Lower Bounds from State Space Relaxations for Concave Cost Network Flow Problems

In this paper we obtain Lower Bounds (LBs) to concave cost network flow problems. The LBs are derived from state space relaxations of a dynamic programming formulation, which involve the use of non-injective mapping functions guaranteing a reduction on the cardinality of the state space. The general state space relaxation procedure is extended to address problems involving transitions that go across several stages, as is the case of network flow problems. Applications for these LBs include: estimation of the quality of heuristic solutions; local search methods that use information of the LB solution structure to find initial solutions to restart the search (Fontes et al., 2003, Networks, 41, 221–228); and branch-and-bound (BB) methods having as a bounding procedure a modified version of the LB algorithm developed here, (see Fontes et al., 2005a). These LBs are iteratively improved by penalizing, in a Lagrangian fashion, customers not exactly satisfied or by performing state space modifications. Both the penalties and the state space are updated by using the subgradient method. Additional constraints are developed to improve further the LBs by reducing the searchable space. The computational results provided show that very good bounds can be obtained for concave cost network flow problems, particularly for fixed-charge problems.     

New formulation and relaxation to solve a concave-cost network flow problem

In this paper we study a minimum cost, multicommodity network flow problem in which the total cost is piecewise linear, concave of the total flow along the arcs. Specifically, the problem can be defined as follows. Given a directed network, a set of pairs of communicating nodes and a set of available capacity ranges and their corresponding variable and fixed cost components for each arc, the problem is to select for each arc a range and identify a path for each commodity between its source and destination nodes so as to minimize the total costs. We also extend the problem to the case of piecewise nonlinear, concave cost function. New mathematical programming formulations of the problems are presented. Efficient solution procedures based on Lagrangean relaxations of the problems are developed. Extensive computational results across a variety of networks are reported. These results indicate that the solution procedures are effective for a wide range of traffic loads and different cost structures. They also show that this work represents an improvement over previous work made by other authors. This improvement is the result of the introduction of the new formulations of the problems and their relaxations.     

A particle swarm optimization-based hybrid algorithm for minimum concave cost network flow problems

Traditionally, minimum cost transshipment problems have been simplified as linear cost problems, which are not practical in real applications. Some advanced local search algorithms have been developed to solve concave cost bipartite network problems. These have been found to be more effective than the traditional linear approximation methods and local search methods. Recently, a genetic algorithm and an ant colony system algorithm were employed to develop two global search algorithms for solving concave cost transshipment problems. These two global search algorithms were found to be more effective than the advanced local search algorithms for solving concave cost transshipment problems. Although the particle swarm optimization algorithm has been used to obtain good results in many applications, to the best of our knowledge, it has not yet been applied in minimum concave cost network flow problems. Thus, in this study, we employ an arc-based particle swarm optimization algorithm, coupled with some genetic algorithm and threshold accepting method techniques, as well as concave cost network heuristics, to develop a hybrid global search algorithm for efficiently solving minimum cost network flow problems with concave arc costs. The proposed algorithm is evaluated by solving several randomly generated network flow problems. The results indicate that the proposed algorithm is more effective than several other recently designed methods, such as local search algorithms, genetic algorithms and ant colony system algorithms, for solving minimum cost network flow problems with concave arc costs.     

A dynamic programming approach for solving single-source uncapacitated concave minimum cost network flow problems

In this paper, we describe a dynamic programming approach to solve optimally the single-source uncapacitated minimum cost network flow problem with general concave costs. This class of problems is known to be NP-Hard and there is a scarcity of methods to solve them in their full generality. The algorithms previously developed critically depend on the type of cost functions considered and on the number of nonlinear arc costs. Here, a new dynamic programming approach that does not depend on any of these factors is proposed. Computational experiments were performed using randomly generated problems. The computational results reported for small and medium size problems indicate the effectiveness of the proposed approach.     

An algorithm for the min concave cost flow problem

A method is presented to solve that class of network flow problems, which may be formulated as one source - multiple destination minimum cost flow problems with concave costs. The global optimum is searched using a branch and bound procedure, in which the enumeration scheme is based on a characterization of the optimal solution set, while linear relaxations of the original problem provide lower bounds.     

Linear Approximations in a Dynamic Programming Approach for the Uncapacitated Single-Source Minimum Concave Cost Network Flow Problem in Acyclic Networks

We consider minimum concave cost flow problems in acyclic, uncapacitated networks with a single source. For these problems a dynamic programming scheme is developed. It is shown that the concave cost functions on the arcs can be approximated by linear functions. Thus the considered problem can be solved by a series of linear programs. This approximation method, whose convergence is shown, works particularly well, if the nodes of the network have small degrees. Computational results on several classes of networks are reported.     

