# Posbist Reliability Theory for Typical Systems with Multicomponents

Authors: Jiaojiao Ren; Cong Wu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper investigates Posbist reliability theory for typical systems with multicomponents. This theory is based on possibility and binary-state assumptions. It works well under situations of insufficient data and epistemic uncertainty. When the lifetime of every component of a typical system is assumed unrelated and a normal, strictly convex fuzzy variable in some possibility space, the Posbist reliability is concretely derived where the exactly value can be obtained at any moment, unlike existing ones only given by a lower and upper bound. We numerically compare the Posbist reliability between typical systems of four types: series, parallel, series-parallel, and parallel-series. Moreover, numerical safety evaluations are given based on possibilistic and probabilistic models.


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## Body

## 1. Introduction

Failures are always inevitable in many industrial systems. To measure the uncertainty of failures, the probabilistic reliability method, defining reliability by probability, is a remarkably successful tool [1–7]. As for the complex structure in practical engineering, natural language is widely used to express the fuzzy information. Another useful theory, the fuzzy reliability theory, was developed. This theory is more advantageous than the probabilistic reliability method, especially in situations that suffer lack of necessary sample data, subjective information, or epistemic uncertainty.The fuzzy reliability theory is based on the fuzzy mathematics theory [8], and the possibility theory [9], employing fuzzy states and possibility rather than binary states and probability, was presented by Onisawa in [10]. Later, the theory was further developed to be three branches: the Posbist reliability theory based on possibility and binary states [11–13]; Profust reliability theory based on probability and fuzzy states [14–16]; and Posfust reliability theory based on possibility and fuzzy states [17]. Afterwards, a new possibilistic reliability index definition was proposed [18]. The fuzzy reliability theory was then extended to structural reliability analysis [19] and reliability analysis of system with fuzzy random variables [20]. In the recent past, a variant of the theory, based on chance measure, to assess mechanism reliability under fuzzy and random uncertainties, was published [21].In the field of reliability theory, the Posbist reliablity theory has drawn plenty of attention because it is particularly reliable when applied to those systems about which only vague data, small samples, and spare statistical charateristics are availble. By adopting the traditional definition of reliability, Posbist reliability models of typical systems had been constructed in [11, 12]. However, typical systems in [11, 12] were required to be irreparable. The authors in [13] established Posbist reliability models of repairable typical systems. Recently, the Posbist reliability theory was studied again in [22, 23]. Furthermore, some perspectives to define and model reliability using fuzzy sets were given in [24]. However, in most existing results, the considered typical systems only investigated two components with different possibility distribution functions and n components with the same possibility distribution functions. In [25], the possibility distribution function of the lifetime of a system with n different components was obtained. However, there was only a lower bound and a upper bound, i.e., min1≤i≤nμXix≤μXx≤max1≤i≤nμXix, without any concretely results. Furthermore, in [26], the authors considered typical systems with n components where distribution functions of the lifetime of the components were assumed to be different. Moreover, the authors in [26] assumed the lifetime of each component as a symmetrical Gaussian fuzzy variable. Parameter hi in the distribution functions was assumed to be the same. Therefore, for more general cases, a given system has not only a symmetrical Gaussian fuzzy variable but also multiple components with different possibility distribution functions. Obtaining possibility distribution function of the system lifetime and Posbist reliability is a challenging problem.Motivated by abovementioned discussions, due to the existence of insufficient and vague data in practical systems, the problem of Posbist reliability modeling and analysis for typical systems with multicomponents is further investigated in this paper via the possibility theory. The lifetime of components are assumed as unrelated and each a normal, strictly convex fuzzy variable, rather than a Gaussian fuzzy variable. The Posbist reliability theory of typical systems is concretely obtained where the exactly value can be obtained at any moment. Finally, numerical examples are provided to illustrate the feasibility of the obtained results.


## 2. Preliminaries

Based on Posbist reliability theory and the existence theorem of the possibility space, see pages 62–69 in [25], a possibility space U,Φ,Poss is assumed to characterize the uncertainties of failure time of the system and its components by the context of possibility measures. Before proceeding, necessary concepts are introduced as follows:Definition 1.
(pattern space, see [25], page 44). Let U be the universe of discourse. Suppose Φ is the discrete topology on U (that is, the power set or the class of all subsets of U). Then, the pair U,Φ is referred to a pattern space.Definition 2.
(possibility space, see [25], page 44). Let U,Φ be a pattern space. A possibility measure Poss is a mapping from Φ to 0,1 with the following properties:(1)
Poss∅=0.(2)
PossU=1.(3)
For an arbitrary collection of setsAα of Φ (finite, countable, or uncountable), there holds(1)Poss∪αAα=supαPossAα.
Then, the tripletU,Φ,Poss is referred to as a possibility space.(2)πA≜supu∈AπXu,and πXu is the possibility distribution function of ΠX. This number, then, may be interpreted as the possibility that a value of X belongs to A, that is,(3)PossX∈A≜πA≜supu∈AπXu.Definition 3 (possibility measure for classical set, see [9]).
For a universe of discourseU and A is a classical set of U, let ΠX be a possibility distribution associated with a variable X which takes values in U. Then, the possibility measure, πA, is defined as a number in 0,1 given byRemark 1.
The classical set is a set distinguishing from the fuzzy set. In classical set theory, the relationship between an element and a set is certain. However, in fuzzy set theory, the relationship between an element and a set is described by a membership function.Definition 4.
(unrelatedness, see [25], page 49). Suppose U,Φ,Poss is a possibility space. A1,…,An∈Φ are said to be mutually unrelated if for any permutation of the set 1,…,n, denoted by i1,…,in, for 1≤k≤n, there holds(4)PossAi1∩…∩Aik=minPossAi1,…,PossAik.Definition 5.
(system (component) lifetime, see [25], page 194). Let fuzzy variable X be a real-valued function defined on a possibility space, i.e., X:U⟶R+=0,+∞, with the membership function μX:R+⟶0,1, then the lifetime of a system (component) is a nonnegative real-valued fuzzy variable, i.e., X:U⟶R+=0,+∞, with possibility distribution function(5)πXx=μXx=PossX=x,x∈R.
Furthermore,X is said to be normal/convex/strictly convex if the fuzzy set x,μXx;x∈R+ is normal/convex/strictly convex.Definition 6.
(Posbist reliability, see [25], page 194). The Posbist reliability of a system is defined as the possibility that the system performs its assigned functions during a predefined exposure period under a given condition, which is denoted as(6)Ret=PossX>t=supu>tPossX=u=supu>tμXu,t∈R+.Definition 7.
(L-R fuzzy numbers, see [23]). Assume that R˜ is a positive bounded and closed-interval of fuzzy numbers, whose membership function satisfies, ∀α>0 and β>0:(7)μR˜x=Lm−−xα,m−−α≤x≤m−,1,m−≤x≤m+,Rx−m+β,m+≤x≤m++β,0,others,where m+ and m− are upper and lower modal values, respectively. m−,m+ is peak range. α and β are left and right scatters, respectively.
Whenm−=m+=m0, the unique maximum value of the membership function will be achieved at the point m0. The membership function is increasing on interval −∞,m0 and decreasing on interval m0,+∞.
In practical application,Lx and Rx can be chosen as linear or nonlinear function, such as, triangular shape, normal shape, and sharp shape.


