1. Transportation Method Lecture 20 By Dr. Arshad Zaheer

2. RECAP  Transportation model (Minimization)  Illustration (Demand < Supply)  Optimal Solution  Modi Method

3. Maximization Total Demand exceeds Total Capacity (Supply)

4. Maximization Maximization problem may be solved by the use of following method Multiply the given pay off matrix of profits or gain by -1. Then use the transportation technique for minimization to obtain optimal solution. To calculate the total profit or gain multiply the total cost by -1

5. Illustration Maximize the profit for this problem Sources D1D1 D2D2 D3D3 Capacity S1S1 10151215 S2S2 98325 S3S3 1282025 Demand302030 65 80

6. Introduce the fictitious supply to balance at zero profit Sources D1D1 D2D2 D3D3 Capacity S1S1 10151215 S2S2 98325 S3S3 1282025 SfSf 00015 Demand30203080

7. Sources DestinationCapacity D1D2D3 S1 10 Xij 15 Xij 12 Xij 15 S2 9 Xij 8 Xij 3 Xij 25 S3 12 Xij 8 Xij 20 Xij 25 SfSf 0 Xij 0 Xij 0 Xij 15 Demand 302030 80

8. Initial Solution by North West Corner Rule

9. Sources DestinationCapacity D1D2D3 S1 10 15 0 12 0 15 S2 9 15 8 10 3030 25 S3 12 0 8 10 20 15 25 SfSf 0000 0000 0 15 Demand 30203080

10. For maximization we multiply all the profits or gains by -1.

11. Sources DestinationCapacity D1D2D3 S1 -10 15 -15 0 -12 0 15 S2 -9 15 -8 10 -30-30 25 S3 -12 0 -8 10 -20 15 25 SfSf 0000 0000 0 15 Demand 30203080

12. Total Profit Total Cost =15*-10 + 15*-9 + 10*-8 + 10*-8 + 15* -20 = -745 Total Profit=-1*- 745 = 745

13. No of Basic Variables= m+n-1 = 4+3-1 =6 m= No of sources n= No of destinations

14. Sources DestinationCapacity D1D2D3 S1 -10 15 -15 0 -12 0 15 U1= S2 -9 15 -8 10 -30-30 25 U2= S3 -12 0 -8 10 -20 15 25 U3= SfSf 0000 0000 0 15 U4= Demand 30 V1= 20 V2= 30 V3= 80 For calculating shadow cost we need to find the values of U and V variables

15. Equations U1+V1=-10let U2=0 U2+V1=-9U1=-1V1=-9 U2+V2=-8U2=OV2=-8 U3+V2=-8U3=0V3=-20 U3+V3=-20U4=20 U4+V3=0

16. Sources DestinationCapacity D1D2D3 S1 -10 15 -15 0 -9 -12 0 15 U1=-1 S2 -9 15 -8 10 -30-30 25 U2=0 S3 -12 0 -8 10 -20 15 25 U3=0 SfSf 0000 0000 0 15 U4=20 Demand 30 V1=-9 20 V2=-8 30 V3=-20 80 Shadow cost of S1, D3 Vij = (Ui + Vj) –Cij V13 = (U1 + V3) –C13 =(-1-20)-12 =-9 We can calculate all the shadow cost in the same way for others

17. Sources DestinationCapacity D1D2D3 S1 -10 15 6 -15 0 -9 -12 0 15 U1=-1 S2 -9 15 -8 10 -17 -3 0 25 U2=0 S3 3 -12 0 -8 10 -20 15 25 U3=0 SfSf 11 0 0 12 0 0 15 U4=20 Demand 30 V1=-9 20 V2=-8 30 V3=-20 80 We add θ in maximum positive shadow cost to proceed further because our optimal condition is not yet satisfied

18. Sources DestinationCapacity D1D2D3 S1 -10 15 6 -15 0 -9 -12 0 15 U1=-1 S2 -9 15 -8 10 -17 -3 0 25 U2=0 S3 3 -12 0 -8 10-θ -20 15+θ 25 U3=0 SfSf 11 0 0 12 0 0+θ 0 15-θ 15 U4=20 Demand 30 V1=-9 20 V2=-8 30 V3=-20 80

19. Maximum θ = Min (10,15) ` = 10

20. Sources DestinationCapacity D1D2D3 S1 -10 15 -15 0 -12 0 15 U1= S2 -9 15 -8 10 -3 0 25 U2= S3 -12 0 -80-80 -20 25 U3= SfSf 0 10 0505 15 U4= Demand 30 V1= 20 V2= 30 V3= 80

21. Total Cost=15*-10 +15*-9 + 10*-8 + 25*-20 =-865 Total Profit/Gain = -1 * -865 =865

22. Equations U1+V1=-10let U2=0 U2+V1=-9U1=-1V1=-9 U2+V2=-8U2= 0V2=-8 U3+V3=-20U3=-12V3=-8 U4+V2=0U4=8 U4+V3=0

23. Sources DestinationCapacity D1D2D3 S1 -10 15 -15 0 -12 0 15 U1=-1 S2 -9 15 -8 10 -3 0 25 U2=0 S3 -12 0 -80-80 -20 25 U3=-12 SfSf 0 10 0505 15 U4=8 Demand 30 V1=-9 20 V2=-8 30 V3=-8 80 Now we can calculate the shadow costs for all cells

