Mechanical system reliability analysis using a combination of graph theory and Boolean function

Mechanical system reliability analysis using a combination of graphtheory and Boolean function

J. Tang*

Center for Computer-Aided Design, 308 IATL, University of Iowa, Iowa City, IA 52242, USA

Received 12 October 1999; accepted 25 September 2000

Abstract

A new method based on graph theory and Boolean function for assessing reliability of mechanical systems is proposed. The procedure for

this approach consists of two parts. By using the graph theory, the formula for the reliability of a mechanical system that considers the

interrelations of subsystems or components is generated. Use of the Boolean function to examine the failure interactions of two particular

elements of the system, followed with demonstrations of how to incorporate such failure dependencies into the analysis of larger systems, a

constructive algorithm for quantifying the genuine interconnections between the subsystems or components is provided. The combination of

graph theory and Boolean function provides an effective way to evaluate the reliability of a large, complex mechanical system. A numerical

example demonstrates that this method an effective approaches in system reliability analysis. q 2001 Elsevier Science Ltd. All rights

reserved.

Keywords: Mechanical system reliability; Graph theory; Bryant tree; Root vertex

1. Introduction

Reliability has become a key factor in the design and

operation of today's large, complex, and expensive mechan-

ical systems. The integrity of modern mechanical systems is

strongly dependent upon the durability and reliability of the

components. However, reliability theory depends heavily on

an understanding of failure physics modeling and on the

techniques of probability and statistics. Thus, mathematical

reliability models play a very important role in reliability

analysis. In today's reliability analysis perhaps the most

pervasive technique is that of estimating the reliability of

a system in terms of the reliability of its components. In fact,

reliability predictions for complex systems typically begin

with predictions of the probabilities of mission success for

the components in a system, and the component predictions

are combined in accordance with a logic model that

describes how the components interact in a system [11].

Then, the result is a predicted mission success probability

for the system. Therefore, it is of great importance to have

practical algorithms which ef®ciently predict the reliability

of complex systems, and which also give useful design

information with respect to individual units. For this reason

a substantial number of formal approaches, such as Fault

Tree Analysis (FTA) and Failure Mode Effects and Criti-

cality Analysis (FMECA), in the area of reliability, have

been carried out for mechanical systems.

The FTA incorporates the desired consideration for

mechanical systems in terms of the topology of a system

and interactions; therefore, it usually is used as a system

reliability model in ®nding the important modes of failure

in a system, and in the assessment of ®rst occurrence prob-

abilities of the top event of a system. The method is math-

ematically correct; however, it requires extensive

calculations for a complex fault tree. General speaking,

for a sequence of N events there will be 2N branches of

the tree. Although the number may be reduced by eliminat-

ing impossible branches, this computational processing

requirement may still be beyond the capability of available

machines [9,12]. Also, discrepancies still exist between

theoretical reliability estimation and actual failure observed

in practice. Intuitively, it might appear that this poor corre-

lation is because the model is not a good functional repre-

sentation of the real system [3]. In such analysis it is

frequently assumed that the component failure is mutually

independent, whereas in reality, this is often not the case.

Therefore, it is necessary to replace the simple reliability

models with more sophisticated models that take into

account the interactions of component failures. In another

Reliability Engineering and System Safety 72 (2001) 21±30

0951-8320/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved.

PII: S0951-8320(00)00099-5

www.elsevier.com/locate/ress

* Fax: 11-319-335-3380.

E-mail address: [email protected] (J. Tang).

words, FTA identi®es the possible causes of a particular

failure and is useful for troubleshooting at any level from

component to system, while the reliability assessment needs

a high degree of effort [7,9,12]. Since it differs itself from

the approach to the problem and the scope of the analysis,

FTA may be looked upon as an alternative to the use of

reliability block diagrams in determining system reliability

in terms of the corresponding components.

On the other hand, FMECA is one of the most widely

employed techniques for enumerating the possible modes

by which components may fail, and for tracing through the

characteristics and consequences of each mode of failure on

the system as a whole. It allows the assessment of the prob-

ability of a failure occurrence as well as the effect of a fail-

ure. The quantitative assessment permits relative ranking of

failure risks and provides input to other analyses. The

method is an analytical technique that ensures all possible

failure modes of a system have been addressed [7,9,11±13].

It is primarily qualitative in nature, although some estima-

tions of failure probabilities are often included. The empha-

sis in FMECA is usually on the basic physical phenomena

that can cause a device to fail. Therefore, it often serves as a

suitable starting point for enumerating and understanding

the failure mechanisms before the progression of accidents

when they pass through several stages and analyze the

effects of component redundancies on system safety.

Hence, for quantifying system behavior, other approaches,

such as event-tree or FTA, are often combined as a supple-

ment to FMECA methods.

In this paper, a system-reliability model based on graph

theory and Boolean function is proposed to formulate a

system equation, characteristic of the reliability of the

system. The methodology proposed incorporates the graph

theory for system level reliability and Boolean analysis for

interactions. Therefore, this method considers not only the

topology (structure) of the system, but also effectively

incorporates interactions between components. Through

the graph theory, a binary argument can be used for consid-

ering the connection between the components. Hence, an

algorithm, which uses graph theory for the cause and effect

relationship between components and assessing reliability

of a complex system, is proposed. The graph theory is used

to re¯ect the logical relationship among various fault events.

