Reliability and risk optimization of multistate systems with application to port transportation system Текст научной статьи по специальности «Компьютерные и информационные науки»

УПРАВЛ1ННЯ У ТЕХНШНИХ СИСТЕМАХ

УПРАВЛ1ННЯ У ТЕХН1ЧНИХ СИСТЕМАХ

УПРАВЛЕНИЕ В ТЕХНИЧЕСКИХ СИСТЕМАХ

CONTROL IN TECHNICAL SYSTEMS

УДК004.052: 656.61

Kotowrocki K.1, Kwiatuszewska-Sarnecka B.2, Soszynska-Budny J.3

1Dr. hab. Sc., Professor, Professor of Department of Mathematics, Gdynia Maritime University, Gdynia, Poland 2PhD, Assistant Professor, Assistant Professor of Department of Mathematics, Gdynia Maritime University, Gdynia, Poland 3PhD., Associate Professor of Department of Mathematics, Gdynia Maritime University, Gdynia, Poland

RELIABILITY AND RISK OPTIMIZATION OF MULTISTATE SYSTEMS WITH APPLICATION TO PORT TRANSPORTATION SYSTEM

The complexity of technical systems' operation processes and its influence on the changing in time systems' structures and their components' reliability parameters posses a difficulty to first meet in real and then to fix and analyse those structures and reliability parameters. By constructing a joint model of reliability of complex technical systems at variable operation conditions, which links a semi-markov modelling of system operation processes with multi-state approach to system reliability analysis, we find the system's main reliability characteristics. Consequently, we use linear programming to build a model of complex technical systems reliability optimization. We investigate the model's application in marine transport, specifically in reliability and risk optimization of a bulk cargo transportation system. The tools we develop can be used in reliability evaluation and optimization of a very wide class of real technical systems operating at varying conditions that influence their reliability structures and the reliability parameters of their components. Consequently, the tools we developed can be implemented by reliability practitioners from both maritime transport industry and other industrial sectors.

Keywords: reliability function, risk function, operation process, optimization.

NOMENCLATURE

Z(t),Zi is a system operation process; a system operation state; i = 1,2,..., v,

pb is an optimal transient probability;

U is a particular reliability state of the system; u = 1,2,..., z;

T (u) is a lifetime of a system in the reliability state subset {u, u +1,..., z};

[ R (t, u )](b) is a conditional reliability function of a system at the operational state zb ;

R (t, u) is an optimal unconditional reliability function of a system;

|(u) is a mean lifetime of the system in the reliability state subset {u, u +1,..., z};

|(u) is a mean lifetime of the system in the reliability particular state u;

R is a critical state of the multi-state system;

r (t) is an optimal risk function of the multi-state system.

INTRODUCTION

Most real technical systems are very complex because they are composed of large numbers of components and subsystems and have high operating complexity. The complexity of the systems' operation processes and its influence on the changing in time systems' structures and their components' reliability parameters posses a difficulty to first meet in real and then to fix and analyse those structures and reliability parameters. A convenient tool to investigate this problem is a semi-markov [2] modelling of the system operation process linked with a multi-state approach for the system reliability analysis [1, 4, 9-10] and a linear programming for the system reliability optimization [3]. Using this approach, it is possible to find this complex system's main reliability characteristics including the system reliability function, the system mean lifetimes in the reliability states subsets and the system risk function [4, 6, 8]. Having those characteristics it is possible to optimize the system operation process to get optimal values [8]. To this end the linear programming [3] can be applied to maximize the mean value of the system lifetime in the subset of the system reliability states, which are not worse than the system critical reliability state.

© Kolowrocki K., Kwiatuszewska-Sarnecka B., Soszynska-Budny J., 2016 DOI 10.15588/1607-3274-2016-2-14

1 SYSTEM RELIABILITY AT VARIABLE OPERATIONS PROCESS

We suppose that the system has v different operation states during its operation process. Thus, we can define the system operation process Z (t), t e< 0,+to>, as the process with discrete operation states from the set Z = {z1, Z2,..., zv}. Further, we assume that Z(t) is a semi-markov process [2] with its conditional sojourn times 9W at the operation state zb when its next operation state is zi, b, l = 1,2,..., v, b *l. In this case the process Z(t) may be described by:

- the vector of probabilities of the process initial operation states [pb (0)]1XV,

- the matrix of probabilities of the process transitions between the operation states [ p^ ]VXV,

- the matrix of conditional distribution functions [Hbi (t)]VXV of the process sojourn times 0y, b * l, in the operation state zb when the next operation state is zl.

Under these assumptions, the sojourn times mean values are given by

[R (t,u)](b) = P(ljb)(u) > t|Z(t) = Zb),

is the conditional reliability function while the system is at the operational state Zb, b = 1,2,..., v.

Next, we denote the system lifetime in the reliability state subset {u,u +1,..., z} by T(b)(u) and by

[R(t, -)](b)=[1, [R(t,1)](b) [R(t,2)](b) ...,[R(t,z)](b)], where for t e< 0,to), b = 1,2,...,v, u = 1,2,..., z,

[R(t,u)](b) = P(T(b)(u) > t|Z(t) = zb),

is the conditional reliability function of the system while the system is at the operational state zb.

