Indices

t

time period

$c{c}^{\prime }{c}_t$

cells

v

vessel

g

tugboat

$p_i$

path; $p_i=\left(c_1\mathrm{,}\cdots \mathrm{,}{c}_T\right)$

${\omega }_v$

scenario for vessel v; ${\omega }_v=\left(t\mathrm{,}{p}_i\right)$

$\overline{\omega }$

scenario for all vessels $\overline{\omega }=\left({\omega }_1\mathrm{,}\cdots \mathrm{,}{\omega }_v\right)$

Sets

$\mathcal{C}$

set of cells

$\mathcal{V}$

set of vessels

$\mathcal{G}$

set of tugboats

$\mathcal{T}$

set of time period

$\mathcal{F}\left(c\right)\subseteq \mathcal{C}$

set of cells adjacent to cell c

${\mathcal{P}}_{{\omega }_{v}t}$

set of paths for vessel scenario ${\omega }_v=\left(t\mathrm{,}{p}_i\right)$

${\Omega }_v$

set of scenarios for vessel v

$\overline{\Omega }$

set of all possible scenarios $\overline{\Omega }={\Omega }_1\times \cdots \times {\Omega }_v$

Parameters

$K_{{\omega }_v}$

environmental consequence associated with vessel v in scenario ${\omega }_v$

$R_{{\omega }_v}$

failure probability for vessel scenario ${\omega }_v$

$R_{\overline{\omega }}$

probability for scenario $\overline{\omega }=\left({\omega }_1\mathrm{,}\cdots \mathrm{,}{\omega }_v\right)$ , $R_{\overline{\omega }}={\displaystyle {\prod }_{v\in \mathcal{V}}}{R}_{{\omega }_v}$

$Q_{gc{\omega }_v}$

probability of successful hook-up by tugboat g with vessel v, given

tugboat g is in cell c at time of distress call t and vessel v follows

scenario ${\omega }_v=\left(t\mathrm{,}{p}_i\right)\mathrm{,}{p}_i\in {\mathcal{P}}_{{\omega }_{v}t}$

$c_{0g}$

initial position of tugboat g

Variables

$x_{gct}$

binary variable taking the value 1 if tugboat g is in cell c at time t,

0 otherwise

$z_{gc{\omega }_v}$

binary variable takes the value 1 if tugboat g is in cell c at time of distress

call t and is allocated (nearest) to vessel v doing scenario ${\omega }_v$