Permutation Flow Shop Scheduling Research Articles

Learning to Optimize Permutation Flow Shop Scheduling via Graph-Based Imitation Learning

The permutation flow shop scheduling (PFSS), aiming at finding the optimal permutation of jobs, is widely used in manufacturing systems. When solving large-scale PFSS problems, traditional optimization algorithms such as heuristics could hardly meet the demands of both solution accuracy and computational efficiency, thus learning-based methods have recently garnered more attention. Some work attempts to solve the problems by reinforcement learning methods, which suffer from slow convergence issues during training and are still not accurate enough regarding the solutions. To that end, we propose to train the model via expert-driven imitation learning, which accelerates convergence more stably and accurately. Moreover, in order to extract better feature representations of input jobs, we incorporate the graph structure as the encoder. The extensive experiments reveal that our proposed model obtains significant promotion and presents excellent generalizability in large-scale problems with up to 1000 jobs. Compared to the state-of-the-art reinforcement learning method, our model's network parameters are reduced to only 37% of theirs, and the solution gap of our model towards the expert solutions decreases from 6.8% to 1.3% on average. The code is available at: https://github.com/longkangli/PFSS-IL.

A distributed permutation flow-shop considering sustainability criteria and real-time scheduling

Recent advancements in production scheduling have arisen in response to the need for adaptation in dynamic environments. This paper addresses the challenge of real-time scheduling within the context of sustainable production. We redefine the sustainable distributed permutation flow-shop scheduling problem using an online mixed-integer programming model. The proposed model prioritizes minimizing makespan while simultaneously constraining energy consumption, reducing the number of lost working days and increasing job opportunities within permissible limits. Our approach considers machines operating in different modes, ranging from manual to automatic, and employs two real-time scheduling strategies: predictive-reactive and proactive-reactive scheduling. We evaluate two rescheduling policies: continuous and event-driven. To demonstrate the model's applicability, we present a case study in auto workpiece production. We manage model complexity through various reformulations and heuristics, such as Lagrangian relaxation and Benders decomposition for initial optimization as well as four problem-specific heuristics for real-time considerations. For solving large-scale instances, we employ simulated annealing and tabu search metaheuristic algorithms. Our findings underscore the benefits of the predictive-reactive scheduling strategy and the efficiency of the event-driven rescheduling policy. By addressing dynamic scheduling challenges and integrating sustainability criteria, this study contributes valuable insights into real-time scheduling and sustainable production.

No-Wait Flow Shop scheduling problem: a systematic literature review and bibliometric analysis

One of the most widely studied problems in flow shop scheduling is not allowing jobs to wait to be processed at the next stage. This constraint causes the job to be processed immediately at the next stage without waiting, so this problem is popularly called the No-Wait Flow Shop. This article aims to provide a comprehensive review of the No-Wait Flow Shop Scheduling (NWFS) problem based on a survey of published articles from 1999 to 2023. The article review is based on a systematic literature review, and bibliometric analysis is also presented based on the network processed using VOSviewer. One hundred twenty articles were collected from the Scopus database, which was reviewed based on NWFS variants, objective functions, and optimization procedures. The no-wait permutation flow shop scheduling (NWPFS) problem is a variant that researchers have widely investigated. Meta-heuristic procedures are widely applied to solve NWFS problems. In addition, the objective function of minimizing makespan is an objective function that researchers often apply. NWFS research gaps and future research trends are also presented in this paper.

Ensemble evolutionary algorithms equipped with Q‐learning strategy for solving distributed heterogeneous permutation flowshop scheduling problems considering sequence‐dependent setup time

AbstractA distributed heterogeneous permutation flowshop scheduling problem with sequence‐dependent setup times (DHPFSP‐SDST) is addressed, which well reflects real‐world scenarios in heterogeneous factories. The objective is to minimise the maximum completion time (makespan) by assigning jobs to factories, and sequencing them within each factory. First, a mathematical model to describe the DHPFSP‐SDST is established. Second, four meta‐heuristics, including genetic algorithms, differential evolution, artificial bee colony, and iterated greedy (IG) algorithms are improved to optimally solve the concerned problem compared with the other existing optimisers in the literature. The Nawaz‐Enscore‐Ham (NEH) heuristic is employed for generating an initial solution. Then, five local search operators are designed based on the problem characteristics to enhance algorithms' performance. To choose the local search operators appropriately during iterations, Q‐learning‐based strategy is adopted. Finally, extensive numerical experiments are conducted on 72 instances using 5 optimisers. The obtained optimisation results and comparisons prove that the improved IG algorithm along with Q‐learning based local search selection strategy shows better performance with respect to its peers. The proposed algorithm exhibits higher efficiency for scheduling the concerned problems.

