# Transportation - 23 Stepping Stone Method - Part 2 of 3 Evaluating the Improved Solution

Published at : 26 Jan 2021
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#OperationsResearch #Math #Statistics #TransportationProblem #SteppingStoneMethod #OpportunityCost #ImprovementIndex #FreeLecture #FreeStudy #Solution #VideoLecture

Stepping Stone Method
This is a procedure to be followed to find improved solution and the optimal solution in turn of a transportation problem after getting its initial solution by applying suitable method. For this purpose, after verifying that the initial solution satisfies the (m + n - 1) condition, each of the non-basic variable (empty cells) is considered one-by-one. The steps involved are as follows:

Step - 1
Select an empty cell for evaluation. It is advisable to follow a specific order of going by rows or columns one-by-one so that no empty cell is left untouched.

Step - 2
Beginning with this selected cell, trace a closed path back to this original cell via occupied cells and moving with vertical or horizontal moves.

Step - 3
Place a plus (+) sign in the original empty cell and placing alternate minus (-) and plus (+) signs on each corner cell of the closed path.

Step - 4
Calculate Opportunity Cost or Improvement Index (OC / II) by adding together the unit cost amounts found in each cell with plus (+) sign and subtracting the unit cost amounts found in each cell with minus (-) sign.

Step - 5
Repeat the above steps for all empty cells. If all OCs / IIs computed are greater than or equal to zero, we have reached the optimal solution. If not, it is possible to improve the current solution to reach the optimal solution and for that purpose first we need to find an improved solution and check it again by applying all these steps.

How to improve the current solution?
If any one or more OC calculated comes to less than zero, we need to select it and trace the closed loop which we had traced or thought of while calculating this OC. compare the quantities allocated in the cells with minus (-) sign and select the minimum from them. Now add this quantity to the quantities in the cells with plus (+) signs and subtract it from the quantities in the cells with minus (-) sings. This is the improved solution. We need to check this improved solution by applying the above steps for optimality.

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