# A Study of Applying Fuzzy Theory in Simulation-Based Education
## Methodology
### Fuzzy Set Theory  
Fuzzy Set Theory was proposed by [PERSON] in 1965 [33]; since then, fuzzy mathematics became popular and widespread in all kinds of fields. For example, during the decision making process, the decision maker will face various uncertainties. If fuzzy theory is introduced during the evaluatingprocess, it would make the decision result match reality more closely. Thus, this study took these theories and applied them to military helicopter simulator training critical factors. The basic fuzzy set theory concepts are listed as follows:  
Fuzzy numbers:  
[PERSON] and [PERSON] [34] defined any of the fuzzy numbers as a fuzzy subset of the real line. A fuzzy number is a quantity whose values are imprecise, rather than exact as is the case with single-valued numbers. Among the various shapes of fuzzy numbers, triangular fuzzy number and trapezoidal fuzzy number are the most commonly used membership function. If fuzzy number \(\widetilde{\mathbf{A}}\) is a fuzzy set and belongs to membership function:  
\[\mu_{\widetilde{A}}(X):U\rightarrow[0,1],\ x\in X \tag{1}\]  
Triangular Fuzzy Numbers (TFN) has linear representations on its left and right side such that its membership function can be defined as:  
\[\mu_{\widetilde{A}}(x)\left\{\begin{aligned} &\frac{x-l}{m-l},& l \leq x\leq m\\ &\frac{x-u}{m-u},& m\leq x\leq u\cdot\\ & 0,& others\end{aligned}\right. \tag{2}\]  
Fuzzy number algorithms:  
[PERSON] [33] modified AHP fuzzy theory and, based on his \"Extension principle,\" used triangular fuzzy number as every evaluation index. FAHP converts the opinions of experts from previous definite values to fuzzy numbers and membership functions presents triangular fuzzy numbers in paired comparison of matrices to develop FAHP. The triangular fuzzy number can be defined as follows:  
\[\mu_{\widetilde{A1}}(x)=(l_{1},m_{1},u_{1});\ \mu_{\widetilde{A2}}(x)=(l_{2},m_{2},u_{2 });\ l_{i}\leq m_{i}\leq u_{i}.\]  
1. Fuzzy number addition  
\[(l_{1},m_{1},u_{1})(+)(l_{2},m_{2},u_{2})=(l_{1}+l_{2},\ m_{1}+m_{2},\ u_{1}+u_{2}) \tag{3}\]  
2. Fuzzy number subtraction  
\[(l_{1},m_{1},u_{1})(-)(l_{2},m_{2},u_{2})=(l_{1}-u_{2},\ m_{1}-m_{2},\ u_{1}-l_{2}). \tag{4}\]  
3. Fuzzy number multiplication  
\[(l_{1},m_{1},u_{1})(x)(l_{2},m_{2},u_{2})=(l_{1}l_{2},\ m_{1}m_{2},\ u_{1}u_{2 }),\ l_{1}\geq 0,\ l_{2}\geq 0. \tag{5}\]  
4. Fuzzy number division  
\[(l_{1},m_{1},u_{1})(/)(l_{2},m_{2},u_{2})=(l_{1}/u_{2},\ m_{1}/m_{2},\ u_{1}/l_{2}),\ l_{1}\geq 0,\ l_{2}\textgreater 0. \tag{6}\]  
Defuzzification:  
\"Competency and Scenario\" evaluation values are fuzzy numbers. This research employs fuzzy AHP to find the fuzzy preference weights. This is also called \"Defuzzification.\"  
Center of area, COA or Center of gravity, COG:  
\[COG=\frac{\sum\mu(y_{i})\cdot y_{i}}{\sum\mu(y_{i})}. \tag{7}\]Center-of-sums, COS:  
\[COS=\frac{\sum y_{j\times}\int\mu_{j}(y_{j})dy_{j}}{\sum\int\mu_{j}(y_{j})dy_{j}} \tag{8}\]  
Mean of maximum, MOM:  
\[MOM=\sum_{j=1}^{m}\frac{z_{j}}{m} \tag{9}\]  
[PERSON] and [PERSON] [35] used the gravity method to defuzzify the fuzzy weight to obtain the best \"Nonfuzzy\" or so-called \"Best crisp value.\" Supposing \(\widetilde{\mathrm{W}_{i}}=(\mathrm{I_{i}},\mathrm{m_{i}},\mathrm{u_{i}})\), the best \"Nonfuzzy\" value \(\mathrm{NF_{i}}\) will be:  
\[NF_{i}=\left[(u_{i}-l_{i})+(m_{i}-l_{i})\right]/3+l_{i}. \tag{10}\]  
After, the defuzzified fuzzy number could gain a certain value which was based on the weight to be prioritized. At the second stage of this study, the collected and analyzed questionnaire in this study obtained a fuzzy value of the \"Competency and Scenario.\" This study used Equation (10) to conduct defuzzication of the values and converted those values into definite values. Then, we may understand the \"Competency and Scenario's\" weight of military helicopter EBT's decision process.