QUESTION 2: Linear Programming Sensitivity Analysis

A salad dressing supplier to Ottawa area restaurants has been using Linear Programming (LP) for years to determine how much dressing they should produce for every season. In particular, they specialize in producing three kinds of dressing: Dijon, Classic Vinaigrette, and Roasted Garlic. All these dressings require some use of olive oil to produce. Taking this into account, the management team has formulated the following LP model that determines the optimal amount they should produce for each dressing.

X : the amount of Dijon

Y : the amount of Classic Vinaigrette

G : the amount of Roasted Garlic

Maximize Z = 1.2X+1.6Y+1.4G (total profit)

subject to

(Olive oil) 6X+5Y+3G <=300 liters

(Labour) 9X+4Y+5G <=280 minutes

(Machine) 2X+8Y+4G <=320 minutes

X,Y,G>=0

The model has been solved using Excel and the following sensitivity report was generated.

Variable Cells
			Final	Reduced	Objective	Allowable	Allowable
	Cell	Name	Value	Cost	Coefficient	Increase	Decrease
	$B$10	Dijon	0	-0.8	1.2	0.8	1E+30
	$C$10	Classic Vinaigrette	20	0	1.6	0.738461538	0.48
	$D$10	Roasted Garlic	40	0	1.4	0.6	0.3
Constraints
			Final	Shadow	Constraint	Allowable	Allowable
	Cell	Name	Value	Price	R.H. Side	Increase	Decrease
	$F$13	Olive Oil Usage	220	0	300	1E+30	A
	$F$14	Labour Usage	280	0.2	280	120	120
	$F$15	Machine Usage	B	0.1	320	147.6923077	96

Answer all the questions below ACCORDING TO the above sensitivity report.

(a) What is the optimal profit? (2 points)

(b) Two numbers have been removed from the resource sensitivity table by your professor (the letters A and B appear instead of the numbers). What are the correct values of A and B? Justify. (4 points)

A is 80

(c) The supplier learned recently that the price of Olive oil has been dropping significantly, and is wondering if they should take this opportunity to purchase more oils. Provide your answer and JUSTIFY it. (3 points)

(d) The profit of Dijon dressing has been found overly estimated. The company however has difficulty determining how much lower the real profit should be. Does this impact the LP optimal solutions? Justify your answer. (3 points)

(e) Some workers are not happy with their salaries and ask for a raise; otherwise, they will quit and this would reduce the available labour minutes from 280 to 240. Should the company consider the raise? If so, what will be a reasonable amount to pay (in total) to these workers in addition to their original salaries? Justify your answer. (5 points)

(f) What if the profit of ALL three dressings (i.e. Dijon, Classic Vinaigrette, and Roasted Garlic) is now $1.5/unit. Do the optimal values of the decision variables change? What will be the impact on the profit? Justify. (5 points)