Global search algorithms for minimum concave-cost network flow problems

We present algorithms for the single-source uncapacitated version of the minimum concave cost network flow problem. Each algorithm exploits the fact that an extreme feasible solution corresponds to a sub-tree of the original network. A global search heuristic based on random extreme feasible initial solutions and local search is developed. The algorithm is used to evaluate the complexity of the randomly generated test problems. An exact global search algorithm is developed, based on enumerative search of rooted subtrees. This exact technique is extended to bound the search based on cost properties and linear underestimation. The technique is accelerated by exploiting the network structure.     

A parametric algorithm for convex cost network flow and related problems

The convex cost network flow problem is to determine the minimum cost flow in a network when cost of flow over each arc is given by a piecewise linear convex function. In this paper, we develop a parametric algorithm for the convex cost network flow problem. We define the concept of optimum basis structure for the convex cost network flow problem. The optimum basis structure is then used to parametrize v, the flow to be transsshipped from source to sink. The resulting algorithm successively augments the flow on the shortest paths from source to sink which are implicitly enumerated by the algorithm. The algorithm is shown to be polynomially bounded. Computational results are presented to demonstrate the efficiency of the algorithm in solving large size problems. We also show how this algorithm can be used to (i) obtain the project cost curve of a CPM network with convex time-cost tradeoff functions; (ii) determine maximum flow in a network with concave gain functions; (iii) determine optimum capacity expansion of a network having convex arc capacity expansion costs.     

A facility location model with safety stock costs: analysis of the cost of single-sourcing requirements

We consider a supply chain setting where multiple uncapacitated facilities serve a set of customers with a single product. The majority of literature on such problems requires assigning all of any given customer??s demand to a single facility. While this single-sourcing strategy is optimal under linear (or concave) cost structures, it will often be suboptimal under the nonlinear costs that arise in the presence of safety stock costs. Our primary goal is to characterize the incremental costs that result from a single-sourcing strategy. We propose a general model that uses a cardinality constraint on the number of supply facilities that may serve a customer. The result is a complex mixed-integer nonlinear programming problem. We provide a generalized Benders decomposition algorithm for the case in which a customer??s demand may be split among an arbitrary number of supply facilities. The Benders subproblem takes the form of an uncapacitated, nonlinear transportation problem, a relevant and interesting problem in its own right. We provide analysis and insight on this subproblem, which allows us to devise a hybrid algorithm based on an outer approximation of this subproblem to accelerate the generalized Benders decomposition algorithm. We also provide computational results for the general model that permit characterizing the costs that arise from a single-sourcing strategy.     

A Lagrangian Relaxation Heuristic for Capacitated Facility Location with Single-Source Constraints

Facility location models are applicable to problems in many diverse areas, such as distribution systems and communication networks. In capacitated facility location problems, a number of facilities with given capacities must be chosen from among a set of possible facility locations and then customers assigned to them. We describe a Lagrangian relaxation heuristic algorithm for capacitated problems in which each customer is served by a single facility. By relaxing the capacity constraints, the uncapacitated facility location problem is obtained as a subproblem and solved by the well-known dual ascent algorithm. The Lagrangian relaxations are complemented by an add heuristic, which is used to obtain an initial feasible solution. Further, a final adjustment heuristic is used to attempt to improve the best solution generated by the relaxations. Computational results are reported on examples generated from the Kuehn and Hamburger test problems.     

Using shortest paths in some transshipment problems with concave costs

This paper deals with the problem of solving an uncapacitated transshipment problem with either one source and several sinks or one sink and several sources. The cost function of the problem is concave in the amount shipped on each arc and thus local optima are possible. A characterization of adjacent extreme flows in terms of corresponding arborescences is given for this type of networks.This characterization together with shortest path methods is then used to attack the problems of finding local optima and of ranking extreme points.A real-world problem and computational evidence for the usefulness of the method are produced.     

Concave cost minimization on networks

The paper deals with the problem of finding a minimum cost multicommodity flow on an uncapacitated network with concave link costs. Problems of this type are the optimal design of a network in the presence of scale economies and the telpack problem.Two different definitions of local optimality are given and compared both from the point of view of the computational complexity and from the point of view of the goodness of the solution they may provide.A vertex following algorithm to find a local optimum is proposed. The computational complexity of each iteration of the algorithm is O(n³), where n is the number of nodes of the network, and is independent of the differentiability of the objective function.Experimental results obtained from a set of test problems of size ranging from 11 nodes and 23 arcs to 48 nodes and 174 arcs, with number of commodities up to 5, are given.     

A Branch and Bound algorithm for the minimax regret spanning arborescence

The paper considers the problem of finding a spanning arborescence on a directed network whose arc costs are partially known. It is assumed that each arc cost can take on values from a known interval defining a possible economic scenario. In this context, the problem of finding the spanning arborescence which better approaches to that of minimum overall cost under each possible scenario is studied. The minimax regret criterion is proposed in order to obtain such a robust solution of the problem. As it is shown, the bounds on the optimal value of the minimax regret optimization problem obtained in a previous paper, can be used here in a Branch and Bound algorithm in order to give an optimal solution. The computational behavior of the algorithm is tested through numerical experiments. This research has been supported by the Spanish Ministry of Education and Science and FEDER Grant No. MTM2006-04393 and by the European Alfa Project, “Engineering System for Preparing and Making Decisions Under Multiple Criteria”, II-0321-FA.     

