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SAFETY OPTIMIZATION OF A FERRY TECHNICAL SYSTEM IN VARIABLE
OPERATION CONDITIONS
Krzysztof Kolowrocki, Joanna Soszynska-Budny
Gdynia Maritime University, Gdynia, Poland e-mail: katmatkk@am.gdynia.pl, joannas@am.gdynia.pl
ABSTRACT
The general model of the safety of complex technical systems in variable operation conditions linking a semi-Markov modeling of the system operation process with a multi-state approach to system safety analysis and linear programming are applied in maritime transport to safety and risk optimization of a ferry technical system.
1 INTRODUCTION
Most real technical systems are very complex and it is difficult to analyze and optimize their safety. Large numbers of components and subsystems and their operating complexity cause that the evaluation and optimization of their safety is complicated. The complexity of the systems' operation processes and their influence on changing in time the systems' structures and their components' safety characteristics is often very difficult to fix and to analyze. A convenient tool for solving this problem is a semi-markov (Grabski 2002) modeling of the system operation processes linked with a multi-state approach to the system safety analysis (Kolowrocki, Soszynska 2008, Kolowrocki, Soszynska 2009) and a linear programming for the system safety optimization (Kolowrocki, Soszynska 2010). This approach to system safety investigation is based on the multi-state system reliability analysis considered for instance in (Aven 1985, Kolowrocki 2004) and on semi-markov operation processes modeling discussed for instance in (Soszynska 2006, Soszynska 2007). An application of the proposed approach to safety analysis and optimization of maritime ferry technical system is presented in this paper.
2 THE FERRY TECHNICAL SYSTEM SAFETY AND RISK
The considered maritime ferry is a passenger Ro-Ro ship operating in Baltic Sea between Gdynia and Karlskrona ports on regular everyday line. We assume that the ferry is composed of a number of main subsystems having an essential influence on its safety. These subsystems are illustrated in Figure 1.
On the scheme of the ferry presented in Figure 1, there are distinguished its following subsystems:
S1 - a navigational subsystem, S 2 - a propulsion and controlling subsystem,
53 - a loading and unloading subsystem,
54 - a hull subsystem,
55 - an anchoring and mooring subsystem,
56 - a protection and rescue subsystem,
57 - a social subsystem.
In our further analysis of the ferry safety we omit the protection and rescue subsystem S 6 and the social subsystem S7 and we consider its strictly technical subsystems S1, S2, S3, S4 and S5 only, further called the ferry technical system (Kolowrocki, Soszynska 2009).
Figure 1. Subsystems having an essential influence on the ferry technical system safety
We assume that the ferry technical system safety structure and the subsystems and components safety depend on its changing in time operation states (Kolowrocki, Soszynska 2010).
Taking into account the experts' opinion on the operation process of the considered ferry, we distinguish the following as its eighteen operation states:
- an operation state z1 - loading at Gdynia Port,
- an operation state z2 - unmooring operations at Gdynia Port,
- an operation state z3 - leaving Gdynia Port and navigation to "GD" buoy,
- an operation state z4 - navigation at restricted waters from "GD" buoy to the end of Traffic Separation Scheme,
- an operation state z5 - navigation at open waters from the end of Traffic Separation Scheme to "Angoring" buoy,
- an operation state z6 - navigation at restricted waters from "Angoring" buoy to "Verko" Berth at Karlskrona,
- an operation state z7 - mooring operations at Karlskrona Port,
- an operation state z8 - unloading at Karlskrona Port,
- an operation state z9 - loading at Karlskrona Port,
- an operation state z10 - unmooring operations at Karlskrona Port,
- an operation state z11 - ferry turning at Karlskrona Port,
- an operation state z12 - leaving Karlskrona Port and navigation at restricted waters to "Angoring" buoy,
- an operation state z13 - navigation at open waters from "Angoring" buoy to the entering Traffic Separation Scheme,
- an operation state z14 - navigation at restricted waters from the entering Traffic Separation Scheme to "GD" buoy,
- an operation state z15 - navigation from "GD" buoy to turning area,
- an operation state z16 - ferry turning at Gdynia Port,
- an operation state z17 - mooring operations at Gdynia Port,
- an operation state z18 - unloading at Gdynia Port.
