Non-identical parallel machine scheduling using genetic algorithm

Most of the scheduling problems are NP-hard. In the literature, several heuristics and dispatching rules are proposed to solve such hard combinatorial optimization problems and genetic algorithm (GA) ranks among the most preferred ones in view of its characteristics such as high adaptability, near optimization and easy realization. But, even though it is a common problem in the industry, only a small number of studies deal with non-identical parallel machines. In this paper, the authors propose a new “crossover operator” and a new “optimality criterion” in order to adapt the GA to non-identical parallel machine scheduling problem. New algorithm is tested on a numerical example by implementing it in a simulation software and computational results are compared to those obtained with LPT (Longest Processing Time) dispatching rule; results were promising. Findings show that, in addition to its high computational speed for larger scale problem, the GA proposed here fits the non-identical parallel machine scheduling problem of minimizing the maximum completion time (makespan).     

A linguistic approach to non-identical parallel processor scheduling with fuzzy processing times

This study presents an application of non-identical parallel processor scheduling under uncertain operation times. We have been motivated from a real case scheduling problem that contains some uncommon welding operations to be processed by workers in an automotive subcontract company. Here each operator may weld each job but in different processing times depending on learning effect because of operator’s ability and experience, and batch sizes. To determine the crisp operation times in such a fuzzy environment, a linguistic reasoning approach (with a 75-“If- Then” rules) considering the learning effect is proposed in the study. Since the fuzzy linguistic approach allows the representation of expert information more directly and adequately, it can be more possible to make realistic schedules under uncertainty. With the objective to balance the workload among all operators, the longest processing time heuristic algorithm is been used and measured average makespan. For evaluating the effectiveness of this approach, it is compared with the scheduling method that use the random operation times generated from a uniform distribution. Results showed that the proposed fuzzy linguistic scheduling approach has balanced the workload of operators with a standard deviation of 0.37 and improved the Cmax value as 16%. A general conclusion can be drawn the proposed approach is able to generate realistic schedules and especially useful to solve non-identical parallel processor scheduling problem under uncertainty. An important contribution of this study is that Mamdani inference method with learning effect is the first time used to obtain the crisp processing times of non-identical processors by the help of a rule base with expert knowledge.     

A genetic algorithm application using fuzzy processing times in non-identical parallel machine scheduling problem

There are many scheduling problems which are NP-hard in the literature. Several heuristics and dispatching rules are proposed to solve such hard combinatorial optimization problems. Genetic algorithms (GA) have shown great advantages in solving the combinatorial optimization problems in view of its characteristic that has high efficiency and that is fit for practical application [1]. Two different scale numerical examples demonstrate the genetic algorithm proposed is efficient and fit for larger scale identical parallel machine scheduling problem for minimizing the makespan. But, even though it is a common problem in the industry, only a small number of studies deal with non-identical parallel machines. In this article, a kind of genetic algorithm based on machine code for minimizing the processing times in non-identical machine scheduling problem is presented. Also triangular fuzzy processing times are used in order to adapt the GA to non-identical parallel machine scheduling problem in the paper. Fuzzy systems are excellent tools for representing heuristic, commonsense rules. That is why we try to use fuzzy systems in this study.     

Ant colony optimization algorithm for scheduling jobs with fuzzy processing time on parallel batch machines with different capacities

We study the problem of scheduling on parallel batch processing machines with different capacities under a fuzzy environment to minimize the makespan. The jobs have non-identical sizes and fuzzy processing times. After constructing a mathematical model of the problem, we propose a fuzzy ant colony optimization (FACO) algorithm. Based on the machine capacity constraint, two candidate job lists are adopted to select the jobs for building the batches. Moreover, based on the unoccupied space of the solution, heuristic information is designed for each candidate list to guide the ants. In addition, a fuzzy local optimization algorithm is incorporated to improve the solution quality. Finally, the proposed algorithm is compared with several state-of-the-art algorithms through extensive simulated experiments and statistical tests. The comparative results indicate that the proposed algorithm can find better solutions within reasonable time than all the other compared algorithms.     

Modified genetic algorithms for solving fuzzy flow shop scheduling problems and their implementation with CUDA

In this paper we propose an improved algorithm to search optimal solutions to the flow shop scheduling problems with fuzzy processing times and fuzzy due dates. A longest common substring method is proposed to combine with the random key method. Numerical simulation shows that longest common substring method combined with rearranging mating method improves the search efficiency of genetic algorithm in this problem. For application in large-sized problems, we also enhance this modified algorithm by CUDA based parallel computation. Numerical experiments show that the performances of the CUDA program on GPU compare favorably to the traditional programs on CPU. Based on the modified algorithm invoking with CUDA scheme, we can search satisfied solutions to the fuzzy flow shop scheduling problems with high performance.     

