Lagrangian Relaxation Problem Research Articles

Aggregation formulation for on‐site multidepot vehicle scheduling scenario

AbstractThe multidepot vehicle scheduling problem (MDVSP) is a fundamental public transport challenge. To address the large‐scale model and inherent solution symmetry associated with the traditional trip‐to‐trip connection‐based approach for MDVSP, a new trip‐to‐route (T2R) connection‐based approach is proposed. Considering real‐world problem characteristics with numerous trips sharing common origin–destination stations and travel times on one route, this approach aggregates same vehicle possible trip sequences into a T2R connection. Two time‐space network aggregation (TSNA) flow formulation versions, route pair‐based TSNA and station pair‐based TSNA, were constructed. Furthermore, TSNA equivalence under any given decomposition strategy, including first‐in‐first‐out, with the multicommodity network flow (MCNF) model was demonstrated. Given the favorable separable TSNA structure, an alternating direction method of multipliers (ADMM)‐based procedure is proposed to decompose the MDVSP into multiple subproblems that can be linearized and readily solved using commercial solvers. The quality of the solutions was assessed using lower bounds obtained from the Lagrangian relaxation problem. The effectiveness and superiority of the proposed MDVSP models and algorithms were subsequently confirmed using random data sets and real‐world instances.

Solving the multi-compartment vehicle routing problem by an augmented Lagrangian relaxation method

In real-world routing applications such as waste collection or fuel distribution, multiple products that cannot be mixed should be jointly collected or delivered. Multi-compartment vehicles are applied with great flexibility in routing and assignment decisions for these real-world applications. This study concentrates on the multi-compartment vehicle routing problem with fixed compartment size and fixed assignment products to compartments. For this problem, we develop two new formulations in the space–time network and state-space–time network. For the state-space–time network formulation, we dualize the task allocation constraints to the objective function and decompose the Lagrangian relaxed problem (LR) into a series of identical routing subproblems. To produce high-quality feasible solutions, the augmented Lagrangian relaxed problem (ALR) with superior convergence and feasibility is further applied to this problem. The ALR is decomposed into many solvable routing subproblems by alternating direction method of multipliers. A time-dependent dynamic programming algorithm is developed to settle the obtained subproblems of the LR and ALR. The subgradient optimization method solves the Lagrangian dual problems. The solution quality can be evaluated by the relative gap between the global lower and upper bounds at each iteration. In numerical experiments, the proposed method produces solutions with good integrality gaps. Based on Lagrangian relaxation and decomposition techniques, this research provides a novel solving scheme for the multi-compartment vehicle routing problem.

Lagrangian relaxation-based decomposition approaches for the capacitated arc routing problem in the state-space-time network

ABSTRACT We study the capacitated arc routing problem in the state-space-time network. A multi-commodity network flow model is formulated in the state-space-time network. Two Lagrangian relaxation-based decomposition methods are developed to tackle this problem. By dualizing the coupling constraints to the objective function, the Lagrangian relaxed problem (LR) can be decomposed into several routing subproblems in the state-space-time network. Then we extend quadratic penalty terms to the LR and obtain an augmented Lagrangian relaxed problem (ALR). By applying the alternating direction method of multipliers and linearization techniques, we decompose the ALR into a sequence of routing subproblems. We solve these subproblems of LR and ALR by a time-dependent dynamic programming algorithm. Feasible solutions are produced according to the result of the LR or ALR. The Lagrangian multipliers are updated by the subgradient optimization method. We implement the proposed methods in three networks, showing that the ALR-based method has better performance.

Solving the reliability-oriented generalized assignment problem by Lagrangian relaxation and Alternating Direction Method of Multipliers

The well-known generalized assignment problem has many real-world applications. The assignment costs between agents and tasks affected by several factors could be unstable and uncertain. In this paper, we assume that the means and variances of assignment costs are known in advance. The idea is that the decision-maker aims to minimize the total assignment costs not only on average, but also to keep their variability as small as possible. Then, a reliability-oriented model with a nonlinear and concave objective function is formulated, and two decomposition-based methods are systematically developed for this challenging problem. The task allocation constraints are dualized and the Lagrangian relaxed problem is broken into many reliability-oriented knapsack subproblems. By the Lagrangian substitution technique, each subproblem is further decomposed into a standard knapsack problem and a simple univariate concave minimization problem. A lower bound is constructed and multipliers are optimized in dual problems by the subgradient method. Feasible solutions can be generated from the results of these reliability-oriented subproblems by Lagrangian heuristic. To further improve the convergence and solution quality, an Alternating Direction Method of Multipliers (ADMM) based decomposition approach is proposed. The augmented Lagrangian relaxed problem is split into a sequence of subproblems by the block coordinate descent method. To cope with the quadratic terms and the concave terms in these subproblems, linearization and Lagrangian substitution techniques are applied. Feasible solutions are produced from these subproblems, and solution quality can be evaluated by the lower bound provided by the Lagrangian relaxed problem. Numerical experiments are conducted on the test cases transformed from the standard benchmark instances, and the ADMM-based method has superior convergence and solution quality.

