Reliability, risk and availability based optimization of complex technical systems operation processes. Part 2. Application in port transportation Текст научной статьи по специальности «Математика»

RELIABILITY, RISK AND AVAILABILITY BASED OPTIMIZATION OF COMPLEX TECHNICAL SYSTEMS OPERATION PROCESSES

PART 2

APPLICATION IN PORT TRANSPORTATION

K. Kolowrocki, J. Soszynska. •

Gdynia Maritime University, Gdynia, Poland e-mail: [email protected], [email protected]

ABSTRACT

The joint general model of reliability and availability of complex technical systems in variable operation conditions linking a semi-markov modeling of the system operation processes with a multi-state approach to system reliability and availability analysis and linear programming considered in the paper Part 1 are applied in maritime industry to reliability, risk and availability optimization of a port piping oil transportation system.

6 RELIABILITY, RISK AND AVAILABILITY EVALUATION OF A PORT OIL PIPING TRANSPORTATION SYSTEM

The oil terminal in D^bogorze is designated for the reception from ships, the storage and sending by carriages or cars the oil products. It is also designated for receiving from carriages or cars, the storage and loading the tankers with oil products such like petrol and oil. The considered system is composed of three terminal parts A, B and C, linked by the piping transportation systems. The scheme of this system is presented in Figure 1 (Kolowrocki et all. 2008).

The unloading of tankers is performed at the piers placed in the Port of Gdynia. The piers is connected with terminal part A through the transportation subsystem S\ built of two piping lines composed of steel pipe segments with diameter of 600 mm. In the part A there is a supporting station fortifying tankers

pumps and making possible further transport of oil by the subsystem S2 to the terminal part B. The subsystem S2 is built of two piping lines composed of steel pipe segments of the diameter 600 mm. The terminal part B is connected with the terminal part C by the subsystem S3. The subsystem Si is built of one piping line composed of steel pipe segments of the diameter 500 mm and two piping lines composed of steel pipe segments of diameter 350 mm. The terminal part C is designated for the loading the rail cisterns with oil products and for the wagon sending to the railway station of the Port of Gdynia and further to the interior of the country. The oil pipeline system consists three subsystems:

- the subsystem S^ composed of two identical pipelines, each composed of 178 pipe segments of length 12m and two valves,

- the subsystem S2 composed of two identical pipelines, each composed of 717 pipe segments of length 12m and to valves,

- the subsystem S3 composed of three different pipelines, each composed of 360 pipe segments of either 10 m or 7,5 m length and two valves.

The subsystems S^, S2, S, are forming a general port oil pipeline system reliability series structure. However, the pipeline system reliability structure and the subsystems reliability depend on its changing in time operation states (Kolowrocki et all. 2008).

Taking into account the varying in time operation process of the considered system we distinguish the following as its eight operation states:

• an operation state zx - transport of one kind of medium from the terminal part B to part C using two out of three pipelines in subsystem S3,

• an operation state z2 - transport of one kind of medium from the terminal part C (from carriages) to part B using one out of three pipelines in subsystem S3,

• an operation state z3 - transport of one kind of medium from the terminal part B through part A to piers using one out of two pipelines in subsystem S2 and one out of two pipelines in subsystem Si,

• an operation state z4 - transport of two kinds of medium from the piers through parts A and B to part C using one out of two pipelines in subsystem Si, one out of two pipelines in subsystem S2 and two out of three pipelines in subsystem £3,

• an operation state z5 - transport of one kind of medium from the piers through part A to B using one out of two pipelines in subsystem Si and one out of two pipelines in subsystem S2,

• an operation state zn - transport of one kind of medium from the terminal part B to C using two out of three pipelines in subsystem S3, and simultaneously transport one kind of medium from the piers through part A to B using one out of two pipelines in parts Si and one out of two pipelines in subsystem S2,

• an operation state z, - lack of medium transport (system is not working)

• an operation state z8 - transport of one kind of medium from the terminal part B to C using one out of three pipelines in part £3, and simultaneously transport second kind of medium from the terminal part C to B using one out of three pipelines in part £3.