Metaheuristics and exact methods to solve a multiobjective parallel machines scheduling problem

This paper deals with a multiobjective parallel machines scheduling problem. It consists in scheduling n independent jobs on m identical parallel machines. The job data such as processing times, release dates, due dates and sequence dependent setup times are considered. The goal is to optimize two different objectives: the makespan and the total tardiness. A mixed integer linear program is proposed to model the studied problem. As this problem is NP-hard in the strong sense, a metaheuristic method which is the second version of the non dominated sorting genetic algorithm (NSGA-II) is proposed to solve this problem. Since the parameters setting of a genetic algorithm is difficult, a fuzzy logic controller coupled with the NSGA-II (FLC-NSGA-II) is therefore proposed. The role of the fuzzy logic is to better set the crossover and the mutation probabilities in order to update the search ability. After that, an exact method based on the two phase method is also developed. We have used four measuring criteria to compare these methods. The experimental results show the advantages and the efficiency of FLC-NSGA-II.     

A fast recursive algorithm based on fuzzy 2-partition entropy approach for threshold selection

The fuzzy c-partition entropy approach for threshold selection is an effective approach for image segmentation. The approach models the image with a fuzzy c-partition, which is obtained using parameterized membership functions. The ideal threshold is determined by searching an optimal parameter combination of the membership functions such that the entropy of the fuzzy c-partition is maximized. It involves large computation when the number of parameters needed to determine the membership function increases. In this paper, a recursive algorithm is proposed for fuzzy 2-partition entropy method, where the membership function is selected as S-function and Z-function with three parameters. The proposed recursive algorithm eliminates many repeated computations, thereby reducing the computation complexity significantly. The proposed method is tested using several real images, and its processing time is compared with those of basic exhaustive algorithm, genetic algorithm (GA), particle swarm optimization (PSO), ant colony optimization (ACO) and simulated annealing (SA). Experimental results show that the proposed method is more effective than basic exhaustive search algorithm, GA, PSO, ACO and SA.     

An improved artificial bee colony algorithm for flexible job-shop scheduling problem with fuzzy processing time

This study addresses flexible job-shop scheduling problem (FJSP) with fuzzy processing time. An improved artificial bee colony (IABC) algorithm is proposed for FJSP cases defined in existing literature and realistic instances in remanufacturing where the uncertainty of the processing time is modeled as fuzzy processing time. The objectives are to minimize the maximum fuzzy completion time and the maximum fuzzy machine workload, respectively. The goal is to make the scheduling algorithm as part of expert and intelligent scheduling system for remanufacturing decision support. A simple and effective heuristic rule is developed to initialize population. Extensive computational experiments are carried out using five benchmark cases and eight realistic instances in remanufacturing. The proposed heuristic rule is evaluated using five benchmark cases for minimizing the maximum fuzzy completion time and the maximum fuzzy machine workload objectives, respectively. IABC algorithm is compared to six meta-heuristics for maximum fuzzy completion time criterion. For maximum fuzzy machine workload, IABC algorithm is compared to six heuristics. The results and comparisons show that IABC algorithm can solve FJSP with fuzzy processing time effectively, both benchmark cases and real-life remanufacturing instances. For practical remanufacturing problem, the schedules by IABC algorithm can satisfy the requirement in real-life shop floor. The IABC algorithm can be as part of expert and intelligent scheduling system to supply decision support for remanufacturing scheduling and management.     

Position tracking of a 3-PSP parallel robot using dynamic growing interval type-2 fuzzy neural control

Parallel robots have complicated structures as well as complex dynamic and kinematic equations, rendering model-based control approaches as ineffective due to their high computational cost and low accuracy. Here, we propose a model-free dynamic-growing control architecture for parallel robots that combines the merits of self-organizing systems with those of interval type-2 fuzzy neural systems. The proposed approach is then applied experimentally to position control of a 3-PSP (Prismatic–Spherical–Prismatic) parallel robot. The proposed rule-base construction is different from most conventional self-organizing approaches by omitting the node pruning process while adding nodes more conservatively. This helps preserve valuable historical rules for when they are needed. The use of interval type-2 fuzzy logic structure also better enables coping with uncertainties in parameters, dynamics of the robot model and uncertainties in rule space. Finally, the adaptation structure allows learning and further adapts the rule base to changing environment. Multiple simulation and experimental studies confirm that the proposed approach leads to fewer rules, lower computational cost and higher accuracy when compared with two competing type-1 and type-2 fuzzy neural controllers.     

