I'm trying to replicate Zhou's Paper on quantifying UX using Fuzzy Math.
In their model, there is a weight vector $A$ for a set of characteristics. in the paper's test case the characteristics were Effectiveness, Efficiency, and Satisfaction with weights [0.4434, 0.1692, 0.3874], respectively. The method of obtaining these weights was something i was able to replicate.
along with this, a rating matrix $R$ which in the each category has 5 bins of discrete ratings:
[Excellent, Good, Medium, Poor, Very Poor]. So for the category effectiveness, 68.75% of the respondents had an Excellent rating while the remainder had a rating of Good. Giving us [0.6875, 0.3125, 0, 0, 0] the scores for Efficiency and Satisfaction were similarly obtained which gives us:
| Excellent | Good | Medium | Poor | Very Poor | |
|---|---|---|---|---|---|
| Effectiveness | 0.6875 | 0.3125 | 0 | 0 | 0 |
| Efficiency | 0.25 | 0.3125 | 0.0625 | 0.25 | 0.125 |
| Satisfaction | 0.125 | 0.625 | 0.25 | 0 | 0 |
The next step describes getting the appraisal vector $B$ where $B = A ∘ R$ where they describe the $∘$ as the composition operator $M(⋅,⊕)$. I'll directly take a chunk of the paper to show how they described the operator: Zhou (2015), Eq(2)
This is where I fail to replicate the math. It seems to equate $B$ onto two different values. it says $B$ is equal to $A ⋅ R$ but the the same time is also equal to $min(1, A ⋅ R)$. Specifically,
$b_j = \sum_{i=1}^n a_i r_{ij} = min \{1,\sum_{i=1}^n a_i r_{ij}\}$
where the weight vector is
$A = (a_1, a_2, ...)$
and the rating vector is $R = $$ \begin{bmatrix} r_{11} & r_{12} & \cdots & r_{1m}\\ r_{21} & r_{22} & \cdots & r_{2m}\\ r_{31} & r_{32} & \cdots & r_{3m}\\ \vdots & \vdots & & \vdots \\ r_{n1} & r_{n2} & \cdots & r_{nm} \\ \end{bmatrix} $$ $
In the example,
$B = A ∘ R = [0.3956, 0.3711, 0.1758, 0.0156, 0.0313]$
I should note that $b_1$ or the first element of $B$ which has a value of 0.3956 has the same value when i do a simple dot product, but the rest of the values ($b_2$ to $b_5$) have glaringly different values. i also can't seem to place where I'd actually do the $\alpha ⊕ \beta = min(1, \alpha + \beta)$ operation as described in the paper.
can anybody point me in the right direction?
