1. The minimum cost flow problem
Note for instructors: this lecture was developed to follow the two lectures that are labeled Chapter 8
MIT and James Orlin © 2003

2. Network Models Networks 1: Introduction to Networks
Paths, Cycles, Trees
The shortest path problem
Dijkstra’s Algorithm
LP Formulation of Shortest Path Problem
node arc incidence matrix
Networks 2: CPM and extensions
Arcs as precedence relation
CPM algorithm
LP Formulation of CPM
Transpose of Node-Arc Matrix
Networks 3: flows on arcs
Large number of diverse applications
MIT and James Orlin © 2003

3. Outline of Lecture Shortest Path Problem
The shortest path problem is a special case of something known as the minimum cost flow problem
We’ll introduce it soon, but first a couple of special cases.
A transportation problem.
Assignment Problem
MIT and James Orlin © 2003

4. Directed and Undirected Networks2 3 4 1 a b c d e
An Undirected Graph 2 3 4 1 a b c d e
A Directed Graph
Networks are used to transport commodities
physical goods (products, liquids)
communication
electricity, etc.
The field of Network Optimization concerns
optimization problems on networks
MIT and James Orlin © 2003

5. The shortest path problem1 2 3 4 5 6
The shortest path problem
x12
x13
x23
x25
x24
x35
x46
x54
x56 1 = 1 -1 = 1 -1 = 1 -1 = 1 -1 = -1 =
The constraint matrix is the node arc incidence matrix
MIT and James Orlin © 2003

6. Applications of Network Optimization
Physical analog of nodes
Physical analog of arcs
Flow
Communication systems
phone exchanges,
computers,
transmission
facilities, satellites
Cables, fiber optic
links, microwave
relay links
Voice messages, Data, Video transmissions
Hydraulic systems
Pumping stations Reservoirs, Lakes
Pipelines
Water, Gas, Oil, Hydraulic fluids
Integrated computer circuits
Gates, registers, processors
Wires
Electrical current
Mechanical systems
Joints
Rods, Beams, Springs
Heat, Energy
Transportation systems
Intersections, Airports, Rail yards
Highways, Airline routes Railbeds
Passengers,
freight,
vehicles,
operators
MIT and James Orlin © 2003

7. A Transportation Problem
3 warehouses: nodes 1, 2, 3
3 retailers: nodes 4, 5, 6 1 2 3 W 4 5 6 R
Supplies of goods at the warehouses (6, 7 and 2) : Upper bound on flow 6 7 2 4 8 3 1 1
Demand for goods at the warehouses (4, 8 and 3) Lower bound on flow 1 2
Costs on shipping from each warehouse to each retailer 1 1
Objective: send at most the supply, send at least the demand, and minimize cost. 1
What is the linear program for this transportation problem?
MIT and James Orlin © 2003

8. The constraints for the transportation problem1 2 3 4 5 6 W R 7 8
x26
x34
x35
x36
x25
x14
x15
x16
x24 = 6 1 1 = 7 1 = 2 1 = 4 1 = 8 1 = 3
This graph is “bipartite.” That is, the nodes are partitioned into two parts and arcs have one endpoint in each part.
MIT and James Orlin © 2003

9. The constraint matrix is a node-arc incidence matrix in disguise1 2 3 4 5 6 W R 7 8
x26
x34
x35
x36
x25
x14
x15
x16
x24 = 6 1 1 = 7 1 = 2 1 = 4 -1 = -4 1 = 8 -1 = -8 1 = 3 -1 = -3
Multiply the constraints for nodes 4, 5, 6 by -1
The matrix becomes a node-arc incidence matrix
MIT and James Orlin © 2003

10. Features of this transportation problem
The constraint matrix is (or can be made to be) the node arc incidence matrix of the network
If supplies/demands are integral, then the flows are also integral.
If the total supply is equal to the total demand, then all supply and demand constraints hold with equality
Very efficient special purpose solution techniques exist
Applications to shipment of goods
Transportation Problem
MIT and James Orlin © 2003

11. On the integrality Property
The fact that solutions to the transportation problem are integral is an amazing property.
In general, solutions to IP are fractional.
Consider x + y = 1, x – y = Unique solution (.5, .5)
But solutions to this transportation problem are integral.
In general, if there is at most one 1 and at most one –1 in any column of the constraint matrix, then every basic feasible solution is integer (so long as RHS is integral.)
For many applications, we want to restrict variables to be integer valued.
Transportation Problem
MIT and James Orlin © 2003

12. A special case: supplies and demands are all 1.
Note the special structure of the integral solutions.
There will be three arcs with 1 unit of flow. 1 2 3 4 5 6 W R W R 1 4 3 5 1 2 6 1
A solution may be viewed as an “Assignment”
MIT and James Orlin © 2003

13. A special case: supplies and demands are all 1.
Every feasible integral solution has three arcs with a flow of 1. And the LP produces integral solutions. 1 2 3 4 5 6 W R W R 1 4 2 1 5 6 1 3
Each node on the left is “assigned” to a node on the right
MIT and James Orlin © 2003

