Hierarchical probability and risk assessment for K-out-of-N system in hierarchy

Highlights

• The hierarchical systems exist widely in the real world.

• Usage of calculation methods within combinatorics for hierarchical system evaluation.

• Proposal of P-out-of-1 system for unification and normalization of K-out-of-N system.

• Investigation of risks for decision-making of hierarchical system maintenance.

Abstract

In this research, the model of hierarchical structure from universe theory is imported into the general probability system which helps logically and subordinately describing, measuring, and evaluating the hierarchical probabilistic system. To clearly classify the hierarchy, “fayer” is proposed as the unit of hierarchy. Then, discussions on the relationship between hierarchy and system-of-systems and the risks caused by the hierarchy are proposed. This paper divides the problems of hierarchical probability mainly into two classes: the problems of surviving ratio and problems of passing ratio, and also, mixed problems of surviving ratio and passing ratio. Simultaneously, the surviving ratio is called as reliability. System reliability is one marked kind of hierarchical probability for the risk assessment of the hierarchical system. This paper introduces a new reliability system, P-out-of-1 system, which is derived from the K-out-of-N systems in hierarchy. The hierarchical reliability calculation of every fayer of the P-out-of-1 system is different from other fayers. Furthermore, the risk assessment of the hierarchical system for decision making of one maintenance manifests many differences by different descriptions in different fayers, even these descriptions concentrate on the same system.

Introduction

The hierarchical system is a fundamental concept from universe theory [1], [2] and graph theory [3]. For example, “…proton, atom, and molecule…Earth–Moon system, the Solar system, and the Galaxy…”. Atoms have the protons be their own components while themselves are the components of the molecules, and the Solar system contains the Earth–Moon system while itself is a part of the Galaxy. Other cases like taxonomy, the division of the administrative area, the structural dismantling, as well as the work breakdown structure, traffic network can also be categorized into the same framework of hierarchy. However, there are some misunderstandings about this concept. Hierarchy sometimes is regarded as the same idea as for scale [4] when comparing sub-system with the whole system. However, the hierarchical system focuses more on the subordination and logicality of components rather than the description of the scale. The scale can only be treated as one of the characteristics or descriptions of the hierarchy. However, the scale and hierarchy often share the same system structure. A good selection of scale is quite important to form a reasonable hierarchy. Until now, not too many studies have concentrated on the specific concept summary of the hierarchical system though hierarchical system exists in many engineering departments, like the power systems, buildings, computers, offshore facilities, lifelines, integrated circuit, pile group foundation, automotive systems, vehicle, and production lines, etc. Also, the influence of hierarchy to the system risk assessment has not been studied yet. For complex systems consisting a large number of components [5], the data measured for evaluation of these components are often in different levels, for example, some data is measured from subsystems, while some data is from sub-subsystems. And this directly brings about risks in further decision making when people confusing data selection measuring from the hierarchical system. Also, for the need of safety and efficiency of system performance, some concepts, like functional surplus, ultra-stability, and failure-reservation, etc. are needed to be considered in the analysis. To evaluate the system, the viewpoint of probability to evaluate the risk is often mentioned.

Among various kinds of performance evaluation indexes, reliability is of great significance [6], [7]. Reliability for a complex system often means evaluating the capacity of risk defense to keep the eligible function [8] of a huge system with a large number of subsystems/modules/components by target cascading method [9]. For a complex system, it is possible to obtain the failure probability (or named as failing probability in this paper) through hierarchical clustering [10] in parallel and serial [11]. The hierarchical reliability has been used in various applications including circuit [12], software [13], and electric power system [14] and even in human reliability analysis [15] from perspective of composition logic, where the components are often treated as independent [16] and the components’ failure events and interactions among these components are regarded as constant [17]. These systems are more likely to be SoS (system of systems) [18]. Furthermore, in order to obtain a more accurate value of reliability for subsequent decision-making within particular risks in practice, such intricate and sophisticated formulation of failure events should take the correlations among components into consideration [19]; and in a complex system, the risk assessment often means the evaluation through the reliability of the complex system [20]. And the state-of-the-art techniques, like the Fault tree [21], the Bayesian network [22] as well as traditional Response surface modeling (RSM) [23], are used in reliability calculation for hierarchical system evaluation. Furthermore, in terms of people's general knowledge, assuming that the measurement would not lead a change to a macroscopic system, the actual health state of the system is independent to any measurement method and means. Then, when measuring subsystems in different levels and evaluating the system according to the data from measurement, the real health status of the hierarchical system is independent from the evaluations at different levels of the system. That is, according to people's common sense, the evaluation results at various levels should be the same. Recalling the stories of Erwin Schrödinger's Cat and Copenhagen interpretation [24], we have a reason to believe that the knowledge of the probability system based on measurement may bring risks in decision-making. Since that the reliability is investigated from various levels of the system, for instance, some kinds of data are measured in subsystem while some are measured in sub-subsystem, the reliability may have its own risk to be applied in decision making.