## 3. Posbist Reliability Models of Typical Systems

### 3.1. Posbist Reliability of Series System

Given a possibility spaceU,Φ,Poss, we consider a series system consisting of n components, as shown in Figure 1. Suppose X1,X2,…,Xn are the lifetimes of the n components, respectively. Then, the lifetime of system X is X=minX1,X2,…,Xn.Figure 1
The logic block diagram of a series system.Theorem 1.
Given a possibility spaceU,Φ,Poss and a series system consisting of n components, the lifetime of components Xii=1,2,…,n are unrelated and each a normal, strictly convex fuzzy variable, with continuous possibility distribution functions μX1x,μX2x,…,μXnx, respectively. Let X be the system lifetime, with possibility distribution function μXx. Then, there exist n positive numbers a1,a2,…,an (without loss of generality, we assume a1≤a2≤⋯≤an), such that the possibility distribution function of X is(8)μXx=maxμX1x,μX2x,…,μXnx,x≤a1,μX1x,a1<x≤a2,minμX1x,μX2x,a2<x≤a3,…minμX1x,μX2x,…,μXn−1x,an−1<x≤an,minμX1x,μX2x,…,μXnx,x>an.Moreover, the Posbist reliability of the series system is(9)Rst=supu>tμXu=1,t≤a1,μX1t,a1<t≤a2,minμX1t,μX2t,a2<t≤a3,…minμX1t,μX2t,…,μXn−1t,an−1<t≤an,minμX1t,μX2t,…,μXnt,t>an.Proof.
Since the considered system is a series system and the lifetime of each component isXii=1,2,…,n, then(10)X=minX1,X2,…,Xn.
For each componentXi, there exists a unique real number ai∈R+, such that(11)supu≥xμXiu=1,ifx≤ai,μXix,ifx>ai.
Without loss of generality, we assumea1≤a2≤⋯≤an. According to the definition of system lifetime,(12)μXx=PossX=minX1,X2,…,Xn=x=PossX1=x,X2≥x,X3≥x,…,Xn≥x∪PossX1≥x,X2=x,X3≥x,…,Xn≥x∪⋯∪PossX1≥x,X2≥x,X3≥x,…,Xn=x=maxPossX1=x,X2≥x,X3≥x,…,Xn≥x,PossX1≥x,X2=x,X3≥x,…,Xn≥x,…,PossX1≥x,X2≥x,X3≥x,…,Xn=x.
In addition,(13)PossX1≥x,…,Xi−1≥x,Xi=x,Xi+1≥x,…,Xn≥x=Poss∪a≥xX1=a,…,∪a≥xXi−1=a,Xi=x,∪a≥xXi+1=a,…,∪a≥xXn=a=Poss∪a≥xX1=a,…,Xi−1=a,Xi=x,Xi+1=a,…,Xn=a=supa≥xPossX1=a,…,Xi−1=a,Xi=x,Xi+1=a,…,Xn=a.
Due to the unrelatedness ofXi,i∈1,2,…,n, we have(14)PossX1≥x,…,Xi−1≥x,Xi=x,Xi+1≥x,…,Xn≥x=supa≥xminPossX1=a,…,PossXi−1=a,PossXi=x,PossXi+1=a,…,PossXn=a=supa≥xminμX1a,…,μXi−1a,μXix,μXi+1a,…,μXna=minsupa≥xμX1a,…,supa≥xμXi−1a,μXix,supa≥xμXi+1a,…,supa≥xμXna,=min1,…,1,μXix,1,…,1=μXix,x≤a1,minμX1x,1,…,1,μXix,1,…,1=minμX1x,μXix,a1<x≤a2,…minμX1x,μX2x,…,μXix,1,…,1=minμX1x,μX2x,…,μXix,ai−1<x≤ai,minμX1x,μX2x,…,μXix,1,…,1=minμX1x,μX2x,…,μXix,ai<x≤ai+1,minμX1x,μX2x,…,μXi+1x,1,…,1=minμX1x,μX2x,…,μXi+1x,ai+1<x≤ai+2,minμX1x,μX2x,…,μXi+2x,1,…,1=minμX1x,μX2x,…,μXi+2x,ai+2<x≤ai+3,…minμX1x,μX2x,…,μXn−1x,1=minμX1x,μX2x,…,μXn−1x,an−1<x≤an,minμX1x,μX2x,…,μXnx,x>an.
Therefore,(15)μXx=maxminμX1x,supa≥xμX2a,…,supa≥xμXna…,minsupa≥xμX1a,…,supa≥xμXi−1a,μXix,supa≥xμXi+1a,…,supa≥xμXna,…,minsupa≥xμX1a,…,supa≥xμXn−1a,μXnx.
Whenx≤a1,(16)μXx=maxμX1x,μX2x,…,μXnx.
Whena1<x≤a2,(17)μXx=maxμX1x,minμX1x,μX2xminμX1x,μX3x,…,minμX1x,μXnx,=μX1x.
Whena2<x≤a3,(18)μXx=maxminμX1x,μX2x,minμX1x,μX2x,μX3x,minμX1x,μX2x,μX4x,…,minμX1x,μX2x,μXnx=minμX1x,μX2x.
Whena3<x≤a4,(19)μXx=maxminμX1x,μX2x,μX3x,minμX1x,μX2x,μX3x,μX4x,minμX1x,μX2x,μX3x,μX5x,…,minμX1x,μX2x,μX3x,μXnx=minμX1x,μX2x,μX3x.…
Whenai−1<x≤ai,(20)μXx=maxminμX1x,…,μXi−1x,minμX1x,…,μXi−1x,μXix,minμX1x,…,μXi−1x,μXi+1x,…,minμX1x,…,μXi−1x,μXnx=minμX1x,…,μXi−1x.…
Whenan−1<x≤an,(21)μXx=maxminμX1x,μX2x,…,μXn−1x,minμX1x,μX2x,…,μXn−1x,μXnx=minμX1x,…,μXn−1x.
Whenx>an,(22)μXx=minμX1x,μX2x,…,μXnx.
So, we have(23)μXx=maxμX1x,μX2x,…,μXnx,x≤a1,μX1x,a1<x≤a2,minμX1x,μX2x,a2<x≤a3,…minμX1x,μX2x,…,μXn−1x,an−1<x≤an,minμX1x,μX2x,…,μXnx,x>an.
Since the lifetime of componentsXii=1,2,…,n are unrelated and each a normal, strictly convex fuzzy variable, with continuous possibility distribution functions μX1x,μX2x,…,μXnx, respectively, ai is a unique real number such that supu∈R+μXiu=μXiai=1. When u≤ai, μXiu is a strictly monotonically increasing function. When u>ai, μXiu is a strictly monotonically decreasing function. When ai=+∞, μXiu is a strictly monotonically increasing function in the whole domain.
The Posbist reliability of series system can be expressed asRst=supu>tμXu; combining with (11), we have when t≤a1,(24)Rst=supu>tμXu=maxsupy∈t,a1μXy,supy∈a1,a2μXy,…,supy∈an,+∞μXy=maxsupy∈t,a1maxμX1y,μX2y,…,μXny,supy∈a1,a2μX1y,…,supy∈an,+∞minμX1y,μX2y,…,μXny,where(25)supy∈t,a1maxμX1y,μX2y,…,μXny=maxsupy∈t,a1μX1y,supy∈t,a1μX2y,…,supy∈t,a1μXny.
Sincesupy>tμX1y=1t≤a1, then(26)supy>tμX1y=supy∈t,a1∪y∈a1,+∞μX1y=maxsupy∈t,a1μX1y,supy∈a1,+∞μX1y=maxsupy∈t,a1μX1y,μX1a1+=1.
So,(27)supy∈t,a1μX1y=1.
No matter what the remaining terms’ value is, the maximum value of all terms is 1.
Therefore,(28)Rst=1.
Whena1<t≤a2,(29)Rst=supu>tμXu=maxsupy∈t,a2μXy,supy∈a2,a3μXy,…,supy∈an,+∞μXy=maxsupy∈t,a2μX1y,supy∈a2,a3minμX1y,μX2y,…,supy∈an,+∞minμX1y,μX2y,…,μXny=maxsupy∈t,a2μX1y,minsupy∈a2,a3μX1y,supy∈a2,a3μX2y,…,minsupy∈an,+∞μX1y,supy∈an,+∞μX2y,…,supy∈an,+∞μXny,where(30)supy∈t,a2μX1y=μX1t.
If(31)supy∈a2,a3μX1y≤supy∈a2,a3μX2y,then(32)minsupy∈a2,a3μX1y,supy∈a2,a3μX2y=supy∈a2,a3μX1y≤μX1y.
Otherwise,(33)minsupy∈a2,a3μX1y,supy∈a2,a3μX2y=supy∈a2,a3μX2y≤supy∈a2,a3μX1y≤supy∈t,a2μX1y≤μX1y.
Similarly, fori∈3,4,…,n, when i=n, let i+1=+∞ and the interval becomes a open set:(34)minsupy∈ai,ai+1μX1y,supy∈ai,ai+1μX2y,…,supy∈ai,ai+1μXiy≤μXit.
Therefore,(35)Rst=μX1t.…
Whenan−1<t≤an,(36)Rst=supu>tμXu=maxsupy∈t,anμXy,supy∈an,+∞μXy=maxsupy∈t,anminμX1y,μX2y,…,μXn−1y,supy∈an,+∞minμX1y,μX2y,…,μXny=maxminsupy∈t,anμX1y,supy∈t,anμX2y,…,supy∈t,anμXn−1y,minsupy∈an,+∞μX1y,supy∈an,+∞μX2y,…,supy∈an,+∞μXny=minμX1t,μX2t,…,μXn−1t.
Whent>an,(37)Rst=supu>tμXu=supu>tminμX1u,μX2u,…,μXnu=minsupu>tμX1u,supu>tμX2u,…,supu>tμXnu=minμX1t,μX2t,…,μXnt.
Therefore, the Posbist reliability of the series system is(38)Rst=supu>tμXu=1,t≤a1,μX1t,a1<t≤a2,minμX1t,μX2t,a2<t≤a3,…minμX1t,μX2t,…,μXn−1t,an−1<t≤an,minμX1t,μX2t,…,μXnt,t>an.
Specially, when the lifetime of the components are assumed as symmetrical Gaussian fuzzy variables, i.e.,Lx and Rx have a normal distribution. We have the following corollary.Corollary 1.
Given a possibility spaceU,Φ,Poss and a series system consisting of n components, the possibility distribution function of the lifetime of components Xii=1,2,…,n are assumed to be(39)μXix=exp−x−mih2,without loss of generality; we assume m1≤m2≤⋯≤mn; then, the possibility distribution function of the system lifetime X is(40)μXx=exp−x−m1h2.Moreover, the Posbist reliability of the series system is(41)Rst=supu>tμXu=1,t≤m1,exp−t−m1h2,t>m1.Remark 2.
Given a possibility spaceU,Φ,Poss and a series system consisting of n components, assume the possibility distribution function of the lifetime of components Xii=1,2,…,n are identically and independently distributed ones μX1x; then, there exists a real number a, such that μXx=μX1x and Rst=supu>tμXu=1t≤aμX1tt>a.


