Slide 1
Slide 1 text
Network Simplex
Twitter @jojonki
Slide 2
Slide 2 text
Enjoy!
• We will solve an optimal transform problem by Network Simplex method.
• An example and its solution follows the article below.
• Transportation Problem: Transportation Simplex Method by Rodion
Chachura.
• https://geekrodion.medium.com/transportation-problem-balancing-
fc95df5c6677
• I strongly recommend that you read through the article in advance.
Slide 3
Slide 3 text
Example (given)
30
70
50
Supply
Demand
40 30 40 40
10 8 5 4
7 6 6 8
2 2 2 1
Cost
Slide 4
Slide 4 text
North West Corner Rule
30
70
50
Supply
Demand
40 30 40 40
Slide 5
Slide 5 text
North West Corner Rule
0
70
50
Supply
Demand
30
10 30 40 40
Transportation
amount
Slide 6
Slide 6 text
North West Corner Rule
0
60
50
Supply
Demand
30
10
0 30 40 40
Slide 7
Slide 7 text
North West Corner Rule
0
30
50
Supply
Demand
30
10 30
0 0 40 40
Slide 8
Slide 8 text
North West Corner Rule
0
0
50
Supply
Demand
30
10 30 30
0 0 10 40
Slide 9
Slide 9 text
North West Corner Rule
0
0
40
Supply
Demand
30
10 30 30
10
0 0 0 40
Slide 10
Slide 10 text
North West Corner Rule
0
0
0
Supply
Demand
30
10 30 30
10 40
0 0 0 0
Initial basic feasible
solution
Slide 11
Slide 11 text
10 8 5 4
7 6 6 8
2 2 2 1
North West Corner Rule
0
0
0
Supply
Demand
30
10 30 30
10 40
0 0 0 0
Total cost = 30*2 + 10*10 + 30*8 + 30*5 + 10*6 + 40*6 = 930
Slide 12
Slide 12 text
10 8 5 4
7 6 6 8
2 2 2 1
Check feasibility
u1=0
u2=?
u3=?
Supply
Demand
30
10 30 30
10 40
v1=? v2=? v3=? v4=?
Set u1=0
Basic variables
Slide 13
Slide 13 text
10 8 5 4
7 6 6 8
2 2 2 1
Check feasibility
u1=0
u2=?
u3=?
Supply
Demand
30
10 30 30
10 40
v1=2 v2=? v3=? v4=?
Slide 14
Slide 14 text
10 8 5 4
7 6 6 8
2 2 2 1
Check feasibility
u1=0
u2=8
u3=?
Supply
Demand
30
10 30 30
10 40
v1=2 v2=? v3=? v4=?
Slide 15
Slide 15 text
10 8 5 4
7 6 6 8
2 2 2 1
Check feasibility
u1=0
u2=8
u3=?
Supply
Demand
30
10 30 30
10 40
v1=2 v2=0 v3=? v4=?
Slide 16
Slide 16 text
10 8 5 4
7 6 6 8
2 2 2 1
Check feasibility
u1=0
u2=8
u3=?
Supply
Demand
30
10 30 30
10 40
v1=2 v2=0 v3=-3 v4=?
Slide 17
Slide 17 text
10 8 5 4
7 6 6 8
2 2 2 1
Check feasibility
u1=0
u2=8
u3=9
Supply
Demand
30
10 30 30
10 40
v1=2 v2=0 v3=-3 v4=?
Slide 18
Slide 18 text
10 8 5 4
7 6 6 8
2 2 2 1
Check feasibility
u1=0
u2=8
u3=9
Supply
Demand
30
10 30 30
10 40
v1=2 v2=0 v3=-3 v4=-1
Slide 19
Slide 19 text
10 8 5 4
7 6 6 8
2 2 2 1
Check feasibility
u1=0
u2=8
u3=9
Supply
Demand
30
10 30 30
10 40
v1=2 v2=0 v3=-3 v4=-1
-2 -5 -2
3
4 3
Calculate w for all
non-basic variables
Illegal edges which
does not meet the
constraint
Maximum value
→ Entering variable
Slide 20
Slide 20 text
10 8 5 4
7 6 6 8
2 2 2 1
Looping pivot
u1=0
u2=8
u3=9
Supply
Demand
30
10 30 30
10 40
v1=2 v2=0 v3=-3 v4=-1
Specify the loop
Slide 21
Slide 21 text
10 8 5 4
7 6 6 8
2 2 2 1
Looping pivot
u1=0
u2=8
u3=9
Supply
Demand
30
10 30 30
10 40
v1=2 v2=0 v3=-3 v4=-1
Find an odd cell (blue) with
the smallest value
→ 10
Yellow: even cell
Blue: odd cell
Slide 22
Slide 22 text
10 8 5 4
7 6 6 8
2 2 2 1
Looping pivot
u1=0
u2=8
u3=9
Supply
Demand
30
10-10 30 30+10
10-10 40
v1=2 v2=0 v3=-3 v4=-1
Yellow: even cell
Blue: odd cell
0+10
Update values in the loop
Slide 23
Slide 23 text
10 8 5 4
7 6 6 8
2 2 2 1
Looping pivot
u1=0
u2=8
u3=9
Supply
Demand
30
0 30 40
40
v1=2 v2=0 v3=-3 v4=-1
10
Successfully update
the solution!
Total cost = 30*2 + 30*8 + 40*5 + 10*7 + 40*8 = 890 ← 930
0
Drop this cell (smallest
odd cell) from the basic
variables.
Slide 24
Slide 24 text
10 8 5 4
7 6 6 8
2 2 2 1
Again, check feasibility
Supply
Demand
30
30 40
40
10
u1=0
u2=?
u3=?
v1=? v2=? v3=? v4=?
