Network Flow Problems –Maximal Flow Problems

 

Consider the following flow network:

s1

2

3

nks1

k23

k13

k3n

k1n

ks2

k21

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:1. Maximize the flow through a company’s distribution

network from its factories to its customers.2. Maximize the flow through a company’s supply network

from its vendors to its factories.3. Maximize the flow of oil through a system of pipelines.4. Maximize the flow of water through a system of aqueducts.5. Maximize the flow of vehicles through a transportation

network.

Maximal Flow Problems

 

Definitions:Flow network – consists of nodes and arcsSource node – node where flow originatesSink node – node where flow terminateTransshipment points – intermediate nodesArc/Link – connects two nodesDirected arc – arc with direction of flow indicatedUndirected arc – arc where flow can occur in either directionCapacity(kij) – maximum flow possible for arc (i,j)Flow(f ij) – flow in arc (i,j). Forward arc – arcs with flow out of some nodeBackward arc – arc with flow into some nodePath – 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

 

s1

2

3

nff

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

 

s1

2

3

nff

Max Z = fsts) fs1 + fs2 = f1) f13 + f1n = fs1 + f21 2) f21 + f23 = fs2

3) f3n = f13 + f23 n) f = f3n + f1n

0 <= fij <= kij

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.

ji,

Maximal Flow Problems – Conversion to Standard Form

 

s1 1

2

3

n1

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

s2

n2

Maximal Flow Problems – Conversion to Standard Form

 

f

f

Create a “supersource” and “supersink” with arcs fromthe supersource to the original sources and from the original sinks to the supersink. What capacity should we assign tothese new arcs?

 

s1 1

2

3

n1

s2

n2

s

n

Maximal Flow Problems – Conversion to Standard Form

 

s1

2

3

nff

What if there is an undirected arc (flow can occur in either direction)? See arc (1,2).

k12

Maximal Flow Problems – Conversion to Standard Form

 

s1

2

3

nff

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 directionof the larger of the two flows.

k21k12

Maximal Flow Problems – Lingo Solution

Maximal Flow Problems – Excel Solution