Multi-commodity Problem Research Articles

Minimizing Maximum Unmet Demand by Transportations Between Adjacent Nodes Characterized by Supplies and Demands

Suppose we are given a graph with nodes characterized by amounts of supplies and demands of multiple commodities. The amounts of commodities stored at nodes (supplies) are given by positive numbers while those of demands at nodes are given by negative numbers. To meet demands we send commodities from nodes to neighbors by using vehicles, one at each node, with some loading capacity moving to and from neighbors. In this paper we adopt a one-way transportation model in which we just send commodities from a node to one of its neighbors along an edge. When we choose one neighbor at each node, we have a set of trips which naturally define a graph such that each connected component has at most one cycle, which is known as a pseudoforest. We present a linear-time algorithm for deciding whether there is a set of trips that meet all demands using one-way multi-commodity transportations on a pseudoforest with node degrees bounded by a constant. Using the algorithm, we first present an efficient algorithm for finding an optimal set of one-way one-commodity trips that minimize the maximum unmet demand on a pseudoforest, and then extend the idea to a multi-commodity problem on a pseudoforest with node degrees bounded by a constant.

A multi-commodity location-inventory problem in a closed-loop supply chain with commercial product returns

Commercial product returns represent a growing financial and operational concern for many firms, and they consist of true and false failure returns in the real-world business. This paper studies a joint location-inventory problem in a closed-loop system by considering multi-commodity flows and commercial product returns. First, this problem is formulated as a mixed-integer nonlinear programming model. Next, a novel heuristic method based on differential evolution is designed to solve this model efficiently. Finally, numerical experiments and results are presented to validate the solution approach and provide meaningful managerial insights for business managers to improve their supply chain practice in a closed-loop system with false failure returns.

An Improved Differential Evolution Algorithm for a Multicommodity Location-Inventory Problem with False Failure Returns

Customer returns are a common phenomenon in many industries, and they have a significant impact on business organizations and their supply chains. False failure returns are returned products that have no functional or cosmetic defects, and they represent a large body of customer returns in practice. In this paper, we develop a mixed-integer nonlinear programming model to study a multicommodity location-inventory problem in a forward-reverse logistics network. This model minimizes the total cost in this network by considering false failure returns, and it also considers many real-world business scenarios in forward and reverse logistics flows. Moreover, we design a new heuristic approach to solve the model efficiently. Finally, numerical experiments are conducted to validate our solution approach and provide meaningful managerial insights.

Complex vehicle transport problems: taxonomy, new variants, challenges and solution methodology

Due to appeared of several and new alternatives of vehicle routing problems (VRPs) in the last years, the contributions focus on design of traffic flow and transportation systems, vehicle routing, and other important aspects of supply chain management such as dynamic routing, multi commodity problems, inventory management, lot sizing, and lot scheduling. In this paper, we are going to present an extensive review of the general vehicle routing problem then we are going to focus on the problematic approaches in this field. For each one of these problematic, a positioning compared to this state of the art will be developed. In the first part a synthesis of literature review devoted to the complex and logistic transport problems in general and a transport problem with limited fleet and common carrier in particular with stochastic and dynamic vehicle routing problem. In the second part, a description of the vehicle routing problem with mathematical formulations will be presented. The different variants of this problem will be approached.

Step graphs and their application to the organization of commodity flows in networks

The concepts of step graphs and networks in which the edges are divided into ordered classes (shortage classes) are introduced. Each path in such graphs is assigned a tuple (an ordered sequence of nonnegative integers). The tuples are compared lexicographically. The problem of finding the path with the minimum tuple according to this ordering is stated. Functionals are compared using the lexicographic rule. An algorithm for finding the optimal path is described, and the relationship between step networks and weighted networks is investigated. The proposed formalism is applied to the problem of filling a network subject to constraints imposed on communication line capacities with communication flows under the condition that requests for the organization of such flows arrive to the network sequentially in time and the filling strategy must maximize the number of satisfied requests. The idea of an algorithm for the approximate solution of the integer multicommodity problem that uses the concepts of step networks and sequential algorithms is proposed.