In kinds of reliability problems [25], [26], the K-out-of-N system is one common recurring system, existing in a variety of engineering practices. For the evaluation standard, K-out-of-N, a requirement is K<=N, where K and N are integers for the count. There are two kinds of K-out-of-N system namely K-out-of-N:G [27] and K-out-of-N:F [28], in which, the K-out-of-N:G system means at least K components surviving in the entire system of N components, while the K-out-of-N:F system requires that if and only if no more than K components fail in a system of N components. In this paper, the K-out-of-N system just means K-out-of-N:F. Moreover, there are some researches on hierarchical reliability caring about the parameter or parameter relationship between system and sub-system in dynamic changing K in the K-out-of-N system [29], [30], and optimization with sub-systems to reach a high reliability of the system [31], utilizing K-out-of-N system in serial-parallel style [32] or the artificial weighted method [33] to recognize the impact of local systems (sub-systems) on the whole system's changing.

As we know, in discrete engineering systems, the maintenance costs of different levels of elements in hierarchy are not the same. To determine the reliability of all elements in hierarchy, it is very important but quite troublesome in practice. Even though there are some differences between hierarchy and scale, if we can find a standard describing the phenomenon of hierarchy in a system, it is able to understand the system better. Nevertheless, in the system of K-out-of-N, there is a natural standard P = K/N to compute the failure probability, which can also be named as P-out-of-1. In this paper, there are 2 classes of hierarchical probability, (Class 1) surviving ratio and (Class 2) passing ratio, and mixed situation of two classes will be studied by considering two typical kinds of problems (independent, example A, B, C and correlative, example D). The surviving ratio is to evaluate the surviving ability of the function in the lifetime of the system corresponding to the definition of reliability while the passing ratio is about the initial status of satisfaction on the system corresponding to the current quality of the system. Both surviving ratio and passing ratio are about P_c. The problem examples include (A) the adjustment of transportation/storage loss, (B) qualification inspection, (C) the evaluation of integrated circuit, and (D) the evaluation of pile group foundation.

However, to evaluate a system, a unified standard is needed for any system modeling technique. To obtain the comparable results, there is a need of normalization for the evaluation standard which should often be unified in this system modeling technique, for example, the normalization of “K-out-of-N” in hierarchy could be “P-out-of-1″. In this paper, we are not aiming to propose a new evaluation method for complex systems, like the current state of system evaluation techniques, Fault tree or Bayesian networks. Meanwhile, we apply the most basic probability knowledge from Frequentists into the K-out-of-N system in hierarchy to explain and show the influence of the hierarchy in the system evaluation with one kind of artificial standard, e.g. “P-out-of-1″. When surveying the measuring data from different levels, whether the same standard can fit for every level of system and whether there is some risk are the main issues when evaluating the system. Overall, this paper is trying to assess the risk in decision making for P-out-of-1 system using hierarchical (failure) probability, in which, the P-out-of-1 system is proposed as a normalized recurring reliability system as K-out-of-N system in hierarchy.

This paper is split into six sections. After the introduction, the basic concept of hierarchical probability reliability, and the Central Limit theorem of hierarchical probability are introduced in Section 2. The calculation procedures of three basic classes of hierarchical reliabilities and Monte Carlo simulation are presented in Section 3 with details. Then, the worked examples are proposed in Section 4 and the relationship between hierarchical risk assessment and risk estimate for decision-making is provided in Section 5. Finally, conclusions are summarized in Section 6.