Terms Of Minimal Cuts Research Articles

An efficient algorithm for the reliability evaluation of multistate flow networks under budget constraints

Many real-world systems can be modeled by multi-state flow networks (MFNs) and their reliability evaluation features in designing and control of these systems. Considering the cost constraint makes the problem of reliability evaluation of an MFN more realistic. For a given demand value d and a given cost limit c, the reliability of an MFN at level (d, c) is the probability of transmitting at least d units from the source node to the sink node through the network within the cost of c. This article addresses the so-called (d, c)-MC problem, i.e., the problem of reliability evaluation of an MFN with cost constraint in terms of minimal cuts. It presents new results on which a new algorithm is based. This algorithm finds all (d, c)-MC candidates without duplicates and verifies them more efficiently than existing ones. The complexity results for this algorithm and an example of its use are provided. Finally, numerical experiments with R language implementations of the presented algorithm and other competitive algorithms are considered. Both the time complexity analysis and numerical experiments demonstrate the presented algorithm to be more efficient than the fastest competing algorithms in 81.41–85.11% of cases.

An Algorithm to Search for All Minimal Cuts in a Flow Network

Reliability evaluation approaches exploit a variety of tools for system modeling and calculation of reliability indices. Most proposed approaches in the literature are in terms of minimal cuts (MCs) or minimal paths. Hence, finding all the MCs in a network is an attractive problem in network flow and reliability theory. Here, investigating some works related to the MC problem, we first point out a number of defects in certain proposed algorithms, and present certain new techniques to rectify the flaws in two algorithms. Moreover, we present some techniques to improve the algorithms. Using our results, we propose an improved algorithm for finding all the MCs in a flow network. A benchmark network example is used to illustrate the performance of the algorithm. The proposed algorithm is shown to be correct. Finally, after establishing the complexity results, we demonstrate the proposed algorithm to be more efficient than some existing algorithms.

An MP-based approximation algorithm on reliability evaluation of multistate flow networks

In recent decades, multistate two-terminal reliability problem has attracted several researchers, and accordingly many exact and approximation approaches have been proposed in the literature in terms of minimal cuts (MCs) or minimal paths (MPs) to address this problem. Here, an MP-based approximation approach is developed based on exact algorithms. With all the MPs at hand, the approach rearranges the MPs ascendingly with respect to their lengths and then sets the flow on some MPs to be zero which turns to reduce the computing cost in solving the problem. We provide the complexity results, and by employing some benchmarks and one thousand randomly generated networks illustrate that not only in many cases the proposed approach determines very good approximate solutions much faster than the exact algorithms, but also in many other cases it even determines exact solutions significantly faster than the available exact algorithms in the literature. Moreover, the Dolan-Moré performance profile affirms the efficiency of our proposed algorithm. Finally, we state how to compute the system reliability by using the d-MPs, and show that from a very good approximation set of d-MPs, the system reliability is approximated with a very good accuracy.

Assessing Reliability of Multistate Flow Networks Under Cost Constraint in Terms of Minimal Cuts

Assessing the reliability of a multistate flow network (MFN) has been significantly attractive in recent years. Generally, some assumptions such as budget or time limits are considered in this problem to make it more realistic. Considering the cost constraint, the network reliability at level [Formula: see text] is the probability of transmitting at least [Formula: see text] units of product from a source node to a destination node through the network within the cost of [Formula: see text], where [Formula: see text] is a given demand value. Here, the problem of reliability evaluation of an MFN with cost constraint in terms of minimal cuts is considered. Presenting some techniques to improve the solution, a simple improved algorithm is proposed to address the problem. The complexity results are provided which demonstrate our proposed algorithm to be more efficient than some other existing ones in the literature. Some numerical comparisons between our MATLAB implementation of the proposed algorithm here and a recently proposed one are made. This way, computational comparative results on three benchmarks and a thousand random test problems are provided in the sense of performance profile introduced by Dolan and Moré, which show our proposed algorithm to be more efficient than the other.

An approximate approach for reliability evaluation of a multistate flow network in terms of minimal cuts

An approximate approach, based on exact methods, is proposed to assess reliability of the multistate flow networks in terms of minimal cuts (MCs). The method selects some MCs to remove from the reliability evaluation process, reducing the computing cost and resulting in a very good approximation. Several benchmarks have been employed to show the efficiency of the proposed technique. The numerical experiments show that at times our approach finds even the exact solution more than 3 or 4 times faster, and in other cases it obtains a very good approximation more than 114 times faster than the available exact algorithms in the literature.

