Cost Multicommodity Flow Problem Research Articles

A novel method for solving the multi-commodity flow problem on evolving networks

The minimum cost multi-commodity flow (MMCF) problem on evolving networks is the latest challenge to solving the min-cost network flow problem of multi-commodity flowing from multi-source and multi-sink on an evolving network. Previous researches primarily focus on the solution of MMCF problem within static networks. However, researchers ignored the MMCF problem within networks where the structure dynamically changes over time. Evolving networks have a dynamically changing and unpredictable network structure. The existing methods must recalculate when the network structure changes, leading to high computational complexity. This manuscript proposes a dynamic graph computing model for adaptive dynamic path selection and optimization. First, the model selects the path with the least cost for each commodity, assuming the route can transport a specified number of such commodities to find the optimal path faster. Second, we design path selection and flow update strategies for each node in the graph to adapt to changes in the graph structure. At the same time, we can prevent multiple commodity conflicts from exceeding the marginal capacity limit. Finally, the optimal path of each commodity is dynamically pruned based on the original optical path to obtain the tree structure and determine the absolute path to meet the specified transportation quantity of each commodity. Performance analysis results show that our method can efficiently cope with the dynamic changes of the network structure, which provides ideas for solving the multi-commodity network flow problem in dynamic networks.

Minimum cost multicommodity network flow problem in time-varying networks: by decomposition principle

Time-varying network flows (also called dynamic network flows) generalize standard network flows by introducing an element of time. In this paper, we consider the dynamic version of the minimum cost multicommodity flow problem with storage at intermediate nodes. This problem is known to be NP-hard. By using of the flow decomposition theorem in network flows, we propose an efficient model based on dynamic path flows for this problem. For some special structures of the path-flow formulation, we provide an algorithm based on decomposition principle, for solving the proposed model. In the end, the efficiency of the proposed approach is evaluated through a number of experimental tests.

A simulation-optimization approach for the stochastic discrete cost multicommodity flow problem

ABSTRACTThis article addresses a variant of the Discrete Cost Multicommodity Flow (DCMF) problem with random demands, where a penalty is incurred for each unrouted demand. The problem requires finding a network topology that minimizes the sum of the fixed installation facility costs and the expected penalties of unmet multicommodity demands. A two-stage stochastic programming with recourse model is proposed. A simulation-optimization approach is developed to solve this challenging problem approximately. To be precise, the first-stage problem requires solving a specific multi-facility network design problem using an exact enhanced cut-generation procedure coupled with a column generation algorithm. The second-stage problem aims at computing the expected penalty using a Monte Carlo simulation procedure together with a hedging strategy. To assess the empirical performance of the proposed approach, a Sample Average Approximation (SAA) procedure is developed to derive valid lower bounds. Results of extensive computational experiments attest to the efficacy of the proposed approach.

Solving Minimum Cost Multi-Commodity Network Flow Problem Using Lexicographic Goal Programming Approach

In an urban transport system, a dysfunction often occurred as demand for transportation infrastructure exceeds available supply. The result includes traffic congestion, higher travel time and cost, higher emission of harmful gases and general reduction in quality of life. In this research, an attempt was made to minimize travel time on three urban road segments using Lexicographic Goal Programming. The positive and negative deviations from the goals were minimized. A minimum cost multi-commodity network flow problem with multiple objectives was successfully modelled using LINDO 6.1. The modelling technique provided a solution that effectively minimized travel time by 50%.

Fresh Water Supply from Different Sources in the Shipping

This research is based on savings coming from combination of water production and better planning of water bunkering in ports. Such operational research (OR) approach is based on better utilization of ship tank capacity in relation to water production on board. The efficient heuristic algorithm for optimal decisions for N different water sources (e.g. production by evaporation, production by reverse osmosis process and bunkering in ports + bottled water) for the ship with limited water tanks capacity is being developed. Mathematical model is developed on the basis of capacity expansion problem, same as the Minimum Cost Multi-Commodity Flow Problem (MCMCF).

Uncertain minimum cost multicommodity flow problem

This paper investigates the minimum cost multicommodity flow problem with uncertain costs and uncertain capacities. Uncertainty theory is used to deal with indeterminacy factors in uncertain network. An ($$\alpha $$ź, $$\beta $$β)-minimum cost multicommodity flow model is formulated. Some properties of the model are analyzed. An equivalence relationship between the ($$\alpha $$ź, $$\beta $$β)-minimum cost multicommodity flow and the minimum cost multicommodity flow is proved. Then, a global optimal algorithm is developed for finding the ($$\alpha $$ź, $$\beta $$β)-minimum cost multicommodity flow and the minimum cost multicommodity flow, and obtaining the uncertainty distribution of total shipping cost. Finally, two examples are presented to illustrate the algorithm.

