Lagrangian Relaxation Algorithm Research Articles

Solving the service-oriented single-route school bus routing problem: Exact and heuristic solutions

This paper addresses a school bus routing problem, which is classified as a location–allocation–routing problem. The problem consists of selecting pickup locations, allocating students to them, and generating a route that traverses between them. The proposed model is for a single school and a single-route. The objective is to find the subset of pickup stops aiming to minimize the total distance walked by all students from their homes to the respective pickup stops, subject to an upper bound on the route distance of connecting selected stops. We present an exact and heuristic algorithms which are developed based on a layered graph. Computational results are conducted on a series of generated benchmark instances and test data from Norway that demonstrate a good performance of the proposed approach. • Modeling school bus routing operation for schools in area of Innlandet in Norway. • A new problem variation in the school bus routing problem with a focus on students’ convenience. • Showing a trade-off between the total walking distance of students and the length of the bus route. • Determining a route among a set of pickup points, chosen among bus stops and passengers’ home addresses. • Lagrangian relaxation and label setting heuristic algorithms based on layered graph.

Lagrangian heuristic algorithm for green multi-product production routing problem with reverse logistics and remanufacturing

Today, due to increased market competition, the integration of production and distribution decisions into the supply chain leads to efficiency improvements. Therefore, production routing models have been developed to optimize production and distribution, simultaneously. In recent years, since, the product life cycle has become shorter than in the past, product return policies with fast response times, emphasis on return management, deformation and restorage of finished goods have become very significant. In this paper, the multi-product production routing problem with startup costs and environmental considerations has been studied. Furthermore, reverse logistics and remanufacturing decisions have been integrated. After modeling the problem as mixed-integer linear programming, due to its NP-hardness and the successful application of the Lagrangian Relaxation algorithm (LR) in solving complex supply chain problems, this algorithm has been chosen as the solution method. After applying the standard LR algorithm and the improved LR algorithm that its subgradient optimization method was modified, as a heuristic algorithm, the feasiblizer algorithm is also proposed for feasibilization of the solution obtained from the LR algorithms. To validate the model and solution method, firstly, test problems are solved by GAMS, and then the proposed algorithm is applied to test problems. The numerical results show the good performance of the LR algorithm in medium-size test problems. Finally, based on the computational experiments, managerial insights on the problem have been provided.

Joint Scheduling of Vessel Traffic and Pilots in Seaport Waters

In the busiest seaports, vessel traffic and vessel pilotage management play a crucial role in congestion mitigation. The management of vessel traffic and pilotage involves scheduling the vessels for sailing into and out of a seaport and scheduling the pilots for navigating the vessels in the port waters. In this paper, we study the integrated vessel traffic and pilot scheduling problem of a seaport. We manage the vessel traffic by optimizing the utilization of the navigation channels and the utilization of the anchorage areas in the terminal basin and incorporate the decision of pilot scheduling into the decision of vessel traffic management for congestion mitigation and vessel service enhancement. We formulate the problem on a time–space network with vessel- and pilot-dependent arc costs and develop an integer programming model that minimizes the sum of the berthing and departure tardiness cost of vessels, the cost of unsatisfied vessel service requests, and the pilot dispatching cost. For solving the model, we enumerate feasible vessel paths a priori and develop a Lagrangian relaxation algorithm that decomposes the problem into a vessel and pilot path assignment subproblems. Computational performance of the Lagrangian relaxation algorithm is tested on problem instances generated based on the physical layout and operational data of the Waigaoqiao Port in Shanghai.

Dwell Time Allocation Algorithm for Multiple Target Tracking in LPI Radar Network Based on Cooperative Game.

To solve the problem of dwell time management for multiple target tracking in Low Probability of Intercept (LPI) radar network, a Nash bargaining solution (NBS) dwell time allocation algorithm based on cooperative game theory is proposed. This algorithm can achieve the desired low interception performance by optimizing the allocation of the dwell time of each radar under the constraints of the given target detection performance, minimizing the total dwell time of radar network. By introducing two variables, dwell time and target allocation indicators, we decompose the dwell time and target allocation into two subproblems. Firstly, combining the Lagrange relaxation algorithm with the Newton iteration method, we derive the iterative formula for the dwell time of each radar. The dwell time allocation of the radars corresponding to each target is obtained. Secondly, we use the fixed Hungarian algorithm to determine the target allocation scheme based on the dwell time allocation results. Simulation results show that the proposed algorithm can effectively reduce the total dwell time of the radar network, and hence, improve the LPI performance.

