Lagrangian Relaxation Algorithm Research Articles

A Lagrangian relaxation algorithm for stochastic fixed interval scheduling problem with non-identical machines and job classes

This paper deals with operational fixed interval scheduling problems under uncertainty caused by random delays. This stochastic programming problem has a deterministic reformulation based on network flow under the assumption that the machines are identical and the multivariate distribution of random delays follows an Archimedean copula. In this paper, we focus on the problem with heterogeneous machines and jobs classes. The deterministic problem with no delays and more than one machine type has been shown to be NP-hard. We generalize the deterministic reformulation of the stochastic problem with random delays for non-identical (heterogeneous) machines. This formulation for more than one machine type loses an important property of total unimodularity that holds for identical machines reformulation using the network flow problem. Therefore, an algorithm based on the Lagrangian relaxation is derived and implemented together with an efficient upper bounding heuristic. Its efficiency is confirmed by solving a large number of simulated instances.

Integrated optimization of planning and operation of a shared automated electric vehicle system considering the trip selection and opportunity cost

<abstract> <p>Shared autonomous electric vehicle systems (SAEVS) combine autonomous driving technology with shared electric vehicle services to provide advantages over traditional shared vehicle systems, including autonomous vehicle relocation and rapid response to user needs. In this study, we seek to enhance the operational efficiency and profitability of SAEVS by considering trip selection and the potential opportunity cost associated with unmet user demands. An integer linear programming (ILP) model is developed using a spatio-temporal state network to optimize the system design planning (e.g., charging facility, vehicle fleet sizing and distribution) and operational decisions (e.g., vehicle operational relocation and trip selection strategy). To handle the computational complexities of this model, we propose a Lagrangian relaxation (LR) algorithm. The performance of the LR algorithm is evaluated through a case study. The results, along with a parameter sensitivity analysis, reveal several key findings: (ⅰ) Allocating vehicles to stations with concentrated early peak demand, distributing charging facilities to stations with high total demand throughout the day and implementing vehicle relocation after the early demand peak can mitigate uneven vehicle distribution; (ⅱ) Implementing trip selection enhances SAEVS profitability; (ⅲ) Increasing opportunity cost meets user demands but at the expense of reduced profit; (ⅳ) It is recommended that SAEVS be equipped with charging facilities of suitable charging power based on operational conditions.</p> </abstract>

A Lagrangian relaxation algorithm and hybrid genetic algorithm-black widow optimization for perishable products supply chain with sustainable development goals consideration

A Lagrangian relaxation algorithm and hybrid genetic algorithm-black widow optimization for perishable products supply chain with sustainable development goals consideration

Resource Scheduling of Time-Sensitive Services for B5G/6G Connected Automated Vehicles

Due to the shortage of spectrum resources in the Internet of vehicles, the diversified time-sensitive services for B5G/6G connected automated vehicles cannot be effectively transmitted and the existing researches lack efficient and intelligent resource scheduling methods. Therefore, this paper proposes a spectrum resource scheduling model for time-sensitive services, and designs a two-tier joint resource scheduling method based on the age of information and bandwidth requirements under macro base station (MBS) and Road Side Unit (RSU). In the first layer, we propose a Vickery-Clarke-Groves (VCG) enabled auction model to solve the time-sensitive services resource scheduling problem, in order to guarantee the authenticity of vehicle users (VUEs) bidding. In addition, the lagrange relaxation algorithm is utilized to obtain the optimal solution of spectrum resource allocation to ensure the VUEs service delay and achieve low complexity resource scheduling. In the second layer, we design a federated learning based MBS auxiliary communication method to alleviate RSU communication pressure. The communication links of differentiated time-sensitive services are predicted to improve the quality of perception services. Simulation results verify that the proposed algorithms can enhance the VUEs experience effectively compared with the conventional methods, in terms of the delay, throughput and packet loss ratio.

Robust multimodal hub network design for hazmat considering a detour strategy

Most traditional hub networks focus on ordinary cargo transportation and usually limit the number of transshipment occurrences to minimise costs. However, this limitation is not conducive to the transportation of hazmat, which often requires detours to avoid high risks. Therefore, this paper proposes a new multimodal network model considering a detour strategy and discusses the uncertainty in the network from the demand point of view. The method of robust optimisation is used to address the uncertainty in the proposed model, and the Lagrange relaxation algorithm and augmented ɛ-constraint method are used to transform and simplify the model. In addition, simulated data of hazmat transportation in the Yangtze River Delta urban agglomeration are used for a case study. The results show that under the consideration of uncertain demand, changes in the demand uncertainty degree and transit discount factor cause changes in the total cost and risk of the multimodal hub network, thus affecting the carrier’s location and route decisions; the new model can better control and mitigate risk, which is more suitable for hazmat transportation. This study provides a reference for multimodal transport carriers in hazmat transportation decision making.