Lagrangean Heuristic for a Multi-Plant Lot-Sizing Problem with Transfer and Storage Capacities

The paper addresses a multi-item, multi-plant lot-sizing problem with transfer costs and capacity constraints. The problem is reformulated according to a multi-commodity flow formalism, and decomposed, through Lagrangean relaxation, into a master facility location problem and a slave minimal cost multi-commodity flow problem. The decomposition framework gives rise in a natural way to designing a Lagrangean based heuristic. Numerical experiments showing the efficiency of the proposed approach are reported.

Bilevel Programming and Linear Approximation of Solving Minimum Concave Cost Multicommodity Flow Problem

Bilevel Programming and Linear Approximation of Solving Minimum Concave Cost Multicommodity Flow Problem

Bilevel Programming and Linear Approximation of Solving Minimum Concave Cost Multicommodity Flow Problem

Bilevel Programming and Linear Approximation of Solving Minimum Concave Cost Multicommodity Flow Problem

Traffic flow consideration in design of freight distribution system

This study is part of a series of research projects on a distribution system we developed to deal with cases in a state-owned company. It concerns the design of the Public Service Obligation State-owned Company (PSO-SOC) distribution system. The intrinsic features of PSO-SOC are distributing strategic commodities and having subsidies within the cost function. Hence their distribution flow has to be secured under consideration of moving the commodities within road networks that have traffic flow dependency. This paper focuses on the solution of the proposed model which represents traffic flow dependency within a freight distribution network.The mathematical formulation takes the form of a Minimum Cost Multicommodity Flow (MCMF) problem. Traffic flow dependency is incorporated into the model by introducing a coefficient of speed, which is derived from the traffic assignment of ordinary traffic associated with the transportation of the type of freight under consideration The solution of the proposed model is formulated by Network Representation (NR), in which all of the components of the mathematical model are represented in the form of dummy links and nodes added to the original (physical) network. It is to be noted then, that the traffic flow on each road or link is represented by a link performance function (LPF), depicting traffic flow dependent travel time and consequent cost. The MCMF problem of NR is further solved by a Primal–Dual Algorithm.Finally, an illustrative example is exercised to show how the proposed step-wise solution works.

Some concepts of the fuzzy multicommodity flow problem and their application in fuzzy network design

We treat with the Minimal Cost Multicommodity Flow Problem (MCMFP) in the setting of fuzzy sets, by forming a coherent algorithmic framework referred to as a fuzzy MCMFP. Given the character of granular information captured by fuzzy sets, the objective is to find multiple flows satisfying the demands of commodities, by using available supplies consuming the least possible cost. With this regard, the supply and demand of nodes may be presented linguistically; the travel cost and capacity of links can be defined under uncertainty as well. To solve this problem, two efficient algorithms are motivated. In the first, we utilize fuzzy shortest paths and K -shortest paths to generate preferred and absorbing paths, and then we find the flow on them by solving a classic MCMFP. The second algorithm exhibits with fuzzy supply-demand, and employs a total order on trapezoidal fuzzy numbers to reduce the fuzzy MCMFP into four classic MCMFPs. Some examples are solved to demonstrate the performance of the presented methods. Among the various applications of this scheme in providing a suitable interface between the model and physical world, we focus on network design under fuzziness. The granular nature of the description of the future travel demand contributes to the generality of the planning model, and determines a certain perspective from which we will looking at the network.

An extended model and a column generation algorithm for the planar multicommodity flow problem

AbstractWe present a new model and a column generation algorithm for the linear minimum cost multicommodity flow problem. The new model has two differences when compared to the classic formulation based on flows on paths: it has an additional set of variables and an additional set of inequalities. The extra variables are associated with flows on circuits where some arcs are reversed. The extra inequalities assure that an optimal solution to the extended model can be converted to an optimal solution to the path formulation. In order to obtain an optimal solution to the extended model, we use column generation. The extra variables are explicitly considered in the restricted master problem, from the beginning of the column generation process. The subproblem remains a set of shortest path problems. Our present work is related to the use of dual–optimal inequalities in the sense that we extend a model with additional primal variables/dual inequalities. However, our extended model also has additional primal inequalities/dual variables. We present computational results of our method in planar instances and compare them with standard column generation, a bundle method, and a general‐purpose solver. For the tested instances, there is an effective improvement in computational time of the column generation method when the extended model is used. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 50(1), 3–16 2007

Descent direction algorithm with multicommodity flow problem for signal optimization and traffic assignment jointly

Network managers wish to optimize control parameters such as signal setting which are very related to the traffic assignment models. On the other hand traffic assignment patterns as an important instrument for predicting the amount of flow on network links are dependent to control decisions. According to the significance of this concept, some important papers about this mutually relation are reviewed in this paper. Then we implement a nonlinear algorithm on a minimal cost multicommodity flow (MCMF) problem to optimize some control policies subject to optimal flows. Although we take signal times into account, but this approach has a far more reaching application in urban network control and design. We employ a hybrid intelligent algorithm integrating decent direction algorithm and an interior point algorithm in a mutually consistent scheme for obtaining optimal signals and equilibrium flows. An example is given to illustrate the effectiveness of our scheme.