Nevertheless, the structure of the system is not explicitly

evident as the only logical relationship among various fault

events in a mechanical system; the physical interconnection

is also important [3]. Therefore, the problem of how to

evaluate the physical interconnection is induced. In order

to quantify the physical interconnections, a new methodol-

ogy based on a Boolean Equation, Shannon's formula, has

also been proposed in this paper. Consequently, the physical

interconnection is calculated in a straightforward fashion.

Obviously, even though the variables have only two

possible values and functions with these same two values,

there exists the almost unlimited possibility of combining

many functions of many variables through many stages of

modern engineering systems that lends Boolean analysis its

own typical complexity in theory and practice [1]. For

instance, for one variable X only, there are four Boolean

functions, namely two constants, f1�X� � 0; f2�X� � 1,

and two logics, f3�X� � X; and f4�X� � �X: For an n-

component system, it still is a huge computational problem.

However, in the proposed method because the use of

Boolean analysis is limited to examine failure interactions

between the components, followed with estimations of how

to incorporate such failure dependencies into the analysis of

a system, only the partial components will be involved in

each physical interconnection computation. Instead of the

system graph, a sub-graph that contains related components

only will be used. Therefore, the algorithm will deduce the

combinatorially appropriate tree for the system in a rational

level; A decomposition of a system graph will be performed

so that the number of events in a sub-tree will be extremely

reduced. It would bene®t from ef®cient algorithms.

In applying the proposed methodology, the quanti®cation

of mechanical system reliability has been obtained effec-

tively. The proposed method provides not only a new algo-

rithm, but also a strategic point that allows decomposing the

original system as several small subsystems for reducing the

complexity of its analysis. Moreover, various system para-

meters can also be naturally incorporated into graph models,

and existing mathematical results and algorithms in graph

theory and Boolean function can be effectively utilized to an

advantage for failure consideration.

2. Graph theory and Boolean function in reliabilityanalysis

2.1. Mechanical system graph for reliability assessment

As the terminology suggests, see Appendix A [2], a graph

is not usually thought of as an ordered pair, but as a collec-

tion of vertices, some of which are joined by edges. It is then

a natural step to simulate a mechanical system by drawing a

picture of a graph. According to the de®nition, a mechanical

system is one of many real-world objects that can conveni-

ently be described by means of a diagram consisting of a set

of points together with lines joining certain pairs of these

points. In fact, a system is a well-structured, i.e. organized,

set of components. Also, a system can be made up of inter-

facing components. It is easy to de®ne each component,

subsystem, or assembly as a primary element, and a

mechanical system is de®ned as a set of interconnected

elements. Therefore, a network of links can naturally repre-

sent the mechanical system where the components in the

mechanical system could be de®ned as points, and a link

indicated a joining pair using an edge represents a connec-

tion between the components, then it is a graph.

In reliability theory, another graph called a reliability

block diagram is used frequently. From the reliability

block diagram, the system graph is drawn easily. The

J. Tang / Reliability Engineering and System Safety 72 (2001) 21±3022

reliability block diagram is a special kind of pseudo graph.

For example, consider a mechanical system is composed of

eight components, and assume that the system will operate if

any sequence or series of components from C1 to C8 is to

operate. The reliability block diagram is shown in Fig. 1.

The corresponding fault tree and the system graph of this

mechanical system are illustrated as shown in Fig. 2(a) and

(b), respectively.

The reliability graph G consists of eight nodes (compo-

nents) and nine edges (links), see Appendix A [2]. The

nodes and edges constructed a set of vertices, V�G� �{C1;C2;¼;C8}; and a set of edges, E�G� �{e1; e2;¼; e9}: The node Ci represents the component and

the edge ei is the connection between the components. Once

the graph of the mechanical system is obtained, the corre-

sponding adjacency matrix of this system graph is also

formed, it is

A �

0 1 1 0 0 0 0 0

1 0 0 1 0 0 0 0

1 0 0 0 1 1 0 0

0 1 0 0 0 0 1 0

0 0 1 0 0 0 0 1

0 0 1 0 0 0 0 1

0 0 0 1 0 0 0 1

0 0 0 0 1 1 1 0

0BBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCA

This is a symmetric {0,1} matrix of order 8, in which each

row or column corresponds to the component, such that off-

diagonal elements, aij, represent the connection between

components Ci and Cj. It is equal to 1, i.e. aij � 1 if compo-

nent Ci is connected to component Cj, otherwise, aij � 0:

Whereas, aii � 0 for all principal diagonal terms since a

component cannot be connected to itself.

Obviously, the characteristics of the connections between

the components are considered in this matrix; however, the

genuine physical interconnections, the quantitative affec-

tions, have not been re¯ected. Moreover, the reliability of

components themselves was not involved, either. To have a

completed expression for system reliability, these factors

must be involved. Therefore, three new matrices are intro-

duced for this purpose, which are the component connection

matrix V , the component reliability matrix RC, and the

system reliability matrix G .