At the moment because of the luck of sufficient statistical data about the oil terminal operation process it is not possible to estimate its all operational characteristics. However, on the basis the still limited data, given in (Kolowrocki et all. 2008), the transient probabilities pbl from the operation state zb into the operations state z, for ¿,/ = 1,2,...,8, b^l, were preliminary evaluated. Their approximate values are included in the matrix below

0 0 0 0 0.06 0.06 0.86 0.02

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0.125 0 0 0 0 0.125 0.687 0.063

0.4 0 0 0 0.6 0 0 0

0.82 0 0 0 0.16 0 0 0.02

0.67 0 0 0 0 0 0.33 0

Unfortunately, it was not possible yet to determine the matrix of the conditional distribution functions [Hu(t)]g18 of the sojourn times 9bl for ¿,/ = 1,2,...,8, b^l, [1], [6] and further consequently, according to (2.1), it was also not possible to determine the vector [//6(/)]lt8 of the unconditional distribution functions of the sojourn times 9b of this operation process at the operation states zb, b = 1,2,...,8. However, on the basis of the preliminary statistical data coming from experiment it was

possible to evaluate approximately the conditional mean values Mb, =E[0bl], b,J = 1,2,...,8, b^I, of sojourn times in the particular operation states defined by (3). On the basis of the statistical data given in Tables 1-10 in (Kolowrocki et all. 2008) (Appendix 1A) their approximate evolutions are as follows:

Ml5 = 720, M16 =420, M17 = 698.95, M18 = 480, M5l =750, M56 = 564, M57 = 748.7, M58 = 540, M61 = 360, M6} =360, Mn = 975.3, M75 = 872.4,

M78 = 600, M81 = 900, M87 = 420. (43)

Hence, by (2), the unconditional mean lifetimes in the operation states are M, = E[0J = pl5Ml5 + pl6Ml6 + puMxl + plsMls = 0.06- 720 + 0.06- 420+0.86-698.95 +0.02- 480 = 679.1, M5 = E[05 ] = p5lM5l + p56M56 + p51M51 + p5SM5S = 0.125- 750 + 0.125- 564 + 0.687- 748.7 + 0.063- 540 = 712.63, M6 = E[06 ] = p6lM6l + p65M65 =0.4-360 + 0.6-360 = 360, M1 = E[07 ] = pnMn + plsMls + pnMK =0.82-975.3 + 0.16-872.4 + 0.02-600 = 951.33,

M8 =E[0S]= pslMsl + p„M„ =0.67-900+0.3 3-420 = 741.6. (44)

Since from the system of equations (5) given here in the form

[■, , ' n4 > ' n6 ' n7 ' ] 7Zl+7Z2+7Z3+7ZA + 7t5 + 7Z6 + 7t1 + 7Tg = 1,

we get

nx =0.396, n2 =0, ^=0, =0, =0.116, =0.038, n1 =0.435, =0.015, (45)

then the limit values of the transient probabilities pb (?) at the operational states zb, according to (4), are given by

Pi = 0.34, p2 = 0, p3 =0, p4 = 0, ps = 0.1, p6 = 0.02, Pl = 0.53, ps = 0.01. (46)

From the above result, according to (34)-(35), the unconditional multistate (three-state) reliability function of the system is of the form

R3(t,-) =[1, R3(t,l), R3(t,2)1 (47)

with the coordinates given by

R3(t,l) = 0.34-[i?(f,l)](1) + 0-[i?(i,l)]'2) + 0-[i?(i,l)]'3) + 0-[i?(i,l)]'4)

+ 0.01-[i?(i,l)]'5) + 0.02 [R(t, 1)](6) + 0.53 [R(t, 1)](7) + 0.01 [«(/,1)](8)] fori>0, (48)

R3(t, 2) = 0.34-[i?(f,2)]n) + 0{J?(r,2)f,+0{J?(r,2)],3) + 0-[J?(r,2)]'4)

+ 0.1-[i?(i,2)]<5) + 0.02 -[i?(i,2)],6) +0.53 {R(t,2)]m + 0.01-[i?(i,2)f;1 ] for t > 0, (49)

where [tf(i,l)]'A), [R(t,2)f\ b = 1,2,...,8, are fixed in (Kolowrocki et all. 2008).