Using subsidies to stabilize peer-to-peer ridesharing markets with role assignment

Peer-to-peer (P2P) ridesharing is a form of shared-use mobility that has emerged in recent decades as a result of enabling of the sharing economy, and the advancement of new technologies that allow for easy and fast communication between individuals. A P2P ridesharing system provides a platform to match a group of drivers, who use their personal vehicles to travel, with their peer riders who are in need of transportation. P2P ridesharing systems are traditionally categorized as two-side markets, with two mutually exclusive sets of agents, i.e., riders and drivers. Fixing the roles of participants a priori, however, could come at an opportunity cost of missed social welfare/revenue for the system. Consequently, this paper proposes a new market game, and its corresponding mathematical formulation, that outputs matching, role assignment, and pricing. We investigate the stability properties of this market game, and present a mathematical formulation that yields a stable outcome if one exists. Furthermore, we propose a Lagrangian relaxation algorithm to obtain a stable solution for large-scale games with empty cores through subsidizing the system. Using numerical experiments, we demonstrate the benefits of the proposed methodology, and its advantages over previously proposed methods for stabilizing non-bipartite graphs.

Distributed and Asynchronous Active Fault Management for Networked Microgrids

A distributed and asynchronous active fault management (DA-AFM) method is developed to manage networked microgrids' (NMs) performance under balanced or unbalanced grid faults. The DA-AFM aims to (1) enable NMs' fast fault ride-through capabilities, (2) limit the total fault contributions by coordinating heterogeneous microgrids in the NM system, and (3) deploy software-defined networks (SDN) to ensure highly resilient AFM. The problem is formulated in an optimization form that can incorporate various fault management objectives and constraints in a programmable and flexible fashion. The scalability and resiliency of the DA-AFM system are guaranteed by adopting an SDN-enabled distributed and asynchronous surrogate Lagrangian relaxation (DA-SLR) algorithm, which avoids single point of failure, preserves privacy, and eliminates the idle waiting of other subproblems. Case studies are performed on a six-microgrid NM system to validate the effectiveness and efficacy of DA-AFM. Testing results show that DA-AFM has excellent convergence performance, supports plug-and-play, is resilient to communication delay and failures, meets real-time requirements and is scalable.

Continuous Time Formulation and Lagrangian Relaxation Algorithm for the Gate Assignment Problem

Gates are important operating facilities and resources in civil airports. It is a core task in the airport operation management to select reasonable gates for inbound and outbound flights. We present a continuous time formulation with second-order cone programming (SOCP) for the gate assignment problem which allocates flights to available gates to optimize both the transfer time of passengers and the robustness of the airport operations schedules. The problem is formulated as a mixed integer nonlinear program, and then, the quadratic objective that minimizes the walking distance of transferring passengers is linearized, and the objective that minimizes the variance of idle time at the gates is transformed to a second-order cone constraint with a linear objective function. Then, a Lagrangian relaxation algorithm is developed by exploiting the problem structure. Computational tests are carried out to illustrate the efficiency of the model and the algorithms. It is shown that the continuous time formulation is more efficient than the existing model, and the Lagrangian relaxation algorithm can obtain better solutions faster than a commercial solver.

Reliable cross-docking location problem under the risk of disruptions

This paper seeks to develop a reliable network of cross-docks by taking in to account disruption and reliability issues to hedge against heterogeneous risk of cross-docking failure. In real environments, applying a recovery policy can be a feasible strategy to handle disruptions. Hence, in this study, a recovery policy has been addressed in the form of reallocating suppliers to alternative cross-docks or altering the transportation strategy to move shipments. In addition to cross-dock location design, the optimum capacity of opened cross-docks will be determined considering the loads that will be served by each cross-docking center under regular and disruption conditions. A mixed integer nonlinear programming formulation is presented for the problem and is then linearized to present an efficient model. In order to solve it, two Lagrangian relaxation algorithms are designed and tested on 40 problem instances with different values of parameters. The results achieved by GAMS/CPLEX are compared with those of two algorithms and some analyses are performed on the solutions. Moreover, as the case study, the focus has been placed on logistic part of a car-manufacturing company with a vast supply chain network, containing more than 600 suppliers. The logistic strategies have been applied in order to reduce the transportation cost through the supply chain network and diminish the disruption subsequences in such a network. Based on the results, some managerial recommendations are presented.