Application of Modified Benders Decomposition to Single-Stage Multi-Commodity Multi-Period Warehouse Location Problem: An Empirical Investigation

In the past, researchers have applied Bender’s decomposition to distribution problem and used feasibility constraint to speed up the performance of Bender’s decomposition. Further, the application of Branch and Bound to single-stage multi-commodity single-period warehouse location problem (SSMCSPWLP) with strong constraints has shown that they are more effective. It was also shown in the previous research (in the context of Branch and Bound Methodology) that hybrid formulation for the single-stage single-period multi-commodity warehouse location problem yielded superior results. In this paper we apply Benders’ decomposition to strong and weak formulations of single-stage multi-commodity multi-period warehouse location problem (SSMCMPWLP). As suggested in the previous literature we put feasibility constraints in the pure integer sub- problem to speed up the performance of Benders’ decomposition. We also develop an additional cut (constraint that is again added to pure integer sub-problem) and show that it further speeded up Benders’ Decomposition. This research led to the possibility of applying Benders’ Decomposition to the hybrid formulation of SSMCMPWLP in future.

Multicommodity Production Planning: Qualitative Analysis and Applications

We develop a qualitative analysis theory for the convex-cost dynamic multicommodity production planning problem, which can be used, without performing any computational work, to provide invaluable insight to managers when faced with the task of deciding how to respond to changes in problem environment. We first formulate the problem as a multicommodity flow problem with parameters associated with each arc–commodity pair. We then reduce the problem to an equivalent single-commodity flow problem and develop a complete characterization of conformality among production, sales, and inventory activities in various instances of the problem. By combining the conformality characterizations with the monotonicity theory of Granot and Veinott [Granot F, Veinott AF Jr (1985) Substitutes, complements and ripples in network flow. Math. Oper. Res. 10:471–497] for single-commodity problems, we study the effects of changes in problem environment on optimal production, sales, and inventory schedules in the multicommodity problem. Numerous applications are presented and analyzed.

Iterative mass balance method for the multi-commodity flow maximisation problem

There are many multi-commodity maximisation problems present in the literature. The problem addressed in this article deals with the number of sources producing a mixture of multiple commodities with different proportions. There are also separators which separate the commodity flows. However, component failures often occur after the separation of commodities, resulting in a limited flow capacity to carry these commodities. As a result, production from certain number of sources has to be reduced to adjust for the capacity reduction caused by component failures. The objective is to maximise the flow with respect to one particular commodity, while satisfying the capacity constraints and the flow conservation law. This problem can be solved as a linear programming problem. In this article, a new algorithm is proposed for a three-commodity problem, which performs much faster than the simplex method. The three-commodity problem has a close relation to oil and gas production, where three different commodities, namely oil, gas and water are dealt with. This method can be used in a discrete-event simulator for long-term production planning while designing a piped network for the oil field.

Improving an interior‐point algorithm for multicommodity flows by quadratic regularizations

AbstractOne of the best approaches for some classes of multicommodity flow problems is a specialized interior‐point method that solves the normal equations by a combination of Cholesky factorizations and preconditioned conjugate gradient. Its efficiency depends on the spectral radius—in [0,1)—of a certain matrix in the definition of the preconditioner. In a recent work, the authors improved this algorithm (i.e., reduced the spectral radius) for general block‐angular problems by adding a quadratic regularization to the logarithmic barrier. This barrier was shown to be self‐concordant, which guarantees the convergence and polynomial complexity of the algorithm. In this work, we focus on linear multicommodity problems, a particular case of primal block‐angular ones. General results are tailored for multicommodity flows, allowing a local sensitivity analysis on the effect of the regularization. Extensive computational results on some standard and some difficult instances, testing several regularization strategies, are also provided. These results show that the regularized interior‐point algorithm is more efficient than the nonregularized one. From this work it can be concluded that, if interior‐point methods based on conjugate gradients are used, linear multicommodity flow problems are most efficiently solved as a sequence of quadratic ones. © 2011 Wiley Periodicals, Inc. NETWORKS, 2012

