Disjunctive Graph Model Research Articles

Study on the Multi-Equipment Integrated Scheduling Problem of a U-Shaped Automated Container Terminal Based on Graph Neural Network and Deep Reinforcement Learning

Intelligent Guided Vehicles (IGVs) in U-shaped automated container terminals (ACTs) have longer travel paths than those in conventional vertical layout ACTs, and their interactions with double trolley quay cranes (DTQCs) and double cantilever rail cranes (DCRCs) are more frequent and complex, so the scheduling strategy of a traditional ACT cannot easily be applied to a U-shaped ACT. With the aim of minimizing the maximum task completion times within a U-shaped ACT, this study investigates the integrated scheduling problem of DTQCs, IGVs and DCRCs under the hybrid “loading and unloading” mode, expresses the problem as a Markovian decision-making process, and establishes a disjunctive graph model. A deep reinforcement learning algorithm based on a graph neural network combined with a proximal policy optimization algorithm is proposed. To verify the superiority of the proposed models and algorithms, instances of different scales were stochastically generated to compare the proposed method with several heuristic algorithms. This study also analyses the idle time of the equipment under two loading and unloading modes, and the results show that the hybrid mode can enhance the operational effectiveness. of the U-shaped ACT.

A generalized disjunctive graph model for a complex production problem

This work focuses on the modeling of the features of a real-world pharmaceutical production problem, where several limited resources must be gathered and scheduled in order to mix and process active substances. The problem can be modelled as a generalization of the job-shop scheduling problem, in which each job corresponds to an order. Besides the standard precedence and capacity constraints of a classical job-shop scheduling problem, release, no-wait and time interval production constraints are added. Furthermore, the processing of each job task must be assisted by an appropriate number of operators and performed by a specific machine, out of a set of feasible machines, that is temporarily installed in a compatible clean room. An innovative generalized disjunctive graph that models all the constraints and the resource conflicts is presented.

Low-Carbon Flexible Job Shop Scheduling Problem Based on Deep Reinforcement Learning

As the focus on environmental sustainability sharpens, the significance of low-carbon manufacturing and energy conservation continues to rise. While traditional flexible job shop scheduling strategies are primarily concerned with minimizing completion times, they often overlook the energy consumption of machines. To address this gap, this paper introduces a novel solution utilizing deep reinforcement learning. The study begins by defining the Low-carbon Flexible Job Shop Scheduling problem (LC-FJSP) and constructing a disjunctive graph model. A sophisticated representation, based on the Markov Decision Process (MDP), incorporates a low-carbon graph attention network featuring multi-head attention modules and graph pooling techniques, aimed at boosting the model’s generalization capabilities. Additionally, Bayesian optimization is employed to enhance the solution refinement process, and the method is benchmarked against conventional models. The empirical results indicate that our algorithm markedly enhances scheduling efficiency by 5% to 12% and reduces carbon emissions by 3% to 8%. This work not only contributes new insights and methods to the realm of low-carbon manufacturing and green production but also underscores its considerable theoretical and practical implications.

A DQN-based memetic algorithm for energy-efficient job shop scheduling problem with integrated limited AGVs

AGVs have gained significant popularity in various industries. However, the existing literature rarely considers the integrated scheduling of production and logistics on the workshop due to the NP-hard property of both machine scheduling and AGV scheduling. The energy-efficient job shop scheduling problem with limited AGVs is investigated in this paper. A multi-objective memetic algorithm with deep Q-network (DQNMMA) is proposed to minimize the makespan and total energy consumption. In DQNMMA, ten features are selected to describe the current state of the population. This enables the DQN to dynamically adjust the crossover probability according to the population evolution. Formulas for calculating the head length and tail length of each node in the disjunctive graph model are presented for the first time to enable fast and accurate access to the critical paths. Building upon the analysis of critical paths, four problem properties are developed as the foundation for designing six neighborhood operators. Then, a property-based variable neighborhood search strategy is proposed to enhance the exploration capability of the algorithm. Numerous experimental results demonstrate that the proposed approaches can effectively enhance the performance of the algorithm, especially in solving large-scale problems. The comparative analysis with three other state-of-the-art multi-objective algorithms confirms the superiority and effectiveness of the proposed DQNMMA.

