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Transportation-1
Network Problems
• Linear programming has a wide variety of applications
• Network problems
– Special types of linear programs
– Particular structure involving networks
• Ultimately, a network problem can be represented as a
linear programming model
• However the resulting A matrix is very sparse, and
involves only zeroes and ones
• This structure of the A matrix led to the development of
specialized algorithms to solve network problems
Transportation-2
Types of Network Problems
• Shortest Path
Special case: Project Management with PERT/CPM
• Minimum Spanning Tree
• Maximum Flow/Minimum Cut
• Minimum Cost Flow
Special case: Transportation and Assignment Problems
• Set Covering/Partitioning
• Traveling Salesperson
• Facility Location
and many more
Transportation-3
The Transportation Problem
Transportation-4
The Transportation Problem
• The problem of finding the minimum-cost distribution of
a given commodity
from a group of supply centers (sources) i=1,…,m
to a group of receiving centers (destinations) j=1,…,n
• Each source has a certain supply (si)
• Each destination has a certain demand (dj)
• The cost of shipping from a source to a destination is
directly proportional to the number of units shipped
Transportation-5
Simple Network Representation
1
Supply s2
2
…
Supply s1
Destinations
1
Demand d1
2
Demand d2
…
Sources
xij
Supply sm
n
m
Demand dn
Costs cij
Transportation-6
Example: P&T Co.
• Produces canned peas at three canneries
Bellingham, WA, Eugene, OR, and Albert Lea, MN
• Ships by truck to four warehouses
Sacramento, CA, Salt Lake City, UT, Rapid City, SD, and
Albuquerque, NM
• Estimates of shipping costs, production capacities and
demands for the upcoming season is given
• The management needs to make a plan on the least
costly shipments to meet demand
Transportation-7
Example: P&T Co. Map
1
2
1
3
3
2
4
Transportation-8
Example: P&T Co. Data
Warehouse
Supply
Cannery
1
2
3
4
1
$ 464
$ 513
$ 654
$ 867
75
2
$ 352
$ 216
$ 690
$ 791
125
3
$ 995
$ 682
$ 388
$ 685
100
Demand
80
65
70
85
(Truckloads)
(Truckloads)
Shipping cost
per truckload
Transportation-9
Example: P&T Co.
• Network representation
Transportation-10
Example: P&T Co.
• Linear programming formulation
Let xij denote…
Minimize
subject to
Transportation-11
General LP Formulation for Transportation
Problems
Transportation-12
Feasible Solutions
• A transportation problem will have feasible solutions if
and only if
m
n
s  d
i 1
i
j 1
j
• How to deal with cases when the equation doesn’t hold?
Transportation-13
Integer Solutions Property: Unimodularity
• Unimodularity relates to the properties of the A matrix
(determinants of the submatrices, beyond scope)
• Transportation problems are unimodular, so we get the
integers solutions property:
For transportation problems, when every si and dj
have an integer value, every BFS is integer valued.
• Most network problems also have this property.
Transportation-14
Transportation Simplex Method
• Since any transportation problem can be formulated as
an LP, we can use the simplex method to find an optimal
solution
• Because of the special structure of a transportation LP,
the iterations of the simplex method have a very special
form
• The transportation simplex method is nothing but the
original simplex method, but it streamlines the iterations
given this special form
Transportation-15
Transportation Simplex Method
Initialization
(Find initial CPF solution)
Is the
current
CPF
solution
optimal?