Set u1=0
0
Slide 25
Slide 25 text
10 8 5 4
7 6 6 8
2 2 2 1
Again, check feasibility
Supply
Demand
30
30 40
40
10
u1=0
u2=8
u3=5
v1=2 v2=0 v3=-3 v4=3
0
Slide 26
Slide 26 text
10 8 5 4
7 6 6 8
2 2 2 1
Again, check feasibility
Supply
Demand
30
30 40
40
10
u1=0
u2=8
u3=5
v1=2 v2=0 v3=-3 v4=3
-2 -5 2
7
-4
-1
Calculate w for all
non-basic variables
0
Illegal edges which
does not meet the
constraint
Slide 27
Slide 27 text
10 8 5 4
7 6 6 8
2 2 2 1
Again, check feasibility
Supply
Demand
30
30 40
40
10
u1=0
u2=8
u3=5
v1=2 v2=0 v3=-3 v4=3
-2 -5 2
7
-4
-1
0
Maximum value
→ Entering variable
Slide 28
Slide 28 text
10 8 5 4
7 6 6 8
2 2 2 1
Looping pivot
Supply
Demand
30
30 40
40
10
u1=0
u2=8
u3=5
v1=2 v2=0 v3=-3 v4=3
0
Specify the loop
Yellow: even cell
Blue: odd cell
Find an odd cell (blue) with
the smallest value
→ 0
Slide 29
Slide 29 text
10 8 5 4
7 6 6 8
2 2 2 1
Looping pivot
Supply
Demand
30
30 40
40-0
10+0
u1=0
u2=8
u3=5
v1=2 v2=0 v3=-3 v4=3
0-0
Yellow: even cell
Blue: odd cell
0+0
Update values in the loop
Slide 30
Slide 30 text
10 8 5 4
7 6 6 8
2 2 2 1
Looping pivot
Supply
Demand
30
30 40
40
10
u1=0
u2=8
u3=5
v1=2 v2=0 v3=-3 v4=3
0
Successfully update
the solution!
Total cost = 30*2 + 30*8 + 40*5 + 10*7 + 40*8 = 890 ← 890
Not getting worse!
Slide 31
Slide 31 text
10 8 5 4
7 6 6 8
2 2 2 1
Supply
Demand
30
30 40
40
10
0
Again, check feasibility
u1=0
u2=?
u3=?
Set u1=0 v1=? v2=? v3=? v4=?
Slide 32
Slide 32 text
10 8 5 4
7 6 6 8
2 2 2 1
Supply
Demand
30
30 40
40
10
0
Again, check feasibility
u1=0
u2=1
u3=5
v1=2 v2=7 v3=4 v4=3
Slide 33
Slide 33 text
10 8 5 4
7 6 6 8
2 2 2 1
Supply
Demand
30
30 40
40
10
0
Again, check feasibility
u1=0
u2=1
u3=5
v1=2 v2=7 v3=4 v4=3
5 2 2
-7
3
6
Calculate w for all
non-basic variables
Maximum value
→ Entering variable
Slide 34
Slide 34 text
10 8 5 4
7 6 6 8
2 2 2 1
Supply
Demand
30
30 40
40
10
0
Looping pivot
u1=0
u2=1
u3=5
v1=2 v2=7 v3=4 v4=3
Find an odd cell (blue) with
the smallest value
→ 30
Slide 35
Slide 35 text
10 8 5 4
7 6 6 8
2 2 2 1
Supply
Demand
30
40
10
10
30
Looping pivot
u1=0
u2=1
u3=5
v1=2 v2=7 v3=4 v4=3
Update values in the loop
30
Slide 36
Slide 36 text
10 8 5 4
7 6 6 8
2 2 2 1
Supply
Demand
30
40
10
10
30
Looping pivot
u1=0
u2=1
u3=5
v1=2 v2=7 v3=4 v4=3
30
Successfully update
the solution!
Total cost = 30*2 + 40*5 + 30*4 + 10*7 + 30*6 + 10*8 = 710 ← 890
Slide 37
Slide 37 text
10 8 5 4
7 6 6 8
2 2 2 1
Supply
Demand
30
40
10
10
30
u1=0
u2=1
u3=5
v1=2 v2=1 v3=4 v4=3
30
Again, check feasibility
-1 2 2
-7
3
-6
Calculate w for all
non-basic variables
Maximum value
→ Entering variable
Slide 38
Slide 38 text
10 8 5 4
7 6 6 8
2 2 2 1
Supply
Demand
30
40
10
10
30
u1=0
u2=1
u3=5
v1=2 v2=1 v3=4 v4=3
30
Looping pivot
Find an odd cell (blue) with
the smallest value
→ 10
Slide 39
Slide 39 text
10 8 5 4
7 6 6 8
2 2 2 1
Supply
Demand
30
30
10
40
u1=0
u2=1
u3=5
v1=2 v2=1 v3=4 v4=3
30
Looping pivot
Update values in the loop
10
Slide 40
Slide 40 text
10 8 5 4
7 6 6 8
2 2 2 1
Supply
Demand
30
30
10
40
u1=0
u2=4
u3=5
v1=2 v2=1 v3=1 v4=0
30
Again, check feasibility
10
-1 -1 -1
-3
-3
-4
GREAT!
No illegal edges.
This is the optimal
transportation.
Slide 41
Slide 41 text
10 8 5 4
7 6 6 8
2 2 2 1
Supply
Demand
30
30
10
40
u1=0
u2=4
u3=5
v1=2 v2=1 v3=1 v4=0
30
Yay! :D
10
Total cost = 30*2 + 30*5 + 40*4 + 10*7 + 30*6 + 10*6 = 680