Let v ij be the physical interconnection between compo-

nent Ci and component Cj. By substituting aij ± 0 with v ij

in the adjacency matrix, the component connection matrix

V � �vij� is de®ned. For the system shown in Fig. 2, this

matrix is as follows:

V �

0 v12 v13 0 0 0 0 0

v12 0 0 v24 0 0 0 0

v13 0 0 0 v35 v36 0 0

0 v24 0 0 0 0 v47 0

0 0 v35 0 0 0 0 v58

0 0 v36 0 0 0 0 v68

0 0 0 v47 0 0 0 v78

0 0 0 0 v58 v68 v78 0

0BBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCA�2:1�

In addition, assume the system is made up of n components

each with a reliability Ri; the component reliability matrix

J. Tang / Reliability Engineering and System Safety 72 (2001) 21±30 23

C2 C4 C7

C3

C5

C6

C1 C8

Fig. 1. Reliability block diagram of a mechanical system.

(b) System Graph

(a) Fault Tree

e1

e4

e2

e5 e6

e8

e7

e9

e3

C2

C3

C1

C5

C6

C7

C8

C4

Basic Event Basic Event

Combinationof Events

X1 X8

X2 X4 X7 X3

Basic Event

X5 X6

Basic Event

Basic Event

Top Event

Fig. 2. Fault tree, and system graph of a mechanical system: (a) fault tree; and (b) system graph.

RC is then de®ned as follows:

RC �

R1 0 ¼ ¼ 0

0 R2 0 ¼ 0

¼ ¼

0 ¼ ¼ 0 Rn

0BBBBBB@

1CCCCCCA �2:2�

It is a diagonal matrix; its element Ri is a variable of compo-

nent representing the reliability of ith component.

Based on the component connection matrix V and

component reliability matrix RC, the system reliability

matrix G � �g ij� is de®ned by using the following expres-

sion,

G � �gij� � �RCI 2 V� �2:3�where I is an identity matrix. This is the matrix that Gandhi

introduced in his paper [3], and called the Variable Char-

acteristic Reliability Matrix.

A simple mechanical system, in which the components Ci

�i � 1; 2;¼; n�; are connected in series from C1 to Cn, is

shown in Fig. 3(a). Assume it is a chain model [8,12]

composed of n links where the chain will break if any one

link breaks. This implies that failure events in the different

units are independent. The corresponding graph of this

system is a special case, the set of edges is a path from v1

to vn (the case v1 � vn describes a cyclic path, i.e. a cycle),

see Fig. 3(b). The component connection matrix of this

graph is a null matrix, vij � 0 because of the independent

unit assumption. Its system reliability matrix is then a diag-

onal matrix as expressed in Eq. (2.5).

A special kind of graph is called a tree. Note that this is

not a fault tree. There is no fault tree in the strict sense of

graph theory since in a fault tree OR- and AND-gates follow

each other on any path from any input to the output. Every

further layer of gates corresponds to a further step in nesting

depth. Hence, the de®nition of a tree is a connected acyclic

graph [2,6]. An acyclic graph is one that contains no cycles.

In a tree, any two vertices are connected by a unique path.

In a directed tree there is one node with out-going edges

only; it is a root of the tree, the node called the root vertex.

The end nodes with exactly one link ending in them are

called the terminal vertex. In the following section, the

directed tree will be used for analyzing the physical inter-

connections.

2.2. Boolean function and its application on physical

interconnection between components

Boolean variables are restricted to indicator or {0,1} vari-

ables. In other words, it is the variables whose values,

namely 0 and 1, are not free for a wide range of interpreta-

tions. They can be de®ned as binary indicator variables, and

allow for the use of standard algebra to write as Boolean

function [1]. Let X � {X1;X2;¼;Xn} be a set of Boolean

variables. A Boolean function is an assignment on a binary

indicator variable set X to form a mapping from X into

binary indicator or {0,1}. Obviously, Boolean indicator

variables and all their functions are of two variables.

Boolean function is the most important conceptual basis

of reliability theory, where the fault trees are pictures of

Boolean functions describing the superposition of compo-

nent faults to create system faults. It is understood that n

components, C1;C2;¼;Cn; of a system S have binary prop-

erties which can be modeled by indicator variables. In fact, a

component Ci seen as an indivisible unit, can be in either of

two binary states as described by the two integer values 0

and 1 of its associated indicator variable Xi, such as

Xi �0; if component Ci properly

1; if component Ci failures

(�2:4�

Moreover, an elementary state of a system is described by

the (ordered) set of the binary states of all of its n compo-

nents; it corresponds to a binary n-vector. A general state of

a system is described by the set of the elementary states

belonging to it. If the binary state of a system (operate or

failure) is meant, then the binary indicator variable Xf of the

system S is a helpful concept. This implies that all relevant

features of S are expressed by f . Precisely, in system relia-

bility analysis, a Boolean function f is an assignment on a

set of Boolean variable X (components), to form a mapping

from X into {0,1}. Hence, for the system, it also has

Xf � f�X1;X2;¼;Xn� �0 if system S properly

1 if system S failures

(�2:5�

The possible representations of Boolean functions are so

numerous that even some normal forms are not unique.