Additionally, as in (Kolowrocki, Soszynska 2009, 2010), we assume that subsystems Sv, v = 1,2,...,5, of the ferry technical system are composed of five-state, components, and their safety states are 0,1,2,3 and 4. Consequently the components conditional multi-state safety function is the vector (Kolowrocki, Soszynska 2009)
[ s(v)(t, -)r= [1, [ s(v)(t,1)](b),[ s(v)(i ,2)](b ),[s(v)(t ,3)](b),[ s(v) (t ,4)](b) ], b = 1,2,...,18, with the exponential co-ordinates
[s(v)(t ,1)](b) = exp[-[^^v)(1)](b) t], [ s(v) (t, 2)](b) = exp[-[4">(2)](b) t],
[ s(v) (t, 3)](b) = exp[-[4">(3)](b) t], [ s(v) (t, 4)](b) = exp[ -[^(4)](b) t], b = 1,2,...,18,
Further, assuming that the ferry technical system is in the safety state subset {u, u +1,...,4} u = 0,1,2,3,4, if all its subsystems are in this subset o safety states, we conclude that the ferry is five-state series system (Kolowrocki, Soszynska 2009) of subsystems S1, S2, S3, S4, S5 and S6.
The ferry operation process is very regular in the sense that the operation state changes are from the particular state zb, b = 1,2,...,17, to the neighboring state zb+1, b = 1,2,...,17, and from z18
to Zj only. Therefore, the probabilities of transitions between the operation states are given by (Kolowrocki, Soszynska 2009)
'010...00 001...00
[Pbl ] =
' 000...01 100...00
On the basis of statistical data coming from experts the matrix of the density functions of the ferry technical system operation process conditional sojourn times 0bl b,l = 1,2,...,18, defined in (Kolowrocki, Soszynska 2009), can be evaluated.
Next, the mean values Mbl = E[0bl ], b, l = 1,2,...,18, b ^ l, of the system operation process conditional sojourn times dbl in particular operation states can be determined and they are:
M12 = 54.67, M23 = 2.57, M34 = 37.33, M45 = 52.27, M56 = 526.43, M67 = 37.16, M78 = 7.02, M89 = 23.26, M910 = 53.69, M1011 = 2.86, Mm2 = 4.38, M1213 = 24.12, M1314 = 508.60, M1415 = 50.14, M1516 = 34.43, Mieil = 4.59, Mllw = 7.92, M^ = 18.74.
Hence, according to (2) (Kolowrocki, Soszynska 2010), the mean values of the unconditional sojourn times in the operation states are:
Mj = 54.67, M2 = 2.57, M3 = 37.33, M4 = 52.27, M5 = 526.43, M6 = 37.16, M7 = 7.02,
M8 = 23.26, M9 = 53.69, M10 = 2.86, M11 = 4.38, M12 = 24.12, M13 = 508.60, M14 = 50.14,
M15 = 34.43, M16 = 4.59, M17 = 7.92, M18 = 18.74.
Since from the system of equations given in (Kolowrocki, Soszynska 2009, 2010) taking here the form
we get
7Tb = 0.056 for b = 1,2,...,18.
Thus, according to the results contained in (Kolowrocki, Soszynska 2009, 2010), the long term proportion of transients pb at the operational states zb, can be approximated by
p1 = 0.038, p2 = 0.002, p3 = 0.026, p4 = 0.036, p5 = 0.363, p6 = 0.026, p7 = 0.005,
p8 = 0.016, p9 = 0.037, p10 = 0.002, p11 = 0.003, p12 = 0.016, p13 = 0.351, p14 = 0.034,
Under the assumption that the changes of the ferry operation states have an influence on the subsystems Sv, v = 1,2,...,5, components safety and on the ferry technical system safety structures
as well, on the basis of expert opinions and statistical data given in (Soszynska et al. 2009), the ferry technical system safety structures and their components safety functions and the ferry technical system conditional safety functions at different operation states can be determined (Kolowrocki, Soszynska 2010). Namely, in the case when the system operation time is large enough, the unconditional fife-state safety function of the ferry technical system is given by the vector
K Lis = K List Pbi]
bl J18xis
p15 = 0.024, p16 = 0.003, p17 = 0.005, p1s = 0.013.
