1. The Network Simplex Method
Reduced Costs and Node Potentials
Updated 7 April 2008

2. Reduced Costs: Generic Simplex Method
The row 0 coefficients of the Simplex Tableau are known as reduced costs.
The reduced cost of a basic variable is zero.
Non-basic variables with negative reduced cost are eligible to enter the basis.
If all the reduced costs are non-negative, then the Simplex method terminates.
Network Simplex: Part 2

3. Formula to Compute Reduced Costs
Let B be the set of basic variables.
Let AB be the submatrix of A comprised of the columns corresponding to the basic variables.
Let c' be the vector of reduced costs (i.e. row 0 of the tableau).
c' = c – A where  = cB(AB)-1.
 are referred to as the dual multipliers.
Network Simplex: Part 2

4. Formula in the Network Context
Network Simplex: Part 2

5. Formula in the Network Context
Network Simplex: Part 2

6. Reduced Costs: Network Simplex
Let c'ij be the reduced cost of arc (i, j) and i be the potential at node i.
c'ij = cij - i + j where
1 = 0
c'ij = 0 for all basic arcs
Eligible non-basic arcs
Arcs in L are eligible if c'ij < 0
Arcs in U are eligible if c'ij > 0
Network Simplex: Part 2

7. An initial BFS Basic arcs (variables)
Non-basic arcs at their lower bounds
L = {(1,4), (3,5)}
Non-basic arcs at their upper bounds.
U = {(3,4)}
Network Simplex: Part 2

8. Initial Solution xij 4 bi i j L = {(1,4),(3,5)} 2 U = {(3,4)} 6 2 -3 81 3 5 10 -7 5 1 4
Cost = 109 -4
Network Simplex: Part 2

9. Calculating Node Potentials
c'12 = c12 - 1 + 2
c'12 = 2
0 = 5 + 2
2 = -5
cij i j 2 5 2
3 = -4 4 1 3 5
5 = -7
1=0 5 4
4 = -2
Network Simplex: Part 2

10. Calculating Reduced Costs
c'ij i j
2 = -5
c'14=7+0-2=5
c'35=8+4-7=5 2
3 = -4 5 1 3 5
5 = -7
1=0 12 5
(3,4) is the only eligible arc. 4
4 = -2
Network Simplex: Part 2

11. Pivot Step: Decrease flow on (3,4)j
xij bi 4 2
2+
6+ -3
8- 1 3 5 10 -7
5-
1- 4
  1 -4
Network Simplex: Part 2

12. Decreasing the flow on (3,4)
Consider decreasing the flow on (3,4) by 
Cost = (-c13 -c34 - c45 + c12 + c25)  =109 + ( )  = 
Maximum decrease is  = 1.
Flow on (4,5) goes to 0.
Network Simplex: Part 2

13. Decreasing the flow on (3,4)
To bring arc (3,4) in to the basis we need to reduce its flow.
The more we reduce the flow on (3,4), the more we will improve the objective function.
To maintain flow balance, we must
Keep the flow on (1,4) and (3,5) at zero
Decrease flow on (1,3) and (4,5)
Increase flow on (1,2) and (2,5)
When (3,4) becomes basic, (4,5) will become non-basic and join L.
Network Simplex: Part 2

14. New Solution i j xij bi 4 L = {(1,4), (3,5),(4,5)} 2 3 7 -3 7 1 3 5 101 3 5 10 -7 4 4
Cost = 97 -4
Network Simplex: Part 2

15. Recalculate Node Potentials
2 = -5
cij i j 2 5 2
3 = -4 4 1 3 5
5 = -7
1=0 10 4
4 = -14
Network Simplex: Part 2

16. Recalculate Reduced Costs
2 = -5
c'14=7-14
c'ij i j 2
c'35=5+4-7
c'45=5+14-7
3 = -4 2 1 3 5
5 = -7
1=0 -7 12 4
(1,4) becomes basic.
It is in L and c' < 0.
4 = -14
Network Simplex: Part 2

17. Increasing the Flow on (1,4)j
xij bi 4 2 3 7 -3
7- 1 3 5 10 -7
4-
0+ 4
Cost = 97 -4
Network Simplex: Part 2

18. New Solution 4 i j xij bi 2 3 7 -3 3 1 3 5 10 -7 4 4 Cost = 69 -43 1 3 5 10 -7 4 4
Cost = 69 -4
Network Simplex: Part 2

19. Recalculate Node Potentials
2 = -5
cij i j 2 5 2
3 = -4 4 1 3 5
5 = -7
1=0 7 4
4 = -7
Network Simplex: Part 2

20. Recalculate Reduced Costs
2 = -5
c'34 =10+4-7=7
c'ij i j 2
c'35 = 8+4-7=5
c'45 = 5+7-7=5
3 = -4 1 3 5
5 = -7
1=0 4
4 = -7
Network Simplex: Part 2

21. Outline of Network Simplex
Determine a starting BFS. The n-1 basic variables correspond to a spanning tree.
Compute the node potentials.
Compute the reduced costs for the non-basic arcs.
The current solution is optimal if all arcs in L have non-negative reduced cost and all arcs in U have non-positive reduced cost.
Select a non-basic arc (i, j) to enter the basis.
Identify the cycle created by adding (i, j) to the tree. Use flow conservation to determine the new flows on the arcs in the cycle. The arc that first hits its lower or upper limit leaves the basis.
Network Simplex: Part 2