Download Algorithm GENITOR

Optimal allocation of multistate elements in linear
consecutively-connected systems
Gregory Levitin, Senior Member IEEE
The Israel Electric Corporation Ltd., Haifa
Summary & Conclusions - A linear consecutively-connected system consists of N+1
linear ordered positions. M statistically independent multistate elements with different
characteristics are to be allocated at the N first positions. Each element can provide a connection
between the position to which it is allocated and the next few positions. The reliability of the
connection for any given element depends on the position to which it is allocated and on the
number of positions it connects. The system fails if the first position (source) is not connected
with the N+1-th one (sink). An algorithm based on the universal generating function method is
suggested for the linear consecutively-connected system reliability determination. This algorithm
can handle cases in which any number of multistate elements are allocated in the same position
while some positions remain empty. It is shown that in many cases, such uneven allocation
provides greater system reliability than the even one. A genetic algorithm is used as optimization
tool in order to solve the optimal element allocation problem.
Index Terms – System reliability, multistate elements, linear consecutively-connected
systems, universal generating function, genetic algorithm.
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Acronyms
ME
multistate element
LMCCS
linear multistate consecutively-connected system
UGF
universal generating function
GA
genetic algorithm
Notation
R
reliability of LMCCS
N
number of positions in LMCCS
M
number of available elements in LMCCS
ei
i-th ME of LMCCS
Cn
n-th position 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
Si
random state of i-th ME
pik
Pr{Si=k}
Pi
vector representing pmf of state of i-th ME: Pi={pi0,…,piK-1}
Tn
random number of the most remote position to which arc from Cn exists
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 any ME)
2
Un(z) u-function for the group of MEs belonging to En (located at position n)
n
~
U n(z) u-function for subset of MEs  E i
i 1
, 
operators over u-functions
()
(x)=min{x,N+1}.
1. INTRODUCTION
The LMCCS consists of N+1 consecutively 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 Si of ei is a discrete r.v. with pmf:
Pr{Si  k}  p ik ,
K 1
 pik  1.
k 0
Si=k (with 1kK-1) 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}. Si=0 implies the total failure state of ei
(no arcs created by ei 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
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 Si are s-independent.
The system is failed iff 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
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signals that reach the next Si stations. Note that Si is a r.v. dependent on power and availability of
retransmitter amplifiers. 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
probabilities of being in each state 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
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RA=2p2-p22.
(1)
In case B, the LMCCS 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.
In this paper, we consider a generalization of the optimal allocation problem. In this
generalization, the number of MEs is not necessarily equal to number of positions (MN), and an
arbitrary number of elements can be allocated at each position (some positions may be empty).
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 pmf. Section 4 describes the optimization approach used
and its adaptation to the problem formulated. In the fifth section, illustrative examples are
presented 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
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N
 E n  E,
(3)
E i  E j  ø, ij.
(4)
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
M
|En|=  f (h (i), n ),
(5)
i 1
1, x  y
where f ( x, y)  
0, x  y.

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.
For the given set of MEs with specified vectors P1(n),…, PM(n) representing pmf of their
states, the only factor influencing the LMCCS reliability is the allocation of its elements H.
Therefore, the optimal allocation problem can be formulated as follows.
Find vector H* maximizing the LMCCS reliability R:
H*( P1(n),…, PM(n))=arg{R(H, P1(n),…, PM(n))max}.
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(6)
3. LMCCS RELIABILITY ESTIMATION BASED ON A UNIVERSAL GENERATING
FUNCTION
The procedure used in this paper for LMCCS reliability evaluation is based on the universal
z-transform (also called u-function or universal generating function) technique, which was
introduced in [7] and which proved to be very effective for reliability evaluation of different
types of multi-state systems [8-13]. Unlike the recursive algorithm suggested in [4], the ufunction technique can straightforwardly handle cases in which any number of multistate
elements are allocated in the same position while some positions remain empty. The u-function
extends the widely known ordinary moment generating function.
The UGF (u-transform) of a discrete r.v. X is defined as a polynomial
u (z) 
K 1
 pk z x k ,
(7)
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 r.v. Tn be the
number of the most remote position to which an 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 ) .
(8)
k 0
Note that the absence of any ME at position Cn implies that no paths exist from Cn to any
further 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.
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(9)
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
polynomial Un(z) 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
K 1
k 0
j 0
(u in (z), u mn (z))  (  p ik (n )z ( n  k ) ,
K 1 K 1
  pik (n )p mj (n )z
 p mj (n )z (n  j) ) 
max{ ( n  k ), ( n  j)}
(10)
k  0 j 0
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 (for k>0 or j>0).
Note that for any uin(z)
(u0n(z),uin(z))=uin(z).
(11)
One can see that the operator  satisfies the following conditions:
{u1 (z),..., u d (z), u d 1 (z),..., u v (z)}  {{u1 (z),..., u d (z)}, {u d 1 (z),...., u v (z)}}
(12)
for arbitrary d. Therefore, it can be applied recursively to obtain the u-function for an arbitrary
group of MEs allocated at Cn:
U n ( z) 

