A note on solving a system reliability problem

This paper is concerned with solving a constrained system reliability problem from the literature, which is to maximize a nonlinear reliability function subject to two linear constraints. The variables are positive integers, representing the allocations of the stages. By running on an ordinary personal computer the computer program listed in this paper, the best known solution for a widely-known 15-stage example from the literature was obtained thirteen times among the first eighty candidate solutions of one computer run. The computational results presented in this paper suggest that the computer program listed here can be useful as a model for solving the constrained system reliability problem.     

Techniques for Optimum Spares Allocation: A Tutorial Review

In many modern complex systems the problem of achieving high reliability leads to the use of interchangeable modular components accompanied by a stock of spare parts. This paper examines, compares, and assesses several of the techniques presented in the literature for allocating the numbers of spares of each part type to be stocked in order to maximize the system reliability subject to constraints on resources (i.e., weight, volume, cost, etc.). The problem of optimum spares allocation is complicated since resources are consumed in a discrete fashion and the expression for the system reliability is a nonlinear transcendental function. The classical dynamic programming algorithm produces all optimal spares allocations; however, the solution can become computationally intractable even with the aid of a modern high-speed digital computer. In the case of multiple constraints the time problem is vastly exacerbated. In such a case one must turn to a procedure that yields a near-optimal solution in a reasonable amount of computer time. Two approximate methods discussed in this paper are the incremental reliability per pound algorithm and the Lagrange multiplier algorithm. These algorithms are readily adaptable to handle multiple constraints. Computer programs are developed for each of the three optimization algorithms and are utilized to obtain the spares allocation for a few systems. The optimization theory presented is directly applicable to series or parallel systems. A concluding example illustrates how this can be extended to certain series-parallel systems.     

Determining Component Reliability and Redundancy for Optimum System Reliability

The usual constrained reliability optimization problem is extended to include determining the optimal level of component reliability and the number of redundancies in each stage. With cost, weight, and volume constraints, the problem is one in which the component reliability is a variable, and the optimal trade-off between adding components and improving individual component reliability is determined. This is a mixed integer nonlinear programming problem in which the system reliability is to be maximized as a function of component reliability level and the number of components used at each stage. The model is illustrated with three general non linear constraints imposed on the system. The Hooke and Jeeves pattern search technique in combination with the heuristic approach by Aggarwal et al, is used to solve the problem. The Hooke and Jeeves pattern search technique is a sequential search routine for maximizing the system reliability, RS (R, X). The argument in the Hooke and Jeeves pattern search is the component reliability, R, which is varied according to exploratory moves and pattern moves until the maximum of RS (R, X) is obtained. The heuristic approach is applied to each value of the component reliability, R, to obtain the optimal number of redundancies, X, which maximizes RS (R, X) for the stated R.     

A New Technique to Optimize System Reliability

This paper describes an algorithm for solving reliability optimization problems formulated as nonlinear binary programming problems with multiple-choice constraints. These constraints stand for restrictions in which only one variable is assigned to each subset making up the set; thus, they are expressed by equations whose r.h.s. is unity. Different types of methods for achieving high reliability (an increase in component reliability, parallel redundancy, standby redundancy, etc.) can be easily used simultaneously as design alternatives for each subsystem. In order to solve the problem effectively, the Lawler & Bell algorithm is improved by introducing a new lexicographic enumeration order which always satisfies the multiple-choice constraints. The function for obtaining feasible solutions which give first ~ L-th minimum values of the objective function is added to the algorithm in order to make it more useful for decision making. After a numerical example assists in understanding the algorithm, the computational efficiency is compared with that of the Lawler & Bell algorithm.     

Reliability Optimization by Generalized Lagrangian-Function and Reduced-Gradient Methods

Nonlinear optimization problems for reliability of a complex system are solved using the generalized Lagrangian function (GLF) method and the generalized reduced gradient (GRG) method. GLF is twice continuously differentiable and closely related to the generalized penalty function which includes the interior and exterior penalty functions as a special case. GRG generalizes the Wolfe reduced gradient method and has been coded in FORTRAN title ``GREG' by Abadie et al. Two system reliability optimization problems are solved. The first maximizes complex-system reliability with a tangent cost-function; the second minimizes the cost, with a minimum system reliability. The results are compared with those using the Sequential Unconstrained Minimization Technique (SUMT) and the direct search approach by Luus and Jaakola (LJ). Many algorithms have been proposed for solving the general nonlinear programming problem. Only a few have been demonstrated to be effective when applied to large-scale nonlinear programming problems, and none has proved to be so superior that it can be classified as a universal algorithm. Both GLF and GRG methods presented here have been successfully used in solving a number of general nonlinear programming problems in a variety of engineering applications and are better methods among the many algorithms.     

