Lagrangian Relaxation-based Algorithm Research Articles

Improved approximation algorithms for solving the squared metric k-facility location problem

The squared metric k-facility location problem is a frequently encountered generalization of the k-means problem, where a specific cost should be paid for opening each facility. The current best approximation ratio for this problem is 44.473+ϵ, which was obtained using a local search algorithm. We advance the state-of-the-art for the problem by devising a Lagrangian relaxation-based algorithm that achieves an improved approximation guarantee of 36.342+ϵ. Our improvement comes from a new deterministic rounding approach, which exploits the properties of the squared metric.

A Traveler Incentive Program for Promoting Community-Based Ridesharing

Traffic congestion has become a serious issue around the globe, partly owing to single-occupancy commuter trips. Ridesharing can present a suitable alternative for serving commuter trips. However, there are several important obstacles that impede ridesharing systems from becoming a viable mode of transportation, including the lack of a guarantee for a ride back home as well as the difficulty of obtaining a critical mass of participants. This paper addresses these obstacles by introducing a traveler incentive program (TIP) to promote community-based ridesharing with a ride back home guarantee among commuters. The TIP program allocates incentives to (1) directly subsidize a select set of ridesharing rides and (2) encourage a small, carefully selected set of travelers to change their travel behavior (i.e., departure or arrival times). We formulate the underlying ride-matching problem as a budget-constrained min-cost flow problem and present a Lagrangian relaxation-based algorithm with a worst-case optimality bound to solve large-scale instances of this problem in polynomial time. We further propose a polynomial-time, budget-balanced version of the problem. Numerical experiments suggest that allocating subsidies to change travel behavior is significantly more beneficial than directly subsidizing rides. Furthermore, using a flat tax rate as low as 1% can double the system’s social welfare in the budget-balanced variant of the incentive program.

Collaborative real-time optimization strategy for train rescheduling and track emergency maintenance of high-speed railway: A Lagrangian relaxation-based decomposition algorithm

To ensure the efficient operation and low maintenance costs of high-speed railway lines, a collaborative real-time optimization strategy for train rescheduling and track emergency maintenance is proposed in this paper. This strategy not only makes online decisions on the track emergency maintenance, but simultaneously deals with the delays caused by the track emergency and disturbances. Based on a bi-directional railway line, a mixed-integer nonlinear optimization model is established, in which the decision variables mainly include the track maintenance intervention type, the end time of track maintenance, arrival/departure time, arrival/departure orders, stopping plans, and train cancellations. The proposed nonlinear model is converted to an equivalent linear model by a linearization method to reduce the computational burden. Moreover, a Lagrangian relaxation-based decomposition algorithm under a rolling horizon framework is designed to satisfy the real-time performance further. Particularly, the rolling horizon framework divides the whole time horizon into three stages according to the status of the maintenance work, i.e., the maintenance decision-making stage, the maintenance stage, and the maintenance completion stage. Furthermore, the Lagrangian relaxation-based algorithm decomposes the original large-scale optimization problem into several smaller sub-problems, which can be computed in parallel to improve the solution procedure. This designed algorithm not only reduces the computation effort for the real-time implementation, but also realizes online feedback correction and improves the robustness of the control strategy. Several numerical experiments are carried out based on the data of Beijing-Shanghai high-speed railway to demonstrate the feasibility and effectiveness of the proposed strategy.

Robust design of blood supply chains under risk of disruptions using Lagrangian relaxation

Emergency supply of blood in disasters is a crucial task for humanitarian aid. In this paper, we present a bi-objective robust optimization model for the design of blood supply chains that are resilient to disaster scenarios. The proposed two-stage stochastic optimization model aims at minimizing the time and cost of delivering blood to hospitals after the occurrence of a disaster, while considering possible disruptions in blood facilities and transportation routes. A Lagrangian relaxation-based algorithm is developed that is capable of solving large-scale instances of the model. We apply this framework to a real case study of blood banks in Jordan.

An adaptive Lagrangian relaxation-based algorithm for a coordinated water supply and wastewater collection network design problem

The last century has seen an increased prevalence and duration of droughts as well as the water shortage especially in Middle East countries like Iran. This urgent situation in Iran such as Urmia Lake in the west Azerbaijan province is a motivation for us to model a new coordinated water supply and wastewater collection network design problem. Due to the uncertainty, as one of inherent sections of the water supply chain, a two-stage stochastic programming approach is used to formulate the problem. To solve the proposed model, a Lagrangian relaxation-based algorithm formulated by a new adaptive strategy is employed. This algorithm considers both upper and lower bounds of the problem to reach a performance solution. The proposed algorithm is compared with two similar algorithms from the literature to reveal its performance. As such, the efficiency of the proposed model is evaluated by some sensitivity analyses. Finally, a comprehensive discussion is provided to show the main findings and practical insights of this research.

