## 3 Scheduling Tasks of a Welding Robot
### Task Scheduling of the Welding Robot  
The determination of the effective and supporting task times of the robot allowed for the analysis of alternative work scenarios in terms of the considered objective criterion. The \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Type of Movement** & **Segments** & **Designations** & **Time [s]** \\ \hline \multirow{3}{*}{effective} & _AB_, _DE_, _ED_ & \(t_{AB}\), \(t_{DE}\), \(t_{ED}\) & 22.50 \\ \cline{2-4}  & _CB_, _FE_ & \(t_{CB}\), \(t_{FE}\) & 13.50 \\ \hline \multirow{3}{*}{supporting} & _BD_ & \(t_{BD}\) & 12.10 \\ \cline{2-4}  & _BE_ & \(t_{BE}\) & 11.88 \\ \cline{2-4}  & _DC_ & \(t_{DC}\) & 10.27 \\ \cline{2-4}  & _DF_ & \(t_{DF}\) & 5.89 \\ \cline{2-4}  & _EC_, _BF_ & \(t_{EC}\), \(t_{BF}\) & 8.89 \\ \cline{2-4}  & _EF_ & \(t_{EF}\) & 4.52 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Times of individual movements of the robot. Figure 3: Model of the developed workstation and movements analysis. obtained total times of joining the structure with the use of the welding robot are shown in Table 3.  
Upon analyzing the above scenarios of welding robot operation during the joining of structures in one setting, it should be concluded that the most beneficial, from the point of view of the adopted criterion of the objective, is the scenario II with a sequence of movements \(AB\)e-\(BD\)s-\(DE\)e-\(EF\)s-\(FE\)e-\(EC\)s-\(CD\)e (e--effective movement; s--supporting movement). Consequently, the researched sets \(O\) and \(PT\) will be defined as:  
\[PT=\{pt_{1},pt_{2},pt_{3},pt_{4},pt_{5},pt_{6},pt_{7}\}=\{22.50,\,12.10,\,22.50,\,4.52,\,13.50,\,8.89,\,13.50\} \tag{7}\]  
\[O=\{o_{1},o_{2},o_{3},o_{4},o_{5},o_{6},o_{7}\}=\{AB,\,BD,\,DE,\,EF,\,FE,\,EC,\, CB\} \tag{8}\]  
Scenario II is characterized by the minimum value of the expression \(\sum pt_{k}=25.51\) [s], which consequently allows one to obtain the value of the adopted objective criterion at \(C_{max}=97.51\) [s]. Due to the constant nature of the times of the effective tasks (welding joints), it is the times of the supporting tasks that significantly impact the values of the \(C_{max}\) index, i.e., the completion time of all tasks. The benefits of the most favorable scenario of the robot's work are most noticeable when compared with the scenario in which the sequence of movements determines the highest value of the objective function (Figure 4). In order to present the obtained difference, the elongation index of the completion deadline of all tasks \(\Delta C_{max}\), which is a commonly used indicator for evaluating schedules [11], can be used. In the analyzed case:  
\[\Delta C_{max}=C_{maxIII}-C_{maxII}=103.04-97.51=5.53\ [\mathrm{s}] \tag{9}\]  
where:  
\(\Delta C_{max}\)--elongation of completion time of all jobs,  
\(C_{maxIII}\)--the task completion date in scenario III (the least favorable scenario),  
\(C_{maxIII}\)--the task completion date in scenario II (the most favorable). In order to illustrate the differences between the scenarios, they were summarized in the form of Gantt charts (Figure 4). However, it should be noted that the obtained elongation of completion time relates to the robot tasks only in one setting. Therefore, in the case of the whole process, this difference will increase, and for one workpiece, it will equal \(2\cdot\Delta C_{max}=11.06\) [s]. Figure 5 compares the robot task schedules in scenarios II and III. Figure 5 confirms that the analysis of alternative scheduling of the welding robot is visibly beneficial in terms of reducing the process operation time. The obtained time reduction stands at 5.4%.