(1)
s(t, •) = [1, s(t,1), s(t, 2), s(t, 3), s(t, 4)], t > 0,
(2)
where, after considering the values of pb, b = 1,2,...,18, given by (1), its co-ordinates are
s(t, u ) = 0.038 •[ s(t, u )](1) + 0.002 • [s(t, u )](2) + 0.026 • [s(t, u)](3) + 0.036 • [ s(t, u )](4)
+ 0.363 • [ s(t, u)](5) + 0.026 •[ s(t, u)](6) + 0.005 •[ s(t, u)](7) + 0.016 • [ s(t, u)](s)
+ 0.037 • [s(t, u)](9) + 0.002 • [s(t, u)](10) + 0.003 •[s(t, u)](11) + 0.016 [s(t, u)](12)
+ 0.351 •[s(t, u)](13) + 0.034 • [s(t, u)](14) + 0.024 [s(t,u)](15) + 0.003 •[s(t, u)](16)
0.005 •[ $ (t, u )](17) + 0.013 •[ s(t, u)]
(18)
(3)
for t > 0, u = 1,2,3,4, where [s(t, u)](b), b = 1,2,...,18, are the system conditional safety functions at particular operation states zb, b = 1,2,...,18, determined in (Kolowrocki, Soszynska 2010). The safety function of the ferry technical system is presented in Figure 2
Figure 2. Graph of the safety function [s(t, •)] coordinates
From (3), the mean values of the ferry technical system unconditional lifetimes in the safety state subsets {1,2,3,4}, {2,3,4}, {3,4}, {4} respectively are:
/j(1) = 4.70 years,
ju(2) = 0.038 • 6.45 + 0.002 • 2.43 + 0.026 • 3.9 + 0.036 • 3.80 + 0.363 • 3.80 + 0.026 • 3.24 + 0.005 • 2.43 + 0.016 • 7.69 + 0.037 • 7.69 + 0.002 • 2.43 + 0.003 • 3.37 + 0.016 • 3.80 + 0.351 • 3.80 + 0.034 • 3.80 + 0.024 • 3.90 + 0.003 • 3.37 + 0.005 • 2.43 + 0.013 • 6.45 = 4.11 years,
/u(3) = 3.66 years, /(4) = 3.29 years.
(4)
From the above (Kolowrocki, Soszynska 2009, 2010), the mean values of the system lifetimes in the particular safety states are:
^(1) = ju(1) - ju(2) = 0.59, ^(2) = /u(2) - /u(3) = 0.77 years,
/(3) = /(3) -/(4) = 0.45, /(4) = /(4) = 2.29 years.
(5)
If we assume that the critical safety state is r = 2, then the system risk function (Kolowrocki, Soszynska 2009, 2010), is given by
r(t) = 1 - s(t, 2)
(6)
where s(t, 2) is given by (3) for u = 2.
The moment when the system risk function exceeds a permitted level, for instance 8 = 0.05,
is
t= r~1(8) = 0.21 year. (7)
Figure 3. Graph of the ferry technical system risk function r (t) 3 OPTIMIZATION OF THE FERRY TECHNICAL SYSTEM OPERATION PROCESS
Considering the equation (3), it is natural to assume that the system operation process has a significant influence on the system safety. This influence is also clearly expressed in the equation (4), for the mean values of the system unconditional lifetimes in the safety state subsets.
The objective function defined in (Kolowrocki, Soszynska 2010), in this case as the system critical state is r = 2 , takes the form
¡u(2) = p1 • 6.45 + p2• 2.43 + p3 • 3.90 + p4 • 3.80 + p5 • 3.80 + p6 • 3.24 + p7• 2.43 + p^ 7.69 + p9 • 7.69 + p10• 2.43 + p11 • 3.37 + p12 • 3.80 + p13^3.80 + p14 • 3.80 + p15^3.90 + p16• 3.37 + p17• 2.43 + p18 • 6.45. (8)
Since the lower pb and upper pb bounds of the unknown transient probabilities pb, b = 1,2,...,18, coming from experts are respectively:
p1 = 0.0006, p2 = 0.001, p3 = 0.018, p4 = 0.027, p5 = 0.286, p6 = 0.018,
p7 = 0.002, p 8 = 0.001, p 9 = 0.001, p 10 = 0.001, p11 = 0.002, p 12 = 0.013,