Stochastic battery energy storage scheduling considering cell degradation and distributed energy resources

The penetration of distributed energy resources (DERs) greatly raises the risk of distribution network operation such as peak shaving and voltage stability. Battery energy storage (BES) has been widely accepted as the most potential application to cope with the challenge of high penetration of DERs. To cope with the uncertainties and variability of DERs, a stochastic BES scheduling model and its algorithm are proposed. The overall economy is achieved by fully considering the cell degradation, DERs, electricity purchasing, and active power losses. The rainflow algorithm-based cycle counting method is incorporated in the BES scheduling model to capture the cell degradation, greatly extending the expected BES lifetime and achieving a better economy. DER power scenarios are generated to consider the uncertainties and correlations based on the Copula theory. To solve the BES scheduling model, we propose a Lagrangian relaxation-based algorithm, which has a significantly reduced complexity with respect to the existing techniques. For this reason, the proposed algorithm enables much more scenarios incorporated in the BES scheduling model and better captures the DER uncertainties and correlations. Finally, numerical studies for the day-ahead BES scheduling in the IEEE 123-node test feeder are presented to demonstrate the merits of the proposed method. Results show that the actual BES life expectancy of the proposed model has increased to 4.89 times compared with the traditional ones. The problems caused by DERs are greatly alleviated by fully capturing the uncertainties and correlations with the proposed method.

A Lagrangian Relaxation-based Algorithm to Solve a Home Health Care Routing Problem

Nowadays, a rapid growth in the rate of life expectancy can be seen especially in the developed countries. Accordingly, the population of elderlies has been increased. By another point of view, the number of hospitals, retirement homes along with medical staffs has not been grown with a same rate. Hence, Home Health Care (HHC) operations including a set of nurses and patients have been developed recently by both academia and health practitioners to consider elderlies’ preferences willing to receive their cares at their homes instead of hospitals or retirement homes. To alleviate the drawbacks of pervious works and make HHC more practical, this paper introduces not only a new mathematical formulation considering new suppositions in this research area but also a solution approach based on Lagrangian relaxation theory has been employed for the first time. The main strategy of used algorithm aims to fill the gap between the lower bound and upper bound of problem and finds a solution which has both optimality and feasibility properties. By generating a number of numerical examples, results show the performance of the proposed algorithm analyzed by different criteria as well as the efficiency of developed formulation through a set of sensitivity analyses.

A Lagrangian Relaxation-based Algorithm to solve a Collaborative Water Supply Chain Network Design Problem

A Lagrangian Relaxation-based Algorithm to solve a Collaborative Water Supply Chain Network Design Problem

Optimal coordination of air conditioning system and personal fans for building energy efficiency improvement

Air temperature and speed play a critical role in the thermal sensation of comfort felt by occupants, especially in the tropics. It is of great practical interest to coordinate air conditioning and mechanical ventilation (ACMV) system and personal fans so as to enhance building demand response (DR) capability while minimizing energy cost in response to a specific electricity price signal and maintaining a thermal comfort level. In this paper, an optimization problem of coordinating ACMV and personal fans is addressed, which captures the coupling between ACMV and fans. A Lagrangian relaxation-based algorithm is developed to solve the problem by individually solving the subproblems of ACMV and personal fans with Lagrangian multipliers as the coordinated signals. This algorithm can separate the calculation of the cooling effect from the optimization procedure, so we do not have to solve the problem using a non-analytical model for evaluating the cooling effect provided by the fans. The performance of the proposed method is evaluated and validated using experimental and simulation results. Both the results show that coordinating ACMV and fans can substantially enhance building DR capability, save energy cost, and also improve customized thermal comfort microenvironment.