Optimal service zones and frequencies for flexible-route freight deliveries

Freight delivery planning is a critical area in transportation and logistics. This paper presents a planning-level model for jointly optimizing headways and zone sizes for flexible-route freight deliveries. The model analyzes deliveries for a one-to-many demand pattern from one terminal to random destinations in a zone, which are reached through approximated travelling salesman problem (TSP) tours. Then, the problem is reformulated as a Lagrangian relaxation problem to constrain the fleet size to an integer value. The proposed model and closed-form solutions provide direct insights into the relations affecting those decision variables. Results from the base case study and sensitivity analyses indicate how various factors affect optimal service frequencies, delivery zone sizes, and costs of freight deliveries. Before detailed vehicle routing problems (i.e., actual delivery scheduling and operations) are solved, the proposed model should be useful for planning freight delivery systems, regarding critical decisions about geographic coverage of services, delivery frequencies, vehicle capacities, and other system characteristics. The developed model can also be used for pickups with a many-to-one demand pattern and extended to serve many-to-many demand patterns in multiple delivery zones through transfers at a central terminal.

Scheduling of steelmaking-continuous casting process with different processing routes using effective surrogate Lagrangian relaxation approach and improved concave–convex procedure

This paper studies a steelmaking-continuous casting scheduling problem with different processing routes. We model this problem as a mixed-integer nonlinear programming problem. Next, Lagrangian relaxation approach is introduced to solve this problem by relaxing the coupling constraints. Due to the nonseparability in Lagrangian functions, we design an improved concave–convex procedure to decompose the Lagrangian relaxation problem into three tractable subproblems and analyse the convergence of the improved concave–convex procedure under some assumptions. Furthermore, we present an effective surrogate subgradient algorithm with global convergence to solve the Lagrangian dual problem. Lastly, computational experiments on the practical production data show the effectiveness of the proposed surrogate subgradient method for solving this steelmaking-continuous casting scheduling problem.

Adaptive memory red deer algorithm for cross-dock truck scheduling with products time window

PurposeThe cross-docking strategy has a significant influence on supply chain and logistics efficiency. This paper aims to investigate the most suitable and efficient way to schedule the transfer of logistics activities and present a meta-heuristic method of the truck scheduling problem in cross-docking logistics. A truck scheduling problem with products time window is investigated with objectives of minimizing the total product transshipment time and earliness and tardiness cost of outbound trucks.Design/methodology/approachThis research proposed a meta-heuristic method for the truck scheduling problem with products time window. To solve the problem, a lower bound of the problem is built through a novel two-stage Lagrangian relaxation problem and on account of the NP-hard nature of the truck scheduling problem, the novel red deer algorithm with the mechanism of the heuristic oscillating local search algorithm, as well as adaptive memory programming was proposed to overcome the inferior capability of the original red deer algorithm in the aspect of local search and run time.FindingsTheory analysis and simulation experiments on an industrial case of a cross-docking center with a product’s time window are conducted in this paper. Satisfactory results show that the performance of the red deer algorithm is enhanced due to the mechanism of heuristic oscillating local search algorithm and adaptive memory programming and the proposed method efficiently solves the real-world size case of truck scheduling problems in cross-docking with product time window.Research limitations/implicationsThe consideration of products time window has very realistic significance in different logistics applications such as cold-chain logistics and pharmaceutical supply chain. Furthermore, the novel adaptive memory red deer algorithm could be modified and applied to other complex optimization scheduling problems such as scheduling problems considering energy-efficiency or other logistics strategies.Originality/valueFor the first time in the truck scheduling problem with the cross-docking strategy, the product’s time window is considered. Furthermore, a mathematical model with objectives of minimizing the total product transshipment time and earliness and tardiness cost of outbound trucks is developed. To solve the proposed problem, a novel adaptive memory red deer algorithm with the mechanism of heuristic oscillating local search algorithm was proposed to overcome the inferior capability of genetic algorithm in the aspect of local search and run time.

