tutorial-dcdcd3_ or_transportation (1)

Amity School of Engineering & Technology, Noida.Department of Mechanical and Automation Engineering

Operations Research (POE 301 )Assignment – 3 ( Transportation Problem )

Last date of submission: 02.09.2014Group No. – 1

Q1. Solve the following transportation problem:

From

To

A B C Available

I

II

III

50

90

250

30

45

200

220

170

50

1

3

4

Requirement 4 2 2

Q2. The following data is given:

Sources

Destinations

1 2 3 Capacities

1

2

3

2

4

1

2

1

3

3

2

-

10

15

40

Requirement 20 15 30

The cost of shipment from third source to the third destination is not known. How many units should be transported from sources to destinations so that the total cost of transporting all the units to their destinations is a minimum?

Q3. Solve the following transportation problem:

From

To

A B C Available

I

II

III

6

4

1

8

9

2

4

8

6

14

12

5

Demand 6 10 15

Group No. – 2

Q4.Three fertilizer factories X, Y, Z located at different places of the country produce 6, 4 & 5 lac tones of urea respectively. Under the directive of the Central Govt., they are to be distributed to 3 states A, B & C as 5, 3 & 7 lac tones respectively. The transportation cost per tone in rupees is given below:

Factories

States

A B C

X

Y

Z

11

15

20

17

12

12

16

14

15

Q5. A company has three warehouses W1, W2 and W3. It is reqd. to deliver a product from these warehouses to three customers A, B and C. The warehouses have the following units in stock.

Warehouse W1 W2 W3

No. of units 65 42 43And customer requirements are:Customer A B CNo. of units 70 30 50The table below shows the costs of transporting one unit from warehouse to the customer:

Customer

To

W1 W2 W3

A

B

C

5

4

6

7

4

7

8

6

7

Find the optimal transportation cost.Q.6. Consider a transportation problem with m=3 and n=4,where: C11 = 2 C12 =3 C13 =11 C14 = 7 C21 = 1 C22 = 0 C23 = 6 C24 = 1 C31 = 5 C32 = 8 C33 = 15 C34 = 9Suppose S1 = 6, S2 = 1 and S3= 10 whereas D1 = 7, D2 = 5, D3 = 3 and D4 = 2.Apply the transportation simplex method to find out an optimal solution.Group No. – 3Q.7. Solve the following transportation problem:

Source

Destination

1 2 3 4 Available

1

2

3

21

17

32

16

18

27

25

14

18

13

23

41

11

13

19

Requirement 6 10 12 15 43

Q.8. Solve the following transportation problem:

Source

Destination

D1 D2 D3 D4

Available

1

2

3

1

3

4

2

3

2

1

2

5

4

1

9

30

50

20

Requirement 20 40 30 10

Q.9.Find the optimal solution to the following transportation problem:

Source

Destination

D1 D2 D3 D4

Available

1

2

3

23

12

22

27

17

28

16

20

12

18

51

32

30

40

53


The cell entries are transportation costs.Group No. – 4Q.10 A company has three factories Fi ( i = 1,2,3) from which it transports the product to four warehouses W j

(j=1,2,3,4). The net costs of production at tree factories are Rs. 4, 3, 5 respectively. Given the following information on unit costs of transportation, capacities at three factories and of the requirement at four warehouses, find the optimum allocation.

Factory

Unit cost of production

Transportation cost per unit

W1 W2 W3 W4

Available

F1

F2

F3

4

3

5

5

4

7

7

6

6

3

9

4

8

5

5

300

500

200

Requirement 200 30 400 100 1000

Q.11 A company has three cement factories A, B and C, and four area distributors W, X, Y and Z. with identical costs of production at three factories, the only variable cost involved is the transportation cost. The monthly production capacity (in tone) of the three factories, monthly demand of the four distributors, and the transportation cost per tone in Rs from the different factories to different distribution centre are given below:

Factory

Destination

W X Y Z Available

A

B

C

20

45

22

25

50

10

50

15

45

10

40

35

45000

50000

55000

Monthly Demand 50000 40000 30000 30000

Suggest an optimal transport schedule and find the minimum transportation cost.

