Single Scheduling Research Articles

Genetic Programming With Lexicase Selection for Large-Scale Dynamic Flexible Job Shop Scheduling

Dynamic flexible job shop scheduling is a prominent combinatorial optimisation problem with many real-world applications. Genetic programming has been widely used to automatically evolve effective scheduling heuristics for dynamic flexible job shop scheduling. A limitation of genetic programming is the premature convergence due to the loss of population diversity. To overcome this limitation, this work considers using lexicase selection to improve population diversity, which has achieved success on regression and program synthesis problems. However, it is not trivial to apply lexicase selection to genetic programming for dynamic flexible job shop scheduling, since a fitness case (training scheduling simulation) is often large-scale, making the fitness evaluation very time-consuming. To address this issue, we propose a new multi-case fitness scheme, which creates multiple cases from a single scheduling simulation. Based on the multi-case fitness, we develop a new genetic programming algorithm with lexicase selection, which uses a single simulation for fitness evaluation, thus achieving a better balance between the number of cases for lexicase selection and evaluation efficiency. The experiments on a wide range of dynamic scheduling scenarios show that the proposed algorithm can achieve better population diversity and final performance than the current genetic programming parent selection methods and a state-of-the-art deep reinforcement learning method.

Towards Double-Layer Dynamic Heterogeneous Redundancy Architecture for Reliable Railway Passenger Service System

Researchers have proposed the dynamic heterogeneous redundancy (DHR) architecture, which integrates dynamic, heterogeneous, redundant, and closed-loop feedback elements into the system, to fortify the reliability of the railway passenger service system (RPSS). However, there are at least two weaknesses with the common DHR architectures: (1) they need system nodes with enough computing and storage resources; (2) they have hardly considered the reliability of DHR architecture. To this end, this paper proposes a double-layer DHR (DDHR) architecture to ensure the reliability of RPSS. This architecture introduces a set of algorithms, which are optimized co-computation and ruling weight optimization algorithms for the data processing flow of the DDHR architecture. This set improves the reliability of the DDHR architecture. For the evaluation of the reliability of DDHR architecture, this paper also proposes two metrics: (1) Dynamic available similarity metric. This metric does not rely on the overall similarity of the double-layer redundant executor sets but evaluates the similarity of their performance under the specified interaction paths within a single scheduling cycle. The smaller its similarity, the higher its reliability. (2) Scheduling cycle under dual-layer similarity threshold. This metric evaluates the reliability of the RPSS under actual conditions by setting the schedulable similarity thresholds between the same and different layers of the dual-layer redundant executives in the scheduling process. Finally, analog simulation experiments and prototype system building experiments are carried out, whose numerical experimental results show that the DDHR architecture outperforms the traditional DHR architecture in terms of reliability and performance under different redundancy and dynamically available similarity thresholds, while the algorithmic complexity and multi-tasking concurrency performance are slightly weaker than that of the DHR architecture, but can be applied to the main operations of the RPSS in general.

Out-of-order execution enabled deep reinforcement learning for dynamic additive manufacturing scheduling

Additive Manufacturing (AM) has revolutionized the production landscape by enabling on-demand customized manufacturing. However, the efficient management of dynamic AM orders poses significant challenges for production planning and scheduling. This paper addresses the dynamic scheduling problem considering batch processing, random order arrival and machine eligibility constraints, aiming to minimize total tardiness in a parallel non-identical AM machine environment. To tackle this problem, we propose the out-of-order enabled dueling deep Q network (O3-DDQN) approach. In the proposed approach, the problem is formulated as a Markov decision process (MDP). Three-dimensional features, encompassing dynamic orders, AM machines, and delays, are extracted using a ‘look around’ method to represent the production status at a rescheduling point. Additionally, five novel composite scheduling rules based on the out-of-order principle are introduced for selection when an AM machine completes processing or a new order arrives. Moreover, we design a reward function that is strongly correlated with the objective to evaluate the agent’s chosen action. Experimental results demonstrate the superiority of the O3-DDQN approach over single scheduling rules, randomly selected rules, and the classic DQN method. The average improvement rate of performance reaches 13.09% compared to composite scheduling rules and random rules. Additionally, the O3-DDQN outperforms the classic DQN agent with a 6.54% improvement rate. The O3-DDQN algorithm improves scheduling in dynamic AM environments, enhancing productivity and on-time delivery. This research contributes to advancing AM production and offers insights into efficient resource allocation.

