## 2 Modeling of Low-Carbon Scheduling Problem for Ship Segment Painting
### Description of the Problem  
The ship segmental painting low-carbon scheduling problem studied in this paper can be described as follows: There exists a group of segments that arrive at the sandblasting shop and have to pass through the various stages of the sandblasting shop continuously. Paint teams (consisting of equipment and operators) with different processing capabilities exist for each stage. Once the process is completed in the sandblasting shop, the segments are transported by flatbed trucks to the paint shop for painting. The painting team consumes a lot of energy during its operation. Different time and energy consumptions may be required for different segments. Between particular stages, the transportation distances of the segments are different and require different transportation times and energy consumption. The objective is to determine the sequence of segments for each stage and the allocation of painting teams to minimize the time to completion \((MS)\) and the total carbon emissions \((EC)\) to achieve an optimal mix that minimizes the carbon emissions generated by the energy consumption of the equipment. The following assumptions are presented:  
* Each segment must pass through all phases sequentially, with each segment job assigned to a transportation device at a given phase;
* At any given time, a maximum of one segmental painting team may be processed, and a maximum of one segment may be processed on one painting team;
* Interruptions and preemption are not permitted;
* Sections are transported to the paint shop immediately after completion in the sandblasting shop;
* The buffer between the stages of the segmented painting workshop is unrestricted;
* The types of equipment turn on when the paint team is assigned the first segment to begin processing. Only when all segments of the paint team are completed do the types of equipment turn off;
* Segmented jobs must be completely processed by the painting team in the last stage before the next stage can be started;
* Paint teams are permitted idle time, and segments are not to be processed by the paint team until the adjacent operation has been adjusted;  
The relevant symbols are listed in Table 1.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Meaning** \\ \hline \(J\) & Set of segments and \(J=\{1,\ldots,j,\ldots,n\}\) \\ \(I\) & Set of stages and \(I=\{1,\ldots,i,\ldots,s\}\) \\ \(M_{i}\) & Set of painting teams at the stage \(i\) and \\ \(M_{i}=\{1,\ldots,m,\ldots|M_{i}|\}\) \\ \(L\) & A grouping of positions designed to accommodate segments allocated to \\  & individual painting teams and \(L=\{1,\ldots,q,\ldots,n\}\) \\ \(p_{i,j}\) & Segment \(j\) Processing time for the segmental painting team on the stage \(i\) \\  & Setup time from segment \(j\) to \(j^{\prime}\) stage \(i\)\(j=j^{\prime}\) represents that segment \(j\) is the first \\  & segment assigned to a painting team \\ \(f_{m,m^{\prime},j}\) & Transportation from painting team \(m\) on stage \(i\) to paint team \(m^{\prime}\) on stage \(i+1\) \\ \(T^{s}_{i,j^{\prime},i}\) & Setup energy consumption from segment \(j\) to segment \(j^{\prime}\) in stage \(i\). \(j=j^{\prime}\) \\ \(P^{W}_{i}\) & Energy consumption of the painting team in process on stage \(i\) \\ \(P^{I}_{i}\) & Energy consumption of the painting team at idle on stage \(i\) \\ \(P^{F}_{i}\) & Energy consumption of transportation equipment between \(i\) and \(i+1\) \\  & in a given phase \\ \(v_{m,i}\) & Equipment energy utilization for the painting team \(m\) on stage \(i\) \\ \(b_{i,j}\) & Start time for the segment \(j\) during the stage \(i\) \\ \(e_{i,j}\) & Ending time for the segment \(j\) during the stage \(i\) \\ \(B_{i,m,q}\) & The outset time for the segment at the position \(q\) within painting team \(m\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters and variables.