Subgradient Optimization Procedure Research Articles

Joint optimization of train scheduling and maintenance planning in a railway network: A heuristic algorithm using Lagrangian relaxation

Train scheduling and maintenance planning compete for the resources in a railway network. A commonly used way is dealing with maintenance planning first and then train scheduling, or vice versa. In this paper, we propose a joint optimization model for the two problems in a railway network with double-track, where the upstream and downstream trains are independent and a maintenance task on a section cannot be split or disrupted. In order to solve the model, a heuristic algorithm using Lagrangian relaxation is developed. Due to the large number of constraints, we use a dynamic constraint-generation technique in the iterations of the sub-gradient optimization procedure. We apply the model and algorithm to a practical problem in the Chinese railway network, in which some additional trains are inserted into a fixed existing timetable and the maintenance plan on the involved high-speed railway sections is adjusted. The computational results illustrate the effectiveness and efficiency of the proposed model and algorithm.

A Lagrangian decomposition approach to computing feasible solutions for quadratic binary programs

In this paper, we develop a Lagrangian decomposition based heuristic method for general quadratic binary programs (QBPs) with linear constraints. We extend the idea of Lagrangian decomposition by Chardaire and Sutter (Manag Sci 41(4):704–712, 1995) and Billionnet and Soutif (Eur J Oper Res 157(3):565–575, 2004a, Inf J Comput 16(2):188–197, 2004b) in which the quadratic objective is converted to a bilinear function by introducing auxiliary variables to duplicate the original complicating variables in the problem. Instead of using linear constraints to assure the equity between the two types of decision variables, we introduce generalized quadratic constraints and relax them with Lagrangian multipliers. Instead of computing an upper bound for a maximization problem, we focus on lower bounding with Lagrangian decomposition based heuristic. We take advantage of the decomposability presented in the Lagrangian subproblems to speed up the heuristic and identify one feasible solution at each iteration of the subgradient optimization procedure. With numerical studies on several classes of representative QBPs, we investigate the sensitivity of lower-bounding performance on parameters of the additional quadratic constraints. We also demonstrate the potentially improved quality of preprocessing in comparison with the use of a QBP solver.

The strategic berth template problem

A marine container terminal operator may have a situation with excessive calling requests to be served especially when some new service contracts are under consideration. For this situation, we propose a strategic berth template problem (BTPS) that selects the ships among the requesting ones to be served and arrange their berth-windows within a limited planning horizon. The BTPS employs the subgradient optimization procedure, which is an improved version of the procedure that the authors developed for the operational berth allocation problem. A wide variety of numerical experiments indicate the improved subgradient procedure works well for the BTPS.

A misclassification cost risk bound based on hybrid particle swarm optimization heuristic

Linear discriminant analysis models to minimize misclassification cost have recently gained popularity. It is well known that the misclassification cost minimizing linear discriminant analysis problem is an ▪-complete problem that is difficult to solve to optimality for large scale datasets. As a result, heuristic techniques have gained popularity but it is difficult to assess how well these heuristic techniques perform. One way to aid assessment of the performance of heuristic techniques is to establish a lower-bound on the optimal value of misclassification cost. In this paper, we propose and use a hybrid particle swarm optimization (PSO) and Lagrangian relaxation (LR) based heuristic to establish a misclassification cost lower bound (MCLB) for two-group linear classifiers. We use the subgradient optimization procedure to tighten the MCLB. Using simulated and real-world datasets, we test a misclassification cost minimizing linear genetic algorithm classifier and two commercial non-linear classifiers (C5.0 and C&RT) to compare their performances with the MCLB. Our holdout sample tests indicate that the proposed MCLB works well for both linear and non-linear classifiers when class data distributions are normal. Additionally, as misclassification cost asymmetry increases, the proposed MCLB appears to provide better results.

Optimal supplier choice with discounting

This paper investigates a model for pricing the demand for a set of goods when suppliers operate discount schedules based on total business value. We formulate the buyers's decision problem as a mixed binary integer program, which is a generalization of the capacitated facility location problem (CFLP). A branch and bound (BnB) procedure using Lagrangean relaxation and subgradient optimization is developed for solving large-scale problems that can arise when suppliers’ discount schedules contain multiple price breaks. Results of computer trials on specially adapted large benchmark instances of the CFLP confirm that a sub-gradient optimization procedure based on Shor and Zhurbenko's r-algorithm, which employs a space dilation in the direction of the difference between two successive subgradients, can be used efficiently for solving the dual problem at any node of the BnB tree.

