# Example on profit function that contains the maximum of a decision and a random variable

I am looking for an example of a profit function that contains the maximum of a decision and a random variable in this form: $$\max(d,X)$$, where $$d$$ is a decision variable and $$X$$ is a random variable. I can find many profit functions with the minimum operations, that is, with a term like $$\min(d,X)$$, but I cannot find a single application with $$\max(d,X)$$. Note that the sign in front of the term $$\max(d,X)$$ must be positive, because $$-\max(d,X)=\min(-d,-X)$$ and thus I still end up with a minimum operation. In addition, this term must be in a profit function (that can be maximized), not in a cost function (to can be minimized). Does anyone know such an application? Thanks a lot.

• What is a decision variable? Do you have to choose $d$? – VARulle May 31 '20 at 11:59
• Yes, a decision variable is a decision chosen by the decision maker. For example, it can be the price of a product. The profit function can contain many terms, as long as a term is in the form $\max(d,X)$, it is good. – Justin Jun 1 '20 at 17:46
• O.k. - I first thought that the profit function must be of the form $\pi=\max(d,X)$, but I understand it only has to contain this term. – VARulle Jun 1 '20 at 22:23
• Are you looking for an example in the published literature or can I make up some profit function with a motivating "story"? – VARulle Jun 1 '20 at 22:26
• Yes, it only needs to contain such a max term. Any profit function with a "story" is fine. Thanks. – Justin Jun 3 '20 at 1:30