This results in some dif®culty in analysis. Consequently,

there is a need for normal forms, which are unique, called

canonical normal forms. However, Bryant introduced a

class of algorithms for manipulating Boolean functions,

which represents Boolean function as directed acyclic

graphs, called Bryant trees. Based on a Bryant tree, a new

algorithm, which can be used to compute the physical inter-

connection very ef®ciently, has been developed in this

paper. This algorithm is of some importance in reliability

theory. Using this algorithm for manipulating the Boolean

functions, the representation is in a more ef®cient manner,


(a) Reliability Block Diagram (b) System Graph

CnCiC1Ci CnC1

Fig. 3. A mechanical system with component in series: (a) reliability block diagram; and (b) system graph.

and is a canonical form. Then, every function has a unique

representation.

A Bryant tree is composed of terminal and non-terminal

vertices, which are connected by edges. Terminal vertices

have the value of 0 or 1 and non-terminal vertices corre-

spond to the basic vertices (events) connected to other

vertices of a tree. Each non-terminal vertex has a 0-branch,

which represents a basic event non-occurrence (works), and

a 1-branch, which represents a basic event occurrence

(fails). Therefore, all paths through the Bryant tree termi-

nate in one of two states: either a 1-state (corresponds to

system failure), or a 0-state, (success). Fig. 4 depicts a

typical Bryant tree of three basic events.

A Boolean function lends itself in a natural way to a

binary tree representation. The size of the resulting binary

tree is determined by the ordering that have to be given to

the basic events in the tree before the Bryant tree is

constructed. This ordering has further implications for the

analysis. In most cases, a convenient top±down ordering

introduced by Rauzy is used [4]. That is to say, the basic

events that are placed higher up the tree are listed ®rst and

are regarded as being ªless thanº (using the symbol ª , º to

indicate the ordering) those lower down the tree. Once the

basic events in the tree have been given an ordering, the

Bryant tree will be constructed (see the situation shown in

Fig. 4). In this example, the order is X1 , X2 , X3; which is

a rooted tree. It can be seen that the Bryant tree is one of the

most ef®cient methods for Boolean formulae management.

This diagram speci®es the failure logic equation in a form,

which is easier to manipulate. From the Bryant tree both the

qualitative and the quantitative analysis can be achieved. In

addition, a Bryant tree is a graph encoding Shannon's

decomposition of a formula [5]. By using Shannon's decom-

position, the computation of the probability of the root

event, through given the probabilities of relevant events

that affect the root event, becomes more reliable and

straightforward. This is the idea that induces the proposed

algorithm for computing the physical interconnection.

Let f be a Boolean function on X, and Xi be a variable of

X, then Shannon's decomposition theorem indicates for

every variable Xi,

f � �Xi ^ f�1�� _ � �Xi ^ f�0��; �i � 1; 2;¼; n�; �2:6�the two terms �Xi ^ f�1�� and � �Xi ^ f�0�� are disjointed.

To compute the physical interconnection, it is necessary

to treat recursively (with one variable, namely Xi) f�Xi � 1�and f�Xi � 0� as f (X) with Eq. (2.6) for each basic event in

the Bryant tree. By assigning one of the two connected

events (components) as a root and the other a terminal

vertex, respectively, the Bryant tree is formed. Applying

the proposed algorithm, one will end in the terminal event

of this binary tree with a single term by Shannon's decom-

position theorem, working one's way up through the tree,

and obtaining a canonical form of disjointed terms. The

algorithm contains the application of Eq. (2.6) with an ith

vertex not used before as a root vertex, and then simpli®ca-

tion of f�Xi � 1� and f�Xi � 0� by applying the absorption

terms. Therefore, the physical interconnection between the

root event (vertex) and terminal event (vertex) is obtained

directly from the diagram.

2.3. Evaluation of physical interconnection using Boolean

function

By giving an order for all components in a mechanical

system or a corresponding reliability block diagram, a

Bryant tree of the system can be formed. The connection

between any two components of the system can be graphed

as a sub-tree of this Bryant tree. Since the Bryant tree

de®nes the cause of the system failure mode in terms of

the component failure represented by basic events, it

enables the system probability (either reliability or failure

probability) effected by the failure of any basic event to be

calculated based on the provided information. The same

thought is used for evaluating the physical interconnection.

Hence, the sub-tree of the Bryant tree can de®ne the affec-

tions between any two components. The measure signi®es

the contribution that each component makes to another

component failure.

Assign an ordering for all basic events of the system.

Then, de®ning the component that is effected by another

component is a root event, and the other component is a

terminal event. Following the assigned order from the root

event to the terminal event, a sub-tree of the original Bryant

tree is formed. Applying Shannon's formula, Eq. (2.6), to

this tree, the affection of a terminal event to the root event

has been evaluated. The affection of the failure of a terminal

event to the root event is obtained by taking the expectation

of the path, i.e.

vij �Y

i[Path

�kiE�f�X � 1�� _ �1 2 ki�E�f�X � 0�� 2:7�

where k i is the failure probability that event i has occurred,

the product range i should be overall events of the vertex set

that belong to the path, and E is the expectation.