In the case when the system operation time is large enough, the unconditional reliability function of the system is given by

R(t, •) = [1, R(t,1), R(t,2), ..., R(t, z)], t > 0,

where

Mi

bl'-

= E[0bl] = jtdHbl(t), b, l = 1,2,...,v, b * l. (1)

The mean values E[ 0 b] of the unconditional sojourn times 0b are given by

R (t, u) = Z Pb[R(t,u)] b=1

|(b)

(3)

The mean values of the system lifetimes in the reliability state subset {u, u +1,..., z} are

M

= E[0b] = ZPblMbl, b = 1,2,..

l=1

(2)

where Mbl are defined by (1).

Limit values of the transient probabilities at the operation states are given by

pb = lim Pb (t) = ^J^, b = 1,2,..., v,

t^-i» v

ZKiMi l=1

where the probabilities of the vector [rcj ]1XV satisfy the

v

system of equations [%b ] = [^b ][Pbl ] and Z^l = 1 and

l=1

pb(t) = P(Z(t) = zb) , t e< 0,+to), b = 1,2,..., v.

We assume that the system is composed of n independent multistate components Ei, i = 1,2,..., n, and that the changes of the operation process Z(t) states have an influence on both the system components Ei reliability and on the system reliability structure. Consequently, we denote the component Ei lifetime in the reliability state

subset {u,u +1,...,z} by TI<'b')(u) and by

[R (t, -)](b)= [1, [Ri (t,1)](b), [R (t,2)]

i(b)

(b)-

[R (t, z)]

where for t e< 0,to), b = 1,2,., v, u = 1,2,..., z,

|a(u) = E[T(u)] ^ZpbV-b(u), u = 1,2,... b=1

(4)

and the mean values of the system lifetimes in the particular reliability state u, are [4]

|(u) = |(u)-|(u +1), u = 1,2,...,z-1, |(z) = |(z). (5) A probability r(t) = P(s(t) < r | R(0) = z) = P(T(b)(r) < t),

t e (-to , to),

that the system is in the subset of reliability states worse than the critical state r, r e {1,...,z} while it was in the state z at the moment t = 0 is called a risk function of the multi-state system [4].

Under this definition, from (3), we have

r(t) = 1 - R(t,r), t e (-to , to ),

(6)

and if t is the moment when the risk exceeds a permitted level 8 , then

t = r _1(8),

(7)

where r x(t), if it exists, is the inverse function of the risk function r(t).

2 OPTIMAL TRANSIENT PROBABILITIES MAXIMIZING SYSTEM LIFETIME

Considering the equation (3), 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 (4) for the mean values of the system

z

v

ynPAB^IHH^ y TEXHRHHX CHCTEMAX

unconditional lifetimes in the reliability state subsets. From linear equation (4), we can see that the mean value of the system unconditional lifetime |(u), u = 1,2,..., z, is determined by the limit values of transient probabilities p,, b = 1,2,..., v, of the system operation states and the mean values |b(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. Therefore, the system lifetime optimization approach based on the linear programming can be proposed. Namely, we may look for the corresponding optimal values pb of the transient probabilities Pb in the system operation states to maximize the mean value |(u) of the unconditional system lifetimes in the reliability state subsets {u, u +1,..., z} under the assumption that the mean values |b(u) of the system conditional lifetimes in the reliability state subsets are fixed. As a special case of the above formulated system lifetime optimization problem: if r, r = 1,2,..., z, is a system critical reliability state, then we want to find the optimal values pb of the transient probabilities Pb in the system operation states to maximize the mean value | (r) of the unconditional system lifetime in the reliability state subset {r, r +1,,..., z} under the assumption that the mean values |b (r), b = 1,2,., v, of the system conditional lifetimes in this reliability state subset are fixed. More exactly, we formulate the optimization problem as a linear programming model with the objective function of the following linear form

№ ) = I PbVb (r) b=1

(8)

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

I Pb = 1, b=1

Pb <Pb <pb, b = U,..., Vf

(9)

where |b (r), lb(r) > 0, b = 1,2,..., v, 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, (10)

are respectively the lower and upper bounds of the unknown transient probabilities Pb .

Now, we can obtain the optimal solution to the formulated by (8)—(10) 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 (8). First, we arrange the system conditional lifetime mean

values |b(r), b = 1,2,.,v, in non-increasing order

lb1(r) > lb2(r) > . . . >lbv (r), b e {1,2,..., v} for i = 1,2,..., v.

Next, we substitute

xi = Pbt , Xi = A , Xi = Pb for i = 1,2,..., v (11)

and we maximize with respect to xi, i = 1,2,..., v, the linear form (8) that takes the form

= I x^b. (r) i=1

(12)

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

I xi =1, x. < x. < x., i = 1,2,..., V, (13; i=1

where h, (r), (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 < xi < 1 and Xj, 0 < Xj < 1, Xj < Xj, i = 1,2,..., v, (14)

are the lower and upper bounds of the unknown probabilities Xj, i = 1,2,..., v, respectively. We define

x = Ix,, y = 1 - x i=1

(15)

and

I I

I _ ^ c I

x0 = 0, X0 = 0 and X Xj, X Xj for j=1 ¿=1

I = 1,2,..., v. (16)

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

xI - x1 < y

(17)

and we fix the optimal solution that maximize (12) in the following way:

i) if I = 0, the optimal solution is xrj = y + xj and x. = x.

