Section 2.9

Maximum Flow Problems

Minimum Cost Network Flows

Shortest Path Problems

Maximum Flow Problems

Reference: Network Flows by Ahuja, Magnanti, and Orlin

Background

• In a capacitated network, we wish to send as much flow as possible between two special nodes, a source node s and a sink node t, without exceeding the capacity of any arc.

• The problem reportedly rose in relation to the rail networks of the Soviet Union, during the 1950's. The US wanted to know how quickly the Soviet Union could get supplies through its rail network to its satellite states in Eastern Europe.

• In addition, the US wanted to know which rails it could destroy most easily to cut off the satellite states from the rest of the Soviet Union. It turned out that these two problems were closely related, and that solving the max flow problem also solves the min cut problem of figuring out the cheapest way to cut off the Soviet Union from its satellites.

Additional reference: The Importance of Algorithms by Ibackstrom from www.topcoder.com

Assumptions

• The network is directed.• All capacities are nonnegative integers.• The network does not contain a directed path

from node s to node t composed only of infinite capacity arcs.

• Whenever an arc (i,j) belongs to A, arc (j,i) also belongs to A.

• The network does not contain parallel arcs.

Max Flow General Formulation

:( , ) :( , )

,max X

. .

0

0 for each ( , )

t s

ij jiA Aj i j j j

ij

i

ij

s t

X X i V

u AjX i

The standard LP formulation of the Maximum s-t Flow problem adds a dummyReturn arc, (t,s), to the network in question, and then maximizes the flow on thisarc. The capacity of the return arc is unbounded, and the rest of the formulationconsists of balance of flow and capacity constraints.

Network Flow ModelLet denote the set of nodes and denote the set of arcs

in our network . At each node denote the

supply at node , and the demand at node , and define

; we shall assume th

, let

at

i

j

i i i

V A

G i

i d j

s

V s

b d

( ,

0,

that is, the total amount of supply in the network equals

the total demand. The Minimum Cost Network Flow

Problem is given by

min

iV

ij i

i

ij

b

c X

)

:( , ) :( , )

X 0, (

. .

,

,

)

j

j i j k

A

ij ki iA

j

k iA

i

V

i A

X i

j

s t

X b

Importance of Network Models

• Claim 2.1 – Nearly all network models can be formulated as minimum cost network flow problems

• Claim 2.2 – Any problem modeled as a minimum cost network flow problem has an optimal solution that has only integer values, provided each supply and each demand has integral value

College wants to know when to purchase a new copier over the next 6 years.During those years that a copier is not purchased, maintenance must be performedto keep machine running. Maintenance costs are shown below.

Age at Start of Year Maint. Cost for Next Year

0 $2000

1 $3500

2 $6000

3 $9500

4 $14,500

At the start of this school year, a new copier is purchased at a cost of $10,000.The cost (in today’s dollars) of purchasing a machine at the start of each of thenext 5 years is given below.

Year Purchase Cost

1 $10,000

2 $13,000

3 $16,500

4 $20,000

5 $25,000

No trade-in value when a machine is replaced. Determine the years in which a new copiershould be purchased to minimize total cost (purchase + maintenance) for 5 years.

Costs

25

(purchase price in year )

(maintenance costs during years , 1,..., 1)

(resale price in year after years of service).

13 2 3.5 6 24.5

12 15.5 21.5 30.5 43.5

15 18.5 24.5 34

ijc i

i i j

j j i

c

c

18.5 22 28

24 27.5

27

Formulation

12 13 14 15 16

23 24 25 26 12

34 35 36 13 23

45 46 14 24 34

56 15 25 35 45

16 26 36 46 5

(

6

, )

min

. .

1

0

0

0

0

1

0, ( , ) X

i jij ij

A

ij

c X

s t

X X X X X

X X X X X

X X X X X

X X X X X

X X

i

X X X

X X X X X

j A