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

https://doi.org/10.1016/j.ejor.2005.03.024Get rights and content

Abstract

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.

Introduction

One of the motivations for addressing the concave Minimum Cost Network Flow Problem (MCNFP) is its frequent applicability to model practical problems. This can be seen from the wide and diverse application areas in which it has been utilized, see e.g. [16] and the references therein. Other problems without a natural network formulation have been solved efficiently after being reformulated as network optimization problems [8], [14], [20].

The main feature defining the complexity of MCNFPs is the type of cost function for each arc. A discussion of other parameters affecting problem complexity is given in [10]. Concave cost functions in network problems arise in practice due to economic considerations. Economies of scale often exist, leading to a decrease in marginal costs. Fixed-charge costs may arise, for example, in the case of toll charges on a highway, landing fees at an airport, considering a new customer, or a new supply route.

The complexity of concave MCNFPs arises from minimizing a concave function over a convex feasible region, defined by the network constraints, which implies that a local optimum is not necessarily a global optimum. Furthermore, as is also the case for many other types of global optimization problems, there is no simple criterion for deciding whether a local minimum is also a global minimum. Although concave MCNFPs are known to be NP-hard [18] (even for the simplest version, i.e. fixed-charge single source uncapacitated MCNFPs), they do exhibit some special mathematical properties that make them more tractable than general nonlinear MCNFPs [16]. In addition, it has been shown [29] 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 [37].

Most of the work developed on concave MCNFPs considers problems with fixed-charge cost functions, i.e. functions having a fixed cost component and a linear routing cost component, which are a particular case of the more general concave cost functions. A recent work is given in [32] where the authors also discuss specific variants of this problem. Other works [19], [22] consider problems having a routing concave cost component but no fixed cost component.

In this paper, a new Dynamic Programming (DP) formulation for the Single-Source Uncapacitated (SSU) MCNFP with general concave costs is given and an algorithm to solve it exactly is developed. The DP formulation has two main characteristics: (i) no assumption other than separability and additivity is needed, and (ii) it is independent of both the type of cost functions considered and the number of nonlinear arc costs. Hence, different types of cost functions are considered in the computational experiments including linear costs, fixed-charge costs, and cost functions having both a routing concave component and a fixed component. As far as the authors are aware of, the latter type has only been addressed in [4], [11], [12], [13]. The computational results reported support this behaviour of the algorithm and show its effectiveness in solving problems of small to medium size. Furthermore, the proposed approach can also be used to find optimal tree-shaped solutions for other classes of single source MCNFPs, for example, problems with nonlinear cost that are neither convex nor concave and/or with capacity constraints [10].

Section snippets

Overview of existing methods

Existing algorithms for SSU concave MCNFPs can be characterized in terms of the type of problems they solve and whether the solution provided is exact (a global optimum) or approximate (a bound). Although this class of problems is known to be NP-hard, there are special cases arising from imposing additional structure for which polynomial-time or even strongly polynomial-time algorithms have been developed. For a recent survey on concave MCNFPs with a fixed number of nonlinear arcs see [35].

The

Dynamic programming formulation

Dynamic programming (DP) is an effective method to solve combinatorial problems of a sequential nature. It provides a framework for decomposing an optimization problem into a nested family of subproblems. This nested structure suggests a recursive approach for solving the original problem using the solution of the subproblems. The recursion expresses an intuitive principle of optimality for sequential decision processes; that is, once we have reached a particular state, a necessary condition

Computational implementation of the DP algorithm

A pure forward DP algorithm is easily derived from the DP recursion of the SSU concave MCNFP, Eqs. (1), (2), (3). It generates all the states of a particular stage one by one. Such implementation may result in considerable waste of computational effort either when complete enumeration of the state space is not required, or when some states are not feasible, due to additional constraints. In the latter case, the infeasibility of a state is only discovered after it has been generated.

Here, we

Test problems

A set of test problems were randomly generated to computationally evaluate the DP algorithm. The following four types of cost function were used: polynomials of degree 0 (linear problems), degree 1 (fixed-charge problems), and degree 2 (problems having a concave routing component) both with and without a fixed cost component. (We decided to choose polynomial functions, since any smooth function can be easily approximated by a Taylor series.) The cost functions gij are nondecreasing and aij, bij

Computational results

The DP algorithm presented in this paper was implemented in Fortran and computationally evaluated on a 200 MHz Pentium PC with 64 MB of RAM.

Table 2 summarizes the computational performance of the algorithm for all 600 randomly generated test problems. The figures shown in this table were obtained as averages over 30 problem instances of a given problem size and cost function type. Two measures of performance were computed for each problem:

  • Time. Computational time, in minutes, required to find an

Conclusions

In this paper, we consider the Single Source Uncapacitated (SSU) Minimum Cost Network Flow Problem (MCNFP) with general concave costs, in which a subset of arcs must be selected so that flow can be routed to vertices with known demands from a single source at minimum cost.

We developed a new Dynamic programming (DP) methodology for solving optimally the SSU concave MCNFP. The structure of an optimal solution to this problem is known to be a tree rooted at the single source vertex. Therefore, the

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