## 4 Result and Discussion  
In the presented moving window decision-making algorithm, the moving window size \(S_{MW}\) is an important factor that could affect the optimization performance as it defines the time domain where the fouling proceeding is simulated for the algorithm to make cleaning decision at each moving-step. Hence, the effect of \(S_{MW}\) on the optimality of the obtained solution was intensively investigated in this study for the varied HEN cleaning schedule problems with different time horizons ranging from 12 to 84 months. Figure 3 shows the result for the case of \(t_{F}=18\) months (i.e., \(N_{P}=18\)), where one can easily notice that there is an optimal \(S_{MW}\) to make the optimized objective function value (\(Obj_{opt}\)) be minimal (here, the optimal \(S_{MW}=4\)). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & \multicolumn{10}{c}{**Heat Exchanger**} \\ \cline{2-10}  & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** \\ \hline \(F_{h}\times 10^{-5}\) [lb/h] & 1.41 & 0.74 & 4.23 & 4.29 & 2.08 & 4.23 & 2.10 & 1.41 & 2.83 & 2.08 \\ \(F_{c}\times 10^{-5}\) [lb/h] & 7.21 & 7.21 & 7.21 & 7.21 & 7.21 & 7.21 & 7.21 & 6.49 & 6.49 & 6.49 \\ \(C_{ph}\) [BTU/lb \({}^{\circ}\)F] & 0.67 & 0.70 & 0.62 & 0.62 & 0.67 & 0.62 & 0.69 & 0.67 & 0.69 & 0.67 \\ \(C_{pc}\) [BTU/lb \({}^{\circ}\)F] & 0.46 & 0.46 & 0.46 & 0.55 & 0.55 & 0.55 & 0.57 & 0.57 & 0.57 \\ \(A\) [ft\({}^{\circ}\)] & 465 & 287 & 1192 & 1488 & 183 & 546 & 492 & 437 & 885 & 1257 \\ \(a\times 10^{7}\) [ft\({}^{\circ}\)F/BTU] & 1.23 & 1.84 & 1.23 & 1.64 & 3.07 & 2.25 & 3.07 & 3.27 & 3.68 & 3.88 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Model parameters used in cleaning schedule optimization of 10-unit HEN system. Adapted from [PERSON] et al. [18]. Figure 3: The effect of moving window size on the optimized objective function value and the number of cleanings for the case of \(t_{F}=18\) months. The existence of the optimal \(S_{MW}\) can be explained by the trade-off nature inherent in the objective function to be minimized. When \(S_{MW}\) is too small, the fouling proceeding would be simulated to be trivial; hence the moving window algorithm could make less cleaning decisions far from the optimality because the cleaning cost term in the objective function would always be overestimated than the extra fuel cost term. When \(S_{MW}\) is too large, on the other hand, the algorithm would tend to make more frequent cleaning decisions since the extra fuel cost term would more significantly contribute to the objective function due to the severe fouling simulated. [PERSON] and [PERSON] [15] tested a similar moving window technique combined with their MILP approach, but simply concluded that the moving window technique was not effective because making larger moving window size did not guarantee improved performance. However, they did not consider the effect of the moving window size in depth and missed the existence of the optimal \(S_{MW}\). As can be seen in Figure 3, the use of larger \(S_{MW}\) just results in the increase of the total number of cleanings, which does not necessarily mean an improved solution. The existence of the optimal \(S_{MW}\) was always observed for the different time horizon problems tested in this study (see Table 2). When comparing the performance of the moving window method, therefore, its optimal \(S_{MW}\) should be determined first for the given time horizon. In various industries, the HEN systems are usually operated continuously for several years between the whole system shutdowns [21; 22]. Most precedent studies, however, have addressed the HEN cleaning schedule optimization problems only for a relatively short time period [18; 24; 25]. In this study, therefore, we have compared the performance of our moving window decision-making algorithm with its original version using a fixed time horizon [18] for a quite long time period up to 5 years (Table 2).