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# Linear Programming Constraints and Binding

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:

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).

#### Solution Preview

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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*X1A + \$6.25*X2A + \$6.50*X3A+ ...

#### Solution Summary

The expert examines linear programming constraints and bindings for slack/surplus variables.

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