Random Arc Capacities Research Articles

Network design with probabilistic capacities

We consider a network design problem with random arc capacities and give a formulation with a probabilistic capacity constraint on each cut of the network. To handle the exponentially‐many probabilistic constraints a separation procedure that solves a nonlinear minimum cut problem is introduced. For the case with independent arc capacities, we exploit the supermodularity of the set function defining the constraints and generate cutting planes based on the supermodular covering knapsack polytope. For the general correlated case, we give a reformulation of the constraints that allows to uncover and utilize the submodularity of a related function. The computational results indicate that exploiting the underlying submodularity and supermodularity arising with the probabilistic constraints provides significant advantages over the classical approaches. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 71(1), 16–30 2018

The maximum flow problem of uncertain random network

The maximum flow problem is an important problem of network optimization and it covers a wide range of engineering and management applications. The goal of the problem is to find the maximum amount of flow from the source to the sink in a network. This paper investigates two models of the maximum flow of an uncertain random network under the framework of chance theory. The expected value constrained maximum flow and chance constrained maximum flow models with uncertain random arc capacities are proposed. We prove that there exists an equivalence relationship between the uncertain random maximum flow models and the deterministic maximum flow models. Furthermore, some important properties of the models are analyzed, and two algorithms are proposed. Finally, as an illustration, a numerical example is presented to show the effectiveness of the models and algorithms.

An arc-exchange decomposition method for multistage dynamic networks with random arc capacities

Multistage dynamic networks with random arc capacities (MDNRAC) have been successfully used for modeling various resource allocation problems in the transportation area. However, solving these problems is generally computationally intensive, and there is still a need to develop more efficient solution approaches. In this paper, we propose a new heuristic approach that solves the MDNRAC problem by decomposing the network at each stage into a series of subproblems with tree structures. Each subproblem can be solved efficiently. The main advantage is that this approach provides an efficient computational device to handle the large-scale problem instances with fairly good solution quality. We show that the objective value obtained from this decomposition approach is an upper bound for that of the MDNRAC problem. Numerical results demonstrate that our proposed approach works very well.

The dynamic fleet management problem with uncertain demand and customer chosen service level

In this paper, we study a dynamic fleet management problem with uncertain demands and customer chosen service levels. We first show that the problem can be transformed into a dynamic network with partially dependent random arc capacities, and then develop a structural decomposition approach which decomposes the network recourse problem into a series of tree recourse problems (TRPs). As each TRP can be solved by an efficient algorithm, the decomposition approach can solve the problem very efficiently. We conduct numerical experiments to compare its performance with two alternative methods. Numerical experiments show that the performance of our method is quite encouraging.

Computing Multi-State Two-Terminal Reliability Through Critical Arc States That Interrupt Demand

A variety of algorithms have been proposed to compute the NP-hard multi-state two-terminal reliability at demand level <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> (MS2TR <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> ). Most of the methods solve the problem in terms of two to three NP-hard sub-problems. The only existing method that computes MS2TR <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> without requiring all multi-state minimal paths or cuts <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori,</i> excluding the impractical complete enumeration method, and Doulliez & Jamoulle's popular but incorrect decomposition method, see P. Doulliez and E. Jamoulle, “Transportation networks with random arc capacities,” RAIRO, vol. 3, pp. 45-60, 1972, is the decomposition method proposed by Jane & Laih, see C.-C. Jane and Y.-W. Laith, “A practical algorithm for computing multi-state two-terminal reliability,” IEEE Trans. Reliability, vol. 57, pp. 295-302, 2008, recently. In contrast to Jane & Laih's method that computes MS2TR <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> in term of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> -flows, this paper presents a novel decomposition method that computes 1-MS2TR <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> in terms of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> -cuts. Contribution of the proposed method is that it is the first unique method that computes MS2TR <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> by way of cut sets without requiring all multi-state minimal paths or cuts a priori. An example illustrates the proposed algorithm. In addition, we make computational comparisons between the two algorithms in terms of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> -flows, and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> -cuts.

