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SENSITIVITY ANALYSIS OF OPTIMAL REDUNDANCY SOLUTIONS
Igor Ushakov
San Diego, California, USA e-mail: [email protected]
Solving practical optimal redundancy problems, one can ponder: what is the sense of optimizing if input data are taken "from the ceiling"? Indeed, statistical data are so unreliable (especially in reliability problems) that such doubts have a very good ground.
Not found any sources after searching the answer for this question, the author decided to make some investigation of optimal solutions sensitivityunder influence of data scattering.
A simple series system of six units has been considered (see Figure 1). For reliability increasing, one uses a loaded redundancy, i.e. if a main unit k has xk redundant units, its reliability is found from
Pk (Xk) = 1 - (1 - pk) Xk+1
where pk is a probability of failure free operation (PFFO) of a single unit k. And the total cost of xk redundant units is equal to ck-xk, where ck is the cost of a single unit of type k.
12 3 4 5 6
Figure 1. Series system underwent analysis Units' parameters are presented in Table 1. Table 1. Input data
Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6
Pk 0.8 0.8 0.8 0.9 0.9 0.9
Ck 5 5 5 1 1 1
Assumed that units are mutually independent, i.e. system's reliability is defined as
PSystem (Xk,1* k<6) = nPk(Xk)
1 < k < 6
And the total system's redundant units cost is:
^ (x,,1< k<6) =
1< k < 6
Below presented solutions of both problems of optimal redundancy: direct:
min C ( **, i< k < 6) | p ( Xk, i< k < 6) > p *}
1 < Xk
and inverse:
max M**,i< k<6) I cX,i< k<6)< c *}
1 < xk < ro
For finding the optimal solutions, the Steepest Descent Method was applied. For this "base" system the solutions for several sets of parameters are presented for Direct Problem in Table 2 and for Inverse Problem in Table 3. (Numbers are given with high accuracy only for demonstration purposes; in practice, one has to use only significant positions after a row of nines.)
Table 2. Solution for Direct problem
P* x1 x2 x3 x4 x5 x6 Achieved P System C
0.95 3 3 3 3 2 2 0.9559520 52
0.99 4 4 3 3 3 3 0.991187 69
0.995 5 4 4 4 3 3 0.995229 75
0.999 6 5 5 4 4 4 0.999218 93
Table 3. Solution for Inverse problem
C* X3 x4 X5 X6 Achieved C System P
50 3 3 2 2 2 2 46 0.931676
75 4 4 3 3 3 3 75 0.995229
100 5 4 4 4 3 3 99.5 0.999602
The questions of interest are: how optimal solution will change if input data are changed? Two types of experiments have been performed: in the first series of experiments, different unit's costs with fixed probabilities were considered (see Figure 2) and in another one different unit's probabilities with fixed costs were considered (see Figure 3).
Figure 2. Input data for the first series of experiments
Figure 3. Input data for the second series of experiments The results of calculations are as follows:
Table 4. Values of Probabilities of Failure-free operations
0.999 0.995 0.99 0.95
Initial 0.999218 0.99566 0.9922 0.955952
Various C 0.998996 0.99566 0.9922 0.955952
Various P 0.999218 0.99566 0.9922 0.955952
In conclusion , there was performed a Monte Carlo simulation where parameter of the PFFO and cost were changed simultaneously. In this case, parameters of probabilities of each unit were calculated (in Excel) as:
pk=0.8pk+0.4pk*RAND()
and
ck= 0.8ck+0.4*RAND(), i.e. considered a random variation of the values within ±20% limits..