Eco-system optimal time-dependent flow assignment in a congested network

This research addresses the eco-system optimal dynamic traffic assignment (ESODTA) problem which aims to find system optimal eco-routing or green routing flows that minimize total vehicular emission in a congested network. We propose a generic agent-based ESODTA model and a simplified queueing model (SQM) that is able to clearly distinguish vehicles’ speed in free-flow and congested conditions for multi-scale emission analysis, and facilitates analyzing the relationship between link emission and delay. Based on the SQM, an expanded space-time network is constructed to formulate the ESODTA with constant bottleneck discharge capacities. The resulting integer linear model of the ESODTA is solved by a Lagrangian relaxation-based algorithm. For the simulation-based ESODTA, we present the column-generation-based heuristic, which requires link and path marginal emissions in the embedded time-dependent least-cost path algorithm and the gradient-projection-based descent direction method. We derive a formula of marginal emission which encompasses the marginal travel time as a special case, and develop an algorithm for evaluating path marginal emissions in a congested network. Numerical experiments are conducted to demonstrate that the proposed algorithm is able to effectively obtain coordinated route flows that minimize the system-wide vehicular emission for large-scale networks.

Data Sweeper: A Proactive Filtering Framework for Error-Bounded Sensor Data Collection

This paper presents data sweeper—a novel framework that attempts to reduce network traffic for error-bounded data collection in wireless sensor networks. Unlike existing passive filters, a data sweeper migrates in the network and proactively suppresses data updates while maintaining the user-defined error bound. Intuitively, the migration of a data sweeper learns the data change of each sensor node on the fly, which helps to maximize the filtering capacity. We design the data sweeper framework in such a way that it can accommodate diverse query specifications and be easily incorporated into the existing sensor network protocols. Moreover, we develop efficient strategies for query precision maintenance, sweeper migration, and data suppression within the framework. In particular, in order to maximize traffic reduction and adapt to online data updates, a Lagrangian relaxation-based algorithm is proposed for data suppression. Extensive simulations based on real-world traces show that the data sweeper significantly reduces the network traffic and extends the system lifetime under various network configurations.

A Markov Chain-Based Testability Growth Model With a Cost-Benefit Function

In this paper, we propose a Markov chain-based testability growth model (TGM) for the just in-time fix program. This model can help the system designers to manage the testability growth process during system maturation. We also derive a cost-benefit model for allocating test resources to optimize a specified testability metric subject to a constraint on cumulative test cost. Bayesian inference, coupled with a hybrid genetic and particle swarm optimization method, is used to estimate the parameters of the TGM from evolving data, and the resulting model is utilized to track and project the testability metric. A near-optimal Lagrangian relaxation-based algorithm is applied to solve the test resource allocation problem. The testability growth and resource allocation models are validated via simulation examples. Results show that the model and algorithms presented in this paper have the potential to efficiently manage the testability growth problem.

A multi-day activity-based inventory routing model with space–time–needs constraints

We extend activity routing problems to consider ‘needs’ satisfaction over multiple days using an inventory routing problem concept. The resulting inventory-based selective household activity routing problem allows activity type choice, duration choice, activity destination choice, departure time, and scheduling of activities, all within space–time–needs constraints. A Lagrangian relaxation-based algorithm is proposed to solve the model for multiple days. A computational study is conducted. The model can measure several effects: heterogeneity in impacts of travel time changes on different days; the use of dual price as a threshold when needs play a role in the utility maximisation objectives; the possibility for activity participation consolidation due to increased travel time; and non-uniform impacts of travel time increase on utilities across a heterogeneous population. Comparison of the algorithm's performance against a commercial package showed average objective values that differed by only 1.8% or less.

Reliable distribution networks design with nonlinear fortification function

Distribution networks have been facing an increased exposure to the risk of unpredicted disruptions causing significant economic losses. The current literature features a limited number of studies considering fortification of network facilities. In this paper, we develop a reliable uncapacitated fixed-charge location model with fortification to support the design of distribution networks. The model considers heterogeneous facility failure probabilities, one layer of supplier backup, and facility fortification within a finite budget. Facility reliability improvement is modelled as a nonlinear function of fortification investment. The problem is formulated as a nonlinear mixed integer programming model proven to be -hard. A Lagrangian relaxation-based heuristic algorithm is developed and its computational efficiency for solving large-scale problems is demonstrated.

An integrated supply chain network design problem for bidirectional flows

Coordination among supply chains has elicited considerable attention in both academia and industry. This paper investigates an integrated supply chain network design problem that involves the determination of the locations for distribution centers and the assignment of customers and suppliers to the corresponding distribution centers. The problem simultaneously involves the distribution of products from the manufacturer to the customers and the collection of components from the suppliers to the manufacturer via cross-docking at distribution centers. The co-location of different types of distribution centers and coordinated transportation are introduced to achieve cost savings. A Lagrangian relaxation-based algorithm is then developed. Extensive computational experiments show that the proposed algorithm has stable performance and outperforms CPLEX for large-scale problems. An industrial case study is considered and sensitivity analysis is conducted to explore managerial insights. Finally, conclusions are drawn, and future research directions are outlined.

