I have an optimisation problem with decision variables that are multiplied with another (a weighted average is calculated). I'd like to convert it into a linear program. I found this link that explains how to deal with fractional objectives (6.3) which basically (correct me if I'm wrong) substitutes the decision variable with a linear equation and adding more constraints, so once the substituted solution is found the actual solution has to be calculated (as for the example in the link by dividing by $t$). Since now the decision variable is not in the numerical range as it used to be, other calculations which rely on the actual value are invalid (see calculation of $price$).
How is this dealt with in practice. Can this problem be converted into a LP? If so how?
$$ \text{objective:} \max(price * amount) \\ price = \begin{cases} qual > \alpha: a * qual_{i} * qual_{k}, \\ qual > \beta : b - qual \end{cases} \\ amount = \sum x_{i} \\ qual = \frac{\sum (x_{i} * a_{i})}{amount}\\ s.t.: qual >= \beta \\ qual <= \gamma $$