Reliability optomization of complex systems Текст научной статьи по специальности «Компьютерные и информационные науки»

RELIABILITY OPTOMIZATION OF COMPLEX SYSTEMS

Krzysztof Kolowrocki, Joanna Soszynska-Budny.

Gdynia Maritime University, Gdynia, Poland katmatkk@am. gdynia.pl, [email protected]

ABSTRACT

The method based on the results of the joint model linking a semi-Markov modelling of the system operation process with a multistate approach to system reliability and the linear programming are proposed to the operation and reliability optimization of complex technical systems at the variable operation conditions. The method consists in determining the optimal values of limit transient probabilities at the system operation states that maximize the system lifetimes in the reliability state subsets. The proposed method is applied to the operation and reliability optimization of the exemplary technical multistate non-homogeneous system composed of a series-parallel and a series-"m out of F subsystems linked in series that is changing its reliability structure and its components reliability parameters at its variable operation conditions.

1 INTRODUCTION

The complex technical systems reliability improvement and decreasing the risk of exceeding a critical reliability state are of great value in the industrial practice (Kolowrocki, Soszynska-Budny, 2011; Kuo, Prasad, 2000; Kuo, Zuo 2003; Vercellis, 2009). In everyday practice, there are needed the tools that could be applied to improving the reliability characteristics of the multistate systems operating at variable conditions. There are needed the tools allowing for finding the distributions and the expected values of the optimal times until the exceeding by the system the reliability critical state, the optimal system risk function and the moment when the system risk function exceeds a permitted level and allowing for changing their operation processes after comparing the values of these characteristics with their values before their operation processes optimization in order to improve their reliability (Klabjan, Adelman, 2008; Kolowrocki, Soszynska-Budny, 2009, 2010, 2011; Lisnianski, Levitin 2003, Tang, Yin, Xi, 2007).

2 COMPLEX SYSTEM RELIABILITY AND OPERATION PROCESS OPTIMIZATION

Considering the equation (25) (Kolowrocki, Soszynska-Budny, 2013), it is natural to assume that the system operation process has a significant influence on the system reliability. This influence is also clearly expressed in the equation (26) (Kolowrocki, Soszynska-Budny, 2013) for the mean values of the system unconditional lifetimes in the reliability state subsets.

From the linear equation (26) (Kolowrocki, Soszynska-Budny, 2013), we can see that the mean value of the system unconditional lifetime M(u), u = 1,2,..., z, is determined by the limit values of

transient probabilities pb, b = 1,2,...,v, of the system operation process at the operation states given by (8) (Kolowrocki, Soszynska-Budny, 2013) and the mean values Mb(u), b = 1,2,...,v, u = 1,2,...,z, of the system conditional lifetimes in the reliability state subsets {u,u +1,...,z}, u = 1,2,..., z, given by (27) (Kolowrocki, Soszynska-Budny, 2013). Therefore, the system lifetime optimization approach based on the linear programming (Klabjan, Adelman, 2008; Kolowrocki, Soszynska-Budny, 2009, 2010, 2011).

can be proposed. Namely, we may look for the corresponding optimal values pb, b = 1,2,...,v, of the transient probabilities pb, b = 1,2,..., v, of the system operation process at the operation states to maximize the mean value M (u) of the unconditional system lifetimes in the reliability state subsets {u,u +1,..., z}, u = 1,2,..., z, under the assumption that the mean values Mb (u), b = 1,2,...,v, u = 1,2,..., z, of the system conditional lifetimes in the reliability state subsets are fixed. As a special and practically important case of the above formulated system lifetime optimization problem, if r, r = 1,2,..., z, is a system critical reliability state, we may look for the

optimal values pb, b = 1,2,...,v, of the transient probabilities pb, b = 1,2,...,v, of the system

operation process at the system operation states to maximize the mean value M(r) of the

unconditional system lifetime in the reliability state subset {r, r +1,,..., z}, r = 1,2,..., z, under the

assumption that the mean values Mb (r), b = 1,2,..., v, r = 1,2,..., z, of the system conditional

lifetimes in this reliability state subset are fixed. More exactly, we may formulate the optimization problem as a linear programming model with the objective function of the following form

