Yi-Kuei Lin

A simple algorithm for reliability evaluation of a stochastic-flow network with node failure

Abstract This paper addresses a stochastic-flow network in which each arc or node has several capacities and may fail. Given the demand d, we try to evaluate the system reliability that the maximum flow of the network is not less than d. A simple algorithm is proposed firstly to generate all lower boundary points for d, and then the system reliability can be calculated in terms of such points. One computer example is shown to illustrate the solution procedure. Scope and purpose The maximum flow problem is a standard problem in operations research and network analysis (Ford and Fulkerson, Flows in networks. NJ: Princeton University Press, 1962). This paper discusses the maximum flow problem for a stochastic-flow network in which arcs and nodes all have several capacities and may fail. We can evaluate the probability (named system reliability here) that the maximum flow is not less than a given demand, and then treat such a reliability as a system performance index. The purpose of this paper is to provide such a performance index for many real-world systems such as computer systems, telecommunication systems, logistics systems, etc.

Using minimal cuts to evaluate the system reliability of a stochastic-flow network with failures at nodes and arcs

Abstract This paper deals with a stochastic-flow network in which each node and arc has a designated capacity, which will have different lower levels due to various partial and complete failures. We try to evaluate the system reliability that the maximum flow of the network is not less than a demand (d+1). A simple algorithm in terms of minimal cuts is first proposed to generate all upper boundary points for d, and then the system reliability can be calculated in terms of such points. The upper boundary point for d is a maximal vector, which represents the capacity of each component (arc or node), such that the maximum flow of the network is d. A computer example is shown to illustrate the solution procedure.

Extend the quickest path problem to the system reliability evaluation for a stochastic-flow network

From the point of view of quality management, it is an important issue to reduce the transmission time in the network. The quickest path problem is to find the path in the network to send a given amount of data from the source to the sink such that the transmission time is minimized. Traditionally, this problem assumed that the capacity of each arc in the network is deterministic. However, the capacity of each arc is stochastic due to failure, maintenance, etc. in many real-life networks. This paper proposes a simple algorithm to evaluate the probability that d units of data can be sent from the source to the sink through the stochastic-flow network within T units of time. Such a probability is called the system reliability. The proposed algorithm firstly generates all lower boundary points for (d, T) and the system reliability can then be computed in terms of such points.

Reliability of a stochastic-flow network with unreliable branches & nodes, under budget constraints

System reliability evaluation for flow networks is an important issue in quality management. This paper concentrates on a stochastic-flow network in which nodes as well as branches have several possible capacities, and can fail. The possibility is evaluated that a given amount of messages can be transmitted through the stochastic-flow network under the budget constraint. Such a possibility, system reliability, is a performance index for a stochastic-flow network. A minimal path, an order sequence of nodes & branches from the source to the sink without cycles, is used to assign the flow to each component (branch or node). A lower boundary point for (d, C) is a minimal capacity vector, which enables the system to transmit d messages under the budget C. Based on minimal paths, an efficient algorithm is proposed to generate all lower boundary points for (d, C). The system reliability can then be calculated in terms of all lower boundary points for (d, C) by applying the inclusion-exclusion rule. Simulation shows that the implicit enumeration (step 1) of the proposed algorithm can be executed efficiently.

On a multicommodity stochastic-flow network with unreliable nodes subject to budget constraint

From the quality management and decision making view point, reliability and unreliability are important indices to measure the quality level for a stochastic-flow network. In a multicommodity stochastic-flow network with unreliable nodes, the branches and nodes all have several possible capacities and may fail. Different types of the commodity, which are transmitted through the same network simultaneously, compete the capacities of branches and nodes. In this paper we first define the system capacity as a vector for a multicommodity stochastic-flow network with unreliable nodes. Then we design a performance index which is the probability that the upper bound of the system capacity is a given pattern subject to the budget constraint. It can be applied to evaluate the quality level for such a network. A simple approach based on minimal cuts is thus presented to evaluate the performance index.

ON RELIABILITY EVALUATION OF A STOCHASTIC-FLOW NETWORK IN TERMS OF MINIMAL CUTS

ABSTRACT In a stochastic-flow network, each arc has several capacity levels. Given the system demand d, the reliability of such a network for demand d (i.e., the probability that the maximum flow is not less than d) can be computed in terms of d-MCs. Jane, Lin and Yuan have presented an algorithm to generate all d-MCs in terms of minimal cuts. This article proposes another algorithm, which is based on the more publicly accepted “comparison” method, to evaluate the reliability. To compare such two algorithms, one example is illustrated to show how d-MCs are generated differently. Also, the computational complexity is analyzed.

On performance evaluation of ERP systems with fuzzy mathematics

An enterprise resource planning (ERP) system is a complex network composed of various business processes. This paper proposes a method based on stochastic-flow network model to evaluate the performance of an ERP system depending upon the fuzzy linguistic results of the ERP examination of the users involved. The nodes in the network denote the persons responsible for the business tasks during the processes. The arcs between nodes denote the process precedence relationships in the ERP system. When the process starts, the documents are initiated from the source node to its succeeding nodes. Finally, the documents are released in the destination node. Thus, the performance of an ERP system is related to the document flow under the network. The failure of an ERP system is therefore described as in the condition that the flow of the system is under the acceptable level d. By using the fuzzy linguistic results of the ERP examination of the users, we propose a fuzzy linguistic performance index, defuzzified from the probability of maximal flow not less than d, to evaluate the performance of an ERP system. An algorithm is subsequently proposed to generate the performance index, which can be used to assess the system performance either before or after the system going live.

System Reliability Evaluation for a Multistate Supply Chain Network With Failure Nodes Using Minimal Paths

This work devotes to the application of network methods for the reliability of a complex supply chain system, which is a set of several factories with supply-demand relationship. Two characters are considered in the proposed network: 1) nodes, and arcs all have multiple possible capacities, and may fail; and 2) the capacity weight varies with arcs, nodes, and types of commodity. The purpose of this paper is to study the systems reliability, in this case the possibility that a given quantity (d 1,d 2) of two types of commodities can be transmitted from the source factory to the destination factory simultaneously. Such a possibility can be treated as a performance index to measure the quality level of a supply chain network. The flow model is constructed by flow assignments, and capacity vectors. The (d 1,d 2) -MP, which represents the capacity of each arc/node, is a minimal capacity vector meeting the demand constraint. A simple algorithm in terms of minimal paths is first proposed to generate all (d 1,d 2)-MP. The system reliability can then be calculated efficiently in terms of (d 1,d 2)-MP. Time complexity of the proposed algorithm is also analyzed.

Study on the multicommodity reliability of a capacitated-flow network☆

Abstract 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.

Multi-objective optimization for stochastic computer networks using NSGA-II and TOPSIS

Network reliability is a performance indicator of computer/communication networks to measure the quality level. However, it is costly to improve or maximize network reliability. This study attempts to maximize network reliability with minimal cost by finding the optimal transmission line assignment. These two conflicting objectives frustrate decision makers. In this study, a set of transmission lines is ready to be assigned to the computer network, and the computer network associated with any transmission line assignment is regarded as a stochastic computer network (SCN) because of the multistate transmission lines. Therefore, network reliability means the probability to transmit a specified amount of data successfully through the SCN. To solve this multiple objectives programming problem, this study proposes an approach integrating Non-dominated Sorting Genetic Algorithm II (NSGA-II) and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS). NSGA-II searches for the Pareto set where network reliability is evaluated in terms of minimal paths and Recursive Sum of Disjoint Products (RSDP). Subsequently, TOPSIS determines the best compromise solution. Several real computer networks serve to demonstrate the proposed approach.