66 C. Gong et al. In this paper, an attempt is made to incorporate CPT in the corroding ship hull structure maintenance planning to explore the optimal solution to decision-makers’ choices from the perspective of risk perception. To this end, the optimal maintenance is investigated by maximizing the preference characterized by CPT. The design variable selected is the inspection time. It is assumed that ship components are repaired after inspection if their corrosion depth reaches the allowance of the Classification Societies Rules [13]. The uncertainties associated with material strength, geometry, corrosion growth rate, bending resistance model, and external loadings are considered. The distributions of ship life-cycle maintenance and failure costs are computed using Monte Carlo simulation, and CPT preference is evaluated on this basis. 7.2 Cumulative Prospect Theory CPT consists of a value function and a probability weighting function. The value function describes the desirability of the solution outcome (e.g. cost). Contrary to the utility function considering the consistent risk attitude for all outcomes, the value function is convex for losses and concave for gains, in agreement with the observed people’s risk-prone and risk-averse behaviors for losses and gains, respectively. Losses and gains are classified according to a reference point, which represents decision-makers’ expectation of future outcomes [14]. The weighting function is inverse S-shaped, reflecting the fact that people subjectively overweight small probabilities and underweight moderate and high probabilities in the decision-making process [11]. Consider a set of possible monetary outcomes relative to the reference point in ascending order G=(x1, p1; . . . , xn, pn), where pi denotes the probability of the i-th monetary outcome xi (i =1, 2, . . . , n) with n i=1 pi = 1. According to CPT, decision-makers choose an alternative that maximizes the expected prospect value [11]: E[V] = m j =1 π− j v xj + n k =m+1 π+ k v(xk) (7.1) where xj (j =1, . . . , m) ≤0; xk (k =m+1, . . . , n) ≥0; v(•) is the prospect function; π− j (j =1, . . . , m) andπ+ k (j =m+1, . . . , n) are decision weights for losses and gains, respectively. The decision weights are [11] π−1 =w−(p1),π+n =w+(pn) (7.2a) π− j =w− p1 +· · ·+pj −w− p1 +· · ·+pj−1 , if 1 <j ≤m (7.2b) π+ k =w+(pk +· · ·+pn) −w+(pk+1 +· · ·+pn), if m<k <n (7.2c) where w+(·) and w−(·) are the weighting functions for gains and losses, respectively. Tversky and Kahneman [11] suggests probabilities weighting functions w+: [0, 1] →[0, 1] for gains and w−: [0, 1] →[0, 1] for losses, respectively, as w+(·) = (·)b+ (·)b+ +(1−(·))b+ 1/b+ (7.3a) w−(·) = (·)b− (·)b− +(1−(·))b− 1/b− (7.3b) withb− and b+ ∈(0,1). The value function is defined as v(x) = xa, if x ≥0 −λ(−x)β, if x <0 (7.4) whereλspecifies loss aversion degree withλ≥1; α, β∈(0, 1) are the exponential parameters. λis introduced to reflect more sensitivity to losses than gains. Tversky and Kahneman [11] found that α =β =0.88, λ=2.25, b− =0.69, andb+ =0.61.
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