iE n
(u in (z)) 
( n  K 1)

k n
q nk z k .
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 Un(z) (as well
as polynomials uin(z) for individual MEs) can have no more than min{K,N-n+2} terms
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(corresponding to values of Tn{n,n+1,…,min{n+K-1,N+1}).
Consider now the paths starting from C1 that are provided by elements allocated in
subsequent positions. Let r.v. Yn be the number of the farthest position of a path from C1
n
 E i . All the paths provided by MEs from E1 are single arc
provided by MEs belonging to
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 C1
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) :
~
( U n (z))  (
( n  K 1)

j n
j
q njz ) 
( n  K 1)

j n 1
q njz j .
(13)
~
Now having the pmf of r.v. Yn and r.v. Tn+1, represented by U n (z) and Un+1(z), one can
~
determine u-function U n 1 (z) representing pmf of r.v. Yn+1:
~
~
U n 1 (z) =(( U n (z) ),Un+1(z)).
(14)
~
Computing equation (14) recursively, one obtains U N (z) that contains two terms
~
corresponding to YN=N and YN=N+1. ( U N (z) ) has only one term corresponding to the
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probability that the path from C1 to CN+1 exists. The coefficient of this term is equal to LMCCS
reliability R.
The following procedure determines the reliability of LMCCS with the given allocation of
MEs (see Fig. 4).
1. Assign Un(z)=u0n(z)=zn for each n[1,N].
2. For the given vector H for each 1iM, determine uih(i)(z) using Eq. (8) and modify
Uh(i)(z):
Uh(i)(z)=(Uh(i)(z),uih(i)(z)).
~
3. Assign U1 ( z ) =U1(z) and apply in sequence Eq. (14) for n=1,…,N-1.
~
4. Obtain the coefficient of the resulting single term polynomial ( U N (z) ) as the
LMCCS reliability.
In order to illustrate the procedure, we will consider a LMCCS with N=M=K=3, H={1,1,3},
Pr{Si=j}=pij(n) for ME ei allocated at position Cn.
The u-functions of the individual MEs allocated as determined by the vector H are:
u11(z)=p10(1)z1+p11(1)z2+p12(1)z3, u21(z)=p20(1)z1+p21(1)z2+p22(1)z3,
u02(z)=z2,
u33(z)=p30(3)z3+[p31(3)+p32(3)]z4.
The u-functions representing pmf of r.v. T1, T2 and T3 for the groups of MEs allocated at the
same positions are after simplification:
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), U3(z)=u33(z).
The u-functions representing r.v. Y1, Y2 and Y3 are:
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~
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,
~
~
U3 (z) =(( U 2 (z) ),U3(z))=[p12(1)+p22(1)-p12(1)p22(1)]p30(3)z3+
[p12(1)+p22(1)-p12(1)p22(1)][p31(3)+p32(3)]z4.
~
Finally, the LMCCS reliability obtained from U3 (z) is:
~
( U3 (z) )=[p12(1)+p22(1)-p12(1)p22(1)][(p31(3)+p32(3)]z4,
R=[p12(1)+p22(1)-p12(1)p22(1)][p31(3)+p32(3)].
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.
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 [9-25].
It is recognized that GAs have the theoretical property of global convergence [26]. Despite
the fact that their convergence reliability and convergence velocity are contradictory, for most
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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 [27]. More detailed information on
GAs can be found in Goldberg’s comprehensive book [28], and recent developments in GA
theory and practice can be found in books [25, 26]. The steady state version of the GA used in
this paper was developed by Whitley [29]. As reported in [30] 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
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
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fitness function structure, which is especially important when constrained optimization problems
are considered [31].
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 avoids premature
convergence to a local optimum and facilitates jumps in the solution space. 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
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.
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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 reliability of
LMCCS with allocation, represented by the corresponding vector H. To estimate the LMCCS
reliability 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:
P1=2 4 1 4 2 3
P2=1 1 2 3 4 2
O=2 4 2 3 4 3
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The mutation operator moves a 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 MEs 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 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 (the solution fitness for the constrained