### 3.2. Posbist Reliability of Parallel System

Given a possibility spaceU,Φ,Poss, we consider a parallel system consisting of n components, as shown in Figure 2. Suppose X is the system lifetime and X1,X2,…,Xn are the lifetimes of n components, respectively. Then, the lifetime of system X is X=maX1,X2,…,Xn.Figure 2
The logic block diagram of a parallel system.Theorem 2.
Given a possibility spaceU,Φ,Poss and a parallel system consisting of n components, the lifetime of components Xii=1,2,…,n are unrelated and each a normal, strictly convex fuzzy variable, with continuous possibility distribution functions μX1x,μX2x,…,μXnx, respectively. Let the system lifetime be X with possibility distribution function μXx. Then, there exist n positive numbers a1,a2,…,an (without loss of generality, we assume a1≤a2≤⋯≤an), such that the possibility distribution function of X is(42)μXx=minμX1x,μX2x,…,μXnx,x≤a1,minμX2x,μX3x,…,μXnx,a1<x≤a2,minμX3x,μX4x,…,μXnx,a2<x≤a3,…μXnx,an−1<x≤an,maxμX1x,μX2x,…,μXnx,x>an.Moreover, the Posbist reliability of the parallel system is(43)Rpt=supu>tμXu=1,t≤an,maxμX1t,μX2t,…,μXnt,t>an.Proof.
Similar to Theorem1, for each component Xi, there exists a unique real number ai∈R+, such that(44)supu≤xμXiu=μXix,ifx≤ai,1,ifx>ai.
Due to the assumptiona1≤a2≤⋯≤an and combining with the definition of system lifetime, we have(45)μXx=PossX=maxX1,X2,…,Xn=x=maxPossX1=x,X2≤x,X3≤x,…,Xn≤x,PossX1≤x,X2=x,X3≤x,…,Xn≤x,…,PossX1≤x,X2≤x,X3≤x,…,Xn=x,where(46)PossX1≤x,…,Xi−1≤x,Xi=x,Xi+1≤x,…,Xn≤x=supa≤xPossX1=a,…,Xi−1=a,Xi=x,Xi+1=a,…,Xn=a.
Because of the unrelatedness ofXi,i∈1,2,…,n, we have(47)PossX1≤x,…,Xi−1≤x,Xi=x,Xi+1≤x,…,Xn≤x=supa≤xminPossX1=a,…,PossXi−1=a,PossXi=x,PossXi+1=a,…,PossXn=a=supa≤xminμX1a,…,μXi−1a,μXix,μXi+1a,…,μXna=minsupa≤xμX1a,…,supa≤xμXi−1a,μXix,supa≤xμXi+1a,…,supa≤xμXna,=minμX1x,μX2x,…,μXix,…,μXnx,x≤a1min1,μX2x,μX3x,…,μXix,…,μXnx=minμX2x,μX3x,…,μXnx,a1<x≤a2,min1,1,μX3x,μX4x,…,μXix,…,μXnx=minμX3x,μX4x,…,μXnx,a2<x≤a3,…min1,…,1,μXix,…,μXnx=minμXix,μXi+1x,…,μXnx,ai−1<x≤ai,min1,…,1,μXix,…,μXnx=minμXix,μXi+1x,…,μXnx,ai<x≤ai+1,min1,…,1,μXix,1,μXi+2x,…,μXnx=minμXix,μXi+2x,…,μXnx,ai+1<x≤ai+2,min1,…,1,μXix,1,1,μXi+3x,…,μXnx=minμXix,μXi+3x,…,μXnx,ai+2<x≤ai+3,…min1,…,1,μXix,1,…,1,μXnx=minμXix,μXnx,an−1<x≤an,min1,…,1,μXix,1,…,1=μXix,x>an.
Therefore,(48)μXx=maxminμX1x,supa≤xμX2a,…,supa≤xμXna,…,minsupa≤xμX1a,…,supa≤xμXi−1a,μXix,supa≤xμXi+1a,…,supa≤xμXna,…,minsupa≤xμX1a,…,supa≤xμXn−1a,μXnx.
Whenx≤a1,(49)μXx=minμX1x,μX2x,…,μXnx.
Whena1<x≤a2,(50)μXx=maxminμX1x,μX2x,…,μXnx,minμX2x,…,μXnx,…,minμX2x,…,μXnx=minμX2x,…,μXnx.
Whena2<x≤a3,(51)μXx=maxminμX1x,μX3x,…,μXnx,minμX2x,…,μXnx,minμX3x,…,μXnx,…,minμX3x,…,μXnx=minμX3x,…,μXnx.
Whena3<x≤a4,(52)μXx=maxminμX1x,μX4x,…,μXnx,minμX2x,μX4x,…,μXnx,minμX3x,μX4x,…,μXnx,minμX4x,…,μXnx,…,minμX4x,…,μXnx=minμX4x,…,μXnx.…
Whenai−1<x≤ai,(53)μXx=maxminμX1x,μXix,…,μXnx,minμX2x,μXix,…,μXnx,…,minμXi−1x,μXix,…,μXnx,minμXix,…,μXnx,…,minμXix,…,μXnx=minμXix,…,μXnx.…
Whenan−1<x≤an,(54)μXx=maxminμX1x,μXnx,minμX2x,μXnx,…,minμXn−1x,μXnx,μXnx=μXnx.
Whenx>an,(55)μXx=maxμX1x,μX2x,…,μXnx.
So, we have(56)μXx=minμX1x,μX2x,…,μXnx,x≤a1,minμX2x,μX3x,…,μXnx,a1<x≤a2,minμX3x,μX4x,…,μXnx,a2<x≤a3,…μXnx,an−1<x≤anmaxμX1x,μX2x,…,μXnx,x>an.
The Posbist reliability of the parallel system is(57)Rpt=supu>tμXu.
Since the pointai is a unique maximum value point, based on the monotonicity of μXix, one has when t≤a1(58)Rpt=supu>tμXu=supt,a1∪a1,a2∪⋯∪an,+∞μXu=maxsupy∈t,a1minμX1y,μX2y,…,μXny,supy∈a1,a2minμX2y,μX3y,…,μXny,…,supy∈an−1,anμXny,supy∈an,+∞maxμX1y,μX2y,…,μXny=maxminsupy∈t,a1μX1y,…,supy∈t,a2μXny,minsupy∈a1,a2μX2y,…,supy∈a1,a2μXny,…,supy∈an−1,anμXny,maxsupy∈an,+∞μX1y,supy∈an,+∞μX2y,…,supy∈an,+∞μXny=maxminsupy∈t,a1μX1y,supy∈t,a2μX2y,…,supy∈t,anμXny,maxsupy∈an,+∞μX1y,supy∈an,+∞μX2y,…,supy∈an,+∞μXny=1.
Whena1<t≤a2,(59)Rpt=supu>tμXu=supt,a2∪a2,a3∪⋯∪an,+∞μXu=maxsupy∈t,a2minμX2y,μX3y,…,μXny,supy∈a2,a3minμX3y,μX4y,…,μXny,…,supy∈an−1,anμXny,supy∈an,+∞maxμX1y,μX2y,…,μXny=maxminsupy∈t,a2μX2y,…,supy∈t,a2μXny,minsupy∈a2,a3μX3y,…,supy∈a2,a3μXny,…,supy∈an−1,anμXny,maxsupy∈an,+∞μX1y,supy∈an,+∞μX2y,…,supy∈an,+∞μXny=maxminsupy∈t,a2μX2y,ssupy∈t,a3μX3y,…,supy∈t,anμXny,maxsupy∈an,+∞μX1y,supy∈an,+∞μX2y,…,supy∈an,+∞μXny=1.…
Whenan−1<t≤an,(60)Rpt=supu>tμXu=supt,an∪an,+∞μXu=maxsupy∈t,anμXny,supy∈an,+∞maxμX1y,μX2y,…,μXny=1.
Whent>an,(61)Rpt=supu>tμXu=supt,+∞maxμX1y,μX2y,…,μXny=maxμX1t,μX2t,…,μXnt.
Therefore, the Posbist reliability of the parallel system is(62)Rpt=supu>tμXu=1,t≤an,maxμX1t,μX2t,…,μXnt,t>an.Remark 3.
From the lifetime of the parallel system,X=maxX1,X2,…,Xn, and equations (46) and (47), we can know that the property of μXiu what we are focused on is the supremum value when u≤x. For the series system case, due to the lifetime is defined by X=minX1,X2,…,Xn, the property of μXiu what we are focused on is the supremum value when u≥x. This is a big difference between the proof of the theory in series and parallel systems.
Specially, when the lifetime of the components are assumed as symmetrical Gaussian fuzzy variables, i.e.,Lx and Rx have a normal distribution. We have the following corollary.Corollary 2.
Given a possibility spaceU,Φ,Poss and a parallel system consisting of n components, the possibility distribution function of the lifetime of components Xii=1,2,…,n are(63)μXix=exp−x−mih2,without loss of generality; we assume m1≤m2≤⋯≤mn, then the possibility distribution function of the system lifetime X is(64)μXx=exp−x−mnh2,
Moreover, the Posbist reliability of the parallel system is(65)Rpt=supu>tμXu=1,t≤mnexp−t−mnh2,t>mn.Remark 4.
Given a possibility spaceU,Φ,Poss and a parallel system consisting of n components, assume the possibility distribution function of the lifetime of the components Xii=1,2,…,n are identically and independently distributed ones μX1x; then, there exists a real number a, such that μXx=μX1x and Rpt=supu>tμXu=1,t≤a,μX1t,t>a.


