Very easy way to solve degeracy in Transpiortaion Problem

1. Solve Degeneracy in
Transportation Problem
LDC INSTITUTE OF TECHNICAL STDIES, SORAON
PRAYAGRAJ
BY
Dr. Mrinal Kanti Manik
Director
LDC Institute of Technical studies
Soraon, Allahabad, India
mrinalmanik64@gmail.com

2. Degeneracy in Transportation
Problem
 Here I will solve the degeneracy of transportation
problem with one example.
 I have taken the transportation problem having four
different source and five destinations.
•

3. Solution:
Step-1 This problem is balanced
transportation problem as total supply is
equal to total demand.

4. Least cost method has been applied to find
the initial basic feasible solution.
 One can also use any other method to find the
initial basic feasible solution.
 The following solution is obtained.

5. Optimization of the solution using U-V
Method:
• Step-3 Check whether m + n – 1 = total number of allocated
cells.
• In this case m + n – 1 = 4 + 5 – 1 = 8 where as total number of
allocated cells are 7, hence this is the case of degeneracy in
transportation problem.
• Step-4
• So in this case we convert the necessary number (in this case it
is m + n – 1 – total number of allocated cells i.e. 8 – 7 = 1) of
unallocated cells into allocated cells to satisfy the above
condition.

6. • Step-5 To convert unallocated cells into
allocated cells:
 Start from the least value of the unallocated cell.
 Check the loop formation one by one.
 There should be no closed-loop formation.
 Select that loop as a new allocated cell and assign a value ‘Є’.
 The closed loop can be in any form but all the turning point
should be only at allocated cell or at the cell from the loop is
started.

7. • In this problem there are 13 unallocated cells.
 Select the least value (i.e. 5 in this case) from unallocated
cells.
 There are two 5s here so you can select randomly any one.
 Lets select the cell with Є marked
20+ Є

8. Look if there is any closed-loop formation starting
from this cell.
 Step-6 If a closed-loop is drawn from this cell as shown earlier
then it cannot consider for allocation. But here it does not form
any close loop.
 So this cell will be selected and assigned a random value ‘Є’.

9. If the closed loop would have been formed from that cell
then we would try another cell with least value and do
the same procedure and check whether closed loop is
possible or not.
 Considering with Є value of one allocated cell count total number
of allocated cells, that should be 8. here m + n – 1 = 4 + 5 – 1 = 8.
 Then degeneracy of transportation problem is solved
 Calculate cost of the solution as
(20x2+15x3+10x2+30x4+20x15+25x13+5x8=890
 For calculation purpose Є location is taken as zero
 Step-7 Now this solution can be optimized using U-V method.
 We get the below solution after performing optimization using U-
V method.
 Find Ui + Vj = Cij for all allocated cell
 And find penalty Pij = Ui + Vj - Cij for all unallocated cell

10. Initially we assume U1=0 there after from the formula
Ui + Vj = Cij we calculate all value of Ui + Vj

11. Step-8 Here penalty for all unoccupied cell
Pij = Ui + Vj - Cij are shown below
1. C11= 0+12 -10 =2
2. C14= 0-4 -15 =-19
3. C15= 0-2 -9 =-11
4. C21= 6+12 -5 =13 Highest (+)ve value here
5. C22= 6+2 -10 =-2
6. C23= 6+3 -15 =-6
7. C33= 3+3 -14 =-8
8. C34= 3-4 -7 =-8
9. C35= 3-2-15 =-14
10. C41=1 0+12 -20 =2
11. C42=1 0+2 -15 =-3
12. C44=1 0-4 -25 =-19
 For optimality all values of Pij ≤ 0, So again we repeat
the step 6 and that start will be from cell C21

12. After Selecting an occupied cell complete loop and placed
alternative (+)ve and (-)ve value as shown in the figure.
Find out the most minimum vale out of all negative locations.
 Subtract most minimum value from all (-)ve location and add
most minimum value at every (+)ve location of all turning point of
loop

14. Again /Find Ui + Vj = Cij for all allocated cell
and Pij = Ui + Vj - Cij for all unallocated cell
For optimality all values of Pij ≤ 0, Otherwise repeat from
step 6 as follows
**Finally Calculate cost
value35x3+20x5+10x2+10x4+20x5+5x13+25x8=630

15. Thank you