What would you do if you had to turn down a request from a valued customer?

I would be looking for all the possible ways to help the customer to the best, But if I don't have a choice to accommodate customer's request then I would politely refuse the customer by giving him the reason why I can't help. Making the customer understand the logic behind it too. 

If it was for the good of the company & for the rightful reasons, I would professional turn down their request without hesitation, then I would gently own up the decision & account for it professionally to my superiors hence stand my ground to defend my decision & explain my position in the matter. Consequently, I would also consider consulting my superiors about either turning down the request or taking it up, if the superiors tell me to turn it down, I would do it professionally and if the superiors tell me not to turn it down, I would do as they wish. Some modern & good managers or superiors would ask you what you think about the situation, discuss it between you & them and come up with a final decision thus a collective decision would be taken