In [7] (Appendix IB), it is also fixed that the mean values of the system unconditional lifetimes in

the particular reliability state subsets {1,2} and {2} are:

(1) = 0.364, /^(2) = 0.304,

//2(1) = 0.807, ft, (2) = 0.666,

//3(1) = 0.307, //3(2) = 0.218,

//4(1) = 0.079, jii4 (2) = 0.058,

//5(1) = 0.307, ^ (2) = 0.218,

/u6{\) = 0.079, ^ (2) = 0.05$

(1) = 0.110, jU7 (2) = 0.085,

//s(l) = 0.364, //,(2) = 0.079. (50)

After considering (46)-(50) and applying (11), the mean values of the system unconditional lifetimes in the reliability state subsets {1,2} and {2}, before the optimization, respectively are:

Ml) = Pi Mx (1) + Pi M2 (1) + P3M3 (1) + Pa M, (1) + A A (1) + P6 (1) + A /'7 (1) + Ps Ms (1) = 0.34-0.364+ 0.00-0.807 + 0.00-0.307 +0.000.079 +0.100.307 +0.02 0.079

+ 0.53 0.110 + 0.01 0.364 =0.218,

M(2) = px ft (2) + p2 M2 (2) + pm (2) + p4 <u4 (2) + ps ju5 (2) + p6 ju6 (2) + p7 ju7 (2) + ps jus (2) = 0.34- 0.304 + 0.00- 0.666 + 0.00- 0.218 +0.00 0.058 + 0.10-0.218 + 0.02- 0.058

+ 0.53- 0.083 +0.01-0.304 =0.173, (52)

and according to (14), the mean values of the system lifetimes in the particular reliability states u = 1 and u = 2 , before the optimization, respectively are

/7(1) = ju( 1) - ju(2) = 0.045, /7(2) = ¿u(2) = 0.173. (53)

Further, according to (13), the variances and standard deviations of the system unconditional lifetimes in the system reliability state subsets are

<t2(1) = 2 ft R,(t,l)dt -[ju(l)]2 = 0.0520, <x(l) = 0.22% (54)

0

<t2(2) = 2\t R3(t,2)dt - [//(2)]2 = 0.0342, <r(2) = 0.185, (55)

0

where R3(t,l), R3(t,2) are given by (48)-(49) and /u{\), ¿/(2) are given by (51)-(52).

If the critical safety state is r =1, then the system risk function, according to (7), is given by

r(t)= 1-^ (?,1) fori>0, (56)

where Ri (?,1) is given by (48).

Hence, the moment when the system risk function exceeds a permitted level, for instance 8 = 0.05, from (8), is

T=r\S) =0.011 years. (57)

Further, assuming that the oil pipeline system is repaired after its failure and that the time of the system renovation is ignored, applying Theorem 3.1, we obtain the following results:

i) the distribution of the time SN( 1) until the Nth exceeding of reliability critical state 1 of this system, for sufficiently large TV, has approximately normal distribution N(0.21 8A''.0.228a//V ), i.e.,

FW,(M)= P(SN(i)<t) te (-0,00),

ii) the expected value and the variance of the time SN( 1) until the Nth exceeding the reliability critical state 1 of this system take respectively forms

E[SN( 1)] = 0.2187V, D[SN( 1)] = 0.05197V ,

iii) the distribution of the number N(t, 1) of exceeding the reliability critical state 1 of this system up to the moment t,t> 0, for sufficiently large i, is approximately of the form

r0.218iV -t ^ p (0.218(7V + 1)-^ 0.4883V7 Nm) 0.4884V7

P{N{t,\) = N) =FN(0d)(--r)-F„m)( ' . ; ), A' 0.1.2....

iv) the expected value and the variance of the number N(t, 1) of exceeding the reliability critical state 1 of this system at the moment t,t> 0, for sufficiently large i, approximately take respectively forms

H(t,1) = 4.5871, D(t,1) = 5.0095f.