Yes
Stop
No
Move to a better
adjacent CPF solution
Transportation-16
The Transportation Simplex Tableau
Destination
Source
1
2
…
m
Demand
1
2
c11
c12
c21
c22
…
cm1
…
…
…
cm2
d1
…
…
…
d2
…
n
c1n
Supply
ui
s1
c2n
s2
…
cmn
…
sm
dn
Z=
vj
Transportation-17
Prototype Problem
• Holiday shipments of iPods to distribution centers
• Production at 3 facilities,
– A, supply 200k
– B, supply 350k
– C, supply 150k
• Distribute to 4 centers,
–
–
–
–
N, demand 100k
S, demand 140k
E, demand 300k
W, demand 250k
• Total demand vs. total supply
Transportation-18
Prototype Problem
Destination
Source
A
B
C
Dummy
Demand
N
S
E
W
16
13
22
17
14
13
19
15
9
20
23
10
0
0
0
0
100
140
300
Supply
ui
200
350
150
90
250
Z=
vj
Transportation-19
Finding an Initial BFS
• The transportation simplex starts with an initial basic
feasible solution (as does regular simplex)
• There are alternative ways to find an initial BFS, most
common are
– The Northwest corner rule
– Vogel’s method
– Russell’s method (beyond scope)
Transportation-20
The Northwest Corner Rule
• Begin by selecting x11, let x11 = min{ s1, d1 }
• Thereafter, if xij was the last basic variable selected,
– Select xi(j+1) if source i has any supply left
– Otherwise, select x(i+1)j
Transportation-21
The Northwest Corner Rule
Destination
Source
A
B
C
Dummy
Demand
N
S
16
E
13
100
14
22
17
19
15
200
100
13
40
9
Supply
W
20
300
23
10
10
150
0
0
0
0
90
100
140
300
350
150
90
250
Z = 10770
Transportation-22
Vogel’s Method
• For each row and column, calculate its difference:
= (Second smallest cij in row/col) - (Smallest cij in row/col)
• For the row/col with the largest difference, select entry
with minimum cij as basic
• Eliminate any row/col with no supply/demand left from
further steps
• Repeat until BFS found
Transportation-23
Vogel’s Method (1): calculate differences
Destination
Source
A
B
C
Dummy
Demand
diff
N
S
E
W
16
13
22
17
14
13
19
15
9
20
23
10
0
0
0
0
100
140
300
250
9
13
19
10
Supply
diff
200
3
350
1
150
1
90
0
Transportation-24
Vogel’s Method (2): select xDummyE as basic
variable
Destination
Source
A
B
C
Dummy
Demand
diff
N
S
E
W
16
13
22
17
14
13
19
15
9
20
23
10
0
0
0
0
90
100
140
300
250
9
13
19
10
Supply
diff
200
3
350
1
150
1
90
0
Transportation-25
Vogel’s Method (3): update supply, demand
and differences
Destination
Source
A
B
C
Dummy
Demand
diff
N
S
E
W
16
13
22
17
14
13
19
15
9
20
23
10
0
0
0
0
90
100
140
210
250
5
0
3
5
Supply
diff
200
3
350
1
150
1
---
---
Transportation-26
Vogel’s Method (4): select xCN as basic
variable
Destination
Source
A
B
C
Dummy
Demand
diff
N
S
E
W
16
13
22
17
14
13
19
15
9
20
23
10
0
0
0
100
0
90
100
140
210
250
5
0
3
5
Supply
diff
200
3
350
1
150
1
---
---
Transportation-27
Vogel’s Method (5): update supply, demand
and differences
Destination
Source
A
B
C
Dummy
N
S
E
W
16
13
22
17
14
13
19
15
9
20
23
10
0
0
0
100
0
90
Demand
---
140
210
250
diff
---
0
3
5
Supply
diff
200
4
350
2
50
10
---
---
Transportation-28
Vogel’s Method (6): select xCW as basic
variable
Destination
Source
A
B
C
Dummy
N
S
E
W
16
13
22
17
14
13
19
15
9
20
23
10
100
0
50
0
0
0
90
Demand
---
140
210
250
diff
---
0
3
5
Supply
diff
200
4
350
2
50
10
---
---
Transportation-29
Vogel’s Method (7): update supply, demand
and differences
Destination
Source
A
B
C
Dummy
N
S
E
W
16
13
22
17
14
13
19
15
9
20
23
10
100
0
50
0
0
0
90
Demand
---
140
210
200
diff
---
0
3
2
Supply
diff
200
4
350
2
---
---
---
---
Transportation-30
Vogel’s Method (8): select xAS as basic
variable
Destination
Source
A
B
C
Dummy
N
16
S
E
13
W
22
17
140
14
13
19
15
9
20
23
10
100
0
50
0
