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A PRODUCTION AND INVENTORY APPLICATION 323

programming model can be developed from the network by establishing a constraint
for each node and a variable for each arc.
Let x 15 denote the number of square metres of carpet manufactured in quarter 1.
The capacity of the facility is 600 square metres in quarter 1, so the production
capacity constraint is:
x 15 600

Using similar decision variables, we obtain the production capacities for quarters 2–4:

x 26 300
x 37 500
x 48 400

We now consider the development of the constraints for each of the demand
nodes. For node 5, one arc enters the node, which represents the number of square
metres of carpet produced in quarter 1, and one arc leaves the node, which
represents the number of square metres of carpet that will not be sold in quarter 1
and will be carried over for possible sale in quarter 2. In general, for each quarter
the beginning inventory plus the production minus the ending inventory must equal
demand. However, for quarter 1 there is no beginning inventory; thus, the constraint
for node 5 is:

x 15 x 56 ¼ 400

The constraints associated with the demand nodes in quarters 2, 3 and 4 are:

x 56 þ x 26 x 67 ¼ 500
x 67 þ x 37 x 78 ¼ 400
x 78 þ x 48 ¼ 400

Note that the constraint for node 8 (fourth-quarter demand) involves only two
variables because no provision is made for holding inventory for a fifth quarter.
The objective is to minimize total production and inventory cost, so we write the
objective function as:
Min 2x 15 þ 5x 26 þ 3x 37 þ 3x 48 þ 0:25x 56 þ 0:25x 67 þ 0:25x 78

The complete linear programming formulation of the Contois Carpets problem is:

Min 2x 15 þ 5x 26 þ 3x 37 þ 3x 48 þ 0:25x 56 þ 0:25x 67 þ 0:25x 78
s:t:
600
x 15
x 26 300
x 37 500
x 48 400
x 15 x 56 ¼ 400
þ x 56 ¼ 500
x 26 x 67
þ x 67 x 78 ¼ 400
x 37
þ x 78 ¼ 400
x 48
x ij 0 for all i and j

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