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* = 4*X*1 + 5*X*2 + 0.8*X*3 +
2.2*X*4

s.t.

0.5*X*1 + 0.43*X*2 + 0.25*X*3 +
0.18*X*4 ≤ 1700.00

0.9*X*1 + 0.09*X*2 + 0.015*X*3 +
0.29*X*4 ≤ 700.00

0.5*X*2 ≤ 35.00

0.1*X*1 + 0.05*X*4 ≤ 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)**

A factory produces canoes, dinghies, speedboats and catamarans.
The resources required are steel, fibreglass, wood and marine
paint. 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
steel
fibreglass
wood
marine paint
canoes
0.000
27.000
1.000
7.000
$381.80
dinghies
15.000
4.000
2.500
2.100
$153.10
speedboats
36.000
4.000
1.000
2.300
$220.60
catamarans
11.000
64.000...

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Partial of the Excel Solver Sensitivity Report for the LP model
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