The Home Improvement Building Supply Company has received the following order for boards in three lengths:

Length Order $($Quantity$)$

$7$ feet $700$ boards

$9$ feet $1200$ boards

$10$ feet $500$ boards

The company has $25-$foot standard$-$length boards in stock. Therefore, the standard$-$length boards must be cut into the lengths necessary to meet order requirements. Naturally, the company wishes to minimize the number of standard$-$length boards used. The company must, therefore, determine how to cut up the $25-$foot boards in order to meet the order requirements and minimize the number of standard$-$length boards used.

a$.$ Formulate and solve a linear programming model for this problem.

b$.$ When a board is cut in a specific pattern, the amount of board left over is referred to as trim loss. Reformulate and solve the linear programming model for this problem, assuming that the objective is to minimize trim loss rather than to minimize the total number of boards used.