The document discusses the unbalanced transportation problem, which occurs when total supply does not equal total demand, and describes methods to solve it, including the North-West Corner Method, Least Cost Method, and Vogel's Approximation Method. Each method has its advantages and limitations for finding feasible solutions, while also addressing challenges like cost alteration from dummy allocations and inefficiencies in real-life applications. The document emphasizes the importance of these methods in logistics and supply chain management for optimizing resource distribution.

1. Unbalanced transportation problem:
Theory and Solutions
Hemvati Nandan Bahuguna Garhwal University
Submitted by
Shilpa KP
MBA 3rd semester
Submitted to,
Dr, Surendra Kumar Meena
Subject: Operational Research

2. • Introduction
• Unbalanced transportation
problem
• Methods
• VAM
• Least cost method
• North wwest corner method
• Benefits
• Limitations
• Conclusion
Table of
content

3. • The Transportation Problem is a type of optimization problem where the goal is
to determine the most cost-efficient way to transport goods from multiple
suppliers to multiple destinations.
• Types of Transportation Problems:
• 1. Balanced Transportation Problem: Supply equals demand.
• 2. Unbalanced Transportation Problem: Supply does not equal demand.
Introduction to Transportation
Problem

4. What is an Unbalanced Transportation Problem?
• An Unbalanced Transportation Problem arises when the total supply
from all sources is not equal to the total demand at all destinations.
• To handle this, a dummy row or column is added to balance supply
and demand. This ensures the problem can be solved using standard
transportation methods.

5. • Methods to Solve Unbalanced Transportation Problems:
• 1. North-West Corner Method: Simple and systematic allocation of supply.
• 2. Least Cost Method: Allocates resources based on the least transportation
cost.
• 3. Vogel’s Approximation Method (VAM): Prioritizes allocations to
minimize overall cost.
Methods to Solve
Unbalanced Transportation
Problems

6. • The North-West Corner Method is a basic allocation technique used to
generate an initial feasible solution to the transportation problem.
• Unbalanced Case: When supply ≠ demand, a dummy source or destination
is added with a transportation cost of zero to balance the problem.
Process:
Start with the north-west corner of the matrix.
Allocate the minimum value of supply or demand to the cell.
Subtract the allocated amount from the respective supply and demand.
If demand is satisfied, move right. If supply is exhausted, move down.
Continue until all supply and demand are allocated.
North West Corner Method

7. Advantages:
Easy to implement.
No need to calculate penalties or costs initially.
Limitations:
May not yield a cost-effective solution as it ignores
transportation costs.

8. Least Cost Method
The Least Cost Method (LCM) aims to provide a better initial feasible solution by
minimizing costs at each allocation step
Process:
Identify the cell with the lowest transportation cost in the matrix.
Allocate the minimum value of supply or demand to that cell.
Adjust supply and demand by subtracting the allocated value.
Repeat the process with the next least cost cell until all supply and demand are
satisfied.

9. Advantages:
Provides a more cost-efficient solution compared to NWCM.
Simple to implement and understand.
Limitations:
Still a heuristic method and may not guarantee the optimal solution.
Doesn’t consider the overall penalty for allocation decisions.

10. Vogels Approximation
Method
Vogel’s Approximation Method is a refined approach that balances cost efficiency
and resource allocation penalties to provide the best initial feasible solution.
1.Compute penalties for all rows and columns (difference
between the smallest and second smallest costs).
2. Select the row/column with the highest penalty.
3. Allocate as much as possible to the least-cost cell in that
row/column.
4. Adjust supply and demand and recalculate penalties.
5. Repeat until all supply and demand are satisfied.

11. Advantages:
Produces a near-optimal initial solution.
Considers cost and allocation penalties.
Limitations:
More complex and time-consuming compared to NWCM
and LCM.
Penalty calculations require additional effort.

12. • Efficiency
• Flexibility
• Fewer restrictions
• Reduced risk of damage
• Cost effective
• Shortened delivery times
• Safer
Benefi
ts

13. 1. Dummy Rows/Columns Affect Costs:
Adding dummy rows or columns to balance supply and demand may artificially alter the cost matrix.
2. Complexity in Large Problems:
For large-scale problems with many sources and destinations, calculations become time-consuming and
prone to errors.
3. Inefficiency in Real-Life Scenarios:
The assumption of linear costs and exact demand/supply is unrealistic in practical situations like
fluctuating demand.
4. Ignores Other Constraints:
Does not account for constraints like time, perishability of goods, or capacity restrictions in
transportation modes.
5. Limited Applicability:
Primarily focuses on cost minimization, ignoring factors like customer satisfaction or service levels.
6. No Optimal Solutions Guarantee:
The methods may provide feasible solutions but not necessarily the optimal one in all cases.
Limitatio
ns

14. • The Unbalanced Transportation Problem is a common scenario in logistics and supply
chain management.
• By using methods like:
• 1. North-West Corner Method,
• 2. Least Cost Method, and
• 3. Vogel’s Approximation Method (VAM),
• we can balance supply and demand, and determine the most cost-effective way to
distribute goods.
Conclusion