Question

# A factory produces cakes, pies, rolls and danishes. The resources required are flour, sugar, meat and...

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
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.

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)

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