Optimal allocation of multistate elements in linear consecutivelyconnected systems with vulnerable nodes
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: [email protected]
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
multistate element
LMCCS
linear multistate consecutively-connected system
UGF
universal generating function
GA
genetic algorithm
Notation
Pr{x} probability of event x
R
reliability of LMCCS
S
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-)
ei
i-th ME of LMCCS
Cn
n-th position (node) of LMCCS
E
set of all available MEs: E={e1,…,eM}
En
set of MEs located at position n
h(i)
number of position the ME ei is allocated at: eiEh(i)
H
vector representing allocation of MEs in LMCCS: H={h(i),1iM}
K
maximal number of different states of MEs
Gi
random state of i-th ME
pik(n) Pr{Gi=k | eiEn}
PI(n)
vector representing probabilistic state distribution of i-th ME: Pi={pi0(n),…,piK-1(n)}
Tn
random number of the most remote position to which arc from Cn exists
2
Yn
random number of the most remote position to which path from C1 provided by MEs
n
belonging to
 Ei
exists
i 1
qnk
Pr{Tn=k}
uin(z) u-function for i-th ME located at position n (i=0 corresponds to absence of ME)
Un(z) u-function for the group of MEs belonging to En (located at position n)
n
~
(z)
u-function
for
subset
of
MEs
Un
 Ei
i 1
,, 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) Cn, n[1,N+1]. The
first position C1 is the source and the last one CN+1 is the sink (see Figure 1). At each position
C1,…,CN, elements from a set E={e1,…,eM} can be allocated. These elements provide
connections (arcs) between the position in which they are allocated and further positions.
Each element ei has K states, where state Gi of ei is a discrete random value with distribution
depending on location of the element. For an ME located at node Cn:
Pr{G i  k}  p ik (n ),
K 1
 pik (n )  1.
k 0
Gi=k for element ei allocated at position Cn implies that arcs exist from Cn to each of Cn+1,
Cn+2,…,C(n+k), where (x)=min{x,N+1}. Gi=0 implies the total failure state of ei (no arcs exist
from Cn).
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 ei has Ki states and ME
3
em has Km states (KiKm), one can consider both MEs as having K=max{Ki,Km} states while
assigning pik=0 for Kik<K.
All the states Gi are independent.
The system is failed if there is no path from C1 to CN+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 CN+1. Each station Cn (2nN) can have retransmitters generating
signals that reach the next Gi stations. Note that Gi 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 C1,…,CN 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 p0, p1 and p2,
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
RA=2p2-p22.
(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
RB=p2+p1(p1+p2).
(2)
By comparing (1) and (2), one can decide which allocation of the elements is preferable for any
given p1 and p2. Figure 3 presents the decision curve R A=RB on the plane (p1,p2). Observe that
for combinations of p1 and p2 located below the curve, the solution A is preferable while for
combinations of p1 and p2 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 C1 is not destroyed and at least one of the MEs is in
state 2. The system survivability is
SA=(2p2-p22).
(3)
In case B, the LMCCS survives either when the node C1 is not destroyed and ME located in C1 is
in state 2, or if both nodes C1 and C2 are not destroyed, the ME located in C1 is in state 1 and the
ME located in C2 is not in state 0. The system survivability in this case is
SB=p2+2p1(p1+p2).
(4)
By comparing (3) and (4), one can decide which allocation of the elements is preferable for any
given , p1 and p2. Condition SASB can be rewritten as
(2p2-p22)p2+2p1(p1+p2).
and finally as
p2(1-p2)/(p1(p1+p2)).
(5)
Figure 4 presents the maximal values of  for whish SASB as function of variables p1 and
p2. Observe that for given combinations of p1 and p2 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 (p2) belongs to line =p2. Indeed, since p0+p1+p2=1,
the maximal possible value of p1 for each given p2 is 1-p2. Substituting p1 with 1-p2 in Eq. (5)
one obtains p2.
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 En (1nN), i.e. such that
N
 E n  E,
(6)
E i  E j  ø, ij.
(7)
n 1
Each set En, corresponding to LMCCS position Cn, 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 En
as
7
M
|En|= 1(h (i)  n ).
(8)
i 1
In the general case, the probability of being at state k for any ME ei 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 Pi(n)={pi0(n),…,piK-1(n)} for each ME ei 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 P1(n),…, PM(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*( P1(n),…, PM(n))=arg{S(H, P1(n),…, PM(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 1
 pk z x k ,
(10)
k 0
where the variable X has K possible values and pk is the probability that X is equal to xk.
Consider ME ei located at position Cn. In each state k (1k<K), the ME makes Cn connected
with Cn+1,…,C(n+k). The probability of state k for ME ei located at Cn is pik(n). Let random value
Tn be the number of the most remote position to which the arc from Cn exists. The polynomial
uin(z) can represent all the possible states of the ME by relating the probabilities of each state to
the value of Tn in this state:
u in (z) 
K 1
 pik (n )z (n  k ) .
(11)
k 0
Note that the absence of any ME in position Cn implies that no arcs exist from Cn to any
other position. This means that any arc reaching Cn has no continuation with probability 1. In
this case, the corresponding u-function takes the form
u0n(z)=zn.
(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 En using simple algebraic operations on the individual
u-functions of MEs. The composition operator for a pair of MEs ei and em takes the form:
K 1
(u in (z), u mn (z))  (  p ik (n )z
K 1 K 1
k 0
  pik (n )p mj (n )z
k  0 j 0
9
( n  k )
K 1
,
 p mj (n )z (n  j) ) 
j 0
max{ ( n  k ), ( n  j)}
(13)
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 Cn 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 Cn to Cn+1,…,Cmax{(n+k),(n+j)} exist.
Note that for any uin(z)
(u0n(z),uin(z))=uin(z).
(14)
One can see that the operator  satisfies the following conditions:
{u1 (z),..., u k (z), u k 1 (z),..., u v (z)}  {{u1 (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 Cn:

iE n
(u in (z)) 
( n  K 1)

k n
q nk z k .
(16)
This polynomial determines the probabilistic distribution of Tn provided by all the MEs
belonging to En. Observe that, after collecting like terms, the resulting polynomial (as well as
polynomials uin(z) for individual MEs) can have no more than min{K,N-n+2} terms
(corresponding to values of Tn{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 Cn with vulnerability , this means that all
these elements can be destroyed with probability . The probabilities qnk in (16) relate only to
reliability of MEs (internal system reliability). In order to incorporate external destructive factors
into the model, the probabilities qnk should be considered as conditional probabilities that the
node Cn 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
Cn is connected with the corresponding set of nodes is, therefore, equal to (1-)qnk=qnk. If the
node Cn is destroyed its MEs cannot provide connection with any other node (this state
corresponds to term zn in Eq. (16)).
Therefore the u-function of MEs located at vulnerable node Cn takes form
U n (z)  (
( n  K 1)

k n
k
q nk z )  
( n  K 1)

k n
k
n
q nk z  z  
( n  K 1)

k  n 1
q nk z k  (  q nn )z n . (17)
3.4. u-function for the entire LMCCS
Consider now the paths starting from C1 that are provided by elements allocated in
subsequent positions. Let random value Yn be the number of the farthest position of a path from
n
C1 provided by MEs belonging to
 E i . All the paths provided by MEs from E1 are single arc
i 1
paths and, therefore, Y1=T1. The probabilistic distribution of Y1 can be represented by u-function
~
U1 ( z ) which is equal to U1(z).
For an arbitrary pair of adjacent positions Cn and Cn+1, the paths provided by the MEs
n
belonging to
 E i can be continued by arcs provided by MEs belonging to En+1 only if Yn>n (the
i 1
path reaches Cn+1). If this condition is satisfied, the most remote position of a path from C 1
n 1
provided by subset of MEs
 Ei
can be determined as Yn+1=max{Yn,Tn+1}.
i 1
n
In order to consider only the combinations of states of elements from
 Ei
corresponding
i 1
to cases in which paths from C1 to Cn+1 exist (Yn>n), we introduce the following  operator
~
which eliminates the term with Yn=n from polynomial U n (z) :
11
~
( U n (z))  (
( n  K 1)

j n
j
q njz ) 
( n  K 1)

