Lagrangian Relaxation Decomposition Research Articles

Efficient asymmetric reposition strategy of rolling stock for urban rail transit systems

Balancing energy efficiency with high-quality service on urban rail lines with highly asymmetric passenger demand is a crucial yet challenging issue. This paper proposes an efficient asymmetric rolling stock reposition strategy to simultaneously optimize the train timetable, speed profiles and rolling stock circulation, given the passenger demand in the undersaturated direction and timetable in the oversaturated direction. By constructing space-time networks for passengers and rolling stocks, respectively, an integer linear programming model is formulated by considering the tradeoff between the passenger travel time and train energy cost. A rolling horizon scheme incorporating the Lagrangian relaxation decomposition technique is designed, which can be decomposed into two subproblems by relaxing the capacity constraints. The effectiveness of our proposed methodology is validated through numerical experiments on a test network and the Beijing Batong Line, demonstrating that the optimized train circulation plan significantly reduces energy consumption while incurring a minimal increase in travel time.

Multi-periodic train timetabling using a period-type-based Lagrangian relaxation decomposition

To provide passengers with strict regularity of train operation, this research is devoted to modeling and solving the multi-periodic train timetabling problem to simultaneously optimize operation periods, arrival times, and departure times of all period types of trains on a double-track rail network. Based on the construction of a weighted directed graph, a multi-path searching model, namely, a 0-1 linear programming model, is built to minimize the total travel time of all period-types of trains subject to many operational constraints, including station parking capacity and train minimum load factors. After decomposing this model by introducing some Lagrangian multipliers to relax its complicated constraints, a solution algorithm, including a multi-path simultaneous searching sub-algorithm for each period-type of train, is designed to optimize both the feasible and dual solutions, which correspond to the upper and lower bounds, respectively. Finally, the performance, convergence, sensitivity, and practicability of our method are analyzed using many instances on both a small rail network and the high-speed railway between Beijing and Shanghai in China.

Electricity market short‐term risk management via risk‐adjusted probability measures

This study presents an iterative algorithm for modelling the mean-risk model with the conditional value at risk (CVaR). The algorithm is based on the Lagrangian relaxation decomposition, and its main advantage is that it allows removing the coupling between the scenarios due to the constraints used to model the risk. At each stage of the algorithm, a risk-neutral stochastic optimisation problem is solved with the risk-adjusted probabilities that substitute the original ones. The study presents the application of the proposed iterative CVaR algorithm to two different short-term problems where the decision makers are exposed to a high volatility of electricity spot market prices. In the first problem, a time horizon of 1 week is taken into account and a future physical contract is employed as a hedging mechanism. The second problem includes a very detailed formulation of the unit commitment problem. The numerical application is based on realistic data of the Iberian electricity market, where the algorithm has shown a good performance in terms of accuracy and computational time. In addition, this study provides a criterion for selecting the value of the parameters used to implement the CVaR model.

Simultaneous passenger train routing and timetabling using an efficient train-based Lagrangian relaxation decomposition

This paper focuses on the simultaneous passenger train routing and timetabling problem on the rail network consisting of both unidirectional and bidirectional tracks using an efficient train-based Lagrangian relaxation decomposition. We first build an integer linear programming model with many 0–1 binary and non-negative integer decision variables, after then reformulate it as a train path-choice model for providing an easier train-based Lagrangian relaxation decomposition mechanism based on the construction of space-time discretized network extending from node-cell-based rail network. Moreover, through reformulating safety usage interval restrictions with a smaller number of constraints in this reformulated model, the train-based decomposition needs fewer Lagrangian multipliers to relax these constraints. On the basis of this decomposition, a solving framework including a heuristic algorithm is proposed to simultaneously optimize both the dual and feasible solutions. A set of numerical experiments demonstrate the proposed Lagrangian relaxation decomposition approach has better performances in terms of minimizing both train travel time and computational times.

Timetable Optimization for High-Speed Rail with Multiple Operating Periods: Solving Method Based on a Framework of Lagrangian Relaxation Decomposition

By aiming to provide high-speed rail (HSR) passengers with travel convenience in the regularity of train operation, this research is devoted to optimizing a novel type of multiperiod train timetable in which trains operating with the same stop stations, speeds, and headways are classified into one period type. Moreover, trains with different period types have various operating periods on the HSR system. First, the authors propose an optimization model with the aim of minimizing the total travel times of all trains and then further reformulate the model on the basis of a weighted digraph. Through the period type–based decomposition to the reformulated model with Lagrangian relaxation, a solving algorithm that includes a multiflow shortest-path searching algorithm is designed for optimizing the multiperiod train timetable and obtaining a lower bound of the objective to evaluate its quality. Numerical examples illustrate that the solving algorithm has a good convergence and can obtain a satisfactory feasible solution with a small gap between the upper and lower bounds of the objective values.

