What is the method for optimal transportation problem?

Optimal Transportation Problem: A Step-by-Step Guide to Finding the Ideal Solution

Introduction

The optimal transportation problem is a mathematical model used to determine the most efficient way to transport goods from multiple origins to multiple destinations. The goal is to minimize the total transportation cost while satisfying supply and demand constraints.

Method for Optimal Transportation Problem

The following steps outline the method for solving an optimal transportation problem:

1. Formulate the Problem

Define the number of origins and destinations, the supply and demand for each, and the transportation costs between each origin and destination.

2. Find an Initial Basic Feasible Solution

The initial basic feasible solution is a feasible solution to the problem that contains as many non-zero variables as there are constraints. There are various methods for finding an initial basic feasible solution, including:

• Northwest Corner Rule: Start by filling the transportation table from the top-left corner, moving across rows and down columns until all supply and demand is satisfied.
• Vogels Approximation Method: Select the origin-destination pair with the highest difference in transportation costs and allocate as much as possible.

3. Improve the Solution

Once an initial basic feasible solution is found, it can be improved using techniques such as:

• MODI Method: Calculate the minimum opportunity cost of improvement (MODI) for each non-basic variable and select the variable with the highest positive MODI to enter the solution.
• Stepping Stone Method: Identify a closed loop of basic variables and non-basic variables with alternating signs. If the loop contains a negative variable, it can be exchanged with a positive variable to improve the solution.

4. Test for Optimality

After each iteration of improvement, test for optimality by calculating the reduced costs for all non-basic variables. If all reduced costs are non-negative, the solution is optimal.

Example illustrating the process

Consider a transportation problem with the following data:

| Transportation Costs |
|—|—|
| From A to X | 2 |
| From A to Y | 3 |
| From B to X | 1 |
| From B to Y | 2 |

Initial Solution using Northwest Corner Rule:

Improved Solution using MODI Method:

The total transportation cost is 2 150 + 2 250 = 800.

This solution is optimal as all reduced costs are non-negative.

Conclusion

Solving an optimal transportation problem involves finding an initial basic feasible solution and iteratively improving it until optimality is reached. Techniques such as the Northwest Corner Rule, Vogels Approximation Method, MODI Method, and Stepping Stone Method can be used to efficiently solve these problems. By optimizing transportation solutions, businesses can minimize costs and improve efficiency in their supply chain operations.