Evaluating the reliability of a stochastic distribution network in terms of minimal cuts

This paper presents a d-minimal cut based algorithm to evaluate the performance index Rd+1 of a distribution network, defined as the probability that a specified demand d+1 can be successfully distributed through stochastic arc capacities from the source to the destination. To improve the efficiency of solving d-minimal cuts, a novel technique is developed to determine the minimal capacities of arcs. Also, two new judging criteria are proposed to detect duplicate d-minimal cuts. Both theoretical and computational results indicate that our algorithm outperforms the existing methods. Furthermore, a real case study is provided to illustrate the application of the algorithm.

A Comparative Study of Different Approaches for Finding the Upper Boundary Points in Stochastic-Flow Networks

An information system network (ISN) can be modeled as a stochastic-flow network (SFN). There are several algorithms to evaluate reliability of an SFN in terms of Minimal Cuts (MCs). The existing algorithms commonly first find all the upper boundary points (called d-MCs) in an SFN, and then determine the reliability of the network using some approaches such as inclusion-exclusion method, sum of disjoint products, etc. However, most of the algorithms have been compared via complexity results or through one or two benchmark networks. Thus, comparing those algorithms through random test problems can be desired. Here, the authors first state a simple improved algorithm. Then, by generating a number of random test problems and implementing the algorithms in MATLAB, the proposed algorithm is demonstrated to be more efficient than some existing ones in medium-sized networks. The performance profile introduced by Dolan and More is used for analyzing the output of programs.

Characterizing compatibility and agreement of unrooted trees via cuts in graphs.

BackgroundDeciding whether there is a single tree —a supertree— that summarizes the evolutionary information in a collection of unrooted trees is a fundamental problem in phylogenetics. We consider two versions of this question: agreement and compatibility. In the first, the supertree is required to reflect precisely the relationships among the species exhibited by the input trees. In the second, the supertree can be more refined than the input trees.Testing for compatibility is an NP-complete problem; however, the problem is solvable in polynomial time when the number of input trees is fixed. Testing for agreement is also NP-complete, but it is not known whether it is fixed-parameter tractable. Compatibility can be characterized in terms of the existence of a specific kind of triangulation in a structure known as the display graph. Alternatively, it can be characterized as a chordal graph sandwich problem in a structure known as the edge label intersection graph. No characterization of agreement was known.ResultsWe present a simple and natural characterization of compatibility in terms of minimal cuts in the display graph, which is closely related to compatibility of splits. We then derive a characterization for agreement.ConclusionsExplicit characterizations of tree compatibility and agreement are essential to finding practical algorithms for these problems. The simplicity of the characterizations presented here could help to achieve this goal.

Reliability optimization of component assignment problem for a multistate network in terms of minimal cuts

A network constructed by arcs and vertices is a useful tool to model a real-life system, such as a computer/communication, an electric power transmission, a transportation, or a logistics system. Network reliability, the probability to satisfy customers' demand, is a common performance index of such a system. From the quality of service viewpoint, the network reliability optimization is an important objective which many enterprises pursue to maintain their customer satisfaction. Combining with the characteristic of assignment problem, this study is mainly to search for the optimal components assignment with maximal network reliability. A set of components is ready to be assigned to the arcs, each component may have several modes, such as failure, maintenance, or partially reserved by other customers, and thus the network according to any component assignment is multistate. A minimal-cut based genetic algorithm is developed to solve such a problem. In order to show the computational efficiency of the proposed algorithm, a simple computer network and a real one are adopted while comparing with the implicit enumeration method and the approach of random solution generation, respectively.

A stochastic model to study the system capacity for supply chains in terms of minimal cuts

For a single-commodity stochastic flow network, the system capacity is the maximum flow from the source to the sink. We construct a p-commodity stochastic flow network with unreliable nodes, in which branches and nodes all have several possible capacities and may fail, to model a supply chain. Different types of commodities, transmitted through the same network simultaneously, consume the capacities of branches and nodes differently. That is, the capacity weight depends on branches, nodes and types of commodity. We first define the system capacity as a vector and propose a performance index, the probability that the upper bound of the system capacity is a given pattern. Such a performance index can be easily computed in terms of upper boundary states meeting the demand exactly. An efficient algorithm based on minimal cuts is thus presented to generate all upper boundary states. The manager can apply this performance index to measure the transportation level of a supply chain.