Flexible optimal bus‐schedule bridging for metro operation‐interruption

AbstractUnder the sudden interruption of the metro, it is significant to dispatch emergency bridging buses to evacuate stranded passengers to improve linkage management and service reliability. Aiming at the problem of emergency bridging bus scheduling, considering the remaining capacity of conventional buses, passenger tolerance, and site convenience, a flexible combined emergency bridging bus scheduling model is constructed based on the constraints of vehicle capacity, dispatching capacity, and maximum evacuation times of emergency bridging bus, aiming at minimizing the maximum evacuation time and passenger delay. The improved Harris hawk algorithm is used to solve the model, and the evacuation plan of the demand‐responsive and station‐station bridging lines is obtained. The maximum evacuation time is 41 min, and the average delay of stranded passengers is 19 min. The results show that the maximum evacuation time is 10% and 27% less than the fixed combined and single scheduling. The average delay is 16% and 42% less than the fixed combination and traditional single scheduling. The sensitivity analysis of the influencing factors of emergency bridging bus scheduling is conducted. The results show that the flexible combined emergency bridging bus scheduling model constructed in this paper can improve evacuation efficiency and reduce passenger travel delays.

Joint Autoscaling of Containers and Virtual Machines for Cost Optimization in Container Clusters

Autoscaling enables container cluster orchestrators to automatically adjust computational resources, such as containers and Virtual Machines (VMs), to handle fluctuating workloads effectively. This adaptation can involve modifying the amount of resources (horizontal scaling) or adjusting their computational capacity (vertical scaling). The motivation for our work stems from the limitations of previous autoscaling approaches, which are either partial (scaling containers or VMs, but not both) or excessively complex to be used in real systems. This complexity arises from their use of models with a large number of variables and the addressing of two simultaneous challenges: achieving the optimal deployment for a single scheduling window and managing the transition between successive scheduling windows. We propose an Integer Linear Programming (ILP) model to address the challenge of autoscaling containers and VMs jointly, both horizontally and vertically, to minimize deployment costs. This model is designed to be used with predictive autoscalers and be solved in a reasonable time, even for large clusters. To this end, improvements and reasonable simplifications with respect to previous models have been carried out to drastically reduce the size of the resource allocation problem. Furthermore, the proposed model provides an enhanced representation of system performance in comparison to previous approaches. A tool called Conlloovia has been developed to implement this model. To evaluate its performance, we have conducted a comprehensive assessment, comparing it with two heuristic allocators with different problem sizes. Our findings indicate that Conlloovia consistently demonstrates lower deployment costs in a significant number of cases. Conlloovia has also been evaluated with a real application, using synthetic and real workload traces, as well as different scheduling windows, with deployment costs approximately 20% lower than heuristic allocators.

Functions of the interlocking system on mixed traffic lines

These days, automated rail traffic control systems help to lessen the need for human intervention in train traffic management. On the other hand, the various algorithms used by interlocking systems to coordinate the movement of trains belonging to distinct categories result in longer wait times for technological procedures at the station. The possibility of timely routes for trains of multiple categories occurs by employing the method of automatic route preparation by interlocking systems in the organization of the movement of trains of different categories. In interlocking systems, automatic route preparation can be achieved by continuously monitoring train characteristics, train traffic graphs of various train classes, and the condition of equipment involved in the function’s development. Nonetheless, it is predicated on the creation of many models to arrange the specifications of various train classes and train movements inside a single schedule type. In order to prepare the routes ahead of time for the station tracks based on the algorithms for the operation of interlocking systems for various kinds of trains, the paper analyzes the reliance of the traffic composition characteristics.

A two-level charging scheduling method for public electric vehicle charging stations considering heterogeneous demand and nonlinear charging profile

This paper investigates the electric vehicle (EV) charging scheduling problem for public EV charging stations (EVCSs) that can accommodate heterogeneous charging demands, aiming to flatten the aggregate load on the power grid and reduce the peak demand. In contrast to existing works that mainly focus on a single scheduling strategy, a two-level hierarchical charging scheduling method is proposed, which includes an online booking system (OBS) and a pricing-based charging control system (PCCS). Specifically, the implementation of OBS can reduce the service capacity of bookable chargers, whereas PCCS can facilitate the redistribution of charging demand from peak to off-peak time periods. Moreover, the nonlinear charging profile of the battery is incorporated into the model to better reflect the real charging process. A deep reinforcement learning (DRL)-based approach with a discrete Markov decision process (MDP) formulation is designed to find the optimal scheduling solution, where deep Q-network (DQN) and deep deterministic policy gradient (DDPG) are combined to handle the hybrid action space with both discrete and continuous actions. A comprehensive set of experiments is carried out to examine the effectiveness of the developed two-level scheduling scheme. The results indicate that the proposed method improves the scheduling effectiveness compared to single-strategy methods by 6.7% to 49.44%.

