Incremental Minimum Flow Algorithms

The maximum flow and minimum cost-maximum flow problems are both concerned with determining flows through a network between a source and a destination. Both these problems can be formulated as linear programming problems. When given information about a network (network flow diagram, capacities, costs), computing enables one to arrive at a solution to the problem. Once the solution becomes available, it has to be applied to a real world problem. The use of the following computer software in solving these problems will be discussed: R (several packages and functions), specially written Pascal programs and Excel SOLVER. The minimum cost-maximum flow solutions to the following problems will also be discussed: maximum flow, minimum cost-maximum flow, transportation problem, assignment problem, shortest path problem, caterer problem.

The maximum flow and minimum cost-maximum flow problems are both concerned with determining flows through a network between a source and a destination. Maximum flow applies to any problem where the objective is move as many as possible goods/objects/people between two locations via intermediate locations. Maximum flow-minimum cost applies to flow problems where both capacities and costs are involved. When given information about a network (network flow diagram, capacities, costs), computing enables one to arrive at a solution to the problem. Once the solution becomes available, it has to be applied to a real world problem. The use of the following computer software in solving these problems will be discussed: R (several packages and functions), specially written Pascal programs and Excel SOLVER. The minimum cost-maximum flow solutions to the following problems will also be discussed: maximum flow, minimum cost-maximum flow, transportation problem, assignment problem, shortest path problem, caterer problem.

An introduction to dynamic generative networks: Minimum cost flow

In this paper, we present a new strongly polynomial time algorithm for the minimum cost flow problem, based on a refinement of the Edmonds-Karp scaling technique. Our algorithm solves the uncapacitated minimum cost flow problem as a sequence of O(n log n) shortest path problems on networks with n nodes and m arcs and runs in O(n log n(m + n log n)) time. Using a standard transformation, this approach yields an O(m log n(m + n log n)) algorithm for the capacitated minimum cost flow problem. This algorithm improves the best previous strongly polynomial time algorithm, due to Z. Galil and E. Tardos, by a factor of n2/m. Our algorithm for the capacitated minimum cost flow problem is even more efficient if the number of arcs with finite upper bounds, say m′, is much less than m. In this case, the running time of the algorithm is O((m′ + n) log n(m + n log n)).

The open-pit mining problem is to determine the contours of a mine, based on economic data and engineering feasibility requirements, to yield maximum possible net income. This practical problem needs to be solved for very large data sets. In practice, moreover, it is necessary to test multiple scenarios, taking into account a variety of realizations of geological predictions and forecasts of ore value. The industry is experiencing computational difficulties in solving the problem. Yet, the problem is known to be equivalent to the minimum cut or maximum flow problem. For the maximum flow problem there are a number of very efficient algorithms that have been developed over the last decade. On the other hand, the algorithm that is most commonly used by the mining industry has been devised by Lerchs and Grossmann (LG). This algorithm is used in most commercial software packages for open-pit mining. This paper describes a detailed study of the LG algorithm as compared to the maximum flow “push-relabel” algorithm. We propose here an efficient implementation of the push-relabel algorithm adapted to the features of the open-pit mining problem. We study issues such as the properties of the mine and ore distribution and how they affect the running time of the algorithm. We also study some features that improve the performance of the LG algorithm. The proposed implementations offer significant improvements compared to existing algorithms and make the related sensitivity analysis problem practically solvable.

Maximum flow problem on dynamic generative network flows with time-varying bounds

Decision support for truckload carriers in one-shot combinatorial auctions

Given a directed graph G = (N,A) with arc capacities u and a minimum cost flow problem defined on G, the capacity inverse minimum cost flow problem is to find a new capacity vector u' for the arc set A such that a given feasible flow x' is optimal with respect to the modified capacities. Among all capacity vectors u' satisfying this condition, we would like to find one with minimum ||u' - u|| value. We consider two distance measures for ||u' - u||, rectilinear and Chebyshev distances. By reduction from the feedback arc set problem we show that the capacity inverse minimum cost flow problem is NP-hard in the rectilinear case. On the other hand, it is polynomially solvable by a greedy algorithm for the Chebyshev norm. In the latter case we propose a heuristic for the bicriteria problem, where we minimize among all optimal solutions the number of affected arcs. We also present computational results for this heuristic.

