Capacitated-flow Network Research Articles

Probability Distributions on Partially Ordered Sets and Network Interdiction Games

This article poses the following problem: Does there exist a probability distribution over subsets of a finite partially ordered set (poset), such that a set of constraints involving marginal probabilities of the poset’s elements and maximal chains is satisfied? We present a combinatorial algorithm to positively resolve this question. The algorithm can be implemented in polynomial time in the special case where maximal chain probabilities are affine functions of their elements. This existence problem is relevant for the equilibrium characterization of a generic strategic interdiction game on a capacitated flow network. The game involves a routing entity that sends its flow through the network while facing path transportation costs and an interdictor who simultaneously interdicts one or more edges while facing edge interdiction costs. Using our existence result on posets and strict complementary slackness in linear programming, we show that the Nash equilibria of this game can be fully described using primal and dual solutions of a minimum-cost circulation problem. Our analysis provides a new characterization of the critical components in the interdiction game. It also leads to a polynomial-time approach for equilibrium computation.

Theory and applications of an integrated model for capacitated-flow network reliability analysis

To evaluate the performance of engineering systems, this study acts as a bridge between the theories of classical and capacitated-flow network reliabilities and helps integrate them. Unlike existing works, in which a deterministic probability distribution was used for arcs in a network, in this study, the time-dependent reliability function is applied for components in arcs. In particular, the proposed model applies an arbitrary reliability function to evaluate reliability. In arc-level reliability analysis, the components in an arc are characterized by a reliability function, and such components comprise a capacitated-flow arc. Therefore, a certain number of components can be derived from the probability distribution of the capacity provided by an arc. In system-level reliability analysis, an algorithm is used to generate the minimal component vectors (MCV) for the given demand and time constraints. The system reliability can be calculated in terms of the MCVs using the derived capacity probability distribution. Based on the reliability analysis, a maintenance issue with two policies is discussed. Examples, including a large-scale case study of the Taiwan Academic Network, are discussed to validate the correctness, applicability, and scalability of the proposed model.

Method for integrating classical and network reliability: A time varying network reliability evaluation

AbstractMany capacitated flow network papers treat reliability as a time‐invariant property. However, network components degrade over time suggesting that methods to quantify the time‐varying reliability of a network are appropriate. This paper proposes a method to compute the reliability of a multistate capacitated flow network in terms of multistate components characterized by a life distribution. The approach enables the calculation of the probability that demand can be satisfied over period of time despite component degradation. Examples are provided to clarify the steps required and to illustrate the assessments enabled. Based on the classical reliability theory, the Weibull reliability function is employed to characterize the degradation of network components. However, the approach is general and other distributions are also possible. The results indicate that the approach can assist a network supervisor determine reasonable demand rate and time constraint to satisfy demand as well as reason about alternative reliability improvement strategies.

Reliability Evaluation of Production System With In-Line Stockers

A reliability evaluation method is proposed for production systems by applying the capacitated-flow production network (CFPN) model. In particular, this article considers in-line stockers to monitor and to avoid blockage and starvation in the CFPN. First, the minimal capacity that all cells (group of machines) should provide to meet demand is generated. Second, status of stocker is monitored via the proposed “stocker volume check table” to calculate the volume usage of a stocker, and the stocker nonutilization is then obtained. Finally, the system reliability is derived in terms of the minimal capacity vector and stocker nonutilization. An important contribution of this article is to avoid complicated dependency calculation when multiple stockers are considered. A practical case of printed circuit board production is further studied to illustrate the applicability of the proposed method.

A hybrid approach to solve a bi-objective optimization problem of a capacitated-flow network with a time factor

The problem of system reliability evaluation for a capacitated-flow network has been extended by involving the characteristics of the quickest path problem. System reliability is thus defined as the probability that d units of demand can be successfully delivered from a source node to a sink node within time t. System reliability optimization considering the time factor is important for demand transmission, especially for communication systems. However, this factor has not been taken into account in previous research. To this end, in this study, the time factor is employed to adopt a component assignment strategy to maximize the system reliability and minimize the assignment cost. Because the considered problem is bi-objective, a hybrid approach integrating the non-dominated sorting genetic algorithm II, a concept of minimal path, information entropy, and simple additive weighting is developed to determine a set of non-dominated solutions and to subsequently determine the compromise solution from an objective viewpoint. Numerical examples are utilized to demonstrate the computational efficiency of the proposed method.