Dynamic Controllability of Temporal Plans in Uncertain and Partially Observable Environments

The formalism of Simple Temporal Networks (STNs) provides methods for evaluating the feasibility of temporal plans. The basic formalism deals with the consistency of quantitative temporal requirements on scheduled events. This implicitly assumes a single agent has full control over the timing of events. The extension of Simple Temporal Networks with Uncertainty (STNU) introduces uncertainty into the timing of some events. Two main approaches to the feasibility of STNUs involve (1) where a single schedule works irrespective of the duration outcomes, called Strong Controllability, and (2) whether a strategy exists to schedule future events based on the outcomes of past events, called Dynamic Controllability. Case (1) essentially assumes the timing of uncertain events cannot be observed by the agent while case (2) assumes full observability. The formalism of Partially Observable Simple Temporal Networks with Uncertainty (POSTNU) provides an intermediate stance between these two extremes, where a known subset of the uncertain events can be observed when they occur. A sound and complete polynomial algorithm to determining the Dynamic Controllability of POSTNUs has not previously been known; we present one in this paper. This answers an open problem that has been posed in the literature. The approach we take factors the problem into Strong Controllability micro-problems in an overall Dynamic Controllability macro-problem framework. It generalizes the notion of labeled distance graph from STNUs. The generalized labels are expressed as max/min expressions involving the observables. The paper introduces sound generalized reduction rules that act on the generalized labels. These incorporate tightenings based on observability that preserve dynamic viable strategies. It is shown that if the generalized reduction rules reach quiescence without exposing an inconsistency, then the POSTNU is Dynamically Controllable (DC). The paper also presents algorithms that apply the reduction rules in an organized way and reach quiescence in a polynomial number of steps if the POSTNU is Dynamically Controllable. Remarkably, the generalized perspective leads to a simpler and more uniform framework that applies also to the STNU special case. It helps illuminate the previous methods inasmuch as the max/min label representation is more semantically clear than the ad-hoc upper/lower case labels previously used.

BLOOP: Boolean Satisfiability-based Optimized Loop Pipelining

Modulo scheduling is the premier technique for throughput maximization of loops in high-level synthesis by interleaving consecutive loop iterations. The number of clock cycles between data insertions is called the initiation interval (II). For throughput maximization, this value should be as low as possible; therefore, its minimization is the main optimization goal. Despite its long historical existence, modulo scheduling always remained a relevant research topic over the years with many exact and heuristic algorithms available in the literature. Nevertheless, we are able to leverage the scalability of modern Boolean Satisfiability (SAT) solvers to outperform state-of-the-art ILP-based algorithms for latency-optimal modulo scheduling for both integer and rational IIs. Our algorithm is able to compute valid modulo schedules for the whole CHStone and MachSuite benchmark suites, with 99% of the solutions being proven to be throughput optimal for a timeout of only 10 minutes per candidate II. For various time limits, not a single tested scheduler from the state of the art is able to compute more verified optimal solutions or even a single schedule with a higher throughput than our proposed approach. Using an HLS toolflow, we show that our algorithm can be effectively used to generate Pareto-optimal FPGA implementations regarding throughput and resource usage.