fori = 2,3,..., v; (18)

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

x. = x. for i = 1,2,..., I, xI+j = y - xI + xI + xI+j and

xCj = xj for i = I + 2, I + 3,..., v;

(19)

iii) if I = v, the optimal solution is Xj = xt for i = 2,3,..., v. (20) Finally, after making the inverse to (11) substitution, we get the optimal limit transient probabilities

Pb. = x. for i = 1,2,...,v,

(21)

that maximize the system mean lifetime |(r) in the reliability state subset {r,r +1,...,z}, defined by the linear form (8) giving its maximum value in the following form

v

1 (r) = ZPblb (r) for a fixed r e {1,2,..., z} . (22) b=1

From the above, replacing r by 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

¿(u) = ZPb^b (u) for u = 1,2

b=1

(23)

Further, according to (3), the corresponding optimal unconditional multistate reliability function of the system is

R (t,-) = [1, R(t,1),..., R (t, z)\

(24)

where

Rn (t,-) =Z P b [ R(t, u)](b)

b=1

for t > 0 , u = 1,2,..., z, (25)

and by (5) the optimal solutions for the mean values of the system unconditional lifetimes in the particular reliability states are of the form

JI(u) = |(u)-|(u +1), u = 0,1,..., z-1,1 (z) = |(z). (26)

Moreover, considering (6) and (7), the corresponding optimal system risk function and the moment when the risk exceeds a permitted level 8, respectively are given by

r(t) = 1 - R(t,r) for t e (-to,to) and T = r-1(8). (27)-(28)

3 OPTIMAL SOJOURN TIMES OF COMPLEX TECHNICAL SYSTEM OPERATION PROCESS

Replacing the limit transient probabilities pb of the system operation process at the operation states by their optimal values pb and the mean values of the unconditional sojourn times at the operation states by their corresponding unknown optimal values Mib maximizing the mean value of the system lifetime in the reliability states subset {r,r +1,..., z}, we get the system of equations

P b , b = 1,2,..., v.

Zn Mi

l=1

(29)

After simple transformations the above system takes the form

( p1 -¡^M1 + p1n2M 2 +... + p1TivMl v = 0 p2TC1M1 + (p2 -1)^2M2 +... + p2TCVMv = 0

p V%M 1 + p V%2M 2 +... + ( p v-1)^v Mi v = 0,

(30)

where Mb are unknown variables we want to find, pb are optimal transient probabilities and %b are steady probabilities.

Since the system of equations 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 either one or multiple of them, we may find the values of the remaining ones and using this method arrive at the solution of this equation.

Another very useful and much easier applicable in practice tool that can help in planning the operation processes of complex technical systems are the system operation process optimal mean values of the total system

operation process sojourn times 0b at the particular operation states zb, b = 1,2,..., v, during the fixed system operation time 0. They can be obtained by replacing the transient probabilities pb at the operation states zb with their optimal values pb. This results in the following expession

Mib = E[0b] = pbö, b = l,2,■■■, v.

(31)

The knowledge of the optimal values Mib of the mean values of the unconditional sojourn times and the optimal

mean values MMb of the total sojourn times at the particular operation states during the fixed system operation time may be the basis for changing the complex technical systems operation processes in order to ensure that these systems operate both more reliably and more safely. This knowledge may also be useful in these systems operation cost analysis.

4 THE BULK CARGO TRANSPORTATION SYSTEM RELIABILITY AND RISK

The considered bulk cargo terminal placed at the Baltic seaside is designated for storage and reloading of bulk cargo, but its primary activity is loading bulk cargo on board the ships for export. There are two independent transportation systems: the system of reloading rail wagons and the system of loading vessels.

Cargo is brought to the terminal by trains consisting of self-discharging wagons, which are discharged to a hopper and then by means of conveyors transported into one of four storage tanks (silos). Loading of fertilizers from storage tanks on board the ship is done by means of special reloading system which consists of several belt conveyors and one bucket conveyor which allows the transfer of bulk cargo in a vertical direction. Researched system is a system of belt conveyors, referred to as the transport system.

In the conveyor reloading system we distinguish three bulk cargo transportation subsystems, the belt conveyors S1, S2 and S The conveyor loading system is composed of six bulk cargo transportation subsystems, the dosage conveyor S4, the horizontal conveyor S5, the horizontal conveyor S6, the sloping conveyors S7, the dosage conveyor with buffer S8, the loading system S9.

The bulk cargo transportation subsystems are built, respectively:

- the subsystem S1 : 1 rubber belt, 2 drums, set of 121 bow rollers, set of 23 belt supporting rollers,

- the subsystem S2: 1 rubber belt, 2 drums, set of 44 bow rollers, set of 14 belt supporting rollers,

z

ynPABMHHS y TEXHRHHX CHCTEMAX

- the subsystem S3: 1 rubber belt, 2 drums, set of 185 bow rollers, set of 60 belt supporting rollers,

- the subsystem S4: 3 identical belt conveyors, each composed of 1 rubber belt, 2 drums, set of 12 bow rollers, set of 3 belt supporting rollers,

- the subsystem S5 : 1 rubber belt, 2 drums, set of 125 bow rollers, set of 45 belt supporting rollers,

- the subsystem S6: 1 rubber belt, 2 drums, set of 65 bow rollers, set of 20 belt supporting rollers,

- the subsystem S7: 1 rubber belt, 2 drums, set of 12 bow rollers, set of 3 belt supporting rollers,

- the subsystem S8: 1 rubber belt, 2 drums, set of 162 bow rollers, set of 53 belt supporting rollers,

- the subsystem S9: 3 rubber belts, 6 drums, set of 64 bow rollers, set of 20 belt supporting rollers.