Further, assuming that the oil pipeline system is repaired after its failure and that the time of the system renovation is not ignored and it has the mean value ¿/„(1) = 0.005 and the standard deviation ct0 (1) = 0.005, applying Theorem 3.2, we obtain the following results:

i) the distribution function of the time 5^(1) until the Nth system's renovation, for sufficiently large TV, has approximately normal distribution N(0.223N,0.2279V7V), i.e.,

= (N) = t- 0 2237V)

F (t,l) = P(SN(l)<t) = Fm0l)(——jJ-), te(-~>,°o),N = 1,2,...,

ii) the expected value and the variance of the time SN(1) until the Nth system's renovation take respectively forms

E[S„ (1)] = 0.223N , D[SN (1)] = 0.05197V ,

iii) the distribution function of the time SN(l) until the Nth exceeding the reliability critical state 1 of this system takes form

W) - t- 0.2237V + 0.005

V0.05197V -0.000025'

F (t,l) = P(SN(l)<t) = FNm) (—¡==========), te N = 1,2,...,

iv) the expected value and the variance of the time SN(l) until the Nth exceeding the reliability critical state 1 of this system take respectively forms

¿IS*(1)] = 0.2187V + 0.005(7V -1), D[SN(1)] = 0.05197V + 0.000025(7V -1),

v) the distribution of the number N(t,V) of system's renovations up to the moment t,t> 0, is of the

f 0.2237V -t 0.223(7V +1) - t ^

0.482V7 ) Nm)( 0.482V7

PW,D = N) =FNm)(--—) -FNm)( ■ r' ) N = 1,2,...

vi) the expected value and the variance of the number 7V(?,1) of system's renovations up to the moment t,t> 0, take respectively forms

H(t,1) = 4.484i, D(t, 1) = 4.68i,

vii) the distribution of the number N(t, 1) of exceeding the reliability critical state 1 of this system up to the moment ?, ? > 0, is of the form

/>(»(,,) = ») ^mil,a223Wr'-0005)-F,,,,/223^1'""000^ N - 1,2.....

0.482V?+ 0.005 0.482V?-0.005

viii) the expected value and the variance of the number N(t, 1) of exceeding the reliability critical state 1 of this system up to the moment ?, ? > 0, are respectively given by

H(t,\) = t+o°2™5 , D(?,l) = 4.68(? +0.005),

ix) the availability coefficient of the system at the moment t is given by the formula

A{t,l) = 0.9776, ?>0,

x) the availability coefficient of the system in the time interval <?,? + r), r>0, is given by the formula

A(t, r,l) = 4.484J^3 (t,l)dt, ?> 0, r > 0.

X

7 RELIABILITY, RISK AND AVAILABILITY OPTIMIZATION OF A PORT OIL PIPING TRANSPORTATION SYSTEM

The objective function (15), in this case as the critical state is r = 1, takes the form

¿/(1) = Pi -0.364 + p2 0.807 + p3 0.307 + p4- 0.079 + p5 0.307

+ p6 ■ 0.079 + p7 ■ 0.110 + ps ■ 0.364. (58)

The lower pb and upper pb bounds of the unknown limit transient probabilities ph, b = 1,2,...,8, coming from experts are respectively:

p, =0.25, p2 =0.01, p3 =0.01. p4 =0.01, ps =0.08, p6 =0.01, p7 =0.40, p1 =0.01; p1 =0.50, p2 =0.05, p3 =0.05, p4 =0.05,ps =0.20, p6 =0.05, p1 =0.75, p1 =0.05. Therefore, according to (16)-(18), we assume the following bound constraints

ipb=l (59)

4=1

0.25 < pl < 0.50, 0.01 < p2 < 0.05, 0.01 < p3 < 0.05, 0.01 < pA < 0.05, 0.08 < p5 < 0.20, 0.01 < p6 < 0.05, 0.40 < p7 < 0.75, 0.01 <ps< 0.05. (60)

Now, before we find optimal values pb of the limit transient probabilities pb, b =1,2,..., v, that maximize the objective function (58), we arrange the system conditional lifetime mean values ¿/¿(1), b = 1,2,..., 8, in non-increasing order

M2 (1) > Ml (1) > Ms (1) > (1) > Ms (1) > M7 (1) > M4 (1) > M6 (1).

