Reliability Optimization of Complex Systems Using Genetic Algorithm Under Criticality Constraint SAMER HAMED*
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BELAL AYYOUB
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NAWAL ALZABIN
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Computer Engineering Deptartement, Faculty of Engineering Technology Al-Balqa‟a Applied University Amman-JORDAN
Abstract: Reliability designers often try to achieve highly reliable systems. The issue of reliability optimization of complex (non-series, non-parallel) systems is investigated in this paper. Maximization of system reliability is usually subject to link‟s criticality and cost constraints. This is referred to as reliability optimization problem (ROP). A procedure, that determines the maximal reliability of the complex system, is developed and proposed in this paper, with the system links are maximized according to their criticalities. A criticality-based Genetic Algorithm is developed and used to evaluate the optimal reliability of complex systems. It is then employed to reach the optimal solution of such systems by minimizing their total cost, and increasing their overall reliability. Numerical experiments are conducted to show the effectiveness and robustness of the proposed approach when applied to complex systems. Key-Words-: System reliability, Complex systems, Criticality-based Genetic Algorithm, Link‟s criticality.
1 INTRODUCTION
System reliability is defined as the probability of the system to perform its intended function for a specified period of time under certain stated conditions [1]. Many modern systems, both hardware and software, are characterized by a high degree of complexity. Enhancement of the reliability of such systems requires the definition of certain techniques and models that leads to achieve optimum design of the system itself. This paper presents a new metaheuristic-based algorithm that aims to tackle the reliability problem of the system. This is required to identify the system configuration that maximizes its overall reliability, while taking into account a set of resource constraints. Estimating the reliability of the system is an important and challenging problem for system engineers [2]. It is challenging since current estimation techniques require a high level of background in system reliability analysis, and thus familiarity with the system. Traditionally, engineers estimate the reliability of systems by understanding the way by which the different links of the system interact to guarantee higher system reliability. Based on this understanding, a graphical model (usually in the form of a fault tree, a reliability block diagram or a network graph) is typically used to show how system links interaction affects its behavior. Once the graphical model is developed, different analysis methods, such as minimal cut sets, minimal path sets, Boolean truth tables, etc., can be used to quantitatively represent system reliability [3][4]. The estimated reliability characteristics of the links in the system are then introduced into the mathematical model in order to obtain system-level reliability. The above traditional perspective aims to provide accurate predictions about the system reliability using historical or test data. This approach is proved to be valid whenever * Corresponding Author 1
the system success or failure behavior is well understood. In their paper, Yinong Chen, Zhongshi He, Yufang Tian [5], system reliability is classified in to topological and flow reliability. The system is considered to consist of a set of computing nodes and links between the nodes. It is also assumed that the links are reliable, while the nodes may fail with certain probability. In the current paper, it is assumed that the links are subject to failure in a topological reliability. Ideally, one would like to generate system design algorithms that consider, as input, the characteristics of system links, as well as its criteria, and produce, as output, an optimal system design. This is known as system synthesis [6], which is very difficult to achieve. Instead, a system that is already designed is considered in order to try to improve its design by maximizing the links reliability which will maximize the over all system reliability. In the most theoretical reliability problems, two basic methods are followed to improve the reliability of the system. The first is by improving the reliability of each system links, and the second is by adding redundant links [7]. Since the second method is more expensive, the first one is followed in this paper. The aim of this paper is to obtain the optimal system reliability design with the following constrains: (a) Basic linear-cost reliability relationship is used for each system component [7]. (b) Criticality of links [8]. The designer of the system should take this into consideration before building a reliable system. However, the position of a component plays an important role for its criticality, defined as criticality index. Reliability increase will go toward the most critical component. Links‟ criticality can be derived from its failure effects to system reliability failure. 2 SYSTEM RELIABILITY PROBLEM 2.1 Literature View