An Improved Lagrange Particle Swarm Optimization Algorithm and Its Application in Multiple Fault Diagnosis

The fault rate in equipment increases significantly along with the service life of the equipment, especially for multiple fault. Typically, the Bayesian theory is used to construct the model of faults, and intelligent algorithm is used to solve the model. Lagrangian relaxation algorithm can be adopted to solve multiple fault diagnosis models. But the mathematical derivation process may be complex, while the updating method for Lagrangian multiplier is limited and it may fall into a local optimal solution. The particle swarm optimization (PSO) algorithm is a global search algorithm. In this paper, an improved Lagrange-particle swarm optimization algorithm is proposed. The updating of the Lagrangian multipliers is with the PSO algorithm for global searching. The difference between the upper and lower bounds is proposed to construct the fitness function of PSO. The multiple fault diagnosis model can be solved by the improved Lagrange-particle swarm optimization algorithm. Experiment on a case study of sensor data-based multiple fault diagnosis verifies the effectiveness and robustness of the proposed method.

A hybrid genetic and Lagrangian relaxation algorithm for resource-constrained project scheduling under nonrenewable resources

Scheduling under nonrenewable resources is one of the challenging issues in project scheduling problems. There are many cases where the projects are subject to some nonrenewable resources. In most of the literature, nonrenewable resources are assumed to be available in full amount at the beginning of the project. However, in practice, it is prevalent that these resources are procured along the project horizon. This paper studies the generalized resource-constrained project scheduling problem (RCPSP) where, in addition to renewable resources, nonrenewable resources are considered, such as budget or consuming materials by the project activities. As the problem is NP-hard, some sub-algorithm elements are developed, which can be used in the structure of inexact approaches for solving the problem. These elements include constraint propagation, priority rules, schedule generation schemes, and local search improvement procedures. Also, a lower bounding algorithm is developed based on the Lagrangian Relaxation (LR) approach, and the problem is optimized by the Genetic Algorithm (GA). The hybrid GA–LR​ algorithm produces a result reasonably near to optimum solutions. Comprehensive computational experiments based on standard project scheduling problems are performed to evaluate these developments. The experiments showed the performance and robustness of the proposed algorithm.

A Lagrangian heuristic algorithm for sustainable supply chain network considering CO2 emission

Manufacturing companies encounter pressure related to not only cost and service level but also certain limitations based on the constraints of energy saving and emission reduction. This study investigates production planning for a sustainable supply chain, for which factors such as the limits for CO2 emission, stochastic demands, service level, and inventory capacities are considered. We build a mixed-integer programming model and propose a Lagrangian relaxation (LR) algorithm to solve the large-scale production planning problem in a cost-efficient way. Based on the historical data of HY Automobile Co. in China and 120 randomly generated instances, we find that the LR algorithm can generate near-optimal solutions with less than a 1% difference as compared to CPLEX solutions. The algorithm can find a solution within 5 min for a case with a large problem size, whereas CPLEX cannot find the solution in an hour. At the same service level, the cost decreases with the initial inventory capacity and CO2 emission limit and increases with the demand and number of the types of products. The cost gap that is caused by increasing these parameters becomes larger when the service level is higher. This finding indicates that the parameters’ values are important to the company when a high service level is required.

A probability first memetic algorithm for the dynamic multiple-fault diagnosis problem with non-ideal tests

The factorial hidden Markov modeling framework, has been proposed in the literature as an effective approach for intermittent fault diagnosis applications. Moreover, the dynamic multiple fault diagnosis problem (DMFD) with non-ideal tests is computationally challenging. Several methodologies have been introduced for the solution of the DMFD problem, namely: the Lagrangian relaxation algorithm, the deterministic simulated annealing and the block coordinate ascent. However, to the best of our knowledge, no evolutionary algorithm including analysis of DMFD characteristics has ever been attempted. This research paper, presents a probability first memetic algorithm (PFMA) for solving the DMFD combinatorial optimization problem, by considering input probability matrices. PFMA improves the performance of the genetic algorithm, with less computational cost. This is achieved by exploiting input probability information, using a local search strategy and a crossover operator. Computational results show that the proposed algorithm finds the optimal solutions on small-scale stochastic synthesis models. Additionally, high-quality solutions are obtained on medium and large-scale cases. The algorithm has been tested successfully on a real-world application. The experimental results highlight the efficiency of the genetic algorithm, following a reasonable local search strategy, towards the solution of an NP-hard combinatorial optimization problem.