Next, according to (19), we substitute

xl = pn = 0.00, xn = pl = 0.34, x, = ps = 0.01, x4 = p3 = 0.00, x5 = pi = 0.10,

x6 = p7 = 0.53, x7 = p4 = 0.00, x8 = p6 = 0.02 (61)

x. = 0.01, x, = 0.95 for i = 1,2,..., v (62)

and we maximize with respect to xf, i = 1,2,...,8, the linear form (52) that according to (20) takes the form

¿/(1) =x1 -0.807 +x2 -0.364 +x, 0.364 +x4-0.307 + x5 -0.307 +x6-0.110

+ x7-0.079 +x8-0.079 (63)

with the following bound constraints

¿x,=l, (64)

/=i

0.01 < x1 < 0.05, 0.25 < x2 < 0.50, 0.01 < x, < 0.05, 0.01 < x4 < 0.05, 0.08 < x5 < 0.20, 0.40 < x6 < 0.75, 0.01 < x7 < 0.05, 0.01 < x8 < 0.05. (65)

where

Xj = 0.01, x2 = 0.25, x3 = 0.01, x4 = 0.01, jc5 =0.08, x6 = 0.40 , x7 =0.01, x8 =0.01;

Xj = 0.05, x2 = 0.50, x3 = 0.05, x4 =0.05, x5 =0.20, x6 =0.75, x7 =0.05, xs=0.05. (66)

are lower and upper bounds of the unknown limit transient probabilities xt, i = 1,2,..., 8, respectively.

According to (24), we find

x = ix, = 0.78, y = 1 - x = 1 - 0.78 = 0.22 (67)

/=i

and according to (25), we find

-0 n -0 -0 A

x =0, x =0, x -x =0,

X1 =0.01 x1 =0.05, x'-x1 =0.04 x2 = 0.26 x2 = 0.55, x2 - x2 = 0.29,

x8 = 0.78 x8 = 1.70, x8 -x8 = 0.92. (68)

From the above, as according to (67), the inequality (26) takes the form

x1 - x1 < 0.22, (69)

then it follows that the largest value I e {0,1,...,8} such that this inequality holds is I = 1. Therefore, we fix the optimal solution that maximize (63) according to the rule (28). Namely, we get

Xj = Xj =0.05,

x2 = y-x1 + x1 + x2 =0.22-0.05 + 0.01 + 0.25 = 0.43, (70)

x3 = x3 = 0.01, x4 = x4 = 0.01, xs = xs = 0.08, x6 = x6 = 0.40 , x7 = x7 =0.01, xs = xs =0.01.

(71)

Finally, after making the inverse to (61) substitution, we get the optimal limit transient probabilities

pn = xl = 0.05, pl = xn = 0.43, ps = x, = 0.01, = x4 = 0.01, p5 = x5 = 0.08,

p7=x.6 = 0.40, pt = x7 = 0.01, p0=xs= 0.01 (72)

that maximize the system mean lifetime in the reliability state subset {1,2} expressed by the linear form (58) giving, according to (31) and (72), its optimal value

¿(1) = px ■ 0.364 + p2 ■ 0.807 + p3 ■ 0.307 + pA-0.079 + p5 0.307 + p6 ■ 0.079

+ /yO.llO + /y 0.364 =0.43-0.364 +0.05-0.807 +0.01-0.307

+ 0.01-0.079 +0.08-0.307 +0.01-0.079+0.400.110 +0.01-0.364=0.274. (73)

Further, according to (32), substituting the optimal solution (72) in (52), we obtain the optimal solutions for the mean values of the system unconditional lifetimes in the reliability state subset {2}

¿{2) = pj • 0.304 + p2 • 0.666 + p3 ■ 0.218 + p4-0.058 + p5 -0.218 + p6 ■ 0.058

+ /y 0.085 + ps -0.304 =0.43-0.304 +0.05-0.666 +0.01-0.218

+ 0.01-0.058 +0.08-0.218 +0.01-0.058 + 0.40-0.085 +0.01-0.079=0.220. (74)

and according to (36), the optimal solutions for the mean values of the system unconditional lifetimes in the particular reliability states

