Question

For the following provide (a) algebraic formulation, (b) optimized model using Excel-Solver, and (c) statement of the optimal solution.

Paper Company plant produces rolls of paper of various types for its customers. One product type is rolls of wrapping paper in several different standard widths, as follow: 12, 15, 20, 24, 30, or 40 inches. The various-width rolls are produced by slicing 60-inch wide rolls in plant. For a given week, company waits for all its customer orders to come in and then decides how to slice its 60-inch rolls to satisfy that week’s orders, to minimize the waste (i.e., the part of the rolls sliced that’s not used for customers’ orders that week). [For example if customer orders in a particular week totaled six 15-inch wide rolls and two 40-inch wide rolls, the orders could be satisfied by appropriately slicing three 60-inch wide rolls, leaving only two 5-inch waste rolls.]

company wants to use the minimum number of 60-inch rolls to meet each week’s customer orders, minimizing waste – with waste being defined as any part of rolls that is not being shipped out to meet that week’s customer orders for which rolls were sliced.

company needs to determine how many 60-inch rolls are to be used to meet this week’s customer orders, totaling 48 12-inch wide rolls, 19 15-inch wide rolls, 22 20-inch wide rolls, 32 24-inch wide rolls, 14 30-inch wide rolls, and seven 40-inch wide rolls

Answer 1

Solution:

the various cutting alternatives for given types of output rolls:

	Cutting Alternative	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22
Rolls (inches)	12			1					1	1	0	1			1		2	3		1	2	3	5
	15		1					2	1		1	0		1		2	1		4	3	2	1
	20	1					1				1	1	3	2	2	1	1	1
	24					1				2	1	1
	30				2	1	1	1	1
	40	1	1	1
Waste		0	5	8	0	6	10	0	3	0	1	4	0	5	8	10	1	4	0	3	6	9	0
Decision Variable		X1	X2	X3	X4	X5	X6	X7	X8	X9	X10	X11	X12	X13	X14	X15	X16	X17	X18	X19	X20	X21	X22

Decision Variable:

x_i be number of 60 inch rolls cut using cutting alternatives i, where i = 1, 2, 3, 4, 5, … , 22

Objective Function:

The objective is to minimize waste from 60-inch rolls to be produced by alternative cutting methods.

Objective function is formulated as:

Min Z. = 0x₁ + 5x₂ + 8x₃ + 0x₄ + x₅ + … + 9x₂₁ + 0x₂₂

Subject to:

Minimum requirement of 12, 15, 20, 24, 32, and 40 inch rolls are 48, 19, 22, 32, 14, and 7 respectively.

Total units of 12 inch roll produced by cutting alternatives should be at least 48 units:

X₈ + x₉ + x₁₁ + x₁₄ + 2x₁₆ + 3x₁₇ + x₁₉ + 2x₂₀ + 3x₂₁ + 5x₂₂ ≥ 48

Similarly, for other types of rolls.

All X_i are integers.

Excel Model:

Total number of 60” rolls required = 47

Total waste = 51”