Operational research problem(Vogel Approximation)

In summary, VAM is a heuristic method for finding an initial feasible solution in transportation problems. It may result in different solutions due to ties in penalty values, but the optimal solutions should have the same cost. The Kuhn-Tucker conditions will be satisfied for any optimal solution. In VAM, if two costs in the same row or column are the same, the penalty for that row or column will be calculated using the regular method of subtracting the smallest unit cost from the next smallest cost.
  • #1
prashant_ora
1
0
Sir,
Suppose we are asked to find the basic feasible solution for maximizing transportation cost using Vogel approximation method (VAM). We then write the row penalty and column penalty. Suppose there is tie between 2 penalty values, which should be taken first? I have this doubt because I get 2 different solutions in each case.
If there is a tie we would take that penalty corresponding to which there is minimum cost. If there is a tie again in the minimum cost then we would allocate in the cell where maximum can be allocated, and again if there is a tie, then what?
If I choose randamoly then the ansawer would be different . So tell me what should I do.
 
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  • #2
I think you meant to find the minimum transportation cost (or maximize profit). In any case, you should not be surprised that a heuristic method for allocating an initial feasible solution can result in different solutions.

If you iterate to get an optimal solution, these optimal solutions should have the same cost (or benefit). But it is possible that the optimal solution set includes a polyhedral face of the feasible solution space, or just an edge. If so there are multiple optimal solutions.

It is hard to know how to explain to you the underlying reality without knowing what tools (and/or textbooks) you are using. Just remember that the Kuhn-Tucker conditions will be satisfied for any optimal solution, which means that an optimal solution is also a feasible solution for both the problem and its dual.
 
  • #3
If two costs in the same row or column are the same will the penalty of that row or column be zero? or will it be calculated using the regular method i.e. by subtracting the smallest unit cost from the next smallest cost?
 
  • #4
The Vogel's approximation method (VAM) usually produces an optimal or near- optimal starting solution. One study found that VAM yields an optimum solution in 80 percent of the sample problems tested.
 
  • #5


I understand your confusion with the VAM method and the potential for getting different solutions in the case of a tie. In this situation, it is important to follow a systematic approach to ensure consistency in your results. If there is a tie between two penalty values, the one corresponding to the minimum cost should be selected first. If there is a tie again in the minimum cost, then we should allocate in the cell where the maximum amount can be allocated. If there is still a tie, then a random selection can be made. However, it is important to note that this random selection may result in different solutions each time the problem is solved. Therefore, it is recommended to follow a consistent approach, such as always selecting the first tie or always selecting the last tie, to ensure reproducibility in your results. Ultimately, the decision on how to handle ties in VAM should be based on the specific problem at hand and the desired outcome. It is always important to clearly document your approach and any assumptions made in your analysis.
 

1. What is the purpose of using Vogel Approximation in operational research problems?

Vogel Approximation is a mathematical method used in operational research to find the most efficient solution to a transportation problem. This method aims to minimize the total cost of transportation by considering the difference between the two smallest costs in each row and column of a transportation matrix.

2. How does Vogel Approximation differ from other methods used in operational research?

Vogel Approximation is considered to be a more accurate and efficient method compared to other methods such as the Northwest Corner Rule and Least Cost Method. This is because it takes into account the differences between the two smallest costs in each row and column, rather than just the smallest cost.

3. What are the limitations of using Vogel Approximation?

One limitation of Vogel Approximation is that it can only be used for minimization problems, where the goal is to minimize the total cost. It also assumes that the transportation costs are linear and do not change with the amount of goods being transported.

4. How is Vogel Approximation applied in real-world situations?

Vogel Approximation can be applied in various real-world situations such as supply chain management, inventory control, and transportation planning. It can help businesses find the most cost-effective way to transport goods from one location to another.

5. What are some techniques for improving the accuracy of Vogel Approximation?

To improve the accuracy of Vogel Approximation, some techniques that can be used include adjusting the transportation costs to account for any possible nonlinear relationships, using a larger transportation matrix, and performing sensitivity analysis to test the robustness of the solution.

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