New Heuristic Algorithm Based on the Lagrangian Relaxation for a Real-World High-Speed Railway Timetabling Problem

High-speed railway (HSR) systems have been developing rapidly in China and other countries throughout the past decade; as a result, efficiently operating these large-scale systems has presented new challenges to the railway industry. A high-quality train timetable should satisfy transportation demands with the best possible benefit. This study presents a scheduling model for double-track HSR lines based on a train service plan for timetable optimization. First, we construct a space–time network and assign each resource a usage cost at every discrete instant of time to maximize the total profit of a train timetable as a whole. Second, we propose a heuristic method based on the Lagrangian relaxation algorithm to solve the model by updating the resource usage costs according to the conflicts caused by trains in each iteration. Finally, we consider the Beijing–Shanghai HSR line as a real-world application of the methodology. In each of the four cases tested, we obtain an approximate optimal solution with a duality gap of less than 8% within 120 s. The results show that the proposed model and heuristic offer an efficient and promising means of addressing HSR timetable problems.

Lagrangean relaxation algorithms for fixed-charge capacitated relay network design

We consider a fixed-charge capacitated relay network design problem (CRNDP), which has widespread applications in the design of telecommunication and transportation networks, where relay points are presented as signal regenerators, truckload driver and/or tractor exchange stations or electrical vehicle recharging station. Our CRNDP integrates the key features of single allocation hub location problem (SAHLP) and capacitated network design problem (CNDP). Typical range restrictions in truckload transportation, electrical vehicles as well as for keeping signal strength in telecommunications, and further limitations of flow on the links of an underlying network in these mediums motivate the use of distance constraints and link capacity constraints in the design of a relay network for this diverse set of applications. To efficiently solve our model, we propose a Lagrangean relaxation algorithm, where the upper bounds are enhanced by a heuristic during the iterations and the lower bound is improved by the introduction of surrogate constraints. The extensive computational studies are conducted to examine the performance of the algorithm and the results show that the solution method we propose can efficiently solve CRNDP and obtain good-quality solutions, even for very large size instances.

Redesign of the sustainable wheat-flour-bread supply chain network under uncertainty: An improved robust optimization

With the passage of time, the grains supply chain may go out of optimality and be disrupted, so it is necessary to redesign the supply chain to avoid incurring irreparable costs. Because of the occurring grain supply chain sub-optimization event as time passes, this paper redesigns a wheat-flour-bread supply chain network considering sustainable development under uncertainty. For this reason, both facility and transaction networks are investigated, and newly redesigned networks are presented. First, a new definition of sustainable development pillars in supply chains is proposed, and then a four-objective mixed-integer linear programming model is presented. Due to uncertainties in the economic, environmental, social, and technical parameters of the presented model, a robust optimization method is developed to deal with them. Then, an augmented weighted Tchebycheff method, followed by the Lagrangian relaxation algorithm, is applied for solving the four-objective model. Next, a real-case study is performed to appraise the efficiency of the model. Finally, a sensitivity analysis is conducted on the critical parameters of the model to analyze the behavior of different objective functions. With the model’s solution, the redesign of the supply chain provides an enhancement in the economic, environmental, social, infrastructural, and final objective functions by about 18, 23, 31, 37, and 20%, respectively.

The time-dependent electric vehicle routing problem with drone and synchronized mobile battery swapping

Logistics enterprises are moving towards high efficiency and green. As an emerging technology, unmanned aerial vehicle (UAV, also known as drone) is very useful to cope with flexible and fast delivery. Electric vehicles (EVs), as new energy vehicles, are the technical support for green logistics development. This paper considers the last-mile delivery system in which a fleet of homogeneous EVs, equipped with a UAV each, serve a set of customers, and meanwhile battery swapping vehicles (BSVs) offer mobile battery swapping service for EVs. A mixed integer programming (MIP) model is formulated to minimize the sum of depreciation cost, driver wages and energy consumption cost of EVs, UAVs, and BSVs, by considering the nonlinear energy consumption of each vehicle, time-dependent speeds of EVs and BSVs and monetary time value on depreciation cost. A large neighborhood search algorithm based on Q-learning (LNS-QL) is designed in which Q-learning is used to select a combination of tailored destroy and repair operators to solve three sub-problems, (1) the UAV multi-visit problem, (2) the EV and UAV synchronized scheduling problem, (3) the EV and BSV synchronized mobile battery swapping problem. Besides, this paper also designs Lagrangian relaxation algorithm for comparison with LNS-QL. Extensive experiments verify the correctness of the model by testing small-scale instances. Further analysis demonstrates the accuracy and stability of LNS-QL on medium and large-scale instances as well as computational time and verifies the effectiveness of the mobile battery swapping mode in the cooperative delivery system of EVs and UAVs. Actual cases analyze the characteristics of EV delivery from the cost and emissions perspectives by comparing with FV delivery.