A regional biogeography-based optimization algorithm for the distributed assembly permutation flow-shop scheduling problem with fuzzy processing time

For solving the distributed assembly flow shop scheduling problem with fuzzy processing time (FDAPFSP), a regional biogeography-based optimization algorithm (RBBO) is proposed to minimize the maximum fuzzy completion time. The mathematical model is provided. In RBBO, all habitats are divided into regions based on the habitat suitability index, and the habitats of each region are subject to cross-regional migration and replacement procedures. A critical factory optimization strategy is developed to enhance local search capability. Taguchi method is used to determine the parameters of RBBO. In ten FDAPFSP instances, comparative testing of RBBO algorithm with various heuristic and swarm intelligence algorithms are conducted. The computation results show that in ten FDAPFSP cases, the proposed RBBO outperforms other algorithms in nine out of ten FDAPFSP cases.

A Q-learning memetic algorithm for energy-efficient heterogeneous distributed assembly permutation flowshop scheduling considering priorities

Most studies on distributed assembly permutation flowshop scheduling do not consider product priorities and factory heterogeneity. This causes delays in critical products and cannot reflect the real-world production situation. This paper focuses on the energy-efficient heterogeneous distributed assembly permutation flowshop scheduling considering priorities (EHDAPFS-P) to minimize total tardiness and total energy consumption simultaneously. Unlike traditional models, factory heterogeneity and product priorities are considered to better reflect the production environment and customer satisfaction in real-world situations. Then, a Q-learning memetic algorithm (QLMA) is proposed to solve this problem: (i) a high-quality initial population is obtained using a hybrid initialization strategy that combines four problem-specific heuristics; (ii) six efficient neighborhood structures are tailored to guide the population to converge to the promising areas; (iii) the most useful neighborhood structure is selected among the six structures using the Q-learning algorithm to accelerate the convergence, thus maximizing the cumulative and future improvements according to the population state; and (iv) an energy-saving strategy is developed to optimize the total energy consumption without deteriorating the total tardiness. The proposed QLMA is compared with seven state-of-the-art algorithms on 261 benchmark instances to demonstrate its superiority or at least competitiveness.

A Knowledge-Guided End-to-End Optimization Framework Based on Reinforcement Learning for Flow Shop Scheduling

Designing an effective and efficient end-to-end optimization framework with good generalization for shop scheduling is an emerging topic in the informational manufacturing system. Existing end-to-end frameworks have achieved satisfactory results for COPs such as traveling salesman problem and vehicle routing problem. However, the performances of these methods in solving complex COPs such as shop scheduling need to be improved. In this paper, a knowledge-guided end-to-end optimization framework based on reinforcement learning is proposed to solve the permutation flow shop scheduling problem (PFSP). Firstly, a new policy network is designed based on the problem characteristics to deal with different scales of PFSPs and achieve iterative end-to-end generation. Secondly, an improved policy-based reinforcement learning algorithm by using the knowledge accumulated during the training process is designed to enhance the training quality. Thirdly, a knowledge-guided improvement strategy is introduced through the cooperation of local search and supervised learning to improve the learning of the policy. Simulation results and comparisons show that the knowledge-guided end-to-end optimization framework can obtain better results than different kinds of commonly-used optimization methods in limited computation time for solving the PFSP.

