Gregory Levitin
Reliability Department, Planning, Development and Technology Division,
Bait Amir, The Israel Electric Corporation Ltd., P.O. Box 10, Haifa, 31000 Israel
E mail: levitin@iec.co.il
Abstract
A linear multistate consecutively-connected system (LMCCS) consists of N+1 linear ordered positions (nodes). M statistically independent multistate elements (MEs) with different characteristics are to be allocated at the N first positions. Each element can provide a connection between the position in which it is llocated and the next few positions. The reliability of the connection for any given element depends on the position in which it is allocated and on the number of positions it connects. The system fails if the first position (source) is not connected with N+1-th one (sink).
Each system node with all the MEs allocated at this node can be destroyed by external impact with a given probability. The system survivability is defined as the probability that at least one path exists from the source to the sink when both external impacts and internal failures can cause MEs unavailability.
The algorithm for finding the ME distribution providing the greatest possible LMCCS survivability is suggested. A genetic algorithm is used as the optimization tool. Illustrative examples are presented.
Keywords: System survivability ; multistate elements; linear consecutively-connected systems; universal generating function; genetic algorithm.
1

Acronyms
ME
LMCCS
UGF
GA multistate element linear multistate consecutively-connected system universal generating function genetic algorithm
R
S
Notation
Pr{x} probability of event x reliability of LMCCS survivability of LMCCS
N number of positions (nodes) in LMCCS
M number of MEs in LMCCS
 node vulnerability
 probability that node survives during the system operation time (

=1-

) e i i-th ME of LMCCS
C n n-th position (node) of LMCCS
E set of all available MEs: E={e
1
,…,e
M
}
E n set of MEs located at position n h(i) number of position the ME e i
is allocated at: e i

E h(i)
H vector representing allocation of MEs in LMCCS: H ={h(i),1
 i

M}
K maximal number of different states of MEs
G i random state of i-th ME p ik
(n) Pr{G i
=k | e i

E n
}
P
I
(n) vector representing probabilistic state distribution of i-th ME: P i
={p i0
(n),…,p iK-1
(n)}
T n random number of the most remote position to which arc from C n
exists
2

Y n random number of the most remote position to which path from C
1
provided by MEs belonging to i n


1
E i
exists q nk
Pr{T n
=k} u in
(z) u-function for i-th ME located at position n (i=0 corresponds to absence of ME)
U n
(z) u-function for the group of MEs belonging to E n
(located at position n)
~
U n
(z) u-function for subset of MEs i n


1
E i

,

,

operators over u-functions

(

)

(x)=min{x,N+1}
1(

) 1(x)=1 if x is true, 1(x)=0 if x is false
1.
Introduction
The LMCCS consists of N+1 consequently ordered positions (nodes) C n
, n

[1,N+1]. The first position C
1
is the source and the last one C
N+1 is the sink (see Figure 1). At each position
C
1
,…,C
N
, elements from a set E={e
1
,…,e
M
} can be allocated. These elements provide connections (arcs) between the position in which they are allocated and further positions.
Each element e i
has K states, where state G i
of e i
is a discrete random value with distribution depending on location of the element. For an ME located at node C n
:
Pr{ G i
 k }
 p ik
( n ),
K k

1


0 p ik
( n )

1 .
G i
=k for element e i
allocated at position C n
implies that arcs exist from C n
to each of C n+1
,
C n+2
,…,C

(n+k)
, where

(x)=min{x,N+1}. G i
=0 implies the total failure state of e i
(no arcs exist from C n
).
Note that though different MEs can have different number of states, one can define the same number of states for all the MEs without loss of generality. Indeed, if ME e i
has K i
states and ME
3

e m
has K m
states (K i

K m
), one can consider both MEs as having K=max{K i
,K m
} states while assigning p ik
=0 for K i
 k<K.
All the states G i
are independent.
The system is failed if there is no path from C
1
to C
N+1
.
An example of the LMCCS is a set of radio relay stations with a transmitter allocated at C
1 and a receiver allocated in C
N+1
. Each station C n
(2
 n

N) can have retransmitters generating signals that reach the next G i
stations. Note that G i
is a random value dependent on power and availability of retransmitter amplifiers as well as on signal propagation conditions. The aim of the system is to provide propagation of a signal from transmitter to receiver.
The LMCCS was first introduced by Hwang & Yao [1] as a generalization of linear consecutive-k-out-of-n:F system and linear consecutively-connected system with 2-state elements, studied by Shanthikumar [2,3]. Algorithms for LMCCS reliability evaluation were developed by Hwang & Yao [1], Kossow & Preuss [4] and Zuo & Liang [5]. The problem of optimal element allocation in LMCCS was first formulated by Malinowski & Preuss in [6]. In this problem, elements with different characteristics should be allocated in positions C
1
,…,C
N
in such a way that maximizes the LMCCS reliability. A multi-start local search algorithm was suggested for solving this problem.
In all the mentioned works, only the systems with M=N are considered in which only one
ME is located in each position. In many cases, even for M=N, greater reliability can be achieved if some of MEs are gathered in the same position providing redundancy (in hot standby mode) than if all the MEs are evenly distributed between all the positions.
Consider, for example, the simplest case in which two identical MEs should be allocated within LMCCS with N=2. Each ME has three states: state 0 (total failure), state 1 in which the
ME is able to connect the position in which it is located with the next position and state 2 in which the ME is able to connect position it is located in with the next two positions. The
4