System reliability maximization for a computer network by finding the optimal two-class allocation subject to budget

In the real world, a computer/communication system is usually modeled as a capacitated-flow network since each transmission line (resp. facility) denoted by an edge (resp. node) has multiple capacities. System reliability is thus defined to be a probability that d units of data are transmitted successfully from a source node to a sink node. From the perspective of quality management, system reliability is a critical performance indicator of the computer network. This paper focuses on maximizing system reliability for the computer network by finding the optimal two-class allocation subject to a budget, in which the two-class allocation is to allocate exactly one transmission line (resp. facility) to each edge (resp. node). In addition, allocating transmission lines and facilities to the computer network involves an allocation cost where the cost for allocating a transmission line depends on its length. For solving the addressed problem, a genetic algorithm based method is proposed, in which system reliability is evaluated in terms of minimal paths and state-space decomposition. Several experimental results demonstrate that the proposed algorithm can be executed in a reasonable time and has better computational efficiency than several popular soft computing algorithms.

Reliability of a production system with intersectional lines

This article addresses reliability evaluation for a production system with intersectional lines and multiple reworking actions, where the reliability is the probability of demand satisfaction. First, the production system is constructed as a capacitated-flow production network by the revised graphical transformation and decomposition techniques. Capacity analysis is implemented to determine the input flow of each workstation subsequently. Second, two algorithms, including a general algorithm, are proposed to generate all minimal capacity vectors that workstations should provide to satisfy demand. The reliability is derived in terms of such vectors accordingly. A real-world case of printed circuit board production system is utilized to demonstrate the performance evaluation procedure.

Simulation approach to estimate the system reliability of a time-based capacitated flow network susceptible to correlated failures

This paper constructs a time-based capacitated flow network (TCFN) model to analyze the performance of a computer network, considering correlation between the failures of all the physical lines comprising the edges of the network. This model allows us to assess global-scale events, including correlated failures in local and neighboring edges that can cause widespread damage and degrade TCFN performance. A simulation approach is proposed to quantify the probability that a given amount of data can be sent from an origin to destination through this TCFN within a time constraint. This probability that the network satisfies both a specified level of demand and time constraint is referred to as system reliability. The results of several experimental including a practical computer network demonstrate that correlation can positively or negatively impact reliability depending on the demand and time constraint. The modeling approach therefore promotes design and optimization studies to ensure that a network will be highly reliable despite correlated failures.

An optimal capacity assignment for the robust design problem in capacitated flow networks

This paper proposes an exact algorithm to solve the robust design problem in a capacitated flow network in which each edge has several possible capacities. A capacitated flow network is popular in our daily life. For example, the computer network, the power transmission network, or even the supply chain network are capacitated flow networks. In practice, such network may suffer failure, partial failure or maintenance. Therefore, each edge in the network should be assigned sufficient capacity to keep the network functioning normally. The robust design problem (RDP) in a capacitated flow network is to search for the minimum capacity assignment of each edge such that the network still survived even under the edge’s failure. However, how to optimally assign the capacity to each edge is not an easy task. Although this kind of problem was known of NP-hard, this paper proposes an efficient exact algorithm to search for the optimal solutions for such a network and illustrates the efficiency of the proposed algorithm by numerical examples.

On transmission time through k minimal paths of a capacitated-flow network

The quickest path problem is to minimize the transmission time for sending a specified amount of data through a single minimal path. Two deterministic attributes are involved herein; the capacity and the lead time. However, in many real-life networks such as computer systems, urban traffic systems, etc., the arc capacity should be multistate due to failure, maintenance, etc. Such a network is named a capacitated-flow network. The minimum transmission time is thus not a fixed number. This paper is mainly to evaluate system reliability that d units of data can be transmitted through k minimal paths under time constraint T. A simple algorithm is proposed to generate all minimal capacity vectors meeting the demand and time constraints. The system reliability is subsequently computed in terms of such vectors. The optimal k minimal paths with highest system reliability can further be derived.