Pricing and market segmentation in an uncertain supply chain

This paper addresses a new location-allocation-pricing problem in designing a three-level uncertain supply chain network with stochastic price-sensitive demands. Using the market segmentation problems, a supply-chain network is developed with two distribution channels that consist of Brick & Mortar and online markets, when some demand leakages occur from the market with a higher price. Due to the lack of physical observation of products in online markets, a return policy is used. So, demand behavior is analyzed in terms of the pricing and return policy. The problem established location, allocation, order quantities, pricing and refund price decisions to optimize the total profit of the chain. Furthermore, it is formulated as a mixed-integer non-linear programming (MINLP) model and solved by a Lagrangian relaxation algorithm. The numerical study and computational results indicate the efficiency and effectiveness of the proposed algorithm.

Popularity-Based and Version-Aware Caching Scheme at Edge Servers for Multi-Version VoD Systems

Recently, many video-on-demand (VoD) providers have begun storing multiple versions of the same video to offer multiple-quality video services with different bitrates to users, called multi-version VoD. To improve users’ quality of experience (QoE), it is a good idea to cache videos at edge servers in multi-version VoD systems. However, determining which versions of which videos should be cached or replaced in an edge server is still a major challenge for a multi-version VoD system because of its limited cache storage. In this paper, we propose a popularity-based and version-aware caching scheme (PVCS) at edge servers for multi-version VoD systems. First, based on video popularity, we formulate cache placement as a knapsack problem under constraints such as the cache storage and transcoding computation of the edge server, which aims to maximize the cache hit ratio. Second, we use the transcoding relations among versions to calculate a version-aware caching profit when caching a certain version or multiple versions of a video. The version-aware caching profit is the basis for the subsequent cache replacement algorithm. Third, we propose two algorithms, the video cache placement (VCP) algorithm and the video cache replacement (VCrP) algorithm, to solve the cache placement and replacement problems respectively. VCP utilizes the Lagrangian relaxation algorithm to decide which video files should be cached initially, and VCrP decides which video files cached at the edge server will be replaced dynamically based on the version-aware profit. In this way, the PVCS can improve the cache hit ratio and decrease the average start-up delay. Our simulation results have shown that the PVCS outperforms the other schemes in terms of the cache hit ratio and the average start-up delay.

A Combined Benders Decomposition and Lagrangian Relaxation Algorithm for Optimizing a Multi-Product, Multi-Level Omni-Channel Distribution System

The development of supply chain distribution systems from single- to multi-channel networks for delivering items to end customers has effected many changes in the retail sector. Following the adoption of multi-channel distribution strategies and rapid development of relevant technologies, the Omni-channel approach can yield significant benefits and facilitate trade with customers. This paper aims to optimize a multi-product, multi-level Omni-channel distribution network and shipping flows of products within the network under uncertain conditions. A multi-objective mathematical model is developed that minimizes the costs of supply chain while maximizing customer satisfaction over different scenarios. In order to solve the proposed model, a combined algorithm is developed based on Benders Decomposition (BD) and Lagrangian Relaxation (LR). The presented model and solution approach is implemented in a case study of a distribution system, a large e-commerce startup and online store. Five different scenarios with various service levels are investigated and the numerical results are discussed compared to previous findings. The efficiency of the proposed combined BD-LR solution algorithm is also demonstrated. The results obtained from the case study show that higher service levels are correlated with higher levels of customer satisfaction and lower cost of the system.

Vehicle Air Conditioner Temperature Regulation-based Orderly Charging Strategy for Electric Vehicles

Due to high power load peaks and long durations in extreme weather conditions, as well as large-scale unscheduled electric vehicles charging, it may cause “peak plus peak” loading in local areas of the power grid, moreover, lead to transformer substations overload operation in local areas, which is not good for distribution safe and stable operation of the power grid. Considering the driver’s willingness, this study proposes an orderly charging strategy based on the temperature control of electric vehicle on-board air conditioner. The proposed strategy first constructs the temperature control model for electric vehicle cabins with considering the temperature of entire electric vehicle and the special characteristics of urban roads. Then, multi-objective function is formed with minimum user charging cost and variance of grid load fluctuations. Finally, electric vehicles orderly charging period is optimized by combining with electric vehicle travel chains, charge constraints, time-of-use electricity prices, and substation capacity constraints. The results show that temperature-controlled dispatching can effectively reduce the electric vehicles disorderly charging load. It can serve the power grid “cut peaks and fill valleys” and smooth the overall grid load curve using an improved Lagrange relaxation algorithm to optimize electric vehicles charging period.