A Parallel Solution for the Resilient Operation of Power Systems in Geomagnetic Storms

Optimal operation strategies are essential for maintaining a secure, economic, and resilient operation of power systems in dealing with geomagnetic storms. However, practical efforts for attaining such optimal operations are facing significant modeling and solution challenges that stem from the computational complexity of the geomagnetic disturbance (GMD) mitigation problem. To overcome these challenges, this paper proposes a parallel solution approach to improve the computational efficiency. In this approach, the mixed-integer nonlinear programming (MINLP) problem of mitigating geomagnetic disturbance is split into a nonlinear AC optimal power flow subproblem (ACOPF) and a mixed-integer linear DC geomagnetically-induced current (GIC-DC) mitigation subproblem. The solution mitigates the negative effects of geomagnetic storms by introducing transformer neutral and transmission line series GIC blocking devices, transmission switching, and a host of other operation planning schemes which can enhance the power system resilience. Multiple cases on the 6-bus, 18-bus and 150-bus power systems are studied to show the effectiveness and efficiency of the proposed parallel modeling and solution approach. The studied cases also show that the proposed parallel solution approach offers a promising performance which is based on the applications of augmented Lagrangian relaxation and auxiliary problem principle methods.

Regression and extrapolation guided optimization for production–distribution with ship–buy–exchange options

The production–distribution problem relates to the location of manufacturing plants and warehouses, and to the distribution of multiple commodities from plants to warehouses and from warehouses to customers. Motivated by industrial practices, we extend herein the problem by considering ship–buy–exchange options. Given the complexity of the studied problems, the branch-and-bound method embedded in commercial solvers cannot obtain favorable solutions for medium- and large-sized instances. Herein, we propose a new regression and extrapolation guided optimization method for the solution of these problems. This method obtains the solution values related to the Lagrangian and uncapacitated relaxation problems and performs a regression analysis with these identified solutions and the optimal values to generate the optimal solution patterns. The generated knowledge pertaining to these optimal patterns is extended from small- to large-sized problems using the extrapolation and quadratic regression approaches. When solving large-sized problems, the method employs the extrapolated knowledge in a local search procedure to fix some of the variables and combines an optimization method to solve the reduced-sized problems. We compare this optimization method with other existent methods in the literature. The computational results indicate that with the use of the same computational resources, our optimization method can identify supply chain decisions at lower costs.

Fully distributed multi‐area economic dispatch method for active distribution networks

Distributed generators can be integrated in geographically distributed areas and microgrids of active distribution networks (ADNs), and can operate independently. On the basis of the alternating direction method of multipliers, the authors describe an efficient fully distributed algorithm to solve multi-area economic dispatch problems in ADNs without requiring a central coordinator. The physical interpretation of the distributed algorithm and a proof for its convergence are both given. Network losses are taken into account by exploiting the features of ADNs. Two other popular distributed algorithms, including Lagrangian relaxation and auxiliary problem principle, are also implemented and compared with numerical tests. They discuss numerical results that demonstrate the effectiveness and efficiency of the algorithm.

Layered and Distributed Charge Load Dispatch of Considerable Electric Vehicles

Cooperation between controllable loads such as electrical vehicles (EVs) and wind power is regarded as a promising way to promote the integration of wind power. A novel layered and distributed charging load dispatch mechanism is proposed for the control of thousands of EVs in this paper. Based on the Lagrangian Relaxation and Auxiliary Problem Principle, the dispatch framework is developed, consisting of layers of system operator, generation units/EV aggregators, and EVs, and the cooperation between the generation and EVs is considered. Furthermore, the necessity of EV aggregators is analyzed, and the function of them is stressed. Compared with existing distributed methods, the proposed method is proper for large populations of EVs and gains an advantage in reducing generation cost directly. In addition, it is with a wider application scope such as problems with coupled constraints. The case study on IEEE-RTS verifies the method is feasible and valid and the charge load dispatch based on it reduces generation cost and wind power spillage.

Multi-echelon supply chain network design in agile manufacturing

In this paper, we consider a supply chain network design problem in an agile manufacturing scenario with multiple echelons and multiple periods under a situation where multiple customers have heavy demands. Decisions in our supply chain design problem include selection of one or more companies in each echelon, production, inventory, and transportation. We formulate the problem integrating all decisions to minimize the total operational costs including fixed alliance costs between two companies, production, raw material holding, finished products holding, and transportation costs under production and transportation capacity limits. A Lagrangian heuristic is proposed in this paper. Optimizing a Lagrangian relaxation problem provides a lower bound, while a feasible solution is generated by adjustment techniques based on the solution of subproblems at each iteration. Computational results indicate the high quality solutions with less than 5% optimality gap are provided quickly by the approach in this paper. Further, compared to initiative managerial alternatives, an improvement of 15% to 25% is not unusual in certain cases for the proposed approach.

