1. PGT(MATHS), ZIET,MUMBAI
TRANSPORTATION
PROBLEM
PRESENTED BY
M.SRINIVASAN
PGT(MATHS), ZIET,MUMBAI

2. TRANSPORTATION PROBLEM
The cheapest way of transporting material from one place to another place is determined in Transportation Problem

3. As the cost of transport is to be minimized, the objective function will be Min type
The objective function and constraints are obtained by drawing the flow diagram

4. There is a factory located at each of the two places P and Q
There is a factory located at each of the two places P and Q. A certain commodity is delivered to each of the three depots situated at A,B and C. The weekly requirements are, 5, 5 and 4 units. The production capacity of the factories P and Q are 8 and 6 units respectively.
The cost of transportation per unit is given below:
COST IN RUPEES A B C P 16 10 15 Q 12 TO
FROM
How many units should be transported from each factory to each depot in order that the transportation cost is minimum? Formulate the above linear programming problem mathematically.

5. There are 02 factories and 03 depotsQ B C

6. A P Q B C x y 8-(x+y) P has maximum of 8 units
Let x units are transported to A, y units to B and 8 – (x + y) to C A x P y Q B
8-(x+y) C

7. A P Q B C Q has maximum of 6 units
The requirements of the depots A, B and C are 5, 5 and 4 A x
5-x P y
5-y Q B
8-(x+y)
6-(5-x+5-y)
x+y-4 C

8. A P Q B C x 5-x 16 10 5-y y 12 10 x+y-4 8-(x+y) 10 15 COST IN RUPEES A

9. A P Q B C Total cost = x – 7 y + 190 x 5-x 16 10 5-y y 12 10 x+y-4
The total transportation cost is
16x + 10 y + 15(8 – x – y)+ 10(5 – x) + 12(5 – y) + 10(x + y – 4)
Total cost = x – 7 y + 190 A x
5-x 16 10 P y
5-y Q B 12 10
x+y-4
8 -x - y 10 15 C

10. A P Q B C OBJECTIVE FUNCTION Min z = x – 7 y + 190 x 5-x 16 10 5-y y12 10
x+y-4
8 -x - y 10 15 C

11. ALL COMMODITIES TRANSPORTED ARE GREATER THAN OR EQUAL TO ZERO
To get the constraints
ALL COMMODITIES TRANSPORTED ARE GREATER THAN OR EQUAL TO ZERO

12. Constraints
x ≥ 0
5-x ≥ 0
y ≥ 0
5-y ≥ 0
x+y-4 ≥ 0
8-x-y ≥ 0

13. Constraints
x ≥ 0
x ≤ 5
y ≤ 5
y ≥ 0
x + y ≥ 4
x + y ≤ 8

14. Min z = x – 7 y + 190 x + y ≤ 8 x + y ≥ 4 x ≥ 0 x ≤ 5 y ≥ 0 y ≤ 5
THE LPP FOR THE TRANSPORTATION PROBLEM IS
Min z = x – 7 y + 190
SUBJECT TO THE CONSTRAINTS
x + y ≤ 8
x + y ≥ 4
x ≥ 0
x ≤ 5
y ≥ 0
y ≤ 5

15. y axis
x = 5
y = 5
x + y = 8
x + y = 4
X axis

16. FEASIBLE REGION OF LPP

18. THE VALUE OF THE OBEJCTIVE FUNCTION AT THE
CORNER POINTS ARE
POINT (x , y)
Value of z
(4,0)
194
(5,0)
195
(5,3)
174
(3,5)
158
(0,5)
155
(0,4)
162
Z is minimum at x = 0 and y = 5

19. Z is minimum at x = 0 and y = 5
The optimal transportation strategy will be to deliver 0,5 and 3 units from the factory P and 5,0 and 1 units from the factory Q to the depots A, B and C.
The minimum transportation charge is Rs.155

20. A brick manufacturer has two depots A and B which stocks of and bricks respectively. He receives orders from three builders P,Q and R for 15000, and bricks respectively. The cost in Rupees of transporting 1000 bricks to the builders from the depots are given below:
COST IN RUPEES P Q R A 40 20 30 B 60 TO
FROM
How should the manufacturer fulfill the orders so as to keep the cost of transportation minimum?

21. P A B Q R OBJECTIVE FUNCTION Min z = 3x – 30 y + 1800 x 15-x 40 2060 20
x+y-15
30 -x - y 40 30 R

22. Min z = 30x – 30y + 1800 x + y ≤ 30 x + y ≥ 15 x ≥ 0 x ≤ 15 y ≥ 0
THE LPP FOR THE TRANSPORTATION PROBLEM IS
Min z = 30x – 30y
SUBJECT TO THE CONSTRAINTS
x + y ≤ 30
x + y ≥ 15
x ≥ 0
x ≤ 15
y ≥ 0
y ≤ 20

23. THE VALUE OF THE OBEJCTIVE FUNCTION AT THE
CORNER POINTS ARE
POINT (x , y)
Value of z
(15,0)
2250
(15,15)
1800
(10 , 20)
1500
(0,20)
1200
(0,15)
1350
Z is minimum at x = 0 and y = 20

24. Z is minimum at x = 0 and y = 20
The optimal transportation strategy will be to deliver 0,20 and 10 thousand bricks to builders P, Q, R from depot A and 15,0 and 5 thousand of bricks to builders P, Q and R from the depot B
The minimum transportation charge is Rs.1200

25. An oil company has two depots A and B with capacities 7000 and 4000 liters respectively. The company has to supply oil to three petrol pumps D, E and F whose requirements are 4500, 3000 and 3500 liters. The distance in kms between the depots and the petrol pumps is given below:
From / to A B D 7 1 E 6 4 F 3 2
Assuming the transportation cost of 10 liters of oil if Rs.1 per km, how should the delivery be scheduled in order that the transportation cost is mimimum? What is the minimum cost?

26. D x
4500-x 7 3 A y
3000-y B E 60 6
x+y-3500
7000 -x - y 40 3 F

27. Min z = 3x + y + 39500 x + y ≤ 7000 x + y ≥ 3500 x ≥ 0 x ≤ 4500 y ≥ 0
THE LPP FOR THE TRANSPORTATION PROBLEM IS
Min z = 3x + y
SUBJECT TO THE CONSTRAINTS
x + y ≤ 7000
x + y ≥ 3500
x ≥ 0
x ≤ 4500
y ≥ 0
y ≤ 3000
Z is minimum at x = 500 and y = 3000