ECE 307 – Techniques for Engineering

Decisions

Networks and Flows

George Gross

Department of Electrical and Computer Engineering

University of Illinois at Urbana-Champaign

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

1

NETWORKS AND FLOWS

A network is a system of lines or channels

connecting different points

Examples abound in nearly all aspects of life:

electrical systems

communication networks

airline webs

local area networks

distribution systems

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

2

NETWORKS AND FLOWS

The network structure is also common to many

other systems that at first glance are not

necessarily viewed as networks

distribution system consisting of

manufacturing plants, warehouses and retail

outlets

matching problems such as work to people,

assignments to machines and computer

dating

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

3

NETWORKS AND FLOWS

river systems with pondage for electricity

generation

mail delivery networks

project management of multiple tasks in a

large undertaking such as construction or a

space flight

We consider a broad range of network and

network flow problems

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

4

THE TRANSPORTATION PROBLEM

The basic idea of the transportation problem is

illustrated with the problem of distribution of a

specified homogenous product from several

sources to a number of localities at least cost

We consider a system with m warehouses, n

markets and links between them with the specified

costs of transportation

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

5

⎧

⎪1

⎪

⎪

⎪

⎪2

⎪

⎪

⎨

⎪

⎪i

⎪

⎪

⎪

⎪

⎪m

⎩

supply

demand

a1

b1

a2

..

.

..

.

transportation

links with

costs

b2

ci j

bj

ai

c i j = ∞ whenever

am

warehouse i cannot

ship to market j

bn

..

.

..

.

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

⎫

1 ⎪⎪

⎪

⎪

⎪

2⎪

⎪

⎪

⎬

⎪

j⎪

⎪

⎪

⎪

⎪

n⎪

⎪

⎭

markets

warehouses

THE TRANSPORTATION PROBLEM

6

THE TRANSPORTATION PROBLEM

all the supply comes from the m warehouses; we associate the index i = 1, 2, … , m

with a warehouse

all the demand is at the n markets; we

associate the index j = 1, 2, … , n with a

market

shipping costs c i j for each unit from the

warehouse i to the market j

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

7

THE TRANSPORTATION PROBLEM

The transportation problem is to determine the

optimal shipping schedule that minimizes shipping

costs for the set of m warehouses to the set of

n markets : the quantities shipped from the

warehouse i to each market j

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

8

LP FORMULATION OF THE

TRANSPORTATION PROBLEM

The decision variables are

xij

= quanity shipped from warehouse i to market j

i = 1, 2, ... , m

j = 1, 2, ... , n

The objective function is

min

m

n

cij xij

∑

∑

i =1 j =1

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

9

LP FORMULATION OF THE

TRANSPORTATION PROBLEM

The constraints are:

n

xij

∑

j =1

m

xij

∑

i =1

≤ ai

i = 1, 2, ... , m

≥ bj

j = 1, 2, ... , n

i = 1, 2, ... , m

xij ≥ 0

j = 1, 2, ... , n

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

10

LP FORMULATION OF THE

TRANSPORTATION PROBLEM

Note that feasibility requires

m

ai

∑

i =1

≥

n

bj

∑

j =1

When

m

ai

∑

i =1

=

n

bj

∑

j =1

every available unit of supply at the m warehouses is shipped to meet all the demands of the

n markets; this problem is known as the standard

transportation problem

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

11

STANDARD TRANSPORTATION

PROBLEM

min

m

n

cij

∑

∑

i =1

j =1

xij

s .t .

n

xij

∑

j =1

=

i = 1, ... , m

m

xij

∑

i =1

ai

=

bj

j = 1, ... , n

xij ≥ 0

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

12

TRANSPORTATION PROBLEM

EXAMPLE

market j

w/h i

W1

W2

W3

demands

M1

M2

M3

M4

x 11

x 12

x 13

x 14

c 11

x 21

c 12

x 22

c 21

x 31

x 23

x 32

x 33

x 24

c 24

x 34

c 33

c 32

b2

c 14

c 23

c 22

c 31

b1

c 13

b3

supplies

c 34

a1

a2

a3

b4

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

13

STANDARD TRANSPORTATION

PROBLEM

The standard transportation problem has

m n variables x i j

m + n equality constraints

Since

m

n

xi j

∑

∑

i =1 j =1

=

m

ai

∑

i =1

=

n

bj

∑

j =1

there are at most (m + n – 1) independent constraints and consequently at most (m + n – 1)

independent variables x i j

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

14

TRANSPORTATION PROBLEM

EXAMPLE

market j

w/h i

M1

W1

2

10

W3

M3

M4

ai

3

2

W2

bj

M2

2

1

5

8

4

7

5

7

4

3

8

6

6

4

4

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

15

THE LEAST – COST RULE

PROCEDURE

This scheme is used to generate an initial basic

feasible solution which has no more than

(m + n – 1) positive valued basic variables

The key idea of the scheme is to select, at each

step, the variable x i j with the lowest shipping costs

c i j as the next basic variable

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

16

APPLICATION OF THE LEAST – COST

RULE

c 14 is the lowest c i j and we select x 14 as a basic

variable

We choose x 14 as large as possible without

violating any constraints:

min { a 1 , b 4 } = min { 3 , 4 } = 3

We set x 14 = 3 and

x 11 = x 12 = x 13 = 0

We delete row 1 from any further consideration

since all the supplies from W 1 are exhausted

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

17

APPLICATION OF THE LEAST – COST

RULE

market j

w/h i

M1

M2

M3

M4

3

W1

2

W2

2

2

10

W3

bj

ai

1

4

5

8

3

7

5

7

4

6

6

3

4

8

4

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

18

APPLICATION OF THE LEAST – COST

RULE

The remaining demand at M 4 is

4–3 =1

which is the value for the modified demand at M 4

We again apply the criterion selection for the reduced

tableau: c 24 is the lowest-valued c i j with i = 2, j = 4

and we select x 24 as a basic variable

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

19

APPLICATION OF THE LEAST – COST

RULE

We choose x 24 as large as possible without

violating any constraints:

min { a 2 , b 4 } = min { 7 , 1 } = 1

and we set x 24 = 1 and

x 34 = 0

We delete column 4 from any further consideration since all the demand at M 4 is exhausted

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

20

APPLICATION OF THE LEAST – COST

RULE

The remaining supply at W 2 is

7–1 =6,

which is the value for the modified supply at W 2

We repeat these steps until we find the nonzero

basic variables and obtain a basic feasible solution

In the reduced tableau,

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

21

APPLICATION OF THE LEAST – COST

RULE

market j

M1

w/h i

M2

M3

ai

4

W2

6

8

10

5

0

W3

5

6

7

bj

4

3

6

4

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

22

APPLICATION OF THE LEAST – COST

RULE

pick x 23 to enter the basis

set

x 23 = min { 6, 4 } = 4

and set x 33 = 0

eliminate column 3 and reduce the supply at

W 2 to

6–4 = 2

For the reduced tableau

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

23

APPLICATION OF THE LEAST – COST

RULE

market j

M1

M2

ai

w/h i

0

W2

10

8

2

3

W3

5

7

bj

4

6

3

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

24

APPLICATION OF THE LEAST – COST

RULE

pick x 32 to enter the basis

set

x 32 = min { 3, 5 } = 3

and set x 22 = 0

eliminate column 2 in the reduced tableau

and reduce the supply at W 3 to

5–3 = 2

The last reduced tableau is

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

25

Decisions

Networks and Flows

George Gross

Department of Electrical and Computer Engineering

University of Illinois at Urbana-Champaign

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

1

NETWORKS AND FLOWS

A network is a system of lines or channels

connecting different points

Examples abound in nearly all aspects of life:

electrical systems

communication networks

airline webs

local area networks

distribution systems

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

2

NETWORKS AND FLOWS

The network structure is also common to many

other systems that at first glance are not

necessarily viewed as networks

distribution system consisting of

manufacturing plants, warehouses and retail

outlets

matching problems such as work to people,

assignments to machines and computer

dating

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

3

NETWORKS AND FLOWS

river systems with pondage for electricity

generation

mail delivery networks

project management of multiple tasks in a

large undertaking such as construction or a

space flight

We consider a broad range of network and

network flow problems

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

4

THE TRANSPORTATION PROBLEM

The basic idea of the transportation problem is

illustrated with the problem of distribution of a

specified homogenous product from several

sources to a number of localities at least cost

We consider a system with m warehouses, n

markets and links between them with the specified

costs of transportation

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

5

⎧

⎪1

⎪

⎪

⎪

⎪2

⎪

⎪

⎨

⎪

⎪i

⎪

⎪

⎪

⎪

⎪m

⎩

supply

demand

a1

b1

a2

..

.

..

.

transportation

links with

costs

b2

ci j

bj

ai

c i j = ∞ whenever

am

warehouse i cannot

ship to market j

bn

..

.

..

.

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

⎫

1 ⎪⎪

⎪

⎪

⎪

2⎪

⎪

⎪

⎬

⎪

j⎪

⎪

⎪

⎪

⎪

n⎪

⎪

⎭

markets

warehouses

THE TRANSPORTATION PROBLEM

6

THE TRANSPORTATION PROBLEM

all the supply comes from the m warehouses; we associate the index i = 1, 2, … , m

with a warehouse

all the demand is at the n markets; we

associate the index j = 1, 2, … , n with a

market

shipping costs c i j for each unit from the

warehouse i to the market j

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

7

THE TRANSPORTATION PROBLEM

The transportation problem is to determine the

optimal shipping schedule that minimizes shipping

costs for the set of m warehouses to the set of

n markets : the quantities shipped from the

warehouse i to each market j

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

8

LP FORMULATION OF THE

TRANSPORTATION PROBLEM

The decision variables are

xij

= quanity shipped from warehouse i to market j

i = 1, 2, ... , m

j = 1, 2, ... , n

The objective function is

min

m

n

cij xij

∑

∑

i =1 j =1

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

9

LP FORMULATION OF THE

TRANSPORTATION PROBLEM

The constraints are:

n

xij

∑

j =1

m

xij

∑

i =1

≤ ai

i = 1, 2, ... , m

≥ bj

j = 1, 2, ... , n

i = 1, 2, ... , m

xij ≥ 0

j = 1, 2, ... , n

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

10

LP FORMULATION OF THE

TRANSPORTATION PROBLEM

Note that feasibility requires

m

ai

∑

i =1

≥

n

bj

∑

j =1

When

m

ai

∑

i =1

=

n

bj

∑

j =1

every available unit of supply at the m warehouses is shipped to meet all the demands of the

n markets; this problem is known as the standard

transportation problem

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

11

STANDARD TRANSPORTATION

PROBLEM

min

m

n

cij

∑

∑

i =1

j =1

xij

s .t .

n

xij

∑

j =1

=

i = 1, ... , m

m

xij

∑

i =1

ai

=

bj

j = 1, ... , n

xij ≥ 0

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

12

TRANSPORTATION PROBLEM

EXAMPLE

market j

w/h i

W1

W2

W3

demands

M1

M2

M3

M4

x 11

x 12

x 13

x 14

c 11

x 21

c 12

x 22

c 21

x 31

x 23

x 32

x 33

x 24

c 24

x 34

c 33

c 32

b2

c 14

c 23

c 22

c 31

b1

c 13

b3

supplies

c 34

a1

a2

a3

b4

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

13

STANDARD TRANSPORTATION

PROBLEM

The standard transportation problem has

m n variables x i j

m + n equality constraints

Since

m

n

xi j

∑

∑

i =1 j =1

=

m

ai

∑

i =1

=

n

bj

∑

j =1

there are at most (m + n – 1) independent constraints and consequently at most (m + n – 1)

independent variables x i j

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

14

TRANSPORTATION PROBLEM

EXAMPLE

market j

w/h i

M1

W1

2

10

W3

M3

M4

ai

3

2

W2

bj

M2

2

1

5

8

4

7

5

7

4

3

8

6

6

4

4

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

15

THE LEAST – COST RULE

PROCEDURE

This scheme is used to generate an initial basic

feasible solution which has no more than

(m + n – 1) positive valued basic variables

The key idea of the scheme is to select, at each

step, the variable x i j with the lowest shipping costs

c i j as the next basic variable

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

16

APPLICATION OF THE LEAST – COST

RULE

c 14 is the lowest c i j and we select x 14 as a basic

variable

We choose x 14 as large as possible without

violating any constraints:

min { a 1 , b 4 } = min { 3 , 4 } = 3

We set x 14 = 3 and

x 11 = x 12 = x 13 = 0

We delete row 1 from any further consideration

since all the supplies from W 1 are exhausted

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

17

APPLICATION OF THE LEAST – COST

RULE

market j

w/h i

M1

M2

M3

M4

3

W1

2

W2

2

2

10

W3

bj

ai

1

4

5

8

3

7

5

7

4

6

6

3

4

8

4

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

18

APPLICATION OF THE LEAST – COST

RULE

The remaining demand at M 4 is

4–3 =1

which is the value for the modified demand at M 4

We again apply the criterion selection for the reduced

tableau: c 24 is the lowest-valued c i j with i = 2, j = 4

and we select x 24 as a basic variable

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

19

APPLICATION OF THE LEAST – COST

RULE

We choose x 24 as large as possible without

violating any constraints:

min { a 2 , b 4 } = min { 7 , 1 } = 1

and we set x 24 = 1 and

x 34 = 0

We delete column 4 from any further consideration since all the demand at M 4 is exhausted

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

20

APPLICATION OF THE LEAST – COST

RULE

The remaining supply at W 2 is

7–1 =6,

which is the value for the modified supply at W 2

We repeat these steps until we find the nonzero

basic variables and obtain a basic feasible solution

In the reduced tableau,

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

21

APPLICATION OF THE LEAST – COST

RULE

market j

M1

w/h i

M2

M3

ai

4

W2

6

8

10

5

0

W3

5

6

7

bj

4

3

6

4

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

22

APPLICATION OF THE LEAST – COST

RULE

pick x 23 to enter the basis

set

x 23 = min { 6, 4 } = 4

and set x 33 = 0

eliminate column 3 and reduce the supply at

W 2 to

6–4 = 2

For the reduced tableau

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

23

APPLICATION OF THE LEAST – COST

RULE

market j

M1

M2

ai

w/h i

0

W2

10

8

2

3

W3

5

7

bj

4

6

3

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

24

APPLICATION OF THE LEAST – COST

RULE

pick x 32 to enter the basis

set

x 32 = min { 3, 5 } = 3

and set x 22 = 0

eliminate column 2 in the reduced tableau

and reduce the supply at W 3 to

5–3 = 2

The last reduced tableau is

© 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

25

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