搬送を考慮した分散協調型生産スケジューリング手法に関する研究

In recent production system, transportation among processing machines should be included to generate effective schedules. In this study, we focus on Lagrangian Decomposition and Coordination (LDC) method, as one of the distributed cooperative scheduling methods, and apply it to job shop scheduling problem with the transportation. We propose a new method using LDC to solve the production and transport vehicles scheduling problem, in this paper. In the proposed method, firstly we solve Lagrangian relaxation problem and obtain the completion time of each job. Secondly, we apply a heuristic which allocates transport tasks to vehicles, and decides routes of vehicles in the list scheduling which generates a feasible schedule. We show the effectiveness of the proposed method in several simulation experiments.

A Lagrangian Relaxation Method for Crew and Vehicle Rescheduling of Railway Passenger Transportation and its Application

We propose a method for solving the crew rescheduling problem (CRP) and the vehicle rescheduling problem (VRP) based on the Lagrangian relaxation method. The CRP/VRP is formulated as an integer programming problem on the basis of a network flow modeling approach from which a Lagrangian relaxation problem is constructed by relaxing the constraint that links multiple resources. Using two procedures that generate the upper and lower bounds of the primal problem, both of which utilize an efficient shortest path algorithm for the directed acyclic graph (DAG), the proposed method gradually improves the gap between the upper and lower bounds while updating Lagrangian multipliers. Experimental results of real-world vehicle rescheduling data from Japanese railway lines indicated that the proposed method generated feasible solutions that were confirmed to be fairly close to the optimal solutions according to the gap between the upper and lower bounds, and also clarified the quality of the other method's solution by using the gap, which could lead to streamlining and sophisticating real-world rescheduling related activities.

Optimal Mobile Switching Center Positioning and Cells Assignment Using Lagrangian Heuristic

This paper deals with the configuration of a wireless network with the aim of minimizing the overall cost of both operation and network installation. The trade-off between the operation cost and the installation cost is the key consideration when designing cellular telecommunication networks, and can save costs and improve the performance of the network. In this paper, we propose an integrated framework for selecting Mobile Switching Center (MSC) among the candidate MSCs and assigning Base Stations (BSs) to the selected MSCs with the objective function of minimizing the cost of MSC setup, BS to MSC cabling, as well as the cost of handover. Capacity constraint for the selected MSC is also considered in the problem. The problem is expressed in an integer programming model and the Lagrangian relaxation method is proposed to solve the problem by dualizing some constraints. The Lagrangian relaxed problem is decomposed into subproblems that can be resolved optimally. The Lagrangian heuristic algorithm is suggested to find feasible solutions to the original problem. Computational experiments are performed to test the effectiveness and efficiency of the proposed heuristic algorithm. In the experiments, Lagrangian bounds on the optimal solution are used to show the effectiveness of the algorithm. The results of the proposed algorithm are also compared with those of some conventional meta-heuristics, Tabu search (TS) and Genetic algorithm (GA). The computational experiments show that the performance of the proposed heuristics is satisfactory in both the speed and the quality of the solution generated.

A Lagrangian relaxation approach for a large scale new variant of capacitated clustering problem

This paper studies a new variant of capacitated clustering problem (VCCP). In the VCCP, p facilities which procure a raw material from a set of suppliers are to be located among n potential sites (n>p) such that the total cost of assigning suppliers to the facilities and opening such facilities is minimized. Each supplier has a limited supply volume and each facility has a minimum supply requirement that must be satisfied by assigning enough suppliers to the facility. Each supplier can be assigned to at most one facility. When a supplier is assigned to a facility, the former will supply its all available volume to the latter. In order to solve the VCCP, a Lagrangian relaxation approach (LR) with two phases of dual optimization, the subgradient deflection in the first phase and the standard subgradient method in the second phase, is proposed. In the approach, the assignment constraints are relaxed. The resulting Lagrangian relaxed problem can be decomposed into a set of independent knapsack problems, which can be solved to optimality efficiently. At each Lagrangian iteration, a feasible solution is constructed from that of the Lagrangian relaxed problem by applying a greedy algorithm. Finally, the best feasible solution found so far is improved by a simple tabu search algorithm. Numerical tests on random instances show that the proposed LR can produce a tight lower bound and a high quality feasible solution for all instances with up to 4000 suppliers, 200 potential sites, and 100 plants to locate.

