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Network Flow Problems – Maximal Flow ProblemsPowerPoint Presentation

Network Flow Problems – Maximal Flow Problems

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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.

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.

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

Maximal Flow Problems – LP Formulation

f

1

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.

Maximal Flow Problems – LP Formulation

f

1

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.

Maximal Flow Problems – Conversion to Standard Form

What if there are multiple sources and/or multiple sinks?

n1

s1

1

n2

3

s2

2

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

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

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

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