probabilities of being in each state don't depend on MEs allocation and are p
0
, p
1
and p
2
, respectively. There are two possible allocations of the MEs within the LMCCS (figure 2):
A. Both MEs are located in the first position.
B. The MEs are located in the first and second positions.
In case A, the LMCCS succeeds if at least one of the MEs is in state 2 and the system reliability is
R
A
=2p
2
-p
2
2 . (1)
In case B, the LCCS succeeds either when the ME located in the first position is in state 2 or if it is in state 1 and the second element is not in state 0. The system reliability in this case is
R
B
=p
2
+p
1
(p
1
+p
2
). (2)
By comparing (1) and (2), one can decide which allocation of the elements is preferable for any given p
1
and p
2
. Figure 3 presents the decision curve R
A
=R
B
on the plane (p
1
,p
2
). Observe that for combinations of p
1
and p
2
located below the curve, the solution A is preferable while for combinations of p
1
and p
2
located above the curve, solution A provides lower system reliability than solution B.
When a system operates in battle conditions or is affected by a corrosive medium or other hostile environment, the ability of a system to tolerate both the impact of external factors (attack) and internal causes (accidental failures or errors) should be considered. The measure of this ability is referred to as system survivability.
An external factors usually cause failures of group of system elements sharing some common resource (allocated within the same protective casing, having the same power source, gathered geographically etc.). Therefore adding more redundant parallel elements will improve a system reliability but will not be effective from a vulnerability standpoint without sufficient separation between elements [7]. In LMCCS all the MEs located at the same node can be destroyed by a single external impact.
5

When the LMCCS nodes are vulnerable the allocation providing the greatest system survivability can change. In order to estimate the effect of the node vulnerability on the optimal
ME allocation one has to include this parameter into LMCCS survivability model.
Consider the same simplest LMCCS model (figure 2) in which each node (with all the MEs it contains) can be destroyed with probability

.
In case A, the LMCCS survives if node C
1
is not destroyed and at least one of the MEs is in state 2. The system survivability is
S
A
=

(2p
2
-p
2
2
). (3)
In case B, the LMCCS survives either when the node C
1
is not destroyed and ME located in C
1
is in state 2, or if both nodes C
1
and C
2
are not destroyed, the ME located in C
1 is in state 1 and the
ME located in C
2 is not in state 0. The system survivability in this case is
S
B
=
 p
2
+
 2 p
1
(p
1
+p
2
). (4)
By comparing (3) and (4), one can decide which allocation of the elements is preferable for any given

, p
1
and p
2
. Condition S
A

S
B
can be rewritten as

(2p
2
-p
2
2
)
 p
2
+
 2 p
1
(p
1
+p
2
). and finally as
 p
2
(1-p
2
)/(p
1
(p
1
+p
2
)). (5)
Figure 4 presents the maximal values of

for whish S
A

S
B
as function of variables p
1
and p
2
. Observe that for given combinations of p
1
and p
2
the values of

located below the curve correspond to cases when the solution A is preferable while the values of

located above the curve correspond to cases when solution A provides lower system survivability than solution B.
Note that the end point of each curve

(p
2
) belongs to line

=p
2.
Indeed, since p
0
+p
1
+p
2
=1, the maximal possible value of p
1
for each given p
2
is 1-p
2
. Substituting p
1
with 1-p
2
in Eq. (5) one obtains
 p
2
.
6

This paper presents an algorithm for optimal allocation of MEs in LMCCS in which the nodes vulnerability is taken into account. The algorithm finds the allocation of arbitrary number
M of MEs with given state probability distributions (depending on MEs allocation) which maximizes system survivability.
To evaluate the reliability of LMCCS with redundant MEs the procedure based on the use of a universal generating function is developed. A genetic algorithm based on principles of evolution is used as an optimization engine. The integer string solution encoding technique is adopted to represent element allocation in the GA.
Section 2 of the paper presents a formulation of the optimal allocation problem. Section 3 describes the technique used for evaluating the LMCCS reliability for the given allocation of different MEs with the specified state distributions. Section 4 describes the optimization approach used and its adaptation to the problem formulated. The fifth section contains illustrative examples in which the best-obtained allocation solutions are presented for two different LMCCSs.
2.
Problem formulation
The MEs allocation problem can be considered as a problem of partitioning a set E of M elements into a collection of N mutually disjoint subsets E n
(1
 n

N), i.e. such that n
N


1
E n

E ,
E i

E j

, i
 j.
(6)
(7)
Each set E n
, corresponding to LMCCS position C n
, can contain from 0 to M elements. The partition of the set E can be represented by the vector H ={h(i),1
 i

