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The initial basic feasible solution can be obtained by using one of several methods. We will consider only the North-West corner rule of developing an initial solution. Other methods can be found in standard texts on linear programming.
The procedure for constructing an initial basic feasible solution selects the basic variables one at a time. The North-West corner rule begins with an allocation at the top left-hand corner of the tableau and proceeds systematically along either a row or a column and make allocations to subsequent cells until the bottom right-hand corner is reached, by which time enough allocations will have been made to constitute an initial solution.
The procedure for constructing an initial solution using the North-West corner rule is as follows:
The next step is to determine whether the current allocation at any stage of the solution process is optimal. We will present one of the methods used to determine optimality of and improve a current solution. The method derives its name from the analogy of crossing a pond using stepping stones. The occupied cells are analogous to the stepping stones, which are used in making certain movements in this method.
The five steps of the Stepping-Stone Method are as follows:
If all the cost index values obtained for all the currently unoccupied cells are non-negative, then the current solution is optimal. If there are negative values the solution has to be improved.
This means that an allocation is made to one of the empty cells (unused routes) and the necessary adjustments in the supply and demand effected accordingly.