2 Least Cost Method The procedure is as follows Step 1 Determine the smallest cost in the cost matrix of the transportation table Let it be Cij Allocate x j min a b in the ij cell Step 2 If x j a cross off ith row of the transportation table and decrease b by a If x b cross off jh column of the transportation table and decrease a by bj If Xij a bj cross off either the first row or the first column but not both Step 4 Repeat steps 1 and 2 for the resulting table Reduce transportation table until all requirements are satisfied Problem Find an initial basic feasible solution to the following TP using Least Cost Method 1 2 3 F 2 Solution Step 1 0 is the minimum element which appears in 1 2 and 3 1 cells For the First allocation in X 2 min 15 15 15 Cross out first row or first column 1 3 4 3 2 3 2 3 2 3 I 2 3 4 10 0 20 11 15 2 12 7 9 20 25 Step 2 Second allocate in 3 1 cell X 1 min 5 5 5 12 3 4 12 7 3 0 14 16 18 5 5 15 15 10 10 0 20 H 12 7 9 20 0 5 15 15 1 Step 3 Third allocate in 2 2 cell X 2 min 25 0 0 2 3 4 2 Solution 14 16 18 5 14 5 0 15 10 Step 4 Fourth allocate in 2 3 cell X23 min 25 15 15 7 9 20 25 14 16 18 0 0 15 10 B 4 3 Step 5 Fifth allocate in 3 4 cell X34 min 0 10 0 4 1 9 20 25 16 18 5 15 10 4 2 20 10 18 10 9 20 25 16 18 5 20 10 Step 6 Sixth allocate in 2 4 cell X24 min 10 10 10 0 The initial basic feasible solution is given by x12 15 X31 5 x22 0 X23 15 X34 0 X24 10 1 3 4 10 1 2 d 3 A A P 50 30 220 1 P 90 45 170 3 P 250 250 50 4 4 2 2 8 1 2 1 10 0 10 5 10 15 12 25 0 14 16 18 Minimum transportation cost 15 9 10 20 Rs 335 Exercise 1 Using Least Cost Method find out the initial basic feasible solution to the following TP 0 3 Vogel s Approximation Method VAM 20 The procedure is as given below Step 1 For each row of the transportation table find the smallest cost and next to it Determine the difference between them for each row Write them within brackets along the side of the table 21 Step 2 For each column do the same step as in step 1 Step 3 Identify the row or column with the largest difference Let the greatest difference correspond to p and let Cij be the smallest cost in the ith row Allocate x min a b in that cell If xa cross off ith row of the transportation table and decrease b by a If x b cross off jth column of the transportation table and decrease a by bj If xa b cross off either the first row or the first column but not both 3 4 2 3 Step 4 Repeat steps 1 2 and 3 for the resulting table Reduce transportation table until all requirements are satisfied 0 Problem Find an initial basic feasible solution to the following TP using Vogel s Approximation Method 2 12 7 9 20 25 2 15 22 0 14 16 18 5 5 15 15 10 11 15 4 0 20 11 15 10 7 20 11 Step 1 Here the maximum difference is 14 which corresponds to third row of the table For the First allocation in X min 5 5 5 Cross out third column or first column 9 20 0 23 15 10 2 12 7 9 20 25 2 3 a 14 16 18 5 14 15 15 1 10 7 7 7 23