UW Smokestacks has a chimney which needs cleaning regularly. Soot builds up on the inside of the chimney at a rate of 1cm per month. The cost of cleaning when the soot has thickness x cm is 13 + x2 hundred dollars. Cleaning can occur at the end of any month. The chimney is clean at present (i. e. at the start of month 1). According to the contract with the cleaners, it must be cleaned precisely nine times in the next 48 months, one of which times must be at the end of month 48. Also, at least three cleanings must occur in the first 15 months.
It is needed to determine which months the cleaning should occur, to minimize total cost. Formulate a nonlinear mathematical program to solve this problem, such that all constraints are either linear or integrality constraints.