Approximation Algorithms for Nonuniform Buy-at-Bulk Network Design

Buy-at-bulk network design problems arise in settings where the costs for purchasing or installing equipment exhibit economies of scale. The objective is to build a network of cheapest cost to support a given multicommodity flow demand between node pairs. We present approximation algorithms for buy-at-bulk network design problems with costs on both edges and nodes of an undirected graph. Our main result is the first poly-logarithmic approximation ratio for the non-uniform problem that allows different cost functions on each edge and node; the ratio we achieve is $O(\log^4 h)$, where $h$ is the number of demand pairs. In addition we present an $O(\log h)$ approximation for the single sink problem. Poly-logarithmic ratios for some related problems are also obtained. Our algorithm for the multicommodity problem is obtained via a reduction to the single source problem using the notion of junction trees. We believe that this presents a simple yet useful general technique for network design problems.

The Multi-Commodity Source Location Problems and the Price of Greed

Given a graph G = (V;E), we say that a vertex subset S V covers a vertex v2 V if the edge-connectivity between S and v is at least a given integer k, and also say that S covers an edge vw2 E if v and w are both covered. We propose the multi-commodity source location problem, which is such that given a vertex- and edge-weighted graph G, p players each select q vertices, and obtain a prot that is the total over all players of the weight of each player’s covered vertices and edges. However, vertices selected by one player cannot be selected by the other players. The goal is to maximize the total prots of all players. We show that the price of greed, which indicates the ratio of the total prot of cooperating players to that of selsh players based on an ordered strategy, is tightly bounded by minfp;qg. Also when k = 2, we obtain tight bounds for vertex-unweighted trees when sources are located on the leaves.

A new strategy for the undirected two-commodity maximum flow problem

We address the two-commodity maximum flow problem on undirected networks. As a result of a change of variables, we introduce a new formulation that solves the problem through classical maximum flow techniques with only one-commodity. Therefore, a general strategy, based on this change of variables, is defined to deal with other undirected multi-commodity problems. Finally, we extend the single objective problem to a bicriteria environment. We show that the set of efficient solutions of the biobjective undirected two-commodity maximum flow problem is the set of alternative optimum solutions of the undirected two-commodity maximum flow problem. In addition, we prove that the set of efficient extreme points in the objective space has, at most, cardinality two.

Multicommodity flow problems with a bounded number of paths: A flow deviation approach

AbstractWe propose a modified version of the Flow Deviation method of Fratta, Gerla, and Kleinrock to solve multicommodity problems with minimal congestion and a bounded number of active paths. We discuss the approximation of the min–max objective function by a separable convex potential function and give a mixed‐integer nonlinear model for the constrained routing problem. Heuristic control of the path generation is then embedded in the original algorithm based on the concepts of cleaning the quasi inactive paths and reducing the flow width for each commodity. Numerical experiments show the validity of the approach for realistic medium‐sized networks associated with routing problems in broadband communication networks with multiple protocols and label‐switched paths (MPLS). © 2006 Wiley Periodicals, Inc. NETWORKS, Vol. 49(1), 80–89 2007

Availability-differentiated service provisioning in free-space optical access networks

Feature Issue on Optical Wireless Communications (OWC)Weather conditions have a significant impact on availability of free-space optical (FSO) access networks. Under fair weather conditions, such networks can use long-range FSO links that span over 1 km. Under severe weather conditions, e.g., dense fog, only the short-range FSO links in the range of 200–400m remain operational. We propose an availability-differentiated networking scheme that supports two classes of service availability in an FSO network consisting of both long-range and short-range FSO links. Under severe weather conditions, the network topology is reconfigured to use only the short-range FSO links. Owing to the reduced network capacity in the reconfigured network, lower-availability services (LAS) will be preempted by higher-availability services (HAS). LAS and HAS have different revenue values. Given a set of hybrid LAS and HAS requests, we maximize the total revenue of LAS + HAS by selectively provisioning a portion of the service requests in the FSO access network. We formulate the problem into a common terminal multicommodity problem and develop an optimal algorithm to solve this problem in polynomial time.

Solving Difficult Multicommodity Problems with a Specialized Interior-Point Algorithm

Due to recent advances in the development of linear programming solvers, some of the formerly considered difficult multicommodity problems can today be solved in few minutes, even faster than with specialized methods. However, for other kind of multicommodity instances, general linear solvers can still be quite inefficient. In this paper we will give an overview of the current state-of-the-art in solving large-scale multicommodity problems, comparing an specialized interior-point algorithm with CPLEX 6.5 in the solution of difficult multicommodity problems of up to 1 million of variables and 300,000 constraints.

