Operations Research| Richa SaxenaTransportation Problems1SYBSc B Session 3 TRANSPORTATION PROBLEMVOGEL’S APPROXIMATION METHOD (VAM)or Penalty Method: This method works as follows. 1.Obtain the difference between pair of minimum cost values for each of the rows and columns. 2.Select the largest of the cost differences and choose the least-cost cell in that row/column. Consider the supply and the demand at the plant and market involved. The lower of these two is allocated in the cell chosen, and the plant or market whichever is satisfied is deleted. Both are deleted if they are both satisfied due to equal demand and supply. In case there is a tie in the largest cost difference values, the one corresponding to which the larger number of units can be allocated is selected. 3.Calculate the cost differences again for the reduced problem and proceed in the same manner as above. Repeat until all allocations are made. In a given problem, one of these methods is used to find the initial solution. Usually, the north-west corner rule is not used because it does not consider the cost while making allocations. The other two methods tend to provide an initial solution with a comparatively lower cost. Even in these two, the penalty method usually gives better results.(Vohra, 2010) Testing Optimality of the Solution Once the initial solution is obtained, it is tested for optimality. For testing the solution for optimality, it must have? + ? − 𝟏, where m is the number of rows, and n is the number of columns, number of occupied cells. If the number is less, the solution is termed as degenerate. Degeneracy is discussed later. For all rows and columns, ??and ??values are obtained. Each row and each column is assigned one value in such a way that, corresponding to every occupied cell, the row ??and the column ??adds up to the cost value, 𝑐??For this, one of the rows/columns is assigned a value arbitrarily. Usually, the first-row value u1value is set equal to zero and other values are determined one by one, using this and the other successively derived values. Once all these values are obtained, calculate ∆??= ??+ ??–𝑐??. Now, if all ∆??≤0, then the solution is optimal, otherwise not. A positive ∆??in a cell indicates that allocating goods in that cell can reduce cost. Thus, if a cell 2 to 3 in the matrix has ∆??= 3, it means that every unit allocated to this route, that is, every unit sent from plant 2 to market 3 would save cost at the rate of Rs 3 per unit.