Math Models of OR: Network Flow Problemseaton.math.rpi.edu/.../lecture18/18A_networkflowproblemsbeamer.pdf · Simplex and network ﬂow problems We’ve looked at several kinds of

Math Models of OR:Network Flow Problems

John E. Mitchell

Department of Mathematical SciencesRPI, Troy, NY 12180 USA

November 2018

Mitchell Network Flow Problems 1 / 21

Introduction

Outline

1 Introduction

2 Transportation problem

3 Minimum cost network flow

4 Minimum cost network flow with capacities

5 Shortest path

6 Max flow and min cut

7 Sensitivity analysis


Introduction

Simplex and network flow problemsWe’ve looked at several kinds of network flow problems, all of whichare linear optimization problems. We’ve looked at using the simplexalgorithm to solve most of them, exploiting the structure of the problemto make it more efficient.

1 Find an initial basic feasible solution (a spanning tree for many ofthese problems).

2 Use complementary slackness to find a dual solution (dual istypically underdetermined and can be found using a chainreaction).

3 Check dual feasibility in the constraints corresponding to thenonbasic variables.

4 If feasible: STOP with optimality.5 Else, Find the simplex direction to bring in a nonbasic variable

with a negative reduced cost (usually a cycle in the network).6 Find the steplength so one of the basic variables leaves the basis.7 Update, and return to Step 2.


Introduction








Introduction








Introduction








Introduction








Introduction








Introduction








Transportation problem

Outline

1 Introduction




5 Shortest path





Transportation problemsOnly source nodes and destination nodes, with each source linkedto each destination.Each edge has cost cij and is uncapacitated.Dual variables ui for source i and vj for destination j .Dual constraints: ui + vj ≤ cij for all edges (i , j).

s3

s2

s1

d3

d2

d1c11 = 2

c12 = 4

c13 = 3c21 = 1

c22 = 5

c23 = 2

c31 = 1

c32 = 1

c33 = 620

20

20

20

30

10

supply sources destinations demand




s3

s2

s1

d3

d2

d1c11 = 2

c12 = 4

c13 = 3c21 = 1

c22 = 5

c23 = 2

c31 = 1

c32 = 1

c33 = 620

20

20

20

30

10





s3

s2

s1

d3

d2

d1c11 = 2

c12 = 4

c13 = 3c21 = 1

c22 = 5

c23 = 2

c31 = 1

c32 = 1

c33 = 620

20

20

20

30

10





s3

s2

s1

d3

d2

d1c11 = 2

c12 = 4

c13 = 3c21 = 1

c22 = 5

c23 = 2

c31 = 1

c32 = 1

c33 = 620

20

20

20

30

10




Optimal solution

s3

s2

s1

d3

d2

d1x11 = 10

x12 = 10

x13 = 0x23 = 20

x32 = 20

Optimal dual solution:

First set u1 = 0, thenui + vj = cij for basic xij

210 410 30 u1 = 01 5 220 u2 = −11 120 6 u3 = −3v1 = 2 v2 = 4 v3 = 3


Minimum cost network flow

Outline

1 Introduction




5 Shortest path






Allow more general network structure.Each edge has cost cij and is uncapacitated.Each node i has dual variable yi .Dual constraints: yi − yj ≤ cij for edge (i , j) leading from i to j .

1

2

3

4

5c12 = 2

c13 = 6

c24 = 1

c 32=

5

c35 = 3

c43 = 2c45

= 6

Node 1: Supply 10Node 2: Supply 10Node 3: Supply 20Node 4: Transshipment nodeNode 5: Demand 40





1

2

3

4

5c12 = 2

c13 = 6

c24 = 1

c 32=

5

c35 = 3

c43 = 2c45

= 6






1

2

3

4

5c12 = 2

c13 = 6

c24 = 1

c 32=

5

c35 = 3

c43 = 2c45

= 6






1

2

3

4

5c12 = 2

c13 = 6

c24 = 1

c 32=

5

c35 = 3

c43 = 2c45

= 6




Solution

1

2

3

4

5c12 = 2, x12 = 10

c13 = 6

c24 = 1, x24 = 20

c32 = 5

c35 = 3, x35 = 40

c43 = 2, x43 = 20

c45 = 6

Find the dual variables.

y1 − y2 = 2y2 − y4 = 1y3 − y5 = 3y4 − y3 = 2

=⇒

Set y5 = 0:

=⇒ y3 = 3=⇒ y4 = 5=⇒ y2 = 6=⇒ y1 = 8


Minimum cost network flow with capacities

Outline

1 Introduction




5 Shortest path





Problems with edge capacities

Now allow capacities uij on the edges.Use network simplex with upper bounds: variable can be nonbasicat upper or lower bound.Need dual variables yi for each node i and also dual variables wijfor each edge for the upper bound constraint.Explicitly introduced dual variables zij for the lower boundconstraints also, so the dual constraints have the formyi − yj + zij − wij = cij .Complementary slackness:

1 If xij basic then zij = 0, wij = 0, yi − yj = cij .2 If xij = 0 and nonbasic then wij = 0, zij ≥ 0, and yi − yj ≤ cij .3 If xij = uij and nonbasic then zij = 0, wij ≥ 0, and yi − yi ≥ cij .






































Example problem

1

2

3

4

5

6

triples are (cost cij , capacity uij , flow xij )

(2,3,3) UB

(3,7,5) B (7,4,0) LB

(5,6,3) B

(7,4,3) B

(3,2,0) LB

(7,5,5) UB

(5,3,2) B(9,8,3) B

x13 basic implies y1 − y3 = 3x24 basic implies y2 − y4 = 5x34 basic implies y3 − y4 = 5x35 basic implies y3 − y5 = 7x56 basic implies y5 − y6 = 9

Setting y6 = 0 then givesy5 = 9, y3 = 16, y1 = 19,y4 = 11, y2 = 16.


Shortest path

Outline

1 Introduction




5 Shortest path




Shortest path

Shortest path problems

Have a general network with edge costs cij . Want to find the shortestpath between two specific nodes s and t . Can model as a network flowproblem:

Node s has supply 1, node t has demand 1.All other nodes are transshipment nodes.

Optimal solution is integral and hence binary, provided all costs arenonnegative.

Later: we will see a dynamic programming approach to this problem.


Shortest path

Example problem

s

2

3

4

tc12 = 2

c13 = 6

c24 = 1

c 32=

5

c35 = 3

c43 = 2

c45= 6

Source: node sSink: node t

Node s: supply 1Node t : demand 1

s

2

3

4

tc12 = 2, x12 = 1

c13 = 6

c24 = 1, x24 = 1

c32 = 5

c35 = 3, x35 = 1

c43 = 2, x43 = 1

c45 = 6


Max flow and min cut

Outline

1 Introduction




5 Shortest path





Max Flow problems

Have single source s and single sink t .Have capacities uij on the edges, but no edge costs.Solve the problem using augmenting paths to push flow from sto t .Can represent as linear optimization problem. Dual problem hasinterpretation as a MAXCUT problem.Can transform to a minimum cost network flow problem withcapacities: Add an edge from t to s with cost −1 and capacity∞;all other edges have cost 0. Nodes are all transshipment.



Max Flow problems




Max Flow problems




Max Flow problems




Max Flow problems




Formulating as a min cost network flow problem

s

1

2 3

t

(15,15)

(17,10)

(12,0) (8,0)(16,15)

(10,10)(13,10) edge labels:

(uij , xij)

Equivalent to a minimum cost network flow problem with upper bounds:

s

1

2 3

t

(0,15,15)

(0,17,10)(0,12,0) (0,8,0)

(0,16,15)

(0,10,10)(0,13,10)

(−1,+∞,25)

edge labels:(cij ,uij , xij)


Sensitivity analysis

Outline

1 Introduction




5 Shortest path





Sensitivity analysisIf supplies and demands change: If can adjust flow along only thebasic arcs and still stay feasible, then we stay optimal. Since dualvariables stay the same from complementary slackness, we maintaindual feasibility.

s3

s2

s1

d3

d2

d1x11 = 10+t

x12 = 10−t

x13 = 0x23 = 20

x32 = 20+t

+t

+tOptimal dual solution:

First set u1 = 0, thenui + vj = cij for basic xij

210+t 410−t 30 u1 = 01 5 220 u2 = −11 120+t 6 u3 = −3v1 = 2 v2 = 4 v3 = 3



Change in optimal value

Optimal dual solution:210+t 410−t 30 u1 = 01 5 220 u2 = −11 120+t 6 u3 = −3v1 = 2 v2 = 4 v3 = 3

Still dual feasible.Still primal feasible and hence optimal provided −10 ≤ t ≤ 10.Optimal value decreases by 1 unit for each one unit increase in t :

from duality, increase is t(u3 + v1) = −t ;from the edge costs, increase is t(2− 4 + 1) = −t .


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