Automated taxis’ dial-a-ride problem with ride-sharing considering congestion-based dynamic travel times

In this paper, we study the dial-a-ride problem of ride-sharing automated taxis (ATs) in an urban road network, considering the traffic congestion caused by the ATs. This shared automated mobility system is expected to provide a seamless door-to-door service for urban travellers, much like what the existing transportation network companies (TNC) do, but with decreased labour cost and more flexible relocation operations due to the vehicles’ automation. We propose an integer non-linear programming (INLP) model that optimizes the routing of the ATs to maximize the system profit, depending on dynamic travel times, which are a non-linear function of the ATs’ flows. It is important to involve traffic congestion in such a routing problem since for a growing number of ATs circulating in the city their number will lead to delays. The model is embedded within a rolling horizon framework, which divides a typical day into several horizons to deal with the real-time travel demand. In each horizon, the routing model is solved with the demand at that interval and assuring the continuity of the trips between horizons. Nevertheless, each horizon model is hard to solve given its number of constraints and decision variables. Therefore, we propose a solution approach based on a customized Lagrangian relaxation algorithm, which allows identifying a near-optimal solution for this difficult problem. Numerical experiments for the city of Delft, The Netherlands, are used to demonstrate the solution quality of the proposed algorithm as well as obtaining insights about the AT system performance. Results show that the solution algorithm can solve the proposed model for hard instances. Ride-sharing makes the AT system more capable to provide better service regarding delay time and the number of requests that can be attended by the system. The delay penalty on the profit objective function is an effective control parameter on guaranteeing the service quality while maintaining system profitability.

An integrated model for aircraft routing and crew scheduling: Lagrangian Relaxation and metaheuristic algorithm

Airline optimization is a significant problem in recent researches and airline industrial as it can determine the level of service, profit and competition status of the airline. Aircraft and crew are expensive resources that need efficient utilization. This paper focuses simultaneously on two major issues including aircraft maintenance routing and crew scheduling. Several key issues such as aircraft replacement, fairly night flights assignment and long-life aircrafts are considered in this model. We used the flight hours as a new framework to control aircraft maintenance. At first, an integrated mathematical model for aircraft routing and crew scheduling problems is developed with the aim of cost minimization. Then, Lagrangian relaxation and Particle Swarm Optimization algorithm (PSO) are used as the solution techniques. To evaluate the efficiency of solution approaches, model is solved with different numerical examples in small, medium and large sizes and compared with GAMS output. The results show that Lagrangian relaxation method provides better solutions comparing to PSO and also has a small gap to optimum solution.

Applying available-to-promise (ATP) concept in mixed-model assembly line sequencing problems in a Make-To-Order (MTO) environment: problem extension, model formulation and Lagrangian relaxation algorithm

Mixed-model assembly line is known to be a special case of production lines where variety of product models similar to product characteristics are assembled. This article addresses available-to-promise (ATP) in mixed-model assembly line sequencing problems in a Make-To-Order environment in two stages. First, the customers are prioritized based on their corresponding profit values and a decision support system for order acceptance/rejection based on ATP is developed. By implementing this concept and developing a mathematical model, delivery quantity and date in a planning horizon are determined based on the inventories in the stock. The second stage is solving a mixed binary mathematical model to sequence accepted orders suitably according to demands due dates that guarantees the orders are not released too late or too early. The problem simultaneously considers following objectives: minimizing the total tardiness and earliness costs based on the determined priority of orders and minimizing the utility work and idle time of workers in the production line. An algorithm based on Lagrangian relaxation is developed for the problem, and tested in terms of solution quality and computational efficiency. To validate the performance of the proposed algorithm, various test problems in small size are solved using the CPLEX solver, and compared with the Lagrangian relaxation method. Finally, the proposed model is solved in large size problems to analyze the model performance. The drawback of the CPLEX is that it could not solve large problem instances in reasonable time. For the small sized problem, there is approximately 1% duality gap for the Lagrangian relaxation method. The maximum duality gap in the Lagrangian relaxation method for the large sized problem is always kept below 4% while the average computing time is very reasonable. Therefore, according to the results obtained from test problems, the developed Lagrangian relaxation method proved to be the suitable method for this problem.

Reliable facility location design with round-trip transportation under imperfect information Part I: A discrete model

In this paper, we propose a discrete model to investigate reliable location design with round-trip transportation under imperfect information. The unique feature of this problem is that under each disruption scenario, a customer’s outbound and inbound trips are different when she travels to obtain a service. The discrete model is formulated as a compact non-linear integer programming problem and solved efficiently by a customized Lagrangian relaxation algorithm. Numerical experiments find that the discrete model performs well for small and medium-scale problem instances. Sensitivity analyses reveal the impacts of several parameters on both the cost components and the optimal facility layouts.