Eq. (2.7) indicates the physical interconnection values,

v ij, can be expressed as the product of the failure affections

of the path that start from the root event to the terminal event

through the Bryant tree. The path through the Bryant tree is

found simply by including all vertices from the root vertex


X1

X2 X3

1

10 01

1

1 0

0

0

Non-terminal vertex(root vertex)

terminal vertex

1 branch0 branch1 branch0 branch

terminal vertex

Fig. 4. A Bryant tree for three variables (vertices).

to a terminal 1-node of the terminal vertex. It should include

all events, which lie on the 1-branch and the 0-branches for

each of the basic events. Note that basic events, which lie on

a 0-branch, are indicated in the paths as �Xi: For instance,

consider the interaction between event X3 and event X1, i.e.

the physical interconnection v 13, of a four basic event

model. The Bryant tree is shown in Fig. 5. The ordering is

X1 , X2 , X3 , X4; and the root and terminal vertices are

X1 and X3, respectively. The path through the Bryant tree of

the system is as X1�X2X3: The root event measure v 13 can

then be calculated as

v13 � P�X1�X2X3� � kx1

�1 2 kx2�kx3

�2:8�Since the Bryant tree and Shannon's formula are employed,

it gives a unique path for each pair of basic events. The

physical interconnection to be determined is in a very ef®-

cient calculation procedure.

3. Basic concept of assessing system reliability using thegraph theory and Bryant tree

For system success, all n-components must operate

successfully. The reliability of the system is then the inter-

section of each component success [10].

PS � P�X1 > X2 > X3 > ¼ > Xn�� P�X1�P�X2uX1�P�X3uX1X2�¼P�XnuX1X2X3¼Xn21�

�3:1�where Xi is signi®ed as the successful event of component i,

and P�XiuX1X2¼Xi21� is the conditional probability, which

is the reliability of component i evaluated under that compo-

nents 1; 2;¼; i 2 1 are operating. For instance, a simple n-

component system that is considered as a series system

under the independence assumption is shown in Fig. 3(a).

The model can be compared to a chain composed of n links

where the chain will fail if the failure appears in any one

link. From Eq. (3.1), the system reliability can be expressed

as

PS �Yn

i�1

Ri �3:2�

where Ri is the reliability of the ith component. On the other

hand, by applying the graph theory, the reliability graph of

this system is actually a path, also see Fig. 3(b). The compo-

nent connection matrix V is written in the following form:

V �

0 v12 0

v12 0 v23 0

v23 0 v34

v34 ´ ´

0 ´ ´ ´ v�n21�n

0 v�n21�n 0

0BBBBBBBBBBBBB@

1CCCCCCCCCCCCCA�3:3�

In this case, the corresponding Bryant tree is pictured as

shown in Fig. 6.

From the Bryant tree, Fig. 6, and the independent assump-

tion, the genuine physical interconnection between compo-

nents, v ij will be zero, i.e.

v�i21�i � 0; �i � 2;¼; n� �3:4�It can be seen that in certain case, even two components are

connected the genuine physical interconnection can be zero.

Hence, the connection matrix V is null matrix. The system

reliability matrix G of this graph is a diagonal matrix,

Eq. (2.2). Obviously, its corresponding determinant is calcu-

lated by the following expression:

det�G� �Yn

i�1

Ri �3:5�

Comparing Eqs. (3.2) and (3.5), it can be seen that the two

equations have exactly the same results. Thus,

PS � det�G� �3:6�Spontaneously, the conclusion is that the system reliability

can be calculated by using the corresponding determinant of

its system reliability matrix if all components in the system

are independent.

Secondly, for the case, the mechanical system is consid-

ered as only the ®rst component connected with the rest of

the system in series, that is, the ®rst component is indepen-

dent with rest components, and leave the remain connection

arbitrary. Let P�S21� be the reliability of the remaining

subsystem �S 2 1� that consists of all components except

the ®rst one. From Eq. (3.2), the system reliability, then is

PS � R1P�S21� �3:7�where R1 is the ®rst component reliability. While with the


X4X3

X2

X1

11 0

0

0

0

0

0 1

1

1

1

Fig. 5. The Bryant tree of a four event model.

1 1

0... ...Xi-1 Xi+1Xi

11

0

0 Xn

0

00

...1

X1 00

...1 1111

Fig. 6. The Bryant tree of a series mechanical system.

graph theory, the system reliability matrix of this system can

be expressed as

G �R1 0

0 G�S21�

!�3:8�

where G�S21� is a sub-matrix of matrix G that is formed by

deleting the ®rst row and the ®rst column from the original

matrix G , it is the system reliability matrix of the subsystem

�S 2 1�: Assume that Eq. (3.6) for �n 2 1� component

system is true. Then, the reliability of subsystem �S 2 1�;P�S21� is equal to a minor of the original matrix G , i.e.

det�G�S21��: The reliability of the system is the determinant

of the system reliability matrix G ,

PS � det�G�S21�� R1 det�G�S21�� R1P�S21� �3:9�From Eqs. (3.7) and (3.8), it is clear that the results obtained

from two different methods are also the same. That is, the

system reliability is also equal to the determinant of the

system reliability matrix.

Generally, if the mechanical system is considered as the

®rst component it is not only connected with one of the rest

components but also connected with other components, say

the ith component. It can be proved that the above conclu-

sion is still held in this case. Assume that due to the fact

component 1 is connected with component i, the intercon-

nection between ®rst component and ith component is v 1i.