0
0
90
Demand
---
140
210
200
diff
---
0
3
2
Supply
diff
200
4
350
2
---
---
---
---
Transportation-31
Vogel’s Method (9): update supply, demand
and differences
Destination
Source
A
B
C
Dummy
N
16
S
E
13
W
22
17
140
14
13
19
15
9
20
23
10
100
0
50
0
0
0
90
Demand
---
---
210
200
diff
---
---
3
2
Supply
diff
60
5
350
4
---
---
---
---
Transportation-32
Vogel’s Method (10): select xAW as basic
variable
Destination
Source
A
B
C
Dummy
N
16
S
E
13
W
22
17
140
60
14
13
19
15
9
20
23
10
100
0
50
0
0
0
90
Demand
---
---
210
200
diff
---
---
3
2
Supply
diff
60
5
350
4
---
---
---
---
Transportation-33
Vogel’s Method (11): update supply, demand
and differences
Destination
Source
A
B
C
Dummy
N
16
S
E
13
W
22
17
140
60
14
13
19
15
9
20
23
10
100
0
50
0
0
0
90
Demand
---
---
diff
---
---
210
Supply
diff
---
---
350
4
---
---
---
---
140
Transportation-34
Vogel’s Method (12): select xBW and xBE as
basic variables
Destination
Source
A
B
C
Dummy
Demand
diff
N
16
S
E
13
Supply
diff
60
---
---
140
---
W
22
17
140
14
13
19
15
210
9
20
23
10
100
0
50
0
0
0
90
-----
-----
---
---
---
---
---
--Z = 10330
Transportation-35
Optimality Test
• In the regular simplex method, we needed to check the
row-0 coefficients of each nonbasic variable to check
optimality and we have an optimal solution if all are  0
• There is an efficient way to find these row-0 coefficients
for a given BFS to a transportation problem:
– Given the basic variables, calculate values of dual variables
• ui associated with each source
• vj associated with each destination
using cij – ui – vj = 0 for xij basic, or ui + vj = cij
(let ui = 0 for row i with the largest number of basic variables)
– Row-0 coefficients can be found from c’ij=cij-ui-vj for xij nonbasic
Transportation-36
Optimality Test (1)
Destination
Source
A
B
C
Dummy
Demand
N
16
S
13
E
22
17
140
14
13
19
20
23
0
200
140
350
50
150
10
100
0
60
15
210
9
Supply
W
0
0
90
90
100
140
300
ui
250
vj
Transportation-37
Optimality Test (2)
• Calculate ui, vj using cij – ui – vj = 0 for xij basic
(let ui = 0 for row i with the largest number of basic variables)
Destination
Source
A
B
C
Dummy
N
S
16
E
13
Supply
ui
60
200
0
140
350
-2
50
150
-7
90
-21
W
22
17
140
14
13
19
15
210
9
20
23
10
100
0
0
0
0
90
Demand
100
140
300
250
vj
16
13
21
17
Transportation-38
Optimality Test (3)
• Calculate c’ij=cij-ui-vj for xij nonbasic
Destination
Source
A
B
C
Dummy
N
S
16
E
13
14
22
13
9
0
210
350
-2
50
150
-7
90
-21
9
0
5
140
10
14
0
0
15
23
100
200
1
2
20
60
17
19
0
ui
W
140
0
Supply
0
90
8
4
Demand
100
140
300
250
vj
16
13
21
17
Transportation-39
Optimal Solution
Sources
Supply = 200
A
Destinations
B
C
S
Demand = 140
E
Demand = 300
(shortage of 90)
W
Demand = 250
210
140
Supply = 150
Demand = 100
140
60
Supply = 350
N
100
50
Cost Z = 10330
Transportation-40
An Iteration
• Find the entering basic variable
– Select the variable with the largest negative c’ij
• Find the leaving basic variable
– Determine the chain reaction that would result from increasing
the value of the entering variable from zero
– The leaving variable will be the first variable to reach zero
because of this chain reaction
Transportation-41
Initial Solution Obtained by the Northwest
Corner Rule
Destination
Source
A
B
C
Dummy
N
S
16
E
13
22
13
40
20
300
150
9
0
-1
10
10
12
0
2
15
23
-2
0
3
19
-2
9
17
100
100
14
W
0
2
90
-4
Demand
100
140
300
250
vj
16
13
19
15
Supply
ui
200
0
350
0
150
-5
90
-15
Transportation-42
Iteration 1
Destination
Source
A
B
C
Dummy
N
S
16
E
13
22
17
100
100
14
13
19
300
40
9
W
20
23
15
+
10
10
150
0
0
0
+
0
-
?