Models for a traveling purchaser problem with additional side-constraints

The traveling purchaser problem (TPP) is the problem of determining a tour of a purchaser that needs to buy several items in different shops such that the total amount of travel and purchase costs is minimized. Motivated by an application in machine scheduling, we study a variant of the problem with additional constraints, namely, a limit on the maximum number of markets to be visited, a limit on the number of items bought per market and where only one copy per item needs to be bought. We present an integer linear programming (ILP) model which is adequate for obtaining optimal integer solutions for instances with up to 100 markets. We also present and test several variations of a Lagrangian relaxation combined with a subgradient optimization procedure. The relaxed problem can be solved by dynamic programming and can also be viewed as resulting from applying a state space relaxation technique to a dynamic programming formulation. The Lagrangian based method is combined with a heuristic that attempts to transform relaxed solutions into feasible solutions. Computational results for instances with up to 300 markets show that with the exception of a few cases, the reported differences between best upper bound and lower bound values on the optimal solutions are reasonably small.

Lagrangean relaxation based heuristics for lot sizing with setup times

We consider a lot sizing problem with setup times where the objective is to minimize the total inventory carrying cost only. The demand is dynamic over time and there is a single resource of limited capacity. We show that the approaches implemented in the literature for more general versions of the problem do not perform well in this case. We examine the Lagrangean relaxation (LR) of demand constraints in a strong reformulation of the problem. We then design a primal heuristic to generate upper bounds and combine it with the LR problem within a subgradient optimization procedure. We also develop a simple branch and bound heuristic to solve the problem. Computational results on test problems taken from the literature show that our relaxation procedure produces consistently better solutions than the previously developed heuristics in the literature.

A location-routing problem for the conversion to the “click-and-mortar” retailing: The static case

The static conversion from brick-and-mortar retailing to the hybrid click-and-mortar business model is studied from the perspective of distribution logistics. Retailers run warehouses and brick- and- mortar stores to meet the demand of their walk-in customers. When they decide to operate on the Web as an e- tailer, also click- and- mortar stores are needed which can serve both walk-in and online customers. While the distance between home and the nearest open store is used as a proxy measure for walk-in customers, a quality of service (QoS) guarantee for online customers is timely delivery of their orders. We describe and solve a static location-routing based problem for companies that embrace the clicks-and-bricks strategy in their retail operations. An augmented Lagrangian relaxation method embedded in a subgradient optimization procedure generates lower bounds, whereas a heuristic method finds feasible solutions. The performance of the Lagrangian-based solution method is tested on a number of randomly generated test problems.

Geração de padrões de cortes bidimensionais guilhotinados restritos via programação dinâmica e busca em grafo-e/ou

Um método heurístico para geração de padrões de cortes bidimensionais guilhotinados restritos, baseado no método exato de Christofides e Hadjiconstantinou (1995) foi proposto em Silveira e Morabito (2002). O método combina uma relaxação do espaço de estados de uma formulação de programação dinâmica, um procedimento do tipo otimização do subgradiente e uma heurística de factibilização. Neste trabalho, o método de Silveira e Morabito é modificado com a utilização de uma heurística de factibilização mais efetiva que a anterior, e com uma abordagem de busca em grafo-e/ou para geração de boas soluções iniciais. Resultados computacionais de exemplos da literatura e gerados aleatoriamente indicam que o método refinado tem desempenho bem superior ao anterior, e é competitivo diante de outros métodos propostos na literatura.

Modeling and solving the rooted distance-constrained minimum spanning tree problem

In this paper we discuss models and methods for solving the rooted distance constrained minimum spanning tree problem which is defined as follows: given a graph G = ( V , E ) with node set V = { 0 , 1 , … , n } and edge set E , two integer weights, a cost c e and a delay w e associated with each edge e of E , and a natural (time limit) number H , we wish to find a spanning tree T of the graph with minimum total cost and such that the unique path from a specified root node, node 0, to any other node has total delay not greater than H . This problem generalizes the well known hop-constrained spanning tree problem and arises in the design of centralized networks with quality of service constraints and also in package shipment with service guarantee constraints. We present three theoretically equivalent modeling approaches, a column generation scheme, a Lagrangian relaxation combined with subgradient optimization procedure, both based on a path formulation of the problem, and a shortest path (compact) reformulation of the problem which views the underlying subproblem as defined in a layered extended graph. We present results for complete graph instances with up to 40 nodes. Our results indicate that the layered graph path reformulation model is still quite good when the arc weights are reasonably small. Lagrangian relaxation combined with subgradient optimization procedure appears to work much better than column generation and seems to be a quite reasonable approach to the problem for large weight, and even small weight, instances.