This means that component i will contribute a certain

portion to the system reliability. It can be simulated as an

image system in which the ith component with reliability

�21��11i� g1i is connected with the rest of the system in

series. The image system reliability will then be

�21��11i�g1iP�S2i�; where P�S2i� is the reliability of a

�S 2 i� sub-system, a system containing all components

except the ith component, whose system reliability matrix

is G�S2i�; (a sub-matrix by eliminating the ®rst row and the

ith column of the original system reliability matrix G ). Here,

the rest of the subsystem reliability given by Eq. (3.6) is also

assumed. Since g1i � 2v1i; it contributes the term

�21�iv1iP�S2i� to system reliability. Therefore, system

reliability is obtained as

PS � R1P�S21� 1 �21�iv1iP�S2i� �3:10�If the graph theory is used for the same model, the system

reliability matrix G will include the interactions; that is, the

off-diagram terms, g i1 and g 1i, will be non-zero. And, they

are equal to the physical interconnections between the

components C1 and Ci, i.e. v i1 and v 1i, respectively. The

determinant of the system reliability matrix will be

PS � det�G� � R1A11 1 g1iA1i � R1 det�G�S21��1 �2v1i�

� �21��11i� det�G�S21��

� R1P�S21� 1 �21�iv1iP�S2i��3:11�

where A11 and A1iare the cofactors (the determinant by delet-

ing ®rst row and ®rst/ith column, respectively) of the

elements R1 and g 1i of the det(G ), respectively. Also, the

same results are obtained. The conclusion is also true for

this case.

Generally speaking, if two arbitrary components Ci and Cj

in a system are connected to each other, the interconnection

is v ij. By exchanging row 1 and i in the system reliability

matrix, the above conclusion can be proved using the same

method. Therefore, the conclusion can be expanded to the

general case where the system in which their components

are interactive with each other.

To sum up, if there are certain connections between the

components in a system, then the off-diagonal terms that

coincide with these connections of the system reliability

matrix will have certain corresponding non-zero items as

their effective numbers, namely physical interconnections.

The effect of these interactions will then be included in the

determinant. This means that the interaction has been

involved in the system reliability. Eventually, the system

reliability considering the interaction between the compo-

nents then is evaluated directly by the following expression:

PS �Xn

j�1

gijAij�for 1 # i # n; expansion about the ith row�

�Xn

i�1

gijAij

�for 1 # j # n; expansion about the jth column�

�XN[S

sign�N�g1j1g2j2

¼gnjn

�3:12�where g ij is the element of matrix G , i.e.

g ij �Ri i � j

2vij i ± j

(�3:13�

and Aij is the cofactor of element g ij of the det(G ). In the last

summation of Eq. (3.12), N is a permutation j1; j2;¼; jn of

the set S � {1; 2;¼; n}, hence the summation ranges over

all permutations. The function sign(N) is as follows:

sign�N� �

1 if the number of inversions in

sequence j1; j2;¼; jn of N is even

21 if the number of inversions in

sequence j1; j2;¼; jn of N is odd

8>>>>><>>>>>:�3:14�

It can also be seen that in each term of the last summation

of Eq. (3.12), the row subscripts (the ®rst subscripts) are in

their natural order, whereas the column subscripts (the

second subscripts) are in the order j1; j2;¼; jn: Since

the permutation j1; j2;¼; jn is merely a rearrangement of

the numbers from 1 to n, it has no repeats. Thus, each

term in this expression is a product of n elements of the


system reliability matrix, each with its appropriate sign,

with exactly one element from each row and exactly one

element from each column. Hence, every connection will be

included but never performed twice. It is clear that no matter

how many connections are involved all effect of each

component has been included.

4. An application of the methodology

Consider a mechanical system S, which consists of six

components, denoted by Ci. Each component is assumed as

a basic event Xi. The structure of this system expressed by

using a reliability block diagram is shown in Fig. 7. The

corresponding system graph is depicted in Fig. 8.

Assume the reliability of each component in the system S

is Ri, and let v ij be the physical interconnection between

component Ci and Cj. The component reliability matrix RC,

and the adjacency matrix A of this graph and its corre-

sponding component connection matrix V are as follows,

respectively:

RC �

R1 0 0 0 0 0

0 R2 0 0 0 0

0 0 R3 0 0 0

0 0 0 R4 0 0

0 0 0 0 R5 0

0 0 0 0 0 R6

0BBBBBBBBBBBB@

1CCCCCCCCCCCCA;

A �

0 1 0 1 0 0

1 0 1 0 0 0

0 1 0 0 1 1

1 0 0 0 1 1

0 0 1 1 0 0

0 0 1 1 0 0

0BBBBBBBBBBBB@

1CCCCCCCCCCCCA;

V �

0 v12 0 v14 0 0

v12 0 v23 0 0 0

0 v23 0 0 v35 v36

v14 0 0 0 v45 v46

0 0 v35 v45 0 0

0 0 v36 v46 0 0

0BBBBBBBBBBBB@

1CCCCCCCCCCCCA

The reliability matrix is then expressed as

R �

R1 2v12 0 2v14 0 0

2v12 R2 2v23 0 0 0

0 2v23 R3 0 2v35 2v36

2v14 0 0 R4 2v45 2v46

0 0 2v35 2v45 R5 0

0 0 2v36 2v46 0 R6

0BBBBBBBBBBBB@

1CCCCCCCCCCCCA�4:1�

The system reliability can then be calculated by evaluat-

ing the determinant of the reliability matrix R. By expanding

the determinant of this matrix, the formula that is used for

evaluating reliability of system S can be written in the

following form:

R � R1R2R3R4R5R6 2 R1R2R3R5�v46�2 2 R1R2R3R6�v45�2

2 R1R2R4R6�v35�2 2 R1R2R4R5�v36�2

2 R3R4R5R6�v12�2 2 R1R4R5R6�v23�2

2 R2R3R5R6�v14�2 1 R1R2�v35�2�v46�2 1 R1R2�v36�2

� �v45�2 1 R1R5�v23�2�v46�2 1 R1R6�v23�2�v45�2

1 R3R5�v12�2�v46�2 1 R3R6�v12�2�v45�2 1 R4R5�v12�2

� �v36�2 1 R4R6�v12�2�v35�2 1 R2R5�v14�2�v36�2

1 R2R6�v14�2�v35�2 1 R5R6�v14�2�v23�2

2 2R1R2v35v36v45v46 2 2R5v12v14v23v36v46

2 2R6v12v14v23v35v45 2 �v12�2�v35�2�v46�2

2 �v12�2�v36�2�v45�2 1 2�v12�2v35v36v45v46

�4:2�The next step is to calculate the physical interconnection

v ij. The physical interconnections that need to be evaluated

in this system are v 12, v 14, v 23, v 35, v 36, v 45, and v 46.

Hence, basic events X1, X2, X3, and X4 will be selected as

root vertices, and X2, X3, X4, X5, X6 will be selected as term-

inal vertices. To form a Bryant tree for using Shannon's

formula to calculate physical interconnection, an ordering

for the events of the system have to be assigned ®rst.


C1 C2 C3

C4

C5

C6

Fig. 7. Reliability block diagram of the system.

C2

C3

C1

C5

C6

C4

Fig. 8. Graph of system structure.

According to a system structure, the six basic events are

two-level, therefore, take the ordering for the events as a

top±down ordering. Hence, the ordering for the basic events

is X1 , X2 , X3 , X4 , X5 , X6:

Fig. 9 shows the Bryant trees for different basic events as

a root vertex in the order of precedence. Since components

C1, C2, and C3 are connected in series, each failure will have

the same affection on the system failure. Hence, the Bryant

trees for these three basic events as a root vertex are the

same, see Fig. 9(a). The binary tree that is used to evaluate

the physical interconnections is the sub-tree of the whole

Bryant tree. Fig. 9(b) shows one of these sub-trees, path sub-

tree X4 to X6. Therefore, the Bryant trees shown in Fig. 9

indicate all cases of this model.

In these Bryant trees, every path through the tree starts

from the root vertex and proceeds down through the

diagram to a terminal vertex. Paths that terminate at a 1-

vertex yield a subset, which is a failure set. However, only

basic events for which the path leaves its vertex on a 1-

branch on the way to a terminal 1-vertex are included in

the subsets. Therefore, to calculate physical interconnection

v ij, the root and the terminal vertices should be selected as

Xi and Xj, respectively. For every physical interconnection,

the paths (subsets) are listed in Table 1.

The physical interconnections then can be calculated

using following expressions:

v12 � �1 2 kx1�kx2

v23 � �1 2 kx2�kx3

v14 � kx1kx4

v35 � kx3�1 2 kx4

�kx5

v36 � kx3�1 2 kx4

�kx5kx6

v45 � �1 2 kx4�kx5

v46 � �1 2 kx4�kx5

kx6

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

�4:3�

Obviously, the affections for each root event can then be

calculated using Eq. (4.3).

Given the reliability of each component in this system,

which are listed as in Table 2, the system reliability can then

be calculated.

By using these data, the k i is calculated. Table 1 lists the

corresponding path productQk 1, substituting these data

into Eqs. (4.3) and (4.2), the system reliability is evaluated,

PS � 0:885483:

As can be seen in the above example, the computation for

using the proposed methodology is very simple and straight-

forward. In reality, the physical interconnections are always

calculated ®rst. Instead of expanding the determinant as Eq.

(4.2), the most standard computer codes is used for the

evaluation of the determinant det(G ). Therefore, the compu-

tation will be more ef®cient. Obviously, the proposed

method will be very helpful in the application of reliability

analysis of mechanical system design.

5. Conclusions

Traditional FTA and FMECA techniques for mechanical

system reliability analysis are tedious for the complex

systems. Moreover, they can only provide an approximate

result in some cases. To improve the ef®ciency of the analysis

even further, a new analysis method of mechanical system

reliability based on graph theory and Boolean function has

been studied in this paper. The use of graph theoretic modeling

to perform the mechanical system reliability is a reliable and

ef®cient approach. The method is also very simple to use since

it is actually an application of matrix algebra. However, the

quantitative interconnections must be provided before the use

of graph modeling. Therefore, this paper also extends the use

of a Boolean function concept to quantify these interconnec-

tions through a Bryant tree. Based on the study, an ef®cient

reliability assessment method is developed. The algorithm

facilitates the direct calculation of the system reliability. The

mathematical approach is not novel; its potential for engineer-

ing application has been explored signi®cantly. The proposed

method has also been shown that analysis procedures based

on graph theory and a binary tree to represent the system

failure logic can produce a more realistic model for problems.