77(1) = /'/(1)-/X2) = 0.054, 77(2) = /'/(2) = 0.220. (75)

Moreover, according to (34)-(35) and (47)-(49), the corresponding optimal unconditional multistate reliability function of the system is of the form

•) = [1, %{t,\),%{t, 2)],

with the coordinates given by

R3(t,\) =0.43-[i?(f,l)]a) + 0.05 •[i?(i,l)f)+0.01 •[i?(i,l)]'3) +0.01-[J?(r,l)]'4) + 0.08{R(t,V)f> +0.01-[^(?,l)f)+0.40-[^(?,l)]'7) +0.01-[^(i,l)],S)], (77)

R3(t, 2) = 0.43-[tf(i,2)]a) + 0.05 •[^(?,2)]'2)+0.01 •[^(?,2)]'3) +0.01-[^(?,2)]'4)

+ 0m-[R(t,2)f> + 0.01 {R(t,2)](6> +0.40 {R(t,2)]n> + 0.01-[i?(i,2)f > ] (78)

for t>0, where [i?(i,l)](i), [R(t,2)f\ b = 1,2,...,8, are fixed in [7].

Further, according to (13) and (32)-(33), the corresponding optimal variances and standard deviations of the system unconditional lifetime in the system reliability state subsets are

(1) = 2)l R3 (t,\)dt - [//(l)]2 = 0.084, <t(1) = 0.289, (79)

<t2(2) = 2J? R3(t,2)dt-[jii(2)]2 =0.056, ct(2) = 0.237, (80)

0

where R3(t,\), R3(t,2) are given by (77)-(78) and /7(1), /7(2) are given by (73)-(74).

If the critical safety state is r =1, then the optimal system risk function, according to (7) and (37), is

given by

r(t)= 1-^3^,1) fori>0, (81)

where R3(t, 1) is given by (77).

Hence and considering (38), the moment when the optimal system risk function exceeds a permitted level, for instance 8 = 0.05, is

r= r\8) = 0.19 years.

(82)

Replacing //(r) by /7(1) given by (73) and a(r) by cr(l) given by (79) in the expressions for the renewal systems characteristics pointed in Theorem 1 and Theorem 2, we get their corresponding optimal values pointed below.

Under the assumption that the oil pipeline system is repaired after its failure and that the time of the system renovation is ignored, we obtain the following optimal results:

i) the distribution of the time 5^,(1) until the Nth exceeding of reliability critical state 1 of this system, for sufficiently large TV, has approximately normal distribution A<'(0.274A<'.0.289V/V), i.e.,

t - 0 77AN

FiN)(t,\) = P(SN(\) < t) =Fm0l){ Q28 ), re (-0,00),

ii) the expected value and the variance of the time SN (1) until the Nth exceeding the reliability critical state 1 of this system take respectively forms

E[SN (1)] = 0.274TV, D[SN (1)] = 0.0847V ,

iii) the distribution of the number TV(r, 1) of exceeding the reliability critical state 1 of this system up to the moment t,t> 0, for sufficiently large i, is approximately of the form

r 0.2747V-r 0.274(7V + l)-r>

0.552it ) 0.552V7

P(N (r,l) = TV) =FN(0d)(--—)-F„m)( ■ r' ), TV = 0,1,2,....

iv) the expected value and the variance of the number TV(r,l) of exceeding the reliability critical state 1 of this system at the moment t,t> 0, for sufficiently large i, approximately take respectively forms

H(t,\) = 3.649r, Z)(r,l) = 4.06r.

Under the assumption that the oil pipeline system is repaired after its failure and that the time of the system renovation is not ignored and it has the mean value ¿/„(1) = 0.005 and the standard deviation ct0 (1) = 0.005, we obtain the following optimal results:

i) the distribution function of the time SN(l) until the Nth system's renovation, for sufficiently large TV, has approximately normal distribution TV(0.279TV,0.289VtV) , i.e.,

iN) = <t-0219N)>

0.289VTV

F (t,1) = P(Sn(l)<t) = FV(0;1)( re (-oo,oo)5 N = 1,2,...

ii) the expected value and the variance of the time 5^(1) until the Nth system's renovation take respectively forms

