## 3 AUED-Based FPSO (AUFPSO) with NTE  
This study used the AUED method in the proposed FPSO-NTE algorithm to assist in the search for the best combination of the four constant coefficients. The three main steps of AUED method are initializing, performing ten-level uniform layout experiments, and calculating the parameter range for the next ten-level uniform layout. The initialization step includes selection of the parameters to be optimized, their ranges, and the solution accuracy. At this time, a suitable ten-level uniform layout, the output, a stepwise ratio, and the stop condition are also selected. The second main step can then be executed. Ten levels of the parameter range and ten levels of the solution accuracy are defined, and the ten levels are assigned to the ten-level uniform layout. The ten-level uniform layout experiments can then be performed, and the experimental results are recorded. After the ten-level uniform layout experiments in this stage are completed, the best combination obtained according to the output is an optimal or near-optimal value. The range for each parameter is then calculated according to the best combination and the stepwise ratio. The second and third main steps are repeated until the stop condition is met. For a clear understanding of how the AUED method is applied in the proposed FPSO-NTE algorithm, the detailed steps of the method are given below. 1. Initialization of the AUED method in the proposed FPSO-NTE algorithm  
where \(PARA\_NO\) is the total number of parameters; \(LB\) and \(UB\) are the upper and lower bounds for each experimental parameter, respectively; \(LT\) and \(UT\) are temporary values of \(LB\) and \(UB\), respectively; \(LEVEL\) is the level value; \(EXP\_NO\) is the total number of experiments in the uniform layout; \(BEST\) is the best parameter value for the best combination obtained by the uniform layout experiments in this stage; and \(SWR\) is a stepwise ratio. The following example demonstrates the use of the updated algorithm when the upper and lower bounds for each parameter are initially set to 2 and 0, respectively, the number of parameters was 4, and the number of experiments was 10. The first ten-level uniform layout experiments indicated that the best combination [\(P_{1}\)\(P_{2}\)\(P_{3}\)\(P_{4}\)] was [0.2553 0.4514 0.5556 1.2455]. The stepwise ratio was set to 0.8. Here, the first parameter value was used to explain how to calculate a new range for the next ten-level uniform layout experiments stage. When the first ten-level uniform layout experiments stage was completed, \(LB\) and \(UB\) were 0 and 2, respectively. At first, \(LT\) and \(UT\) were 0 and 2, respectively, due to \(LB\) and \(UB\). The best parameter was 0.2553, and the stepwise ratio was 0.8. Therefore, \(LB=0.2553-(2-0)\times 0.8\div 2=-0.5447\) and \(UB\)\(=0.2553-(2-0)\times 0.8\div 2=1.0553\). However, since \(LB\) was lower than \(LT\), \(LB\) must be corrected to \(LT\), and the new range for the first parameter was 0 to 1.0553. This was because the original range was set to 0 to 2. Therefore, we could know that \(LB\) for the next uniform layout experiments must be equal to or more than \(LT\). In the same way, \(UB\) for the next uniform layout experiments must have been equal to or less than \(UT\). Next, the range was divided into ten levels, and a discrete value was calculated for each level. The discrete values were calculated as follows: \(0+((1.0553-0)\div(10-1)\times(1-1))\) = 0 for the first level; \(0+((1.0553-0)\div(10-1)\times(2-1))=0.1173\) for the second level, and so on. The discrete value for the tenth level was calculated as \(0+((1.0553-0)\div(10-1)\times(10-1))=1.0533\). Table 3 shows the levels obtained after the updated algorithm was executed in this instance.