7 Application of Cumulative Prospect Theory to Optimal Inspection Decision-Making for Ship Structures 69 (b) for hogging condition Mw,max =0.19CwvL 2 BCb KNm (7.13) For a period of one year, in-service ships experience many peak cycles. It follows that the annual maximum wave-induced bending moment is described by Gumbel distribution FMwe (mwe) =exp −exp − mwe −λ0 θ0 k (7.14) where λ0 and θ0 are the characteristic value and scale parameter of Gumbel distribution, respectively. The parameters of Gumbel distribution are [18] λ0 =w ln acTr Tw k and θ0 = w k ln acTr Tw 1−k k (7.15) where Tr is the considered reference time, i.e., Tr =1 year; Tw is the average wave period, i.e., Tw =8.0 s; ac is a factor accounting for the time fraction of a load condition, e.g., ac =0.35 for a full load condition. 7.4 Ship Life-Cycle Cost Within the considered service life, the life-cycle cost of ship hull girder is C0 = nin l =1 Cin (1+v0) Tin,l + nrep l =1 Crep (1+v0) Trep,l + nf l =1 Cf (1+v0) Tf,l (7.16) where v0 denotes the discount rate; nin, nrep, and nf are the number of inspections, repair, and failure events, respectively; Tin,l, Trep,l, and Tf,l are the time of the lth inspection, repair, and failure, respectively; Cin, Crep, and Cf are the costs of inspection, repair, and ship failure, respectively. In the CPT assessment, the overall monetary cost is expressed as CR =R− C0,where Rdenotes the reference cost. The cost of repairing a ship structural element includes material, welding, labor, and electricity costs, while the failure cost includes the direct cost of the ship structure and cargo loss, as well as indirect costs of environment pollution and spillage clean-up, and loss of human life. A detailed repair cost estimate is presented in [19]. 7.5 Optimization Framework 7.5.1 Repair Policy The detailed inspection of the ship hull girder is periodically conducted on an empirical basis of five years [20]. For inspection, the ship is transported to a dry dock and the net thickness of the ship’s steel plates are measured by ultrasound equipment. Classification Societies specifies that corroded structural elements are replaced with new ones if the renewal criterion is satisfied [13]: tm < tas-built −twas, where tas-built is the as-built thickness; tm is the measured thickness; twas is the double side waste allowance, rounded up the nearest 0.5 mm. The as-built thickness is the sum of the structural design thickness plus corrosion addition. Corrosion addition adds a safety margin against corrosion. Since distinct environment exists across the mid-ship section, the corrosion growth rates of structural elements of the deck, sides, and bottom plating are different. To account for this, corrosion wastage allowance given by Classification Societies is specific to the location of the mid-ship section [13].
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