# A Treatise on Reconnoitering the Suitability of Fuzzy MARCOS for Assessment of Conceptual Designs
## 2 Methodology
### Fuzzy MARCOS  
In order to implement the fuzzy MARCOS model, the first step is to create an extended fuzzy matrix containing the best (\(Cd^{h}\)) and worst (\(Cd^{m}\)) design concepts based on the beneficial (\(B_{df}\)) and cost (\(C_{df}\)) categories of design features. The best and worst design concepts created in this case will represent the ideal and anti-ideal design concepts, respectively. The best and worst design concepts can be obtained from Equations (11) and (12), respectively. The matrix containing the best and worst design concepts can be obtained by rewriting Equation (10) as presented in Equation (13). \[Cd^{b}=\left\{ \begin{array}{c}Min\Big{[}\widetilde{A}\Big{]}_{n}^{m}\; \forall\;m\in B_{df}\\ \\ Max\Big{[}\widetilde{A}\Big{]}_{n}^{m}\;\forall\;m\in C_{df}\end{array} \right. \tag{11}\]  
\[Cd^{w}=\left\{ \begin{array}{c}Max\Big{[}\widetilde{A}\Big{]}_{n}^{m}\; \forall\;m\in B_{df}\\ \\ Min\Big{[}\widetilde{A}\Big{]}_{n}^{m}\;\forall\;m\in C_{df}\end{array} \right. \tag{12}\]  
The cumulative fuzzy matrix \((\widetilde{C}_{I})\) can be obtained by summing the elements of the weighted matrix. This is obtainable from Equation (17). The cumulative fuzzy matrix is necessary for estimating the utility degree of the design alternatives \(\left[\widetilde{U}_{d}^{I}\right]_{n}\). The utility degree of the design alternatives is a function of the cumulative matrices of the best and worst design. Hence, the utility degree can be expressed in terms of best \(\left[\widetilde{U}_{d}^{I}\right]_{n}^{+}\) and worst \(\left[\widetilde{U}_{d}^{I}\right]_{n}^{-}\) design scenarios as presented in Equations (18) and (19), respectively. The next step is to compute the fuzzy utility matrix \(\left[\widetilde{T}\right]_{n}\). The fuzzy utility matrix is a summation of the utility degrees for the best and worst scenario of the design concepts as presented in Equation (20). Further, the fuzzy utility matrix is necessary for determining a new fuzzy number \(\left[\widetilde{T}\right]_{n}^{new}\), which is the maximum of the utility matrix as presented in Equation (21). This new fuzzy number will be defuzzified using Equation (2) in order to compute the utility functions in relation to the best \(F\left[\widetilde{U}_{d}^{I}\right]_{n}^{+}\) and worst \(F\left[\widetilde{U}_{d}^{I}\right]_{n}^{-}\) design alternatives as presented in Equations (22) and (23), respectively. The next step is to defuzzify the IFNs for the best and worst utility degree scenarios and the best and worst utility functions. This is necessary for the determination of a crisp value for the overall utility function for the design concepts, as presented in Equation (24). \[\widetilde{C}_{I}=\sum_{m=1}^{m=m}[\widetilde{v}]_{n}^{m} \tag{17}\]  
\[\left[\widetilde{U}_{d}^{I}\right]_{n}^{+}=\frac{\widetilde{C}_{I}}{\widetilde {C}_{I}^{b}} \tag{18}\]  
\[\left[\widetilde{U}_{d}^{I}\right]_{n}^{-}=\frac{\widetilde{C}_{I}}{\widetilde {C}_{I}^{w}} \tag{19}\]  
In Equations (18) and (19), \(\widetilde{C}_{I}^{b}\) and \(\widetilde{C}_{I}^{w}\) are the cumulative fuzzy matrix for the best and worst designs, respectively.