Solving Multicommodity Flow Problems by an Approximation Scheme

We study an approximation scheme to solve minimum cost multicommodity flow problems to a relative accuracy of $\varepsilon\in (0,1]$. The proposed scheme, which we shall call Algorithm ${\cal A}$, is a bisection-based procedure, so it maintains an interval that contains the optimal value. At each iteration, Algorithm ${\cal A}$ defines a block angular sharing problem that is solved to a relative accuracy of $O(\varepsilon)$. The computed solution defines the current approximation to an optimal solution and it is used by Algorithm ${\cal A}$ to throw away half of the current interval. It is shown that when Algorithm ${\cal A}$ no longer shrinks the current interval, the current solution solves the minimum cost multicommodity flow problem to the given accuracy $\varepsilon$. To compute approximate solutions to the sharing problems that Algorithm ${\cal A}$ defines, we propose using Algorithm ${\cal L}$ from [J. Villavicencio and M. D. Grigoriadis, Approximate Lagrangian decomposition with a modified Karmarkar logarithmic potential, in Network Optimization, Lecture Notes in Econom. and Math. Systems 450, Springer, Berlin, 1997, pp. 471--485], but we replace its fixed small step size by one computed using inexact line searches. This modification allows large steps in Algorithm ${\cal L}$. We show that the coordination complexities of Algorithm ${\cal L}$ and Algorithm ${\cal L}$ with inexact line searches are the same if these algorithms are applied to the block angular sharing problems defined by Algorithm ${\cal A}$. This result is also true for block angular sharing problems with convex blocks and with coupling constraints given by nonnegative linear functions.

Dynamic space allocation for temporary storage

Methods for assigning a temporary storage inventory, which arrives dynamically at storage areas, to storage space are discussed. It was assumed that no storage location was dedicated to a single type of inventory and that inventory positioned at a storage area could be relocated to another storage area. An attempt was made to minimize the total transportation, storage and handling costs of inventory for a given schedule of arrivals and departures. It is shown that the space allocation problem can be formulated as a minimum cost multi-commodity flow problem. A subgradient optimization algorithm was applied to solve the problem. A numerical experiment was conducted to evaluate the performance of the algorithm.

Discrete Cost Multicommodity Network Optimization Problems and Exact Solution Methods

We first introduce a generic model for discrete cost multicommodity network optimization, together with several variants relevant to telecommunication networks such as: the case where discrete node cost functions (accounting for switching equipment) have to be included in the objective; the case where survivability constraints with respect to single-link and/or single-node failure have to be taken into account. An overview of existing exact solution methods is presented, both for special cases (such as the so-called single-facility and two-facility network loading problems) and for the general case where arbitrary step-increasing link cost-functions are considered. The basic discrete cost multicommodity flow problem (DCMCF) as well as its variant with survivability constraints (DCSMCF) are addressed. Several possible directions for improvement or future investigations are mentioned in the concluding section.

A primal partitioning approach for single and non-simultaneous multicommodity flow problems

In the first part of this paper, we study the minimum linear cost multicommodity flow problem where the given traffic demand is satisfied through routes having less than a given maximum number of edges. We propose a simplex-based solution method which uses a refined primal partitioning of the basis matrix. Our program has been tested on some real world instances given by the national French telecommunication operator (France Telecom) and also on randomly generated multicommodity flow problems. The second part of our paper generalizes the techniques described previously to handle additional survivability constraints. Major telecommunication operators want to design multicommodity survivable networks, i.e., which are able to route all traffic demands even if any node or edge is damaged. We present a simple model of this problem based on non-simultaneous multicommodity flows.

A branch-and-bound method for multicommodity location with balancing requirements

We present a branch-and-bound algorithm for the multicommodity location-allocation problem with balancing requirements. Although the formulation displays a number of characteristics common to classical location models, it also unveils, due to the presence of the balancing requirements, a network flow structure that is most favorable to efficient algorithmic developments. In particular, tight bounds may be efficiently computed by using a reformulation of the weak relaxation of the problem as a minimum cost multicommodity flow problem. We also develop and analyse various branching criteria, and show that the branching rules which from the most efficient branch-and-bound procedure for the present problem are quite different from those used to solve classical location problems. The experimentation has been conducted both on several randomly generated problems and on a large scale application.

A heuristic method for optimizing an intercity data transmission network

This paper deals with the problem of finding the minimum cost flow pattern for a large intercity data transmission network which provides private lines between pairs of cities and which is subject to capacity constraints. This problem can be formulated as a minimum cost multicommodity flow problem. The most efficient linear and integer programming approaches yield programs with memory requirements proportional to the square of the problem size, making impractical their use for large scale problems. In this paper a heuristic method is described which produces suboptimal solutions with smaller execution time requirements, and memory requirements linear with the problem size. Experimental results are presented comparing the proposed method with a Linear Programming approach.

On the solution of a partially structured nonlinear program

AbstractA class of partially structured nonlinear programming problems, containing the capacitated nonlinear minimum cost multicommodity flow problem, is considered. Such problems, although large, can often be solved efficiently and with minimal computational storage by gradient projection methods.