### 3.3. Posbist Reliability of a Series-Parallel System

Given a possibility spaceU,Φ,Poss, we consider a series-parallel system consisting of m systems, each system is a parallel system consisting of n components, as shown in Figure 3. Suppose X is the system lifetime, X1,X2,…,Xm are the lifetimes of m parallel subsystems, and X11,X12,…,X1n,X21,X22,…,X2n,…,Xm1,Xm2,…,Xmn, are the lifetimes of the mn components, respectively. Then, the lifetime of system X is X=minmaxX11,X12,…,X1n,maxX21,X22,…X2n,…,maxXm1,Xm2,…,Xmn=minX1,X2…,Xm.Figure 3
The logic block diagram of a series-parallel system.Theorem 3.
Given a possibility spaceU,Φ,Poss and a series-parallel system consisting of mn components, the lifetime of components Xiji=1,2,…,m;j=1,2,…,n are unrelated and each a normal, strictly convex fuzzy variable, with continuous possibility distribution functions μX11x,μX12x,…,μX1nx,…,μXmnx, respectively. Let the system lifetime be X with possibility distribution function μXx and each parallel subsystem lifetime be Xi with possibility distribution function μXixi=1,2,…,m. Then, there exist mn positive numbers a11,a12,…,a1n,a21,a22,…,a2n,…,amn (without loss of generality, we assume a11≤a12≤⋯≤a1n; a21≤a22≤⋯≤a2n; …; am1≤am2≤⋯≤amn and a1n≤a2n≤⋯≤amn), such that the possibility distribution function of X is(66)μXx=maxμX1x,μX2x,…,μXmx,x≤a1n,μX1x,a1n<x≤a2n,minμX1x,μX2x,a2n<x≤a3n,…minμX1x,μX2x,…,μXm−1x,am−1n<x≤amn,minμX1x,μX2x,…,μXmx,x>amn,where(67)μXix=minμXi1x,μXi2x,…,μXinx,x≤ai1,minμXi2x,μXi3x,…,μXinx,ai1<x≤ai2,minμXi3x,μXi4x,…,μXinx,ai2<x≤ai3…μXinx,ain−1<x≤ain,maxμXi1x,μXi2x,…,μXinx,x>ain,i=1,2,…,m,. Moreover, the Posbist reliability of the series-parallel system is(68)Rspt=supu>tμXu=1,t≤a1n,μX1t,a1n<t≤a2n,minμX1t,μX2t,a2n<t≤a3n,…minμX1t,μX2t,…,μXm−1t,am−1n<t≤amn,minμX1t,μX2t,…,μXmt,t>amn.Proof.
The considered system can be regarded as a series system consisting ofm systems, each system is a parallel system. For each component Xiji=1,2,…,m;j=1,2,…,n, there exists a unique real number aij∈R+, such that(69)supu≤xμXiju=μXijx,ifx≤aij,1,ifx>aij.
According to Theorem2, the possibility distribution function μXixi=1,2,…,m of each parallel system is(70)μXix=minμXi1x,μXi2x,…,μXinx,x≤ai1,minμXi2x,μXi3x,…,μXinx,ai1<x≤ai2,minμXi3x,μXi4x,…,μXinx,ai2<x≤ai3,…μXinx,ain−1<x≤ain,maxμXi1x,μXi2x,…,μXinx,x>ain.
For each possibility distribution functionμXix and a real number ain, we have(71)supu≥xμXiu=1,ifx≤ain,μXix,ifx>ain.
Based on Theorem1, the possibility distribution function of X is(72)μXx=maxμX1x,μX2x,…,μXmx,x≤a1n,μX1x,a1n<x≤a2n,minμX1x,μX2x,a2n<x≤a3n,…minμX1x,μX2x,…,μXm−1x,am−1n<x≤amn,minμX1x,μX2x,…,μXmx,x>amn,,and the Posbist reliability of the series-parallel system is(73)Rspt=supu>tμXu=1,t≤a1n,μX1t,a1n<t≤a2n,minμX1t,μX2t,a2n<t≤a3n,…minμX1t,μX2t,…,μXm−1t,am−1n<t≤amn,minμX1t,μX2t,…,μXmt,t>amn.
Specially, when the lifetime of the components are assumed as symmetrical Gaussian fuzzy variables, i.e.,Lx and Rx have a normal distribution. We have the following corollary.Corollary 3.
Given a possibility spaceU,Φ,Poss and a series-parallel system consisting of mn components, the possibility distribution function of the lifetime of components Xiji=1,2,…,m;j=1,2,…,n are(74)μXijx=exp−x−m¯ijh2.LetM1=maxm¯11,m¯12,…,m¯1n, M2=maxm¯21,m¯22,…,m¯2n, …, Mm=maxm¯m1,m¯m2,…,m¯mn, and M=minM1,M2,…,Mm; then, the possibility distribution function of the system lifetime X is(75)μXx=exp−x−Mh2.Moreover, the Posbist reliability of the series-parallel system is(76)Rspt=supu>tμXu=1,t≤M,exp−t−Mh2,t>M.Remark 5.
Given a possibility spaceU,Φ,Poss and a series-parallel system consisting of mn components, assume the possibility distribution function of the lifetime of the components Xiji=1,2,…,m;j=1,2,…,n are identically and independently distributed ones μX11x; then, there exists a real number a, such that μXx=μX11x and(77)Rspt=supu>tμXu=1,t≤a,μX11t,t>a.


### 3.4. Posbist Reliability of a Parallel-Series System

Given a possibility spaceU,Φ,Poss, we consider a parallel-series system consisting of m systems, each system is a series system consisting of n components, as shown in Figure 4. Suppose X is the system lifetime, X1,X2,…,Xm are the lifetimes of m series systems and X11,X12,…,X1n,X21,X22, …,X2n,…,Xm1,Xm2,…,Xmn are the lifetimes of the mn components, respectively. Then, the lifetime of system X is X=maxminX11,X12,…,X1n,minX21,X22,…,X2n,…,minXm1,Xm2,…,Xmn=maxX1,X2,…,Xm.Figure 4
The logic block diagram of a parallel-series system.Theorem 4.
Given a possibility spaceU,Φ,Poss and a parallel-series system consisting of mn components, the lifetime of components Xiji=1,2,…,m;j=1,2,…,n are unrelated and each a normal, strictly convex fuzzy variable, with continuous possibility distribution functions μX11x,μX12x,…,μX1nx,…,μXmnx, respectively. Let the system lifetime be X with possibility distribution function μXx and each series system lifetime be Xi with possibility distribution function μXixi=1,2,…,m. Then, there exist mn positive numbers a11,a12,…,a1n,a21,a22,…,a2n,…,amn (without loss of generality, we assume a11≤a12≤⋯≤a1n; a21≤a22≤⋯≤a2n; …; am1≤am2≤⋯≤amn and a11≤a21≤⋯≤am1), such that the possibility distribution function of X is(78)μXx=minμX1x,μX2x,…,μXmx,x≤a11,minμX2x,μX3x,…,μXmx,a11<x≤a21,minμX3x,μX4x,…,μXmx,a21<x≤a31,…μXmx,am−11<x≤am1,maxμX1x,μX2x,…,μXmx,x>am1,where(79)μXix=maxμXi1x,μXi2x,…,μXinx,x≤ai1,μXi1x,ai1<x≤ai2,minμXi1x,μXi2x,ai2<x≤ai3,…minμXi1x,μXi2x,…,μXin−1x,ai−n<x≤ain,minμXi1x,μXi2x,…,μXinx,x>ain,i=1,2,…,m.Moreover, the Posbist reliability of the parallel-series system is(80)Rpst=supu>tμXu=1,t≤am1,maxμX1t,μX2t,…,μXmt,t>am1.Proof.
The considered system can be regarded as a parallel system consisting ofm systems, each system is a series system. For each component Xiji=1,2,…,m;j=1,2,…,n, there exists a unique real number aij∈R+, such that(81)supu≥xμXiju=1,ifx≤aij,μXijx,ifx>aij.
According to Theorem1, the possibility distribution function μXixi=1,2,…,m of each series system is(82)μXix=maxμXi1x,μXi2x,…,μXinx,x≤ai1,μXi1x,ai1<x≤ai2,minμXi1x,μXi2x,ai2<x≤ai3,…minμXi1x,μXi2x,…,μXin−1x,ai−1n<x≤ain,minμXi1x,μXi2x,…,μXinx,x>ain.
For each possibility distribution functionμXix and a real number ai1, we have(83)supu≤xμXiu=μXix,ifx≤ai1,1,ifx>ai1.
Based on Theorem2, the possibility distribution function of X is(84)μXx=minμX1x,μX2x,…,μXmx,x≤a11,minμX2x,μX3x,…,μXmx,a11<x≤a21,minμX3x,μX4x,…,μXmx,a21<x≤a31,…μXmx,am−11<x≤am1,maxμX1x,μX2x,…,μXmx,x>am1,and the Posbist reliability of the parallel-series system is(85)Rpst=supu>tμXu=1,t≤am1,maxμX1t,μX2t,…,μXmt,t>am1.
Specially, when the lifetime of components are assumed as symmetrical Gaussian fuzzy variables, i.e.,Lx and Rx are selected as the normal shape. We have the following corollary.Corollary 4.
Given a possibility spaceU,Φ,Poss and a parallel-series system consisting of mn components, the possibility distribution function of the lifetime of components Xiji=1,2,…,m;j=1,2,…,n are(86)μXijx=exp−x−m¯ijh2.LetM^1=minm¯11,m¯12,…,m¯1n, M^2=minm¯21,m¯22,…,m¯2n, …, M^m=minm¯m1,m¯m2,…,m¯mn and M^=maxM1,M2,…,Mm; then, the possibility distribution function of the system lifetime X is(87)μXx=exp−x−M^h2.Moreover, the Posbist reliability of the parallel-series system is(88)Rpst=supu>tμXu=1,t≤M^,exp−t−M^h2,t>M^.Remark 6.
Given a possibility spaceU,Φ,Poss and a parallel-series system consisting of mn components, assume the possibility distribution function of the lifetime of components Xiji=1,2,…,m;j=1,2,…,n are identically and independently distributed ones μX11x; then, there exists a real number a, such that μXx=μX11x and Rpst=supu>tμXu=1,t≤a,μX11t,t>a..Remark 7.
Assume that the components of each system are identically and independently distributed ones and components between each system are different; combining Remarks2, 4, 5, and 6, one can obtain that the Posbist reliability of each system is independent of the number of components. Moreover, if components between each system are also assumed to be the same, the Posbist reliability of each system is the same.