M (r) = ± pbMb (r) (1)

b=1

for a fixed r e {1,2,..., z} and with the following bound constraints

p < Pb < pb, b = 1,2,...,v,

(2)

I Pb = 1, (3)

b=1

where

Mb (r), Mb (r) > 0, b = 1,2,..., v, (4)

are fixed mean values of the system conditional lifetimes in the reliability state subset {r, r +1,..., z} and

Pb, 0 < Pb < 1 and Pb, 0 < Pb < 1, pb < Pb, b = 1,2,...,v, (5)

are lower and upper bounds of the unknown transient probabilities Pb, b = 1,2,..., v, respectively. Now, we can obtain the optimal solution of the formulated by (1)-(5) the linear programming problem, i.e. we can find the optimal values Pb of the transient probabilities Pb, b = 1,2,...,v, that maximize the objective function given by (1).

First, we arrange the system conditional lifetime mean values Mb (r), b = 1,2,..., v, in non-increasing order

Mh (r) > MH (r) > . . . > Mbv (r), where b. e{1,2,...,v} for i = 1,2,..., v. Next, we substitute

xi = Pb, , = Pbi , = Pbi for i = 1,2,...,v (6)

and we maximize with respect to x., i = 1,2,..., v, the linear form (1) that after this transformation takes the form

M (r ) = £ xtMh (r )

i=i

(7)

for a fixed r g {1,2,..., z} with the following bound constraints

x. < x. < x., i = 1,2,..., v, (8)

Ï x. = 1,

i=1

(9)

where

Mbt (r), Mbt (r) > 0, i = 1,2,..., v,

are fixed mean values of the system conditional lifetimes in the reliability state subset {r, r +1,..., z} arranged in non-increasing order and

x., 0 < x. < 1 and x., 0 < x. < 1, x < x., i = 1,2,..., v, (10)

^ i i' i ' i i' 1111

are lower and upper bounds of the unknown probabilities x., i = 1,2,..., v, respectively. To find the optimal values of x., i = 1,2,..., v, we define

x = i xi, €=1 - x (11)

i=1

and

x0 = 0, xc0 = 0 and x1 =Y,xi, xl for I = 1,2,...,v. (12)

i=1 i i=1 i

Next, we find the largest value I g {0,1,...,v} such that

x1 - x1 < €

(13)

and we fix the optimal solution that maximize (7) in the following way:

i) if I = 0, the optimal solution is

x1 = € + x1 and x. = x. for i = 2,3,...,v; (14)

ii) if 0 < I < v, the optimal solution is

x = x for i = 1,2,..., I, x= € - x1 + x1 + x and xx = x

i i 5 5 5 5 I + 1 J- I + 1 i i

Krzysztof Kolowrocki & Joanna Soszynska-Budny - RELIABILITY OPTIMIZATION OF COMPLEX SYSTEMS , n, RT&AT# 04 (31)

(Vol.8) 2013, December

for i = I + 2, I + 3,...,v; (15)

iii) if I = v, the optimal solution is

x. = x. for i = 1,2,...,v. (16)

Finally, after making the inverse to (6) substitution, we get the optimal limit transient probabilities

Pb = x. for i = 1,2,...,v, (17)

that maximize the system mean lifetime in the reliability state subset {r, r +1,..., z}, defined by the linear form (1), giving its maximum value in the following form

M(r) = I PbMb (r) (18)

b=1

for a fixed r e {1,2,..., z}.