14. The Assignment Problem
3 persons: nodes 1, 2, 3
3 tasks: nodes 4, 5, 6 1 2 3 P 4 5 6 T
Each person must be assigned to a task 1 2 1 1
Each task has a person assigned
Utility of assigning a person to a task
Objective: meet constraints while maximizing total utility
Assignment Problem
Formulate this as an Integer Program. (Require all variables to be integer).
MIT and James Orlin © 2003

15. Assignment Problem
Suppose that n persons (labeled 1 to n) are assigned to n tasks (also labeled 1 to n)
Let xij = 1 if person i is assigned to task j
Let xij = 0 if person i is not assigned to task j
Let uij be the utility of assigning person i to task j
Formulate the assignment problem
MIT and James Orlin © 2003

16. The Assignment Problem
In general the LP formulation is given as
Minimize
Each supply is 1
Each demand is 1
MIT and James Orlin © 2003

17. The Transportation Problem
In general the LP formulation is given as
Minimize
All arcs are from a node in S to a node in D, and uncapacitated.
S: Supply nodes
D: Demand nodes
MIT and James Orlin © 2003

18. Comments on the Assignment Problem
The Air Force has used this for assigning thousands of people to jobs.
This is a classical problem. Research on the assignment problem predates research on LPs.
Very efficient special purpose solution techniques exist.
10 years ago, Yusin Lee and J. Orlin solved a problem with 2 million nodes and 40 million arcs in ½ hour.
MIT and James Orlin © 2003

19. Some More on the Assignment Problem1 7 2 8 3 9 4 10 5 11 6 12
There are 6! different assignments.
There are n! different assignments in an n x n assignment problem.
There are n! different assignments in an n x n assignment problem.
Diverse applications.
MIT and James Orlin © 2003

20. An Application of the Assignment Problem
Suppose that there are moving targets in space.
You can identify each target as a pixel on a radar screen.
Given two successive pictures, identify how the targets have moved.
This may be the most efficient way of tracking items.
MIT and James Orlin © 2003

21. The minimum cost flow problem
Assignment problem, transportation problem, and the shortest path problem.
All are efficiently solvable
All have constraint matrices that are the node-arc incidence matrix (or are equivalent)
All have integer value solutions when solved by simplex
We next consider the minimum cost flow problem
includes assignment, transportation, and shortest path as special cases
Has a constraint matrix that is the node arc incidence matrix
Has integer valued solutions when solved by simplex
MIT and James Orlin © 2003

22. The Minimum Cost Flow Problem
Network G = (N, A)
A network with costs,
capacities, supplies, demands
Node set N = {1, 2, 3, 4} 3 4 -5 -2
Arc set A = {(1,2), (1,3), (2,3), (2,4), (3,4)} 1 2 3 4
-$3,
$8,
$7,
$3,
$2, 6 5 2 4 7
Capacities uij on arc (i,j)
e.g., u12 = 6
Cost cij on arc (i,j)
e.g., c12 = $3
Supply/demand bi for node i.
e.g. b1 = 4 (supply)
b3 = -5 (demand)
Send flow in each arc Satisfy “supply/demand” constraints” Satisfy flow bound constraints Minimize cost
MIT and James Orlin © 2003

23. The Minimum Cost Flow Problem1 2 3 4
-$3,
$8,
$7,
$3,
$2, -5 -2
A network with costs,
capacities, supplies, demands 6 5 7
Let xij be the flow on arc (i,j).
Min z = the cost of sending flow
s.t. Flow out of i - Flow into i = bi for each i
0  xij  uij for all arcs (i,j) in A
MIT and James Orlin © 2003

24. Supply/Demand constraints -1 1 = 32 3 4
-$3,
$8,
$7,
$3,
$2, -5 -2 6 5 7
The LP Formulation
x12
x13
x23
x24
x34 -z -3 8 7 3 2 = 1 1 = 4
Supply/Demand constraints -1 1 = 3 -1 1 = -5 -1 = -2
0  x12  6,  x13  5,  x23  2,
0  x24  4,  x34  7,
Lower and upper bounds on variables
The constraint matrix is the node arc incidence matrix
MIT and James Orlin © 2003

25. On Network Flow Problems
If the constraint matrix is a node-arc incidence matrix, then the LP is a network flow problem.**
The solution values are all integer valued.
** Upper bound constraints on variables may be treated similarly to lower bound constraints, and are not viewed as part of the constraint matrix.
MIT and James Orlin © 2003

26. The Minimum Cost Flow Problem
Let xij be the flow on arc (i,j).
Minimize the cost of sending flow
s.t. Flow out of i - Flow into i = bi
0  xij  uij
Minimize
s.t.
0  xij  uij for all (i,j)
MIT and James Orlin © 2003

27. Some Information on the Min Cost Flow Problem
Reference text: Network Flows: Theory, Algorithms, and Applications by Ahuja, Magnanti, and Orlin
This is one of the few problems for which there is a provably efficient implementation of the simplex algorithm (Orlin [1997])
Basic feasible solutions of a minimum cost flow problem are integer valued (assuming that the data is integer valued)
Very efficient solution techniques in practice
15.082J/6.885J: Network Optimization
MIT and James Orlin © 2003

