Lagrangian Relaxation-based Approach Research Articles

An exact method for vessel emission monitoring with a ship-deployed heterogeneous fleet of drones

Using drones to monitor vessel emissions has received a lot of attention currently because of its advantages in ensuring compliance with the regulations of Emission Control Areas (ECAs) in the shipping industry. In the existing literature, drones are assumed to be identical and deployed at fixed base stations, resulting in sub-optimal or even infeasible solutions for vessel inspections. This paper investigates a variant of the vessel emission monitoring problem in an ECA, in which drones are deployed on Coastal Patrol Ships (CPSs) to extend their monitoring coverage of vessels, and each drone may have a different flight speed and endurance. We formulate the problem on a time–space network, based on which a mixed-integer linear programming model is presented, aiming to maximize the inspection benefit of the vessels. An exact method is proposed for solving the model. It relies on a structure-based reformulation of the original model, as well as structure enumeration, column generation, and row generation. Computational results show that this exact method can solve large-scale instances with up to 100 vessels, three CPSs, and six drones within one hour, and outperforms the Gurobi solver, the Lagrangian relaxation-based approach developed in the existing literature, and the policies adopted in reality.

An ADMM-based dual decomposition mechanism for integrating crew scheduling and rostering in an urban rail transit line

The crew planning problem is a key step in the urban rail transit (URT) planning process and has a critical impact on the operational efficiency of a URT line. In general, the crew planning problem consists of two subproblems, crew scheduling and crew rostering, which are usually solved in a sequential manner. Such an approach may, however, lead to a poor-quality crew plan overall. We therefore study the integrated optimization of crew scheduling and crew rostering and propose an effective dual decomposition approach. In particular, we formulate the integrated problem as an integer programming model using a space–time-state network representation, where the objective of the model is to minimize the weighted sum of total travel cost and penalties associated with imbalances in the workloads of the crew members. Then, an Alternating Direction Method of Multipliers (ADMM)-based dual decomposition mechanism that decomposes the model into a set of independent crew member-specific subproblems is introduced, where each of these subproblems is efficiently solved by a tailored time-dependent shortest path algorithm. To improve the performance of ADMM approach, two enhancement strategies are also designed to accelerate convergence. A set of real-life instances based on a rail transit line in Chengdu, China, is used to verify the effectiveness of the proposed model and algorithm. Computational results show that the ADMM-based approach with enhancements significantly outperforms a conventional Lagrangian Relaxation-based approach, yielding improved convergence and significantly smaller optimality gaps. Finally, on a set of real-life instances, the proposed ADMM-based approach with enhancements obtains an optimality gap of, on average, 4.2%. This is substantially better than Lagrangian Relaxation, which provides optimality gaps of, on average, 34.73%.

Optimization-Based Resource Management Algorithms with Considerations of Client Satisfaction and High Availability in Elastic 5G Network Slices.

A combined edge and core cloud computing environment is a novel solution in 5G network slices. The clients’ high availability requirement is a challenge because it limits the possible admission control in front of the edge cloud. This work proposes an orchestrator with a mathematical programming model in a global viewpoint to solve resource management problems and satisfying the clients’ high availability requirements. The proposed Lagrangian relaxation-based approach is adopted to solve the problems at a near-optimal level for increasing the system revenue. A promising and straightforward resource management approach and several experimental cases are used to evaluate the efficiency and effectiveness. Preliminary results are presented as performance evaluations to verify the proposed approach’s suitability for edge and core cloud computing environments. The proposed orchestrator significantly enables the network slicing services and efficiently enhances the clients’ satisfaction of high availability.

A resource-oriented decomposition approach for train timetabling problem with variant running time and minimum headway

ABSTRACT Solving the practical train timetabling problem under complex real-life train operation environment is challenging. This article addresses the train timetabling problem considering the variant parameters (i.e. running time and minimum headway) depending on stop-decisions. Based on a resource-oriented decomposition representation of safety headway, the train timetabling is modeled by cumulative flow variables considering the variant parameters depending on stop-decisions. A Lagrangian relaxation-based approach (LR) is used to decompose the combinatorial train timetabling problem into train-independent shortest path sub-problems, which can be solved simultaneously by parallel computation by relaxing the capacity constraint. A capacity assessment-based heuristic is proposed for improving the feasibility reparing of LR solutions. The solution quality and efficiency are analyzed employing the real-life operational data of Wuhan to Guangzhou high-speed railway in China. The benefits of the improved heuristic and parallel computation are demonstrated in contrast with the existed approach.

