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J.Soszyska, P.Dziula, M.Jurdziski, K.KoowrockiOn multi-state safety analysis in shipping - RTA # 3-4, 2007, December - Special Issuep 5 = 0.374, p 6 = 0.372.(35)We assume that the ship subsystems S ,ii = 1,2,...,6,are its five-state components, i.e. z = 4, with the multistatesafety functions( )s b( )( t,⋅)= [1, s b( )( t,1),s b( )( t,2),..., s b( t,z)],i i i ib =1,2,...,6, i = 1,2,...,6,with exponential co-ordinates different in various shipoperation states z , b =1,2,...,6.bAt the operation states z1and z 2 , i.e. at the cargoloading and un-loading state the ship is built ofn1 = n2= 4 subsystems S3,S4, S5and S6forminga series structure shown in Figure 4.Figure 4. The scheme of the ship structure at theoperation states z1and z 2We assume that the ship subsystems S ,ii = 3,4,5,6,are its five-state components, i.e. z = 4, having themulti-state safety functions( )s b( )i( t,⋅)= [1, s b ( )i( t,1),s b( )i( t,2),s b ( t,3),i = 3,4,5,6, b = 1,2,( )is b ( t,4)i],with exponential co-ordinates, for b = 1,2,respectively given by:- for the loading subsystem S 3( )s b ( )( ,1) = exp[−0.06t], s b ( ,2)= exp[−0.07t],3t3t( )s b ( )( ,3) = exp[−0.08t], s b ( ,4)= exp[−0.09t],3t- for the hull subsystem S 43t( )s b ( )( ,1) = exp[−0.03t], s b ( ,2)= exp[−0.04t],4t4t( )s b ( )4( t,3)= exp[−0.06t], s b4( t,4)= exp[−0.07t],- for the protection and rescue subsystem S 5( )s b ( )( ,1) = exp[−0.10t], s b ( ,2)= exp[−0.12t],5t5t( )s b ( )( ,3) = exp[−0.15t], s b ( ,4)= exp[−0.16t],5tS 3 S 4 S 5 S 65t- for the anchor and mooring subsystem S 6( )s b ( )( ,1) = exp[−0.06t], s b ( ,2)= exp[−0.08t],6t6t( )s b ( )( ,3) = exp[−0.10t], s b ( ,4)= exp[−0.12t].6t6tAssuming that the ship is in the safety state subsets{ u , u + 1,..., z}, u = 1,2,3,4 , if all its subsystems are inthis safety state subset, according to Definition 1 andDefinition 4, the considered system is a five-stateseries system. Thus, by Corollary 3, after applying(10)−(11), we have its conditional safety functions inthe operation states z1and z2respectively forb = 1,2, given by( b)s4 ( t,⋅)( b)( b)= [1, s4( t,1),s4( t,2),( b)( b)s4( t,3),s4( t,4)],t ≥ 0, b = 1,2,wheressss( b)4t( b)4t( ,1) = exp[−(0.06 + 0.03 + 0.10 + 0.06)t]= exp[ −0.25t],( ,2) = exp[-(0.07 + 0.04 + 0.12 +0.08)t]( b)4t( b)4t= exp[ −0.31t],( ,3) = exp[−(0.08 + 0.06 + 0.15 + 0.10)t]= exp[ −0.39t],( ,4) = exp[−(0.09 + 0.07 + 0.16 + 0.12)t]= exp[ −0.44t]for t ≥ 0, b = 1, 2 .The expected values and standard deviations of theship conditional lifetimes in the safety state subsetscalculated from the above result, according to (16)-(17), for b = 1,2,are:(b)(b)m (1) ≅ 4.00, (2)(b)m (4)≅ 2.27 years,(b)(b)σ (1) ≅ 4.00, (2)(b)m ≅ 3.26, m (3)≅ 2.56,(b)σ ≅ 3.26, σ (3)≅ 2.56,- 47 -