28. Baseball Elimination Problem
Bos
Chi
Atl.
Denver 82 79 77 76
Games Won
Games Left 9 4 8 5
Bos
Chi
Atl.
Den. -- 1 7 3
Has Chicago already been eliminated from winning in this hypothetical season finale? Has Atlanta?
MIT and James Orlin © 2003

29. Modeling as an optimization problem
xAB = number of wins by Atlanta over Boston
7- xAB = number of wins by Boston over Atlanta
xAD = number of wins by Atlanta over Denver
1- xAD = number of wins by Denver over Atlanta
xBC = …
xBD = …
xCD = …
Bos
Chi
Atl.
Den. -- 1 7 3 7 1
MIT and James Orlin © 2003

30. Bos
Chi
Atl.
Denver 82 79 77 76
Games Won
Games Left 9 4 8 5
Determining whether Chicago can win the championship, assuming Chicago wins all of its games, and thus has 83 wins. 8 6 7 AB AD BD .
RHS
Bos
7-xAB+
+xBD  1
Chi --
Atl.
xAB
+xAD  6
Den.
1-xAD +
1 - xBD  7
MIT and James Orlin © 2003
and also upper and lower bound constraints.

31. The constraint matrix is a node-arc incidence matrixAB AD BD
Bos
Atl.
Den.
7-xAB+
+xBD
xAB
+xAD
1-xAD +
1 - xBD . 
RHS 1 6 7
xAB
xAD
xBD .
RHS
Bos -1 +1  -6
Atl. 1  6
Den. -1  5
The baseball elimination problem is a minimum cost flow problem in disguise.
MIT and James Orlin © 2003

32. Another Application of the Minimum Cost Flow Problem
Warehouses 5 4
Plants 2 3 1
Customers 6 7
200
300
100
Supplies
400
180
Demands
Ship from suppliers to customers, possibly through warehouses, at minimum cost to meet demands.

33. Matrix Fill In 16 11 20 6 Row sums ? 1 2 5 3 ? 19 13 17 7 Column sums
Is it possible to fill in the missing data with non-negative integers and have the row sums and column sums as specified? If so, how large or small can be the element in position 2,3?
How is this a transportation problem?
MIT and James Orlin © 2003

34. Transportation Problem
Matrix Fill In
entry in (i,j) is “flow”
Rows
Columns
Minimize entry (2,3) ? 1 2 5 3 5 1 ? 1 16 9 13 9 12 1 1 2 3 4 5 1 2 3 4 5 12 7 9 4 2 ? ? ? 1 16 13 1 1 ? ? ? 11 13 9 ? ? 2 5 1 20 12 1 1 3 ? ? 6 1 13 9 19 12 7 13 12 17 7 4
Determine the row and column sums for unknown entries.
Formulate a transportation problem
Transportation Problem
MIT and James Orlin © 2003

35. Conclusions.
Advantages of the transportation problem and the minimum cost flow problem
Integer solutions
Very fast solution methods
Extremely common in modeling
MIT and James Orlin © 2003

36. Multicommodity Flows: a brief glimpse at an important problem
Send 6 units of commodity 1 from node 1 to node 4.
$5, 4
$2, 4
$6, 4
$0, 8 5 1 2 3 4 5 6 5
Send 3 units of commodity 2 from node 3 to node 6.
Two minimum cost flows on a shared network 3 3
The costs and capacities are given.
What is the minimum cost multicommodity flow?
MIT and James Orlin © 2003

37. Multicommodity Flows $5, 4 6 1 4 6 $0, 8 $0, 8 $2, 4 2 5 $0, 8 $0, 8
Send 5 units of commodity 1 from node 1 to node 4.
Send 3 units of commodity 2 from node 3 to node 6. 4
$5, 4 6 1 4 6 2
$0, 8
$0, 8
$2, 4 2 5 2
$0, 8
$0, 8
$6, 4 3 6 3 3 1
The shared arc capacities make the problem much more complex. Optimum flows may be fractional.
MIT and James Orlin © 2003

38. Multicommodity flow Problems can be large
Consider a communication network with 1000 nodes, and 10,000 arcs.
Suppose each node wants to send a different message to each other node.
1,000,000 messages.
1,000,000 commodities
10 billion variables. (10,000 variables per commodity)
MIT and James Orlin © 2003

39. Multicommodity Flow Problems are important in practice
Manufacturing Networks
Road and Rail Transportation Networks
Air Transportation Networks
Communication Networks
MIT and James Orlin © 2003

40. Final comments on multicommodity flows
We mention them because they are important extensions of the minimum cost flow problem
Much harder to solve in practice
Lots of applications
Lots of Research at MIT
Cynthia Barnhart
Jim Orlin
MIT and James Orlin © 2003

41. Summary Minimum Cost Flows shortest paths transportation problem
assignment problem
plus more
important subproblem of many optimization problems, including multicommodity flows
MIT and James Orlin © 2003