Urban layout optimization in a city network under an extended quadratic assignment problem framework

For urban planning, transportation system in a region is essentially a result of the activity participation among geographically distributed locations. Facing congestion, pollution and other serious challenges, Metropolitan Planning Organizations (MPOs) need to offer effective decision-making supports to urban planning, especially urban layout optimization solutions. By mathematically defining the urban layout problem as a joint optimization of (a) activity space (AS)-to-location assignment, (b) demand distribution and (c) traffic flow assignment, this paper constructs an urban layout model by extending the quadratic assignment problem (QAP). In comparison with the standard model, our proposed framework considers possible flow distribution between different types of urban amenities, subject to the region-level demand control total constraint for certain activity types. A Maximal AS-to-location flow (MaxASLF) based heuristic algorithm and a Lagrangian relaxation-based approach are designed to solve the proposed models. Various examples and discussions are provided to examine the effectiveness of the proposed methods.

Collaborative optimization for metro train scheduling and train connections combined with passenger flow control strategy

For high-frequency metro lines, the excessive travel demand during the peak hours brings a high risk to metro system and a low comfort to passengers, so it is important to consider passenger flow control when designing the metro train scheduling strategy. This paper presents a collaborative optimization method for metro train scheduling and train connections combined with passenger control strategy on a bi-directional metro line. Specifically, the dynamic equations for the train headway and train passenger loads along the metro line, the turnaround operations and the entering/exiting depot operations are considered simultaneously. The proposed collaborative optimization problem is formulated as a mixed integer nonlinear programming model to realise the trade-off among the utilization of trains, passenger flow control strategy and the number of awaiting passengers at platforms, which is further reformulated into mixed integer linear programming (MILP) model. To handle the complexity of this MILP model, a Lagrangian relaxation-based approach is designed to decompose the original problem into two small subproblems, which reduces the computational burden of the original problem and can efficiently find a good solution of the train schedule and train connections problem combined with passenger flow control strategy. The numerical experiments are implemented to investigate the effectiveness of the proposed model and approach, which shows that the proposed model is not sensitive to uncertain passenger demand. Under the proposed collaborative optimization approach, the number of train service connections and the crowding inside stations and carriages with the proper passenger flow control strategy can be evidently balanced, and thereby the operation efficiency and safety of the metro lines are effectively improved.

A co-opetitive framework for the hub location problems in transportation networks

In this paper, we present the first model of co-opetition for a Hub Location Problem between two logistics service provider (LSPs) companies where the mother company is the owner of infrastructure. The LSPs would like to cooperate with each other by establishing joint edges with limited capacities connecting their service networks. Such services are in form of pendulum services (a direct service between two points) between nodes of different networks. Additional market can be generated as a result of joining the two networks. At the same time, a competition is taking place between the two operators to increase their share from the additional market generated. In order to solve this problem, we propose a matheuristic approach combining a local search algorithm and a Lagrangian relaxation-based approach. In our matheuristic algorithm, the neighbourhood solutions are evaluated using a Lagrangian relaxation-based approach. Numerical results of applying the proposed algorithm on a real case study of the problem are presented.

Power‐based transmission constrained unit commitment formulation with energy‐based reserve

Staircase transmission constrained unit commitment (TCUC) formulations have been used for decades and are considered as energy-based TCUC formulations (EbUC). Recently, it has been reported that EbUC formulations may result in undeliverable energy schedules. Some power-based TCUC formulations (PbUC) have been proposed to address this problem. However, exact continuous time-based power outputs of units are usually approximated in these PbUC formulations and ramp-rate capacities of units are not accurately modelled. Besides, the implicit coupling between ramp-rate constraints and spinning reserves is neglected in both EbUC and PbUC formulations, which may cause that scheduled spinning reserve levels are unavailable in some time periods. To address these problems, a new PbUC formulation is proposed in this study with continuous time-based power outputs. It is found that the formulation can be transformed into an equivalent mathematical program and a Lagrangian relaxation-based approach is presented. Numerical testing is performed for a 10-unit system and an IEEE 24-bus system. Results show that the model and the solution algorithm in this study are effective.