j n 1
q njz j .
(18)
Now, having the distribution of random value Yn and random value Tn+1, represented by
~
~
U n (z) and Un+1(z) respectively, one can determine u-function U n 1 (z) representing distribution
of random value Yn+1:
~
~
U n 1 (z) =(( U n (z) ),Un+1(z)).
(19)
~
Applying in sequence equation (19), one obtains U N (z) containing two terms corresponding
~
to YN=N and YN=N+1. ( U N (z) ) has only one term corresponding to the probability that the
path from C1 to CN+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 Un(z)=u0n(z)=zn for each n[1,N].
2. For the given h(i) for each 1iM (vector H), determine uih(i)(z) using Eq. (11) and
modify Uh(i)(z):
Uh(i)(z)=(Uh(i)(z),uih(i)(z)).
3. Modify Uh(i)(z) using operator  incorporating node vulnerability:
Uh(i)(z)=(Uh(i)(z)).
~
3. Assign U1 ( z ) =U1(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{Gi=j}=pij(n) for ME ei allocated at position Cn.
Case A corresponds to ME allocation represented by the vector HA={1,1}. The u-functions
of the individual MEs allocated as determined by the vector HA are:
u11(z)=p10(1)z1+p11(1)z2+p12(1)z3, u21(z)=p20(1)z1+p21(1)z2+p22(1)z3,
The u-functions representing distributions of random values T1 and T2 for the groups of MEs
allocated at the same positions are:
(u11(z),u21(z))=p10(1)p20(1)z1+[p10(1)p21(1)+p11(1)(1-p22(1))]z2+[p12(1)+p22(1)-p12(1)p22(1)]z3,
U1(z)=((u11(z),u21(z)))= [+p10(1)p20(1)]z1+[p10(1)p21(1)+p11(1)(1-p22(1))]z2+
[p12(1)+p22(1)-p12(1)p22(1)]z3,
U2(z)=(u02(z))=(z2)=(+)z2=z2.
The u-functions representing random values Y1 and Y2 are:
~
U1 ( z ) =U1(z),
~
( U1 ( z ) )=[p10(1)p21(1)+p11(1)(1-p22(1))]z2+[p12(1)+p22(1)-p12(1)p22(1)]z3,
~
~
U 2 (z) =(( U1 ( z ) ),U2(z))= [p10(1)p21(1)+p11(1)(1-p22(1))]z2+[p12(1)+p22(1)-p12(1)p22(1)]z3,
~
( U 2 (z) )=[p12(1)+p22(1)-p12(1)p22(1)]z3.
~
Finally, the LMCCS survivability obtained from ( U 2 (z) ) is:
SA=[p12(1)+p22(1)-p12(1)p22(1)].
If p12(1)=p22(1)=p2 as in Eq. (3),
SA=(2p2-p22).
Case B corresponds to ME allocation represented by the vector HB={1,2}. The u-functions
of the individual MEs allocated as determined by the vector HB are:
u11(z)=p10(1)z1+p11(1)z2+p12(1)z3, u22(z)=p20(2)z2+[p21(2)+p22(2)]z3,
The u-functions representing distribution of random values T1 and T2 are:
13
U1(z)=(u11(z))=[+p10(1)]z1+p11(1)z2+p12(1)z3,
U2(z)=(u22(z))=[+p20(2)]z2+[p21(2)+p22(2)]z3,
The u-functions representing random values Y1 and Y2 are:
~
U1 ( z ) =U1(z),
~
( U1 ( z ) )=p11(1)z2+p12(1)z3,
~
~
U 2 (z) =(( U1 ( z ) ),U2(z))=p11(1)[+p20(2)]z2+{2p11(1)[p21(2)+p22(2)]
+p12(1)[+p20(2)]+2p12(1)[p21(2)+p22(2)]}z3,
~
( U 2 (z) )={2p11(1)p21(2)+2p11(1)p22(2)+p12(1)+2p12(1)p20(2)
+2p12(1)p21(2)+2p12(1)p22(2)}z3.
~
Since =1- the LMCCS survivability obtained from ( U 2 (z) ) is:
SB=p12(1)+2{p11(1)p21(2)+p11(1)p22(2)-p12(1)+p12(1)(p20(2)+p21(2)+p22(2)]}=
p12(1)+2{p11(1)p21(2)+p11(1)p22(2)}.
If the MEs are identical i.e. p11(1)=p21(2)=p1 and p12(1)=p22(2)=p2 as in Eq. (4),
SB=p2+2p1(p1+p2).
4. Optimization technique
Finding the optimal ME allocation in LMCCS is a complicated combinatorial optimization
problem having NM 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 Pm. 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 Nrep 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 Nc 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 NM 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NS300, 500Nrep10000, 5Nc200 and 0Pm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 NS=100, Nrep=2000, Nc=200 and Pm=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
e1
e2
e3
e4
e5
e6
e7
e8
e9
e10
e11
e12
e13
0
1
No of state
2
3
4
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
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
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
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
0.00
0.35
0.00
0.00
0.00
0.00
0.60
0.00
0.00
0.00
0.00
0.00
0.90
Table 2. Solutions of the first allocation problem.
Node
Solution A
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
C11
C12
C13
S for =0
13
5
1
4
12
8
7
2
9
3
11
6
10
0.58756
ME allocation
Solution B
Solution C Solution D
13
6
5
1
12
3
11
2
8
7
4
9
10
0.59201
4,9,12
13
1,3,11
2,7,8
5,6,10
0.72319
23
13
1,2,11
5,6,12
4,9
7,10
3,8
0.70762
Solution E
13
2
11
3,4
1,6,12
5,9
7,10
8
0.648004
Table 3. MEs' state distributions for the second allocation problem.
Positions
1, 2, 3, 8, 13
6, 7, 10, 11,
14, 18, 19, 20
4, 5, 9, 12,
15, 16, 17
Elements
e1-e4
e5-e8
e9-e12
e13-e16
e1-e4
e5-e8
e9-e12
e13-e16
e1-e4
e5-e8
e9-e12
e13-e16
No of state
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
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
Table 4. Solutions of the second allocation problem.
Position
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
C11
C12
C13
C14
C15
C16
C17
C18
C19
C20
S for =0.00
S for =0.06
ME allocation
Solution A
Solution B
4, 14
9, 11
1,7
12
8
3
10
5
13, 15
6, 16
2
0.96291
0.69539
10
11
1
13
9
8
12
7
4
2
6
15
14
16
5
3
0.93653
0.78428
24
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