1. Network Flow Problems – Maximal Flow Problems
Consider the following flow network:
k1n
ks1 1 n s
k13
k21
k3n 3
ks2 2
k23
The objective is to ship the maximum quantity of a commodity
from a source node s to some sink node n, through a series of arcs while being constrained by a capacity k on each arc.

2. Maximal Flow Problems Examples:
Maximize the flow through a company’s distribution network from its factories to its customers.
Maximize the flow through a company’s supply network from its vendors to its factories.
Maximize the flow of oil through a system of pipelines.
Maximize the flow of water through a system of aqueducts.
Maximize the flow of vehicles through a transportation network.

3. Maximal Flow Problems Definitions:
Flow network – consists of nodes and arcs
Source node – node where flow originates
Sink node – node where flow terminate
Transshipment points – intermediate nodes
Arc/Link – connects two nodes
Directed arc – arc with direction of flow indicated
Undirected arc – arc where flow can occur in either direction
Capacity(kij) – maximum flow possible for arc (i,j)
Flow(f ij) – flow in arc (i,j).
Forward arc – arcs with flow out of some node
Backward arc – arc with flow into some node
Path – series of nodes and arcs between some originating and some terminating node
Cycle – path whose beginning and ending nodes are the same

4. Maximal Flow Problems – LP Formulation1 n f s 3 2
Objective: Maximize Flow (f)
Constraints:
1) The flow on each arc, fij, is less than or
equal to the capacity on each arc, kij.
2) Conservation of flow at each node.
Flow in = flow out.

5. Maximal Flow Problems – LP Formulation1 n f s 3
Max Z = f st
s) fs1 + fs2 = f
1) f13 + f1n = fs1 + f21
2) f21 + f23 = fs2
3) f3n = f13 + f23
n) f = f3n + f1n
0 <= fij <= kij 2
Objective: Maximize Flow (f)
Constraints:
The flow on each arc, fij, is less than or
equal to the capacity on each arc, kij.
Conservation of flow at each node.
Flow in = flow out.

6. Maximal Flow Problems – Conversion to Standard Form
What if there are multiple sources and/or multiple sinks? n1 s1 1 n2 3 s2 2

7. Maximal Flow Problems – Conversion to Standard Form
Create a “supersource” and “supersink” with arcs from
the supersource to the original sources and from the original
sinks to the supersink. What capacity should we assign to
these new arcs? n1 f s1 n 1 f s n2 3 s2 2

8. Maximal Flow Problems – Conversion to Standard Form
What if there is an undirected arc (flow can occur in either
direction)? See arc (1,2). f 1 n f s
k12 3 2

9. Maximal Flow Problems – Conversion to Standard Form
Create two directed arcs with the same capacity. Upon solving
the problem and obtaining flows on each arc, replace the two
directed arcs with a single arc with flow | fij – fji |, in the direction
of the larger of the two flows. f 1 n f s
k21
k12 3 2

10. Maximal Flow Problems – Lingo Solution

11. Maximal Flow Problems – Excel Solution