The usual objective for a minimum$-$cost flow problem is

to minimize the total cost of sending the available supply through the network to satisfy the given demand.

to minimize the amount of flow leaving the supply node in the network or the amount of flow entering the demand

node in the network.

to minimize the number of paths between a supply node and a demand node in the network.

to minimize the cost of generating network flows at the supply nodes and cost of using network flows at the demand

nodes