Write Linear program and AMPL code for the problem.

45" 2-6. The output of a paper mill consists of standard rolls 110 inches (110") wide, which are cut into smaller rolls to meet orders. This week there are orders for rolls of the following widths: Width Orders 20" 48 35 50" 24 55" 10 8 The owner of the mill wants to know what cutting patterns to apply so as to fill the orders using the smallest number of 110" rolls. (a) A cutting pattern consists of a certain number of rolls of each width, such as two of 45" and one of 20", or one of 50" and one of 55" (and 5" of waste). Suppose, to start with, that we consider only the following six patterns: Width 1 2 3 75" 20" 45" 50" 55" 75" 0 1 0 0 1 2 0 0 0 3 0 0 1 1 0 4 2 0 0 1 0 S 6 1 3 0 1 0 0 0 0 10 How many rolls should be cut according to each pattern, to minimize the number of 110" rolls used? Formulate and solve this problem as a linear program, assuming that the number of smaller rolls produced need only be greater than or equal to the number ordered. (b) Re-solve the problem, with the restriction that the number of rolls produced in each size must be between 10% under and 40% over the number ordered. (c) Find another pattern that, when added to those above, improves the optimal solution