## 3.1. Posbist Reliability of Series System

Given a possibility spaceU,Φ,Poss, we consider a series system consisting of n components, as shown in Figure 1. Suppose X1,X2,…,Xn are the lifetimes of the n components, respectively. Then, the lifetime of system X is X=minX1,X2,…,Xn.Figure 1
The logic block diagram of a series system.Theorem 1.
Given a possibility spaceU,Φ,Poss and a series system consisting of n components, the lifetime of components Xii=1,2,…,n are unrelated and each a normal, strictly convex fuzzy variable, with continuous possibility distribution functions μX1x,μX2x,…,μXnx, respectively. Let X be the system lifetime, with possibility distribution function μXx. Then, there exist n positive numbers a1,a2,…,an (without loss of generality, we assume a1≤a2≤⋯≤an), such that the possibility distribution function of X is(8)μXx=maxμX1x,μX2x,…,μXnx,x≤a1,μX1x,a1<x≤a2,minμX1x,μX2x,a2<x≤a3,…minμX1x,μX2x,…,μXn−1x,an−1<x≤an,minμX1x,μX2x,…,μXnx,x>an.Moreover, the Posbist reliability of the series system is(9)Rst=supu>tμXu=1,t≤a1,μX1t,a1<t≤a2,minμX1t,μX2t,a2<t≤a3,…minμX1t,μX2t,…,μXn−1t,an−1<t≤an,minμX1t,μX2t,…,μXnt,t>an.Proof.
Since the considered system is a series system and the lifetime of each component isXii=1,2,…,n, then(10)X=minX1,X2,…,Xn.
For each componentXi, there exists a unique real number ai∈R+, such that(11)supu≥xμXiu=1,ifx≤ai,μXix,ifx>ai.
Without loss of generality, we assumea1≤a2≤⋯≤an. According to the definition of system lifetime,(12)μXx=PossX=minX1,X2,…,Xn=x=PossX1=x,X2≥x,X3≥x,…,Xn≥x∪PossX1≥x,X2=x,X3≥x,…,Xn≥x∪⋯∪PossX1≥x,X2≥x,X3≥x,…,Xn=x=maxPossX1=x,X2≥x,X3≥x,…,Xn≥x,PossX1≥x,X2=x,X3≥x,…,Xn≥x,…,PossX1≥x,X2≥x,X3≥x,…,Xn=x.
In addition,(13)PossX1≥x,…,Xi−1≥x,Xi=x,Xi+1≥x,…,Xn≥x=Poss∪a≥xX1=a,…,∪a≥xXi−1=a,Xi=x,∪a≥xXi+1=a,…,∪a≥xXn=a=Poss∪a≥xX1=a,…,Xi−1=a,Xi=x,Xi+1=a,…,Xn=a=supa≥xPossX1=a,…,Xi−1=a,Xi=x,Xi+1=a,…,Xn=a.
Due to the unrelatedness ofXi,i∈1,2,…,n, we have(14)PossX1≥x,…,Xi−1≥x,Xi=x,Xi+1≥x,…,Xn≥x=supa≥xminPossX1=a,…,PossXi−1=a,PossXi=x,PossXi+1=a,…,PossXn=a=supa≥xminμX1a,…,μXi−1a,μXix,μXi+1a,…,μXna=minsupa≥xμX1a,…,supa≥xμXi−1a,μXix,supa≥xμXi+1a,…,supa≥xμXna,=min1,…,1,μXix,1,…,1=μXix,x≤a1,minμX1x,1,…,1,μXix,1,…,1=minμX1x,μXix,a1<x≤a2,…minμX1x,μX2x,…,μXix,1,…,1=minμX1x,μX2x,…,μXix,ai−1<x≤ai,minμX1x,μX2x,…,μXix,1,…,1=minμX1x,μX2x,…,μXix,ai<x≤ai+1,minμX1x,μX2x,…,μXi+1x,1,…,1=minμX1x,μX2x,…,μXi+1x,ai+1<x≤ai+2,minμX1x,μX2x,…,μXi+2x,1,…,1=minμX1x,μX2x,…,μXi+2x,ai+2<x≤ai+3,…minμX1x,μX2x,…,μXn−1x,1=minμX1x,μX2x,…,μXn−1x,an−1<x≤an,minμX1x,μX2x,…,μXnx,x>an.
Therefore,(15)μXx=maxminμX1x,supa≥xμX2a,…,supa≥xμXna…,minsupa≥xμX1a,…,supa≥xμXi−1a,μXix,supa≥xμXi+1a,…,supa≥xμXna,…,minsupa≥xμX1a,…,supa≥xμXn−1a,μXnx.
Whenx≤a1,(16)μXx=maxμX1x,μX2x,…,μXnx.
Whena1<x≤a2,(17)μXx=maxμX1x,minμX1x,μX2xminμX1x,μX3x,…,minμX1x,μXnx,=μX1x.
Whena2<x≤a3,(18)μXx=maxminμX1x,μX2x,minμX1x,μX2x,μX3x,minμX1x,μX2x,μX4x,…,minμX1x,μX2x,μXnx=minμX1x,μX2x.
Whena3<x≤a4,(19)μXx=maxminμX1x,μX2x,μX3x,minμX1x,μX2x,μX3x,μX4x,minμX1x,μX2x,μX3x,μX5x,…,minμX1x,μX2x,μX3x,μXnx=minμX1x,μX2x,μX3x.…
Whenai−1<x≤ai,(20)μXx=maxminμX1x,…,μXi−1x,minμX1x,…,μXi−1x,μXix,minμX1x,…,μXi−1x,μXi+1x,…,minμX1x,…,μXi−1x,μXnx=minμX1x,…,μXi−1x.…
Whenan−1<x≤an,(21)μXx=maxminμX1x,μX2x,…,μXn−1x,minμX1x,μX2x,…,μXn−1x,μXnx=minμX1x,…,μXn−1x.
Whenx>an,(22)μXx=minμX1x,μX2x,…,μXnx.
So, we have(23)μXx=maxμX1x,μX2x,…,μXnx,x≤a1,μX1x,a1<x≤a2,minμX1x,μX2x,a2<x≤a3,…minμX1x,μX2x,…,μXn−1x,an−1<x≤an,minμX1x,μX2x,…,μXnx,x>an.
Since the lifetime of componentsXii=1,2,…,n are unrelated and each a normal, strictly convex fuzzy variable, with continuous possibility distribution functions μX1x,μX2x,…,μXnx, respectively, ai is a unique real number such that supu∈R+μXiu=μXiai=1. When u≤ai, μXiu is a strictly monotonically increasing function. When u>ai, μXiu is a strictly monotonically decreasing function. When ai=+∞, μXiu is a strictly monotonically increasing function in the whole domain.
The Posbist reliability of series system can be expressed asRst=supu>tμXu; combining with (11), we have when t≤a1,(24)Rst=supu>tμXu=maxsupy∈t,a1μXy,supy∈a1,a2μXy,…,supy∈an,+∞μXy=maxsupy∈t,a1maxμX1y,μX2y,…,μXny,supy∈a1,a2μX1y,…,supy∈an,+∞minμX1y,μX2y,…,μXny,where(25)supy∈t,a1maxμX1y,μX2y,…,μXny=maxsupy∈t,a1μX1y,supy∈t,a1μX2y,…,supy∈t,a1μXny.
Sincesupy>tμX1y=1t≤a1, then(26)supy>tμX1y=supy∈t,a1∪y∈a1,+∞μX1y=maxsupy∈t,a1μX1y,supy∈a1,+∞μX1y=maxsupy∈t,a1μX1y,μX1a1+=1.
So,(27)supy∈t,a1μX1y=1.
No matter what the remaining terms’ value is, the maximum value of all terms is 1.
Therefore,(28)Rst=1.
Whena1<t≤a2,(29)Rst=supu>tμXu=maxsupy∈t,a2μXy,supy∈a2,a3μXy,…,supy∈an,+∞μXy=maxsupy∈t,a2μX1y,supy∈a2,a3minμX1y,μX2y,…,supy∈an,+∞minμX1y,μX2y,…,μXny=maxsupy∈t,a2μX1y,minsupy∈a2,a3μX1y,supy∈a2,a3μX2y,…,minsupy∈an,+∞μX1y,supy∈an,+∞μX2y,…,supy∈an,+∞μXny,where(30)supy∈t,a2μX1y=μX1t.
If(31)supy∈a2,a3μX1y≤supy∈a2,a3μX2y,then(32)minsupy∈a2,a3μX1y,supy∈a2,a3μX2y=supy∈a2,a3μX1y≤μX1y.
Otherwise,(33)minsupy∈a2,a3μX1y,supy∈a2,a3μX2y=supy∈a2,a3μX2y≤supy∈a2,a3μX1y≤supy∈t,a2μX1y≤μX1y.
Similarly, fori∈3,4,…,n, when i=n, let i+1=+∞ and the interval becomes a open set:(34)minsupy∈ai,ai+1μX1y,supy∈ai,ai+1μX2y,…,supy∈ai,ai+1μXiy≤μXit.
Therefore,(35)Rst=μX1t.…
Whenan−1<t≤an,(36)Rst=supu>tμXu=maxsupy∈t,anμXy,supy∈an,+∞μXy=maxsupy∈t,anminμX1y,μX2y,…,μXn−1y,supy∈an,+∞minμX1y,μX2y,…,μXny=maxminsupy∈t,anμX1y,supy∈t,anμX2y,…,supy∈t,anμXn−1y,minsupy∈an,+∞μX1y,supy∈an,+∞μX2y,…,supy∈an,+∞μXny=minμX1t,μX2t,…,μXn−1t.
Whent>an,(37)Rst=supu>tμXu=supu>tminμX1u,μX2u,…,μXnu=minsupu>tμX1u,supu>tμX2u,…,supu>tμXnu=minμX1t,μX2t,…,μXnt.
Therefore, the Posbist reliability of the series system is(38)Rst=supu>tμXu=1,t≤a1,μX1t,a1<t≤a2,minμX1t,μX2t,a2<t≤a3,…minμX1t,μX2t,…,μXn−1t,an−1<t≤an,minμX1t,μX2t,…,μXnt,t>an.
Specially, when the lifetime of the components are assumed as symmetrical Gaussian fuzzy variables, i.e.,Lx and Rx have a normal distribution. We have the following corollary.Corollary 1.
Given a possibility spaceU,Φ,Poss and a series system consisting of n components, the possibility distribution function of the lifetime of components Xii=1,2,…,n are assumed to be(39)μXix=exp−x−mih2,without loss of generality; we assume m1≤m2≤⋯≤mn; then, the possibility distribution function of the system lifetime X is(40)μXx=exp−x−m1h2.Moreover, the Posbist reliability of the series system is(41)Rst=supu>tμXu=1,t≤m1,exp−t−m1h2,t>m1.Remark 2.
Given a possibility spaceU,Φ,Poss and a series system consisting of n components, assume the possibility distribution function of the lifetime of components Xii=1,2,…,n are identically and independently distributed ones μX1x; then, there exists a real number a, such that μXx=μX1x and Rst=supu>tμXu=1t≤aμX1tt>a.