E[SN (1)] = 0.279TV, D[Sn (1)] = 0.084TV ,

iii) the distribution function of the time SN( 1) until the Nth exceeding the reliability critical state 1 of this system takes form

m - r-0.279TV +0.005

V0.084TV-0.000025'

F (r,l) = P(SN(l)<t) =F, ( / ' ' ). re N = 1,2,...,

iv) the expected value and the variance of the time 5^(1) until the Nth exceeding the reliability critical state 1 of this system take respectively forms

E[SN (1)] = 0.2747V + 0.005(7V - 1), D[SN (1)] = 0.083527V + 0.000025(TV - 1), v) the distribution of the number N(t,\) of system's renovations up to the moment t,t> 0, is of the

0 2797V-/1 0 219(N + \)-t

vi) the expected value and the variance of the number A'(7,1) of system's renovations up to the moment t,t> 0, take respectively forms

H(t,\) = 3.584f, D(t,\) = 3.868f,

vii) the distribution of the number N{t,\) of exceeding the reliability critical state 1 of this system up to the moment t,t> 0, is of the form

nw ,1)=») ^,,„.(0279A')-'-0005)-w0279^1'"'"0005»-^2.....

0.549V^ + 0.005 0.49Vi +0.005

viii) the expected value and the variance of the number 7V(f,l) of exceeding the reliability critical state 1 of this system up to the moment t,t> 0, are respectively given by

H<yt,V> ~ t 0 27905 ' W) = 3-868(i + 0 005)'

ix) the availability coefficient of the system at the moment t is given by the formula

A(t,1) = 0.982, t> 0,

x) the availability coefficient of the system in the time interval < t,t + t), t > 0, is given by the formula

A(t,r,l) = 3.584J^3 (t,l)dt, t> 0, r > 0.

X

To obtain the optimal mean sojourn times in the particular operation states maximizing the mean lifetime of the port oil piping transportation system we substitute the optimal limit transient probabilities pb determined by (72) and probabilities nb determined by (45) into the system of equation (40) and we get its following form

-0.2257271^ +0.049887Vf5 + 0.016347\46 + 0.187057kT7 + 0.006457Vfs = 0

0.0198^ +0.00587WT5 + 0.00197\46 + 0.021757Vf7 + 0.00075 Ms = 0 0.00396M1 + 0.001 \6M5 + 0.000387\46 + 0.004357Vf7 + 0.00015 Ms = 0 0.00396 M1 + 0.001 \6M5 + 0.000387\46 + 0.004357Vf7 + 0.00015 Ms = 0 0.03168M! -0.106727Vf5 + 0.00304M6 + 0.03487Vf7 + 0.00127Vfs = 0

0.00396^ + 0.00116M5 - 0.03762M6 + 0.00435M7 + 0.00015MS = 0 0.1584M! +0.0464M5 +0.0152M6 - 0.261 M7+ 0.006M8 = 0 0.00396^ + 0.00116 Ms + 0.00038M6 + 0.00435M7 - 0.01485MS= 0.

Since the above system is homogeneous then it has nonzero solutions when the determinant of the system equations main matrix is equal to zero, i.e. if its rank is less than 8. Moreover, in this case the solutions are ambiguous.

Since the second equation multiplied by five gives the third equation and the third and fourth equations are identical, then after omitting two of them (the second and the third ones), we have

-0.22572^ + 0.04988M, + 0.01634M6 + 0.18705M7 + 0.00645Ms=0 0.00396^ + 0.00116M5 + 0.00038M6+0.00435M? +0.00015MS = 0 0.03168^ -0.10672M5 + 0.00304M6 + 0.0348M7 +0.0012Ms=0 0.00396^ + 0.00116M5 - 0.03762M6 +0.00435M7 + 0.00015MS = 0

0.1584^ +0.0464M5 +0.0152M6 -0.26lM7 +0.006M8=0 0.00396^ + 0.00116Ms + 0.00038M6 + 0.00435M7 - 0.01485MS = 0. (84)

As we are looking for nonzero solutions, we omit the second equation and we get

-0.22572M! +0.04988M5 +0.01634M6 +0.18705M7 + 0.00645MS = 0 0.03168^ -0.10672M5 + 0.00304M6+ 0.0348M7 +0.0012Ms=0 0.00396^ + 0.00116M5 - 0.03762M6 +0.00435M7 + 0.00015MS = 0

0.1584^ + 0.0464 M5 + 0.0152M6-0.261M7 +0.006MS=0 0.00396Mx +0.00116M5 +0.00038M6+ 0.00435M? - 0.01485MS = 0.

