Transportation Problem
and Related Topics
Transportation problem : Narrative representation
There are 3 plants, 3 warehouses.
Production of Plants 1, 2, and 3 are 100, 150, 200 respectively.
Demand of warehouses 1, 2 and 3 are 170, 180, and 100 units
respectively.
Transportation costs for each unit of product is given below
1
Plant 2
3
1
12
14
15
Warehouse
2
11
12
11
3
13
16
12
Formulate this problem as an LP to satisfy demand at minimum
transportation costs.
Transportation Problem and Related Topics
Ardavan Asef-Vaziri
June-2013
2
Data for the Transportation Model
Plant 1
Warehouse 1
Plant 2
Plant 3
Warehouse 2
Warehouse 3
• Quantity demanded at each destination
• Quantity supplied from each origin
• Cost between origin and destination
Transportation Problem and Related Topics
Ardavan Asef-Vaziri
June-2013
3
Data for the Transportation Model
Supply Locations
150
100
Plant 1
200
Plant 2
$13
Plant 3
$15
$12
$11
$11
$14
Warehouse 1
$12
$16
Warehouse 2
$12
Warehouse 1
Demand Locations
Transportation Problem and Related Topics
Ardavan Asef-Vaziri
June-2013
4
Transportation problem I : decision variables
100
150
1
2
x11
x12
170
x13
x21
x23
x31
200
1
3
x33
x22
2
x32
Transportation Problem and Related Topics
3
Ardavan Asef-Vaziri
June-2013
180
100
5
Transportation problem I : decision variables
x11 = Volume of product sent from P1 to W1
x12 = Volume of product sent from P1 to W2
x13 = Volume of product sent from P1 to W3
x21 = Volume of product sent from P2 to W1
x22 = Volume of product sent from P2 to W2
x23 = Volume of product sent from P2 to W3
x31 = Volume of product sent from P3 to W1
x32 = Volume of product sent from P3 to W2
x33 = Volume of product sent from P3 to W3
Minimize Z = 12 x11 + 11 x12 +13 x13 + 14 x21 + 12 x22 +16 x23
+15 x31 + 11 x32 +12 x33
Transportation Problem and Related Topics
Ardavan Asef-Vaziri
June-2013
6
Transportation problem I : supply and demand
constraints: equal only of Total S = Total D
x11 + x12 + x13 = 100
x21 + x22 + x23 =150
x31 + x32 + x33 = 200
x11 + x21 + x31 = 170
x12 + x22 + x32 = 180
x13 + x23 + x33 = 100
x11, x12, x13, x21, x22, x23, x31, x32, x33  0
Transportation Problem and Related Topics
Ardavan Asef-Vaziri
June-2013
7
Transportation problem I : supply and demand
constraints: ≤ for S, ≥ for D always correct
x11 + x12 + x13 ≤ 100
x21 + x22 + x23 ≤ 150
x31 + x32 + x33 ≤ 200
x11 + x21 + x31 ≥ 170
x12 + x22 + x32 ≥ 180
x13 + x23 + x33 ≥ 100
x11, x12, x13, x21, x22, x23, x31, x32, x33  0
Transportation Problem and Related Topics
Ardavan Asef-Vaziri
June-2013
8
Origins
s1
1
s2
2
si
We have a set of ORIGINs
Origin Definition: A source of material
- A set of Manufacturing Plants
- A set of Suppliers
- A set of Warehouses
- A set of Distribution Centers (DC)
In general we refer to them as Origins
i
sm
There are m origins i=1,2, ………., m
m
Each origin i has a supply of si
Transportation Problem and Related Topics
Ardavan Asef-Vaziri
June-2013
9
Destinations
We have a set of DESTINATIONs
1
Destination Definition: A location
with a demand for material
d1
d2
2
- A set of Markets
- A set of Retailers
- A set of Warehouses
- A set of Manufacturing plants
In general we refer to them as Destinations
di
j
dn
There are n destinations j=1,2, ………., n
Each origin j has a supply of dj
Transportation Problem and Related Topics
n
Ardavan Asef-Vaziri
June-2013
10
Transportation Model Assumptions
There is only one route between each pair of origin and
destination
Items to be shipped are all the same
for each and all units sent from origin i to destination j there is a
shipping cost of Cij per unit
Transportation Problem and Related Topics
Ardavan Asef-Vaziri
June-2013
11
Cij : cost of sending one unit of product from origin i
to destination j
1
C21
2
C22
C11
C12
1
2
C2n
C1n
i
m
j
Use Big M (a large number) to
eliminate unacceptable routes and
allocations.