Using Lagrangian Relaxation Decomposition With Heuristic to Integrate the Decisions of Cell Formation and Parts Scheduling Considering Intercell Moves

Cell formation and parts scheduling are two important correlated processes in a cellular manufacturing system; however, the decisions involved in these processes are typically made individually. Determining how to integrate these decisions effectively to pursue a productive and lower cost system has become an important issue. This paper focuses on providing an effective solution to integrate the decisions of cell formation and parts scheduling, while considering intercell moves by using a Lagrangian relaxation decomposition method. A mixed integer nonlinear programming mathematical model (CFPSP) is proposed to determine which part families and machine groups are assigned to cells and in which sequence the parts are processed in the machines to minimize the total tardiness penalty cost. To effectively solve the model, a Lagrangian relaxation decomposition method with a heuristic (LRDH) is developed. Using the LRDH, the CFPSP model is solved by decomposing the model into two subproblems, i.e., the cell formation subproblem (CFPSP-FD) and the parts scheduling subproblem (CFPSP-SD). After linearizing the CFPSP-FD model, the subproblem CFPSP-FD is solved by the MIP optimizer CPLEX. A scatter search approach is developed to solve the subproblem CFPSP-SD. Combined with the Lagrange multipliers, the CFPSP-SD model takes into consideration the assignment of part families and the associated machine groups to each cell, when it sequences the processing of the parts on each machine in cells. An illustration of the application of the CFPSP model in an electronic appliance cellular manufacturing enterprise in China is presented.

Integrated model for software component selection with simultaneous consideration of implementation and verification

One important objective of component-based software engineering is the minimization of the development cost of software products. Thus, the costs of software component implementation and verification, which may involve substantial expenses while under development, should be reduced. In addition, the costs for these processes should not be considered individually, but in an integrated manner, to further reduce development cost. In the current paper, an integrated decision model is proposed to assist decision-makers in selecting reuse scenarios for components used for implementation and in simultaneously determining the optimal number of test cases for verification. An objective of the model is the minimization of development cost, while satisfying the required system and reliability requirements. The Lagrange relaxation decomposition (LRD) method with heuristics was developed to solve integrated decision problems. Based on LRD, the nonlinear model is condensed into a 0–1 knapsack problem for the subproblem on reuse scenario selection and an integer knapsack problem for the subproblem on the determination of the optimal number of tests. Combined with the Lagrange multiplier-determined heuristic, the proposed algorithm can determine the global optimum solution. Simulations of varying sizes for problems and sensitivity analyses were conducted, and the results indicate that LRD is more effective than previous methods in determining global optimal solutions for the integrated decision problem.

Decentralized State Estimation and Bad Measurement Identification: An Efficient Lagrangian Relaxation Approach

This paper proposes a decentralized state-estimation approach that relies on an elaborated instance of the Lagrangian relaxation decomposition technique. The proposed algorithm does not require a central coordinator but just to moderate interchanges of information among neighboring regions, and exploits the structure of the problem to achieve a fast and accurate convergence. Additionally, a decentralized bad measurement identification procedure is developed, which is efficient and robust in terms of identifying bad measurements within regions and in border tie-lines. The accuracy and efficiency of the proposed procedures are assessed by a large number of simulations, which allows drawing statistically sound conclusions.

Heuristics for Joint Decisions in Production, Transportation, and Order Quantity

An attempt is made to tackle joint decisions in assigning production, lot size, transportation, and order quantity for single and multiple products in a production-distribution network system with multiple suppliers and multiple destinations. The approach hinges on providing an optimized solution to the joint decision model (JDM) through a two-layer decomposition (TLD) method that combines several heuristics. By combining the Lagrange multipliers and introducing a number of artificial variables into the two-layer decomposition, a Lagrange relaxation decomposition (LRD) method with heuristics is developed to solve multiproduct joint decision problems (JDM-M). Using the LRD, the JDM-M model is solved by decomposing into two subproblems in two layers. The first layer is the joint decisions in assigning production, transportation flow, and lot size (APLS-TF) using the assignment heuristic AH-M. The second layer is the joint decisions in transportation and order quantity (TOQ-M) using a revised BH heuristic. Combined with Lagrange multipliers, the APLS-TF model takes into consideration the transportation costs together with production costs when it assigns annual production among suppliers. In essence, the algorithm assigns annual production simultaneously with annual transportation flows. Simulations on different sizes of problems and problems with large variances in data have shown that the LRD is effective, and in general more effective than the TLD.

Multi-area coordinated decentralized DC optimal power flow

This paper provides a framework to carry out a multi-area optimal power flow in a coordinated decentralized fashion. A DC nonlinear optimal power flow model is used. Losses are incorporated through additional loads based on cosine approximations. The model makes it possible the independent optimal dispatch of each area while the global economical optimum of the whole electric energy system is achieved. This is possible by means of the Lagrangian relaxation decomposition procedure. Optimal energy pricing rates for the energy traded through the interconnections are derived. The developed algorithm can be run in parallel either to carry out numerical simulations or in an actual multi-area electric energy system.

98/00375 A combination of the genetic algorithm and Lagrangian relaxation decomposition techniques for the generation unit commitment problem

98/00375 A combination of the genetic algorithm and Lagrangian relaxation decomposition techniques for the generation unit commitment problem

A combination of the genetic algorithm and lagrangian relaxation decomposition techniques for the generation unit commitment problem

Unit commitment involves the scheduling of generators in a power system in order to meet the requirements of a given load profile. An analysis of the basis for combining the genetic algorithm (GA) and Lagrangian relaxation (LR) methods for the unit commitment problem is presented. It is shown that a robust unit commitment algorithm can be obtained by combining the global search property of the genetic algorithm with the ability of the Lagrangian decomposition technique to handle all kinds of constraints such as pollution, unit ramping and transmission security.