(85)

From the above we get nonzero solutions in case when the rank of the main matrix is not greater than 4. In our case, since the above system of equations is satisfied by any values of M2, M3 and M4, than after considering expert opinions, it is sensible to assume

and in order to get 4 nonzero solutions of the system of equations (85) to fix one of the remaining unknown variables for instance, according to (44), assuming

(83)

M2 =480, M3 =1440, M4 = 480,

(86)

M, =360.

(87)

After this the system of equations (85) takes the form

-0.22572^+ 0.04988M5+ 0.18705 M7+ 0.00645A/g = -5.8824 0.03168M!- 0.10672M5+ 0.0348M7+ 0.0012Ms = -1.0944 0.00396^+0.00116^+0.00435^+ 0.00015MS= 13.5432 0.1584^+ 0.0464M5- 0.261 M7+ 0.006M8 = -5.472

0.00396^+0.00116^+0.00435^-0.01485^ = -0.1368.

(88)

Next, after subtracting the third equation from the fifth equation, we get

-0.22572^+0.04988^+0.18705^+0.00645^ = -5.8824 0.03168^-0.10672^+0.0348^ + 0.0012MS =-1.0944 0.1584^+ 0.0464M5- 0.261 M7+ 0.006M8 = -5.472

-0.015MS=-13.68. (89)

The solutions of the above system of equations are

Mx = 330, M5 = 210, M7 = 280, M8 = 912. (90)

Hence and considering (86) and (87), we get the following final solution of the equation (83)

Mx =330, M2 =480, M3 = 1440, M4 =480, M5 = 210, M6 = 360, M7 = 280, M8 = 912. (91)

Now, substituting in (41) the above mean values Mb of the system unconditional sojourn times in the particular operation states and the known probabilities pbl of the system operation process transitions between the operation states given in the matrix (42), we may look for the optimal values Mbl of the mean values of the system conditional sojourn times in the particular operation states that maximizing the mean lifetime of the port oil piping transportation system in the reliability states subset {1,2}. The optimal values Mbl, bj = 1,2,...,8, b^l, should to satisfy the following obtained this way system of equations

0.06 Ml5+ 0.06 M16 + 0.86 M17 + 0.02M1S = 330

Pl\M2\ + Pl2M22 + PliM2i + P24M24 + P25M 25 + P 26M 26 + P 21M 27 + P 2ZM 2Z = 480 PnMn + P32M32 + />33^33 + />34M34 + P35M35 + P36M36 + P 37M 37 + PlSM3S= 1440

p4lM4l + p42M42 + p4iM4i + p44M44 + P45M45 + P46M46 + P41M41 + P4SM4S= 480 0.125 M5l+ 0.125 M56 + 0.687M57 + 0.063 M58 = 210 0.4M61+ 0.6M65 = 360 0.82M71-0.16M75 + 0.02M78 = 280 0.67M81+ 0.33M87 = 912.

Unfortunately, the solution of the above system of equations are ambiguous 8 CONCLUSION

The joint general model of reliability and availability of complex technical systems in variable operation conditions linking a semi-markov modeling of the system operation processes with a multi-state approach to system reliability and availability analysis constructed in the paper Part 1 was applied to reliability evaluation of the port oil piping transportation system. The main reliability and availability characteristics were evaluated and maximized after its operation process optimization.

Acknowledgements

The paper describes part of the work in the Poland-Singapore Joint Research Project titled "Safety and

Reliability of Complex Industrial Systems and Processes" supported by grants from the Poland's Ministry of

Science and Higher Education (MSHE grant No. 63/N-Singapore/2007/0) and the Agency for Science,

Technology and Research of Singapore (A*STAR SERC grant No. 072 1340050).

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