# Expert System Based on Integrated Fuzzy AHP for Automatic Cutting Tool Selection
## 2 Proposed Methodology
### Proposed Fuzzy AHP Method for the Multcase Selection Process  
The main idea of the proposed fuzzy AHP is to combine multiple selection cases in only one model. The conventional fuzzy AHP method has only one output for each case, while the proposed fuzzy AHP has many inputs. The output data also differ depending on the values of the input data. The basic process of the proposed fuzzy AHP calculation is the same as the conventional, one but the hierarchy structure is different. The detailed process is described in the following steps. * Step 1: Constructing the hierarchical structure for the problem  
The hierarchical structure, as shown in Figure 3, has many levels including, goal, selection criteria, and evaluation criteria alternatives. The evaluation criteria have the same meaning as the criteria in the conventional fuzzy AHP, where all the criteria and sub-criteria are fixed. However, in the integrated fuzzy AHP, each selection criterion has different choices, known as the sub-criteria or \"sub-sub-criteria\" of the main selection criteria. They are also the input information for the selection process and are the most important criteria. The conventional fuzzy AHP is suitable for making one decision but is not useful for the integrated problem, where there are multiple different cases of input information, such as the cutting tool selection problem. The next three steps are similar to those used in the conventional fuzzy AHP [13]. These are described in detail. * Step 2: Generating the comparison pairwise matrix based on experts' judgments. The decision maker compares the criteria or alternatives using linguistic terms. The triangular fuzzy membership function in Figure 4 is defined by a lower limit \(a_{ij}\), an upper limit \(b_{ij}\), and a value  
Figure 3: Hierarchy structure for the integrated fuzzy analytical hierarchy process (AHP). \(m_{ij}\), where \(a_{ij}<m_{ij}<b_{ij}\), as shown in Equation (1). Table 1 gives the importance scales for both the AHP and the triangular fuzzy AHP. The \(\delta\) and \(\delta^{\prime}\) values depend on the fuzzy scale. The most popular values of \(\delta\) and \(\delta^{\prime}\) are 1 and 0, respectively. \[\overline{\mu}_{A}(x)= \begin{pmatrix}0&&x<a_{ij}\\ \frac{x-a_{ij}}{m_{ij}-a_{ij}}&&a_{ij}\leq x\leq m_{ij}\\ \frac{b_{ij}-x}{b_{ij}-m_{ij}}&&m_{ij}\leq x\leq b_{ij}\\ 0&&x>b_{ij}\end{pmatrix} . \tag{1}\]  
The comparison matrix \(\overline{S}\) is presented as in Equation (2), where a fuzzy number is represented with three points as \(\widetilde{s_{ij}}=\left(a_{ij},\ m_{ij},\ b_{ij}\right)\), \(\widetilde{s_{ji}}=1/\widetilde{s_{ij}}\ =\left(1/b_{ij},\ 1/m_{ij},\ 1/a_{ij}\right)\), \(\widetilde{s_{ii}}=\left(1,\ 1,\ 1\right)\), in which \(i\), \(j=1,\ 2,\ 3\),..., \(n\):  
\[\overline{S}= \begin{bmatrix}1&\widetilde{s_{2}}&\widetilde{s_{3}}&\ldots& \widetilde{s_{in}}\\ 1/\widetilde{s_{2}}&1&\widetilde{s_{23}}&\ldots&\widetilde{s_{2n}}\\ 1/\widetilde{s_{3}}&1/\widetilde{s_{23}}&\ldots&&\\ \ldots&&\ldots&&\\ 1/\widetilde{s_{in}}&1/\widetilde{s_{2n}}&&1\end{bmatrix} . \tag{2}\]  
The pairwise comparison value for evaluation criteria depends on the experts' judgments, as normal. A pairwise comparison between choices (choice 1.1, choice 1.2, etc.) is determined according