Question

Problem 8-31 (Algorithmic)

Gulf Coast Electronics is ready to award contracts for printing its annual report. For the past several years, the four-color annual report has been printed by Johnson Printing and Lakeside Litho. A new firm, Benson Printing, inquired into the possibility of doing a portion of the printing. The quality and service level provided by Lakeside Litho has been extremely high; in fact, only 0.5% of Gulf Coast’s annual reports have had to be discarded because of quality problems. Johnson Printing has also had a high quality level historically, producing an average of only 1% unacceptable reports. Because Gulf Coast Electronics has had no experience with Benson Printing, it estimated Benson’s defective rate to be 10%. Gulf Coast would like to determine how many reports should be printed by each firm to obtain 75,000 acceptable-quality reports. To ensure that Benson Printing will receive some of the contract, management specified that the number of reports awarded to Benson Printing must be at least 0.1% of the volume given to Johnson Printing. In addition, the total volume assigned to Benson Printing, Johnson Printing, and Lakeside Litho should not exceed 30,000, 50,000, and 50,000 copies, respectively. Because of the long-term relationship with Lakeside Litho, management also specified that at least 30,000 reports should be awarded to Lakeside Litho. The cost per copy is $2.45 for Benson Printing, $2.5 for Johnson Printing, and $2.75 for Lakeside Litho.

(a)	Formulate and solve a linear program for determining how many copies should be assigned to each printing firm to minimize the total cost of obtaining 75,000 acceptable-quality reports.
	If required, round your answers to three decimal places.
	For subtractive or negative numbers use a minus sign even if there is a + sign before the blank.

	Let B = number of copies done by Benson Printing
	Let J = number of copies done by Johnson Printing
	Let L = number of copies done by Lakeside Litho

Max	B	+	J	+	L
s.t.
	B					≤		Benson
			J			≤		Johnson
					L	≤		Lakeside
	B	+	J	+	L	=		# useful reports
	B	+	J			≥		Benson - Johnson %
					L	≥		Minimum Lakeside
B, J, L ≥ 0

Optimal Solution:
B
J
L

(b)	Suppose that the quality level for Benson Printing is much better than estimated. What effect, if any, would this quality level have?
	The input in the box below will not be graded, but may be reviewed and considered by your instructor.

(c)	Suppose that management is willing to reconsider its requirement that Lakeside Litho be awarded at least 30,000 reports. What effect, if any, would this consideration have?
	The input in the box below will not be graded, but may be reviewed and considered by your instructor.

Answer 1

(a)

Min	2.45 B	+	2.50 J	+	2.75 L
s.t.
	1 B					≤	30000	Benson
			1 J			≤	50000	Johnson
					1 L	≤	50000	Lakeside
	0.90 B	+	0.99 J	+	0.995 L	=	75000	# useful reports
	1 B	+	- 0.001 J			≥	0	Benson - Johnson %
					1 L	≥	30000	Minimum Lakeside

Optimal Solution

B = 45.565
J = 45564.64
L = 30000.00

(b)

Benson can be given more share of business and the cost will also be less than the previous solution because printing from Benson being less costly.

(c)

This is also a binding constraint removal of which will lead to further reduction in the cost.