The freight routing problem of time-definite freight delivery common carriers

The freight routing problem of time-definite common carriers is to minimize the sum of handling and transportation costs, while meeting service commitments and operational restrictions. There are two types of operational restrictions, capacity and directed in-tree rooted at each destination. Directed in-tree implicitly implies that there is a singular path for each origin–destination pair. The routing problem is an integrality constrained multi-commodity problem with side constraints. In this research, we study two approaches, the Lagrangian relaxation (LR) and implicit enumeration algorithm with ε-optimality (IE- ε). We use the third largest time-definite freight delivery common carrier in Taiwan for our numerical test. The result shows that the IE- ε outperforms the LR, both quantitatively and qualitatively. In addition, two major shortcomings of the LR approach are shown: it may fail to find any feasible solutions even though they exist, and it cannot determine whether the feasible set is empty or not.

Tabu Search With Exact Neighbor Evaluation For Multicommodity Location With Balancing Requirements

In this paper, we present a tabu search heuristic for solving the multicommodity location problem with balancing requirements. This heuristic improves upon a previous implementation by performing an exact, rather than approximate, evaluation of neighboring solutions. It also includes a number of refinements, in particular, a new initialization procedure and enhanced neighborhood reduction techniques. The heuristic is shown to be very effective, as it identifies the optimal solution on every instance taken from a set of randomly generated problems. It also finds optimal or near-optimal solutions on a set of large-size instances derived from an actual application, with computation times that show little variance as compared with the currently best known exact method.

On design of a survivable network architecture for dynamic routing: Optimal solution strategy and an efficient heuristic

We investigate network planning and design under volatile conditions of link failures and traffic overload. Our model is a non-simultaneous multi-commodity problem, with any particular two link failure being considered as one scenario. We show that the optimal solution model is not practically solvable for real-world problems and hence an efficient heuristic is provided which is O( n 6) faster than the optimal model and is based on synthesizing a modified maximum spanning tree using an algorithm due to Gomory and Hu. The output of this procedure is then used to solve a much smaller linear program. Simulation results indicate that the heuristic is near optimal for problems with up to 40 nodes.

Disconnecting sets in single and two-terminal-pair networks

We consider mixed networks, which may include both directed and undirected edges. For a nontrivial vertex subset S, an S-disconnecting set is a set of edges and vertices which intersects every path from any vertex in S to any vertex not in S. Given nonnegative edge and vertex costs, we show that the minimum cost of an S-disconnecting set defines a submodular function. This implies that the set of all S inducing minimum-cost disconnecting sets is the set of closures of a binary relation, thus extending Picard-Queyranne's (1980) result on ordinary minimum cuts. We apply this result to two-pair multicommodity problems in undirected networks, extending Hu's (1963) result to disconnecting sets that may include vertices as well as edges. These results and a result of Provan and Shier (1994) may be used for generating all sets S that induce such minimum-cost disconnecting sets and ranking such sets in order of corresponding costs, for both one-pair problems in mixed networks and two-pair problems in undirected networks. © 1996 John Wiley & Sons, Inc.

An approximate max-flow min-cut relation for undirected multicommodity flow, with applications

In this paper, we prove the first approximate max-flow min-cut theorem for undirected multicommodity flow. We show that for a feasible flow to exist in a multicommodity problem, it is sufficient that every cut's capacity exceeds its demand by a factor ofO(logClogD), whereC is the sum of all finite capacities andD is the sum of demands. Moreover, our theorem yields an algorithm for finding a cut that is approximately minimumrelative to the flow that must cross it. We use this result to obtain an approximation algorithm for T. C. Hu's generalization of the multiway-cut problem. This algorithm can in turn be applied to obtain approximation algorithms for minimum deletion of clauses of a 2-CNF≡ formula, via minimization, and other problems. We also generalize the theorem to hypergraph networks; using this generalization, we can handle CNF≡ clauses with an arbitrary number of literals per clause.