M}, where h(i) is the number of the subset to which element i belongs. One can easily obtain the cardinality of each subset E n as
7

|E n
|= i
M


1
1 ( h ( i )
 n ) .
(8)
In the general case, the probability of being at state k for any ME e i
can depend on the position in which the ME is located (in the example presented above, each relay station can have different conditions of signal propagation and, therefore, different probability of connecting to the next stations even when using identical equipment). This can be taken into account by introducing vector-function P i
(n)={p i0
(n),…,p iK-1
(n)} for each ME e i
located in position n for
1
 n

N.
Each node can be destroyed with probability

. The node destruction means simultaneous transition of all its MEs to state 0.
For the given set of MEs with specified vectors P
1
(n),…, P
M
(n) representing distribution of their states and for the given node vulnerability

, the only factor influencing the LMCCS survivability is the allocation of its elements H . Therefore, the optimal allocation problem can be formulated as follows.
Find vector H* maximizing the LMCCS survivability S:
H* ( P
1
(n),…,
P
M
(n))=arg{S( H , P
1
(n),…,
P
M
(n))
 max}. (9)
3.
LMCCS survivability estimation based on a universal generating function
The procedure used in this paper for LMCCS survivability evaluation is based on the universal z-transform (also called u-function or UGF) technique, which was introduced in [8] and which proved to be very effective for reliability evaluation of different types of multi-state systems [7,9-13]. The u-function extends the widely known ordinary moment generating function.
8

3.1. u-function for individual ME
The UGF of a discrete random value X is defined as a polynomial u ( z )
 k
K

1


0 p k z x k , (10) where the variable X has K possible values and p k
is the probability that X is equal to x k
.
Consider ME e i
located at position C n
. In each state k (1
 k<K), the ME makes C n
connected with C n+1
,…,C

(n+k)
. The probability of state k for ME e i
located at C n
is p ik
(n). Let random value
T n
be the number of the most remote position to which the arc from C n
exists. The polynomial u in
(z) can represent all the possible states of the ME by relating the probabilities of each state to the value of T n
in this state: u in
( z )
 k
K



1
0 p ik
( n ) z

( n
 k )
.
(11)
Note that the absence of any ME in position C n
implies that no arcs exist from C n
to any other position. This means that any arc reaching C n
has no continuation with probability 1. In this case, the corresponding u-function takes the form u
0n
(z)=z n
. (12)
3.2. u-function for group of MEs allocated at the same position
A composition operator

is introduced in order to obtain the u-function of a subsystem containing a number of MEs located at the same position. This operator determines the ufunction for a group of MEs belonging to E n
using simple algebraic operations on the individual u-functions of MEs. The composition operator for a pair of MEs e i
and e m
takes the form:

( u in
( z ), u mn
( z ))

K k


1
0
K

1
  j

0 p ik

(
K k

1


0 p ik
( n ) p mj
( n ) z

( n
( n ) z max{
 k )
,
K

1
 j

0 p mj

( n
 k ),

( n

( n ) z

( n
 j ) } j )
)

(13)
9

The resulting polynomial relates probabilities of each of the possible combinations of states of the two MEs (obtained by multiplying the probabilities of corresponding states of each ME) with the number of the most remote position to which the arc from C n
exists when the MEs are in the given states. It can be easily seen that when one ME is in state k and the second ME is in state j,the arcs from C n
to C n+1
,…,C max{

(n+k),

(n+j)}
exist.
Note that for any u in
(z)

(u
0n
(z),u in
(z))=u in
(z).
One can see that the operator

satisfies the following conditions:
(14)

{ u
1
( z ),..., u k
( z ), u k

1
( z ),..., u v
( z )}
 
{

{ u
1
( z ),..., u k
( z )},

{ u k

1
( z ),...., u v
( z )}} (15) for arbitrary k. Therefore, it can be applied in sequence to obtain the u-function for an arbitrary group of MEs allocated at C n
:
 i

E n
( u in
( z ))


( n k


K
 n

1 ) q nk z k
.
(16)
This polynomial determines the probabilistic distribution of T n
provided by all the MEs belonging to E n
. Observe that, after collecting like terms, the resulting polynomial (as well as polynomials u in
(z) for individual MEs) can have no more than min{K,N-n+2} terms
(corresponding to values of T n
 {n,n+1,…,min{n+K,N+1}).
3.3. Incorporating node vulnerability into corresponding u-function
When MEs are gethered within a single node C n
with vulnerability

, this means that all these elements can be destroyed with probability

. The probabilities q nk
in (16) relate only to reliability of MEs (internal system reliability). In order to incorporate external destructive factors into the model, the probabilities q nk
should be considered as conditional probabilities that the node C n
is connected with the corresponding set of nodes under assumption that the node survives external attacks during the system operation time. Unconditional probability that node
10

C n
is connected with the corresponding set of nodes is, therefore, equal to (1-

)q nk
=
 q nk
. If the node C n
is destroyed its MEs cannot provide connection with any other node (this state corresponds to term z n in Eq. (16)).
Therefore the u-function of MEs located at vulnerable node C n
takes form
U n
( z )
 
(

( n k


K
 n

1 ) q nk z k
)
 