A distributed dispatching method for the brokerage of truckload freights

A distributed dispatching method is suggested, which can be applied to brokerage systems for truckload freights. A mathematical model is formulated for the dispatching problem. It is shown that the problem can be decomposed into a master problem and multiple sub-problems by the Lagrangean relaxation. It is also shown that the subgradient optimization procedure can be applied to the distributed dispatching framework. The effects of various practical considerations on the performance of the suggested dispatching method are tested numerically.

Um método heurístico baseado em relaxação Lagrangiana para o problema de carregamento de paletes do produtor

Neste trabalho desenvolvemos um método heurístico, baseado em relaxação Lagrangiana e surrogate, para resolver o problema de carregamento de paletes do produtor. Tal problema consiste em arranjar o máximo número de caixas por camada sobre o palete, otimizando, assim, o aproveitamento da superfície do palete. Aplicamos um método de redução do problema e uma heurística Lagrangiana no procedimento de otimização do subgradiente. Comparamos as soluções encontradas com soluções produzidas por outros métodos da literatura e com soluções utilizadas em situações práticas. Para isso, utilizamos dados obtidos na literatura e dados reais fornecidos por uma transportadora.

A Heuristic Method for the Set Covering Problem

We present a Lagrangian-based heuristic for the well-known Set Covering Problem (SCP). The algorithm was initially designed for solving very large scale SCP instances, involving up to 5,000 rows and 1,000,000 columns, arising from crew scheduling in the Italian Railway Company, Ferrovie dello Stato SpA. In 1994 Ferrovie dello Stato SpA, jointly with the Italian Operational Research Society, organized a competition, called FASTER, intended to promote the development of algorithms capable of producing good solutions for these instances, since the classical approaches meet with considerable difficulties in tackling them. The main characteristics of the algorithm we propose are (1) a dynamic pricing scheme for the variables, akin to that used for solving large-scale LPs, to be coupled with subgradient optimization and greedy algorithms, and (2) the systematic use of column fixing to obtain improved solutions. Moreover, we propose a number of improvements on the standard way of defining the step-size and the ascent direction within the subgradient optimization procedure, and the scores within the greedy algorithms. Finally, an effective refining procedure is proposed. Our code won the first prize in the FASTER competition, giving the best solution value for all the proposed instances. The algorithm was also tested on the test instances from the literature: in 92 out of the 94 instances in our test bed we found, within short computing time, the optimal (or the best known) solution. Moreover, among the 18 instances for which the optimum is not known, in 6 cases our solution is better than any other solution found by previous techniques.

Using Variable Redefinition for Computing Lower Bounds for Minimum Spanning and Steiner Trees with Hop Constraints

We use variable redefinition (see R. Martin, 1987. Generating Alternative Mixed-Integer Programming Models Using Variable Redefinition, Operations Research 35, 820–831) to strengthen a multicommodity flow (MCF) model for minimum spanning and Steiner trees with hop constraints between a root node and any other node. Hop constraints model quality of service constraints. The Lagrangean dual value associated with one Lagrangean relaxation derived from the MCF formulation dominates the corresponding LP value. However, the lower bounds given after a reasonable number of iterations of the associated subgradient optimization procedure are, for several cases, still far from the theoretical best limit. Martin's variable redefinition technique is used to obtain a generalization of the MCF formulation whose LP bound is equal to the previously mentioned Lagrangean dual bound. We use a set of instances with up to 100 nodes, 50 basic nodes, and 350 edges for comparing an LP approach based on solving the LP relaxation of the new model with the equivalent Lagrangean scheme derived from MCF.

