Question

Wildcat Paper Mills, Inc., has two paper plants, one in Cincinnati and one in Syracuse. Warehouse facilities are located in Albany and Erie. Distributors are located in Boston, New York City, and Philadelphia. Paper units manufactured at a plant must be shipped to a warehouse for inspection before they arrive at a distributor. Its management wants to find the optimal shipping arrangement that can lead to the minimal shipping cost that meets distributors' demands. The plant capacities and distributor demands for the next month are as follows:

Plant	Capacity (units)
Cincinnati	300
Syracuse	275

Distributor	Demand (units)
Boston	250
New York	125
Philadelphia	150

The unit transportation cost between any pair of locations is summarized below:  

	Cincinnati	Syracuse	Albany	Erie	Boston	New York City
Cincinnati	*
Syracuse	$     4.25	*
Albany	$     3.00	$         2.50	*
Erie	$     5.50	$         2.75	$ 2.70	*
Boston	$     6.00	$         2.25	$ 3.25	$ 0.75	*
New York City	$     4.00	$         1.45	$ 1.25	$ 5.55	$ 6.25	*
Philadelphia	$     2.15	$         3.45	$ 2.55	$ 1.85	$ 1.95	$    2.05

Define the decision variables and use them to list the linear program for this problem.

Answer 1

Decision variables: Let Xij be the quantity to ship from origin node to destination node.

where, plants nodes {1,2}, warehouses nodes {3,4} and distributor nodes {5,6,7}

Linear program is following

Min 3X13+5.5X14+2.5X23+2.75X24+4.25(X12+X21)+2.7(X34+X43)+3.25X35+1.25X36+2.55X37+0.75X45+5.55X56+1.85X57+6.25(X56+X65)+1.95(X57+X75)+2.05(X67+X76)

s.t.

X12+X13+X14-X21 <= 300

X21+X23+x24-X12 <= 275

X13+X23+X43-X34-X35-X36-X37 = 0

X14+X24+X34-X43-X45-X46-X47 = 0

X35+X45+X65+X75-X56-X57 = 250

X36+X46+X56+X76-X65-X67 = 125

X37+X47+X57+X67-X75-X76 = 150

All variables >= 0