( n k


K
 n

1 ) q nk z k   z n  

( n k



K n

1

1 ) q nk z k 
(
   q nn
) z n
.
(17)
3.4. u-function for the entire LMCCS
Consider now the paths starting from C
1
that are provided by elements allocated in subsequent positions. Let random value Y n
be the number of the farthest position of a path from
C
1
provided by MEs belonging to i n


1
E i
. All the paths provided by MEs from E
1
are single arc paths and, therefore, Y
1
=T
1
. The probabilistic distribution of Y
1
can be represented by u-function
~
U
1
( z ) which is equal to U
1
(z).
For an arbitrary pair of adjacent positions C n
and C n+1
, the paths provided by the MEs belonging to i n


1
E i can be continued by arcs provided by MEs belonging to E n+1 only if Y n
>n (the path reaches C n+1
). If this condition is satisfied, the most remote position of a path from C
1 provided by subset of MEs i n

1


1
E i
can be determined as Y n+1
=max{Y n
,T n+1
}.
In order to consider only the combinations of states of elements from i n


1
E i
corresponding to cases in which paths from C
1
to C n+1
exist (Y n
>n), we introduce the following

operator which eliminates the term with Y n
=n from polynomial
~
U n
( z ) :
11


(
~
U n
( z ))
 
(

( n

K
 j
 n

1 ) q nj z j
)


( n j


 n
K

1

1 ) q nj z j
.
(18)
Now, having the distribution of random value Y n
and random value T n+1
, represented by
~
U n
( z ) and U n+1
(z) respectively, one can determine u-function
~
U n

1
( z ) representing distribution of random value Y n+1
:
~
U n

1
( z ) =

(

(
~
U n
( z ) ),U n+1
(z)). (19)
Applying in sequence equation (19), one obtains
~
U
N
( z ) containing two terms corresponding to Y
N
=N and Y
N
=N+1.

(
~
U
N
( z ) ) has only one term corresponding to the probability that the path from C
1
to C
N+1
exists. The coefficient of this term is equal to LMCCS survivability S.
3.5. Algorithm for determination of LMCCS survivability
The following procedure determines the survivability of LMCCS with the given allocation of
MEs.
1.
Assign U n
(z)=u
0n
(z)=z n
for each n

[1,N].
2.
For the given h(i) for each 1
 i

M (vector H) , determine u ih(i)
(z) using Eq. (11) and modify U h(i)
(z):
U h(i)
(z)=

(U h(i)
(z),u ih(i)
(z)).
3. Modify U h(i)
(z) using operator

incorporating node vulnerability:
U h(i)
(z)=

(U h(i)
(z)).
3.
Assign
~
U
1
( z ) =U
1
(z) and apply in sequence Eq. (19) for n=1,…,N-1.
4.
Obtain the coefficient of the resulting single term polynomial

(
~
U
N
( z ) ) as the
LMCCS survivability.
3.6. Simple example
12

In order to illustrate the procedure, consider the LMCCS with N=M=2 and K=3 presented in
Fig. 2. Pr{G i
=j}=p ij
(n) for ME e i
allocated at position C n
.
Case A corresponds to ME allocation represented by the vector H
A
={1,1}. The u-functions of the individual MEs allocated as determined by the vector H
A
are: u
11
(z)=p
10
(1)z 1 +p
11
(1)z 2 +p
12
(1)z 3 , u
21
(z)=p
20
(1)z 1 +p
21
(1)z 2 +p
22
(1)z 3 ,
The u-functions representing distributions of random values T
1
and T
2
for the groups of MEs allocated at the same positions are:

(u
11
(z),u
21
(z))=p
10
(1)p
20
(1)z
1
+[p
10
(1)p
21
(1)+p
11
(1)(1-p
22
(1))]z
2
+[p
12
(1)+p
22
(1)-p
12
(1)p
22
(1)]z
3
,
U
1
(z)=

(

(u
11
(z),u
21
(z)))= [

+
 p
10
(1)p
20
(1)]z
1
+[p
10
(1)p
21
(1)+
 p
11
(1)(1-p
22
(1))]z
2
+

[p
12
(1)+p
22
(1)-p
12
(1)p
22
(1)]z 3 ,
U
2
(z)=

(u
02
(z))=

(z 2 )=(

+

)z 2 =z 2 .
The u-functions representing random values Y
1
and Y
2
are:
~
U
1
( z ) =U
1
(z),

(
~
U
1
( z ) )=

[p
10
(1)p
21
(1)+p
11
(1)(1-p
22
(1))]z 2 +

[p
12
(1)+p
22
(1)-p
12
(1)p
22
(1)]z 3 ,
~
U
2
( z ) =

(

(
~
U
1
( z ) ),U
2
(z))=

[p
10
(1)p
21
(1)+p
11
(1)(1-p
22
(1))]z
2
+

[p
12
(1)+p
22
(1)-p
12
(1)p
22
(1)]z
3
,

(
~
U
2
( z ) )=

[p
12
(1)+p
22
(1)-p
12
(1)p
22
(1)]z 3 .
Finally, the LMCCS survivability obtained from