Taking into account the operation process of the considered system we distinguish the following as its three operation states:

- an operation state zj - loading fertilizers from rail wagons on board the ship is done using S1, S2, S3, S6, S7, S8 and S9 subsystems.

- an operation state z2 - discharging rail wagons to storage tanks or hall when subsystems S1, S2 and S3, are used,

- an operation state z3 - loading fertilizers from storage tanks or hall on board the ship is done by using S4, S5, S6, S7, S8 and S9, subsystems.

The limit values of the bulk cargo transportation systems operation process transient probabilities p,(t) at the operation states z,, b = 1,2,3, determined in [5], on the basis of data coming from experts are

P! = 0.2376, p2 = 0.6679, p3 = 0.0945.

(32)

Further, assuming that the system is in the reliability state subset {u,u+1,.,z} if all its subsystems are in this subset of reliability states, we conclude that the bulk cargo transportation system is a series system [4] of subsystems S1, S2, S3, S6, S7, S8 and S9 .

Under the assumption that changes of the bulk cargo transportation system operation states have an influence on both the subsystem Sj reliability and the entire reliability structure [8], on the basis of expert opinions and statistical data given in [9], [10], the bulk cargo transportation system reliability structures and their components reliability functions at different operation states can be determined.

Additionally, we assume that subsystems Sj, i = 1,2,3,...,9, are composed of four-state exponential components, with the reliability functions

[Rj (t, -)](b)= [1, [R (t, 1)](b), [R (t, 2)](b), [Rj (t, 3)](b)], t e<0,to), b = 1,2,3, u = 1,2,3.

At the operation state z1 , at loading of fertilizers from

rail wagons on board the ship, the system is composed of seven non-homogenous series subsystems S1, S2, S3, S6, S7, S8, and S9 forming a series structure.

The conditional reliability function of the system while it is at the operation state z1 is given by

[R(t, -)](1) = [1, [R(t, 1)](1) , [R(t, 2)](1), [R(t, 3)](1) ], where

[R(t, u)](1) = [R147(t, u)]® [R61(t, u)](1) [R24g(t, u)](1) [R88(t, u)](1)

[R18(t, u)](1)[R218(t, u)](1)[R93(t, u)](1)for t e< 0,to), u = 1,2,3,

[R(t, 1)](1) = exp[-74.426t], [R(t, 2)](1) = exp[-93.472t],

[ R (t,3)](1) = exp[-150.206t]. (33)-(35)

The expected values of the conditional lifetimes in the reliability state subsets at the operation state z1 , calculated from the above result given by (33)-(35), are:

lj(1) s 0.013, |j(2) s 0.011, |j(3) s 0.007 years, (36)

and further, using (5), it follows that the conditional lifetimes in the particular reliability states at the operation state z1 are:

|1(1) s 0.002, |1(2) s 0.004, |1(3) s 0.007 years.

At the operation state z2, i.e. at the state of discharging rail wagons to storage tanks or hall, the system is built of three subsystems S1, S 2 and S3 forming a series structure [4].

The conditional reliability function of the bulk cargo transportation system at the operation state z2 is given by

[R(t, -)](2) = [1,[R(t, 1)](2), [R(t, 2)](2), [R(t, 3)](2) ], where

[ R(t, u )](2)= [R147 (t, u)](2) [Rj1(t, u)](2) [R248 (t, u)](2) for t e< 0,to), u = 1,2,3,

i.e.

[R(t,1)](2) = exp[-39.563t], [R(t,2)](2) = exp[-49.663t],

[ R(t,3)](2) = exp[-64.280t]. (37)-(39)

The expected values of the conditional lifetimes in the reliability state subsets at the operation state z2 , calculated from the above result given by (37)-(39), are:

12(1) s0.025, |2(2) 0.020, 0.016 years,

(40)

and further, using (5), it follows that the conditional lifetimes in the particular reliability states at the operation state z2 are:

l2(1) s0.005, |2(2) s 0.004, |2(3) s 0.016 years.

At the operation state z3, i.e. at the loading of fertilizers from storage tanks or hall on board, the bulk cargo transportation system is built of six subsystems one seriesparallel subsystem S4 and five series subsystems S5, S6, S7, S8, S9 forming a series structure [4].

i.e

The conditional reliability function of the system while it is at the operation state z3 is given by

[R(t, -)](3) = [1, [R(t, 1)](3), [R(t, 2)](3), [R(t, 3)](3) ], where

[R(t, u^H^sft {Rra(t, u)](3) • [Rgg(t,u)](3) •[R8(t,u)](3) •[%8(t, u)](3) •[R93(t, u)](3) for t e<0,to), u = 1,2,3,

i.e.