An algorithm to generate all upper boundary points for [formula omitted] in terms of minimal cuts

This paper discusses a stochastic-flow network from single-commodity case to multicommodity case. We propose a performance index, namely the probability that the upper bound of the system capacity is a given vector subject to the budget constraint, to evaluate the quality level for such a network. A simple approach based on minimal cuts is presented to generate the all upper boundary points for the demand d subject to the budget B in order to evaluate the performance index.

System performance of a two-commodity stochastic-flow network with cost attributes in terms of minimal cuts

In the single-commodity case, the system capacity of a stochastic-flow network is the maximum flow from the source to the sink. We concentrate on a two-commodity stochastic-flow network (each arc has several possible capacities and may fail) in which each arc has both capacity and cost attributes. Two types of commodities are transmitted through the same network simultaneously and compete for the capacities. The system capacity is defined as a pattern, and a performance index, the probability that the upper bound of the system capacity equals a given pattern under the budget constraint, is proposed to evaluate the system performance. Such a performance index can be computed in terms of the maximal capacity vectors which satisfy both the demand and budget constraints. A simple approach based on minimal cuts is proposed to generate all such maximal capacity vectors.

A two-commodity multistate flow network with capacity weight varying with edges, nodes and types of commodity

In a single-commodity flow network, the system capacity is the maximum flow from the source to the destination. The reliability and unreliability based on system capacity are two appropriate performance indices to measure the quality level of a flow network. This paper extends the performance evaluation problem to a two-commodity multistate flow network composed of multistate components (edges and nodes). Both types of commodity, which are transmitted through the same network simultaneously, consume the capacities of edges and nodes differently. That is, the capacity weight varies with types of commodity, edges and nodes. The maximal total flow is not suitable to be treated as the system capacity in two-commodity case. Hence, we first define the system capacity as a 2-tuple vector and then propose a performance index, the probability that the upper bound of the system capacity is a given pattern. Such a performance index that is used to measure the quality level of a supply–demand system can be easily computed in terms of upper boundary vectors. An efficient algorithm in terms of minimal cuts is thus presented to generate all upper boundary vectors.

Performance index of a stochastic-flow network with node failure under the budget constraint

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 probability that the maximum flow of the network is equal to the demand d without exceeding the budget B. Such a probability is treated as a performance index to measure the quality level of the network. A simple algorithm in terms of minimal cuts is first proposed to generate all upper boundary points for (d, B), and then the probability that the maximal flow is less than or equal to d can be calculated in terms of such points. The upper boundary point for (d, B) is a maximal vector, which represents the capacity of each component (arc or node), such that the maximum flow of the network under the budget B is d. The performance index can be calculated by repeating the proposed algorithm to obtain all upper boundary points for (d−1, B). A computer simulation example is shown to illustrate the solution procedure.

Unreliability evaluation for a limited-flow network with failed nodes subject to the budget constraint

This paper concentrates on a limited-flow network in which each node and branch has a designated capacity, which will have different lower levels due to various partial and complete failures. We try to evaluate the system unreliability that the maximum flow of the network is less than or equal to the demand d without exceeding the budget B. First, a simple algorithm in terms of minimal cuts is proposed to generate all ( d, B)-MCs in order to evaluate the system unreliability. A computer example is shown to illustrate the solution procedure.

Evaluate the performance of a stochastic-flow network with cost attribute in terms of minimal cuts

This paper proposes a performance index to measure the quality level of a stochastic-flow network in which each node has a designated capacity, which will have different lower levels due to various partial and complete failures. The performance index is the probability that the maximum flow of the network equals the demand d without exceeding the budget b. A simple algorithm in terms of minimal cuts is first proposed to generate all upper boundary points for ( d, b), and then the probability that the maximum flow is less than or equal to d can be calculated in terms of such points. The upper boundary point for ( d, b) is a maximal vector representing the capacity of each arc such that the maximum flow of the network under the budget b is d. The performance index can be calculated by repeating the proposed algorithm to obtain all upper boundary point for ( d−1, b). A benchmark example is shown to illustrate the solution procedure.

STUDY ON THE PERFORMANCE INDEX FOR A MULTICOMMODITY STOCHASTIC-FLOW NETWORK IN TERMS OF MINIMAL CUTS

ABSTRACT From the point of view of quality management, it is an important issue to develop some performance indexes for a flow network. This paper studies the performance indexes for a stochastic-flow network in which each arc has several possible capacities. For the case that multiple types of commodity are transmitted through the same stochastic-flow network, this paper defines the system capacity as a vector and then proposes a new performance index. An algorithm in terms of minimal cuts is proposed to evaluate such a performance index.

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

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.

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.