Multi‐scale coordinated optimal dispatch method of electricity‐thermal‐hydrogen integrated energy systems

AbstractTo achieve carbon neutrality, renewable energy‐based power systems and hydrogen are increasingly being promoted. A novel electricity‐thermal‐hydrogen integrated energy system that combines new energy generation, multi‐source load, and multiple energy storage is proposed by the authors. To address uncertainties in new energy output, and issues of untimely unit regulation response and large planning tracking errors, a multi‐scale scheduling method based on model predictive control (MPC) was proposed. In the day‐ahead dispatching stage, an optimal economic dispatching model was established with the lowest system operation cost as the optimisation objective. The model considers equipment investment, operation, maintenance, and peak‐to‐valley differences in electricity prices. In the intraday dispatching stage, an MPC‐based intraday rolling optimisation correction strategy was proposed to cope with contact line power fluctuations caused by prediction errors of new energy and multi‐source load. This strategy combines time‐domain rolling and feedback correction of the real‐time system state to eliminate the influence of uncertainty factors in the microgrid. The MPC‐based intraday rolling optimal scheduling model was established in the form of a discrete state space and transformed into a quadratic planning problem to improve the efficiency and accuracy of the model solution. Finally, a typical microgrid was used as an example to verify the effectiveness of the proposed method. Results show that the contact line tracking error can be within 0.025 kW, and the single scheduling time was within 0.14 s.

Методика адаптивного регулирования и управления тяговыми ресурсами на полигоне железных дорог

In the context of introduction of railway test site technologies for organization of train traffic, the issue of eliminating the main reasons for inefficiency of freight transportation is acute - uncontrolled delays of locomotives under technological operations, difficulty in forming a single schedule, instability of the daily transportation plan and lack of a suitable train at the time of formation at the locomotive stations. It is necessary to optimize the specific processes of organizing and managing the fleet of traction resources. The methodological framework of the research is based on modern perception about provisions for the synthesis of correct adaptive mechanisms with identification, principles of comprehensive assessment, organizational and economic action of intelligent and automated transport systems of organization and management in production, the methods of expert assessments, Monte Carlo, Delphi, graphic and analytical planning and evaluation of indicators are used, which enable to organize the supply of traction and locomotive crews at the request of carriers by independent commercial structures and individuals on a competitive basis. The mechanism of predictive reduction of losses in the management of traction resources at the railway test site, ensuring the fulfillment of established parameters of the use of locomotives, is given, and the method of adaptive regulation and management of traction resources is proposed, which allows analyzing the production and economic indicators of the use of traction to develop and evaluate the proposed management solutions to improve efficiency of a transportation process.

A dynamic scheduling method with Conv-Dueling and generalized representation based on reinforcement learning

In modern industrial manufacturing, there are uncertain dynamic disturbances between processing machines and jobs which will disrupt the original production plan. This research focuses on dynamic multi-objective flexible scheduling problems such as the multi-constraint relationship among machines, jobs, and uncertain disturbance events. The possible disturbance events include job insertion, machine breakdown, and processing time change. The paper proposes a conv-dueling network model, a multidimensional state representation of the job processing information, and multiple scheduling objectives for minimizing makespan and delay time, while maximizing the completion punctuality rate. We design a multidimensional state space that includes job and machine processing information, an efficient and complete intelligent agent scheduling action space, and a compound scheduling reward function that combines the main task and the branch task. The unsupervised training of the network model utilizes the dueling-double-deep Q-network (D3QN) algorithm. Finally, based on the multi-constraint and multi-disturbance production environment information, the multidimensional state representation matrix of the job is used as input and the optimal scheduling rules are output after the feature extraction of the conv-dueling network model and decision making. This study carries out simulation experiments on 50 test cases. The results show the proposed conv-dueling network model can quickly converge for DQN, DDQN, and D3QN algorithms, and has good stability and universality. The experimental results indicate that the scheduling algorithm proposed in this paper outperforms DQN, DDQN, and single scheduling algorithms in all three scheduling objectives. It also demonstrates high robustness and excellent comprehensive scheduling performance.

Large-Scale Machine Learning Cluster Scheduling via Multi-Agent Graph Reinforcement Learning

Efficient scheduling of distributed deep learning (DL) jobs in large GPU clusters is crucial for resource efficiency and job performance. While server sharing among jobs improves resource utilization, interference among co-located DL jobs occurs due to resource contention. Interference-aware job placement has been studied, with white-box approaches based on explicit interference modeling and black-box schedulers with reinforcement learning. In today’s clusters containing thousands of GPU servers, running a single scheduler to manage all arrival jobs in a timely and effective manner is challenging, due to the large workload scale. We adopt multiple schedulers in a large-scale cluster/data center, and propose a multi-agent reinforcement learning (MARL) scheduling framework to cooperatively learn fine-grained job placement policies, towards the objective of minimizing job completion time (JCT). To achieve topology-aware placements, our proposed framework uses hierarchical graph neural networks to encode the data center topology and server architecture. In view of a common lack of precise reward samples corresponding to different placements, a job interference model is further devised to predict interference levels in face of various co-locations, for training of the MARL schedulers. Testbed and trace-driven evaluations show that our scheduler framework outperforms representative scheduling schemes by more than 20% in terms of average JCT, and is adaptive to various machine learning cluster topologies.

