A dynamic bounding algorithm for approximating multi-state two-terminal reliability

This paper modifies Jane and Laih’s (2008) exact and direct algorithm to provide sequences of upper bounds and lower bounds that converge to the NP-hard multi-state two-terminal reliability. Advantages of the modified algorithm include (1) it does not require a priori the lower and/or upper boundary points of the network, (2) it derives a series of increasing lower bounds and a series of decreasing upper bounds simultaneously, guaranteed to enclose the exact reliability value, and (3) trade-off between accuracy and execution time can be made to ensure an exact difference between the upper and lower bounds within an acceptable time. Examples are analyzed to illustrate the bounding algorithm, and to compare the bounding algorithm with existing algorithms. Computational experiments on a large network are conducted to realize the performance of the bounding algorithm.     

Practical sequential bounds for approximating two-terminal reliability

With increasing emphases on better and more reliable services, network systems have incorporated reliability analysis as an integral part in their planning, design and operation. This article first presents a simple exact decomposition algorithm for computing the NP-hard two-terminal reliability, which measures the probability of successful communication from specified source node to sink node in the network. Then a practical bounding algorithm, which is indispensable for large networks, is presented by modifying the exact algorithm for obtaining sequential lower and upper bounds on two-terminal reliability. Based on randomly generated large networks, computational experiments are conducted to compare the proposed algorithm to the well-known and widely used edge-packing approximation model and to explore the performance of the proposed bounding algorithm. Computational results reveal that the proposed bounding algorithm is superior to the edge-packing model, and the trade-off of accuracy for execution time ensures that an exact difference between upper and lower bounds on two-terminal reliability can be obtained within an acceptable time.     

Redundant paths and reliability bounds in gamma networks

Multistage Interconnection Networks (MINs) are network systems providing fast and efficient communications at a reasonable cost. A gamma network is a specific class of MINs, which provides redundant paths in the system. In a gamma network, information from source nodes is transmitted through a specific set of routes to destination nodes. Reliability of an MIN is used as a measure of system’s ability to transform information from input to output devices. Due to the complexity of network configuration and availability of redundant paths, reliability bounds to estimate the exact reliability of a gamma network is proposed. A numerical example of an 8 × 8 gamma network is presented to demonstrate the accuracy of the reliability bounds. When the lower bound reliability provides sufficient assurance that the system will be operational at some specified time and closely approximates the exact reliability, then no further effort for obtaining the exact reliability expression is necessary.     

Incorporating transfer reliability into equilibrium analysis of railway passenger flow

Passenger’s transfer route choice behavior is one of the prominent research topics in the field of railway transportation. Existing traffic assignment approaches do not properly account for passenger’s expectation for transfer reliability. In this study, the transfer reliability is explicitly defined and a multi-class user equilibrium model is established, given which passengers choose the minimal-cost path based on their expected reliability thresholds. In particular, a path-based traffic assignment algorithm which combines a k-shortest path algorithm and the method of successive averages is proposed. The validity of the proposed approach is verified by an illustrative example. Using the proposed modeling approach, it is possible to determine the passenger’s collective route choice behavior based on the user equilibrium pattern. Moreover, the railway timetables can be evaluated and optimized based on the cost-based level of service estimation.     

Symbolic models for infinite networks of control systems: A compositional approach

This paper presents a compositional framework for the construction of symbolic models for a network composed of a countably infinite number of finite-dimensional discrete-time control subsystems. We refer to such a network as infinite network. The proposed approach is based on the notion of alternating simulation functions. This notion relates a concrete network to its symbolic model with guaranteed mismatch bounds between their output behaviors. We propose a compositional approach to construct a symbolic model for an infinite network, together with an alternating simulation function, by composing symbolic models and alternating simulation functions constructed for subsystems. Assuming that each subsystem is incrementally input-to-state stable and under some small-gain type conditions, we present an algorithm for orderly constructing local symbolic models with properly designed quantization parameters. In this way, the proposed compositional approach can provide us a guideline for constructing an overall symbolic model with any desired approximation accuracy. A compositional controller synthesis scheme is also provided to enforce safety properties on the infinite network in a decentralized fashion. The effectiveness of our result is illustrated through a road traffic network consisting of infinitely many road cells.     