Lagrangian heuristics for the two-echelon, single-source, capacitated facility location problem

Facility location problems form an important class of integer programming problems, with application in the distribution and transportation industries. In this paper we are concerned with a particular type of facility location problem in which there exist two echelons of facilities. Each facility in the second echelon has limited capacity and can be supplied by only one facility (or depot) in the first echelon. Each customer is serviced by only one facility in the second echelon. The number and location of facilities in both echelons together with the allocation of customers to the second-echelon facilities are to be determined simultaneously. We propose a mathematical model for this problem and consider six heuristics based on Lagrangian relaxation for its solution. To solve the dual problem we make use of a subgradient optimization procedure. We present numerical results for a large suite of test problems. These indicate that the lower-bounds obtained from some relaxations have a duality gap which frequently is one third of the one obtained from traditional linear programming relaxation. Furthermore, the overall solution time for the heuristics are less than the time to solve the LP relaxation.

Depth-First Solutions for the Deliveryman Problem on Tree-Like Networks: An Evaluation Using a Permutation Model

The Deliveryman Problem, or DMP (also referred to as the Traveling Repairman Problem TRP), requires a single traveling service unit with a specified home location to serve a set of customers, each of which has a weight or priority. The objective is to find a tour which minimizes a weighted average of the times the customers wait for service. This paper examines the effectiveness of depth-first routes for the DMP on tree and cactus networks. Depth-first routes involve traversing the tree or cactus network in depth-first manner. The best depth-first solution for the DMP can be found very quickly. The evaluation of depth-first routes on general tree and cactus networks is based on a novel formulation for the DMP which employs permutation decision variables and includes as constraints necessary optimality conditions based on network structure. We are unaware of any previous explicit use of these permutation variables in the routing literature. This formulation leads to a Lagrangian relaxation which can be solved as a single-machine minimum weighted-completion time problem with forest precedence constraints. This subproblem is embedded in a subgradient optimization procedure and lower bounds on the costs of optimal DMP solutions are calculated. Computational tests were carried out to determine the quality of the depth-first solutions and the sharpness of the permutation lower bounds. Problem characteristics considered were problem size, variability in arc lengths, variability in customer weights, size of cycles and percentage of nodes in cycles.

A Heuristic for Purchasing of Sheet Metal Decisions Based on Mathematical Programming

An operational research project at the Oskarshamn plant of Saab Scania AB Truckdivision Sweden is reported. The decision problem concerns the product mix of sheet metal purchased. A mathematical model that approximates the real-world problem is derived, the solution of which includes a p-median problem. This optimization problem is solved using Lagrangean relaxation and a subgradient optimization procedure. Numerical results for problems of real-world size and structure are reported and indicate satisfactory efficiency.

A Heuristic for Purchasing of Sheet Metal Decisions Based on Mathematical Programming

An operational research project at the Oskarshamn plant of Saab Scania AB Truckdivision Sweden is reported. The decision problem concerns the product mix of sheet metal purchased. A mathematical model that approximates the real-world problem is derived, the solution of which includes a p-median problem. This optimization problem is solved using Lagrangean relaxation and a subgradient optimization procedure. Numerical results for problems of real-world size and structure are reported and indicate satisfactory efficiency.

Topological design of wide area communication networks

This paper develops a model for designing a backbone network. It assumes the location of the backbone nodes, the traffic between the backbone nodes and the link capacities are given. It determines the links to be included in the design and the routes used by the origin destination pairs. The objective is to obtain the least cost design where the system costs consist of connection costs and queueing costs. The connection costs depend on link capacity and queueing costs are incurred by users due to the limited capacity of links. The Lagrangian relaxation embedded in a subgradient optimization procedure is used to obtain lower bounds on the optimal solution of the problem. A heuristic based on the Lagrangian relaxation is developed to generate feasible solutions.

A Lagrangean heuristic for the pk-median dynamic location problem

We present a Lagrangean heuristic for the p k -median dynamic location problem (DLP- p k ). In this problem clients must be supplied from p k facilities in period k, k = 1, 2, …, K, with the objective of minimizing installation/relocation and transportation costs. DLP- p k may be formulated as a 0–1 integer programming problem. In this paper we develop two Lagrangean relaxations for the problem and a Lagrangean heuristic algorithm based on these two relaxations and on two alternate procedures to find primal feasible solutions. The computational performance of the method is analyses for problems of up to the size 50 clients × 50 potential facility location sites × 7 time periods. We also investigate thhe effect of using in the subgradient optimization procedure step size rules that are an alternative to the classical approach suggested by Held, Wolfe and Crowder.