From the expression (18) for the maximum mean value M (r) of the system unconditional lifetime in the reliability state subset {r,r +1,...,z}, replacing in it the critical reliability state r by the reliability state u, u = 1,2,..., z, we obtain the corresponding optimal solutions for the mean values of the system unconditional lifetimes in the reliability state subsets {u, u +1,..., z} of the form

M(u) = I PbMb (u) for u = 1,2,..., z. (19)

b =1 b b

Further, according to (24)-(25) (Kolowrocki, Soszynska-Budny, 2013), the corresponding optimal unconditional multistate reliability function of the system is the vector

R(t,) = [1, R(t ,1),..., R (t, z)], (20)

with the coordinates given by

R(t,u) = TPb[R(t,u)](b) for t > 0, u = 1,2,...,z. (21)

b=1

And, by (29) (Kolowrocki, Soszynska-Budny, 2013), the optimal solutions for the mean values of the system unconditional lifetimes in the particular reliability states are

M(u) = M(u) -M(u +1), u =,1,...,z -1, M(z) = Ml(z). (22)

Moreover, considering (30) and (31) (Kolowrocki, Soszynska-Budny, 2013), the corresponding optimal system risk function and the optimal moment when the risk exceeds a permitted level 5, respectively are given by

r(t)= 1 - R(t, r), t > 0,

and

i = r l(5), (24)

where R(t, r) is given by (21) for u = r and r _1(t), if it exists, is the inverse function of the optimal risk function r(t).

Replacing in (8) (Kolowrocki, Soszynska-Budny, 2013) the limit transient probabilities pb of the system operation process at the operation states by their optimal values pb, maximizing the mean value M(r) of the system lifetime in the reliability states subset {r,r +1,...,z} defined by (1) and the mean values mb of the unconditional sojourn times at the operation states by their corresponding unknown optimal values m b, we get the system of equations

7r m

pb = , b = 1,2,...,v. (25)

Y,nlni I

l=1

After simple transformations the above system takes the form

(p 1 - 1Km 1 + p xn2 m 2 +... + p lnv?h v = o p 2nx m 1 + (p 2 -1)^2 m 2 +... + p 2nv m v = o

(26)

p^m 1 + pn2m2 +... + (pv - 1)nvmv = o,

where mb are unknown variables we want to find, pb are optimal transient probabilities determined by (17) and nb are steady probabilities determined by (9) (Kolowrocki, Soszynska-Budny, 2013).

Since the system of equations (26) is homogeneous and it can be proved that the determinant of its main matrix is equal to zero, then it has nonzero solutions and moreover, these solutions are ambiguous. Thus, if we fix some of the optimal values mb of the mean values mb of the unconditional sojourn times at the operation states, for instance by arbitrary fixing one or a few of them, we may find the values of the remaining once and this way to get the solution of this equation.

Having this solution, it is also possible to look for the optimal values mbl of the mean values mbl of the conditional sojourn times at the operation states using the following system of equations

tphl™u = mh, b = 1,2,...,v, (27)

i=1

obtained from (7) (Kolowrocki, Soszynska-Budny, 2013) by replacing mb by mb and mbl by mbl, were pbl are known probabilities of the system operation process transitions between the operation states zb i z, b,l = 1,2,...,v, b ^ l, defined by (2) (Kolowrocki, Soszynska-Budny, 2013). Another very useful and much easier to be applied in practice tool that can help in planning the operation processes of the complex technical systems are the system operation process optimal

mean values of the total system operation process sojourn times &b at the particular operation states zb, b = l,2,...,v, during the fixed system operation time в, that can be obtain by the replacing in the formula (10) (Kolowrocki, Soszynska-Budny, 2013) the transient probabilities pb at the operation states zb by their optimal values pb and resulting in the following expession

m = Ц$ь] = Рв, b = 1,2,...,v. (28)

The knowledge of the optimal values mb of the mean values of the unconditional sojourn times and the optimal values mы of the mean values of the conditional sojourn times at the operation states and the optimal mean values Mb of the total sojourn times at the particular operation states during the fixed system operation time may by the basis for changing the complex technical systems operation processes in order to ensure these systems operation more reliable.