Transportation Problem and Related Topics
Ardavan Asef-Vaziri
June-2013
n
12
Xij : Units of product sent from origin i to destination j
x11
1
x12
x21
2
1
2
x22
x2n
i
x1n
j
n
m
Transportation Problem and Related Topics
Ardavan Asef-Vaziri
June-2013
13
The Problem
1
2
The problem is to determine how
much material is sent from each
origin to each destination, such
that all demand is satisfied at the
minimum transportation cost
1
2
i
j
n
m
Transportation Problem and Related Topics
Ardavan Asef-Vaziri
June-2013
14
The Objective Function
1
1
If we send Xij units
2
2
from origin i to destination j,
its cost is Cij Xij
We want to minimize
i
j
Z   Cij xij
n
m
Transportation Problem and Related Topics
Ardavan Asef-Vaziri
June-2013
15
Transportation problem I : decision variables
100
150
1
2
x11
x12
170
x13
x21
x23
x31
200
1
3
x33
x22
2
x32
Transportation Problem and Related Topics
3
Ardavan Asef-Vaziri
June-2013
180
100
16
Transportation problem I : supply and demand
constraints
x11 + x12 + x13
=100
+x21 + x22 + x23
x11
+ x21
x12
+ x22
x13
=150
+x31 + x32 + x33
=200
+ x31
=170
+ x32
+ x23
=180
+ x33
= 100
In transportation problem. each variable Xij appears only in two
constraints, constraints i and constraint m+j, where m is the
number of supply nodes. The coefficients of all the variables in
the constraints are 1.
Transportation Problem and Related Topics
Ardavan Asef-Vaziri
June-2013
17
Our Task
Our main task is to formulate the problem.
By problem formulation we mean to prepare a tabular
representation for this problem.
Then we can simply pass our formulation ( tabular
representation) to EXCEL.
EXCEL will return the optimal solution.
What do we mean by formulation?
Transportation Problem and Related Topics
Ardavan Asef-Vaziri
June-2013
18
Cost Table
Cost Table
Plant 1
Plant 2
Plant 3
Warhouse1 Warhouse2 Warhouse3
12
11
13
14
12
16
15
11
12
`
Decision Variable Table
Warhouse1 Warhouse2 Warhouse3
Plant 1
Plant 2
Plant 3
Transportation Problem and Related Topics
Ardavan Asef-Vaziri
June-2013
19
Right Hand Side (RHS)
Truck
Railroad
Airplane
LHS
RHS
San Siego
1000
0
3000
4000
≥
4000
Transportation Problem and Related Topics
Norfolk
2000
500
0
2500
≥
2500
Pensacola
0
2500
0
2500
≥
2500
Ardavan Asef-Vaziri
LHS
RHS
3000 ≤ 3000
3000 ≤ 3000
3000 ≤ 3000
142000
June-2013
20
Left Hand Side (RHS), and Objective Function
Decision Variable Table
Warhouse1
Warhouse2
Warhouse3
Plant 1
Plant 2
Plant 3
170
0
0
=SUM(B11:B13)
180
0
0
=SUM(C11:C13)
0
0
100
=SUM(D11:D13)
RHS
170
180
100
Transportation Problem and Related Topics
Ardavan Asef-Vaziri
=SUM(B11:D11)
=SUM(B12:D12)
=SUM(B13:D13)
=SUMPRODUCT(B5:D7,B11:D13)
June-2013
RHS
100
150
200
21
≤ for Supply, ≥ for Demand unless Some Equality
Requirement is Enforced
Decision Variable Table
Warhouse1 Warhouse2 Warhouse3
Plant 1
Plant 2
Plant 3
0
0
0
≥
≥
≥
RHS
170
180
100
Transportation Problem and Related Topics
Ardavan Asef-Vaziri
June-2013
RHS
≤ 100
≤ 150
≤ 200
0
0
0
0
22
≤ for Supply, ≥ for Demand unless Some Equality
Requirement is Enforced
Decision Variable Table
Warhouse1 Warhouse2 Warhouse3
Plant 1
Plant 2
Plant 3
0
0
0
≥
≥
≥
RHS
170
180
100
Transportation Problem and Related Topics
Ardavan Asef-Vaziri
June-2013
RHS
≤ 100
≤ 150
≤ 200
0
0
0
0
23
Optimal Solution
Decision Variable Table
RHS
Warhouse1 Warhouse2 Warhouse3
Plant 1
100
0
0
100 ≤ 100
Plant 2
70
80
0
150 ≤ 150
Plant 3
0
100
100
200 ≤ 200
170
180
100
5440
≥
≥
≥
RHS
170
180
100
Extra Credit. How the colors were generated and what they mea?
