Problem:
A fertilizer manufacturer has to fill supply contracts to its two main customers (650 tons to customer A and 800 tons to customer B). It can meet this demand by shipping existing inventory from any of its three warehouses. Warehouse 1 (W1) has 400 tons of inventory on hand, Warehouse 2(W2) and Warehouse 3 (W3) has 600 tons. The company would like to arrange the shipping for the lowest cost possible, where the per-ton transit costs are as follows:
|
|
W 1
|
W 2
|
W 3
|
|
Customer A
|
$7.50
|
$6.25
|
$6.50
|
|
Customer B
|
$6.75
|
$7.00
|
$8.00
|
Write the objective function and the constraint in equations. Let Xij = tons shipped from warehouse i to customer j, and so on. For example, X1A= tons shipped from warehouse 1 to customer A.
The objective function, the LP model =
Minimize Z = $7.50______ + $6.50_______ + (shipping cost to customer A)
$6.75______ +$7.00 ________ (shipping cost to customer B)
Subject to: _______ Tons shipped to customer A
_______ Tons shipped to customer B
_______ Tons shipped from warehouse 1
_______ Tons shipped from warehouse 2
_______ Tons shipped from warehouse 3
∀Xij ≥0 Non negativity condition
Using software the linear programming problem was solved and the following sensitivity report was obtained:
Adjustable Cells:
|
Variable
|
Find Value
|
Reduced Cost
|
Objective Coefficient
|
Allowable Increase
|
Allowable Decrease
|
|
X 1A
|
0
|
1.50
|
$7.50
|
1E+30
|
1.50
|
|
X 2A
|
100
|
0.00
|
$6.25
|
0.25
|
0.75
|
|
X 3A
|
550
|
0.00
|
$6.50
|
0.75
|
0.25
|
|
X 1B
|
400
|
0.00
|
$6.75
|
0.50
|
1E+30
|
|
X 2B
|
400
|
0.00
|
$7.00
|
0.75
|
0.50
|
|
X 3B
|
0
|
0.75
|
$8.00
|
1E+30
|
0.75
|
Constraints:
|
Name
|
Final Value
|
Shadow price
|
Constraint RH Side
|
Allowable Increase
|
Allowable Decrease
|
|
C1
|
650
|
6.50
|
650
|
50
|
550
|
|
C2
|
800
|
7.25
|
800
|
50
|
400
|
|
C3
|
400
|
-0.50
|
400
|
400
|
50
|
|
C4
|
500
|
-0.25
|
500
|
550
|
50
|
|
C5
|
550
|
0.00
|
600
|
1E+30
|
50
|
Based on the information given in the sensitivity reports,
The number of constraints that are binding = ______
For the non binding constraint, the amount of slack/surplus variable value =______
For variable X 3A, the range of optimality is from 6.25 to ______ (round your response to two decimal places).
If to customer A, 10 less tons are supplied, the impact of this on the objective value =______ (round your response to two decimal places).
If to customer B, 10 less tons are supplied, the impact of this on the objective value =________ (round your response to two decimal places).