3 APPLICATION

We consider a series system S composed of the subsystems S1 and S2, with the scheme showed in Figures 1-3 (Kolowrocki, Soszynska-Budny, 2013). This system reliability structure and its components reliability parameters depend on its changing in time operation states with arbitrarily fixed the number of the system operation process states v = 4 and their influence on the system reliability indicated in Sections 2-3 (Kolowrocki, Soszynska-Budny, 2013) where its main reliability characteristics are predicted.

To find the optimal values of those system reliability characteristics, we conclude that the objective function defined by (1), in this case, as the exemplary system critical state is r = 2, according to (89) (Kolowrocki, Soszynska-Budny, 2013), takes the form

M(2) = p1 ■ 25.00 + p2-14.88 + p3 • 13.04 + p4 • 7.04. (29)

Arbitrarily assumed, the lower pb and upper pb bounds of the unknown optimal values of transient probabilities pb, b = 1,2,3,4, respectively are:

p1 = 0.201, p2 = 0.03, p3 = 0.245 . p4 = 0.309; p1 = 0.351, p 2 = 0.105, p3 = 0.395, p4 = 0.459. Therefore, according to (2)-(3), we assume the following bound constraints 0.201 < p1 < 0.351, 0.030 < p2 < 0.105,

0.245 < p3 < 0.395, 0.309 < p4 < 0.459. (30)

i pb = 1, (31)

b =1

Now, before we find optimal values pb of the transient probabilities pb, b = 1,2,3,4, that maximize the objective function (29), w arrange the system conditional lifetime mean values Mb (2), b = 1,2,3,4, in non-increasing order

M 1(2) > M2 (2) > M3(2) > M4(2).

Further, according to (6), we substitute

= p^ = p 2 ' X3 = Pз, X 4 = P 4 ' (32)

and

Xl = p J = 0.201, x2 = p 2 = 0.030, x3 = p3 = 0.245, x4 = p 4 = 0.309; (33)

xl = p j = 0.351, x2 = p 2 = 0.105, x3 = p3 = 0.395, x4 = p 4 = 0.459, (34)

and we maximize with respect to x., i = 1,2,3,4, the linear form (29) that according to (7)-(9) takes the form

M(2) = x ■ 25.00 + x2-14.88 + x3 • 13.04 + x4 • 7.04, (35)

with the following bound constraints

0.201 < x1 < 0.351, 0.030 < x2 < 0.105,

0.245 < x3 < 0.395, 0.309 < x4 < 0.459. (36)

i x. = 1. (37)

i=1

According to (11), we calculate

x = £ x = 0.785, € = 1 - x = 1 - 0.785 = 0.215 (38)

i=1 '

and according to (12), we determine

x0 = 0, x0 = 0, x0 -3c0 = 0, x1 = 0.201, x1 = 0.351, x1 - x1 = 0.150, x2 = 0.231, x2 = 0.456, x2 - x2 = 0.225, x3 = 0.476. x3 = 0.851, x3 -x3 = 0.375,

x4 = 0.785 x4 = 1.31 x4 - x4 = 0.525. (39)

From the above, as according to (38), the inequality (13) takes the form

x7 - x7 < 0.215, (40)

it follows that the largest value I e {0,1,2,3,4} such that this inequality holds is I = 1. Therefore, we fix the optimal solution that maximize linear function (35) according to the rule (15). Namely, we get

x1 = x1 = 0.351,

x 2 = € - x1 + x1 + x 2 = 0.215 - 0.351 + 0.201 + 0.030 = 0.095,

x3 = x3 = 0.245, x4 = x4 = 0.309. (41)

Finally, after making the inverse to (32) substitution, we get the optimal transient probabilities

P1 = x1 = 0.351, P 2 = x2 = 0.095, P3 = x3 = 0.245, P 4 = x4 = 0.309, (42)

that maximize the exemplary system mean lifetime M(2) in the reliability state subset {2,3} expressed by the linear form (29) giving, according to (18) and (42), its optimal value

