Heuristic solutions for general concave minimum cost network flow problems (original) (raw)
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European Journal of Operational Research, 2006
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.
Algorithms for the single-source uncapacitated minimum concave-cost network flow problem
Journal of Global Optimization, 1991
We investigate algorithms, applications, and complexity issues for the single-source uncapacitated (SSU) version of the minimum concave-cost network flow problem (MCNFP). We present applications arising from production planning, and prove complexity results for both global and local search. We formally state the local search algorithm of Gallo and Sodini [5], and present alternative local search algorithms. Computational results are provided to compare the various local search algorithms proposed and the effects of initial solution techniques.
A Lagrangean heuristic for the capacitated concave minimum cost network flow problem
European Journal of Operational Research, 1994
We propose a heuristic solution technique for the capacitated concave minimum cost network flow problem based on a Lagrangean dualization of the problem. Despite its dual character the algorithm guarantees the generation of primal feasible solutions which are local optima and therefore candidates of being the global optimum. The Lagrangean dual is solved by a subgradient search procedure and provides a lower bound to the optimal value. The lower bound is, in general, stronger than the one obtained by a linear approximation of the original problem. It can be used as a judgement of the quality of the solution or in a branch and bound procedure. Computational results from randomly generated problems are presented.
Concave minimum cost network flow problems solved with a colony of ants
Journal of Heuristics, 2012
In this work we address the Single-Source Uncapacitated Minimum Cost Network Flow Problem with concave cost functions. This problem is NP-hard, therefore we propose a hybrid heuristic to solve it. Our goal is not only to apply an Ant Colony Optimization (ACO) algorithm to such a problem, but also to provide an insight on the behaviour of the parameters in the performance of the algorithm. The performance of the ACO algorithm is improved with the hybridization of a local search procedure. The core ACO procedure is used to mainly deal with the exploration of the search space, while the Local Search is incorporated to further cope with the exploitation of the best solutions found. The method we have developed has proven to be very efficient while solving both small and large size problem instances. The problems we have used to test the algorithm were previously solved by other authors using other population based heuristics. Our algorithm was able to improve upon some of their results in terms of solution quality, proving that the HACO algorithm is a very good alternative approach to solve these problems. In addition, our algorithm is substantially faster at achieving these improved solutions. Furthermore, the magnitude of the reduction of the computational requirements grows with problem size.
Solving Concave Network Flow Problems
2012
The Minimum Cost Network Flow Problem (MCNFP) includes a wide range of combinatorial optimization problems. Many applications exist for MCNFPs for instance supply chains, logistics, production planning, communications and transportations. Concave costs are, in many applications, more realistic than linear ones because of the association of prices with economies of scale. When concave costs are introduced in MCNFPs, then the difficulty to solve them increases and they become NP-Hard. Solution methods developed for these problems comprise both exact and approximate algorithms, the latter ones usually of a heuristic type. What we propose to do in this work is to present an overview of the past and most recent literature published on the subject.
Minimum concave-cost network flow problems: Applications, complexity, and algorithms
Annals of Operations Research, 1990
We discuss a wide range of results for minimum concave-cost network flow problems, including related applications, complexity issues, and solution techniques. Applications from production and inventory planning, and transportation and communication network design are discussed. New complexity results are proved which show that this problem is NP-hard for cases with cost functions other than fixed charge. An overview of solution techniques for this problem is presented, with some new results given regarding the implementation of a particular branch-and-bound approach.
On Minimum Concave Cost Network Flow Problems
2008
Minimum concave Cost Network Flow Problems (MCNFPs) arise naturally in many practical applications such as communication, transporta- tion, distribution, and manufacturing, due to economic considerations. In addition, it has been shown that every MCNFP with general nonlinear cost functions can be transformed into a concave MCNFP on an expanded network. It must also be noted, that multiple source and capacitated networks can be transformed into single source and uncapacitated networks. The main feature defining the complexity of MCNFPs is the type of cost function for each arc. Concave MCNFPs are known to be NP-hard even for the simplest version (i.e. fixed-charge single source and uncapacitated). The review presented in this work describes several approaches to the design of Single Source Uncapacitated (SSU) flow networks involving concave costs.
A novel experimental analysis of the minimum cost flow problem
International Journal of Engineering Transaction A: …, 2009
In the GA approach the parameters that influence its performance include population size, crossover rate and mutation rate. Genetic algorithms are suitable for traversing large search spaces since they can do this relatively fast and because the mutation operator diverts the method away from local optima, which will tend to become more common as the search space increases in size. GA's are based in concept on natural genetic and evolutionary mechanisms working on populations of solutions in contrast to other search techniques that work on a single solution. An important aspect of GA's is that although they do not require any prior knowledge or any space limitations such as smoothness, convexity or unimodality of the function to be optimized, they exhibit very good performance in most applications. The minimum cost flow problem is formulated as genetic algorithm and simulated annealing. This paper shows genetic algorithms and simulated annealing are much easier to implement for solving transportation problems compared with constructing mathematical programming formulations. Finally, a new empirical study for the effect of parameters on the rate of convergence of the GA and SA are demonstrated.
Minimum cost network flows: Problems, algorithms, and software
Yugoslav Journal of Operations Research, 2013
We present a wide range of problems concerning minimum cost network flows, and give an overview of the classic linear single-commodity Minimum Cost Network Flow Problem (MCNFP) and some other closely related problems, either tractable or intractable. We also discuss state-of-the-art algorithmic approaches and recent advances in the solution methods for the MCNFP. Finally, optimization software packages for the MCNFP are presented.
Networks, 1994
A new algorithm for generating structured, minimum cost network flow problems (transshipment, transportation, assignment, and shortest path) with known optimal solutions is described. The procedure is based on developing problems around an optimal basis so that the characteristics of solutions can be controlled. Computational tests show that the problems generated are as difficult to solve as are those produced by the commonly used generator NETGEN, while allowing the user a much greater degree of control over the resulting problems.