# Machine Selection for Inventory Tracking with a Continuous Intuitionistic Fuzzy Approach
## 4 CINFU AHP&CODAS Methodology  
Figure 1 illustrates the steps of the proposed CINFU AHP&CINFUSU CODAS methodology. RQ 1, RQ 2, and RQ 3 are answered after Stage 3, Stage 2, and sensitivity analysis, respectively. Stage 1 starts by clearly expressing the study's objective, which, in this case, is selecting the optimal machine for inventory tracking. This stage includes understanding the necessity for machine selection, summarizing the factors affecting this alternative, and establishing the context within which the selection must occur (e.g., operational requirements, budget constraints, technological compatibility, etc.). * \"Determine Criteria\" and \"Determine Alternatives\" can be applied in parallel. Criteria are set to evaluate the alternatives. These criteria should be thorough and contain all factors suitable for machine selection. Criteria determination is essential since it forms the basis for a structured and objective comparison of alternatives. * All possible machine options that can be used for inventory tracking are listed. The set \(A_{i}=\{A_{1},A_{2},\cdots,A_{m}\}\) having \(i=1,2,\cdots,m\) alternatives is evaluated by decision criteria \(n\) of set \(C_{j}=\{C_{1},C_{2},\cdots,C_{n}\}\), with \(j=1,2,\cdots,n\). The vector set representing the criteria's weight is defined as \(\rho_{j}=(\rho_{1},\rho_{2},\cdots,\rho_{n})\), where \(\rho_{j}>0\) and \(\sum_{j=1}^{n}\rho_{j}=1\). The experts in specific subjects who make decisions are called experts \(E_{t},t=1,2,\cdots,T\). * Stage 2 produces the pairwise comparison matrix using linguistic expressions given in Table 1 with Equation (21). Then, the appropriate IVIFNs are created by replacing the defined \(\zeta\) values (\(0<\zeta<1\)) with the linguistic operations using Equation (22). \[\widetilde{\widetilde{\Psi}}_{t}= \begin{array}{l}C_{1}\\ \vdots\\ C_{n}\end{array} \left[ \begin{array}{cccc}\langle EEI;(0.5,0.5)\rangle&\cdots& \langle\tilde{\psi}_{1 nt},(\zeta_{1 nt},\zeta_{2 nt})\rangle\\ \vdots&\ddots&\vdots\\ \langle\tilde{\psi}_{n1t},(\zeta_{n1t},\zeta_{n2t})\rangle&\cdots&\langle EEI; (0.5,0.5)\rangle\end{array} \right] \tag{21}\]  
\[\widetilde{\widetilde{\Psi}}_{[t]}= \begin{array}{l}C_{1}\\ \vdots\\ C_{n}\end{array} \left[ \begin{array}{cccc}\langle[\mu_{1 nt}^{L},\mu_{n1t}^{U} ],[\phi_{1 nt}^{L},\phi_{n1t}^{U}]\rangle&\cdots&\langle[\mu_{1 nt}^{L},\mu_{nt}^{ U}],[\phi_{1 nt}^{L},\phi_{1 nt}^{U}]\rangle\\ \vdots&\ddots&\vdots\\ \langle[\mu_{n1t}^{L},\mu_{n1t}^{U}],[\phi_{n1t}^{L},\phi_{n1t}^{U}]\rangle& \cdots&\langle[\mu_{mnt}^{L},\mu_{mt}^{U}],[\phi_{mnt}^{L},\phi_{mnt}^{U}] \rangle\end{array} \right] \tag{22}\]  
* The consistencies of the IVIF pairwise comparison matrices for each expert are checked. Equation (13) calculates crisp values corresponding to IVIFNs in each expert's pairwise comparison matrices. Then, defuzzified matrices are used to check the consistency considering [PERSON]'s consistency rule [18]. Figure 1: Proposed methodology. * [Step 6: The IVIFWG aggregation operator given by Equation (20) is utilized to compute the aggregated IVIF pairwise comparison matrix \(\widetilde{\Psi}=(\widetilde{\Psi}_{\ell^{\prime}_{j}})_{n\times n}\). The IVIFN for comparing criterion \(C_{i}\) to criterion \(C_{j}\) is represented in the aggregated IVIF pairwise comparison matrix \(\widetilde{\Psi}=(\widetilde{\Psi}_{\ell^{\prime}_{j}})_{n\times n}\), where \(\widetilde{\Psi}_{ij}=\left\langle\left[\mu_{\ell^{\prime}_{j}}^{L},\mu_{ \ell^{\prime}_{j}}^{U}\right]\cdot\left[\theta_{\ell^{\prime}_{j}}^{L},\theta _{\ell^{\prime}_{j}}^{U}\right]\right\rangle\). * [Step 7: The score judgment matrix \(\tilde{R}=(r_{\ell^{\prime}_{j}})_{n\times n}\) is computed by Equation (23). * [Step 8: The interval multiplicative matrix \(\widetilde{T}=(\tau_{\ell^{\prime}_{j}})_{n\times n}\) is computed by Equation (24). * [Step 9: Equation (25) is applied for computing interval \(\widehat{\omega}_{i}\) of each criterion to obtain the priority vector of the interval multiplicative matrix.