Question

Gaermont Paper manufactures paper and in turn sells it to commercial vendors. The company manufactures a 20-inch-wide "standard" roll of paper. However, not all orders are necessarily for this width. The company often receives orders for narrower rolls. To meet those requests, the narrower rolls are cut from the standard rolls. For the following month, the company has committed orders for the next number of rolls.

Ancho del rollo	demanda
12 in.	800
8 in	500
5 in	200
4 in	400

Gaermont Paper would like to determine the minimum number of standard rolls that will be required to meet this demand. Create an appropriate PL model for the problem.
a) (10 points) Clearly define the decision variables.
b) (15 points) Using the decision variables defined in the previous paragraph, formulate, without solving, a
Linear scheduling problem that Gaermont Paper can use to meet demand with the minimum number of standard rolls.

Answer 1

Lets determine various combination of cutting alternatives in which 12 in, 8 in, 5 in, 4 in rolls can be cut from 20 in standard roll. The combination and losses/waste due to the various combinations are as follows:

Cutting Pattern (i)	12 in (n1)	8 in (n2)	5 in (n3)	4 in (n4)	Losses (in) (20 – (12*n1 + 8*n2 + 5*n3 + 4*n4)
1	1	1	0	0	0
2	1	0	1	0	(20 – (12*1 + 8*0 + 5*1 + 4*0) = 3
3	1	0	0	2	0
4	0	2	0	1	0
5	0	1	1	1	3
6	0	0	4	0	0
7	0	0	3	1	1
8	0	0	2	2	2
9	0	0	1	3	3
10	0	0	0	5	0

Decision Variable:

x_i be number of 20 in rolls cut using cutting alternatives i, where i = 1, 2, …, 10.

Objective Function:

The objective is to minimize number of the units of 20 in roll by minimizing the losses to be produced by alternatives.

Objective function is formulated as:

Min Z. = 0x₁ + 3x₂ + 0x₃ + 0x₄ + 3x₅ + 0x₆ + 1x₇ + 2x₈ +3x₉ + 0x₁₀

Subject to:

Minimum requirement of 12 in, 8 in, 5 in, and 4 in rolls are 800, 500, 200, and 400 units respectively:

Total units of 12 in reels produced by cutting alternatives should be at least 800 units:

1x₁ + 1x₂ + 1x₃ + 0x₄ + 0x₅ + 0x₆ + 0x₇ + 0x₈ + 0x₉ + 0x₁₀ ≥ 800

Similarly for 8, 5, and 4 inch rolls:

1x₁ + 0x₂ + 0x₃ + 2x₄ + 1x₅ + 0x₆ + 0x₇ + 0x₈ + 0x₉ + 0x₁₀ ≥ 500   ( 8 in roll)

0x₁ + 1x₂ + 0x₃ + 0x₄ + 1x₅ + 4x₆ + 3x₇ + 2x₈ + 1x₉ + 0x₁₀ ≥ 200   ( 5 in roll)

0x₁ + 0x₂ + 2x₃ + 1x₄ + 1x₅ + 0x₆ + 1x₇ + 2x₈ + 3x₉ + 5x₁₀ ≥ 400   ( 4 in roll)

Non negativity constraint: All x_i≥ 0