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.
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$\begingroup$ What is a decision variable? Do you have to choose $d$? $\endgroup$– VARulleMay 31, 2020 at 11:59
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$\begingroup$ 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. $\endgroup$– JustinJun 1, 2020 at 17:46
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$\begingroup$ 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. $\endgroup$– VARulleJun 1, 2020 at 22:23
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$\begingroup$ Are you looking for an example in the published literature or can I make up some profit function with a motivating "story"? $\endgroup$– VARulleJun 1, 2020 at 22:26
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$\begingroup$ Yes, it only needs to contain such a max term. Any profit function with a "story" is fine. Thanks. $\endgroup$– JustinJun 3, 2020 at 1:30
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