Question

A factory produces cakes, pies, rolls and danishes. The resources required are flour, sugar, meat and fruit. The following table shows

the per-unit resource quantities required for each product,

the price we sell each product for,

the per-unit resource prices, and

the resource quantities available to us for purchase:

Products	Resources				Selling price
	flour	sugar	meat	fruit
cakes	0.500	0.900	0.000	0.100	$8.40
pies	0.430	0.090	0.500	0.000	$20.21
rolls	0.250	0.015	0.000	0.000	$1.45
danishes	0.180	0.290	0.000	0.050	$3.99
Price per unit	$2.50	$1.50	$28.00	$18.00
Amount available	1700.000	700.000	35.000	2200.000

After calculating the per-unit profits, we can formulate the linear program to maximise profit as follows:

Max P = 4X1 + 5X2 + 0.8X3 + 2.2X4
s.t.
0.5X1 + 0.43X2 + 0.25X3 + 0.18X4 ≤ 1700.00
0.9X1 + 0.09X2 + 0.015X3 + 0.29X4 ≤ 700.00
0.5X2 ≤ 35.00
0.1X1 + 0.05X4 ≤ 2200.00
Xi ≥ 0.

When this linear program is solved in Excel, the following Sensitivity Report is produced:

Variable Cells

Cell	Name	Final Value	Reduced Cost	Objective Coefficient	Allowable Increase	Allowable Decrease
$B$5	Cakes	0	-2.660458453	4	2.660458453	1E+30
$B$6	Pies	70	0	5	1E+30	3.250573066
$B$7	Rolls	5149.068768	0	0.8	2.091152074	0.686206897
$B$8	Danishes	2125.737822	0	2.2	13.26666667	0.853793103

Constraints

Cell	Name	Final Value	Shadow Price	Constraint R.H. Side	Allowable Increase	Allowable Decrease
$B$14	Flour	1700	2.851002865	1700	9891.766667	1239.327586
$B$15	Sugar	700	5.816618911	700	1996.694444	593.506
$B$16	Meat	35	6.501146132	35	1656.24424	35
$B$17	Fruit	106.2868911	0	2200	1E+30	2093.713109

Now suppose that we decrease the price of danishes to $2.90 per unit. From the Sensitivity Report, what is the most you can say about the effect on the optimal production plan and profit?

Select one:

a. The optimal production plan will change. The optimal profit will be between $6828.82 and $7330.94.

b. The optimal production plan will not change. The optimal profit will be $6313.89.

c. The optimal production plan will not change. The optimal profit will be $8064.57.

d. The optimal production plan will not change. The optimal profit will be $9145.88.

e. The optimal production plan will change. The optimal profit will be between $5808.47 and $7330.94.

f. The optimal production plan will not change. The optimal profit will be $8993.98.

Answer 1

Answer:

a. The optimal production plan will change. The optimal profit will be between $6828.82 and $7330.94.

Explanation:

The reduced cost of danishes is 0 and the allowable increase is 13.26666667 and allowable decrease is 0.853793103.

This means that optimal values wont change if the price is not increased by 13.26666667 or decreased 0.853793103.

But, we are decreasing the selling price by = 3.99 - 2.90 = $1.09

That means unit profit will change by $1.09. Therefore the optimal solution will also change.

New profit for danishes = 2.90 -1.09= 1.11

Now, the minimum value of objective function with the optimal solution will be :

= 70*5 + 5149.068768*0.8 + 2125.737822*1.11 = 6828.82.

Therefore the asnwer is a)