In a textile plant, cloth is manufactured in rolls of length L. Defects sometimes occur along the length of the cloth. Consider a specific roll with (N - 1) defects appearing at distances y1, y2, . . . , yN-1 from the start of the roll (yi+1 > yi for all i). Denote the start of the roll by y0, the end by yN.
The roll is cut into pieces for sale. The value of a piece depends on its length and the number of defects. Let
v(x, m) = Value of a piece of length x having m defects.
Assume that all cuts are made through defects and that such cutting removes the defect. Specify how to determine where to cut the cloth to maximize total value.