The final results for this case are presented in Tables 5 - 8. Table 5. Results of Monte Carlo simulations for P*=0.999
No. P* = 0.999
P C Xl X2 X3 x4 X5 X6
1 0.999352 100 6 6 6 4 4 4
2 0.999218 102 6 6 6 5 4 4
3 0.999313 102 6 6 6 4 4 4
4 0.999212 97 5 6 6 4 4 4
5 0.999182 102 6 6 6 4 4 4
6 0.999171 97 6 6 5 4 4 4
7 0.999171 103 6 6 6 4 5 4
8 0.999596 100 6 6 6 4 4 4
9 0.999526 100 6 6 6 4 4 4
10 0.999399 100 6 6 6 4 4 4
Table 6. Results of Monte Carlo simulations for P*=0.995
No. P* = 0.995
P C Xl X2 X3 x4 X5 X6
1 0.995478 84 5 5 5 3 3 3
2 0.996755 85 5 4 4 4 3 3
3 0.995026 85 5 4 5 4 3 3
4 0.996777 79 4 5 5 3 3 3
5 0.996777 84 5 5 5 3 3 3
6 0.995525 79 5 5 4 3 3 3
7 0.996732 85 5 5 5 3 4 3
8 0.996732 85 5 5 5 3 4 3
9 0.995645 84 5 5 5 3 3 3
10 0.99567 84 5 5 5 3 3 3
Table 7. Results of Monte Carlo simulations for P*=0.99
No. P* = 0.99
P C Xl X2 X3 x4 X5 X6
1 0.990147 69 4 4 4 3 3 3
2 0.990965 70 4 4 4 4 3 3
3 0.990229 70 4 4 4 4 3 3
4 0.99185 69 4 4 4 3 3 3
5 0.990389 71 4 4 4 4 4 3
6 0.99107 69 4 4 4 3 3 3
7 0.992185 74 5 4 4 3 3 3
8 0.990422 71 4 4 4 3 4 3
9 0.990893 71 5 4 4 3 3 3
10 0.990466 69 4 4 4 3 3 3
Table 7. Results of Monte Carlo simulations for P*=0.95
No. P* = 0.95
P C Xl X2 X3 x4 X5 X6
1 0.950045 52 3 3 3 3 2 2
2 0.955842 52 3 3 3 3 2 2
3 0.951936 52 3 3 3 3 2 2
4 0.951711 54 3 3 3 2 2 2
5 0.957883 50 3 3 3 3 3 2
6 0.951908 51 3 3 3 2 2 2
7 0.962227 51 3 3 3 2 2 2
8 0.962227 51 3 3 3 3 3 2
9 0.95261 50 3 3 3 3 2 3
10 0.950393 52 3 3 3 3 2 2
Analysis of data presented in Tables 5-8 shows relatively significant difference in numerical results (see Figure 4).
Figure 4. Deviation of maximum and minimum values of probability of failure-free operation in results of Monte Carlo simulation
However, the problem is not in coincidence of final values of PFFO or cost. The problem is: how change of parameters influences on the optimal values of x1, x2, ... .
However, one can observe that even with a system of six units (redundant groups) a visual analysis of sets (x^ x2, ..., x6) is extremely difficult and, at the same time, deductions based on some averages or deviations of various xk are almost useless.
The author was forced to invent some kind of a special presentation of sets of xk's. Since there is no official name for such kind of graphical presentation, it is called "multiple polygons" (in Russian "My.nbTH-3Be3gorpaMMbi»). On such multiple polygon there are numbers of "rays" corresponding to the number of redundant of units (groups). Each ray has several levels corecponded to the number of calculated redundant units for considered case (see Figure 5).
X. X
0
/0 \\
i 1 Ü I \ aJ J \\YMJ l ■ i 1 L
r n J ! j
* 1 O o x.
? Î1
m
\\\N-Ï/V /
V Jf y /
V^sTL^Î/ —
x „
i As
Figure 5. Multiple polygon axes with numbered levels
These multiple polygons give a perfect visualization of "close-to-optimal" solutions and characterize observed deviation of particular solutions. Such multiple polygons for considered example are given in Figure 6. (Here bold lines re used for connection xk obtained as optimal solution for units with parameters given in Table 1.)
P* = 0.999
P* = 0.999
P* = 0.999
P* = 0.999
Figure 6. Deviations of optimal solutions for randomly variation of parameters from the optimal solution obtained for parameters given in Table 1
Thus, one can notice that input parameters variation may influence enough significantly enough on probability of failure-free operation and the total system cost from run to run of Monte Carlo simulation though optimal solution remains more or less stable.