## 3.2. Posbist Reliability of Parallel System

Given a possibility spaceU,Φ,Poss, we consider a parallel system consisting of n components, as shown in Figure 2. Suppose X is the system lifetime and X1,X2,…,Xn are the lifetimes of n components, respectively. Then, the lifetime of system X is X=maX1,X2,…,Xn.Figure 2
The logic block diagram of a parallel system.Theorem 2.
Given a possibility spaceU,Φ,Poss and a parallel system consisting of n components, the lifetime of components Xii=1,2,…,n are unrelated and each a normal, strictly convex fuzzy variable, with continuous possibility distribution functions μX1x,μX2x,…,μXnx, respectively. Let the system lifetime be X with possibility distribution function μXx. Then, there exist n positive numbers a1,a2,…,an (without loss of generality, we assume a1≤a2≤⋯≤an), such that the possibility distribution function of X is(42)μXx=minμX1x,μX2x,…,μXnx,x≤a1,minμX2x,μX3x,…,μXnx,a1<x≤a2,minμX3x,μX4x,…,μXnx,a2<x≤a3,…μXnx,an−1<x≤an,maxμX1x,μX2x,…,μXnx,x>an.Moreover, the Posbist reliability of the parallel system is(43)Rpt=supu>tμXu=1,t≤an,maxμX1t,μX2t,…,μXnt,t>an.Proof.
Similar to Theorem1, for each component Xi, there exists a unique real number ai∈R+, such that(44)supu≤xμXiu=μXix,ifx≤ai,1,ifx>ai.
Due to the assumptiona1≤a2≤⋯≤an and combining with the definition of system lifetime, we have(45)μXx=PossX=maxX1,X2,…,Xn=x=maxPossX1=x,X2≤x,X3≤x,…,Xn≤x,PossX1≤x,X2=x,X3≤x,…,Xn≤x,…,PossX1≤x,X2≤x,X3≤x,…,Xn=x,where(46)PossX1≤x,…,Xi−1≤x,Xi=x,Xi+1≤x,…,Xn≤x=supa≤xPossX1=a,…,Xi−1=a,Xi=x,Xi+1=a,…,Xn=a.
Because of the unrelatedness ofXi,i∈1,2,…,n, we have(47)PossX1≤x,…,Xi−1≤x,Xi=x,Xi+1≤x,…,Xn≤x=supa≤xminPossX1=a,…,PossXi−1=a,PossXi=x,PossXi+1=a,…,PossXn=a=supa≤xminμX1a,…,μXi−1a,μXix,μXi+1a,…,μXna=minsupa≤xμX1a,…,supa≤xμXi−1a,μXix,supa≤xμXi+1a,…,supa≤xμXna,=minμX1x,μX2x,…,μXix,…,μXnx,x≤a1min1,μX2x,μX3x,…,μXix,…,μXnx=minμX2x,μX3x,…,μXnx,a1<x≤a2,min1,1,μX3x,μX4x,…,μXix,…,μXnx=minμX3x,μX4x,…,μXnx,a2<x≤a3,…min1,…,1,μXix,…,μXnx=minμXix,μXi+1x,…,μXnx,ai−1<x≤ai,min1,…,1,μXix,…,μXnx=minμXix,μXi+1x,…,μXnx,ai<x≤ai+1,min1,…,1,μXix,1,μXi+2x,…,μXnx=minμXix,μXi+2x,…,μXnx,ai+1<x≤ai+2,min1,…,1,μXix,1,1,μXi+3x,…,μXnx=minμXix,μXi+3x,…,μXnx,ai+2<x≤ai+3,…min1,…,1,μXix,1,…,1,μXnx=minμXix,μXnx,an−1<x≤an,min1,…,1,μXix,1,…,1=μXix,x>an.
Therefore,(48)μXx=maxminμX1x,supa≤xμX2a,…,supa≤xμXna,…,minsupa≤xμX1a,…,supa≤xμXi−1a,μXix,supa≤xμXi+1a,…,supa≤xμXna,…,minsupa≤xμX1a,…,supa≤xμXn−1a,μXnx.
Whenx≤a1,(49)μXx=minμX1x,μX2x,…,μXnx.
Whena1<x≤a2,(50)μXx=maxminμX1x,μX2x,…,μXnx,minμX2x,…,μXnx,…,minμX2x,…,μXnx=minμX2x,…,μXnx.
Whena2<x≤a3,(51)μXx=maxminμX1x,μX3x,…,μXnx,minμX2x,…,μXnx,minμX3x,…,μXnx,…,minμX3x,…,μXnx=minμX3x,…,μXnx.
Whena3<x≤a4,(52)μXx=maxminμX1x,μX4x,…,μXnx,minμX2x,μX4x,…,μXnx,minμX3x,μX4x,…,μXnx,minμX4x,…,μXnx,…,minμX4x,…,μXnx=minμX4x,…,μXnx.…
Whenai−1<x≤ai,(53)μXx=maxminμX1x,μXix,…,μXnx,minμX2x,μXix,…,μXnx,…,minμXi−1x,μXix,…,μXnx,minμXix,…,μXnx,…,minμXix,…,μXnx=minμXix,…,μXnx.…
Whenan−1<x≤an,(54)μXx=maxminμX1x,μXnx,minμX2x,μXnx,…,minμXn−1x,μXnx,μXnx=μXnx.
Whenx>an,(55)μXx=maxμX1x,μX2x,…,μXnx.
So, we have(56)μXx=minμX1x,μX2x,…,μXnx,x≤a1,minμX2x,μX3x,…,μXnx,a1<x≤a2,minμX3x,μX4x,…,μXnx,a2<x≤a3,…μXnx,an−1<x≤anmaxμX1x,μX2x,…,μXnx,x>an.
The Posbist reliability of the parallel system is(57)Rpt=supu>tμXu.
Since the pointai is a unique maximum value point, based on the monotonicity of μXix, one has when t≤a1(58)Rpt=supu>tμXu=supt,a1∪a1,a2∪⋯∪an,+∞μXu=maxsupy∈t,a1minμX1y,μX2y,…,μXny,supy∈a1,a2minμX2y,μX3y,…,μXny,…,supy∈an−1,anμXny,supy∈an,+∞maxμX1y,μX2y,…,μXny=maxminsupy∈t,a1μX1y,…,supy∈t,a2μXny,minsupy∈a1,a2μX2y,…,supy∈a1,a2μXny,…,supy∈an−1,anμXny,maxsupy∈an,+∞μX1y,supy∈an,+∞μX2y,…,supy∈an,+∞μXny=maxminsupy∈t,a1μX1y,supy∈t,a2μX2y,…,supy∈t,anμXny,maxsupy∈an,+∞μX1y,supy∈an,+∞μX2y,…,supy∈an,+∞μXny=1.
Whena1<t≤a2,(59)Rpt=supu>tμXu=supt,a2∪a2,a3∪⋯∪an,+∞μXu=maxsupy∈t,a2minμX2y,μX3y,…,μXny,supy∈a2,a3minμX3y,μX4y,…,μXny,…,supy∈an−1,anμXny,supy∈an,+∞maxμX1y,μX2y,…,μXny=maxminsupy∈t,a2μX2y,…,supy∈t,a2μXny,minsupy∈a2,a3μX3y,…,supy∈a2,a3μXny,…,supy∈an−1,anμXny,maxsupy∈an,+∞μX1y,supy∈an,+∞μX2y,…,supy∈an,+∞μXny=maxminsupy∈t,a2μX2y,ssupy∈t,a3μX3y,…,supy∈t,anμXny,maxsupy∈an,+∞μX1y,supy∈an,+∞μX2y,…,supy∈an,+∞μXny=1.…
Whenan−1<t≤an,(60)Rpt=supu>tμXu=supt,an∪an,+∞μXu=maxsupy∈t,anμXny,supy∈an,+∞maxμX1y,μX2y,…,μXny=1.
Whent>an,(61)Rpt=supu>tμXu=supt,+∞maxμX1y,μX2y,…,μXny=maxμX1t,μX2t,…,μXnt.
Therefore, the Posbist reliability of the parallel system is(62)Rpt=supu>tμXu=1,t≤an,maxμX1t,μX2t,…,μXnt,t>an.Remark 3.
From the lifetime of the parallel system,X=maxX1,X2,…,Xn, and equations (46) and (47), we can know that the property of μXiu what we are focused on is the supremum value when u≤x. For the series system case, due to the lifetime is defined by X=minX1,X2,…,Xn, the property of μXiu what we are focused on is the supremum value when u≥x. This is a big difference between the proof of the theory in series and parallel systems.
Specially, when the lifetime of the components are assumed as symmetrical Gaussian fuzzy variables, i.e.,Lx and Rx have a normal distribution. We have the following corollary.Corollary 2.
Given a possibility spaceU,Φ,Poss and a parallel system consisting of n components, the possibility distribution function of the lifetime of components Xii=1,2,…,n are(63)μXix=exp−x−mih2,without loss of generality; we assume m1≤m2≤⋯≤mn, then the possibility distribution function of the system lifetime X is(64)μXx=exp−x−mnh2,
Moreover, the Posbist reliability of the parallel system is(65)Rpt=supu>tμXu=1,t≤mnexp−t−mnh2,t>mn.Remark 4.
Given a possibility spaceU,Φ,Poss and a parallel system consisting of n components, assume the possibility distribution function of the lifetime of the components Xii=1,2,…,n are identically and independently distributed ones μX1x; then, there exists a real number a, such that μXx=μX1x and Rpt=supu>tμXu=1,t≤a,μX1t,t>a.