Multiobjective Optimization for Adaptive Offloading in Distributed Multiuser MIMO Cell-Free 6G Networks

By offloading some computational tasks to the edge server, edge computing can help relieve the increasing computation burden of mobile users and improve the quality of experience of users. In this paper, we investigate the computation offloading in edge computing-enabled multiuser cell-free (CF) multi-input multi-output (MIMO) networks with multiple building baseband units (BBUs). The user association, transmit covariances, CPU-cycle frequencies, and allocated bandwidths are jointly optimized to minimize two objectives: the energy consumption of mobile devices (MDs) and execution delay of tasks. We formulate it as a vector optimization problem and adopt the scalarization technique to transform it into a scalar mixed integer nonlinear programming (MINLP). Then, to solve the MINLP, we introduce user sorting into the framework of branch and bound and propose a sorted MD association (MDA) method. Two key issues are tackled in the proposed MDA, i.e., the lower bound of the MINLP and the updation of incumbent solution. For the first issue, we present a Lagrangian dual relaxation algorithm based on subgradient projection, while the second one involving another nonconvex minimization problem, we propose a successive convex approximation (SCA) based algorithm to solve it. Finally, extensive simulations demonstrate that the proposed method is able to reduce the system cost significantly and achieve better system performance by comparing with the benchmark schemes.

High-speed railway-based fast logistics service network design problem

Different from the conventional logistics service network design problem, we design a fast logistics service network based on high-speed railway. An integrative optimization model which is applicable for solving practical problems is established. This paper simultaneously considers three subproblems: Train timetabling, freight flow assignment and electrical multiple units (EMU) routing plan, in which the objectives are simultaneous to minimize the total train travel time, the operation cost and transportation cost of freight transport, the number of freight EMU and the number of maintenance tasks. The constraints imposed in the model include space-time path resource assignment restriction, node operation capability, train safety interval time, train connection time restriction, freight service time window, train loading capacity restriction and EMU routing restriction. Based on the thoughts of divide and conquer, the original problem is decomposed by using the decomposition mechanism of the Lagrange relaxation algorithm to solve the integrated optimization model. To verify the feasibility and effectiveness of the model and algorithm proposed in this paper, a case study is conducted based on Harbin Dalian high-speed railway.

A Lagrangian Method for Calculation of Passing Capacity on a Railway Hub Station

This research paper proposes a Lagrangian method to address the passing capacity of the calculation problem (PCCP) for a hub station in a high-speed railway (HSR) system. The passing capacity of a hub station is critical for determining the train timetable and maximizing the number of trains that can operate on different lines. The objective of this study is to determine the maximum number of trains that can pass through, start at, or end at a hub station. To achieve this objective, a mathematical model was introduced to solve the PCCP. The model was decomposed into two parts using a Lagrangian relaxation algorithm. The first part of the model was a simple train arrival problem (TAP) that reflected the timing of trains at the hub station with simultaneous arrival and departure time constraints. The second part of the model was a train spatio-temporal routing problem (TSRP) that aimed to solve the shortest spatio-temporal path of trains with free conflict with the train’s trajectory. A real instance was provided to demonstrate the feasibility of the proposed approach and the effectiveness of the Lagrangian method. The results showed that the proposed method can efficiently solve the PCCP and maximize the passing capacity of a hub station in an HSR system.

