Download the maximization of a serial system`s reliability

International Workshop
Advanced Researches in Computational Mechanics and
Virtual Engineering 
18 – 20 October 2006, Brasov, Romania
THE MAXIMIZATION OF A SERIAL SYSTEM’S RELIABILITY
USING THE LAGRANGE MULTIPLIERS METHOD
Eugenia SECARĂ
Transilvania University of Brasov, Brasov, ROMANIA, [email protected]
Abstract: The paper presents the modality in which the reliability of a serial system can be maximized by using the Lagrange
function.
Keywords: reliability, Lagrange function, reliability maximization
Taking into consideration the theoretical basis presented in the paper “Optimizing the systems
reliability using the Lagrange functions” the objective is the maximization of a serial system’s reliability:
Rs ( x) 
N
x
 1  (1  r j ) j 
(1)
j 1
complying with the following linear restriction:
g ( x) 
N
 cjxj  C
(2)
j 1
where xj ≥ 1, j = 1, 2, 3, 4 (round numbers).
The cj , rj constants range the following set of values:
j
cj
rj
1
1.2
0.8
Table 1: Constants values
2
3
2.3
3.4
0.7
0.75
4
4.5
0.85
Therefore C = 61.7
The Lagrange function can be defined as follows:
m
L( x,  )  f ( x)   i  gi ( x)
(3)
i 1
where λi, i = 1, 2, …, m represents the Lagrange multipliers and x j, represents the allocations of the
system reserves, j = 1, 2, …, N.
In the case of a single restriction problem it is sufficient to introduce just one multiplier for creating the
free restrictions Lagrange auxiliary function.
Knowing that max{Rs(x)} is equivalent to min{- Rs(x)}, the Lagrange function can be written as
follows:
N
 N

x
L( x,  )    1  q j j     c j x j  C 


  j 1
j 1 

(4)
International Workshop
Advanced Researches in Computational Mechanics and
Virtual Engineering 
18 – 20 October 2006, Brasov, Romania
The solution which minimizes the objective function, considering relation no. (4), represents the
solution for the one restriction maximization problem.
The stationary conditions are:
x
xj
L qk k ln qk N 

 1  q j    ck , k  1, 2,..., N

xk
1  qk j 1 
(5)
L N
  cjxj C  0
 j 1
(6)
Usually the allocated systems have a high reliability, so that it can be approximated:
N
xj
 1  q j 
j 1 

x
1  qk k
1
(7)
Using relations (5) and (7) it results:
xk 
 ck
1
ln
, k  1, 2,..., N
ln qk  ln qk
(8)
Using relations (5) and (8) it results:
N
C 
ln  
cj
ln
j 1 ln q j
N cj
cj
 ln q j
S
(9)

j 1 ln q j
Calculating relation (9) it results λ = es.
Introducing λ into relation (8) there are obtained the real values for the optimum:
xk 
ck 
1 
 S  ln

ln qk 
 ln qk 
(10)
Using relation (3) it can be determined the optimum allocation of the reservation by rounding the real
solution. Unfortunately this method is often inefficient.
It is recommendable that the determination of the optimum allocation using round numbers to be done
by an exhaustive search among the peaks of the unit side hypercube that contains the real numbers solution for
the problem.
The result of the testing around the real optimum solution is the generation of a non-dominant
allocations family [1] using a sequence decision process which guarantees the obtaining of an optimum round
numbers solution.
The optimum real solution is determined using relations (9) and (10):
x = 5.79; 6.96; 5.57; 4.26
Rs(x) = 0.999093
g(x) = 61.00 for λ = 2.20 × 10 -4.
International Workshop
Advanced Researches in Computational Mechanics and
Virtual Engineering 
18 – 20 October 2006, Brasov, Romania
REFERENCES
[1] Popenţiu Fl.: Optimizarea fiabilităţii sistemelor prin tehnica alocaţiilor, Teză de doctorat, I.P. Bucureşti
[2] Secară E: Teză de doctorat, Universitatea „Transilvania” din Brasov
[3] Tillman F. A. and others: Optimization of systems reliability, Marcel Dekker Inc., New York