Prescribing and sales of intramammary antimicrobials in Ireland in 2019 and 2020: the role of milk purchasers

BackgroundIn Ireland between 2008 and 2022, intramammary antimicrobial (AM) products could be prescribed by a veterinary practitioner under what was known as Schedule 8 (or remote) prescribing. Under this prescribing route, an annual herd visit was not required when criteria were met as outlined in Animal Remedies Regulation 2007 to 2017 (statutory instruments No. 786/2007 and 558/2017). Under this prescribing route, the responsibilities of the milk purchaser, the farmer and the veterinary practitioner were each outlined, and a written mastitis control programme (MCP) was required. Milk purchasers implemented MCPs on participating farms (so-called MCP herds) with support from veterinary practitioner(s) who undertook Schedule 8 prescribing of intramammary AM tubes. This study seeks a clearer understanding of the role of milk purchasers in the prescribing and sale of intramammary AM products in Ireland during 2019 and 2020, whilst this Regulation was in force. Specifically, the study sought insights into the role of milk purchasers in the prescribing and sale of intramammary AM products in the Irish dairy industry during 2019 and 2020, using anonymised and highly aggregated milk purchaser data. The study also provided insights into milk quality among supplying herds during this period.MethodsFor this study, we had access to anonymised, highly aggregated data from all milk purchasers that operated a MCP on at least some of their supplying herds during 2019 or 2020. Data collection was undertaken by the Department of Agriculture, Food and Marine. Data analysis was primarily descriptive.ResultsData were available on 11 milk purchasers (64.7% of all) and 13,251 supplying herds. Of these, 52% were MCP herds. The quality of milk from supplying herds varied significantly by month, year and milk purchaser. During 2019 and 2020, there was a single Schedule 8 prescriber (a private veterinary practitioner prescribing intramammary AMs as part of a MCP), on average, for 549.3 herds. The sale of intramammary AM products through milk purchasers represented 15.2% and 26.9% of national sales in in-lactation and dry cow tubes, respectively. There was an overall 2% increase in sales through milk purchasers between 2019 and 2020. Few European Medicines Agency (EMA) category B (‘Restrict’) intramammary AM products were sold by milk purchasers. For both in-lactation and dry cow tubes, there was a statistically significant association between EMA classification and route of sale (through milk purchasers or otherwise).ConclusionsThe study findings provide important insights into mastitis control and intramammary AM stewardship in the Irish dairy industry. Significant differences between milk purchasers were observed in the quality of milk, as measured through somatic cell count (SCC) values, from supplying herds. This warrants further research. In the context of intramammary AM prescribing, veterinary oversight under the Animal Remedies Regulation 2007 to 2017 was very limited during 2019 and 2020. There were also significant associations between EMA classification and route of sale during 2019 and 2020, reinforcing the need for Irish veterinary practitioners to move away from EMA category B intramammary AMs. Higher quality data are needed to address important industry questions. Specifically it is recommended that national bulk tank SCC data are made available for public good research. Past experiences with Schedule 8 prescribing (no longer permitted from 28 January 2022) may influence current efforts towards improved intramammary AM stewardship.

An Adaptive Reinforcement Learning-Based Scheduling Approach with Combination Rules for Mixed-Line Job Shop Production

Affected by economic globalization and market diversification, more manufacturing enterprises realize that large-scale production cannot adapt to the current market environment. The new trend of multivariety customized mixed-line production brings a higher level of disturbances and uncertainties to production planning. Traditional methods cannot be directly applied to the classic flexible job shop scheduling problem (FJSP). Therefore, this paper presents an adaptive scheduling method for mixed-line job shop scheduling. First, the scheduling problem caused by combined processing constraints is studied and transformed by introducing the definition of virtual operation. According to the situation of the coexistence of trial-production and batch production, the disturbance processing mechanism is established. And a scheduling decision model is established based on contextual bands (CBs) in reinforcement learning to overcome the shortcoming of poor performance of traditional single dispatching rule strategy. Through continuous trial and error learning, each scheduler can select the most suitable scheduling rules according to the environment state. Finally, we benchmark the performance of the scheduling algorithm with scheduling methods based on a variety of single scheduling rules. The results show that the proposed algorithm not only improves the performance in the mixed production scheduling problem but also effectively copes with emergency trial-production orders.