Many methods have been reported to improve system reliability. Tillman, Hwang, and Kuo [9] provided a survey of optimal system reliability. They divided optimal system reliability models into series, parallel, series-parallel, parallel-series, standby, and complex classes. They also categorized optimization methods into integer programming, dynamic programming, linear programming, geometric programming, generalized Lagrangian functions, and heuristic approaches. The authors concluded that many algorithms have been proposed but only a few have been demonstrated to be effective when applied to large-scale nonlinear programming problems. Also, none has proven to be generally superior. Fyffe, Hines, and Lee [10] provided a dynamic programming algorithm for solving the system reliability allocation problem. As the number of constraints in a given reliability problem increases, the computation required for solving the problem increases exponentially. In order to overcome these computational difficulties, the authors introduced the Lagrange multiplier to reduce the dimensionality of the problem. To illustrate their computational procedure, the authors used a hypothetical system reliability allocation problem, which consists of fourteen functional units connected in series. While their formulation provides a selection of links, the search space is restricted to consider only solutions where the same component type is used in parallel. Nakagawa and Miyazaki [11] proposed a more efficient algorithm. In their algorithm, the authors used surrogate constraints obtained by combining multiple constraints into one constraint. In order to demonstrate the efficiency of their algorithm, they also solved 33 variations of the Fyffe problem. Of the 33 problems, their algorithm produces optimal solutions for 30 of them. Misra and Sharma [12] presented a simple and efficient technique for solving integer-programming problems such as the system reliability design problem. The algorithm was based on function evaluations and a search limited to the boundary of resources. In the nonlinear programming approach, Hwang, Tillman and Kuo [13] used the generalized Lagrangian function method and the generalized reduced gradient method to solve nonlinear optimization problems for reliability of a complex system. They first maximize complex-system reliability with a tangent cost-function and then minimize the cost with minimum system reliability. The same authors also present a mixed integer programming approach to solve the reliability problem [14]. They maximize the system reliability as a function of component 2
reliability level and the number of links at each stage. Using a genetic algorithm (GA) approach, Coit and Smith [15], provide a competitive and robust algorithm to solve the system reliability problem. The authors use a penalty guided algorithm which searches over feasible and infeasible regions to identify a final, feasible optimal, or near optimal, solution. The penalty function is adaptive and responds to the search history. The GA performs very well on two types of problems: redundancy allocation as originally proposed by Fyffe, et al., and randomly generated problems with more complex configurations. For a fixed design configuration and known incremental decreases in component failure rates and their associated costs, Painton and Campbell [16] also used a GA based algorithm to find a maximum reliability solution to satisfy specific cost constraints. They formulate a flexible algorithm to optimize the 5th percentile of the mean time-between-failure distribution. Many researches also concerned with genetic algorithm as a solution for their problems, such as TSP (traveling salesman. Lin and Mitsuo [17] propose a new shortest path routing algorithm by using a priority-based Genetic Algorithm (priGA) approach in OSPF. Their Numerical experiments showed that the proposed GA approach gave better performance than the recent research on the SPR problem. In [18] Jiang, Sarker and Abbass introduced a non-stationary version of the classical traveling salesman problem (NTSP). They have developed a GA based methodology for solving NTSP problems. They have solved 30 randomly generated NTSP instances using two different crossover operators. In this paper, genetic algorithm optimization will be modified and adapted in order to consider the links criticality, which will then be continuously improved until reaching the optimal system reliability. 2.2 Genetic Algorithm Optimization Approach
In GA, the most critical issue while encoding and decoding between chromosomes, is the legality of a chromosome, which refers to the phenomenon of whether a chromosome represents a solution to a given problem. The illegality of chromosomes originates from the nature of encoding techniques. A simple one cut-point crossover operation usually yields illegal offspring. Since an illegal chromosome cannot be decoded to a solution, it cannot be evaluated. Repairing techniques are usually adopted to convert an illegal chromosome to a legal one. For this purpose, a new criticalitybased crossover algorithm is taken into consideration. Such an algorithm will be developed and numerically tested in this paper.
3 METHODOLOGY 3.1 Problem Definition
In this sub-section, the parameters used in the developed model are defined as follows:. Rs : System reliability Pi : Reliability of link i. Qi : Probability of failure of link i. n : Total number of links. ICRi : Index of criticality measure. Ct : Total cost of links. Ci : Cost of link i. Ct(max) : Maximum cost for system improvement Pi(min) : Minimum reliability value of link i. Pi(max) : Maximum reliability value of link i. 3.2 The Proposed Algorithm
The adapted genetic approach tries to find the optimal path by increasing the system reliability with minimum cost. The adapted algorithm steps are illustrated as follows:
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Generating the initial population Computing the fitness of each path Computing the overall system reliability and total cost. While Rs