M(2) = Pl - 25.00 + P2 -14.88 + P3 • 13.04 + P4 • 7.04

= 0.351- 25.00 + 0.095 -14.88 + 0.245 -13.04 + 0.309 ■ 7.07 = 15.56. (43)

Substituting the optimal solution (42) into the formula (19), we obtain the optimal solution for the mean values of the exemplary system unconditional lifetimes in the reliability state subsets {1,2,3} and {3},that are as follows

M(1) = Pl ■ 27.78 + P2 -16.27 + P3 • 14.82 + P4 ■ 7.72

= 0.351-27.78 + 0.095-16.27 + 0.245-14.82 + 0.309 - 7.72 = 17.31, (44)

M(3) = Pl - 22.73 + P2 -13.71 + P3 -11.48 + P4 - 6.47

= 0.351-22.73 + 0.095-13.71 + 0.245-11.48 + 0.309 - 6.47 = 14.09 (45)

and according to (22), the optimal values of the mean values of the system unconditional lifetimes in the particular reliability states 1, 2 and 3, respectively are

M (1) = Mi(1) - Mi(2) = 1.75, M (2) = Mi(2) - Mi(3) = 1.47,

M (3) = Mi (3) = 14.09. (46)

Moreover, according to (20)-(21), the corresponding optimal unconditional multistate reliability function of the system is of the form

R(t, -) = [1, R(t, 1), R(t, 2), R(t, 3) ], t > 0, (47)

with the coordinates given by

R(t,1) = 0.351 • [ R(t,1)](1) + 0.095 • [ R(t ,1)](2) + 0.245 • [ R(t ,1)](3)

+ 0.309 • [R(t,1)](4) for t > 0, (48)

R (t, 2) = 0.351 • [ R(t ,2)](1) + 0.095 • [ R(t,2)](2) + 0.245 • [ R(t,2)](3)

+ 0.309 • [R(t,2)](4) for t > 0, (49)

R(t, 3) = 0.351 • [R(t,3)](1) + 0.0095 • [R(t,3)](2) + 0.245-[R(t,3)](3)

+ 0.309• [R(t,3)](4) for t > 0, (50)

where [R(t,1)](b), [R(t,2)](b), [R(t,3)](b), b = 1,2,3,4, are fixed in Section 3 (Kolowrocki, Soszynska-Budny, 2013).

The graph of the exemplary system optimal reliability function R(t, •) given by (47)-(50) is presented in Figure 1.

Fig. 1. The graph of the exemplary system optimal reliability function R (t, •) coordinates

As the critical reliability state is r =2, then the exemplary system optimal system risk function, according to (23), is given by

r(t) = 1 - R(t, 2) for t > 0, (51)

where R (t ,2) is given by (49).

Hence and considering (24), the moment when the optimal system risk function exceeds a permitted level, for instance 5 = 0.025, is

i = r-1 (5) s 2.55. (52)

Fig. 2. The graph of the exemplary system optimal risk function r(t)

Substituting the exemplary operation process optimal transient probabilities at operation states

P1 = 0.351, P2 = 0.095, P3 = 0.245, P4 = 0.309,

determined by (42) and the steady probabilities nl = 0.236, n2 = 0.169, tT3 = 0.234, tT4 = 0.361,

determined by (17) in Section 2 (Kolowrocki, Soszynska-Budny, 2013) into (26), we get the following system of equations with the unknown optimal mean values mb of the exemplary system operation process unconditional sojourn times at the operation states we are looking for

- 0.153164mm t + 0.059319m2 + 0.082134 m3 + 0.126711m4 = 0

0.02242ml -0.152945m2 + 0.02223m3 + 0.034295m4 = 0

0.05782mmj + 0.041405m2 - 0.17667m3 + 0.088445m4 = 0

0.072924mwj + 0.052221m2 + 0.072306m3 - 0.249451m4 = 0. (53)

The determinant of the main matrix of the above homogeneous system of equations is equal to zero and therefore there are non-zero solutions of this system of equations that are ambiguous and dependent on one or more parameters. Thus, we may fix some of them and determine the remaining ones. To show the way of solving this system of equations, we may suppose that we are arbitrarily interested in fixing the value of m4 and we put

m4 = 400.