Reliability Optimization of a Series-Parallel System

A mathematical model is formulated for optimizing the reliability of a system subject to given linear constraints; the system has several stages in series; each stage has parallel redundancy to improve the reliability. Part I shows a new way to transform the model of constrained optimization to a saddle point problem by using Lagrange multipliers. Conditions are derived for maximizing the reliability function; Newton's method is used to solve the resulting multidimensional nonlinear algebraic equations. Further modifications are provided to avoid inverting the large Jacobian matrices; therefore this method is practical for large systems. Part II shows how to transform the model of constrained optimization to a multistage decision process and uses the Maximum principle to arrive at the optimal decision. This approach is easy to apply, formulate, and program. The solution can be obtained without fear of nonconvergence (very often experienced with earlier methods) besides providing considerable saving in computer time. Design alternatives can be easily considered.     

Redundancy Optimization Through Simplex Pattern Search

Optimal allocation of redundancy in a series system with separately maintained repairable subsystems subject to multiple constraints is investigated for maximizing system reliability using simplex pattern search and separable programming. The constraints, besides the usual ones, include one based on s-expected busy periods of the maintenance facilities. A numerical example illustrates the method.     

Exact approach to reliability of wireless mesh networks with directional antennas

The paper is devoted to modeling and optimization of reliable wireless mesh networks that employ directional antennas. We introduce two mixed-integer programming formulations that allow to simultaneously characterize routing patterns and transmission schedules. The first model allows for maximizing the minimal flow in a network. The second model involves reliability constraints and aims at minimizing the number of used directional antennas. In both cases locations of mesh routers are known. However, the number of installed radio interfaces and their directions are subject to optimization. We discuss a way of solving a cost minimization problem based on the introduced characterization, and present an extensive numerical study that illustrates the efficiency of the solution algorithm. We also provide an algorithm capable of verifying feasibility of obtained solutions. Moreover, in rare cases of failed verification, the algorithm provides additional constraints that should be added to the problem.     

Systems Reliability Subject to Multiple Nonlinear Constraints

A simple computational procedure has been developed for maximizing reliability of multistage parallel systems subject to multiple nonlinear constraints. It appears that the procedure can be applied to a variety of optimization problems with separable objective and multiple constraint functions.     

A recurrent neural network for nonlinear convex optimization subject to nonlinear inequality constraints

This paper presents a novel recurrent neural network for solving nonlinear convex programming problems subject to nonlinear inequality constraints. Under the condition that the objective function is convex and all constraint functions are strictly convex or that the objective function is strictly convex and the constraint function is convex, the proposed neural network is proved to be stable in the sense of Lyapunov and globally convergent to an exact optimal solution. Compared with the existing neural networks for solving such nonlinear optimization problems, the proposed neural network has two major advantages. One is that it can solve convex programming problems with general convex inequality constraints. Another is that it does not require a Lipschitz condition on the objective function and constraint function. Simulation results are given to illustrate further the global convergence and performance of the proposed neural network for constrained nonlinear optimization.     

Optimal apportionment of reliability and redundancy in seriessystems under multiple objectives

A multiobjective reliability apportionment problem for a series system with time-dependent reliability is presented. The resulting mathematical programming formulation determines the optimal level of component reliability and the number of redundant components at each stage. The problem is a multiobjective, nonlinear, mixed-integer mathematical programming problem, subject to several design constraints. Sequential unconstrained minimization techniques in conjunction with heuristic algorithms are used to find an optimum solution. A generalization of the problem in view of inherent vagueness in the objective and the constraint functions results in an ill-structured reliability apportionment problem. This multiobjective fuzzy optimization problem is solved using nonlinear programming. The computational procedure is illustrated through a numerical example. The fuzzy optimization techniques can be useful during initial stages of the conceptual design of engineering systems where the design goals and design constraints have not been clearly identified or stated, and for decision making problems in ill-structured situations     

An algorithm to solve integer programming problems: An efficient tool for reliability design

In many reliability design problems, the decision variables can only have integer values. The redundancy allocation is an example of one such problem; others include spare parts allocation, or repairmen allocation, which necessitate an integer programming formulation. In other words, integer programming plays an important role in system reliability optimization. In this paper, an algorithm is presented which provides an exact, simple and economical solution to any general class of integer programming problems and thereby offers reliability designers an efficient tool for system design. The algorithm can be used effectively to solve a wide variety of reliability design problems. The scope of use of this algorithm is also indicated and the procedure is illustrated by an example.     

Redundancy Allocation for Series-Parallel Systems Using Integer Linear Programming

We consider the problem of maximizing the reliability of a series-parallel system given cost and weight constraints on the system. The number of components in each subsystem, and the choice of components are the decision variables. In this paper, we propose an integer linear programming approach that gives an approximate feasible solution, close to the optimal solution, together with an upper bound on the optimal reliability. We show that integer linear programming is a useful approach for solving this reliability problem. The mathematical programming model is relatively simple. Its implementation is immediate by using a mathematical programming language, and integer linear programming software. And the computational experiments show that the performance of this approach is excellent based on a comparison with previous results.      

