# Linear Programming-Based Non-Probabilistic Reliability Bounds Method for Series Systems
## 1 Introduction  
Because of the requirement for only the boundary information of structural uncertain parameters, the non-probabilistic convex model is more applicable for managing issues with limited information than the probability model. Non-probabilistic convex models mainly include interval models [1; 2], ellipsoid models [3; 4; 5], parallelepiped models [6; 7], and super-ellipsoid models [8; 9]. In recent years, several reliability methods based on non-probabilistic convex models have been developed for evaluating the safety degree of structures with convex models. These models principally include the robust reliability index of [PERSON] et al. [10; 11] and [PERSON] et al. [12]; the non-probabilistic safety factor of [PERSON] and [PERSON] [13]; the non-probabilistic reliability index of [PERSON] et al. [14; 15], [PERSON] and [PERSON] [16], [PERSON] et al. [17], and [PERSON] et al. [18]; and the non-probabilistic reliability of [PERSON] et al. [19; 20] and [PERSON] et al. [21]. Due to the perfect similarity with their probabilistic counterparts, the non-probabilistic reliability index and reliability have gained more attention and have been applied to reliability analysis and the reliability-based design optimization of structures. Most of the abovementioned studies were focused on the reliability of an individual component. However, a structural system generally consists of a set of interconnected components. The task of system reliability is to provide a methodology to determine or at least estimate the safety degree of an entire system and to build a bridge between the reliability of a component and the reliability of the system. By representing uncertain parameters as interval variables, [PERSON] et al. proposed a system reliability measure based on the minimum non-probabilistic reliability index of all possible failure modes [22]. Likewise, [PERSON] et al. subsequently suggested a system safety measure based on the non-probabilistic set-theoretic model [23]. These preceding studies focused on series systems and ignored the dependence between the component states. [PERSON] et al. formulated a Monte Carlo simulation (MCS) for both series and parallel systems [21] in which the dependence described above could be further considered. [PERSON] et al. [24] proposed a method for calculating system non-probabilistic reliability index by using the Kriging model instead of a time-consuming simulation. [PERSON] et al. [25] proposed a non-probabilistic systematic reliability method(NSRM) for the reliability assessment of series-parallel systems where an equivalence-based approach is used to determine the equivalent limit state functions of parallel subsystems. In a recent work of the authors of this paper, a non-probabilistic reliability bounds method (NRBM) for series systems was proposed [26]. The NRBM could provide the analytical solution to estimate the system reliability and therefore had a higher efficiency than the MCS. Nevertheless, it should be emphasized that the NRBM still had some drawbacks. First, its precision depended on the specific order of components. Second, there was no guarantee that it would yield the narrowest possible bounds of the non-probabilistic failure degree. In this paper, a novel linear programming-based non-probabilistic reliability bounds method (LPNRBM) is proposed to overcome the above drawbacks. This method is both old and new. That is, the method is chronologically old because its probabilistic version was proposed by [PERSON] and [PERSON] [27] in 2003. However, the method has been revised to investigate the reliability of a series system with convex model uncertainty, which inevitably has some non-probabilistic characteristics. The main innovations of the proposed method lie in the following aspects. First, it is independent of the ordering of the components and can provide the narrowest possible bounds for the given information.