A genetic algorithm with critical path-based variable neighborhood search for distributed assembly job shop scheduling problem

Production scheduling in distributed manufacturing systems has become an active research field, where large-sized complicated products, such as airplanes and ships, are taken as the primary focus. This paper investigates a distributed assembly job shop scheduling problem (DAJSP) which consists of two production phases. The first stage processes components in several job shops, and the second stage assembles the processed parts into final products. First, a mixed integer linear programming (MILP) model is established to describe the problem with minimizing maximum completion time and find optimal schedules for small-scale scenarios. Afterwards, a genetic algorithm with variable neighborhood search (GA-VNS) is proposed to address more complex instances, which adopts the genetic algorithm as the main framework and employs the variable neighborhood search for exploration. A problem-specific three-vector encoding scheme is designed to represent three decision-making processes of the DAJSP accordingly. To improve candidate solutions, a disjunctive graph model for DAJSP is formulated and three critical path-based neighborhood structures which directly perform on encoding vectors are designed. Numerical experiments are conducted on four groups of instances with different scales and the experimental results demonstrate the effectiveness of the proposed MILP model and GA-VNS. To sum up, the proposed GA-VNS shows the best performance on 30 instances out of 40 instances, while the superior stability has also been proved by statistical tests. In addition, two complicated DAJSP cases are abstracted from an enterprise for fabricating large complex components to further validate the GA-VNS.

Flexible job-shop scheduling with transportation resources

This paper addresses an extension of the flexible job-shop scheduling problem where transportation resources are explicitly considered when moving jobs from one machine to another. Operations should be assigned to and scheduled on machines and vehicles and the routes of vehicles should be determined. We extend the classical disjunctive graph model to include transportation operations and exploit the graph in an integrated approach to solve the problem. We propose a metaheuristic using a neighborhood function that allows a large set of moves to be explored. As the exact computation of the makespan of every move is time-consuming, we present a move evaluation procedure that runs in constant time (which does not depend on the size of the instance) to choose a promising move in the neighborhood of a solution. This move evaluation procedure is used in a tabu search framework. Computational results show the efficiency of the proposed approach, the quality of the move evaluation procedure and the relevance of explicitly modeling transportation resources. New benchmark instances are also proposed.

Multi-Objective Production and Scheduling Optimization of Offshore Wind Turbine Steel Pipe Piles Based on Improved Hesitant Fuzzy Method

This paper investigates the multi-objective optimization problem in the production of offshore wind turbine steel pipe piles (OWTSPP). Considering the particularity of the steel pipe pile production process, it is divided into a flexible flow shop scheduling (FFSS) stage and an open parallel shop scheduling (OPSS) stage, respectively. Mathematical models are established for each stage, and the critical path and production time information are obtained using a disjunctive graph model. Due to the inability of existing empirical scheduling methods to balance production goals, an improved Pythagorean hesitant fuzzy method (IPHFM) is proposed to solve the multi-objective optimization problem in steel pipe pile production. Specifically, the maximum completion time, machine total load, and total completion time are taken as optimization objectives. The improved Lagrange multiplier method with penalty terms is used to handle the constraints and objective functions, and a Lagrange objective function is generated. Then, the Lagrange objective function matrix is obtained by normalization and same-scale processing, and an algorithm is designed to obtain the Pareto front solution set. Finally, this paper compares the optimal scheduling plans under the empirical scheduling method and the improved method. The results show that the improved method can significantly improve production efficiency in both small-scale and large-scale production, with improvements of 15.7% and 22.16%, respectively.

A novel MILP model for job shop scheduling problem with mobile robots

The mobile robot is the essential equipment for automated logistics in the intelligent workshop, but the literature on shop scheduling rarely considers transport resources. This paper studies the integrated scheduling of machines and mobile robots, which can facilitate the efficiency of production systems. For the job shop scheduling problem with mobile robots (JSPMR), the existing mathematical models are too complex to obtain the optimal solution in an efficient time. Therefore, a novel mixed integer linear programming (MILP) model is proposed to minimize the makespan. Firstly, in view of the property of the problem, a disjunctive graph model is modified to describe the relationship between transport and processing tasks. Secondly, a more accurate and simplified MILP is proposed based on the modified disjunctive graph model. Two related proofs are given to prove the proposed model satisfies all special situations. Thirdly, the proposed MILP is tested on the well-known benchmark, including 82 instances. The proposed model is the first MILP model to obtain optimal solutions for all instances. Finally, 40 larger-scale instances are presented based on a real-world engineering case and used to validate the performance of models further. The comparison results verify the effectiveness and superior computational performance of the proposed model.