(a) (b)

X4

X6

X5

0

0

0

00

1

1

1

1

10

0

X1

X2X3

11

1

00

0

0

0

0

0

0

X4

1

1

X6

X5

0

1

1

1

0

0

X4

1

1

X6

X5

0

1

1

1

0

0

X4

1

1

X6

X5

0

1

1

1

0

0

Fig. 9. The Bryant trees and a sub-tree of the model.

This paper contributes to the development of a methodol-

ogy for the evaluation of system reliability. In this paper, a

constructive method for evaluating the reliability interac-

tions between the components based on a Boolean function

has also been provided. An example illustrates the method,

and the results obtained are quite logical and encouraging.

In addition, the analysis time can be compared, and the

proposed method is favorable.

Appendix A

Basic Graph Theory

A graph G is an ordered triple

G � �V�G�;E�G�;fG�: �A:1�It consists of a nonempty set V(G) of vertices, a set E(G),

disjoint from V(G), of edges,

V�G� � {v1; v2;¼; vn}

E�G� � {e1; e2;¼; em};

(�A:2�

and an incidence function fG that associates with each edge

of G and unordered pair of (not necessarily distinct) vertices

of G. The incidence function fG is de®ned by

fG�ei� � vjvk; i � 1; 2;¼;m; j; k � 1; 2;¼; n �A:3�namely, if ej is an edge, and vi and vj are vertices such that

fG�ej� � vivj; then ej is said to join vi and vj; the vertices vi

and vj are called the end points of ej. In the most practical

situation a graph is just a set of ªlinksº called edges

e1;¼; ei;¼; em; where ei is described by a pair of ªnodesº,

i.e. ei � �vjvk�:To any graph G there corresponds an n £ n matrix called

an adjacency matrix, A�G� � �aij�; in which aij is the number

of {0,1}, and indicates the existence of the edges joining vi

and vj. Since joining with the vertex itself is not considered

here, for term aij � 1 if and only if i ± j; it is always the off-

diagonal term in the matrix. Thus the adjacency matrix is a

square symmetric matrix. Fig. A1 shows a general graph in

which there are six vertices and nine edges.

In this graph, the set of vertices is V � �v1; v2;¼; v6�; the

set of edges is E � �e1; e2;¼; e9�; and the incidence func-

tions are fG�e1� � v1v2; fG�e2� � v1v3;¼;fG�e9� � v5v6:

The corresponding adjacency matrix is as follows:

A �

0 1 1 0 0 0

1 0 0 1 1 0

1 0 0 1 1 0

0 1 1 0 1 1

0 1 1 1 0 1

0 0 0 1 1 0

0BBBBBBBBBBBB@

1CCCCCCCCCCCCA:

References

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tion. IEEE Trans Comput 1986;C-35(8):677±91.

[2] Bollobas B. Modern graph theory. New York: Springer, 1998.

[3] Gandhi OP, Agrawal VP, Shishodia KS. Reliability analysis and

evaluation of system. Reliab Engng System Safety 1991;32:283±305.

[4] Rauzy A. New algorithm for fault tree analysis. Reliab Engng System

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[5] Shannon RM, Andrews JD. New approaches to evaluating fault trees.

Proceedings ESREL`95, 1995. p. 241±54.

[6] Chung FRK, Grakman BL, Coppersmith D. On trees containing all

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graph. New York: Wiley, 1980.

[7] Bloch HP, Geitner FK. An introduction to machinery reliability

assessment. New York: Van Nostrand Reinhold, 1990.

[8] Martin P. Aspects of mechanical system reliability. In: Libberton GP,

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and New York, 1988. p. 93±102.

[9] Henley EJ, Kumamoto H. Probabilistic risk assessment: reliability

engineering, design, and analysis. New York: IEEE Press, 1992.

[10] Martin P. Consequential Failure in Mechanical System. Reliab Engng

1982;3:23±45.

[11] Frankel EG. System reliability and risk analysis. Boston: Kluwer,

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[13] Carter ADS. Mechanical reliability. London: Macmillan, 1986.


ν1

ν2

ν3

ν4

ν5

ν6

e1

e2

e3e4e5

e6

e7

e8

e9

Fig. A1. An example of a graph.

Table 1

The connection related subsets

Connection v12 v 14 v23 v 35 v 36 v45 v 46

Subsets �X1X2 X1X4�X2X3 X3

�X4X5 X3�X4X5X6

�X4X5�X4X5X6Q

k i 1.96 £ 1022 8 £ 1024 1.96 £ 1022 1.92 £ 1024 1.92 £ 1026 9.6 £ 1023 9.6 £ 1025

Table 2

The component reliability of the system

Component C1(X1) C2(X2) C3(X3) C4(X4) C5(X5) C6(X6)

Reliability 0.98 0.98 0.98 0.96 0.99 0.99

Mechanical system reliability analysis using a combination of graph theory and Boolean function

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