Using Conditional formatting.
Green if the decision variable is >0
Red if the constraint is binding LHS = RHS
Transportation Problem and Related Topics
Ardavan Asef-Vaziri
June-2013
24
Example: Narrative Representation
We have 3 factories and 4 warehouses.
Production of factories are 100, 200, 150 respectively.
Demand of warehouses are 80, 90, 120, 160 respectively.
Transportation cost for each unit of material from each origin to
each destination is given below.
1
Origin 2
3
Destination
1
2
3
4
7
7
12
3
8
8
10
16
4
1
8
5
Formulate this problem as a transportation problem
Transportation Problem and Related Topics
Ardavan Asef-Vaziri
June-2013
25
Excel : Data
Transportation Problem and Related Topics
Ardavan Asef-Vaziri
June-2013
26
The Assignment Problem : Example
11 repairmen and 10 tasks. The time (in minutes) to complete
each job by each repairman is given below.
Cost Table
Repairman
1
2
3
4
5
6
7
8
9
10
11
Task
Time of task j if done by repairman i
1 2 3 4 5 6 7 8 9 10
40 40 45 30 45 35 50 20 45 30
30 50 30 30 35 30 55 30 55 40
50 20 30 55 30 40 55 25 30 20
35 40 35 55 35 20 45 55 45 45
45 35 50 30 35 20 55 35 40 20
30 35 50 35 45 35 50 30 55 40
50 55 35 40 45 25 55 35 45 35
20 40 40 25 45 55 35 30 40 40
20 20 45 50 20 50 50 30 25 50
20 40 40 35 20 40 40 30 50 35
45 50 55 30 50 35 55 50 45 40
Assign each task to one repairman in order to minimize to total
repair time by all the repairmen.
In the assignment problem, all RHSs are 1. That is the only
difference with the transportation problem,.
Transportation Problem and Related Topics
Ardavan Asef-Vaziri
June-2013
27
The Assignment Problem : Example
Decision Variables
1
1 0
2 0
3 0
4 0
Repairman
5 0
6 1
7 0
8 0
9 0
10 0
11 0
1
≥
1
2
0
0
1
0
0
0
0
0
0
0
0
1
≥
1
3
0
1
0
0
0
0
0
0
0
0
0
1
≥
1
Transportation Problem and Related Topics
Task
4 5
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 1
1 0
1 1
≥ ≥
1 1
6
0
0
0
1
0
0
0
0
0
0
0
1
≥
1
7
0
0
0
0
0
0
0
1
0
0
0
1
≥
1
8
1
0
0
0
0
0
0
0
0
0
0
1
≥
1
9 10
0 0 1
0 0 1
0 0 1
0 0 1
0 1 1
0 0 1
0 0 0
0 0 1
1 0 1
0 0 1
0 0 1
1 1 250
≥ ≥
1 1
Ardavan Asef-Vaziri
June-2013
≤
≤
≤
≤
≤
≤
≤
≤
≤
≤
≤
1
1
1
1
1
1
1
1
1
1
1
28