## 4 Proposed Method
### Solving the Proposed Set Covering Model  
Typically, the classical GSRSP-BT model is quite difficult to solve directly via exact methods. The classical model must be reformulated into a set covering model that can find the route within a specified time window to allow possible solutions to be found with faster computational times. This problem has two objective functions: to minimize the total amount of CO2e emissions and to minimize the total travel time. \(S\) is defined as a set of time periods (\(s\in S\)) and \(S_{\text{typ}}\) is a set of time periods for job sequence \(v\) operating in port p (\(s_{\text{typ}}\in S_{\text{typ}}\), \(S_{\text{typ}}\subset S\)). The model assumes that the travel times and service times are the same in each period. In this step, we developed the port capacity constraint to avoid time clashes when barge loading and unloading in port p in period s using binary parameters \(\eta_{\text{typ}}^{load}\) and \(\eta_{\text{typ}}^{unload}\) for multiple berths in a port. The proposed set covering model is as follows:  
\[Minimize\ f_{1}=\sum_{r\in r}COE_{r}x_{r} \tag{12}\]  
\[Minimize\ f_{2}=\sum_{r\in R}TT_{r}x_{r} \tag{13}\]  
subject to  
\[\sum_{r\in R}x_{r}n_{ri}=1\quad\forall i\in I \tag{14}\]  
\[\sum_{r\in R}x_{r}\leq N \tag{15}\]  
\[\sum_{r\in R}\sum_{p\in V_{r}}x_{r}\eta_{\text{typ}}^{load}+\sum_{r\in R}\sum _{p\in V_{r}}x_{r}\eta_{\text{typ}}^{unload}\leq K_{sp}\quad\forall p\in P,\,\forall s_{\text{typ}}\in S \tag{16}\]  
\[x_{r}\in\{0,1\}\quad\forall\,r\in R \tag{17}\]  
where  
\[\eta_{\text{typ}}^{load}=\left\{\begin{array}{l}1,\left(st_{p_{o}}^{load} \right)\leq s_{\text{typ}}\leq\left(ct_{p_{x}}^{load}\right)\\ 0,\text{Otherwise}\end{array}\right. \tag{18}\]  
\[\eta_{\text{typ}}^{unload}=\left\{\begin{array}{l}1,\left(st_{p_{o}}^{unload }\right)\leq s_{\text{typ}}\leq\left(ct_{p_{o}}^{unload}\right)\\ 0,\text{Otherwise}\end{array}\right. \tag{19}\]  
\[K_{sp}=\sum_{k_{p}\in K_{p}}\sum_{\in V_{r}}\tau_{\text{tsk}_{p}}\quad\forall p \in P,\forall s\in S \tag{20}\]  
\[\tau_{\text{tsk}_{p}}=\left\{\begin{array}{l}1,r_{k_{p}}\leq s_{\text{typ}} \leq l_{k_{p}}\\ 0,\text{Otherwise}\end{array}\right. \tag{21}\]  
Equations (12) and (13) are objective functions that are intended to minimize the total CO2e emissions and the total travel time. Constraint (14) enables the formation of a transportation plan to transport goods according to the requirements of job \(i\). Constraint (15) ensures that the number of barges used does not exceed the carrier's maximum number of barges. Constraint (16) ensures that the total number of shipments specified at port p for loading and unloading does not exceed the loading capacity of port p (the number of berths in port p in period s) in period s. We developed this constraint for use specifically with the proposed bi-objective set covering model based on route representation. Constraint (17) ensures that the decision variable is binary valued. Equations (18) and (19) determine the binary parameter used to describe period s when the barge is docked at port p for transport job sequence \(v\) in route \(r\). Equations (20) and (21) determine binary parameters (\(\eta_{\text{typ}}^{load}\) and \(\eta_{\text{typ}}^{unload}\)) that take a value of 1 if barge loading/unloading occurs at port p for transport job sequence v in route r in period s, and these binary parameters are used in constraint (16). The binary parameter \(\left(s_{pv}\right)\) is obtained from the feasible route generation. The Pareto frontier is gained using the \(\varepsilon\)-constraint method proposed by [PERSON] and [PERSON] [42], which includes the following steps.