## 3.3. Posbist Reliability of a Series-Parallel System

Given a possibility spaceU,Φ,Poss, we consider a series-parallel system consisting of m systems, each system is a parallel system consisting of n components, as shown in Figure 3. Suppose X is the system lifetime, X1,X2,…,Xm are the lifetimes of m parallel subsystems, and X11,X12,…,X1n,X21,X22,…,X2n,…,Xm1,Xm2,…,Xmn, are the lifetimes of the mn components, respectively. Then, the lifetime of system X is X=minmaxX11,X12,…,X1n,maxX21,X22,…X2n,…,maxXm1,Xm2,…,Xmn=minX1,X2…,Xm.Figure 3
The logic block diagram of a series-parallel system.Theorem 3.
Given a possibility spaceU,Φ,Poss and a series-parallel system consisting of mn components, the lifetime of components Xiji=1,2,…,m;j=1,2,…,n are unrelated and each a normal, strictly convex fuzzy variable, with continuous possibility distribution functions μX11x,μX12x,…,μX1nx,…,μXmnx, respectively. Let the system lifetime be X with possibility distribution function μXx and each parallel subsystem lifetime be Xi with possibility distribution function μXixi=1,2,…,m. Then, there exist mn positive numbers a11,a12,…,a1n,a21,a22,…,a2n,…,amn (without loss of generality, we assume a11≤a12≤⋯≤a1n; a21≤a22≤⋯≤a2n; …; am1≤am2≤⋯≤amn and a1n≤a2n≤⋯≤amn), such that the possibility distribution function of X is(66)μXx=maxμX1x,μX2x,…,μXmx,x≤a1n,μX1x,a1n<x≤a2n,minμX1x,μX2x,a2n<x≤a3n,…minμX1x,μX2x,…,μXm−1x,am−1n<x≤amn,minμX1x,μX2x,…,μXmx,x>amn,where(67)μXix=minμXi1x,μXi2x,…,μXinx,x≤ai1,minμXi2x,μXi3x,…,μXinx,ai1<x≤ai2,minμXi3x,μXi4x,…,μXinx,ai2<x≤ai3…μXinx,ain−1<x≤ain,maxμXi1x,μXi2x,…,μXinx,x>ain,i=1,2,…,m,. Moreover, the Posbist reliability of the series-parallel system is(68)Rspt=supu>tμXu=1,t≤a1n,μX1t,a1n<t≤a2n,minμX1t,μX2t,a2n<t≤a3n,…minμX1t,μX2t,…,μXm−1t,am−1n<t≤amn,minμX1t,μX2t,…,μXmt,t>amn.Proof.
The considered system can be regarded as a series system consisting ofm systems, each system is a parallel system. For each component Xiji=1,2,…,m;j=1,2,…,n, there exists a unique real number aij∈R+, such that(69)supu≤xμXiju=μXijx,ifx≤aij,1,ifx>aij.
According to Theorem2, the possibility distribution function μXixi=1,2,…,m of each parallel system is(70)μXix=minμXi1x,μXi2x,…,μXinx,x≤ai1,minμXi2x,μXi3x,…,μXinx,ai1<x≤ai2,minμXi3x,μXi4x,…,μXinx,ai2<x≤ai3,…μXinx,ain−1<x≤ain,maxμXi1x,μXi2x,…,μXinx,x>ain.
For each possibility distribution functionμXix and a real number ain, we have(71)supu≥xμXiu=1,ifx≤ain,μXix,ifx>ain.
Based on Theorem1, the possibility distribution function of X is(72)μXx=maxμX1x,μX2x,…,μXmx,x≤a1n,μX1x,a1n<x≤a2n,minμX1x,μX2x,a2n<x≤a3n,…minμX1x,μX2x,…,μXm−1x,am−1n<x≤amn,minμX1x,μX2x,…,μXmx,x>amn,,and the Posbist reliability of the series-parallel system is(73)Rspt=supu>tμXu=1,t≤a1n,μX1t,a1n<t≤a2n,minμX1t,μX2t,a2n<t≤a3n,…minμX1t,μX2t,…,μXm−1t,am−1n<t≤amn,minμX1t,μX2t,…,μXmt,t>amn.
Specially, when the lifetime of the components are assumed as symmetrical Gaussian fuzzy variables, i.e.,Lx and Rx have a normal distribution. We have the following corollary.Corollary 3.
Given a possibility spaceU,Φ,Poss and a series-parallel system consisting of mn components, the possibility distribution function of the lifetime of components Xiji=1,2,…,m;j=1,2,…,n are(74)μXijx=exp−x−m¯ijh2.LetM1=maxm¯11,m¯12,…,m¯1n, M2=maxm¯21,m¯22,…,m¯2n, …, Mm=maxm¯m1,m¯m2,…,m¯mn, and M=minM1,M2,…,Mm; then, the possibility distribution function of the system lifetime X is(75)μXx=exp−x−Mh2.Moreover, the Posbist reliability of the series-parallel system is(76)Rspt=supu>tμXu=1,t≤M,exp−t−Mh2,t>M.Remark 5.
Given a possibility spaceU,Φ,Poss and a series-parallel system consisting of mn components, assume the possibility distribution function of the lifetime of the components Xiji=1,2,…,m;j=1,2,…,n are identically and independently distributed ones μX11x; then, there exists a real number a, such that μXx=μX11x and(77)Rspt=supu>tμXu=1,t≤a,μX11t,t>a.


## 3.4. Posbist Reliability of a Parallel-Series System

Given a possibility spaceU,Φ,Poss, we consider a parallel-series system consisting of m systems, each system is a series system consisting of n components, as shown in Figure 4. Suppose X is the system lifetime, X1,X2,…,Xm are the lifetimes of m series systems and X11,X12,…,X1n,X21,X22, …,X2n,…,Xm1,Xm2,…,Xmn are the lifetimes of the mn components, respectively. Then, the lifetime of system X is X=maxminX11,X12,…,X1n,minX21,X22,…,X2n,…,minXm1,Xm2,…,Xmn=maxX1,X2,…,Xm.Figure 4
The logic block diagram of a parallel-series system.Theorem 4.
Given a possibility spaceU,Φ,Poss and a parallel-series system consisting of mn components, the lifetime of components Xiji=1,2,…,m;j=1,2,…,n are unrelated and each a normal, strictly convex fuzzy variable, with continuous possibility distribution functions μX11x,μX12x,…,μX1nx,…,μXmnx, respectively. Let the system lifetime be X with possibility distribution function μXx and each series system lifetime be Xi with possibility distribution function μXixi=1,2,…,m. Then, there exist mn positive numbers a11,a12,…,a1n,a21,a22,…,a2n,…,amn (without loss of generality, we assume a11≤a12≤⋯≤a1n; a21≤a22≤⋯≤a2n; …; am1≤am2≤⋯≤amn and a11≤a21≤⋯≤am1), such that the possibility distribution function of X is(78)μXx=minμX1x,μX2x,…,μXmx,x≤a11,minμX2x,μX3x,…,μXmx,a11<x≤a21,minμX3x,μX4x,…,μXmx,a21<x≤a31,…μXmx,am−11<x≤am1,maxμX1x,μX2x,…,μXmx,x>am1,where(79)μXix=maxμXi1x,μXi2x,…,μXinx,x≤ai1,μXi1x,ai1<x≤ai2,minμXi1x,μXi2x,ai2<x≤ai3,…minμXi1x,μXi2x,…,μXin−1x,ai−n<x≤ain,minμXi1x,μXi2x,…,μXinx,x>ain,i=1,2,…,m.Moreover, the Posbist reliability of the parallel-series system is(80)Rpst=supu>tμXu=1,t≤am1,maxμX1t,μX2t,…,μXmt,t>am1.Proof.
The considered system can be regarded as a parallel system consisting ofm systems, each system is a series system. For each component Xiji=1,2,…,m;j=1,2,…,n, there exists a unique real number aij∈R+, such that(81)supu≥xμXiju=1,ifx≤aij,μXijx,ifx>aij.
According to Theorem1, the possibility distribution function μXixi=1,2,…,m of each series system is(82)μXix=maxμXi1x,μXi2x,…,μXinx,x≤ai1,μXi1x,ai1<x≤ai2,minμXi1x,μXi2x,ai2<x≤ai3,…minμXi1x,μXi2x,…,μXin−1x,ai−1n<x≤ain,minμXi1x,μXi2x,…,μXinx,x>ain.
For each possibility distribution functionμXix and a real number ai1, we have(83)supu≤xμXiu=μXix,ifx≤ai1,1,ifx>ai1.
Based on Theorem2, the possibility distribution function of X is(84)μXx=minμX1x,μX2x,…,μXmx,x≤a11,minμX2x,μX3x,…,μXmx,a11<x≤a21,minμX3x,μX4x,…,μXmx,a21<x≤a31,…μXmx,am−11<x≤am1,maxμX1x,μX2x,…,μXmx,x>am1,and the Posbist reliability of the parallel-series system is(85)Rpst=supu>tμXu=1,t≤am1,maxμX1t,μX2t,…,μXmt,t>am1.
Specially, when the lifetime of components are assumed as symmetrical Gaussian fuzzy variables, i.e.,Lx and Rx are selected as the normal shape. We have the following corollary.Corollary 4.
Given a possibility spaceU,Φ,Poss and a parallel-series system consisting of mn components, the possibility distribution function of the lifetime of components Xiji=1,2,…,m;j=1,2,…,n are(86)μXijx=exp−x−m¯ijh2.LetM^1=minm¯11,m¯12,…,m¯1n, M^2=minm¯21,m¯22,…,m¯2n, …, M^m=minm¯m1,m¯m2,…,m¯mn and M^=maxM1,M2,…,Mm; then, the possibility distribution function of the system lifetime X is(87)μXx=exp−x−M^h2.Moreover, the Posbist reliability of the parallel-series system is(88)Rpst=supu>tμXu=1,t≤M^,exp−t−M^h2,t>M^.Remark 6.
Given a possibility spaceU,Φ,Poss and a parallel-series system consisting of mn components, assume the possibility distribution function of the lifetime of components Xiji=1,2,…,m;j=1,2,…,n are identically and independently distributed ones μX11x; then, there exists a real number a, such that μXx=μX11x and Rpst=supu>tμXu=1,t≤a,μX11t,t>a..Remark 7.
Assume that the components of each system are identically and independently distributed ones and components between each system are different; combining Remarks2, 4, 5, and 6, one can obtain that the Posbist reliability of each system is independent of the number of components. Moreover, if components between each system are also assumed to be the same, the Posbist reliability of each system is the same.