A meta-heuristic approach for solving the Urban Network Design Problem

This paper proposes an optimisation model and a meta-heuristic algorithm for solving the urban network design problem. The problem consists in optimising the layout of an urban road network by designing directions of existing roads and signal settings at intersections. A non-linear constrained optimisation model for solving this problem is formulated, adopting a bi-level approach in order to reduce the complexity of solution methods and the computation times. A Scatter Search algorithm based on a random descent method is proposed and tested on a real dimension network. Initial results show that the proposed approach allows local optimal solutions to be obtained in reasonable computation times.     

A penalty function heuristic for the resource constrained shortest path problem

The resource constrained shortest path problem (RCSP) consists of finding the shortest path between two nodes of an assigned network, with the constraint that traversing an arc of the network implies the consumption of certain limited resources. In this paper we propose a new heuristic for the solution of the RCSP problem in medium and large scale networks. It is based on the extension to the discrete case of the penalty function heuristic approach for the fast ε-approximate solution of difficult large-scale continuous linear programming problems. Computational experience on test instances has shown that the proposed penalty function heuristic (PFH) is very effective in the solution of medium and large scale RCSP instances. For all the tests reported it provides very good upper bounds (in many cases the optimal solution) in less than 26 iterations, where each iteration requires only the computation of a shortest path.     

Maximizing the Net Present Value of a Project Under Resource Constraints Using a Lagrangian Relaxation Based Heuristic with Tight Upper Bounds

Resource-constrained project scheduling under a net present value objective attracts growing interest. Because this is an NP-hard problem, it is unlikely that optimum solutions can be computed for large instances within reasonable computation time. Thus, heuristics have become a popular research field. Up to now, however, upper bounds are not well researched. Therefore, most researchers evaluate their heuristics on the basis of a best known lower bound, but it is unclear how good the performance really is. With this contribution we close this gap and derive tight upper bounds on the basis of a Lagrangian relaxation of the resource constraints. We also use this approach as a basis for a heuristic and show that our heuristic as well as the cash flow weight heuristic proposed by Baroum and Patterson yield solutions very close to the optimum result. Furthermore, we discuss the proper choice of a test-bed and emphasize that discount rates must be carefully chosen to give realistic instances.     

Algorithmic improvements on dynamic programming for the bi-objective {0,1} knapsack problem

This paper presents several methodological and algorithmic improvements over a state-of-the-art dynamic programming algorithm for solving the bi-objective {0,1} knapsack problem. The variants proposed make use of new definitions of lower and upper bounds, which allow a large number of states to be discarded. The computation of these bounds are based on the application of dichotomic search, definition of new bound sets, and bi-objective simplex algorithms to solve the relaxed problem. Although these new techniques are not of a common application for dynamic programming, we show that the best variants tested in this work can lead to an average improvement of 10 to 30 % in CPU-time and significant less memory usage than the original approach in a wide benchmark set of instances, even for the most difficult ones in the literature.     

Modelling redundancy allocation for a fuzzy random parallel-series system

Due to subjective judgment, imprecise human knowledge and perception in capturing statistical data, the real data of lifetimes in many systems are both random and fuzzy in nature. Based on the fuzzy random variables that are used to characterize the lifetimes, this paper studies the redundancy allocation problems to a fuzzy random parallel-series system.Two fuzzy random redundancy allocation models (FR-RAM) are developed through reliability maximization and cost minimization, respectively. Some properties of the FR-RAM are obtained, in which an analytical formula of reliability with convex lifetimes is derived and the sensitivity of the reliability is discussed. To solve the FR-RAMs, we first address the computation of reliability. A random simulation method based on the derived analytical formula is proposed to compute the reliability with convex lifetimes. As for the reliability with nonconvex lifetimes, the technique of fuzzy random simulation together with the discretization method of fuzzy random variable is employed to compute the reliability, and a convergence theorem of the fuzzy random simulation is proved. Subsequently, we integrate the computation approaches of the reliability and genetic algorithm (GA) to search for the approximately optimal redundancy allocation of the models. Finally, some numerical examples are provided to illustrate the feasibility of the solution algorithm and quantify its effectiveness.     

