Question

The objective of the transportation problem is to minimize the total cost of shipments, which is expressed the sum of the products shipment costs and amounts form the warehouses to the cities as follows.

Answer 1

The given solution is not optimal solution.

in the transportation problems there are three methods to calculate initial feasible solution.

1. northwest corner method (NWC)

2. Least cost method (LCM)

3. Vogel's approximation method (VAM)

the transportation cost of initial feasible solution through vogel approximation method will be least when compare to other two methods.

in the given question least cost method has been applied.

now VAM method shall be applied.

Algorithm for Vogel’s approximation method is as follows:
(i) Calculate penalties for each row and column by taking the difference between the smallest cost and next highest cost available in that row/column. If there are two smallest costs, then the penalty is zero.
(ii) Select the row/column, which has the largest penalty and make an allocation in the cell having the least cost in the selected row/column. If two or more equal penalties exist, select one where a row/column contains minimum unit cost. If there is again a tie, select one where maximum allocation can be made.
(iii) Delete the row/column, which has satisfied the supply and demand.
(iv) Repeat steps (i) and (ii) until the entire supply and demands are satisfied.
(v) Obtain the initial basic feasible solution.

solution:-

BOSTON NY CH IN SUPPLY PENALTY

DT 15⁷⁵    20¹⁷⁵ 16 21 250 1 1 5 1

PT 25 13⁵⁵ 5⁵ 11⁷⁰ 130 6 6 2 2

BF 15 15 7²³⁵ 17 235 8

DEMAND 75 230 240 70

PENALTY 0 2 2 6

10 7 11 10

10 7 10

7 10

according to above solution LEAST TRANSPORT COST is

DT to boston 15*75= 1125

DT to NY 20*175= 3500

PT to NY 13*55= 715

PT to CH 5*5= 25

PT to IN 11*70= 770

BF to CH 7*235= 1645

total least cost 7780