Performance evaluation for the logistics system in case that capacity weight varies from arcs and types of commodity

Most of Taiwan high-technology companies establish their factories in China to enhance the logistics systems. In order to show and improve the company's competitive advantage, it is an important issue to measure the performance of the logistics system. Hence, we try to propose an index to evaluate the performance of the logistics system between China and Taiwan by constructing a two-commodity capacitated-flow network. In the proposed model, each arc has both the capacity and cost attributes. Each type of commodity consumes the capacity of each arc differently. That is, the capacity weight varies from the arcs and types of commodity. It is known that the system capacity is the maximum flow from the source to the sink for a single-commodity capacitated-flow network. However, the conventional performance indexes proposed for a single-commodity case are not suitable for a two-commodity case. Hence, we define the system capacity as a vector for a two-commodity capacitated-flow network. Then a performance index, the probability that the upper bound of the system capacity equals a given pattern under the budget constraint, is proposed to measure the performance of logistics systems. An efficient algorithm based on minimal cuts is presented to evaluate such a performance index.

Reliability of a computer network in case capacity weight varying with arcs, nodes and types of commodity

The computer network can be modeled as a capacitated-flow network. This paper concentrates on a two-commodity capacitated-flow network with three characters: (1) nodes as well as arcs have multiple possible capacities and may fail, (2) each component (arc/node) has both capacity and cost attributes; and (3) the capacity weight varies with arcs, nodes and types of commodity (or named file). We study the possibility that a given quantity of two types of files can be transmitted through this network simultaneously under the budget constraint. Such a possibility is named the system reliability which is a performance index to measure the quality level of supply demand systems such as computer, telecommunication, electric-power transmission and transportation systems. The approach of minimal paths is applied to describe the relationship among flow assignments and capacity vectors. A simple algorithm in terms of minimal paths is proposed to evaluate the system reliability.

Flow reliability of a probabilistic capacitated-flow network in multiple node pairs case

This article mainly generalizes the flow (or transportation) reliability problem for a directed capacitated-flow network in which the capacity of each arc ai has the values 0<1<2<⋯<Mi from s (source) to t (sink) case to a multiple node pairs case. Given the demands for all specified node pairs simultaneously in the network, a simple algorithm is proposed first to find out the family of all lower boundary points for such demands in terms of minimal paths. The flow reliability, the probability that the system allows the flow satisfying the demands simultaneously, can be calculated in terms of such lower boundary points. The overall-terminal flow reliability, one source to multiple sinks flow reliability and multiple sources to one sink flow reliability can be calculated as special cases.

Overall-terminal reliability of a stochastic capacitated-flow network

For a stochastic and directed capacitated-flow network in which the capacity of each arc has several possible values, this article generalizes the system reliability problem from single source node and single sink node cases to an overall-terminal case. Given the demand for each node pair simultaneously, a simple algorithm is proposed first to generate all lower boundary points for such demands in terms of minimal paths. The lower boundary point is a vector denoting the current capacity of each arc. The system reliability, the probability that the system satisfies the demands simultaneously, can be calculated in terms of such lower boundary points by applying the inclusion-exclusion method.

Study on the multicommodity reliability of a capacitated-flow network

Traditionally, many researchers solved the multicommodity maximum flow problem by assuming that the arcs of the flow network are deterministic. When the arcs are stochastic (i.e., the capacity of each arc has several values), this article studies how to calculate the probability that a capacitated-flow network with a unique source mode satisfies a demand ( d 1, d 2,…, d p ) at the unique sink node, where d k is the demand of commodity k. Such a probability is named the multicommodity reliability and is dependent on capacities of arcs. One solution procedure is proposed to evaluate the multicommodity reliability, which includes two parts: an algorithm to generate all ( d 1 , d 2 ,…, d p )-MPs and a method to calculate the multicommodity reliability in terms of ( d 1, d 2,…, d p )-MPs. Two illustrative examples are given.