Substituting the above value into the system of equations (53), we get

- 0.153164«, + 0.059319m2 + 0.082134m3 = -50.6844

0.02242m1 - 0.152945m2 + 0.02223m3 = -13.7180

0.05782m«, + 0.041405m2 - 0.17667m3 = -35.3780

0.072924« j + 0.052221«2 + 0.072306 m 3 = 99.7804

and we solve it with respect to m 1, m2 and m3, after omitting its last equation. This way obtained solutions that satisfy (53), are

« = 689, m 2 = 261, m3 = 487, m4 = 400.

(54)

It can be seen that these solution differ much from the values ml, m2, m3 and m4 estimated in Section 2 (Kolowrocki, Soszynska-Budny, 2013) and given by (13)-(16) (Kolowrocki, Soszynska-Budny, 2013).

Having these solutions, it is also possible to look for the optimal values mbl of the mean values mbl of the exemplary system operation process conditional sojourn times at operation states. Namely, substituting the values mb instead of mb, the probabilities

[Pi ] =

0 0.22 0.32 0.46

0.20 0 0.30 0.50

0.12 0.16 0 0.72

0.48 0.22 0.30 0

of the exemplary system operation process transitions between the operation states given by (11) in Section 2 (Kolowrocki, Soszynska-Budny, 2013) and replacing mbl by mbl in (27), we get the following system of equations

0.22m 12 + 0.32m 13 + 0.46m 14 = 689

0.20m 21 + 0.30m23 + 0.50th24 = 261

0.12m 31 + 0.16m32 + 0.72m34 = 487

0.48m41 + 0.22 42 + 0.30m 14 = 400

(55)

with the unknown optimal values mbl we want to find.

As the solutions of the above system of equations are ambiguous, then we fix some of them, say that because of practically important reasons, and we find the remaining ones. For instance:

- we fix in the first equation m12 = 200, m13 = 500 and we find m14 = 1054;

- we fix in the second equation m 21 = 100, m 23 = 100 and we find m24 = 422;

- we fix in the third equation m31 = 900, m32 = 500 and we find m 34 = 415;

- we fix in the fourth equation m41 = 300, m42 = 500 and we find m43 = 487. (56)

It can be seen that these solutions differ much from the mean values of the exemplary system conditional sojourn times at the particular operation states before its operation process optimization given by (12) (Kolowrocki, Soszynska-Budny, 2013).

Another very useful and much easier to be applied in practice tool that can help in planning the operation process of the exemplary system are the system operation process optimal mean values of the total sojourn times at the particular operation states during the system operation time that by the same assumpion as in Section 2 (Kolowrocki, Soszynska-Budny, 2013) is equel to 9 = 1year = 365 days. Under this assumption, after aplying (28), we get the optimal values of the exemplary system operation process total sojourn times at the particular operation states during 1 year

m = E = P19 = 0.341 • 365 = 124.5, m€2 = E[#2] = P2 9 = 0.105 • 365 = 38.3, m3 = E[#3 ] = P3 9 = 0.245 • 365 = 89.4,

m4 = E [ #4] = P 49 = 0.309 • 365 = 112.8, (57)

that differ much from the values of mfij, mS2, mfi3, mS4, determined by (19) in Section 2 (Kolowrocki, Soszynska-Budny, 2013).

In practice, the knowledge of the optimal values of mb mbl and j£b given respectively by (54),

(56), (57), can be very important and helpful for the system operation process planning and improving in order to make the system operation more reliable.

4 CONCLUSION

Presented in this paper tool is useful in reliability and operation optimization of a very wide class of real technical systems operating at the varying conditions that have an influence on changing their reliability structures and their components reliability parameters. The results can be interesting for reliability practitioners from various industrial sectors.

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