(
~
U
2
( z ) ) is:
S
A
=

[p
12
(1)+p
22
(1)-p
12
(1)p
22
(1)].
If p
12
(1)=p
22
(1)=p
2
as in Eq. (3),
S
A
=

(2p
2
-p
2
2
).
Case B corresponds to ME allocation represented by the vector H
B
={1,2}. The u-functions of the individual MEs allocated as determined by the vector H
B
are: u
11
(z)=p
10
(1)z 1 +p
11
(1)z 2 +p
12
(1)z 3 , u
22
(z)=p
20
(2)z 2 +[p
21
(2)+p
22
(2)]z 3 ,
The u-functions representing distribution of random values T
1
and T
2
are:
13

U
1
(z)=

(u
11
(z))=[

+
 p
10
(1)]z 1 +
 p
11
(1)z 2 +
 p
12
(1)z 3 ,
U
2
(z)=

(u
22
(z))=[

+
 p
20
(2)]z
2
+

[p
21
(2)+p
22
(2)]z
3
,
The u-functions representing random values Y
1
and Y
2
are:
~
U
1
( z ) =U
1
(z),

(
~
U
1
( z ) )=
 p
11
(1)z 2 +
 p
12
(1)z 3 ,
~
U
2
( z ) =

(

(
~
U
1
( z ) ),U
2
(z))=
 p
11
(1)[

+
 p
20
(2)]z
2
+{
 2 p
11
(1)[p
21
(2)+p
22
(2)]
+
 p
12
(1)[

+
 p
20
(2)]+
 2 p
12
(1)[p
21
(2)+p
22
(2)]}z 3 ,

(
~
U
2
( z ) )={
 2 p
11
(1)p
21
(2)+
 2 p
11
(1)p
22
(2)+
 p
12
(1)+
 2 p
12
(1)p
20
(2)
+
 2 p
12
(1)p
21
(2)+
 2 p
12
(1)p
22
(2)}z
3
.
Since

=1-

the LMCCS survivability obtained from

(
~
U
2
( z ) ) is:
S
B
=
 p
12
(1)+
 2
{p
11
(1)p
21
(2)+p
11
(1)p
22
(2)-p
12
(1)+p
12
(1)(p
20
(2)+p
21
(2)+p
22
(2)]}=
 p
12
(1)+
 2
{p
11
(1)p
21
(2)+p
11
(1)p
22
(2)}.
If the MEs are identical i.e. p
11
(1)=p
21
(2)=p
1
and p
12
(1)=p
22
(2)=p
2
as in Eq. (4),
S
B
=
 p
2
+
 2 p
1
(p
1
+p
2
).
4.
Optimization technique
Finding the optimal ME allocation in LMCCS is a complicated combinatorial optimization problem having N
M
possible solutions. An exhaustive examination of all these solutions is not realistic even for a moderate number of positions and elements, considering reasonable time limitations. As in most combinatorial optimization problems, the quality of a given solution is the only information available during the search for the optimal solution. Therefore, a heuristic search algorithm is needed which uses only estimates of solution quality and which does not require derivative information to determine the next direction of the search.
14

The recently developed family of genetic algorithms is based on the simple principle of evolutionary search in solution space. GAs have been proven to be effective optimization tools for a large number of applications. Successful applications of GAs in reliability engineering are reported in [7,9-17].
It is recognized that GAs have the theoretical property of global convergence [18]. Despite the fact that their convergence reliability and convergence velocity are contradictory, for most practical, moderately sized combinatorial problems, the proper choice of GA parameters allows solutions close enough to the optimal one to be obtained in a short time.
4.1.
Genetic Algorithm
Basic notions of GAs are originally inspired by biological genetics. GAs operate with
"chromosomal" representation of solutions, where crossover, mutation and selection procedures are applied. "Chromosomal" representation requires the solution to be coded as a finite length string. Unlike various constructive optimization algorithms that use sophisticated methods to obtain a good singular solution, the GA deals with a set of solutions (population) and tends to manipulate each solution in the simplest manner.
A brief introduction to genetic algorithms is presented in [19]. More detailed information on
GAs can be found in Goldberg’s comprehensive book [20], and recent developments in GA theory and practice can be found in books [17,18].
The steady state version of the GA used in this paper was developed by Whitley [21]. As reported in [22] this version, named GENITOR, outperforms the basic “generational” GA. The structure of steady state GA is as follows:
Algorithm GENITOR
1. Generate an initial population of Ns randomly constructed solutions (strings) and evaluate their fitness. (Unlike the “generational” GA, the steady state GA performs the evolution search
15

