TRANSPORTATION

Definition

Problems relating to transportation of same items from different centers to various destinations, so that overall transportation cost is minimizes , are called transportation problem (TP).

Problems are classified into

1. Balanced TP problem
2. Unbalanced TP problem

The optimal solution for the given transportation problem cannot be found out directly. We have to find out the initial solution and based on it we have to optimize. In other words the process of finding optimal solution consists of two stages viz

1. Initial solution
2. From initial solution to optimal solution

Initial solution (VAN method)

1. Verify the given TP problem is balanced, if not make it balance

2. Construct a table, indicating the unit transportation cost, demand & supply

3. Examine the first row identify the least transportation cost and the 2nd best least transportation cost, calculate the difference. Indicate it at the right hand side of the first row. Similarly calculate for the remaining rows & indicate. These entries are called unit penalties.

4. Follow the same procedure for the columns, these are called column wise penalties

5. Identify the largest penalty and encircled it, examine the rows/columns against the encircled entry. Select the box having least cost, lying in that row/column. Perform the following

   1. If row total is less indicates that total in that box and exhaust that rows and modify the column total
   2. If column total is less then put it in that box and exhaust that column and modify the row total
   3. If row total and column total are same put it in that box and exhaust that column simultaneously (whenever we exhaust simultaneously the initial solution becomes non-basic (degeneracy)

6. Now examine the remaining rows and calculate row penalties. Similarly calculate column penalties for the remaining columns

7. Select the largest penalty and encircles. Identify the encircled entry occurs. Select the box having least cost. Examine its row total and column total. Perform a, b, c as earlier stated.

8. Repeat this process until all rows and column are exhausted

9. Count the no of boxes filled in , verify it is equal to (m+n-1) where m=no of rows & n= no of columns in the VAN table, If equal the initial solution is basic, for unequal the solution is non-basic.

• Breaker rules:
  1. In a row, column if the two boxes have same least cost the difference is zero
  2. If there is a tie in the largest penalty, select the one that gives cheapest cost
  3. If again there is a tie in the least cost then select the cost where more quantities can be loaded into it.
  4. If still there is a tie, select arbitrarily
• Important points
  1. If any entries are shown as NIL then those entries should be replaced by infinity (∞)
  2. If the initial solution is non-basic then it should be made into basic, then only we can find optimal sol.
• Optimization
  1. Construct a table (T1) and indicate the initial basic solution
  2. Construct another table (T2) and indicate the unit transportation cost for the allocated cells. Based on the cost calculate a set of Ui & Vj values
  3. For each empty box calculate (Ui + Vj) and indicating those empty boxes, by encircling. These encircled values indicate the allowable cost for those empty boxes.
  4. Construct a table (T3) and calculate the difference between the cost given in the problem and encircled entry. These difference are called dij or Δij
  5. Make out decision in the following manners.
     1. If all the entries in the T3 and are non-zero positive then initial solution itself is optimum and unique
     2. If all the entries in T3 are positive and some entries are zeros, then also the initial solution itself is optimum but not unique. The member of zeros so many optimal solution with same optimal cost can be determined.
     3. If one or more entries in T3 are –ve then the initial sol. Is not optimum. It should be revised using LOOP TRANSPORTATION