexible flow shop scheduling problem considering limited buffers and step-deteriorating jobs, where there are multiple non-identical parallel machines. A mixed integer programming model is proposed, with the criterion of minimizing the makespan and total tardiness simultaneously. To handle this problem, an effective hybrid meta-heuristic algorithm, named GVNSA, is developed based on genetic algorithm (GA), variable neighborhood search (VNS) and simulated annealing (SA). In the algorithm, with a two-dimensional matrix encoding scheme, the NEH (Nawaz–Enscore–Ham) heuristic and bottleneck elimination method are implemented to determine the initial population. A three-level rolling translation approach is designed for decoding. To balance the exploration and exploitation abilities, three effective steps are executed: 1) partial matching crossover and mutation strategy based on multiple neighborhood search structures are imposed on the GA operators; 2) a VNS with SA is introduced to re-optimize some individuals from GA, where four neighborhood structures are constructed; 3) a modified CDS (Campbell–Dudek–Smith) heuristic is embedded to disturb population in the mid-iteration. Numerical experiments are carried out on test problems with different scales. Computational results demonstrate that the proposed GVNSA can obtain higher quality solutions in comparison with other heuristics and meta-heuristics existing in literature.

An iterated local search procedure for the job sequencing and tool switching problem with non-identical parallel machines

In this paper, a new generalization of the uniform job sequencing and tool switching problem is presented. It considers non-identical parallel machines, which differ in tool magazine capacity and setup times. Applications of the problem can be found in the metal working or semiconductor manufacturing industries when the jobs require different tool sets for processing. The paper provides a Mixed Integer Programming formulation for the job sequencing and tool switching problem with machine-dependent processing and tool switching times under the consideration of three conflicting objectives (minimizing the total flowtime, minimizing the makespan and minimizing the total number of tool switches). Different efficient construction heuristics and Iterated Local Search methods are proposed and evaluated, using 640 new and publicly available instances. The results of the construction heuristics, the Iterated Local Search schemes and a Mixed Integer Programming Solver are discussed. The extensive computational experiments show the merit of the perturbation strategy in order to overcome local optima. It is shown that certain methods are more or less suitable depending on the objective function.

Unrelated parallel machine scheduling with new criteria: Complexity and models

In this paper, a scheduling problem on non-identical parallel machines with auxiliary resources and sequence-dependent and machine-dependent setup times is studied. This problem can be found in various manufacturing contexts, and in particular in workshops of wafer manufacturing facilities. Three different criteria are defined and analyzed: The number of products completed before the end of a given time horizon, the weighted sum of completion times and the number of auxiliary resource moves. The first criterion is maximized, while the two others are minimized. The first and the third criteria are not classical in scheduling theory, but are justified in industrial settings. The complexity of the problem with each of the new criteria is characterized. Integer linear programming models are also proposed and numerical experiments are conducted to analyze their behavior.

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This research deals with a scheduling problem for two-stage flow line, where each stage of the line consists of non-identical parallel machines. We first present a mixed integer linear programming formulation for the problem, and using this formulation, we can easily find the optimal solutions for small-sized problems with any commercial optimization software. However, since the problem is NP-hard and the size of a real problem is large in general, two genetic algorithm based heuristics are proposed to solve the practical big-size problems in a reasonable computational time. And then, to assess the performances of the algorithms, we conduct some computational experiment, from which we found both the heuristic algorithms show very good performances.

Scheduling parallel-batching processing machines problem with learning and deterioration effect in fuzzy environment

In this paper, a problem of scheduling jobs with different sizes and fuzzy processing times (FPT) on non-identical parallel batch machines to minimize makespan is investigated. Moreover, the processing time (PT) of each batch is subject to the location-based learning and total-PT-based deterioration effect. Since this is an NP-hard combinatorial optimization problem, an improved intelligent algorithm based on fruit fly optimization algorithm (IFOA) is proposed. To verify the performance of the algorithm, the IFOA is compared with three state-of-the-art algorithms. The comparative results demonstrate that the proposed IFOA outperforms the other compared algorithms.