Case Study # 4: Optimizing Production Mix for Outdoor Furniture Manufacturing at Patio Paradise, Inc. (Total marks: 25)

Patio Paradise, Inc. is a manufacturer of outdoor furniture that offers three different products: a lounge chair, a sectional sofa, and a dining table. The manufacturing process of these products involves two departments, cutting and assembly. Each product must go through both cutting and assembly processes, which have a limited number of machine hours available: 1000 hours for cutting and 1200 hours for assembly. Patio Paradise, Inc. is also limited by the available material due to a shortage in supply. Currently, the company only has 2000 lbs of wood to work with. The net profit that Patio Paradise receives from the manufacture and sale of one unit of each product is quite lucrative: $3 for a lounge chair, $3 for a sectional sofa, and $5 for a dining table.

The OR manager at Patio Paradise model the problem as a linear program to decide on the optimal product mix to maximize its profits and solved it with Excel to generate the following Sensitivity Analysis Report:

Variable cells

cell	name	Final value	Reduced cost	Objective coefficient	Allowable increase	Allowable decrease
$C$13	Lounge chair	700	0	3	2	0.78
$D$13	Sectional sofa	0	-1.38	3	1.38	1E+30
$E$13	Dinning table	133.33	0	5	1.75	2

Constraints

cell	name	Final value	Shadow price	Constraint R.H. side	Allowable increase	Allowable decrease
$F$15	Cutting	1000	1.17	1000	200	466.67
$F$16	Assembly	866.67	0	1200	1E+30	333.33
$F$17	Wood	2000	0.8	2000	555.56	333.33

Use the Sensitivity Report above to answer the following questions. You must clearly show your working.

a) (5 points) What is the optimal solution? What is total profit that Patio Paradise anticipates by producing this mix?

b) (3 points) What is the value of one more unit of cutting time? of assembly time? of wood?

c) (5 points) A local distributor has offered to sell Patio Paradise, Inc. some additional wood for $0.70/lb. Should Patio Paradise buy it? If yes, how much would the firm's profit increase if they bought 550 lbs. and used it in an optimal fashion?

d) (2 points) The R&D department has been redesigning the sectional sofa to make it more profitable. If it can sell one unit of this sofa with a unit profit of $4, is it profitable to produce wood under the new design?

e) (5 points) Patio Paradise, Inc. has a chance to sell some of its capacity in cutting at cost of $1.50/hour. If it sells 200 hours at that price, how will this affect profits?

f) (5 points) If the unit profit on chairs were to decrease to $2.40, what would be the optimal solution and what profits would this production plan give?