A dynamic dispatching problem for autonomous mine trucks in open-pit mines considering endogenous congestion

The introduction of autonomous mine trucks can improve the efficiency, productivity, and safety of open-pit mines, but may require more reasonable dynamic truck dispatching system to guide routes and schedules than manual driven mine trucks. In practical operations, for security consideration, only one autonomous mine truck is allowed to pass through each intersection at a time, which may lead to endogenous congestion at intersections. In this paper, we first introduce the autonomous mine trucks and endogenous congestion at intersections in the dynamic truck dispatching problem. We propose the concept of road resource which denotes the time-extended intersection and give the time–space network with road resources. The rolling horizon framework is applied to model the dynamic feature. In each horizon, the dispatching of autonomous mine trucks is formulated as a mixed integer linear programming model on the time–space network with road resources. To solve the dispatching problem in each horizon, we propose a Lagrangian relaxation algorithm to decompose the problem into a series of shortest path problem, which is solved by the designed label setting algorithm. Moreover, to yield high-quality feasible solution in each iteration of Lagrangian relaxation algorithm, a repairing algorithm with an adaptive initialization strategy is designed. A case study of an open-pit mine in China is involved to show the efficiency of the designed model with road resources and the proposed algorithm. The discussions about the productions, waiting time, travel distance, and serving time are given by sensitivity analysis.

A user-knowledge vector space reconstruction model for the expert knowledge recommendation system

Expert Knowledge Recommendation System (EKRS) is an intelligent research assistance system. The system is formed by mapping two sets of conceptual spaces through Institutional Repository (IR) and Core Resource Dataset (CRD) in 2018. The user knowledge pattern matching (UKPM) of EKRS has problems such as uncertain user knowledge text matching, slow update of expert knowledge, and inability to accurately track user knowledge. This paper establishes a user knowledge vector space reconstruction model (UKVSM) through the following steps to solve the above problems. Firstly, the text feature items of IR and CRD are reconstructed and the depth and density correction coefficient matrix of the original node of the text semantic meaning is calculated based on the similarity of feature items of the semantic layer. Secondly, in order to improve the efficiency of UKPM exact matching, the Lagrangian relaxation algorithm (LRA) is used to optimize the two sets of knowledge matching strategies. Finally, the real data set is extracted from the EKRS platform, and the model and algorithm proposed in this paper are tested and verified respectively, and compared with other methods. Experiments show that reconstruction model can improve the accuracy of user knowledge task assignment in EKRS, while LRA can improve the efficiency of model solving.

Designing intelligent public parking locations for autonomous vehicles

Looking for parking spaces in crowded areas can be a stressful and time-consuming challenge. With the arrival of technologies like autonomous vehicles (AVs), users can park far away from their destinations. AVs can move from the city center, which has high congestion, into a parking spot outside the area, and thus have more flexibility in choosing the parking location. This study aims to provide a rent bid for the daytime parking of AVs that considers urban land use to evaluate parking strategies possibly chosen by AV users. We determine an actual parking demand function by integrating individual preferences into a p-median problem that controls user optimality. Combining all these together, a novel dynamic optimization formulation is proposed to design the location of parking facilities for AVs that examines the driver's parking preference including rent bid, waiting time for searching parking lots, and travel costs. A Lagrangian relaxation (LR) algorithm is presented to solve the stochastic parking location problem that integrates a reliability strategy to balance demand and supply for parking spaces. The average gap between the exact and the LR solution is about 6% which is very reasonable. We provide a detailed case study of our model with real data generated from daily household travel in New York City. The results show that the average parking price is reduced by 34%, and the average empty trips by AVs decrease by 22% using the proposed public parking strategy.

A Mixed-Integer Linear Formulation for a Dynamic Modified Stochastic p-Median Problem in a Competitive Supply Chain Network Design

Background: The Dynamic Modified Stochastic p-Median Problem (DMS-p-MP) is an important problem in supply chain network design, as it deals with the optimal location of facilities and the allocation of demand in a dynamic and uncertain environment. Methods: In this research paper, we propose a mixed-integer linear formulation for the DMS-p-MP, which captures the key features of the problem and allows for efficient solution methods. The DMS-p-MP adds two key features to the classical problem: (1) it considers the dynamic nature of the problem, where the demand is uncertain and changes over time, and (2) it allows for the modification of the facility locations over time, subject to a fixed number of modifications. The proposed model uses robust optimization in order to address the uncertainty of demand by allowing for the optimization of solutions that are not overly sensitive to small changes in the data or parameters. To manage the computational challenges presented by large-scale DMS-p-MP networks, a Lagrangian relaxation (LR) algorithm is employed. Results: Our computational study in a real-life case study demonstrates the effectiveness of the proposed formulation in solving the DMS p-Median Problem. The results show that the number of opened and closed buildings remains unchanged as the time horizon increases due to the periodic nature of our demand. Conclusions: This formulation can be applied to real-world problems, providing decision-makers with an effective tool to optimize their supply chain network design in a dynamic and uncertain environment.