within the same population improving its average fitness by replacing worst solutions with better ones).
2. Select two solutions randomly and produce a new solution (offspring) using a crossover procedure that provides inheritance of some basic properties of the parent strings in the offspring.
The probability of selecting the solution as a parent is proportional to the rank of this solution.
(All the solutions in the population are ranked by increasing order of their fitness). Unlike the fitness-based parent selection scheme, the rank-based scheme reduces GA dependence on the fitness function structure, which is especially important when constrained optimization problems are considered [23].
3. Allow the offspring to mutate with given probability P m
. Mutation results in slight changes in the offspring structure and maintains diversity of solutions. This procedure facilitates jumps in the solution space, which helps the GA to avoid premature convergence to a local optimum [20].
The positive changes in the solution code created by the mutation can be later propagated throughout the population via crossovers.
4. Decode the offspring to obtain the objective function (fitness) values. These values are a measure of quality, which is used in comparing different solutions.
5. Apply a selection procedure that compares the new offspring with the worst solution in the population and selects the one that is better. The better solution joins the population and the worse one is discarded. If the population contains equivalent solutions following the selection process, redundancies are eliminated and, as a result, the population size decreases. Note that each time the new solution has sufficient fitness to enter the population, it alters the pool of prospective parent solutions and increases the average fitness of the current population. The average fitness increases monotonically (or, in the worst case, does not vary) during each genetic cycle (steps 2-5).
6. Generate new randomly constructed solutions to replenish the population after repeating steps 2-5 N rep
times (or until the population contains a single solution or solutions with equal
16

quality). Run the new genetic cycle (return to step 2). In the beginning of a new genetic cycle, the average fitness can decrease drastically due to inclusion of poor random solutions into the population. These new solutions are necessary to bring into the population new "genetic material" which widens the search space and, like a mutation operator, prevents premature convergence to the local optimum.
7. Terminate the GA after N c
genetic cycles.
End_Algorithm
The final population contains the best solution achieved. It also contains different nearoptimal solutions, which may be of interest in the decision-making process.
4.2. Solution representation and basic GA procedures
To apply the genetic algorithm to a specific problem, one must define a solution representation and decoding procedure, as well as specific crossover and mutation procedures.
As it was shown in section 2, any arbitrary M-length vector H with elements h(i) belonging to the range [1,N] represents a feasible allocation of MEs. Such vectors can represent each one of the possible N
M
different solutions. The fitness of each solution is equal to the survivability of
LMCCS with allocation, represented by the corresponding vector H . To estimate the LMCCS survivability for the arbitrary vector H , one should apply the procedure presented in section 3.
The random solution generation procedure provides solution feasibility by generating vectors of random integer numbers within the range [1,N]. It can be seen that the following crossover and mutation procedures also preserve solution feasibility.
The crossover operator for given parent vectors P1 , P2 and the offspring vector O is defined as follows: first P1 is copied to O , then all numbers of elements belonging to the fragment between a and b positions of the vector P2 ( where a and b are random values, 1
 a<b

M) are copied to the corresponding positions of O . The following example illustrates the crossover procedure for M=6, N=4:
17

P1 = 2 4 1 4 2 3
P2 = 1 1 2 3 4 2
O = 2 4 2 3 4 3
The mutation operator moves the randomly chosen ME to the adjacent position (if such a position exists) by modifying a randomly chosen element h(i) of H using rule h(i)=max{h(i)-1,1} or rule h(i)=min{h(i)+1,N} with equal probability. The vector O in our example can take the following form after applying the mutation operator :
O = 2 3 2 3 4 3 .
5.
Illustrative examples
5.1. Two ME allocation problems.
Consider the ME allocation problem presented in [6], in which N=M=13 and reliability characteristics of MEs do not depend on their allocation. All the MEs have two states with nonzero probabilities. The state probability distributions of the MEs are presented in Table 1.
In order to compare the solution presented in [6] (solution A) with the solution obtained by the GA (solution B), we first solve the allocation problem for node vulnerability

=0 when allocation of no more than one ME at each position is allowed. This is done by imposing a penalty on the solutions in which more than one ME is allocated in the same position. Both solutions are presented in Table 2. One can see that solution B which was obtained by the GA is better. This solution considerably improves when all the limitations on the ME allocation are removed. Observe that the best solution obtained by the GA (solution C), in which only 5 out of
13 positions are occupied by the MEs, provides much greater reliability than solution B. (Note that when

=0 the system survivability is equal to its reliability since only internal factors can cause the system failure).
The ME allocation solutions obtained for

=0.06 and

=0.3 (solutions D and E respectively) are also presented in Table 2. In Figure 5 the three optimal solutions obtained for different values
18

of node vulnerability are presented in the form of graphs. In these graphs maximal length arcs provided by each ME in operable condition are depicted.
Note that each solution provides the greatest possible system survivability only for the given value of

. When

varies, the optimal ME allocation changes. One can see the LMCCS survivability obtained for solutions C, D and E as function of variable node vulnerability in
Figure 6.
Observe that the greater the node vulnerability, the greater the number of occupied nodes in the optimal solution. Indeed, by increasing ME separation the system tries to compensate its increasing vulnerability.
In the second example, LMCCS consists of N=20 positions and M=16 MEs. There are four groups of identical MEs (four elements in each group). The ME state distributions depend on the
ME allocation. All the positions are divided into three groups, such that the positions belonging to the same group have the same influence on the MEs state distributions. The probabilistic distributions of MEs states are presented in Table 3.
Two ME allocation solutions obtained by the GA for

=0 and

=0.06 (solutions A and B respectively) are presented in Table 4. Observe that in the solution B all the MEs are totally separated. While for invulnerable system (