On stochastic programs over trees with partially dependent arc capacities

AbstractWe study the new problem of stochastic programs over trees with dependent random arc capacities. This problem can be used as a subproblem in decomposition methods that solve multi‐stage networks with independent random arc capacities and random travel times. An efficient algorithm is provided to compute the expected total cost. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 50(2), 157–163 2007

A two-stage stochastic programming framework for transportation planning in disaster response

This study proposes a two-stage stochastic programming model to plan the transportation of vital first-aid commodities to disaster-affected areas during emergency response. A multi-commodity, multi-modal network flow formulation is developed to describe the flow of material over an urban transportation network. Since it is difficult to predict the timing and magnitude of any disaster and its impact on the urban system, resource mobilization is treated in a random manner, and the resource requirements are represented as random variables. Furthermore, uncertainty arising from the vulnerability of the transportation system leads to random arc capacities and supply amounts. Randomness is represented by a finite sample of scenarios for capacity, supply and demand triplet. The two stages are defined with respect to information asymmetry, which discloses uncertainty during the progress of the response. The approach is validated by quantifying the expected value of perfect and stochastic information in problem instances generated out of actual data.

An Algorithm for Multistage Dynamic Networks with Random Arc Capacities, with an Application to Dynamic Fleet Management

We consider the class of multistage dynamic networks with random arc capacities a framework that is well suited to model dynamic fleet management problems. We propose a successive convex approximation approach that produces an approximation to the expected recourse function which captures the future effects of current decisions under uncertainty. This method decomposes the network in each stage into tree subproblems, whose expected recourse functions are easy to obtain. We also compare this method with two alternative methods on a set of dynamic fleet management problems. The numerical results show that this method is superior to the two alternative methods.

A network recourse decomposition method for dynamic networks with random arc capacities

AbstractWe study a class of two‐stage dynamic networks with random arc capacities where the decisions in the first stage must be made before realizing the random quantities in the second stage. The expected total cost of the second stage is a function of the first‐stage decisions, known as the expected recourse function, which is generally intractable. This paper proposes a new method to approximate the expected recourse function as a convex, separable function of the supplies to the second stage. First, the second‐stage network is decomposed using Lagrangian relaxation into tree subproblems whose expected recourse functions are numerically tractable. Subgradient optimization is then used to update the Lagrange multipliers to improve the approximations. Numerical experiments show that this structural decomposition approach can produce high‐quality approximations of the expected recourse functions for large stochastic networks. © 1994 by John Wiley &amp; Sons, Inc.

Stochastic programs over trees with random arc capacities

AbstractIn this paper, we study a special class of stochastic programs where the recourse problem consists of trees with random arc capacities, which we call tree recourse. We develop an efficient procedure to obtain the expected recourse function exactly for single‐stage problems. This procedure is used in a backward recursion to find the exact expected recourse function for multistage trees. We show that for large trees n‐stage networks can be solved more easily than can a single‐stage network with n‐levels. We also show that the expected recourse functions for single‐stage and n‐stage formulations of the same network can be quite similar. Finally, we illustrate the potential of using the tree recourse problems in a more general stochastic network problem. © 1994 by John Wiley &amp; Sons, Inc.

Restricted Recourse Strategies for Dynamic Networks with Random Arc Capacities

We consider a class of multistage stochastic programming problems that can be formulated as networks with random arc capacities. Large problems have proved intractable using exact methods and hence various approximations have been proposed, ranging from approximating the recourse function to sampling a small number of scenarios to capture future uncertainties. We explore the use of specialized recourse strategies that are not as general as network recourse but nonetheless capture some of the important tradeoffs. These new recourse strategies allow us to develop approximations to the recourse function that can be used to solve problems with thousands of random variables. Given these approximations, classical optimization methods can be used. The concept of hierarchical recourse is introduced and used to synthesize and generalize earlier notions of nodal recourse and cyclic recourse.

