The **cost per unit** shipped from Node *i*
to Node *j* is *c _{ij}*, and the

Node

For the following picture, Node 2 has a net stock
position of - 10, which we write at the node. You need to ship 10 units
to Node 2 to satisfy the requirement at that city. There is a conservation
of flow constraint for each node. That is, the total amount remaining at
a node plus the amount shipped out of it equals the amount initially at
the node plus the amount shipped into it. Because the net stock position
of Node 2 is - 10, you need to leave 10 units there by reallocating units
from nodes that have an excess.

As an example, let's write the conservation of flow constraint for Node 2. Node 2 is connected to Node 1 by an incoming arc and to Nodes 3 and 5 by outgoing arcs. If there are no units initially at Node 2 but we desire to drop off 10 units, the conservation of flow constraint is

*x _{2,3} + x_{2,5} + 10 = x_{1,2}*

The constraint says that the total of the outgoing material plus the 100 units that must be dropped off has to equal the total of the incoming material. When written in the standard form, the constraint is

*x _{2,3} + x_{2,5} - x_{1,2
}= 10*

The arc capacity constraints are

*x _{ij} <= m_{ij}*

Sometimes, an arc has a minimum amount that is greater
than 0. For example, you might make a policy decision to ship at least
some minimal amount between two cities to maintain good relations with
a transportation service. Or you might have a contract specifying a minimum
shipment. If the minimum is greater than 0, let *n _{ij}* represent
it for Variable

*x _{ij} => n_{ij}*

Let's relate the above terminology to the real situation
of grain and cereal products. The nodes are farms, distribution centers,
mills, and bakeries. The arcs are transportation links connecting the nodes.
Some of the nodes are connected by boats, some by trains, and some by trucks.
The arcs have costs for shipping products. The boats, trains, and trucks
have capacity limits. Gathered grain is reallocated from farms to distribution
centers that serve mills. The net stock position is negative at distribution
centers that require grain. The net stock position is positive at farms
that have excess product (grain) available. This network model maximizes
the profit for grain distributing and processing. Profit maximization is
the objective because the model is also used for pricing decisions. The
total revenue is not fixed and a price change affects the demand for the
agricultural products.

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