[R(t, 1)](3) =exp[-57.7581] -3exp[-55.0071] +3exp[-52.2561],(41)

[R(t, 2)](3) = exp[-70.974t]-3exp[-68.018t] +3exp[-65.062t] (42)

[R(t, 3)](3) = exp[-89.416t]-3exp[-86.140t] +3exp[-82.864t].(43)

The expected values of the conditional lifetimes in the reliability state subsets at the operation state z3, calculated from the above result given by (41)-(43), are:

|3(1) = 0.020, |3(2) = 0.016, |3(3) = 0.013 years, (44)

and further, using (5), it follows that the conditional lifetimes in the particular reliability states at the operational state z3 are:

|3(1) = 0.004, |3(2) = 0.003, |3(3) = 0.013 years.

In the case when the system operation time is large enough, the unconditional reliability function of the bulk cargo transportation system is given by the vector

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

where, according to (3) and after considering the values of

pb, b = 1,2,3, given by (32), its co-ordinates are as follows:

R(t,u) = p1 •[R(t,u)](1) + p2 • [R(t,u)](2) + p3 • [R(t,u)](3) (45)

for t > 0, u = 1,2,3, where [R(t,u)](1) and [R(t,u)](2) and [ R(t, u )](3) are respectively given by (33)-(35) and (37)-(39) and (41)-(43), i.e.

R(t, 1) = 0.6679exp[-39.563t] + 0.0945exp[-74.426t] + + 0.2376[exp[-57.758t ] - 3exp[-55.007t ] + 3exp[-52.256t ]] ,(46)

R (t ,2) = 0.6679 exp[-93.472t ] + 0.0945exp[-49.663t ] + + 0.2376[exp[-70.974t ] - 3exp[-68.018t ] + 3exp[-65.062t ]],(47)

R(t,3) = 0.6679 exp[-150.206t ] + 0.0945exp[-64.280t ] +

+ 0.0945[exp[-89.416t ] - 3exp[-86.140t ] + 3exp[-82.864t ]].(48)

The mean values of the system unconditional lifetimes in the reliability state subsets, according to (4) are respectively:

i (1) = 0.016, |(2) = 0.013, |(3) = 0.009. (49) The mean values of the system lifetimes in the particular reliability states, (5), are

1(1) = |(1) -1(2) = 0.003, | (2) = |(2) -1(3) = 0.004,

1(3) = |(3) = 0.009.

If the critical reliability state is r = 2, then the system risk function, according to (6), is given by r(t) = 1 - [0.6679 exp[-93.4721] +0.0945 exp[-49.663t] +

+0.2376(exp[-70.974t] -3exp[-68.018t] +3exp[-65.0621])] for t >0. (50)

Hence, the moment when the system risk function (Fig. 1) exceeds a permitted level, for instance 8 = 0.05, from (7), is

t = r-1( 8 ) = 0.000627 years. (51)

The system risk function

1,0 i 0,9 -0,8 -0,7 -0,6 -£ 0,5 -0,4 - I 0,3 -0,2 -0,1 -

0,0 -,-,-,-,-,-,-,

0,00 0,05 0,10 0,15 0,20 0,25 0,30

t

Figure 1 - The graph of the port bulk cargo transportation system risk function

5 OPTIMIZATION OF THE BULK CARGO TRANSPORTATION SYSTEM OPERATION PROCESS

In our case, as the critical state is r = 2, then considering the expression for |(2), the objective function (8), takes the form

I(2) = pr 0.011 + p2• 0.020 + p3 • 0.016 = 0.013 years. (52)

The lower pb and upper pb bounds of the unknown transient probabilities pb, b = 1,2,3, coming from experts, respectively are [6]:

p = 0.150, p 2 = 0.005, p3 = 0.015,

p = 0.850, p 2 = 0.120, p3 = 0.390.

Therefore, according to (9)-(10), we assume the following bound constraints

3

Zpb =1 0.250 < p < 0.850, b=1

0.005 < p2 < 0.150, 0.050 < p3 < 0.550.

Now, before we find optimal values pb of the transient probabilities pb, b = 1,2,3, that maximize the objective function (53), we arrange the system conditional lifetimes mean values (2), b = 1,2,3, in non-increasing order

12(2) > I3(2) > I1(2).

Next, according to (11), we substitute

x1 = p2 = 0.0945, x2 = p3 = 0.2376, x3 = p1 = 0.6679, (53)

ynPÄB^IHH^ y TEXHRHHX CHCTEMAX

X! = 0.005, X2 = 0.050, X3 = 0.250, X1 = 0.150,

x2 = 0.550, X3 = 0.850, (54)

where Xj and Xj are lower and upper bounds of the unknown limit transient probabilities Xj, i = 1,2,3, respectively and we maximize with respect to Xj, j = 1,2,3, the linear form (52) that according to (13) takes the form

l(2) = X! • 0,011 + x2 • 0,020 + X3 • 0,016, (55) with the following bound constraints

Z X = 1, j=1

0.005 < xx < 0.12, 0.015 < x2 < 0.390, 0.150< x3 < 0.850. (56)

According to (15)-(17), we calculate and fix the optimal solution that maximizes linear function (55) according to the rule (19). Namely, we get

X! = X! = 0.120 , X2 = X2 = 0.390,

X3 = 0.830 - 0.490 + 0.150 = 0.490.

