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  1. Suppose we have a network with node set $N = \{1,\dots, n\}$ and arc set $E$. $(i, j) \in E$ if there is a link between node $i$ and node $j$. We need to send $L$ commodities from their respective origin to destination node. The multicommodity flow problem has the following specifications:

(a) There are $L$ origin-destination (O-D) pairs of nodes $(s_1, t_1), \dots,(s_L, t_L)$, and $d^k$ is the amount of flow that must be sent from $s_k$ to $t_k$ for $k = 1, \dots, L$.

(b) $u_{i,j}$ is the capacity (shared by all commodities) on the arc $(i, j)$ for any $(i, j) \in E$.

(c) $c^k_{i,j}$ is the cost of sending one unit of commodity $k$ through arc $(i, j)$. Write down an optimization model for the above problem to minimize the total cost.

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This is the multicommodity flow problem. A formulation is given here.

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