## 4. Numerical Examples

### 4.1. The Posbist Reliability of the Typical System

We are in a position to compare the Posbist reliability of a series system, parallel system, series-parallel system, and parallel-series system. Each system is consisting of four components. The chosen parameter values arem111=140, m212=160, m321=180, m422=200, h111=50, h212=70, h321=90, and h422=110. The possibility distribution function of the lifetime of each component is(89)μX111x=exp−x−140502,μX212x=exp−x−160702,μX321x=exp−x−180902,μX422x=exp−x−2001102.When the considered systems are series and parallel system, the lifetime of components areX1, X2, X3, and X4 with continuous possibility distribution functions μX1x, μX2x, μX3x, and μX4x. When the considered systems are series-parallel and parallel-series system, the lifetime of the components are X11, X12, X21, and X22 with continuous possibility distribution functions μX11x, μX12x, μX21x, and μX22x. From (89), we have μX1x=μX11x, μX2x=μX12x, μX3x=μX21x, and μX4x=μX22x. Let t=170hour, the Posbist reliability of each system is calculated as follows.(1) The series system: from (9), the Posbist reliability for the series system consisting of four components is(90)Rst=1,t≤140,exp−x−140502,140<t≤160,minexp−x−140502,exp−x−160702,160<t≤180,minexp−x−140502,exp−x−160702,exp−x−180902,180<t≤200,minexp−x−140502,exp−x−160702,exp−x−180902,exp−x−2001102,t>200.Whent=170hour, we can obtain(91)Rs170=minexp−170−140502,exp−170−160702=0.5488.(2) The parallel system: from (43), the Posbist reliability for the parallel system consisting of four components is(92)Rpt=1,t≤200,maxexp−x−140502,exp−x−160702,exp−x−180902,exp−x−2001102,t>200.Whent=170hour, we can obtain(93)Rp170=1.(3) The series-parallel system: from (73), the Posbist reliability for the series-parallel system consisting of four components is(94)Rspt=1,t≤160,μX1t,160<t≤200,minμX1t,μX2t,t>200,where(95)μX1x=minexp−140−x502,exp−160−x702,x≤140,exp−160−x702,140<x≤160,maxexp−x−140502,exp−x−160702,x>160,μX2x=minexp−180−x902,exp−200−x1102,x≤180,exp−200−x1102,180<x≤200,maxexp−x−180902,exp−x−2001102,x>200.Whent=170hour, we can obtain(96)Rsp170=maxexp−170−140502exp−170−160702=0.8669.(4) The parallel-series system: from (80), the Posbist reliability for the parallel-series system consisting of four components is(97)Rpst=1,t≤180,maxμX1t,μX2t,t>180,where(98)μX1x=maxexp−140−x502,exp−160−x702,x≤140,exp−x−140502,140<x≤160,minexp−x−140502,exp−160−x702,x>160,μX2x=maxexp−180−x902,exp−200−x1102,x≤180,exp−x−180902,180<x≤200,minexp−x−180902,exp−x−2001102,x>200.Whent=170hour, we can obtain(99)Rps170=1.Whent=170hour, the Posbist reliability for each system is shown in Table 1.The Posbist reliability for different systems consisting of four components at various points is shown in Figure 5.Table 1
The Posbist reliability for different systems of four components.
SeriesParallelSeries-parallelParallel-seriesPosbist reliability0.548810.86691Figure 5
The Posbist reliability for different systems at various points (hours).From Table1 and Figure 5, we have, at the same instant, for different systems consisting of the same components, the Posbist reliability of the parallel system is no less than the Posbist reliability of the parallel-series system. The Posbist reliability of the parallel-series system is no less than the Posbist reliability of the series-parallel system. The Posbist reliability of the series-parallel system is no less than the Posbist reliability of the series system.


### 4.2. A Comparison of Posbist and Probabilistic Reliability Models for Safety Evaluations

For the systems consisting of same components, at the same instant, safety evaluation is different between Posbist and probabilistic reliability models.F⋅ represents the failure event of system; p and π represent the failure probability and failure possibility, respectively. We consider the series system, parallel system, series-parallel system, and parallel-series system consisting of mn components. For the series-parallel system, each parallel system consists of n components. For the parallel-series system, each series system consists of n components. Assume the components are identically and independently distributed ones. For the series system, Fsp=1−1−pmn and Fsπ=π. For the parallel system, Fpp=pmn and Fpπ=π. For the series-parallel system, Fspp=1−1−pnm and Fspπ=π. For the parallel-series system, Fpsp=1−1−pnm and Fpsπ=π. Therefore, let p=0.6 and π=0.5, and the safety evaluation of each system is shown in Figure 6.Figure 6
A comparison of Posbist and probabilistic reliability models for safety evaluations.
(a)(b)(c)(d)From Figure6, for each system, the possibility of failure Fπ is a constant. However, the probability of failure Fp changes. For the series system, as shown in (a) of Figure 6, the probability of failure Fp increases with the increasement of the number of components mn. For the parallel system, as shown in (b) of Figure 6, the probability of failure Fp decreases with the increasement of the number of components mn. For the series-parallel system, as shown in (c) of Figure 6, if the system components’ number n is fixed, the probability of failure Fp increases with the increasement of the number of m; if the number m is fixed, the probability of failure Fp decreases with the increasement of the system components number of n. For the parallel-series system, as shown in (d) of Figure 6, if the system components’ number n is fixed, the probability of failure Fp decreases with the increasement of the number of m; if the number m is fixed, the probability of failure Fp increases with the increasement of the system components number of n. Therefore, for different cases, a suitable model should be selected [27].


## 4.1. The Posbist Reliability of the Typical System

We are in a position to compare the Posbist reliability of a series system, parallel system, series-parallel system, and parallel-series system. Each system is consisting of four components. The chosen parameter values arem111=140, m212=160, m321=180, m422=200, h111=50, h212=70, h321=90, and h422=110. The possibility distribution function of the lifetime of each component is(89)μX111x=exp−x−140502,μX212x=exp−x−160702,μX321x=exp−x−180902,μX422x=exp−x−2001102.When the considered systems are series and parallel system, the lifetime of components areX1, X2, X3, and X4 with continuous possibility distribution functions μX1x, μX2x, μX3x, and μX4x. When the considered systems are series-parallel and parallel-series system, the lifetime of the components are X11, X12, X21, and X22 with continuous possibility distribution functions μX11x, μX12x, μX21x, and μX22x. From (89), we have μX1x=μX11x, μX2x=μX12x, μX3x=μX21x, and μX4x=μX22x. Let t=170hour, the Posbist reliability of each system is calculated as follows.(1) The series system: from (9), the Posbist reliability for the series system consisting of four components is(90)Rst=1,t≤140,exp−x−140502,140<t≤160,minexp−x−140502,exp−x−160702,160<t≤180,minexp−x−140502,exp−x−160702,exp−x−180902,180<t≤200,minexp−x−140502,exp−x−160702,exp−x−180902,exp−x−2001102,t>200.Whent=170hour, we can obtain(91)Rs170=minexp−170−140502,exp−170−160702=0.5488.(2) The parallel system: from (43), the Posbist reliability for the parallel system consisting of four components is(92)Rpt=1,t≤200,maxexp−x−140502,exp−x−160702,exp−x−180902,exp−x−2001102,t>200.Whent=170hour, we can obtain(93)Rp170=1.(3) The series-parallel system: from (73), the Posbist reliability for the series-parallel system consisting of four components is(94)Rspt=1,t≤160,μX1t,160<t≤200,minμX1t,μX2t,t>200,where(95)μX1x=minexp−140−x502,exp−160−x702,x≤140,exp−160−x702,140<x≤160,maxexp−x−140502,exp−x−160702,x>160,μX2x=minexp−180−x902,exp−200−x1102,x≤180,exp−200−x1102,180<x≤200,maxexp−x−180902,exp−x−2001102,x>200.Whent=170hour, we can obtain(96)Rsp170=maxexp−170−140502exp−170−160702=0.8669.(4) The parallel-series system: from (80), the Posbist reliability for the parallel-series system consisting of four components is(97)Rpst=1,t≤180,maxμX1t,μX2t,t>180,where(98)μX1x=maxexp−140−x502,exp−160−x702,x≤140,exp−x−140502,140<x≤160,minexp−x−140502,exp−160−x702,x>160,μX2x=maxexp−180−x902,exp−200−x1102,x≤180,exp−x−180902,180<x≤200,minexp−x−180902,exp−x−2001102,x>200.Whent=170hour, we can obtain(99)Rps170=1.Whent=170hour, the Posbist reliability for each system is shown in Table 1.The Posbist reliability for different systems consisting of four components at various points is shown in Figure 5.Table 1
The Posbist reliability for different systems of four components.
SeriesParallelSeries-parallelParallel-seriesPosbist reliability0.548810.86691Figure 5
The Posbist reliability for different systems at various points (hours).From Table1 and Figure 5, we have, at the same instant, for different systems consisting of the same components, the Posbist reliability of the parallel system is no less than the Posbist reliability of the parallel-series system. The Posbist reliability of the parallel-series system is no less than the Posbist reliability of the series-parallel system. The Posbist reliability of the series-parallel system is no less than the Posbist reliability of the series system.


## 4.2. A Comparison of Posbist and Probabilistic Reliability Models for Safety Evaluations

For the systems consisting of same components, at the same instant, safety evaluation is different between Posbist and probabilistic reliability models.F⋅ represents the failure event of system; p and π represent the failure probability and failure possibility, respectively. We consider the series system, parallel system, series-parallel system, and parallel-series system consisting of mn components. For the series-parallel system, each parallel system consists of n components. For the parallel-series system, each series system consists of n components. Assume the components are identically and independently distributed ones. For the series system, Fsp=1−1−pmn and Fsπ=π. For the parallel system, Fpp=pmn and Fpπ=π. For the series-parallel system, Fspp=1−1−pnm and Fspπ=π. For the parallel-series system, Fpsp=1−1−pnm and Fpsπ=π. Therefore, let p=0.6 and π=0.5, and the safety evaluation of each system is shown in Figure 6.Figure 6
A comparison of Posbist and probabilistic reliability models for safety evaluations.
(a)(b)(c)(d)From Figure6, for each system, the possibility of failure Fπ is a constant. However, the probability of failure Fp changes. For the series system, as shown in (a) of Figure 6, the probability of failure Fp increases with the increasement of the number of components mn. For the parallel system, as shown in (b) of Figure 6, the probability of failure Fp decreases with the increasement of the number of components mn. For the series-parallel system, as shown in (c) of Figure 6, if the system components’ number n is fixed, the probability of failure Fp increases with the increasement of the number of m; if the number m is fixed, the probability of failure Fp decreases with the increasement of the system components number of n. For the parallel-series system, as shown in (d) of Figure 6, if the system components’ number n is fixed, the probability of failure Fp decreases with the increasement of the number of m; if the number m is fixed, the probability of failure Fp increases with the increasement of the system components number of n. Therefore, for different cases, a suitable model should be selected [27].


## 5. Conclusion

The problem of Posbist reliability theory for typical systems with multiple different components has been further investigated via the possibility theory. The results have been given when typical systems withn different components. Moreover, our results have been concretely derived where the exactly value can be obtained at any moment, unlike existing ones only given by a lower and upper bound. Finally, numerical examples have been provided to compare the Posbist reliability between typical systems of four types: series, parallel, series-parallel, and parallel-series. Moreover, numerical safety evaluations have been given based on possibilistic and probabilistic models.



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