Evacuation planning for disaster responses: A stochastic programming framework

Some disasters such as earthquakes, floods and hurricanes may result in evacuation for people in an affected area. This paper focuses on finding the a priori evacuation plans by considering side constraints and scenario-based stochastic link travel times and capacities. Hence a stochastic programming framework is developed so as to provide a reorganization of the traffic routing for a disaster response. Considering the different preferences of decision-makers, three evaluation criteria are introduced to formulate the objective function. Crisp linear equivalents for different evacuation strategies are further deduced to simplify solution methodologies. A heuristic algorithm combining the Lagrangian relaxation-based approach with K-shortest path techniques is designed to solve the expected disutility model. The experimental results indicate that the algorithm can solve large-scale instances for the problem of interest efficiently and effectively.

Distributed State Estimation of Hybrid AC/HVDC Grids by Network Decomposition

State estimation (SE) for bulk power systems is traditionally formulated and executed as one integrated problem in a control center energy management system. This simultaneous approach may present challenges when the concerned network consists of different types of systems operated by different entities (independent system operator, transmission system operator, or utility). A leading example is provided by hybrid ac/high voltage dc (HVDC) grids. We propose a distributed solution method where the system network is decomposed into several subsystems on which SE problems are executed separately and are iteratively coordinated through updated boundary information. Hybrid ac/HVDC grids naturally decompose into ac and HVDC subsystems. The proposed method uses a Lagrangian relaxation-based approach and block-wise Gauss–Seidel solution technique to arrive at a solution. The paper describes the method in the context of HVDC grids based on voltage source converter (VSC) technology. It presents a unified modeling framework for monopole/bipole converter configurations, parallel converters, converter-less dc buses, and wind farm connections. Illustrations and test cases are drawn using hybrid ac/HVDC grids formed by IEEE 39 and 300 bus ac grid networks along with monopole and bipole VSC HVDC grids.

Supply chain network design with unreliable suppliers: a Lagrangian relaxation-based approach

This paper deals with the integrated facility location and supplier selection decisions for the design of supply chain network with reliable and unreliable suppliers. Two problems are addressed: (1) facility location/supplier selection; and (2) facility location/supplier reliability. We first consider the facility location and supplier selections problem where all the suppliers are reliable. The decisions concern the selection of suppliers, the location of distribution centres (DCs), the allocation of suppliers to DCs and the allocation of retailers to DCs. The objective is to minimise fixed DCs location costs, inventory and safety stock costs at the DCs and ordering costs and transportation costs across the network. The introduction of inventory costs and safety stock costs leads to a non-linear NP-hard optimisation problem. To solve this problem, a Lagrangian relaxation-based approach is developed. For the second problem, a two-period decision model is proposed in which selected suppliers are reliable in the first period and can fail in the second period. The corresponding facility location/supplier reliability problem is formulated as a non-linear stochastic programming problem. A Monte Carlo optimisation approach combining the sample average approximation scheme and the Lagrangian relaxation-based approach is proposed. Computational results are presented to evaluate the efficiency of the proposed approaches.

Frequency-regulating reserve constrained unit commitment for an isolated power system

The purpose of unit commitment (UC) for electric utilities is to determine the optimal thermal unit on/off statuses and their MW generations over the scheduled time horizon. The UC problem is formulated to minimize the total generation cost, while the load demand, reserve requirements, and unit constraints are satisfied. Among the UC constraints, an adequate provision of reserve is important to ensure the security of the power system and the frequency-regulating reserve is essential to bring the system frequency back to acceptable level following the loss of a sizable online unit within seconds. In this paper, the authors present and solve a mixed-integer linear programming (MILP)-based UC problem including the frequency-regulating reserve (FRR) constraints to determine the optimal FRR requirements and unit MW schedules for an isolated power system. Simulation results are then compared with those obtained by Lagrangian relaxation-based approach and by the current Taipower operation practice. It is observed that favorable reserve and unit MW schedules are obtained by the proposed method while the system security is maintained.