Multiprocessor Task Assignment with Fuzzy Hopfield Neural Network Clustering Technique

Most scheduling applications have been demonstrated as NP-complete problems. A variety of schemes are introduced in solving those scheduling applications, such as linear programming, neural networks, and fuzzy logic. In this paper, a new approach of first analogising a scheduling problem to a clustering problem and then using a fuzzy Hopfield neural network clustering technique to solve the scheduling problem is proposed. This fuzzy Hopfield neural network algorithm integrates fuzzy c-means clustering strategies into a Hopfield neural network. This investigation utilises this new approach to demonstrate the feasibility of resolving a multiprocessor scheduling problem with no process migration and constrained times (execution time and deadline). Each process is regarded as a data sample, and every processor is taken as a cluster. Simulation results illustrate that imposing the fuzzy Hopfield neural network onto the proposed energy function provides an appropriate approach to solving this class of scheduling problem.       

Parallel machine total tardiness scheduling with a new hybrid metaheuristic approach

This work proposes a hybrid metaheuristic (HMH) approach which integrates several features from tabu search (TS), simulated annealing (SA) and variable neighbourhood search (VNS) in a new configurable scheduling algorithm. In particular, either a deterministic or a random candidate list strategy can be used to generate the neighbourhood of a solution, both a tabu list mechanism and the SA probabilistic rule can be adopted to accept solutions, and the dimension of the explored neighbourhood can be dynamically modified. The considered class of scheduling problems is characterized by a set of independent jobs to be executed on a set of parallel machines with non-zero ready times and sequence dependent setups. In particular, the NP-hard generalized parallel machine total tardiness problem (GPMTP) recently defined by Bilge et al. [A tabu search algorithm for parallel machine total tardiness problem. Computers & Operations Research 2004;31:397–414], is faced. Several alternative configurations of the HMH have been tested on the same benchmark set used by Bilge et al. The results obtained highlight the appropriateness of the proposed approach.     

Single machine scheduling problems under position-dependent fuzzy learning effect with fuzzy processing times

In this paper, we consider single machine scheduling problems under position-dependent fuzzy learning effect with fuzzy processing times. We study three objectives which are to minimize makespan, total completion time and total weighted completion time. Furthermore, we show that these three problems are polynomially solvable under position-dependent fuzzy learning effects with fuzzy processing times. In order to model the uncertainty of fuzzy model parameters such as processing time and learning effect, we use an approach called likelihood profile that depends on the possibility and necessity measures of fuzzy parameters. For three objective functions, we build Fuzzy Mixed Integer Nonlinear Programming (FMINP) models using dependent chance constrained programming techniques for the same predetermined confidence levels. Furthermore, we present polynomially solvable algorithms for different confidence levels for these problems.     

Dynamic task scheduling modeling in unstructured heterogeneous multiprocessor systems

An algorithm is proposed for scheduling dependent tasks in time-varying heterogeneous multiprocessor systems, in which computational power and links between processors are allowed to change over time. Link contention is considered in the multiprocessor scheduling problem. A linear switching-state space-modeling paradigm is introduced to enable theoretical analysis from a system engineering perspective. Theoretical analysis of this model shows its robustness against changes in processing power and link failure. The proposed algorithm uses a fuzzy decision-making procedure to handle changes in the multiprocessor system. The efficiency of the proposed algorithm is illustrated by several random experiments and comparison against a recent benchmark approach. The results show up to 18% average improvement in makespan, especially for larger scale systems.     

A fast estimation of distribution algorithm for dynamic fuzzy flexible job-shop scheduling problem

Due to the complicated circumstances in workshop, most of the conventional scheduling algorithms fail to meet the requirements of instantaneity, complexity, and dynamicity in job-shop scheduling problems. Compared with the static algorithms, dynamic scheduling algorithms can better fulfill the requirements in real situations. Considering that both flexibility and fuzzy processing time are common in reality, this paper focuses on the dynamic flexible job-shop scheduling problem with fuzzy processing time (DfFJSP). By adopting a series of transforming procedures, the original DfFJSP is simplified as a traditional static fuzzy flexible job-shop problem, which is more suitable to take advantage of the existing algorithms. In this paper, estimation of distribution algorithm (EDA) is brought into address the post-transforming problem. An improved EDA is developed through making use of several elements omitted in original EDA, including the historical-optimal solution and the standardized solution vectors. The improved algorithm is named as fast estimation of distribution algorithm (fEDA) since it performs better in convergence speed and computation precision, compared with the original EDA. To sum up, the ingenious transformation and the effective fEDA algorithm provide an efficient and practical way to tackle the dynamic flexible fuzzy job-shop scheduling problem.     

