@mmcdara Wow!!! you have actually helped me done justice to this problem, the updated version is superb and works well, so clearly explained and it has broadened my knowledge of Maple optimization. i thank you very much, Its great to have someone like you on this platform. you clearly understood my problem without much information! you are a genius!!!

@mmcdara Thank you very much for your help, yes, the Cost_rate_1(K, A, M) is to be minimized by providing the best values of  K,A and M that gives the minimum cost rate.

I will make corrections to the code as suggested. thanks a million, i sincerely appreciate you