Transportation.ppt which is helpful in OR

transportation is the chapter in operations research subject

1. TRANSPORTATION PROBLEMSTRANSPORTATION PROBLEMS

2. Methods to solve Transportation Problems
Initial Basic Feasible
Solutions (IBFS)
Optimality Tests
1)NWCM
2)LCEM
3)VAM
1)Stepping stone Method
2)MODI Method

3. 1) North-West Corner Method (NWCM)1) North-West Corner Method (NWCM)
Step1: Select the upper left (north-west) cell of the
transportation matrix and allocate the maximum possible
value to X11 which is equal to min(a1,b1).
Step2:
•If allocation made is equal to the supply available at the
first source (a1 in first row), then move vertically down to
the cell (2,1).
•If allocation made is equal to demand of the first
destination (b1 in first column), then move horizontally to
the cell (1,2).
•If a1=b1 , then allocate X11= a1 or b1 and move to cell (2,2).
Step3: Continue the process until an allocation is made in
the south-east corner cell of the transportation table.

4. NWCMNWCM

5. ExampleExample:: Solve the Transportation Table to find Initial Solve the Transportation Table to find Initial
Basic Feasible Solution using NWCM.Basic Feasible Solution using NWCM.
Total Cost =19*5+30*2+30*6+40*3+70*4+20*14Total Cost =19*5+30*2+30*6+40*3+70*4+20*14
= Rs. 1015 = Rs. 1015
Supply
19 30 50 10
5 2
70 30 40 60
6 3
40 8 70 20
4 14
Demand 34
S1
S2
S3
7
9
18
5 8 7 14
D1 D2 D3 D4

6. 2) Least Cost Method (LCEM)2) Least Cost Method (LCEM)
Step1: Select the cell having lowest unit cost in the entire
table and allocate the minimum of supply or demand values
in that cell.
Step2: Then eliminate the row or column in which supply or
demand is exhausted. If both the supply and demand
values are same, either of the row or column can be
eliminated.
In case, the smallest unit cost is not unique, then select the
cell where maximum allocation can be made.
Step3: Repeat the process with next lowest unit cost and
continue until the entire available supply at various sources
and demand at various destinations is satisfied.

7. LCEMLCEM

8. Supply
70
2
40
3
Demand 345
S2 2
S3 3
D1
Supply
70 40 60
40 70 20
7
Demand 34
S3 10
5 7 7
S2 9
D1 D3 D4
Supply
70 40
7
40 70
Demand 34
9
3S3
5 7
S2
D1 D3
Supply
19 30 50 10
70 30 40 60
40 8 70 20
8
Demand 34
S3 18
5 8 7 14
S1 7
S2 9
D1 D2 D3 D4
Supply
19 50 10
7
70 40 60
40 70 20
Demand 34
7
9
S1
S2
S3 10
5
D1 D3 D4
7 14

9. The total transportation cost obtained by this methodThe total transportation cost obtained by this method
= 8*8+10*7+20*7+40*7+70*2+40*3= 8*8+10*7+20*7+40*7+70*2+40*3
= Rs.814= Rs.814
Here, we can see that the Here, we can see that the Least Cost MethodLeast Cost Method involves a involves a
lower cost than the lower cost than the North-West Corner MethodNorth-West Corner Method..

10. 3) Vogel’s Approximation Method (VAM)3) Vogel’s Approximation Method (VAM)
Step1: Calculate penalty for each row and column by taking the
difference between the two smallest unit costs. This penalty or
extra cost has to be paid if one fails to allocate the minimum
unit transportation cost.
Step2: Select the row or column with the highest penalty and
select the minimum unit cost of that row or column. Then,
allocate the minimum of supply or demand values in that cell. If
there is a tie, then select the cell where maximum allocation
could be made.
Step3: Adjust the supply and demand and eliminate the satisfied
row or column. If a row and column are satisfied
simultaneously, only of them is eliminated and the other one is
assigned a zero value.Any row or column having zero supply
or demand, can not be used in calculating future penalties.
Step4: Repeat the process until all the supply sources and
demand destinations are satisfied.

11. VAMVAM

12. SupplyRow Diff.
19 30 50 10
70 30 40 60
40 8 70 20
8
Demand 34
Col.Diff.
9
10
12
21 22 10 10
D1 D2 D3
S1
S2
S3
5 8 7
D4
14
7
9
18
SupplyRow Diff.
19 50 10
5
70 40 60
40 70 20
Demand 34
Col.Diff.
D1 D3 D4
S1 7
S2 9
S3 10
5 7 14
21 10 10
9
20
20
SupplyRow Diff.
50 10
40 60
70 20
10
Demand 34
Col.Diff. 10 10
40
20
50
D3 D4
S1 2
S2 9
S3 10
7 14
SupplyRow Diff.
40 60
7 2
Demand 34
Col.Diff.
20
7 2
S2 9
D3 D4
SupplyRow Diff.
50 10
2
40 60
Demand 34
Col.Diff.
20
10 50
40
7 4
S1 2
S2 9
D3 D4

13. The total transportation cost obtained by this methodThe total transportation cost obtained by this method
= 8*8+19*5+20*10+10*2+40*7+60*2= 8*8+19*5+20*10+10*2+40*7+60*2
= Rs.779= Rs.779
Here, we can see that Here, we can see that Vogel’s Approximation MethodVogel’s Approximation Method
involves the lowest cost than involves the lowest cost than North-West Corner MethodNorth-West Corner Method
and and Least Cost MethodLeast Cost Method and hence is the most preferred and hence is the most preferred
method of finding initial basic feasible solution.method of finding initial basic feasible solution.

14. MODI MethodMODI Method
RIM Requirement: m+n-1=No. of occupied cells
m=No. of columns
n=No. of rows

15. MODI MethodMODI Method

16. MODI MethodMODI Method

17. MODI MethodMODI Method

18. MODI MethodMODI Method

19. MODI MethodMODI Method
If Opportunity cost is negative then select largest negative from
all and take the loop of that cell.

20. MODI MethodMODI Method
Then select the least negative from that loop and subtract it from the units
having negative sign and add it to the units having positive sign.

21. MODI MethodMODI Method

22. Problems in TransportationProblems in Transportation
1)Balanced Problem
2)Unbalanced Problem
3)Minimization Problem
4)Maximization Problem
5)Non Degenerate Problem
6)Degeneracy