=0) the reliability of the solution B is slightly lower than the reliability of the solution A, for

=0.06 the solution B provides system survivability increase by 12.7% over solution A.
5.2. Computational Effort and Algorithm Consistency
The C language realization of the algorithm was tested on a Pentium II PC on a set of 20 randomly generated problems with 10

N

25 and 10

M

30. Different GA parameters were tested in the ranges 50

N
S

300, 500

N rep

10000, 5

N c

200 and 0

P m

1. The chosen combination of parameters that provided the GA convergence to the best achieved solutions by
19

the shortest time in 75% of the tests is N
S
=100, N rep
=2000, N c
=200 and P m
=1. The time taken to obtain the best-in-population solution (time of the last modification of the best solution obtained) for the GA with the chosen parameters did not exceed 200 seconds. When this GA was tested on the first and second problems presented in section 5.1 the times taken to obtain the best-inpopulation solutions were 40 seconds and 90 seconds respectively.
To demonstrate the consistency of the suggested algorithm, we repeated the GA 100 times with different starting solutions (initial population) for the second problem (M=16, N=20). The coefficient of variation was calculated for fitness values of best-in-population solutions obtained during the genetic search by different GA search processes. The variation of this index during the
GA procedure is presented in Fig. 7. One can see that the standard deviation of the final solution fitness does not exceed 1.7 % of its average value.
6.
Conclusions
The paper presents an algorithm for evaluating survivability of linear multistate consecutively-connected system in which any system node can be destroyed by external impact.
An example of such system is a set of vulnerable radio relay stations in which multistate retransmitters with different characteristics are allocated.
The problem of finding an allocation of the retransmitters among the system nodes that provides the greatest system survivability is formulated. The procedure for solving this problem is suggested which uses a genetic algorithm as the optimization tool.
It is shown that in many cases greater survivability can be achieved if some of retransmitters are gathered in the same node providing redundancy than if all the retransmitters are evenly distributed among all the nodes. The greater the node vulnerability, the greater the number of occupied nodes in the optimal solution since by increasing retransmitter separation the system tries to compensate its increasing vulnerability.
20

References
[1] F. Hwang, Y. Yao, Multistate consecutively-connected systems , IEEE Transactions on Reliability, vol. 38, 1989, pp. 472-474.
[2] J. Shanthikumar, A recursive algorithm to evaluate the reliability of a consecutive-k-out-of-n:F system, IEEE Transactions on Reliability, vol. R-31, 1982, pp. 442-443.
[3] J. Shanthikumar, Reliability of systems with consecutive minimal cutsets , IEEE Transactions on
Reliability, vol. R-36, 1987, pp. 546-550.
[4] A. Kossow, W. Preuss, Reliability of linear consecutively-connected systems with multistate components, IEEE Transactions on Reliability, vol. 44, 1995, pp. 518-522.
[5] M. Zuo, M. Liang, Reliability of multistate consecutively-connected systems, Reliability Engineering
& System Safety, vol. 44, 1994, pp. 173-176.
[6] J. Malinowski, W. Preuss, Reliability increase of consecutive-k-out-of-n:F and related systems through components' rearrangement, Microelectronics and Reliability , vol. 36, 1996, pp. 1417-1423.
[7] G. Levitin, A. Lisnianski, Optimal separation of elements in vulnerable multi-state systems,
Reliability Engineering & System Safety , vol. 73, 2001, pp. 55-66.
[8] I. A. Ushakov, Universal generating function, Sov. J. Computing System Science, vol. 24, No 5,
1986, pp. 118-129.
[9] G. Levitin, A. Lisnianski, H. Beh-Haim, D. Elmakis, Redundancy Optimization for Series-parallel
Multi-state Systems, IEEE Transactions on Reliability , vol. 47, 1998, pp. 165-172.
[10] G. Levitin, Multistate series-parallel system expansion-scheduling subject to availability constraints,
IEEE Transactions on Reliability , vol. 49, 2000, pp. 71-79.
[11] G. Levitin, A. Lisnianski, A new approach to solving problems of multi-state system reliability optimization, Quality and Reliability Engineering International , vol. 47, 2001, pp. 93-104.
[12] G. Levitin, A. Lisnianski, Reliability optimization for weighted voting system, Reliability
Engineering & System Safety , vol. 71, 2001, pp. 131-138.
[13] G. Levitin, A. Lisnianski, Structure Optimization of Multi-state System with Two Failure Modes,
Reliability Engineering & System Safety , vol. 72, 2001, pp. 75-89.
[14] L. Painton and J. Campbell, Genetic algorithm in optimization of system reliability , IEEE Trans.
Reliability , 44, 1995, pp. 172-178.
[15] D. Coit and A. Smith, Reliability optimization of series-parallel systems using genetic algorithm,
IEEE Trans. Reliability , 45, 1996, pp. 254-266.
[16] M. Gen and J. Kim, GA-based reliability design: state-of-the-art survey , Computers & Ind. Engng ,
37, 1999, pp. 151-155.
[17] M. Gen and R. Cheng, Genetic Algorithms and engineering optimization , John Wiley & Sons, New
York, 2000.
[18] T. B  ck, Evolutionary Algorithms in Theory and Practice. Evolution Strategies. Evolutionary
Programming. Genetic Algorithms, Oxford University Press, 1996.
[19] S. Austin, An introduction to genetic algorithms, AI Expert , 5, 1990, pp. 49-53.
[20] D. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning , Addison Wesley,
Reading, MA, 1989.
[21] D. Whitley, The GENITOR Algorithm and Selective Pressure: Why Rank-Based Allocation of
Reproductive Trials is Best. Proc. 3th International Conf. on Genetic Algorithms. D. Schaffer, ed., pp.
116-121. Morgan Kaufmann, 1989.
[22]. G. Syswerda, A study of reproduction in generational and steady-state genetic algorithms, in G.J.E.
Rawlings (ed.), Foundations of Genetic Algorithms , Morgan Kaufmann, San Mateo, CA, 1991.
[23]. D. Powell, M. Skolnik, Using genetic algorithms in engineering design optimization with non-linear constraints , Proc. Of the fifth Int. Conf. On Genetic Algorithms , Morgan Kaufmann, 1993, pp. 424-431.
21