An improved iterated greedy algorithm for distributed mixed no-wait permutation flowshop problems with makespan criterion

The distributed permutation flowshop scheduling is a critical issue in various industries, involving jobs allocation and scheduling among multiple flowshops. This paper extends the research to explore the Distributed Mixed No-Wait Permutation Flowshop Scheduling Problems (DMNWPFSP) with minimizing makespan. The innovation lies in an optimized mathematical model, hybrid heuristic algorithms, an improved iterated greedy algorithm (IIG), and high-quality solutions. Extensive experimental results demonstrate the effectiveness and superiority of the proposed IIG in terms of scheduling quality, computational efficiency, and robustness compared to existing approaches. The outcomes of this work contribute to the field of distributed flowshop scheduling, providing valuable insights for practitioners seeking to enhance production efficiency and competitiveness.

Parameter less hybrid IG-Jaya approach for permutation flow shop scheduling problem

Parameter less hybrid IG-Jaya approach for permutation flow shop scheduling problem

An Optimization Model for Smart and Sustainable Distributed Permutation Flow Shop Scheduling

Smart production scheduling has gained significant attention due to advancements in industrial informatics and technologies that enable the monitoring, control, and adaptation of task scheduling in response to disruptive events. These events can include machine breakdowns, variations in task processing times, and the arrival of new or unexpected tasks. Concurrently, sustainable production scheduling aims to optimize task scheduling by considering economic, environmental, and social factors. This paper introduces a novel optimization model for the development of smart and sustainable production scheduling in a distributed permutation flow shop. The proposed model aims to minimize the makespan while simultaneously limiting the number of lost working days and energy consumption. It also strives to increase job opportunities within acceptable limits. To evaluate the proposed model, we conduct numerical simulations using various examples and a real-case study focusing on auto workpiece production. The results demonstrate the superior performance of the proposed model. Sensitivity analyses are performed to assess the model's ability to deal with disruptions and uncertainties while satisfying economic, environmental, and social considerations.

An adaptive dynamic neighborhood crow search algorithm for solving permutation flow shop scheduling problems

An adaptive dynamic neighborhood crow search algorithm for solving permutation flow shop scheduling problems

A self-learning knowledge-based MOEA/D for distributed heterogeneous assembly permutation flowshop scheduling with batch delivery

With the requirements of just-in-time and make-to-order, it is necessary to consider the batch delivery in distributed heterogeneous assembly flow shop scheduling as well as the minimization of inventory and tardiness costs. Hence, this work proposes a new mixed integer linear programming model and a self-learning knowledge-based multi-objective evolutionary algorithm based on decomposition. In this algorithm, 4 problem-specific properties are extracted and embedded in the designed solution representation, forward-reverse objective calculation and local search operators. A new hybrid initialization with six knowledge-based NEH heuristics and a random generation is developed to generate better initial solutions. A self-learning selection strategy is developed to coordinate the above knowledge-based local search operators. The experimental study, based on 900 small and 540 large instances, demonstrates that the designed components such as hybrid initialization and self-learning selection strategy are effective, and the proposed algorithm is superior to 6 existing algorithms in the literature in solving this new problem.

A modified Aquila optimizer algorithm for optimization energy-efficient no-idle permutation flow shop scheduling problem

Increasing energy consumption has faced challenges and pressures for modern manufacturing operations. The production sector accounts for half of the world's total energy consumption. Reducing idle machine time by em­ploying No-Idle Permutation Flow Shop Scheduling (NIPFSP) is one of the best decisions for reducing energy consumption. This article modifies one of the energy consumption-solving algorithms, the Aquila Optimizer (AO) algo­rithm. This research contributes by 1) proposing novel AO procedures for solving energy consumption problems with NIPFSP and 2) expanding the literature on metaheuristic algorithms that can solve energy consumption problems with NIPFSP. To analyze whether the AO algorithm is optimal, we compared by using the Grey Wolf Optimizer (GWO) algorithm. It com­pares these two algorithms to tackle the problem of energy consumption by testing four distinct problems. Comparison of the AO and GWO algorithm is thirty times for each case for each population and iteration. The outcome of comparing the two algorithms is using a t-test on independent samples and ECR. In all case studies, the results demonstrate that the AO algorithm has a lower energy consumption value than GWO. The AO algorithm is there­fore recommended for minimizing energy consumption because it can produce more optimal results than the comparison algorithm.

