Linear Programming using Microsoft Solver
Consider the following example that demonstrates optimization of transportation.
There are production facilities in Battle Creek, Cherry Creek, and Dee Creek with annual capacities of 500 units, 400 units, and 600 units, respectively. The annual demands at warehouses in Worchester, Dorchester, and Rochester are 300 units, 700 units, and 400 units, respectively. The table below gives the unit transportation costs between the production facilities and the warehouses. Worchester 
Dorchester 
Rochester 

Battle Creek 
$20/unit 
$30/unit 
$13/unit 
Cherry Creek 
$10/unit 
$5/unit 
$17/unit 
Dee Creek 
$15/unit 
$12/unit 
$45/unit 
This problem can be modeled as a linear programming model as follows:
Decision Variables
Xbw = # of units to be transported from Battle Creek to Worchester
Xcw = # of units to be transported from Cherry Creek to Worchester
Xdw = # of units to be transported from Dee Creek to Worchester
Xbd = # of units to be transported from Battle Creek to Dorchester
Xcd = # of units to be transported from Cherry Creek to Dorchester
Xdd = # of units to be transported from Dee Creek to Dorchester
Xbr = # of units to be transported from Battle Creek to Rochester
Xcr = # of units to be transported from Cherry Creek to Rochester
Xdr = # of units to be transported from Dee Creek to Rochester
Objective Function
Minimize total annual transportation cost ($):
= 20*Xbw + 10*Xcw + 15*Xdw + 30*Xbd + 5*Xcd + 12*Xdd + 13*Xbr + 17*Xcr + 45*Xdr
Constraints
Demand Constraints
Xbw + Xcw + Xdw ≥ 300 (demand at Worchester)
Xbd + Xcd + Xdd ≥ 700 (demand at Dorchester)
Xbr + Xcr + Xdr ≥ 400 (demand at Rochester)
Capacity Constraints
Xbw + Xbd + Xbr ≤ 500 (capacity at Battle Creek)
Xcw + Xcd + Xcr ≤ 400 (capacity at Cherry Creek)
Xdw + Xdd + Xdr ≤ 600 (capacity at Dee Creek)
NonNegativity Constraints
Xbw, Xcw, Xdw, Xbd, Xcd, Xdd, Xbr, Xcr, and Xdr are ≥ 0
Integer Constraints
Xbw, Xcw, Xdw, Xbd, Xcd, Xdd, Xbr, Xcr, and Xdr are integers
The above model can be solved using the Microsoft Excel Solver tool.
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