Optimisation framework and method for solving the serial dual-shop collaborative scheduling problem

Due to the production process or environment, most products need to go through multiple production workshops, from raw materials to finished products. For example, a large structural part must go through a blanking workshop and a machining workshop. However, the research on the serial multi-shop cooperative scheduling problem is insufficient. Different from the single shop scheduling problem, the multi-shop scheduling problem can be regarded as a scheduling problem composed of multiple sub-problems. Therefore, optimisation processes between subproblems may be the key to solving this kind of problem. This paper focuses on the optimisation framework for the serial multi-shop cooperative scheduling problem. A specific serial dual-shop collaborative scheduling problem is abstracted from the production, whose first and second shop is the hybrid flow shop and the job shop, respectively. A disjunctive graph model of this specific problem is constructed. After that, four different optimisation frameworks are proposed, and testing instances are designed to compare these frameworks. Under different optimisation frameworks, the same tabu search algorithm with an efficient neighbourhood structure is used to optimize this problem. Finally, the optimisation results of different frameworks are compared and the results of each optimisation framework are analyzed.

Auction-based approach with improved disjunctive graph model for job shop scheduling problem with parallel batch processing

The job-shop scheduling problem (JSSP) is encountered in several industries, including the military where heat treatment is applied prior to the machining process in production. This study aims to minimize the overall make-span of a JSSP with parallel batch processing. The problem is formulated as a mixed-integer linear programming model. Feasible solutions are derived from an auction-based approach for forming batches, allocating operation machines, and scheduling. An improved disjunctive graph model is further developed to search for better solutions. We conduct numerical experiments to test a set of benchmark instances. A comparison of the results with those obtained applying other existing algorithms and CPLEX demonstrates the effectiveness and stability of the proposed auction-based approach and improved graph model. Furthermore, a statistical analysis using IBM SPSS shows that the proposed auction-based approach has an absolute advantage in solving medium-scale and large-scale instances of JSSP with batch processing.

Hybrid algorithm based on improved extended shifting bottleneck procedure and GA for assembly job shop scheduling problem

With the makespan as the optimisation goal, we propose a hybrid solving method that combines improved extended shifting bottleneck procedure (i-ESB) and genetic algorithm (GA) for the assembly job shop scheduling problem (AJSSP). Hybrid genetic algorithm (HGA) uses a GA based on operation constraint chain coding to achieve global search and a local search based on an i-ESB. In the design of i-ESB, an extended disjunctive graph model (EDG) corresponding to AJSSP is presented. The calculation method of the operation head and tail length based on EDG is studied, as well as the searching method of key operations. The Schrage algorithm with disturbance is used to solve the single-machine scheduling subproblem. The selection criterion for bottleneck machines is increased. A greedy bottleneck machine re-optimisation process is designed. The effectiveness and superiority of the proposed algorithm are verified by testing and analysing the relevant examples in the literature.

Scheduling of a job-shop problem with limited output buffers

ABSTRACTThis article addresses a job-shop problem with limited output buffers (JS-LOB) with the objective of minimizing the process makespan. An integer nonlinear mathematical programming model is proposed to describe this problem. Based on the model, a two-stage algorithm consisting of obtaining feasible solutions and a local search is proposed to solve the JS-LOB problem. The local search has two operators: the first is a neighbourhood structure based on a disjunctive graph model, and the second is similar to crossover in the genetic algorithm to avoid falling into local optima. Computational results are presented for a set of benchmark tests. The results show the effectiveness of the proposed algorithm and indicate whether the processing time of the job conforms to a uniform distribution. When the proportion between the capacity of the buffer and the number of jobs is larger than 20%, the influence of the buffer becomes very small.

Job Shop Scheduling by Branch and Bound Using Genetic Programming

A classical depth-first branch and bound (BB) method is adopted to minimize the makespan of job shops based on the disjunctive graph model. The engine of the BB is Giffler-Thompson’s active schedule generation method. The performance of the BB method highly depends on the selection of child nodes in earlier branching stages. To support the selection decision, several features of nodes are stored under the BB method, and the correct selection at each branching stage is informed by the mixed-integer linear programming model. The stored data of a test problem instance is analyzed by genetic programming (GP) to generate rules for selecting the correct nodes. The depth-first BB method guided by the generated rules by GP is applied for 42 benchmark instances and exhibits competitive performance when compared with the baseline rule that always selects the child node with the smallest lower bound on makespan.

Metaheuristics for the job-shop scheduling problem with machine availability constraints

Abstract This paper addresses the job-shop scheduling problem in which the machines are not available during the whole planning horizon and with the objective of minimizing the makespan. The disjunctive graph model is used to represent job sequences and to adapt and extend known structural properties of the classical job-shop scheduling problem to the problem at hand. These results have been included in two metaheuristics (Simulated Annealing and Tabu Search) with specific neighborhood functions and diversification structures. Computational experiments on problem instances of the literature show that our Tabu Search approach outperforms Simulated Annealing and existing approaches.