Solving the Distributed Permutation Flow-Shop Scheduling Problem Using Constrained Programming

The permutation flow-shop scheduling problem is a classical problem in scheduling that aims at identifying the optimal sequence of jobs that should be processed in a number of machines in an effort to minimize makespan or some other performance criterion. The distributed permutation flow-shop scheduling problem adds multiple factories where copies of the machines exist and asks for minimizing the makespan on the longest-running location. In this paper, the problem is approached using Constraint Programming and its specialized scheduling features, such as interval variables and non-overlap constraints, while a novel heuristic is proposed for computing lower bounds. Two constraint programming models are proposed: one that solves the Distributed Permutation Flow-shop Scheduling problem, and another one that drops the constraint of processing jobs under the same order for all machines of each factory. The experiments use an extended public dataset of problem instances to validate the approach’s effectiveness. In the process, optimality is proved for many problem instances known in the literature but has yet to be proven optimal. Moreover, a high speed of reaching optimal solutions is achieved for many problems, even with moderate big sizes (e.g., seven factories, 20 machines, and 20 jobs). The critical role that the number of jobs plays in the complexity of the problem is identified and discussed. In conclusion, this paper demonstrates the great benefits of scheduling problems that stem from using state-of-the-art constraint programming solvers and models that capture the problem tightly.

Deterministic constructive vN-NEH[formula omitted] algorithm to solve permutation flow shop scheduling problem with makespan criterion

Scheduling jobs in a permutation flow shop (PFS) environment is one of the most studied problems in scheduling theory and practice. In this paper, we propose a new efficient algorithm to solve the permutation flow shop scheduling problem (PFSP) with the makespan criterion. The proposed algorithm, referred to as vN-NEH+, extends the Nawaz–Enscore–Ham (NEH) heuristic by employing the variable-length list (vN-list) of candidate jobs. Extensive numerical experiments on the standard set of benchmarks show that the proposed vN-NEH+ algorithm is one of the most efficient NEH-based deterministic constructive algorithms in terms of trade-off between solution quality and computing time. Moreover, in contrary to existing scheduling methods, vN-NEH+ produces in a single run a set of good quality solutions that can serve as an initial population in population-based metaheuristics for solving PFSPs. Another advantage of vN-NEH+ is that it can work in parallel mode, which is difficult or even impossible with other NEH-based heuristics. In addition to vN-NEH+, we propose a new ART.NEH (Average Relative Time over NEH) indicator to assess the running time of PFSP algorithms. The main advantage of ART.NEH is its hardware/software and instance size independence, which allows an objective and reliable comparison of computational effort of PFSP algorithms. In addition, the idea behind ART.NEH can be easily adopted to reliably assess time complexity of algorithms for solving various other computational problems.

Hybrid grey wolf optimizer for solving permutation flow shop scheduling problem

SummaryThe permutation flow shop scheduling problem, as a classical problem in the scheduling field, is an NP‐hard problem. However, most of the reported algorithms are difficult to achieve good accuracy and efficiency. To address this problem, a hybrid grey wolf optimizer (HGWO) is proposed in this paper. First, one cooperative initialization strategy is proposed to improve the quality of the initial solution based on the improved Nawaz‐Enscore‐Ham (NEH) method and the tent chaotic map method. Second, a levy flight strategy is introduced to balance the exploitation and exploration of the algorithm for the problem's characteristics. Third, the crossover and mutation strategy, and the critical block exchange based on critical path strategy are proposed to avoid falling into the local optimum. In addition, for the best individual, the variable neighborhood descent strategy is proposed to enhance the convergence accuracy of the algorithm. To verify the performance of the proposed algorithm, three different types of instances are selected for comparison experiments with other existing methods, and the experimental results show that the proposed HGWO outperforms other comparison algorithms in solving the problem.