A Lagrangian Based Branch-and-Bound Algorithm for Production-transportation Problems

We propose a branch-and-bound algorithm of Falk–Soland's type for solving the minimum cost production-transportation problem with concave production costs. To accelerate the convergence of the algorithm, we reinforce the bounding operation using a Lagrangian relaxation, which is a concave minimization but yields a tighter bound than the usual linear programming relaxation in O(mn log n) additional time. Computational results indicate that the algorithm can solve fairly large scale problems.     

无容量限制的最小费用流问题

本文研究了无容量限制的带固定费用和可变费用的单物资和二物资的最小费用流问题,并分别给出了多项式算法.最后应用该算法,计算了一个二物资的最小费用流问题的实例.     

On Solving Quickest Time Problems in Time-Dependent, Dynamic Networks

In this paper, a pseudopolynomial time algorithm is presented for solving the integral time-dependent quickest flow problem (TDQFP) and its multiple source and sink counterparts: the time-dependent evacuation and quickest transshipment problems. A more widely known, though less general version, is the quickest flow problem (QFP). The QFP has historically been defined on a dynamic network, where time is divided into discrete units, flow moves through the network over time, travel times determine how long each unit of flow spends traversing an arc, and capacities restrict the rate of flow on an arc. The goal of the QFP is to determine the paths along which to send a given supply from a single source to a single sink such that the last unit of flow arrives at the sink in the minimum time. The main contribution of this paper is the time-dependent quickest flow (TDQFP) algorithm which solves the TDQFP, i.e. it solves the integral QFP, as defined above, on a time-dependent dynamic network, where the arc travel times, arc and node capacities, and supply at the source vary with time. Furthermore, this algorithm solves the time-dependent minimum time dynamic flow problem, whose objective is to determine the paths that lead to the minimum total time spent completing all shipments from source to sink. An optimal solution to the latter problem is guaranteed to be optimal for the TDQFP. By adding a small number of nodes and arcs to the existing network, we show how the algorithm can be used to solve both the time-dependent evacuation and the time-dependent quickest transshipment problems. This revised version was published online in August 2006 with corrections to the Cover Date.     

带有异常点的平方度量设施选址问题

传统的设施选址问题一般假设所有顾客都被服务,考虑到异常点的存在不仅会增加总费用(设施的开设费用与连接费用之和),也会影响到对其他顾客的服务质量.研究异常点在最终方案中允许不被服务的情况,称之为带有异常点的平方度量设施选址问题.该问题是无容量设施选址问题的推广.问题可描述如下:给定设施集合、顾客集,以及设施开设费用和顾客...     

Valid inequalities for separable concave constraints with indicator variables

We study valid inequalities for optimization models that contain both binary indicator variables and separable concave constraints. These models reduce to a mixed-integer linear program (MILP) when the concave constraints are ignored, or to a nonconvex global optimization problem when the binary restrictions are ignored. In algorithms designed to solve these problems to global optimality, cutting planes to strengthen the relaxation are traditionally obtained using valid inequalities for the MILP only. We propose a technique to obtain valid inequalities that are based on both the MILP constraints and the concave constraints. We begin by characterizing the convex hull of a four-dimensional set consisting of a single binary indicator variable, a single concave constraint, and two linear inequalities. Using this analysis, we demonstrate how valid inequalities for the single node flow set and for the lot-sizing polyhedron can be “tilted” to give valid inequalities that also account for separable concave functions of the arc flows. We present computational results demonstrating the utility of the new inequalities for nonlinear transportation problems and for lot-sizing problems with concave costs. To our knowledge, this is one of the first works that simultaneously convexifies both nonconvex functions and binary variables to strengthen the relaxations of practical mixed-integer nonlinear programs.     

Lower bounds for the two-stage uncapacitated facility location problem

In the two-stage uncapacitated facility location problem, a set of customers is served from a set of depots which receives the product from a set of plants. If a plant or depot serves a product, a fixed cost must be paid, and there are different transportation costs between plants and depots, and depots and customers. The objective is to locate plants and depots, given both sets of potential locations, such that each customer is served and the total cost is as minimal as possible. In this paper, we present a mixed integer formulation based on twice-indexed transportation variables, and perform an analysis of several Lagrangian relaxations which are obtained from it, trying to determine good lower bounds on its optimal value. Computational results are also presented which support the theoretical potential of one of the relaxations.     

A capacity constrained single-facility dynamic lot-size model

A computationally efficient algorithm for a multi-period single commodity production planning problem with capacity constraints is developed. The model differs from earlier well-known studies involving concave cost functions in the introduction of production capacity constraints which need not be equal in every period. The objective is to find an optimal production schedule that minimizes the total production and inventory costs. Backlogging is not allowed. The structure of the optimal solution is characterized and then used in an efficient algorithm.