Hybrid co-evolutionary particle swarm optimization and noising metaheuristics for the delay constrained least cost path problem

This paper presents a co-evolutionary particle swarm optimization (PSO) algorithm, hybridized with noising metaheuristics, for solving the delay constrained least cost (DCLC) path problem, i.e., shortest-path problem with a delay constraint on the total "cost" of the optimal path. The proposed algorithm uses the principle of Lagrange relaxation based aggregated cost. It essentially consists of two concurrent PSOs for solving the resulting minimization-maximization problem. The main PSO is designed as a hybrid PSO-noising metaheuristics algorithm for efficient global search to solve the minimization part of the DCLC-Lagrangian relaxation by finding multiple shortest paths between a source-destination pair. The auxiliary/second PSO is a co-evolutionary PSO to obtain the optimal Lagrangian multiplier for solving the maximization part of the Lagrangian relaxation problem. For the main PSO, a novel heuristics-based path encoding/decoding scheme has been devised for representation of network paths as particles. The simulation results on several networks with random topologies illustrate the efficiency of the proposed hybrid algorithm for the constrained shortest path computation problems.

An exact algorithm for the fixed-charge multiple knapsack problem

Abstract We formulate the fixed-charge multiple knapsack problem (FCMKP) as an extension of the multiple knapsack problem (MKP). The Lagrangian relaxation problem is easily solved, and together with a greedy heuristic we obtain a pair of upper and lower bounds quickly. We make use of these bounds in the pegging test to reduce the problem size. We also present a branch-and-bound (B&B) algorithm to solve FCMKP to optimality. This algorithm exploits the Lagrangian upper bound as well as the pegging result for pruning, and at each terminal subproblem solve MKP exactly by invoking MULKNAP code developed by Pisinger [Pisinger, D., 1999. An exact algorithm for large multiple knapsack problems. European Journal of Operational Research 114, 528–541]. As a result, we are able to solve almost all test problems with up to 32,000 items and 50 knapsacks within a few seconds on an ordinary computing environment, although the algorithm remains some weakness for small instances with relatively many knapsacks.

A mathematical programming model and solution for scheduling production orders in Shanghai Baoshan Iron and Steel Complex

In this paper, we investigate the production order scheduling problem derived from the production of steel sheets in Shanghai Baoshan Iron and Steel Complex (Baosteel). A deterministic mixed integer programming (MIP) model for scheduling production orders on some critical and bottleneck operations in Baosteel is presented in which practical technological constraints have been considered. The objective is to determine the starting and ending times of production orders on corresponding operations under capacity constraints for minimizing the sum of weighted completion times of all orders. Due to large numbers of variables and constraints in the model, a decomposition solution methodology based on a synergistic combination of Lagrangian relaxation, linear programming and heuristics is developed. Unlike the commonly used method of relaxing capacity constraints, this methodology alternatively relaxes constraints coupling integer variables with continuous variables which are introduced to the objective function by Lagrangian multipliers. The Lagrangian relaxed problem can be decomposed into two sub-problems by separating continuous variables from integer ones. The sub-problem that relates to continuous variables is a linear programming problem which can be solved using standard software package OSL, while the other sub-problem is an integer programming problem which can be solved optimally by further decomposition. The subgradient optimization method is used to update Lagrangian multipliers. A production order scheduling simulation system for Baosteel is developed by embedding the above Lagrangian heuristics. Computational results for problems with up to 100 orders show that the proposed Lagrangian relaxation method is stable and can find good solutions within a reasonable time.

Towards Strong Duality in Integer Programming

We consider in this paper the Lagrangian dual method for solving general integer programming. New properties of Lagrangian duality are derived by a means of perturbation analysis. In particular, a necessary and sufficient condition for a primal optimal solution to be generated by the Lagrangian relaxation is obtained. The solution properties of Lagrangian relaxation problem are studied systematically. To overcome the difficulties caused by duality gap between the primal problem and the dual problem, we introduce an equivalent reformulation for the primal problem via applying a pth power to the constraints. We prove that this reformulation possesses an asymptotic strong duality property. Primal feasibility and primal optimality of the Lagrangian relaxation problems can be achieved in this reformulation when the parameter p is larger than a threshold value, thus ensuring the existence of an optimal primal-dual pair. We further show that duality gap for this partial pth power reformulation is a strictly decreasing function of p in the case of a single constraint.