90
Demand
100
140
300
250
vj
16
13
19
15
Supply
ui
200
0
350
0
150
-5
90
-15
Transportation-43
End of Iteration 1
Destination
Source
A
B
C
Dummy
Demand
N
S
16
E
13
22
17
13
19
15
40
9
20
210
23
100
10
150
0
0
0
0
140
300
350
150
90
90
100
ui
200
100
100
14
Supply
W
250
vj
Transportation-44
Optimality Test
Destination
Source
A
B
C
Dummy
N
S
16
E
13
22
13
210
23
-2
150
9
0
3
100
10
12
0
2
15
40
20
0
3
19
-2
9
17
100
100
14
W
0
90
6
4
Demand
100
140
300
250
vj
16
13
19
15
Supply
ui
200
0
350
0
150
-5
90
-19
Transportation-45
Iteration 2
Destination
Source
A
B
C
Dummy
N
S
16
-
13
+
13
?
9
W
+ 22
17
- 19
15
100
100
14
E
210
40
20
23
100
10
150
0
0
0
0
90
Demand
100
140
300
250
vj
16
13
19
15
Supply
ui
200
0
350
0
150
-5
90
-19
Transportation-46
End of Iteration 2
Destination
Source
A
B
C
Dummy
Demand
N
S
16
E
13
22
17
13
19
15
40
9
210
20
23
100
10
150
0
0
0
0
140
300
350
150
90
90
100
ui
200
140
60
14
Supply
W
250
vj
Transportation-47
Optimality Test
Destination
Source
A
B
C
Dummy
N
S
16
E
13
22
13
210
23
0
150
9
0
5
100
10
14
0
0
15
2
20
0
1
19
40
9
17
140
60
14
W
0
90
8
4
Demand
100
140
300
250
vj
14
11
19
15
Supply
ui
200
2
350
0
150
-5
90
-19
Z = 10330
Transportation-48
Optimal Solution
Sources
Destinations
60
Supply = 200
A
N
Demand = 100
S
Demand = 140
E
Demand = 300
(shortage of 90)
W
Demand = 250
140
40
Supply = 350
210
B
100
Supply = 150
C
150
Cost Z = 10330
Transportation-49
The Assignment Problem
• The problem of finding the minimum-costly assignment
of a set of tasks (i=1,…,m) to a set of agents (j=1,…,n)
• Each task should be performed by one agent
• Each agent should perform one task
• A cost cij associated with each assignment
• We should have m=n (if not…?)
• A special type of linear programming problem, and
• A special type of transportation problem,
with si=dj= ?
Transportation-50
Prototype Problem
• Assign students to mentors
• Each assignment has a ‘mismatch’ index
• Minimize mismatches
Mentor
Student
Harry
Draco
Goyle
Demand
Snape
McGonagall
Lupin
5
2
3
1
4
5
2
4
4
1
1
Supply
1
1
1
1
Transportation-51
Prototype Problem
• Linear programming formulation
Let xij denote…
Minimize
subject to
Transportation-52
General LP Formulation for Assignment
Problems
Transportation-53
Solving the Assignment Problem
• It is a linear programming problem, so we could use
regular simplex method
• It is a transportation problem, so we could use
transportation simplex method
• However, it has a very special structure, such that it can
be solved in polynomial time
• Many such algorithms exist, but the best known (and one
of the oldest) is the Hungarian Method
Transportation-54
The Hungarian Method
1.
2.
3.
4.
5.
Subtract row minimums from each element in the row
Subtract column minimums from each element in the column
Cover the zeroes with as few lines as possible
If the number of lines = n, then optimal solution is hidden in zeroes
Otherwise, find the minimum cost that is not covered by any lines
1. Subtract it from all uncovered elements
2. Add it to all elements at intersections (covered by two lines)
6. Back to step 3
Transportation-55
The Hungarian Method – Optimal Solution
How to identify the optimal solution:
• Make the assignments one at a time in positions that
have zero elements.
• Begin with rows or columns that have only one zero.
Cross out both the row and the column involved after
each assignment is made.
• Move on to the rows and columns that are not yet
crossed out to select the next assignment, with
preference given to any such row or column that has
only one zero that is not crossed out.
• Continue until every row and every column has exactly
one assignment and so has been crossed out.
Transportation-56
Hungarian Method
Mentor
Mentor
Student
Snape
McG
Lupin
5
1
2
2
4
4
3
5
4
Harry
Draco
Goyle
Student
Snape
McG
Harry
Draco
Goyle
Mentor
Student
Snape
McG
Lupin
Mentor
Lupin
Student
Harry
Harry
Draco
Draco
Goyle
Goyle
Snape
McG
Lupin
Transportation-57