An optimization model with a lagrangian relaxation algorithm for artificial internet of things-enabled sustainable circular supply chain networks

An optimization model with a lagrangian relaxation algorithm for artificial internet of things-enabled sustainable circular supply chain networks

Designing a resilient retail supply network for fresh products under disruption risks.

The retail sector supplies the daily fresh products and increasingly plays a key role in the stability and livability of cities. However, public health events such as COVID-19 have caused frequent product shortages in recent years. The risk of fresh product shortages not only causes retailers to lose profits, but also affects the normal life of residents. In this paper, we address the problem of designing a resilient retail supply network for fresh products under the supply disruption risks and propose a bi-objective mixed-integer programming model. This model can help retailers to select suppliers, distribution centers and transportation routes under different scenarios and implement three resilience strategies, namely, priority supply, multiple sourcing and lateral transshipment. We use the ε -constraint method to transform the multi-objective problem into a single objective model and develop a Lagrangian relaxation algorithm to solve the different scale instances. This model is solved for a real-life case of a supermarket to obtain managerial insights. In the case study, this paper shows the set of Pareto fronts with different inventory periods, maximum allowed decay and decay rate. We calculate the expected total cost under targeted disruption scenarios and evaluate the effectiveness of these resilience strategies when implemented concurrently or separately. Our results identify the most critical suppliers and distribution centers that should be fortified. We elaborate that more resilience strategies are not always better and managers need to take appropriate resilience strategies according to their own problems.

An Efficient Lagrangian Relaxation Algorithm for the Shared Parking Problem

In recent years, with the significant increase in the number of motor vehicles in big cities, parking difficulties have become increasingly prominent. It is crucial for alleviating the problem to tap the potential of idle parking spaces. We study how to make the most of the shared idle parking spaces to provide intelligent and efficient parking services for parking demanders by optimizing the allocation of the parking spaces. First, concerning the rigid requirements of the demanders on the parking rates and the walking distances, a matching model is developed with the objective of maximizing the utilization of shared parking spaces. In this model, the parking time intervals of parking demanders and the available time intervals of shared parking spaces are generalized to real intervals from integer intervals in previous related research. Second, the model is analyzed and proved to be NP-complete. A Lagrangian relaxation algorithm is considered to solve the model efficiently since the relaxation model becomes much easier to solve if a certain set of constraints are dropped. The novel points of the Lagrangian relaxation algorithm include two algorithms BDP (Bottom-up Dynamic Programming) and GH (Greedy Heuristic). The Lagrangian relaxation algorithm repeats three processes with different Lagrangian multiplier determined by the sub-gradient method until the stop-criteria is met: (1) find upper bounds based on algorithm BDP; (2) find lower bounds by algorithm GH. (3) update the Lagrangian multiplier. Finally, we conduct the experiments to simulate a realistic scenario where the commuters share idle parking spaces to the platform and drivers find parking spaces for some daily activities. The results show that our algorithm has high computational efficiency and accuracy. The efficiency advantage is distinct, especially when the computational scale of the problem is large. In addition, the final upper bound obtained by our algorithm is validated to be tight.

Designing a sustainable reverse supply chain network for COVID-19 vaccine waste under uncertainty

The vast nationwide COVID-19 vaccination programs are implemented in many countries worldwide. Mass vaccination is causing a rapid increase in infectious and non-infectious vaccine wastes, potentially posing a severe threat if there is no well-organized management plan. This paper develops a mixed-integer mathematical programming model to design a COVID-19 vaccine waste reverse supply chain (CVWRSC) for the first time. The presented problem is based on minimizing the system's total cost and carbon emission. The uncertainty in the tendency rate of vaccination is considered, and a robust optimization approach is used to deal with it, where an interactive fuzzy approach converts the model into a single objective problem. Additionally, a Lagrangian relaxation (LR) algorithm is utilized to deal with the computational difficulty of the large-scale CVWRSC network. The model's practicality is investigated by solving a real-life case study. The results show the gain of the developed integrated network, where the presented framework performs better than the disintegrated vaccine and waste supply chain models. According to the results, vaccination operations and transportation of non-infectious wastes are responsible for a large portion of total cost and emission, respectively. Autoclaving technology plays a vital role in treating infectious wastes. Moreover, the sensitivity analyses demonstrate that the vaccination tendency rate significantly impacts both objective functions. The case study results prove the model's robustness under different realization scenarios, where the average objective function of the robust model is less than the deterministic model ones’ in all scenarios. Finally, some insights are given based on the obtained results.