# Smart Product Design Process through the Implementation of a Fuzzy Kano-AHP-DEMATEL-QFD Approach
## 3 Proposed methodology
### Analytic Hierarchy Process  
AHP assumes complete aggregation among criteria and develops a linear additive model. The weights and scores are achieved basically by pairwise comparisons between all options with each other [64]. The basic procedure to carry out the AHP consists of the following steps:  
1. _Structuring a decision problem and selection of criteria:_ The first step is to decompose a decision problem into its constituent parts. In its simplest form, this structure comprises a goal or focus at the topmost level, criteria (\(C_{i}\) ), subcriteria (\(SC_{ij}\)) at the intermediate levels, while the lowest level contains the alternatives available for the decision-maker (\(A_{k}\)) (Figure 6).  
Arranging all the components in a hierarchy provides an overall view of the complex relationships and helps the decision-maker to assess whether the elements in each level are of the same magnitude so that they can be compared accurately [64]. An element in a given level does not have to function as a criterion for all the elements in the level below. Each level may represent a different cut at the problem, so the hierarchy does not need to be completed [64]. When constructing hierarchies, it is essential to consider the environment surrounding the problem and to identify the issues or attributes that contribute to the solution, as well as to identify all participants associated with the problem.  
2. Construct a set of pairwise comparison matrices for criteria, sub-criteria, and alternatives (Equation (2)). \[A=\left[\begin{array}{cccc}1&a_{12}&\ldots&a_{1n}\\ a_{21}&1&\ldots&a_{2n}\\ \ldots&\ldots&\ldots&\ldots\\ a_{n1}&a_{n2}&\ldots&1\end{array}\right]\textit{Where, }\ a_{ij}=\ \frac{1}{a_{ji}}\] (2)  
Figure 5: Membership functions [24].  
Figure 6: AHP (Analytic Hierarchy Process) hierarchy scheme.  
3. Perform computations to find the importance of each criterion and alternative (Equation (3)). \[W_{i}=\frac{\left(\prod_{j=1}^{n}a_{ij}\right)^{\frac{1}{n}}}{\sum_{i=1}^{n}\left( \prod_{j=1}^{n}a_{ij}\right)^{\frac{1}{n}}}\,\ i,j=1,\ 2,\ \ldots,\ n\] (3)
4. Calculate the maximum eigenvalue, consistency index, Consistency Ratio (_CR_), and normalized values for criteria and alternatives to ensure that the calculated weights derived from the pairwise comparison matrix are acceptable. Obtaining _CR_ implies the previous calculation of the maximum eigenvalue (\(\lambda\)_max_) by applying Equation (4). \[\lambda_{max}=\frac{\sum_{i=1}^{n}cv_{i}}{n}\] (4)  
Additionally, _CV_ represents the consistency values for factors, and it is defined as: \(CV=\left[cv_{i}\right]_{1\text{\tiny{2}}m}\), where \(cv_{i}=\ \frac{c_{i}}{w_{i}}\) (i = 1, 2, \(\ldots,\) n) (refer to Equation (5)). On the other hand, matrix \(C\) (refer to Equation (6)) denotes an n-dimensional column vector, representing the total of the weighted values for the important intensity of criteria. Then, _CR_ (Consistency Ratio) (refer to Equation (7)) is computed using the _RI_ (Random Consistency Index, depending on the size of the problem) [19]. If the _CR_ \(<\) 0.10, the matrix is considered consistent, and the derived weights are then reliable for supporting decision-making.  
\[CI=\frac{\lambda_{max}-n}{n-1} \tag{5}\]  
\[C=\left[c_{i}\right]_{n\text{\tiny{2}}1},\ i=1,\ 2,\ \ldots,\ n \tag{6}\]  
\[CR=\frac{CI}{RI} \tag{7}\]  
When AHP is applied in group decision-making processes, two different approaches are used, including the aggregation of individual judgments and the aggregation of individual priorities.