# Optimal transportation and linear programming model

*label*Business

*timer*Asked: May 20th, 2013

**Question description**

Optimal transportation of goods in a supply chain is essential because it is important that:

• The total transportation cost is minimized

• The demand at warehouses is satisfied

• The capacity at production facilities is not exceeded

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

How much of the demand at each of the warehouses must be met by each of the production

facilities?

This problem can be modeled as a linear programming model as

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)

Non-Negativity 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