Enunciating utility maximization problem using set theory

Question

I need to enunciate a problem using set theory and I am not sure how to start. The problem goes like this:

You are a car manufacturer and need to decide how many colours to use in your next bash of cars. Assuming that you can only produce one car model but you are able to decide among an infinite range of colour, and the more colours you use the more expensive the batch. On the other side, you have a finite group of people. All of them will buy a car and have different colour preferences (for example, they like red 1, brown 0.5, and blue 0.02 etc). The situation is basically the more colours you use, the more satisfaction among customers but the lower your profits. I know fuzzy sets are part of the equation, but I am not sure how to put everything together formally. Also game theory might be useful?

Any help/ideas would be really appreciated.

Answer 1

Let $C$ be the [infinite] set of all colors, and $P$ be the [finite] set of all people. For all $p \in P$, $f_p: C \rightarrow \mathbb{R}$ is a color-liking function where a higher value means a greater satisfaction with that color.

Assumptions:

• The total color satisfaction for a person is the sum of the satisfactions for each color and the total color satisfaction for $P$ is the sum of the total color satisfactions of all people in $P$.
• The price of coloring the car is dependent only on the number of colors; i.e. the price function is $q: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$.
• The price and the total color satisfaction are geometrically weighted (as opposed to arithmetically, or exponentially, or harmonically, etc.) and that the weights are $w_p$ and $w_c$, respectively.

Then, the task is to choose some colors $D$ for your car, $D \subset C$, such that $$\sqrt[w_p + w_c]{t_p^{w_p}t_c^{w_c}}$$ is minimized, where $$ t_p = \sum_{c \in D} q(c) \\ t_c = \sum_{p \in P} \sum_{c \in D} f_p(c) $$ ($t_p$ is the total cost and $t_c$ is the total color satisfaction.)