Figure Captions
Figure 1: LMCCS structure.
Figure 2: Two possible allocations of MEs in LMCCS with N=M=2.
Figure 3: Comparison of two possible allocations of MEs in LMCCS with N=M=2.
Figure 4: Comparison of two possible allocations of MEs in LMCCS with vulnerable nodes and
N=M=2.
Figure 5: Three ME allocations obtained by the GA for

=0 (C),

=0.06 (D) and

=0.3 (E).
Figure 6: LMCCS survivability as function of node vulnerability for the three ME allocations obtained by the GA for

=0 (C),

=0.06 (D) and

=0.3 (E).
Figure 7: Coefficient of variation of best-in-population solution fitness obtained by 100 different search processes as function of number of crossovers.
22

Table 1. MEs' state distributions for the first allocation problem.
Element
C
5
C
6
C
7
C
8
C
1
C
2
C
3
C
4
C
9
C
10
C
11
C
12
C
13
S for

=0
No of state e
5 e
6 e
7 e
8 e
1 e
2 e
3 e
4 e
9 e
10 e
11 e
12 e
13
0
0.70
0.65
0.60
0.55
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
1
0.00
0.00
0.00
0.00
0.50
0.55
0.00
0.00
0.00
0.75
0.00
0.00
0.00
2
0.00
0.00
0.00
0.45
0.00
0.00
0.00
0.00
0.70
0.00
0.00
0.85
0.00
3
0.30
0.00
0.40
0.00
0.00
0.00
0.00
0.65
0.00
0.00
0.80
0.00
0.00
Table 2. Solutions of the first allocation problem.
Node ME allocation
0.90
4
0.00
0.35
0.00
0.00
0.00
0.00
0.60
0.00
0.00
0.00
0.00
0.00
Solution A Solution B Solution C Solution D Solution E
12
8
7
2
13
5
1
4
9
3
11
6
10
0.58756
12
3
11
2
13
6
5
1
8
7
4
9
10
0.59201
4,9,12
-
13
-
-
-
1,3,11
-
-
2,7,8
-
-
5,6,10
0.72319
13
-
-
-
1,2,11
-
-
5,6,12
4,9
7,10
3,8
-
-
0.70762
13
-
-
2
11
-
3,4
1,6,12
5,9
7,10
8
-
-
0.648004
23

Table 3. MEs' state distributions for the second allocation problem.
C
9
C
10
C
11
C
12
C
13
C
14
C
15
C
16
C
5
C
6
C
7
C
8
C
1
C
2
C
3
C
4
C
17
C
18
C
19
Positions Elements No of state
1, 2, 3, 8, 13
6, 7, 10, 11,
14, 18, 19, 20
4, 5, 9, 12,
15, 16, 17 e
1
-e
4 e
5
-e
8 e
9
-e
12 e
13
-e
16 e
1
-e
4 e
5
-e
8 e
9
-e
12 e
13
-e
16 e
1
-e
4 e
5
-e
8 e
9
-e
12 e
13
-e
16
0
0.03
0.05
0.02
0.05
0.03
0.05
0.05
0.05
0.05
0.08
0.05
0.10
1
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.10
0.30
0.22
0.95
0.05
Table 4. Solutions of the second allocation problem.
2
0.15
0.00
0.05
0.95
0.85
0.10
0.85
0.85
0.65
0.00
0.00
0.85
Position
C
20
S for

=0.00
S for

=0.06
ME allocation
Solution A Solution B
-
8
3
-
10
5
-
13, 15
4, 14
-
9, 11
-
-
1,7
-
12
-
6, 16
2
-
0.96291
2
6
15
14
-
7
4
-
13
9
8
12
10
-
11
1
16
5
3
-
0.93653
0.69539 0.78428
3
0.65
0.10
0.93
0.00
0.12
0.05
0.10
0.00
0.00
0.70
0.00
0.00
4
0.17
0.85
0.00
0.00
0.00
0.80
0.00
0.00
0.00
0.00
0.00
0.00
24