24. Sources DestinationCapacity D1D2D3 S1 -10 15 6 -15 0 3 -12 0 15 U1=-1 S2 -9 15 -8 10 -5 -3 0 25 U2=0 S3 -9 -12 0 -12 -8 0 -20 25 U3=-12 SfSf -1 0 0 10 0505 15 U4=8 Demand 30 V1=-9 20 V2=-8 30 V3=-8 80 shadow costs are still positive so we use θ to proceed further

25. Sources DestinationCapacity D1D2D3 S1 -10 15-θ 6 15 0+θ 3 -12 0 15 U1=-1 S2 -9 15+θ -8 10-θ -5 -3 0 25 U2=0 S3 -9 -12 0 -12 -8 0 -20 25 U3=-12 SfSf -1 0 0 10 0505 15 U4=8 Demand 30 V1=-9 20 V2=-8 30 V3=-8 80

26. Maximum θ = Min (10, 15) ` = 10

27. Sources DestinationCapacity D1D2D3 S1 -10 5 -15 10 -12 0 15 U1= S2 -9 25 -80-80 -3 0 25 U2= S3 -12 0 -8 0 -20 25 U3= SfSf 0 10 0505 15 U4= Demand 30 V1= 20 V2= 30 V3= 80

28. Total Cost = 5*-10 + 10*-15 + 25*-9 + 25*-20 =- 925 Total Gain/Profit= = -1 * -925 = 925

29. Equations U1+V1=-10let U2=0 U1+V2=-15U1=-1V1=-9 U2+V1=-9U2= 0V2=-14 U3+V3=-20U3=-6V3=-14 U4+V2=0U4=14 U4+V3=0

30. Sources DestinationCapacity D1D2D3 S1 -10 5 -15 10 -12 0 15 U1=-1 S2 -9 25 -80-80 -3 0 25 U2=0 S3 -12 0 -8 0 -20 25 U3=-6 SfSf 0 10 0505 15 U4=14 Demand 30 V1=-9 20 V2=-14 30 V3=-14 80 Now the shadow cost for each cell can be calculated easily

31. Sources DestinationCapacity D1D2D3 S1 -10 5 -15 10 -3 -12 0 15 U1=-1 S2 -9 25 -6 -8 0 -11 -3 0 25 U2=0 S3 -3 -12 0 -12 -8 0 -20 25 U3=-6 SfSf 5 0 0 10 0505 15 U4=14 Demand 30 V1=-9 20 V2=-14 30 V3=-14 80 Criteria for optimality is not satisfied so we will proceed further with use of θ

32. Sources DestinationCapacity D1D2D3 S1 -10 5-θ -15 10+θ -3 -12 0 15 U1=-1 S2 -9 25 -6 -8 0 -11 -3 0 25 U2=0 S3 -3 -12 0 -12 -8 0 -20 25 U3=-6 SfSf 5 0 0+θ 0 10-θ 0505 15 U4=14 Demand 30 V1=-9 20 V2=-14 30 V3=-14 80

33. Maximum θ = Min (5, 10) ` = 5

34. Sources DestinationCapacity D1D2D3 S1 -10 0 -15 15 -12 0 15 U1= S2 -9 25 -8 0 -3 0 25 U2= S3 -12 0 -8 0 -20 25 U3= SfSf 0 5 0 5 0505 15 U4= Demand 30 V1= 20 V2= 30 V3= 80

35. Total Cost =15*-15 + 25*-9 + 25*-20 =-950 Total Profit/Gain= =-1 * - 950 =950

36. Equations U1+V2=-15let U2=0 U2+V1=-9U1=-6V1=-9 U3+V3=-20U2= 0V2=-9 U4+V1=0U3=-11V3=-9 U4+V2=0U4=9 U4+V3=0

37. Sources DestinationCapacity D1D2D3 S1 -10 0 -15 15 -12 0 15 U1=-6 S2 -9 25 -8 0 -3 0 25 U2=0 S3 -12 0 -8 0 -20 25 U3=-11 SfSf 0 5 0 5 0505 15 U4=9 Demand 30 V1=-9 20 V2=-9 30 V3=-9 80 Now calculate the shadow costs for non basic cells

38. Sources DestinationCapacity D1D2D3 S1 -5 -10 0 15 -3 -12 0 15 U1=-6 S2 -9 25 -1 -8 0 -6 -3 0 25 U2=0 S3 -8 -12 0 -12 -8 0 -20 25 U3=-11 SfSf 0 5 0 5 0505 15 U4=9 Demand 30 V1=-9 20 V2=-9 30 V3=-9 80 Criteria for optimality has been satisfied as all the shadow costs are non- positive

39. Optimal Distribution S1 ─ ─ ─ ─ > D2 = 15 S2 ─ ─ ─ ─ > D1 = 25 S3 ─ ─ ─ ─ > D3 = 25 Sf ─ ─ ─ ─ > D1 = 5 Sf ─ ─ ─ ─ > D2 = 5 Sf ─ ─ ─ ─ > D3 = 5 Total = 80 Total Gain = 950