p13 = 0.286, p14 = 0.025, p 15 = 0.018, p16 = 0.002, p17 = 0.002, p18 = 0.001,
p1 = 0.056, p 2 = 0.002, p3 = 0.027, p4 = 0.056, p5 = 0.780, p6 = 0.024,
p7 = 0.018, p8 = 0.018, p9 = 0.056, p10 = 0.003, = 0.004, p12 = 0.024,
p13 = 0.780, p14 = 0.043, p15 = 0.024, p16 = 0.004, p17 = 0.007, p18 = 0.018, then we assume, for the objective function defined by (8), the following bounds constraints
0.0006 < p1 < 0.056, 0.001 < p2 < 0.002, 0.018 < p3 < 0.027, 0.027 < p4 < 0.056, 0.286 < p5 < 0.780, 0.018 < p6 < 0.024, 0.002 < p7 < 0.018, 0.001 < p8 < 0.018, 0.001 < p9 < 0.056, 0.001 < p10 < 0.003, 0.002 < p11 < 0.004, 0.013 < p12 < 0.024, 0.286 < p13 < 0.780, 0.025 < p14 < 0.043, 0.018 < p15 < 0.024, 0.002 < p16 < 0.004, 0.002 < p17 < 0.007, 0.001 < p18 < 0.018,
(9)
18
Z pb = 1,
b=1
Now, in order to find the optimal values pb of the transient probabilities pb, b = 1,2,...,18, that maximize the objective function (8), w arrange the system conditional lifetimes mean values /b (2), b = 1,2,...,18, in non-increasing order
/ (2) > /U9 (2) > / (2) > / (2) > /U3 (2) > /U15 (2) > / (2) > /U5 (2) > ^ (2)
> /U13 (2) > (2) > (2) > ^16 (2) > /u6 (2) > ^2 (2) > /un (2) > /uV) (2) > (2),
and we substitute
x1 = p 8 = 0.016, x 2 = p9 = 0.037, x3 = p1 = 0.038, x4 = p18 = 0.013,
x 5 = p3 = 0.026, x6 = p15 = 0.024, x7 = p4 = 0.036, x8 = p 5 = 0.363,
x 9 = p12 = 0.016, x10 = p13 = 0.351, x11 = p14 = 0.034, x12 = p11 = 0.003,
x13 = p16 = 0.003, x14 = p 6 = 0.026, x15 = p 2 = 0.002, x16 = p 7 = 0.005,
x17 = p10 = 0.002, x18 = p17 = 0.005. (10)
Afterwards, we maximize with respect to xi, i = 1,2,...,18, the linear form (8) that after considering the substitution (10) takes the form
/u(2)= x1 • 7.69 + x2 • 7.69 + x3 • 6.45 + x4 • 6.45 + x5 • 3.90 + x6 • 3.90
+ x7 • 3.80 + x8 • 3.80 + x9 • 3.80 + x10 • 3.80 + xn • 3.80 + x12 • 3.37 + x13 • 3.37 + x14 • 3.24 + x15 • 2.43 + x16 • 2.43 + x17 • 2.43 + x18 • 2.43, (11)
with the following bound constraints
0.001 < x1 < 0.018, 0.001 < x2 < 0.056, 0.0006 < x3 < 0.056,
0.001 < x4 < 0.018, 0.018 < x5 < 0.027, 0.018 < x6 < 0.024, 0.027 < x7 < 0.056, 0.286 < x8 < 0.780, 0.013 < x9 < 0.024, 0.286 < x10 < 0.780, 0.025 < x11 < 0.043, 0.002 < x12 < 0.004, 0.002 < x13 < 0.004, 0.018 < x14 < 0.024, 0.001 < x15 < 0.002,
0.002 < x16 < 0.018, 0.001 < x17 < 0.003, 0.002 < x18 < 0.007,
18
Z xi = 1.
i=1
Further, according to the procedure given in (Kolowrocki, Soszynska 2010), we calculate
18
x = Z xt = 0.7046, € = 1 - x = 1 - 0.7046 = 0.2954 (12)
i=1
and we find
x0 = 0, X0 = 0, x0 -x0 = 0, x1 = 0.001 x1 = 0.018, xc1 - x1 = 0.017 x2 = 0.002, x2 = 0.074, x2 - x2 = 0.072, x3 = 0.0026, x3 = 0.13, x3 - x3 = 0.1274, x4 = 0.0036, x4 = 0.148, x4 - x4 = 0.1444, x5 = 0.0216, x5 = 0.175, x5 - x5 = 0.1534, x6 = 0.0396, x6 = 0.199, x6 - x6 = 0.1594, x7 = 0.0666, x7 = 0.255, x7 - x7 = 0.1884,
x8 = 0.3526, x8 = 1.035, x8 - x8 = 0.6824. (13)
From the above, as according to (13), after considering the inequality
x1 - x1 < 0.295, (14)
it follows that the largest value I e {0,1,...,18} such that this inequality holds is I = 7.
