This document discusses the transportation problem in operations research. It introduces the transportation problem as determining how to distribute a single commodity from multiple sources to multiple destinations to minimize costs. Key points covered include: - Each source has a supply capacity and each destination has a demand requirement. The goal is for total supply to equal total demand. - Transportation costs are proportional to the quantity shipped between each origin-destination pair. - The problem can be represented as a network with nodes for sources and destinations and arcs showing possible shipments. - Several methods are presented for finding an initial feasible solution, including the Northwest Corner method, Least Cost method, and Vogel's Approximation method. - An example problem

1. CHAPTER THREE
THE
TRANSPORTATION
PROBLEM

2. Introduction to transportation
problem
2
▰ Transportation deals with the transportation of a commodity (single
product) from ‘m’ sources (origin or supply or capacity centers) to
‘n’ destinations (demand or requirement centers).
▰ Is a special type of LPP where the objective is to minimize the cost
of distributing a product from a number of sources or origins to a
number of destinations.
▰ The objective is to determine the amount to be shifted from each
source to each destination to minimize the total transportation cost

3. Characteristics transportation
model
▰ 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
▰ The model has constraints for supply at each source and
demand at each destination
▰ Total supply should equal with the total demand.
▰ Only one product is transported between routes
3

4. Network Representation
4
▰ Nodes/circles indicate sources and destinations
▰ Arc/arrow represent a routes for distributing a product from
sources to destinations.

5. Procedure to find an optimal solution

6. Transportation table format

7. Mathematical model formulation
 Let 𝑐𝑖𝑗 be the cost of shipping one unit of the commodity from
source” i” to destination “j” for each route. Let 𝑥𝑖𝑗be the units
shipped per route. Then the LPP is stated below.

8. Example
A wheat supplier has contracted to provide wheat for three universities(Hawassa,
Dilla and Wachamo) The wheat can be supplied from three cities( Robe, Assela
and Adama). Given the capacity of each city and demand of each university,
determine how many Kgs of wheat to transport from each source to each
destination on a monthly basis in order to minimize the total cost of
transportation.
City Supply University Demand
1. Adama 150 A. Wachamo 200
2. Assela 175 B. Dilla 100
3. Robe 275 C. Hawassa 300
Total 600 Kgs Total 600 Kgs
Transport cost from city to university (ETB/Kg)
City A. Wachamo B. Dilla C. Hawassa
1. Adama
2. Asella
3. Robe
6
7
4
8
11
5
10
11
12

9. Tableau format

10. Model formulation
10

11. Finding initial feasible solution
▰ Methods of finding initial feasible solution:
▻ The Northwest Corner Method
▻ An Intuitive Approach/Least Cost Method
▻ Vogel’s Approximation/ Penalty Method
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12. 12
The Northwest Corner Method

13. The northwest corner
method(NWC)
Total cost(z )= 6(150) + 8(0) + 10(0) + 7(50) + 11(100) +
11(25) + 4(0) + 5(0) + 12(275) = 5,925

14. 14

15. Least cost method(LCM)
Total cost(z )= 6(0) + 8(25) + 10(125) + 7(0) + 11(0) + 11(25)
+ 4(200) + 5(75) + 12(0) = 4,550

16. 16

17. Vogel’s Approximation/
method (VAM)
Total cost(z )= 6(0) + 8(0) + 10(150) + 7(175) + 11(0) + 11(0)
+ 4(25) + 5(100) + 12(150) = 5,125

18. 18

19. 19