Negative circuits for flows and submodular flows

Decomposition based approaches are recalled from primal and dual point of view. The possibility of building partially disaggregated reduced master problems is investigated. This extends the idea of aggregated-versus-disaggregated formulation to a gradual choice of alternative level of aggregation. Partial aggregation is applied to the linear multicommodity minimum cost flow problem. The possibility of having only partially aggregated bundles opens a wide range of alternatives with different trade-offs between the number of iterations and the required computation for solving it. This trade-off is explored for several sets of instances and the results are compared with the ones obtained by directly solving the natural node-arc formulation. An iterative solution process to the route assignment problem is proposed, based on the well-known Frank Wolfe algorithm. In order to provide a first feasible solution to the Frank Wolfe algorithm, a linear multicommodity min-cost flow problem is solved to optimality by using the decomposition techniques mentioned above. Solutions of this problem are useful for network orientation and design, especially in relation with public transportation systems as the Personal Rapid Transit. A single-commodity robust network design problem is addressed. In this, an undirected graph with edge costs is given together with a discrete set of balance matrices, representing different supply/demand scenarios. The goal is to determine the minimum cost installation of capacities on the edges such that the flow exchange is feasible for every scenario. A set of new instances that are computationally hard for the natural flow formulation are solved by means of a new heuristic algorithm. Finally, an efficient decomposition-based heuristic approach for a large scale stochastic unit commitment problem is presented. The addressed real-world stochastic problem employs at its core a deterministic unit commitment planning model developed by the California Independent System Operator (ISO).

An optimization model which is frequently used to assist decision makers in the areas of resource scheduling, planning, and distribution is the minimum cost multiperiod network flow problem. This model describes network structure decision‐making problems over time. Such problems arise in the areas of production/distribution systems, economic planning, communication systems, material handling systems, traffic systems, railway systems, building evacuation systems, energy systems, as well as in many others. Although existing network solution techniques are efficient, there are still limitations to the size of problems that can be solved. To date, only a few researchers have taken the multiperiod structure into consideration in devising efficient solution methods. Standard network codes are usually used because of their availability and perceived efficiency. In this paper we discuss the development, implementation, and computational testing of a new technique, the forward network simplex method, for solving linear, minimum cost, multiperiod network flow problems. The forward network simplex method is a forward algorithm which exploits the natural decomposition of multiperiod network problems by limiting its pivoting activity. A forward algorithm is an approach to solving dynamic problems by solving successively longer finite subproblems, terminating when a stopping rule can be invoked or a decision horizon found. Such procedures are available for a large number of special structure models. Here we describe the specialization of the forward simplex method of Aronson, Morton, and Thompson to solving multiperiod network network flow problems. Computational results indicate that both the solution time and pivot count are linear in the number of periods. For standard network optimization codes, which do not exploit the multiperiod structure, the pivot count is linear in the number of periods; however, the solution time is quadratic.

Frequency-Domain Modeling and Analysis of Stochastic Flows in Transportation Networks

ABSTRACTThe investigation of flow–ecology relationships constitutes the basis for the development of environmental flow criteria. The need to understand hydrology–ecology linkages in natural systems has increased owing to the prospect of climate change and flow regime management, especially in water‐scarce areas such as Mediterranean basins. Our research quantified the macroinvertebrate community response at family, genus and species level to natural flow regime dynamics in freshwater streams of a Mediterranean semiarid basin (Segura River, SE Spain) and identified the flow components that influence the composition and richness of biotic assemblages. Flow stability and minimum flows were the principal hydrological drivers of macroinvertebrate assemblages, whereas the magnitude of average and maximum flows had a limited effect. Perennial stable streams were characterized by flow sensitive lotic taxa (Ephemeroptera, Plecoptera and Trichoptera) and intermittent streams by predominately lentic taxa (Odonata, Coleoptera, Heteroptera and Diptera). Relatively minor biological changes were recorded for intermediate flow regime classes along a gradient of flow stability. Seasonal variation and minimum flows are key hydrological components that need to be considered for river management and environmental flows in the Segura River basin and other Mediterranean basins. The anthropogenic modification of these parameters, due to both human activities and climate change, would probably lead to significant changes in the structure and composition of communities in perennial stable streams. This would be characterized by a reduction of flow sensitive Ephemeroptera, Plecoptera and Trichoptera taxa and an increase in more resilient Odonata, Coleoptera, Heteroptera and Diptera taxa. Copyright © 2012 John Wiley & Sons, Ltd.

The efficient set of a linear multicriteria programming problem can be represented by a reverse convex constraint of the form g(z)≤0, where g is a concave function. Consequently, the problem of optimizing some real function over the efficient set belongs to an important problem class of global optimization called reverse convex programming. Since the concave function used in the literature is only defined on some set containing the feasible set of the underlying multicriteria programming problem, most global optimization techniques for handling this kind of reverse convex constraint cannot be applied. The main purpose of our article is to present a method for overcoming this disadvantage. We construct a concave function which is finitely defined on the whole space and can be considered as an extension of the existing function. Different forms of the linear multicriteria programming problem are discussed, including the minimum maximal flow problem as an example.

The minimum flow problem in a network is to find a flow with the smallest value so that the flow on each arc is bounded from below by the arc capacity. We present a linear time algorithm for solving the problem in (s,t) planar directed networks.