Finally, according to (21) after making the inverse to (53) substitution, we get the optimal transient probabilities

p1 = X3 = 0.490, p 2 = Xj = 0.120, p3 = X2 = 0.390, (57)

that maximize the system mean lifetime in the reliability state subset {2,3} expressed by the linear form (53) giving, according to (12) and (57), its optimal value

l(2)= p1 • 0,011 + p2 • 0,020 + p3 • 0,016 = 0.49 • 0.011 +

+ 0.12 • 0.020 +0.39 • 0.016 = 0.014. (58)

6 OPTIMAL RELIABILITY CHARACTERISTICS OF THE BULK CARGO TRANSPORTATION SYSTEM

Further, substituting the optimal solution (57) according to (24), we obtain the optimal solution for the mean value of the system unconditional lifetime in the reliability state subset {1,2}, {3} that respectively amounts:

|(1) s 0.0172, |(3) s 0.0104, (59)

and according to (26), the optimal solutions for the mean values of the system unconditional lifetimes in the particular reliability states are

|(1) s 0.0032, |(2) s 0.0036, |j(3) s 0.0104.

Moreover, according to (24)-(25), 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) ] for t > 0,

where according to (3) and after considering the values of p,, its co-ordinates are as follows:

R(t,u)s 0.49 •[R(t,u)](1)+ 0.12 • [R(t,u)](2)+ 0.39 •[R(t,u)](3) for t > 0, u = 1,2,3, (60)

where [R(t,u)](1), [R(t,u)](2), [R(t,u)](3)are respectively given by (33)-(35) and (37)-(39), (41)-(43).

If the critical reliability state is r = 2, then the system risk function, according to (27), is given by

r(t) = 1 -R(t2) = 1 - [0.49 exp[-93.4721] +0.12 exp[-49.6631] + +0.39(exp[-70.974t] -3exp[-68.018t] +3exp[-65.062t]]p

where R(t,2) is given by (60) for u = 2 .

Hence, considering (28), the moment when the optimal system risk function (Fig. 2) exceeds a permitted level, for instance 8 =0.05, is

t = r-1(8) s 0.000676 years. (62)

Comparing the bulk cargo transportation system reliability characteristics after its operation process optimization given by (58)-(62) with the corresponding characteristics before this optimization determined by (45)-(51) justifies this action.

The optimal system risk function

t

Figure 2 - The graph of the port bulk cargo transportation system optimal risk function

7 OPTIMAL SOJOURN TIMES OF BULK CARGO TRANSPORTATION SYSTEM OPERATION PROCESS AT OPERATION STATES

Having the values of the optimal transient probabilities determined by (57), it is possible to find the optimal conditional and unconditional mean values of the sojourn times of the bulk cargo transportation operation process at the operation states and the optimal mean values of the total unconditional sojourn times of the bulk cargo transportation system operation process at the operation states during the fixed operation time as well.

Substituting the optimal transient probabilities at operation states determined in (57) and the steady probabilities

n1 = 0.315, n2 = 0.5, n3 = 0.185, we get the following system of equations

- 0,16065AA1 + 0.245M2 + .0.09065M3 = 0 • 0.0378M1 + (-0.44)M2 + 0.0222M3 = 0. 0.12285M1 + 0.195M2 + (-0.11285)AM3 = 0

with the unknown optimal mean values Mi, of the system unconditional sojourn times in the operation states. Consequently, we get

MM1, M2 = 0.154286A/1, M3 = 1.216216AM1.

Thus, we may fix Mj and determine the remaining ones. In our case, after considering expert opinion, we conclude that it is sensible to assume

M1 = 2.

This way the obtained solutions of the system of equations, are

M1 = 2, M2 = 0.308571, M3 = 2.432432. (64)

It can be seen that these solutions differ substantially from the values M1, M2, M3.

Other very useful and much easier to apply in practice tools that can help in planning the operation process of the technical system are the system operation process optimal mean values of the total sojourn times at the particular operation states during the fixed system operation time 0. Assuming the system operation time 0 = 1 year = 365 days, after aplying (32), we get their values

Mj = E[0j] = pj0 = 0.49 • 365 s 179days, M2 = E[02] = p20 = 0.12 • 365 = 44days, M3 = E[03] = p30 = 0.39• 365 = 142 days. (65)

In practice, the knowledge of the optimal values of Mb and M b given respectively by (64)-(65), is impornat and very helpful in planing and improving the operation process, as it allows for more reliable and safer system operation. From the performed analysis of the results of the bulk cargo transportation system operation process optimization it can be suggested to change the operation process characteristics that result in replacing (or the approaching/convergence to) the unconditional mean sojourn times Mb in the particular operation states before the

optimization by their optimal values Mb after the optimization. The easiest way of reorganizing the system operation process leads to replacing (or the approaching/convergence to) the

total sojourn times, Mb = E[0b], of the bulk cargo transportation system operation process, and in particular operation states during the operation time 0 = 1 year, with their

optimal values Mb = E [0 b ]. CONCLUSIONS

The joint model of reliability of complex technical systems at variable operation conditions linking a semi-markov modelling of the system operation processes with a multistate approach to system reliability analysis was constructed. Next, the final results obtained from this joint model and linear programming were used to build the model of complex technical systems reliability optimization. These tools can be used in reliability evaluation and optimization of a very wide class of real technical systems operating at

varying conditions that influence their reliability structures and the reliability parameters of their components. The practical application of these tools to reliability and risk evaluation and optimization of a technical system of a bulk cargo transportation system, operating in variable conditions, and the results achieved can be implemented by reliability practitioners from both maritime transport industry and other industrial sectors.