Decomposition Is All You Need: Single-Objective to Multi-Objective Optimization towards Artificial General Intelligence

As a new abstract computational model in evolutionary transfer optimization (ETO), single-objective to multi-objective optimization (SMO) is conducted at the macroscopic level rather than the intermediate level for specific algorithms or the microscopic level for specific operators; this method aims to develop systems with a profound grasp of evolutionary dynamic and learning mechanism similar to human intelligence via a “decomposition” style (in the abstract of the well-known “Transformer” article “Attention is All You Need”, they use “attention” instead). To the best of our knowledge, it is the first work of SMO for discrete cases because we extend our conference paper and inherit its originality status. In this paper, by implementing the abstract SMO in specialized memetic algorithms, key knowledge from single-objective problems/tasks to the multi-objective core problem/task can be transferred or “gathered” for permutation flow shop scheduling problems, which will reduce the notorious complexity in combinatorial spaces for multi-objective settings in a straight method; this is because single-objective tasks are easier to complete than their multi-objective versions. Extensive experimental studies and theoretical results on benchmarks (1) emphasize our decomposition root in mathematical programming, such as Lagrangian relaxation and column generation; (2) provide two “where to go” strategies for both SMO and ETO; and (3) contribute to the mission of building safe and beneficial artificial general intelligence for manufacturing via evolutionary computation.

A Reinforcement Learning Approach to Robust Scheduling of Permutation Flow Shop.

The permutation flow shop scheduling problem (PFSP) stands as a classic conundrum within the realm of combinatorial optimization, serving as a prevalent organizational structure in authentic production settings. Given that conventional scheduling approaches fall short of effectively addressing the intricate and ever-shifting production landscape of PFSP, this study proposes an end-to-end deep reinforcement learning methodology with the objective of minimizing the maximum completion time. To tackle PFSP, we initially model it as a Markov decision process, delineating pertinent states, actions, and reward functions. A notably innovative facet of our approach involves leveraging disjunctive graphs to represent PFSP state information. To glean the intrinsic topological data embedded within the disjunctive graph's underpinning, we architect a policy network based on a graph isomorphism network, subsequently trained through proximal policy optimization. Our devised methodology is compared with six baseline methods on randomly generated instances and the Taillard benchmark, respectively. The experimental results unequivocally underscore the superiority of our proposed approach in terms of makespan and computation time. Notably, the makespan can save up to 183.2 h in randomly generated instances and 188.4 h in the Taillard benchmark. The calculation time can be reduced by up to 18.70 s for randomly generated instances and up to 18.16 s for the Taillard benchmark.

Multi-Objective Optimization Flow Shop Scheduling Problem Solving the Makespan and Total Flow Time with Sequence Independent Setup Time

In this paper, a multi-objective permutation flow shop scheduling problem with sequence independent setup time is studied. The bi-objective function that we consider is a linear combination of the makespan and total flow time with a weighted factor for each criterion. We propose a set of exact and approximate methods to minimize the makespan and total flow time in our flow shop optimization problem case. Hence, our main goal is to find the sequence of jobs that minimizes the two criteria of makespan and total flow time. The purpose is to solve this problem, with a mixed integer linear programming model (MILP) and a collection of efficient metaheuristics for different sizes of instances. Moreover, three metaheuristics are used: the Genetic Algorithm (GA), the Iterative Local Search (ILS) algorithm and the Iterated Greedy (IG) algorithm. The three last algorithms GA, ILS and IG are suggested in two ways for exploring the neighborhood. In order to test the efficacy of our resolution approach, different series of instances containing n jobs and m machines are generated randomly ranging from small to relatively large instances. The examination of the suggested simulations allowed us to remark that, for large and medium-scale instances, IG based on the exploration of the neighborhood records the best performances in terms of comparison with the other metaheuristics.

Low-carbon no-idle permutation flow shop schedulling problem: giant trevally optimizer vs African vultures optimization algorithm

Greenhouse gas emissions continue to increase due to increased energy consumption. One of the largest emission-contributing sectors is the manufacturing industry. Therefore, the manufacturing industry is required to minimize carbon emissions. One of the efforts to solve the emission problem is to minimize machine downtime throughout the production procedure, which stands for no-idle permutation flowshop scheduling (NIPFSP). This article uses two metaheuristic algorithms, giant trevally optimizer (GTO) and African vultures optimization algorithm (AVOA), to solve the carbon emission problem. Both algorithms are tested on 3 cases with 30 runs for every population and iteration. To compare the outcomes of each algorithm, an independent sample t-test was employed. The results show that the GTO algorithm has better results than the AVOA algorithm on small and large case data. The findings indicate that both the GTO and AVOA algorithms yield comparable results when applied to medium-sized research datasets, suggesting their effectiveness in such scenarios.