A version of the secretary problem is considered. The ranks of items, 
whose values are independent, identically distributed random variables 
$X_1,X_2,\ldots,X_n$ from a uniform distribution on $[0; 1]$,
are observed sequentially by the grader. He has to select exactly one item, 
when it appears, and receives  a payoff which is a function of the unobserved realization 
of random variable assigned to the item  diminished by some cost.   
The methods of analysis are based on the existence of an embedded Markov
chain and use the technique of backward induction. 
The result is a generalization of the selection model considered by
Bearden~\cite{dea06:cardinal}. 
The asymptotic behaviour of the solution is also investigated.