Commentary on Motschman et al.: Moving behavioral economic demand into the real world means moving beyond single schedules of reinforcement.

Commentary on Motschman et al.: Moving behavioral economic demand into the real world means moving beyond single schedules of reinforcement.

Methodology and analytical approach to investigate the impact of building temperature setpoint schedules

This paper presents a method for generating a large set of stochastically varying temperature setpoint schedules for building performance simulations. It analyses tradeoffs resulting from specific changes to those schedules in terms of three quantities: total electricity consumption per day; maximum hourly electricity consumption per day and predicted percentage dissatisfied (occupant thermal comfort). The method for generating schedules requires a single base schedule as the starting point and, using a few clearly defined parameters, transforms it into a set of schedules that can be used for modelling an existing building stock or for performing parametric studies. Temperature setpoint schedules are generated and simulated for a small office building in a cool-dry climate in three different case studies pertaining to changing temperature setpoint schedules. Tradeoffs between the three output metrics are significant and vary based on the temperature setpoint schedules and the time of the year.

Scheduling of AGVs in Automated Container Terminal Based on the Deep Deterministic Policy Gradient (DDPG) Using the Convolutional Neural Network (CNN)

In order to improve the horizontal transportation efficiency of the terminal Automated Guided Vehicles (AGVs), it is necessary to focus on coordinating the time and space synchronization operation of the loading and unloading of equipment, the transportation of equipment during the operation, and the reduction in the completion time of the task. Traditional scheduling methods limited dynamic response capabilities and were not suitable for handling dynamic terminal operating environments. Therefore, this paper discusses how to use delivery task information and AGVs spatiotemporal information to dynamically schedule AGVs, minimizes the delay time of tasks and AGVs travel time, and proposes a deep reinforcement learning algorithm framework. The framework combines the benefits of real-time response and flexibility of the Convolutional Neural Network (CNN) and the Deep Deterministic Policy Gradient (DDPG) algorithm, and can dynamically adjust AGVs scheduling strategies according to the input spatiotemporal state information. In the framework, firstly, the AGVs scheduling process is defined as a Markov decision process, which analyzes the system’s spatiotemporal state information in detail, introduces assignment heuristic rules, and rewards the reshaping mechanism in order to realize the decoupling of the model and the AGVs dynamic scheduling problem. Then, a multi-channel matrix is built to characterize space–time state information, the CNN is used to generalize and approximate the action value functions of different state information, and the DDPG algorithm is used to achieve the best AGV and container matching in the decision stage. The proposed model and algorithm frame are applied to experiments with different cases. The scheduling performance of the adaptive genetic algorithm and rolling horizon approach is compared. The results show that, compared with a single scheduling rule, the proposed algorithm improves the average performance of task completion time, task delay time, AGVs travel time and task delay rate by 15.63%, 56.16%, 16.36% and 30.22%, respectively; compared with AGA and RHPA, it reduces the tasks completion time by approximately 3.10% and 2.40%.

Mixed Scheduling Model for Limited-Stop and Normal Bus Service with Fleet Size Constraint

Limited-stop service is useful to increase operation efficiency where the demand is unbalanced at different stops and unidirectional. A mixed scheduling model for limited-stop buses and normal buses is proposed considering the fleet size constraint. This model can optimize the total cost in terms of waiting time, in-vehicle time and operation cost by simultaneously adjusting the frequencies of limited-stop buses and normal buses. The feasibility and validity of the proposed model is shown by applying it to one bus route in the city of Zhenjiang, China. The results indicate that the mixed scheduling service can reduce the total cost and travel time compared with the single scheduling service in the case of unbalanced passenger flow distribution and fleet constraints. With a larger fleet, the mixed scheduling service is superior. There is an optimal fleet allocation that minimizes the cost for the system, and a significant saving could be attained by the mixed scheduling service. This study contributed to the depth analysis of the relationship among the influencing factors of mixed scheduling, such as fleet size constraint, departure interval and cost.

Single machine batch scheduling with non-increasing time slot costs

Single machine batch scheduling with non-increasing time slot costs