Integration of routing into a resource-constrained project scheduling problem

The resource-constrained project scheduling problem (RCPSP), one of the most challenging combinatorial optimization scheduling problems, has been the focus of a great deal of research, resulting in numerous publications in the last decade. Previous publications focused on the RCPSP, including several extensions with different objectives to be minimized and constraints to be checked. The present work investigates the integration of the routing, i.e., the transport of the resources between activities into the RCPSP, and provides a new resolution scheme. The two subproblems are solved using an integrated approach that draws on both a disjunctive graph model and an explicit modeling of the routing. The resolution scheme takes advantage of an indirect representation of the solution to define both the schedule of activities and the routing of vehicles. The routing solution is modeled by a set of trips that define the loaded transport operations of vehicles that are induced by the flow in the graph. The numerical experiments prove that the models and the methods introduced in this paper are promising for solving the RCPSP with routing.

Auction-based cooperation mechanism for cell part scheduling with transportation capacity constraint

This paper addresses cell part scheduling (CPS) problem with transportation capacity constraint. In the problem, parts may need to visit different cells, they have to be transferred by automated guided vehicle (AGV). The objective is to minimise the over-all process make-span. An integer nonlinear programming (INLP) model is formulated to allocate the machines, AGVs and schedule all parts. A reasonable transportation mode is presented, and an auction-based heuristic approach is proposed to solve the problem, which focuses on dealing with cooperation between machines and AGVs during processing and transferring parts. The auction consists of two aspects: auction for AGV and auction for machine. In both auctions, AGVs and machines act as auctioneers respectively, and parts act as bidders. A new improved disjunctive graph model is developed to optimise the feasible solutions obtained by auction-based approach. Numerical experiments were conducted to test the auction-based approach and improved disjunctive graph model. The results demonstrate the effectiveness of proposed auction-based approach and improved disjunctive graph model, also indicate influence of the capacity of AGV on scheduling parts.

Solving the flexible job shop scheduling problem with sequence-dependent setup times

This paper addresses the flexible job shop scheduling problem with sequence-dependent setup times and where the objective is to minimize the makespan. We first present a mathematical model which can solve small instances to optimality, and also serves as a problem representation. After studying structural properties of the problem using a disjunctive graph model, we develop a tabu search algorithm with specific neighborhood functions and a diversification structure. Extensive experiments are conducted on benchmark instances. Test results first show that our algorithm outperforms most existing approaches for the classical flexible job shop scheduling problem. Additional experiments also confirm the significant improvement achieved by integrating the propositions introduced in this study.

Extended GRASP for the job shop scheduling problem with total weighted tardiness objective

The paper proposes a heuristic for the job shop scheduling problem with minimizing the total weighted tardiness of jobs as objective. It is built upon the well known metaheuristic GRASP and strengthened with an inclusion of specific local search components. The design is based on an advanced disjunctive graph model which enables capturing solution schedules through a tree graph called critical tree. The tree graph allows for effectively steering a first-descent search algorithm which further incorporates powerful neighborhood operators and a fast move evaluation procedure based on heads updating. Additionally, amplifying and path relinking is adaptively applied to the best schedules discovered. We present computational results of the new heuristic on two famous sets of benchmark instances, we identify ten new best solutions, and we demonstrate the high potential of the approach through a comparison with state-of-the-art methods.

A flow-based tabu search algorithm for the RCPSP with transfer times

In this paper, we propose a tabu search algorithm for the resource-constrained project scheduling problem with transfer times. Solutions are represented by resource flows extending the disjunctive graph model for shop scheduling problems. Neighborhoods are defined by parallel and serial modifications rerouting or reversing flow on certain arcs. This approach is evaluated from a theoretical and experimental point of view. Besides studying the connectivity of different neighborhoods, computational results are presented for benchmark instances with and without transfer times.

Scheduling of no buffer job shop cells with blocking constraints and automated guided vehicles

The blocking job shop (BJS) problem is an extension of a job shop problem with no buffer constraints. It means that after a job is completed on the current machine, it remains on that machine until the next machine becomes available. This paper addresses an extension of the BJS problem, which takes into account transferring jobs between different machines using a limited number of automated guided vehicles (AGV), called a BJS–AGV problem. Two integer non-linear programming (INLP) models are proposed. A two-stage heuristic algorithm that combines an improving timetabling method and a local search is proposed to solve the BJS–AGV problem. A neighborhood structure in the local search is proposed based on a disjunctive graph model. According to the characteristics of the BJS–AGV problem, four principles are proposed to guarantee the feasibility of the search neighborhood. Computation results are presented for a set of benchmarking tests, some of which are enlarged by transportation times between different machines. The numerical results show the effectiveness of the proposed two-stage algorithm.