Therefore, we fix the optimal solution that maximize linear function (11) according to the rule given in (Kolowrocki, Soszynska 2010). Namely, we get
x1 = x1 = 0.018, x2 = x 2 = 0.056, x3 = x3 = 0.056, x 4 = x 4 = 0.018, x5 = x5 = 0.027, x6 = x6 = 0.024, x7 = x7 = 0.056, xx8 = € - x7 + x7 + x8 = 0.2954 - 0.255 + 0.0666 + 0.286 = 0.393, x9 = x9 = 0.013, x10 = x10 = 0.286, xx11 = x11 = 0.025, xx12 = x12 = 0.002, xx13 = x13 = 0.002, x14 = x14 = 0.018, xx15 = x15 = 0.001, xx16 = x16 = 0.002, xx17 = x17 = 0.001, xx18 = x18 = 0.002. Finally, after making the substitution inverse to (10), we get the optimal transient probabilities p 8 = xc1 = 0.018, p9 = x2 = 0.056, p1 = x3 = 0.056, p18 = xx4 = 0.018,
p3 = xx5 = 0.027, p15 = x6 = 0.024, p4 = xx7 = 0.056, p5 = x8 = 0.393, P12 = x9 = 0.013, p13 = xx10 = 0.286, p14 = xx11 = 0.025, pn = xx12 = 0.002,
p16 = xx13 = 0.002, p6 = xc14 = 0.018, p 12 = xx15 = 0.001, p7 = x16 = 0.002,
p 10 = xn = 0.001, p 17 = x18 = 0.002, (15)
that maximize the system mean lifetime in the safety state subset {2,3,4} expressed by the linear form (8) giving its optimal value
//(2) = 0.056 • 6.45 + 0.001-2.43 + 0.027 • 3.90 + 0.056 • 3.80 + 0.393 • 3.80
+ 0.018 • 3.24 + 0.002 • 2.43 + 0.018 • 7.69 + 0.056 • 7.69 + 0.0012.43
+ 0.002 • 3.37 + 0.013 • 3.80 + 0.286 • 3.80 + 0.025 • 3.80 + 0.024 • 3.90
+ 0.002 • 3.37 + 0.002 • 2.43 + 0.018 • 6.45 = 4.27. (16)
4 THE FERRY TECHNICAL SYSTEM OPTIMAL SAFETY CHARACTERISTICS
Further, using the optimal transient probabilities (15), we obtain the optimal solution for the mean value of the system unconditional lifetime in the safety state subset {1,2,3,}, {3,4} and {4}
ß(1) = 4.92, ß(3 =)3.79, ß(4) = 3.42,
(17)
and the optimal solutions for the mean values of the system unconditional lifetimes in the particular safety states 1,2,3 and 4 are as follows
ß(1) = 0.65, ß(2) = 0.48, ß(3) = 0.37, ß(4) = 3.42.
(18)
Moreover, according to (2), the corresponding optimal unconditional multistate safety function of the system is of the form of the vector
s(t,) = [1, s(t,1),s(t,2), s(t,3),s(t,4)], t > 0, with the coordinates given by
s(t,1) = 0.056 •[ s(t, u )](1) + 0.001-[ s(t, u)](2) + 0.027 • [ s(t,u)](3) + 0.056 • [ s(t, u )](4) + 0.393 • [ s(t, u)](5) + 0.018 •[ s(t, u)](6) + 0.002 -[s(t, u)](7) + 0.018 - [s(t, u)](8) + 0.056 • [s(t, u)](9) + 0.001 • [ s(t, u )](10) + 0.002 •[ s(t, u)](11) + 0.013 •[ s(t, u)](12) + 0.286 -[s(t, u)](13) + 0.025 • [s(t, u)](14) + 0.024 •[s(t, u)](15) + 0.002 {s(t,u)](16) + 0.002 •[ s (t, u)](17) + 0.018 •[ s(t, u)](18),
(19)
(20)
for t > 0, u = 1,2,3,4, where [s(t,u)](b), b = 1,2,...,18, are the system conditional safety functions at particular operation states zb, b = 1,2,...,18, given in (Kolowrocki, Soszynska 2010). The safety function of the ferry technical system is presented in Figure 4.
Figure 4. Graph of the optimal safety function [i(t, •)] coordinates
If the critical safety state is r = 2, then the system risk function (Kolowrocki, Soszynska 2010) is given by
r(t) = 1 - s(t,2) for t > 0,
(21)
where s(t,2) is given by (20) for u = 2.
Hence, the moment when the optimal system risk function exceeds a permitted level, for instance 8 = 0.05, is
i = r -1 (5) = 0.22 year.
(22)
Figure 5. The graph of the ferry technical system optimal risk function r(t)
The comparison of the ferry technical system safety characteristics after its operation process optimization given by (16)-(22) with the corresponding characteristics before this optimization determined by (2)-(7) justifies the sensibility of this action.
5 CONCLUSION
The joint model of the safety 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 safety analysis was applied to the maritime ferry technical system safety characteristics evaluation. Next, the final results obtained from this joint model and a linear programming were used to perform this complex technical system safety optimization. These tools practical application to safety and risk evaluation and optimization of a technical system of a ferry operating in variable operation conditions at the Baltic Sea waters and the results achieved are interesting for safety practitioners from maritime transport industry and from other industrial sectors as well.
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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