problem is evaluated as R + M - max | E n | ). Both solutions are presented in Table 2. One can
1 n  N
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. One can see 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 then solution B.
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 pmf of ME states depend on the ME
allocation. All the positions are divided into three groups, such that the positions belonging to the
same groups have the same influence on the MEs state pmf. The probabilistic distributions of
MEs states are presented in Table 3. As in the first example, solutions are obtained for the
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constrained allocation problem in which allocation of no more than one ME in each position is
allowed (solution A) and the allocation problem without constraints (solution B). The obtained
solutions are presented in Table 4. As in the first example, the free allocation when MEs occupy
just 11 positions provides greater reliability than does allocation in which the number of
occupied positions is equal to number of MEs.
5.2. Computational Effort and Algorithm Consistency
The C language realization of the algorithm was tested on a Pentium II PC. The chosen
parameters of GA were NS=100, Nrep=2000, Nc=50 and Pm=1. The time taken to obtain the bestin-population solution (time of the last modification of the best solution obtained) did not exceed
25 seconds for the first problem and 60 seconds for the second one.
To demonstrate the consistency of the suggested algorithm, we repeated the GA 25 times
with different starting solutions (initial population) for the second problem (M=16, N=20) with
and without allocation constraints. 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. 5. One can see
that the standard deviation of the final solution fitness does not exceed 0.8 % of its average value
for the unconstrained problem and 1.75% for the constrained one.
6. POSSIBLE EXTENSIONS OF THE ALGORITHM
LMCCS reliability can be enhanced not just by optimal allocation of a given set of elements,
but also by providing redundancy (including more elements to the system) and/or increasing
element reliabilities. Redundancy and element reliabilitiy enhancement, however, increase the
system cost. Thus, a budget-constrained reliability optimization problem arises. In order to solve
this problem one has to have
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- a function relating number and type of multistate elements to system cost (for redundancy
allocation problem);
- functions relatind the number of consecutive states of elements and probabilities of these states
to cost of the elements (for reliability allocation problem).
The most general formulation of LMCCS reliability optimization problem includes both
choise of the proper multistate elements and their optimal distribution among a LMCCS
positions. Note that the total number of the elements included to the system should also be
determined by the optimization procedure.
Consider a list of available multistate elements characterised by state pmf and cost. The
problem is to choose for each LMCCS position an arbitrary number of elements from this list in
order to achieve the greatest system reliability by the limited cost. Such problem was formulated
in [32] for series-parallel multistate system. Using the algorithm for LMCCS reliability
evaluation presented in this paper and the optimization technique suggested in [32] one can solve
the redundancy-reliability allocation problem for the LMCCS.
17
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20
Figure Captions
Figure 1: LMCCS.
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: Schematic representation of algorithm for LMCCS reliability evaluation.
Figure 5: CV of best-in-population solution fitness obtained by 25 different search processes as
function of number of crossovers.
21
Table 1. MEs' state pmf 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.
Position
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
C11
C12
C13
Reliability
Elements allocation
Solution A
Solution B
Solution C
13
5
1
4
12
8
7
2
9
3
11
6
10
0.58756
13
6
5
1
12
3
11
2
8
7
4
9
10
0.59201
2, 5, 6, 7, 10
8
13
1, 3, 11
4, 9, 12
0.72319
22
Table 3. MEs' state pmf 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
Reliability
Elements allocation
Solution A
Solution B
10
11
3
14
4
6
12
5
2
9
8
15
16
13
7
1
0.94222
4, 14
9, 11
1,7
12
8
3
10
5
13, 15
6, 16
2
0.96291
23
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