TSS Dissertation Abstracts—Abstracts for the 1993 Transportation Science Section Dissertation Prize Competition

Nine dissertations were submitted to the 1993 Transportation Science Section dissertation prize competition. Two honorable mentions were awarded: Raymond K. Cheung (Dynamic Networks with Random Arc Capacities: Solution Methods and Applications) and Rekha Pillai (The Traveling Salesman Subset-tour Problem with One Additional Constraint). Extended abstracts are presented. (1) Dynamic Networks with Random Arc Capacities: Solution Methods and Applications by Raymond K. Cheung; (2) The Traveling Salesman Subset-tour Problem with One Additional Constraint (TSSP + 1) by Rekha Pillai; (3) A Model for Roadway Traffic Control Software by Darcy M. Bullock; (4) L'Optimisation des Frequences d'un Reseau de Transport en Commun by Isabelle Constantin; (5) Learning Heuristics for Repetitive Combinatorial Optimization Problems: With an Application in Train Scheduling by David R. Kraay; (6) Modeling Route Choice Behavior in the Presence of Information Using Concepts from Fuzzy Set Theory and Approximate Reasoning by Tsippy Lotan; (7) Air Carrier, Fare Class, and Flight Demand: An Individual Choice Approach by Kimon E. Proussaloglou; (8) Dynamic Transportation Network Models for Advanced Traveler Information Systems by Bin Ran; (9) Multicommodity Network Flows: The Impact of Formulation on Decomposition by Kimberly Lynn Ross.

Selected Abstracts for the 1990–1991 Transportation Science Section Dissertation Prize Competition

Fourteen dissertations were submitted to the 1990–1991 Transportation Science Section dissertation prize competition. The first prize was awarded by Dr. Linos F. Frantzeskakis for “Dynamic Networks with Random Arc Capacities, with Application to the Stochastic Dynamic Vehicle Allocation Problem.” Three honorable mentions were also awarded. The recipients are listed in alphabetical order: Dr. Saad A. H. Al-Gadhi for “Characterization of Crowd Behavior and Movement,” Dr. Rina Schneur for “Scaling Algorithms for Multicommodity Flow Problems and Network Flow Problems with Side Constraints,” and Dr. Garrett van Ryzin for “Stochastic and Dynamic Vehicle Routing in Euclidean Service Regions.” The abstracts of all 14 theses submitted are reproduced.

Monte Carlo, Control Variates, and Stochastic Ordering

Using prior information about the stochastic orderings between a phenomenon of interest and other ancillary phenomena whose population parameters are known in a Monte Carlo experiment, this paper derives an unbiased control variate estimator and an exact $100 \times (1 - \alpha )$ percent confidence interval for a proportion and, more generally, for a distribution function for finite sample size K. These results substantially improve on the more traditional control variate variance-reduction method for which, in the case of finite K, neither an unbiased point estimator is generally available nor, except in the case of multinormal data, is an exact $100 \times (1 - \alpha )$ percent confidence interval available.The paper also derives upper bounds on the estimator's variance and coefficient of variation and a lower bound on the achievable variance reduction in terms of the known parameters for the ancillary phenomena. Computing these bounds prior to sampling provides useful guidance rarely available with other variance reduction techniques. In the case of estimating a discrete distribution function, the paper also shows how to derive a variance-reduced estimator of the population mean from the estimated distribution function, and gives the variance of this sample mean to order ${1 / K}$. An example illustrates the technique for estimating the distribution of maximal flow in a flow-network problem with random arc capacities.

Bounds On The Expected Maximal Value Of A Class Of Stochastic Network Problems

AbstractA lower bound on the expected maximal cost in a special bipartite network with random supply, demand and arc-capacities is presented. The formulation covers many different stochastic assignment and (bipartite) matching problems.

