پژوهشی درباره روش آرام مرزهای برنامه ریزی خطی برای قابلیت اطمینان سیستم
|کد مقاله||سال انتشار||مقاله انگلیسی||ترجمه فارسی||تعداد کلمات|
|25399||2013||7 صفحه PDF||سفارش دهید||محاسبه نشده|
Publisher : Elsevier - Science Direct (الزویر - ساینس دایرکت)
Journal : Structural Safety, Volume 41, March 2013, Pages 64–72
The linear programming (LP) bounds method was applied for computing bounds on the system reliability of general systems based on the individual component state probabilities and joint probabilities of the states of a small number of components. In the LP bounds method, the bounds of the system reliability can be obtained by using LP. These bounds are useful approximations when exact solutions are costly or unavailable. However, the size of the LP problem determined by the number of design variables and the number of constraints increases exponentially with the number of components. This size problem is the main drawback of the LP bounds method. This paper presents a relaxed linear programming (RLP) bounds method to overcome this drawback of the LP bounds method. The accuracy and efficiency of the RLP bounds method are investigated using numerical examples involving series and parallel systems.
A system, in general, consists of a number of components, and the state of the system depends on the states of its constituent components. The probability that a system is in a particular functioning state (system reliability), or its complement (system failure probability), can then be expressed based on the probability of the component states. Computation of this probability, however, is extremely difficult, particularly when there exists a dependency among the component states and when the number of components is large. The idea of using linear programming (LP) to compute bounds on system reliability was first explored by Hailperin . Kounias and Marin  used the approach to examine the accuracy of some theoretical bounds. Later, specialized versions of this approach were employed in fields such as operations research . Song and Der Kiureghian proposed the linear programming (LP) bounds method for computing the bounds on the failure probability of general systems based on the joint probabilities of the states of k components (when k = 1, these joint probabilities become the individual component state probabilities) [ 4]. The LP formulation has a number of important advantages over other existing methods (e.g., Boole bounds [ 5] or Zhang bounds [ 6]). They include: (a) any “level,” i.e., the number (k) of components considered in the joint probabilities of the states, of information can be used, including equalities and inequalities; (b) the statistical dependency among component states is easily accounted for in terms of their joint probabilities; (c) the method guarantees the narrowest possible bounds for the given information of individual and joint component states probabilities; (d) the method is applicable to a general system, including a system that is neither pure series nor pure parallel and a system for which no theoretical formula exists; and (e) critical components and cut sets within a system can be easily identified [ 7]. There exists, however, a critical drawback in the LP bounds method. The size of the LP problem, which is usually determined by the number of design variables and the number of constraints, increases exponentially with the number of components. For a system with n two-state components, the number of design variables in the LP bounds method is Nd = 2n. When n = 17, Nd = 131 072 and the problem can be solved with ordinary LP solvers on a PC. When n = 100, this number becomes Nd ≈ 1.27 × 1030, which is enormously large. The number of constraints, which depends on the number of design variables and the level of joint state probabilities, also becomes enormously large in the application of the LP bounds method to a large system. This size issue—both the number of design variables and constraints—would be a hindrance in the application of the LP bounds method to a large system. To overcome the size issue of design variables, Der Kiureghian and Song propose a multi-scale approach, whereby the system is decomposed into subsystems and a hierarchy of analysis is performed by considering each subsystem or set of subsystems separately . The decomposition facilitates solution of the system reliability by the LP bounds method, whereby the large LP problems for the entire system is replaced by several LP problems of much smaller size. This facility, however, comes at a cost; the system bounds computed for the decomposed system can be wider than the bounds computed for the intact system with the same level of probability information. This paper proposes a relaxed linear programming (RLP) bounds method to overcome the size problem of the LP bounds method. The RLP bounds method employs the universal generating function to reduce the number of design variables from 2n to n2 − n + 2. The number of constraints can also be reduced substantially. The accuracy and efficiency of the RLP bounds method are investigated using numerical examples involving series and parallel systems.
نتیجه گیری انگلیسی
As an efficient reliability tool for a system with a large number of components, the relaxed linear programming (RLP) bounds method is developed in this paper. The RLP bounds method can be used to estimate the bounds for a series system as well as a parallel system. Like the LP bounds method, the RLP bounds method can use any level of joint failure probabilities, including equalities, inequalities. The accuracy and applicability of the RLP bounds method are investigated along with the LP bounds method and MC simulations using numerical examples. The RLP bounds method provides a result comparable to that of the LP bounds method. The efficiency of the original RLP bound method (RLP1) is further sought for in this paper by disregarding some of the inequality constraints (RLP2) and all of the inequality constraints (RLP3). The bounds estimated by RLP1 are close to the bounds of the LP bounds method, and RLP1 provides more accurate bounds than RLP2 and RLP3 if the bounds are not identical. When the components are equicorrelated and equireliable, RLP3 works as well as RLP1. When the components are not equicorrelated or equireliable, RLP2 considering “important” inequality constraints still works as well as RLP1. Although the authors have some clews for how to determine such important constraints, the theoretical reasoning is still one of the topics of our future works. The incomplete sets of joint failure probabilities can be handled by using the LP+RLP approaches. Because the efficiency and applicability of the LP+RLP approaches are mostly based on that of the LP bounds method, the authors just show its ability to handle the system with incomplete set of joint failure probabilities of k components and not put more focus on that. The RLP bounds method which can handle the incomplete set of joint failure probabilities without any help of the LP bound method is being developed.