Directly performance-constrained template-based layout retargeting and optimization for analog integrated circuits

Due to intrinsic intricacy, layout parasitics exhibit a significant impact on the performance of analog integrated circuits. In this paper a directly performance-constrained parasitic-aware automatic layout retargeting and optimization algorithm is presented. Unlike the conventional sensitivity analysis, a general central-difference based scheme using any simulator for sensitivity computation is deployed. We propose a piecewise sensitivity model to enforce more accurate sensitivity computation during parasitic optimization. Moreover, mixed-integer performance constraints due to parasitics are included in the formulated mixed integer nonlinear programming problem rather than through either indirect parasitic-bound constraints or inaccurate worst-case sensitivities. A graph technique and mixed-integer nonlinear programming are effectively combined to solve the formulated parasitic optimization problem. The automatically generated target layouts can satisfy performance constraints to ensure the desired specifications. The experimental results show that the proposed algorithm can achieve effective retargeting of analog circuits with less layout area and significant reduction in execution time.     

非完备杂波先验知识场景下MIMO-STAP稳健波形设计

本文考虑了色高斯干扰条件下MIMO STAP稳健波形优化问题以提高非完备杂波先验知识条件下多输入多输出（MIMO）雷达体制下空时自适应处理（STAP）最坏情况下探测性能。由于高斯干扰（包括杂波、干扰以及热噪声）场景下最大化系统输出信干噪比（SINR）等价于最大化MIMO STAP检测性能,因而在本文所建立杂波协方差估计误差的模型基础上,总功率发射以及参数不确定凸集约束下,经推导可得稳健波形优化问题。为求解得到的复杂非线性问题,本文提出了一种迭代算法以优化发射波形相关阵（WCM）从而最大化凸不确定集上最差情况下的输出SINR进而改善最差情况下MIMO STAP的检测性能。通过利用对角加载（DL）方法,所提算法中的每个迭代步骤皆可表示为能获得高效求解的半定规划（SDP）问题。与非稳健方法及非相关波形相比,数值实验验证了本文所提方法的有效性。      

基于效用最大化的IEEE802.16带宽分配算法

由于IEEE802.16无线城域网协议并未给出网络带宽分配算法或建议,该文提出将802.16服务流带宽分配纳入统一的对数效用函数模型,使问题转化为效用最优化下的非线性规划(NP)求解。同时针对实际应用的实时性要求,提出了适用于对数效用函数的快速解法,使NP问题可以用线性运算解决,大大降低了计算复杂度。仿真结果表明,效用最优化算法比max-min公平算法在吞吐量和效用上均具有明显的优势,还可以灵活地改变效用函数参数,在不同服务质量(QoS)要求下高效地做出分配。     

Optimal channel assignment in cellular networks

In this paper, we consider the problem of assigning frequencies to mobile terminals in a cellular network. We show that an optimal solution can be obtained by solving a sequence of alternating linear and quadratic maximization programming problems. We address co-channel constraints and adopt as an objective function the maximization of potentially established calls. Our algorithm is fairly general, and does not depend on any special network structure. This study indicates that mathematical programming can be used as an efficient technique for solving the aforementioned problem.     

Reliability‐aware service chaining mapping in NFV‐enabled networks

Network function virtualization can significantly improve the flexibility and effectiveness of network appliances via a mapping process called service function chaining. However, the failure of any single virtualized network function causes the breakdown of the entire chain, which results in resource wastage, delays, and significant data loss. Redundancy can be used to protect network appliances; however, when failures occur, it may significantly degrade network efficiency. In addition, it is difficult to efficiently map the primary and backups to optimize the management cost and service reliability without violating the capacity, delay, and reliability constraints, which is referred to as the reliability‐aware service chaining mapping problem. In this paper, a mixed integer linear programming formulation is provided to address this problem along with a novel online algorithm that adopts the joint protection redundancy model and novel backup selection scheme. The results show that the proposed algorithm can significantly improve the request acceptance ratio and reduce the consumption of physical resources compared to existing backup algorithms.     

Energy‐Efficiency Power Allocation for Cognitive Radio MIMO‐OFDM Systems

This paper studies energy‐efficiency (EE) power allocation for cognitive radio MIMO‐OFDM systems. Our aim is to minimize energy efficiency, measured by “Joule per bit” metric, while maintaining the minimal rate requirement of a secondary user under a total power constraint and mutual interference power constraints. However, since the formulated EE problem in this paper is non‐convex, it is difficult to solve directly in general. To make it solvable, firstly we transform the original problem into an equivalent convex optimization problem via fractional programming. Then, the equivalent convex optimization problem is solved by a sequential quadratic programming algorithm. Finally, a new iterative energy‐ efficiency power allocation algorithm is presented. Numerical results show that the proposed method can obtain better EE performance than the maximizing capacity algorithm.     

Surrogate Constraints Algorithm for Reliability Optimization Problems with Multiple Constraints

This paper presents a surrogate constraints algorithm for solving nonlinear programming, nonlinear integer programming, and nonlinear mixed integer programming problems. The algorithm contains a new technique for generating a succession of vector values of surrogate multiplier (ie, surrogate problems). By using this technique, a computer can keep a polyhedron, which is a vector space of surrogate multipliers to be considered at a certain time, in its memory. Furthermore it can cut the polyhedron by a given hyperplane, and produce the remaining space as the next polyhedron. Simple examples are included.