EMIS 8373: Integer Programming

1
EMIS 8373 Integer
Programming

• Easy Integer Programming Problems Network Flow
  Problems
• updated 11 February 2007

2
The Minimum Cost Network Flow Problem (MCNFP)

• Extremely useful model in OR EM
• Important Special Cases of the MCNFP
• Transportation and Assignment Problems
• Maximum Flow Problem
• Minimum Cut Problem
• Shortest Path Problem
• Network Structure
• BFSs for MCNFP LPs have integer values !!!
• Problems can be formulated graphically

3
Elements of the MCNFP

• Defined on a network G (N,A)
• N is a set of n nodes 1, 2, , n
• Each node i has an associated value b(i)
• b(i) lt 0 gt node i is a demand node with a demand
  for b(i) units of some commodity
• b(i) 0 gt node i is a transshipment node
• b(i) gt 0 gt node i is a supply node with a supply
  of b(i) units

4
Elements of the MNCFP

• A is a set of arcs that carry flow
• Decision variable xij determines the units of
  flow on arc (i,j)
• The arc (i,j) from node i to node j has
• cost cij per unit of flow on arc (i,j)
• upper bound on flow of uij (capacity)
• lower bound on flow of ?ij (usually 0)

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Example MCNFP

• N 1, 2, 3, 4
• b(1) 5, b(2) -2, b(3) 0, b(4) -3
• A (1,2), (1,3), (2,3), (2,4), (3,4)
• c12 3, c13 2, c23 1, c24 4, c34 4
• ?12 2, ?13 0, ?23 0, ?24 1, ?34 0
• u12 5, u13 2, u23 2, u24 3, u34 3

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Graphical Network Flow Formulation
(cij, ?ij, uij)
i
j
arc (i,j)
b(j)
b(i)
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Example MCNFP
-2
(4, 1,3)
(3, 2,5)
2
5
-3
1
4
(1, 0,2)
(4, 0,3)
(2, 0,2)
3
0
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Requirements for a Feasible Flow

• Flow on all arcs is within the allowable bounds
  ?ij ? xij ? uij for all arcs (i,j)
• Flow is balanced at all nodes
• flow out of node i - flow into node i b(i)
• MCNFP find a minimum-cost feasible flow

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LP Formulation of MCNFP
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LP for Example MCNFP
Min 3X12 2 X13 X23 4 X24 4 X34
s.t. X12 X13 5 Node 1
X23 X24 X12 -2 Node 2 X34 X13 -
X23 0 Node 3 X24 - X34
-3 Node 4 2 ? X12 ? 5, 0 ? X13 ? 2,
0 ? X23 ? 2, 1 ? X24 ? 3, 0 ? X34 ?
3,
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Example Feasible Solution
-2
(4, 1,3)
(3, 2,5)
2
5
3
5
-3
1
4
(1, 0,2)
0
0
0
(4, 0,3)
(2, 0,2)
3
Cost 15 12 27
0
12
Optimal Solution for Example
-2
(4, 1,3)
(3, 2,5)
2
3
1
5
-3
1
4
(1, 0,2)
0
2
2
(4, 0,3)
(2, 0,2)
3
Cost 25
0
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Transportation Problems
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Graphical Network Flow Formulation
(cij, uij)
i
j
arc (i,j)
b(j)
b(i)
?ij0
15
Supply Nodes
Demand Nodes
(13, 1)
4
I
-1
(35, 1)
1
F
-1
(42, 1)
(0,1)
2
(0,4)
G
-1
(9, 1)
(0,2)
-3
D
S
-1
Dummy Node
16
Supply Nodes
Demand Nodes
4
I
F
1
2
G
S
-3
Dummy Node
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Shortest Path Problems

• Defined on a Network with two special nodes s
  and t
• A path from s to t is an alternating sequence of
  nodes and arcs starting at s and ending at t
• s,(s,n1),n1,(n1,n2),,(ni,nj),nj,(nj,t),t
• Find a minimum-cost path from s to t

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Shortest Path Example
5
10
1
2
3
s
t
7
1
7
4
1,(1,2),2,(2,3),3 Length 15 1,(1,2),2,(2,4),4,(4
,3) Length 13 1,(1,4),4,(4,3),3 Length 14
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MCNFP Formulation of Shortest Path Problems

• Source node s has a supply of 1
• Sink node t has a demand of 1
• All other nodes are transshipment nodes
• Each arc has capacity 1
• Tracing the unit of flow from s to t gives a path
  from s to t

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Shortest Path as MCNFP
0
(5,1,0)
(10,0,1)
1
2
3
1
-1
(1,0,1)
(7,0,1)
4
(7,0,1)
0
1
0
1
2
3
1
1
0
4
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Shortest Path Example

• In a rural area of Texas, there are six farms
  connected by small roads. The distances in miles
  between the farms are given in the following
  table.
• What is the minimum distance to get from Farm 1
  to Farm 6?

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Graphical Network Flow Formulation
(cij)
i
j
arc (i,j)
b(j)
b(i)
?ij 0, uij1
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Formulation as Shortest Path
0
0
9
2
4
4
8
s
t
5
1
6
4
3
10
1
6
5
-1
2
3
5
0
0
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LP Formulation
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Maximum Flow Problems

• Defined on a network
• Source Node s
• Sink node t
• All other nodes are transshipment Nodes
• Arcs have capacities, but no costs
• Maximize the total flow from s to t

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Example Rerouting Airline Passengers

• Due to a mechanical problem, Fly-By-Night
  Airlines had to cancel flight 162 - its only
  non-stop flight from San Francisco to New York.
• Formulate a maximum flow problem to reroute as
  many passengers as possible from San Francisco to
  New York.

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Data for Fly-by-Night Example
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Network Representation
2
D
C
4
5
s
t
SF
NY
4
6
7
5
H
A
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Graphical Network Flow Formulation
(uij)
i
j
arc (i,j)
b(j)
b(i)
?ij 0 cij 0
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MCNF Formulation of Maximum Flow Problems

• Let arc cost 0 for all arcs
• Add an arc from t to s
• Give this arc a cost of 1 and infinite capacity
• All nodes are transshipment nodes
• Circulation Problem

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Formulation as MCNFP
(0,0,2)
D
C
(0,0,4)
(0,0,5)
SF
NY
(0,0,4)
(0,0,7)
(0,0,6)
(0,0,5)
H
A
(-1,0,?)
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MCNFP Solution
(0,0,2)
D
C
(0,0,4)
(0,0,5)
2
2
4
SF
NY
(0,0,4)
2
(0,0,7)
(0,0,6)
(0,0,5)
5
H
A
7
5
(-1,0,?)
9
33
LP Formulation
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NSC Example

• Max production per month 4,000 tons
• Inventory holding cost 120/ton/month
• Initial inventory 1,000 tons
• Final inventory 1,500 tons

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Network Flow Formulation
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Arc Parameters

• All arcs have ?ij 0 and uij ?
• Arcs (pi, d0) have cost 0.
• Arcs (Ii, di1) and (Ii,Ii1) have cost 120.

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Backorder Cost of 200/unit/month
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Parameters for Backorder Arcs

• All arcs have ?ij 0 and uij ?