REFERENCES

1. Aven T. Reliability evaluation of multi-state systems with multistate components / T. Aven // IEEE Transactions on reliability. -1985. - Vol. 34. - P. 473-479.

2. Grabski F. Semi-Markov Models of Systems Reliability and Operations / F. Grabski. - Warsaw : Systems Research Institute, Polish Academy of Science. - 2015.

3. Klabjan D. Existence of optimal policies for semi-Markov decision processes using duality for infinite linear programming / D. Klabjan, D. Adelman // Siam. J. Control Optim. - 2006. -Vol. 44(6). - P. 2104-2122. DOI: 10.1137/s0363012903437290

4. Kolowrocki K. Reliability of large and complex systems / K. Kolowrocki. - Elsevier, 2014. - 460 p. DOI: 10.1016/b978-0-08-099949-4.00010-6

5. Kolowrocki K. Modelling of operational processes of port bulk cargo system / K. Kolowrocki, B. Kwiatuszewska-Sarnecka, J. Soszynska // Proc. 2nd Summer Safety and Reliability Seminars -SSARS 2008. - Gdacsk-Sopo. - 2008. - Vol. 2. - P. 217-222.

6. Kolowrocki K. Reliability and risk analysis of large systems with ageing components / K. Kolowrocki, B. Kwiatuszewska-Sarnecka // Reliability Engineering & System Safety - 2008. - Vol. 93. -P. 1821-182. DOI: 10.1016/j.ress.2008.03.008

7. Kolowrocki K. Reliability and availability of complex systems / K. Kolowrocki, J. Soszynska // Quality and Reliability Engineering International. - 2006. - Vol. 22, Issue 1. - P. 79-99. DOI: 10.1002/qre.749

8. Kolowrocki K. Reliability and Safety of Complex Technical Systems and Processes: Modeling-Identification-Prediction-Optimization / K. Kolowrocki, J. Soszycska-Budny. - London : Springer, 2011. -405 p. DOI: 10.1007/978-0-85729-694-8

9. Kolowrocki K. Reliability and risk improvement with components quantitative and qualitative redundancy - bulk cargo terminal / K. Koiowrocki, B. Kwiatuszewska-Sarnecka, J. Soszynska // Proc. 9nd Summer Safety and Reliability Seminars (SSARS 2015). -Gdacsk-Sopot, 2015.

10. Kwiatuszewska-Sarnecka B. Analysis of Reservation Efficiency in Series Systems : thesis ... doctor of philosophy / Kwiatuszewska-Sarnecka Bozena. - Gdynia Maritime University, 2003.

11. Kwiatuszewska-Sarnecka B. Reliability Improvement of Large Multi-state Series-parallel Systems / B. Kwiatuszewska-Sarnecka // International Journal of Automation and Computing. - 2006. -Vol. 2. - P. 157-164. DOI: 10.1007/s11633-006-0157-y

12. Lisnianski A. Multi-state System Reliability. Assessment, Optimisation and Applications / A. Lisnianski, G. Levitin. -Singapore : World Scientific Publishing Co., 2003. - 376 p.

13. Meng F. Component-relevancy and characterization in multistate systems / F. Meng // IEEE Transactions on reliability. -1993. - Vol. 42. - P. 478-483. DOI: 10.1109/24.257834

Article was submitted 01.12.2015.

After revision 15.12.2015.

Коловроцкий К.1, Квапишевска-Сарнецка Б.2, Сошинська-Будный Й.3

'Д-р наук, профессор, профессор кафедры математики, Морская Академия в Гдыне, Гдыня, Польша 2Д-р философии, доцент, доцент кафедры математики, Морская Академия в Гдыне, Гдыня, Польша 3Д-р философии, адъюнкт кафедры математики, Морская Академия в Гдыне, Гдыня, Польша

ОПТИМИЗАЦИЯ НАДЕЖНОСТИ И РИСКОВ СИСТЕМ С НЕСКОЛЬКИМИ УСТОЙЧИВЫМИ СОСТОЯНИЯМИ В ПРИЛОЖЕНИИ К ТРАНСПОРТНОЙ СИСТЕМЕ ПОРТА

Сложность процессов работы технических систем и их влияние на изменение во времени структур систем и параметров надежности их компонентов обусловливают сложности при первой встрече в реальности, а затем в фиксации и анализе этих структур и

УПРАВЛ1ННЯ У ТЕХШЧНИХ СИСТЕМАХ

параметров надежности. Путем построения объединенной модели надежности сложных технических систем в различных условиях эксплуатации, связывающей полумарковское моделирование процессов работы системы с подходом нескольких состояний в анализе надежности систем, мы находим основные характеристики надежности системы. Затем мы используем линейное программирование для того, чтобы построить модель оптимизации надежности сложных технических систем. Мы исследуем приложение модели в морском транспорте, в частности, в оптимизации надежности и рисков объемной системы грузоперевозок. Инструменты, разработанные нами, могут быть использованы для оценки надежности и оптимизации очень широкого класса реальных технических систем, работающих в различных условиях, которые влияют на их структуру надежности и параметры надежности их компонентов. Следовательно, разработанные нами инструменты могут быть использованы специалистами-практиками в области надежности как в отрасли морского транспорта, так и в других отраслях промышленности.

