Algorithms for the single-source uncapacitated minimum concave-cost network flow problem (original) (raw)
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We address the single-source uncapacitated minimum cost network flow problem with general concave cost functions. Exact methods to solve this class of problems in their full generality are only able to address small to medium size instances, since this class of problems is known to be NP-Hard. Therefore, approximate methods are more suitable. In this work, we present a hybrid approach combining a genetic algorithm with a local search. Randomly generated test problems have been used to test the computational performance of the algorithm. The results obtained for these test problems are compared to optimal solutions obtained by a dynamic programming method for the smaller problem instances and to upper bounds obtained by a local search method for the larger problem instances. From the results reported it can be shown that the hybrid methodology improves upon previous approaches in terms of efficiency and also on the pure genetic algorithm, i.e., without using the local search procedure.
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.
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Annals of Operations Research, 1990
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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.
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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.
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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.
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Yugoslav Journal of Operations Research, 2013
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A Primal Algorithm for Finding Minimum-Cost Flows in Capacitated Networks With Applications
Bell System Technical Journal, 1982
Algorithms for finding a minimum-cost, single-commodity flow in a capacitated network are based on variants of the simplex method of linear programming. We describe an implementation of a primal algorithm which is fast and can solve large problems. The major ideas incorporated are (i) the sparsity ofthe network is used to reduce the time and computer storage space requirements; (ii) basic solutions are stored compactly as spanning trees of the network; (iii) a candidate stack is used to allow flexible strategies in choosing an arc to enter the basis tree; (iv) the predecessor and thread data structures are used to efficiently traverse the tree and to update the solution at each iteration; (v) rules are implemented to avoid cycling or stalling caused by degeneracy; and (vi)piecewise-linear, convex arc costs are handled implicitly. The Primal Network Flow Convex (PNFC) code implements this algorithm and three examples, from communication networks, that can be solved with PNFC are discussed: (i) solving the area transfer problem; (ii) scheduling the collection of traffic data records; and (iii) planning the placement ofpair-gain systems.
Approximate resolution of a multi-commodity network flow problem with non-convex routing costs
This paper presents an approximate procedure to solve a multi-commodity capacitated network flow problem with concave routing costs, considering also outsourcing, overload and underutilization facility costs. The model is derived from a real NP production, inventory management and transportation problem concerning to the Andalusian Regional Network of Clinical Laboratories (RNCL). In order to speed up the solving procedure, the inherent non-convex modified all-unit discount problem is approximately solved by a sequential two-phase modelling approach. Numerical results obtained from two different instances are reported.
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.