Maximum flow and critical cutset as descriptors of multi-state systems with randomly capacitated components

Let G = ( V, E, s, t) denote a directed network with node set V, arc set E = {1,…, n}, source node s and sink node t. Let Γ denote the set of all minimal s− t cutsets and b 1( τ), …, B n ( τ), the random arc capacities at time τ with known joint probability distribution function. Let Λ(τ) denote the maximum s− t flow at time τ and D( τ), the corresponding critical minimal s− t cutset. Let Ω denote a set of minimal s− t cutsets. This paper describes a comprehensive Monte Carlo sampling plan for efficiently estimating the probability that D(τ)ϵΩ⊂-Γ and x< λ( τ)⩽ y at time τ and the probability that D(τ)ε Ω given that x < Λ( τ) ⩽ y at time τ. The proposed method makes use of a readily obtainable upper bound on the probability that Λ( τ) > x to gain its computational advantage. Techniques are described for computing confidence intervals and credibility measures for assessing that specified accuracies have been achieved. The paper includes an algorithm for performing the Monte Carlo sampling experiment, an example to illustrate the technique and a listing of all steps needed for implementation.

Two new component importance measures for a flow network system

This paper presents two new component importance measures for a flow network system with random arc capacities: 1. (1) system availability improvement potential of the component; 2. (2) expected unutilized capacity of the component. A numerical example is given.

Weapons systems analysis, Part III: warfare gaming, tactics and techniques : Yosef S. Sherif. Microelectron. Reliab.22 (5), 951 (1982)

Let G = (V, E, s, t) denote a directed network with node set V, arc set E = {1,…, n}, source node s and sink node t. Let Γ denote the set of all minimal s−t cutsets and b1(τ), …, Bn(τ), the random arc capacities at time τ with known joint probability distribution function. Let Λ(τ) denote the maximum s−t flow at time τ and D(τ), the corresponding critical minimal s−t cutset. Let Ω denote a set of minimal s−t cutsets. This paper describes a comprehensive Monte Carlo sampling plan for efficiently estimating the probability that D(τ)ϵΩ⊂-Γ and x<λ(τ)⩽y at time τ and the probability that D(τ)ε Ω given that x < Λ(τ) ⩽ y at time τ. The proposed method makes use of a readily obtainable upper bound on the probability that Λ(τ) > x to gain its computational advantage. Techniques are described for computing confidence intervals and credibility measures for assessing that specified accuracies have been achieved. The paper includes an algorithm for performing the Monte Carlo sampling experiment, an example to illustrate the technique and a listing of all steps needed for implementation.

Modular decomposition and reliability computation in stochastic transportation networks having cutnodes

AbstractConsider a flow network having random arc capacities and having associated with each node n a “supply‐demand random variable” Yn whose absolute value equals the supply available at the node when Yn assumes a non‐negative value and the demand required by the node when Yn assumes a nonpositive value. A fundamental problem is the computation of the reliability R, that is, the probability that the random variables will assume values that permit a feasible flow. Upon adapting the graph‐theoretic concepts of “cutnode” and “block,” it is possible to identify a “block‐module,” an independent, nontrivial subnetwork that has one and only one node (the “cutnode”) connected to nodes outside the subnetwork. The reliability of the network will increase by a known factor after a “block‐modular decomposition” that consists of a transformation of the cutnode's supply‐demand random variable and the deletion of the remainder of the block module. Provided the original network possesses at least one block module, R can be determined from a sequence of block‐modular decompositions that reduce the original network to a single node whose reliability is easily computed. Computational experience with a computer implementation of such a decomposition method is reported, and the application of the method to the analysis of electrical power networks is included.

Maximum flow in networks with a small number of random arc capacities

AbstractIn a directed, capacitated network with single source and sink in which the capacities of some of the arcs are random variables, the max‐flow value is also a random variable. I propose a new approach to the problem of computing its distribution and expected value via a corollary of the max‐flow min‐cut theorem, and implement it explicitly for small numbers of random capacities. I consider the error obtained in estimating the expected max‐flow value by setting some of the random capacities equal to their means. When the flows are restructed by lower bounds, I give the probability of existence of a feasible flow.