A cross entropy approach to design of reliable networks

One of the most important parameters determining the performance of communication networks is network reliability. The network reliability strongly depends on not only topological layout of the communication networks but also reliability and availability of the communication facilities. The selection of optimal network topology is an NP-hard problem so that computation time of enumeration-based methods grows exponentially with network size. This paper presents a new solution approach based on cross-entropy method, called NCE, to design of communication networks. The design problem is to find a network topology with minimum cost such that all-terminal reliability is not less than a given level of reliability. To investigate the effectiveness of the proposed NCE, comparisons with other heuristic approaches given in the literature for the design problem are carried out in a three-stage experimental study. Computational results show that NCE is an effective heuristic approach to design of reliable networks.     

Performance assessment and reliability analysis of dependable and distributed computing systems based on BDD and recursive merge

System reliability evaluation, sensitivity analysis, failure frequency analysis, importance measures, and optimal design are important issues that have become research topics for distributed dependable computing. Finding all of the Minimal File Spanning Trees (MFST) and avoiding repeatedly computing the redundant MFSTs have been key techniques for evaluating the reliability of a distributed computing system (DCS) in previous works. However, identifying all of the disjointed MFSTs is difficult and time consuming for large-scale networks. Although existing algorithms have been demonstrated to work well on medium-scale networks, they have two inherent drawbacks. First, they do not support efficient manipulation of Boolean algebra. The sum-of-disjoint-products method used by these algorithms is inefficient when dealing with large Boolean functions. Second, the tree-based partitioning algorithm does not merge isomorphic sub-problems, and therefore, redundant computations cannot be avoided. In this paper, we propose a new efficient algorithm for the reliability evaluation of a DCS based on the recursive merge and the binary decision diagram (BDD). Using the BDD substitution method, we can easily apply our algorithm to a network with imperfect nodes. The experimental results show a significant improvement in the execution time compared to previous works.     

Probability chains: A general linearization technique for modeling reliability in facility location and related problems

In this paper, we propose an efficient technique for linearizing facility location problems with site-dependent failure probabilities, focusing on the unreliable p-median problem. Our approach is based on the use of a specialized flow network, which we refer to as a probability chain, to evaluate compound probability terms. The resulting linear model is compact in size. The method can be employed in a straightforward way to linearize similarly structured problems, such as the maximum expected covering problem. We further discuss how probability chains can be extended to problems with co-location and other, more general problem classes. Additional lower bounds as well as valid inequalities for use within a branch and cut algorithm are introduced to significantly speed up overall solution time. Computational results are presented for several test problems showing the efficiency of our linear model in comparison to existing problem formulations.     

基于鲁棒优化的随机时变网络最优路径研究

交通事故、恶劣天气以及偶发的交通拥堵等都会导致道路交通网络中行程时间的不确定性,极大地影响了道路交通系统的可靠性,同时给日常生活中出行计划的制定以及出行路径的选择带来了不便。因此,本次研究将综合考虑道路交通网络中由于交通流量的全天变化所导致的路径行程时间的时变特征,以及由于事故、天气等不确定因素所导致的路径行程时间的随机特征,并以此作为路网环境的假设条件,对出行路径选择问题进行研究。具体地,首先建立行程时间的动态随机变量,并在此基础上模拟构建了随机时变网络。随后,定义了该网络环境下路径选择过程中所考虑的成本费用,并通过鲁棒优化的方法,将成本费用鲁棒性最强的路径视为最优路径。随后,在随机一致性条件下,通过数学推导证明了该模型可以简化为解决一个确定性时变网络中的最短路径问题。最终,具有多项式时间计算复杂度的改进Dijkstra算法被应用到模型的求解中,并通过小型算例验证模型及算法的有效性。结果表明,本研究中所提出的方法可以被高效率算法所求解,并且不依赖于先验行程时间概率分布的获取,因此对后续的大规模实际城市道路网络应用提供了良好的理论基础。此外,由于具有行程时间随机时变特征的交通网络更接近实际道路情况,因此本次研究的研究成果具有较高的实际意义和应用价值。     

Supporting reliability engineers in exploiting the power of Dynamic Bayesian Networks

