## 3 Problem Definition and Optimization Model
### Model Formulation
#### 3.2.2 Symbol Description  
To promote the foundation of the model of PAVP, the descriptions of the symbols used are listed in Tables 2-4:  
\begin{table}
\begin{tabular}{c l} \hline \hline
**Symbol** & **Descriptions** \\ \hline \(p_{i}\) & Status of valve \(j\) \\ \(qd_{i}\) & Status of pipeline from valve \(i\) to valve \(j\) \\ \(\omega(d_{i})\) & Pipeline path from valve \(i\) to valve \(j\) of vessel \(a\) \\ \(\omega(d_{i})\) & Connection status of pipeline from valve \(i\) to valve \(j\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Symbol descriptions of 0–1 variables.  
\begin{table}
\begin{tabular}{c l} \hline \hline
**Symbol** & **Descriptions** \\ \hline \(A\) & Set of vessels, A = {1, 2, 3,..., a}, \\ \(B\) & Set of berths, B = {1, 2, 3,..., b}, \\ \(C_{b}\) & Set of oil transfer arms of berth \(b\), C\({}_{b}\) = {1, 2, 3,..., c}, \\ \(D\) & Set of pipelines, D = {1, 2, 3,..., d}, \\ \(E\) & Set of tanks, E = {1, 2, 3,..., e}, \\ \(F\) & Set of valves, \\ \(G\) & Set of cargoes, G = {1, 2, 3,..., g}. \\ \hline \hline \end{tabular}
\end{table}
Table 2: Symbol descriptions of set.  
\begin{table}
\begin{tabular}{c l} \hline \hline
**Symbol** & **Descriptions** \\ \hline \(ATA_{a}\) & Actual time of arrival of vessel \(a\) \\ \(C\) & Total cost of vessels of a plan, unit: USD \\ \(C_{1}\) & Cost of oil transfer arms, unit: USD \\ \(C_{2}\) & Cost of pipelines, unit: USD \\ \(C_{3}\) & Cost of valves, unit: USD \\ \(Cf_{i}\) & Cost of operations of valve \(j\), unit: USD/unit \\ \(Cl\) & Fixed cost of pipelines occupation, unit: USD/m \\ \(C_{s}\) & Fixed cost of oil transfer arms handling, unit: USD/unit \\ \(C_{t}\) & Fixed cost of oil transfer arms occupation, unit: USD/h/unit \\ \(D_{s}\) & Pipeline section, unit: m\({}^{2}\) \\ \(ECf_{j}\) & Energy consumption of valve \(j\) \\ \(ETD_{a}\) & Estimated time of departure of vessel \(a\) \\ \(sa_{a}\) & Type of liquid cargo of vessel \(a\) \\ \(Se_{e}\) & Type of liquid cargo of tank \(e\) \\ \(Sd\) & Type of liquid cargo of pipeline \(d\) \\ \(L_{a}\) & Total length of operation pipeline of vessel \(a\), unit: m \\ \(l_{ij}\) & Pipeline length from valve \(i\) to valve \(j\), unit: m \\ \(N_{ca}\) & Quantity of oil transfer arm occupation \\ \(N_{bc}\) & Quantity of oil transfer arm in berth \(b\) \\ \(n_{iF}\) & Operation times of valve \(j\), \(j\)\(\in\)\(F\) \\ \(T\) & Total time of vessels of a plan, unit: h \\ \(T_{c}\) & Preparation and handling time of oil transfer arm, unit: h \\ \(T_{ca}\) & Unloading time of cargo of vessel \(a\), unit: h \\ \(T_{c(in)}\) & The moment when the oil tank \(e\) starts to feed oil \\ \(T_{c(inx)}\) & The moment when the oil tank \(e\) reaches its maximum storage capacity \\ \(T_{c(inx)}\) & The moment when the oil tank \(e\) reaches its minimum storage capacity \\ \(T_{c(out)}\) & The moment when the oil tank \(e\) starts to drain oil \\ \(T_{pa}\) & Pipeline oil circuit emptying time, unit: h \\ \(v\) & Maximum passable flow velocity of pipeline, unit: m/s \\ \(V_{a}\) & Standard flow rate of oil transfer arm of vessel \(a\), unit: m\({}^{3}\)/h \\ \(V_{ca}\) & Unloading actual flow rate of oil transfer arm of vessel \(a\), unit: m\({}^{3}\)/h \\ \(W_{a}\) & Liquid cargo unloading volume of vessel \(a\), unit: m\({}^{3}\) \\ \(W_{eu}\) & Vacant space of tank \(e\), unit: m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Symbol descriptions of parameters.