Ключевые слова: функция надежности, функция риска, рабочий процесс, оптимизация.

Коловроцький К.1, Квашшевска-Сарнецка Б.2, Сошинська-Будний Й.3

'Д-р наук, професор, професор кафедри математики, Морська Академiя в Гдиш, Гдиня, Польща

2Д-р фшософи, доцент, доцент кафедри математики, Морська Академiя в Гдиш, Гдиня, Польща

3Д-р фшософи, ад'юнкт кафедри математики, Морська Академiя в Гдиш, Гдиня, Польща

ОПТИМ1ЗАЦ1Я НАД1ЙНОСТ1 I РИЗИК1В СИСТЕМ З К1ЛЬКОМА СТ1ЙКИМИ СТАНАМИ У ЗАСТОСУВАНН1 ДО ТРАНСПОРТНО1 СИСТЕМИ ПОРТУ

Складшсть процешв роботи техшчних систем та 1хнш вплив на змшу в час структур систем i параметрiв надшност 1хшх компоненпв обумовлюють складнощi при першш зустрiчi у реальности а потсм у фжсаци i аналiзi цих структур i параметрiв надiйностi. Шляхом побудови об'еднано! моделi надiйностi складних технiчних систем в рiзних умовах експлуатацп, що зв'язуе напiвмарковське моделювання процесiв роботи системи з шдходом декiлькох станiв в аналiзi надiйностi систем, ми знаходимо основш характеристики надiйностi системи. Потсм ми використовуемо лiнiйне програмування для того, щоб побудувати модель оптимiзацil надiйностi складних технiчних систем. Ми дослщжуемо застосування моделi в морському транспорту зокрема в оптимiзацil надiйностi та ризикiв об'емно! системи вантажоперевезень. 1нструменти, розробленi нами, можуть бути використанi для оцiнки надшност та оптимiзацil дуже широкого класу реальних техшчних систем, що працюють в рiзних умовах, якi впливають на 1х структуру надiйностi i параметри надiйностi гхшх компонентiв. Отже, розроблеш нами iнструменти можуть бути використаш фах1вцями-практиками в галузi надiйностi як у галузi морського транспорту, так i в шших галузях промисловостi.

Ключовi слова: функщя надiйностi, функцiя ризику, робочий процес, оптимiзацiя.

REFERENCES

1. Aven T. Reliability evaluation of multi-state systems with multistate components, IEEE Transactions on reliability, 1985, Vol. 34, pp. 473-479.

2. Grabski F. Semi-Markov Models of Systems Reliability and Operations. Warsaw, Systems Research Institute, Polish Academy of Science, 2015.

3. Klabjan D., Adelman D. Existence of optimal policies for semi-Markov decision processes using duality for infinite linear programming, Siam. J. Control Optim, 2006, Vol. 44(6), pp. 2104-2122. DOI: 10.1137/s0363012903437290

4. Kolowrocki K. Reliability of large and complex systems. Elsevier, 2014, 460 p. DOI: 10.1016/b978-0-08-099949-4.00010-6

5. Kolowrocki K. Kwiatuszewska-Sarnecka B., Soszynska J. Modelling of operational processes of port bulk cargo system, Proc. 2nd Summer Safety and Reliability Seminars. - SSARS 2008. Gdacsk-Sopot, 2008, Vol. 2, pp. 217-222.

6. Kolowrocki K., Kwiatuszewska-Sarnecka B. Reliability and risk analysis of large systems with ageing components, Reliability Engineering & System Safety, 2008, Vol. 93, pp. 1821-182. DOI: 10.1016/j.ress.2008.03.008

7. Kolowrocki K., Soszynska J. Reliability and availability of complex systems, Quality and Reliability Engineering International, 2006, Vol. 22, Issue 1, pp. 79-99. DOI: 10.1002/qre.749

8. Kolowrocki K., Soszycska-Budny J. Reliability and Safety of Complex Technical Systems and Processes: Modeling-Identification-Prediction-Optimization. London, Springer, 2011, 405 p. DOI: 10.1007/978-0-85729-694-8

9. Kolowrocki K., Kwiatuszewska-Sarnecka B., Soszycska J. Reliability and risk improvement with components quantitative and qualitative redundancy - bulk cargo terminal, Proc. 9nd Summer Safety and Reliability Seminars (SSARS 2015). Gdacsk-Sopot, 2015.

10. Kwiatuszewska-Sarnecka B. Analysis of Reservation Efficiency in Series Systems : thesis ... doctor of philosophy. Gdynia Maritime University, 2003.

11. Kwiatuszewska-Sarnecka B. Reliability Improvement of Large Multi-state Series-parallel Systems, International Journal of Automation and Computing, 2006, Vol. 2, pp. 157-164. DOI: 10.1007/s11633-006-0157-y

12. Lisnianski A., Levitin G. Multi-state System Reliability. Assessment, Optimisation and Applications. Singapore, World Scientific Publishing Co., 2003, 376 p.

13. Meng F. Component-relevancy and characterization in multistate systems, IEEE Transactions on reliability, 1993, Vol. 42, pp. 478-483. DOI: 10.1109/24.257834