Electromagnetism-like mechanism and simulated annealing algorithms for flowshop scheduling problems minimizing the total weighted tardiness and makespan

This paper presents an efficient meta-heuristic algorithm based on electromagnetism-like mechanism (EM), in which has been successfully implemented in a few combinatorial problems. We propose the EM for scheduling the flow shop problem that minimizes the makespan and total weighted tardiness and considers transportation times between machines and stage skipping (i.e., some jobs may not need to be processed on all the machines). To show the efficiency of this proposed algorithm, we also apply simulated annealing (SA) and some other well-recognized constructive heuristics, such as SPT, NEH, (g/2, g/2) Johnson’ rule, EWDD, SLACK, and NEH_EWDD for the given problems. To evaluate the performance and robustness of our proposed EM, we experiment a number of test problems. Our computational results show that our proposed EM in almost all cases outperforms SA and other foregoing heuristics applied to this paper.     

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.     

Parallel machine scheduling in fuzzy environment with hybrid ant colony optimization including a comparison of fuzzy number ranking methods in consideration of spread of fuzziness

This paper studies parallel machine scheduling problems in consideration of real world uncertainty quantified based on fuzzy numbers. Although this study is not the first to study the subject problem, it advances this area of research in two areas: (1) Rather than arbitrarily picking a method, it chooses the most appropriate fuzzy number ranking method based on an in-depth investigation of the effect of spread of fuzziness on the performance of fuzzy ranking methods; (2) It develops the first hybrid ant colony optimization for fuzzy parallel machine scheduling. Randomly generated datasets are used to test the performance of fuzzy ranking methods as well as the proposed algorithm, i.e. hybrid ant colony optimization. The proposed hybrid ant colony optimization outperforms a hybrid particle swarm optimization published recently and two simulated annealing based algorithms modified from our previous work.     

A note on machine scheduling with sum-of-logarithm-processing-time-based and position-based learning effects

Recently, Biskup [2] classifies the learning effect models in scheduling environments into two types: position-based and sum-of-processing-time-based. In this paper, we study scheduling problem with sum-of-logarithm-processing-time-based and position-based learning effects. We show that the single machine scheduling problems to minimize the makespan and the total completion time can both be solved by the smallest (normal) processing time first (SPT) rule. We also show that the problems to minimize the maximum lateness, the total weighted completion times and the total tardiness have polynomial-time solutions under agreeable WSPT rule and agreeable EDD rule. In addition, we show that m-machine permutation flowshop problems are still polynomially solvable under the proposed learning model.     

A meta-heuristic algorithm for the fuzzy assembly line balancing type-E problem

Consideration is given to a single-model assembly line balancing problem with fuzzy task processing times. The problem referred to herein as f-SALBP-E consists of finding a combination of the number of workstations and the cycle time as well as a respective line balance such that the efficiency of the line is maximized. f-SALBP-E is an extension of the classical SALBP-E under fuzziness. First, a formulation of the problem is given with the tasks processing times presented by triangular fuzzy membership functions. Then, since the problem is known to be NP-hard, a meta-heuristic based on a Genetic Algorithm (GA) is developed for its solution. The performance of the proposed solution approach is studied and discussed over multiple benchmarks test problems taken from the open literature. The results demonstrate very satisfactory performance for the developed approach in terms of both solution time and quality.     

A palmer-based continuous fuzzy flexible flow-shop scheduling algorithm

 Flexible flow shops can be thought of as generalizations of simple flow shops. In the past, the processing time for each job was usually assumed to be known exactly, but in many real-world applications, processing times may vary dynamically due to human factors or operating faults. In the past, we demonstrated how discrete fuzzy concepts could easily be used in the Palmer algorithm for managing uncertain flexible-flow-shop scheduling. In this paper, we generalize it to continuous fuzzy domains. We use triangular membership functions for flexible flow shops with more than two machine centers to examine processing-time uncertainties and to make scheduling more suitable for real applications. We first use the triangular fuzzy LPT algorithm to allocate jobs, and then use the triangular fuzzy Palmer algorithm to deal with sequencing the tasks. The proposed method thus provides a more flexible way of scheduling jobs than conventional scheduling methods.