Lagrangian Relaxation-Based Unit Commitment Considering Fast Response Reserve Constraints

Unit commitment (UC) is to determine the optimal unit status and generation level during each time interval of the scheduled period. The purpose of UC is to minimize the total generation cost while satisfying system demand, reserve requirements, and unit constraints. Among the UC constraints, an adequate provision of reserve is important to ensure the security of power system and the fast-response reserve is essential to bring system frequency back to acceptable level following the loss of an online unit within a few seconds. In this paper, the authors present and solve a UC problem including the frequency-based reserve constraints to determine the optimal FRR requirements and unit MW schedules. The UC problem is solved by using Lagrangian Relaxation-based approach and compared with the actual system schedules. It is observed that favorable reserve and unit MW schedules are obtained by the proposed method while the system security is maintained.

A Lagrangian relaxation approach to simultaneous strategic and tactical planning in supply chain design

We study a multi-echelon joint inventory-location model that simultaneously determines the location of warehouses and inventory policies at the warehouses and retailers. The model is formulated as a nonlinear mixed-integer program, and is solved using a Lagrangian relaxation-based approach. The efficiency of the algorithm and benefits of integration are evaluated through a computational study.

Feasible Aggressor-Set Identification Under Constraints for Maximum Coupling Noise

In this paper, we consider the problem of identifying a feasible set of aggressor nets that would induce maximum crosstalk noise or delay pushout on a coupled victim net, under given logical constraints. We present a novel mathematical formulation of this problem and propose a Lagrangian relaxation-based approach for solving it efficiently and optimally. Experimental results show that the proposed approach is run-time efficient by a factor of up to 800 times in comparison to an exhaustive search approach and reduces timing pessimism by up to 36%. We also formulate and solve this problem while considering the noise susceptibility of the victim's receiving gate.

Determining link weight system under various objectives for OSPF networks using a Lagrangian relaxation-based approach

An important traffic engineering problem for OSPF networks is the determination of optimal link weights. Certainly, this depends on the traffic engineering objective. Regardless, often a variety of performance measures may be of interest to a network provider due to their impact on the network. In this paper, we consider different objectives and discuss how they impact the determination of the link weights and different performance measures. In particular, we propose a composite objective function; furthermore, we present a Lagrangian relaxation-based dual approach to determine the link weight system. We then consider different performance measures and discuss the effectiveness of different objectives through computational studies of a variety of network topologies. We find that our proposed composite objective function with Lagrangian relaxation-based dual approach is very effective in meeting different performance measures and is computationally very fast.

Scheduling flexible flow shops with sequence-dependent setup effects

This paper presents a Lagrangian relaxation-based approach for production scheduling of flexible flow shops (PSFFS) where sequence-dependent setup effects are significant. PSPFS is first formulated as a separable integer programming problem with synchronization constraints between production and machine usage. Lagrangian relaxation is then applied, and the scheduling is decomposed into part production and machine scheduling. In these subproblems, there are network flow structures in the equality constraints describing machine status change and production flow balance. Our iterative algorithm applies a minimum cost network flow algorithm to individual subproblems and adopts an efficient surrogate subgradient method to optimize Lagrangian multipliers. A machine availability-searching heuristic finally adjusts the solution to satisfy all synchronization constraints by exploiting the network structure, economic interpretation of Lagrangian multipliers, and the slack time policy. Numerical results of 16 cases, each having 20 test problems, demonstrate that differences between the schedules obtained by our approach and the true optima are on average within 15%, CPU times spent are all less than 17 min on a Pentium-II personal computer. Among the problem dimension factors, the number of part types has the most significant effect on both optimality and computation efficiency. Application of the methodology to daily scheduling of a realistic integrated circuit testing facility of 30 machines takes about 6 min of CPU time to generate a near-optimal solution.