In this paper, we present an approach to reliability modeling and analysis based on the automatic conversion of a particular reliability engineering model, the Dynamic Fault Tree (DFT), into Dynamic Bayesian Networks (DBN). The approach is implemented in a software tool called RADYBAN (Reliability Analysis with DYnamic BAyesian Networks). The aim is to provide a familiar interface to reliability engineers, by allowing them to model the system to be analyzed with a standard formalism; however, a modular algorithm is implemented to automatically compile a DFT into the corresponding DBN. In fact, when the computation of specific reliability measures is requested, classical algorithms for the inference on Dynamic Bayesian Networks are exploited, in order to compute the requested parameters. This is performed in a totally transparent way to the user, who could in principle be completely unaware of the underlying Bayesian Network. The use of DBNs allows the user to be able to compute measures that are not directly computable from DFTs, but that are naturally obtainable from DBN inference. Moreover, the modeling capabilities of a DBN, allow us to extend the basic DFT formalism, by introducing probabilistic dependencies among system components, as well as the definition of specific repair policies that can be taken into account during the reliability analysis phase. We finally show how the approach operates on some specific examples, by describing the advantages of having available a full inference engine based on DBNs for the requested analysis tasks.     

Two-stage network constrained robust unit commitment problem

For a current deregulated power system, a large amount of operating reserve is often required to maintain the reliability of the power system using traditional approaches. In this paper, we propose a two-stage robust optimization model to address the network constrained unit commitment problem under uncertainty. In our approach, uncertain problem parameters are assumed to be within a given uncertainty set. We study cases with and without transmission capacity and ramp-rate limits (The latter case was described in Zhang and Guan (2009), for which the analysis part is included in Section 3 in this paper). We also analyze solution schemes to solve each problem that include an exact solution approach and an efficient heuristic approach that provides tight lower and upper bounds for the general network constrained robust unit commitment problem. The final computational experiments on an IEEE 118-bus system verify the effectiveness of our approaches, as compared to the nominal model without considering the uncertainty.     

A new operational vector approach for time-fractional subdiffusion equations of distributed order based on hybrid functions

This research study deals with the numerical solutions of linear and nonlinear time-fractional subdiffusion equations of distributed order. The main aim of our approach is based on the hybrid of block-pulse functions and shifted Legendre polynomials. We produce a novel and exact operational vector for the fractional Riemann–Liouville integral and use it via the Gauss–Legendre quadrature formula and collocation method. Consequently, we reduce the proposed equations to systems of equations. The convergence and error bounds for the new method are investigated. Six problems are tested to confirm the accuracy of the proposed approach. Comparisons between the obtained numerical results and other existing methods are provided. Numerical experiments illustrate the reliability, applicability, and efficiency of the proposed method.     

A simple algorithm to search for all MCs in networks

Evaluating the network reliability is an important topic in the planning, designing, and control of systems. The minimal cut (MC, an edge set) set is one of the major and fundamental tools for evaluating the network reliability. In this study, an alternative method is given to define a MC using a node set (called MCV). A very simple algorithm based on some intuitive theorems that characterize the structure of the MCV and the relationship between MC and MCV is developed to find the MCs between two special nodes. The proposed algorithm is then generalized to find all MCs between all pairs of nodes. The proposed algorithm is not only easier to understand and implement, but is also better than the existing best-known algorithm. The correctness of the proposed algorithm will be analyzed and proven. One example is illustrated to show how all MCs are generated and verified in a network using the proposed algorithm.     

Interactive Multiple Criteria Decision Making based on preference driven Evolutionary Multiobjective Optimization with controllable accuracy

We present an approach to interactive Multiple Criteria Decision Making based on preference driven Evolutionary Multiobjective Optimization with controllable accuracy.The approach relies on formulae for lower and upper bounds on coordinates of the outcome of an arbitrary efficient variant corresponding to preference information expressed by the Decision Maker. In contrast to earlier works on that subject, here lower and upper bounds can be calculated and their accuracy controlled entirely within evolutionary computation framework. This is made possible by exploration of not only the region of feasible variants - a standard within evolutionary optimization, but also the region of infeasible variants, the latter to our best knowledge being a novel approach within Evolutionary Multiobjective Optimization.To illustrate how this concept can be applied to interactive Multiple Criteria Decision Making, two algorithms employing evolutionary computations are proposed and their usefulness demonstrated by a numerical example.     

Computational algorithms for hierarchically structured project networks

A new approach to large size network management based on a structured network representation is proposed. Definitions of arc-structured and vertex-structured networks are given and an extension of topological sorting and time computation algorithms is described. A generalization of the concept of vertex-structured networks is discussed.