Dynamic origin–destination demand flow estimation under congested traffic conditions

This paper presents a single-level nonlinear optimization model to estimate dynamic origin–destination (OD) demand. The model is a path flow-based optimization model, which incorporates heterogeneous sources of traffic measurements and does not require explicit dynamic link-path incidences. The objective is to minimize (i) the deviation between observed and estimated traffic states and (ii) the deviation between aggregated path flows and target OD flows, subject to the dynamic user equilibrium (DUE) constraint represented by a gap-function-based reformulation. A Lagrangian relaxation-based algorithm which dualizes the difficult DUE constraint to the objective function is proposed to solve the model. This algorithm integrates a gradient-projection-based path flow adjustment method within a column generation-based framework. Additionally, a dynamic network loading (DNL) model, based on Newell’s simplified kinematic wave theory, is employed in the DUE assignment process to realistically capture congestion phenomena and shock wave propagation. This research also derives analytical gradient formulas for the changes in link flow and density due to the unit change of time-dependent path inflow in a general network under congestion conditions. Numerical experiments conducted on three different networks illustrate the effectiveness and shed some light on the properties of the proposed OD demand estimation method.

A Lagrangian relaxation-based algorithm for the allocation of yard cranes for yard activities with different priorities

This paper proposes a mixed integer programming model for the allocation of rail mounted gantry cranes for four basic yard activities with different priorities. The model pays special attention to the typical features of this kind of gantry cranes, such as a restricted traveling range and a limited number of adjustments during loading and discharging operations. In contrast to most of the literature dealing with these four yard activities individually, this paper models them into an integrated problem, whose computational complexity is proved to be NP-hard. We are therefore motivated to develop a Lagrangian relaxation-based heuristic to solve the problem. We compare the proposed heuristic with the branch-and-bound method that uses commercial software packages. Extensive computational results show that the proposed heuristic achieves competitive solution qualities for solving the tested problems.

A Model With a Heuristic Algorithm for Solving the Long-Term Many-to-Many Car Pooling Problem

Long-term car pooling is defined as the sharing of a private vehicle by more than one user who need to reach a destination following a semicommon route between the individuals' points of origin and destination in a specific period. In this paper, we employ a network flow technique to systematically develop a long-term many-to-many car pooling model. The model is formulated as a special integer multiple-commodity network flow problem. A Lagrangian relaxation-based algorithm is also developed to solve the model. The performance of the heuristic algorithm is evaluated by carrying out a case study using real data and suitable assumptions. The test results confirm the usefulness of the model and the heuristic algorithm and that they could be useful in practice.

The Framework and Algorithms for the Survivable Mapping of Virtual Network onto a Substrate Network

Network virtualization serves as an effective method for providing a flexible and highly adaptable shared substrate network to satisfy the diversity of demands. However, the problem of efficiently mapping a virtual network onto a substrate network is intractable, as it is NP-hard. How to guarantee survivability of such a mapping efficiently is another great challenge. In this article, we investigate the survivable virtual network mapping (SVNM) problem and propose two kinds of algorithms for solving this problem efficiently. First, we formulate the model with a minimum-cost objective for the survivable network virtualization problem by mixed integer linear programming. We then devise two kinds of algorithms for solving the SVNM problem efficiently: (1) Lagrangian relaxation-based algorithms including LR-SVNM-M and LR-SVNM-D; (2) Heuristic algorithms including H-SVNM-D and H-SVNM-M. The simulation results show that these algorithms proposed in this study can be used to balance the tradeoff between time efficiency and the mapping cost.

Applying Lagrangian relaxation-based algorithms for airline coordinated flight scheduling problems

Yan and Chen (2007, 2008) employed network flow techniques to construct coordinated scheduling models for passenger- and cargo-transportation, respectively. These models are formulated as mixed integer multiple commodity network flow problems with side constraints (NFPWS) that are characterized as NP-hard. Problem sizes are expected to be huge making the model more difficult to solve than traditional passenger/cargo flight scheduling problems. Therefore, a family of Lagrangian based algorithm is developed to solve the coordinated fleet routing and flight scheduling problems. Numerical tests are performed to evaluate the proposed algorithm using real operating data from two Taiwan airlines. The test results indicate that these solution algorithms are a significant improvement over those obtained with CPLEX. Moreover, the Lagrangian based algorithms are better than the mixed-stop heuristic, consequently they could be useful for allied airlines to solve coordinated fleet routing and flight scheduling problems.