Q.12 A company has three factories at Amethi, Baghpat, and Gwalior and four distribution centre at Allahabad, Mumbai, Calcutta and Delhi. With identical cost of transportation at the three factories the only variable cost involved is transportation cost. The demand at four distribution centers is 5000, 6000 and 2500 tones respectively. The demand at four distribution centers is 6000, 4000, 2000 and 1500 tones respectively. The transportation cost per tone from factories to different places are given below:

Distribution Centre

Factory

Allahabad Mumbai Calcutta Delhi

Amethi

Baghpat

Gwalior

3

7

2

2

5

5

7

2

4

6

3

5

Suggest the optimum transportation schedule and find the minimum cost of transportation.Group No. – 5Q.13 A wholesale company has three warehouses from which supplies are drawn for four retail customers. The company deals in a single product, the supplies of which at each warehouse are:

Warehouse

Distribution Centre

Supply (units) Customer no. Demand (units)

1

2

3

20

28

17

1

2

3

4

15

19

13

18

Conveniently, total supply at the warehouses is equal to total demand from the customers. The following Table gives the transportation costs per unit shipment from each warehouse to each customer.

Warehouse

Customer

1 2 3 4

1

2

3

3

6

7

6

1

8

8

2

3

5

5

9

Determine what supplies to dispatch from each of the warehouses to each customer so as to minimize overall transportation cost.

Q.14 A company spending Rs.1000 on transportation of its units from three plants to four distribution centers. The supply and demand of units with unit cost of transportation are given as:

Plant

Distribution centre

1 2 3 4Capacity (tons)

A

B

C

19

70

40

30

30

10

50

40

60

12

60

20

7

10

18

Demand 5 8 7 15 35

What can be the maximum saving by optimum scheduling?

Q.15.The following table shows all the necessary information on the available supply to each warehouse, the requirement of each market and the unit transportation cost from each warehouse to each market:

Factory

Market

I II III IV Supply

A

B

C

5

4

4

2

8

6

4

1

7

3

6

5

22

15

8

Requirement 7 12 17 9

The shipping clerk has worked out the following schedule from experience: 12 units from A to II, 1 unit from A to III, 9 units from A to IV, 15 units from B to III, 7 units from C to I and 1 unit from C to III.(a) Check and see if the clerk has the optimal schedule.(b) Find the optimal schedule and minimum total shipping cost.(c) If the clerk is approached by a carrier of route C to II who offers to reduce his rate in the hope of getting some business. By how much must the rate be reduced before the clerk should consider giving him an order?Group No. – 6Q.16. A steel company is concerned with the problem of distributing imported ore from three ports to four steel mills situated throughout the country. The supplies of ore arriving at ports are

Ports Tones per week a 20,000 b 38,000 c 16,000Demands at the steel mills are: Steel mills Tones per week A 10,000 B 18,000 C 22,000 D 24,000 Transportation costs are Re. 0.05 per tone per km. The distances between the ports and the steel mills are given below:

Ports

Steel mills

A B C D

a

b

c

50

80

90

60

40

70

100

70

30

50

50

50

Calculate the transportation plan, which will minimize ore distribution cost for the steel company. State the cost of this distribution plan.

Q.17 Solve the following transportation problem to find the minimum transportation cost:

Source

Destination

D1 D2 D3 D4 D5

Available

S1

S2

S3

S4

4

1

7

4

7

4

2

7

3

7

4

2

8

3

7

4

2

8

7

7

4

7

9

2


Q.18. A manufacturer has distribution centers at X, Y, and Z. These centers have availability 40, 20, and 40 units respectively of his product. His retail outlet at A, B, C, D and E requires 25,10,20,30 and 15 resp. The transport cost in Rs. per unit between each centre outlet is given below:

Distribution centre

Retail Outlet

A B C D E

X

Y

Z

55

35

40

30

30

60

40

100

95

50

45

35

50

60

30

Determine the optimal distribution to minimize the cost of transportation.Group No. – 7Q.19 Solve the following transportation problem: To

From

9 12 9 6 9 10 7 3 7 7 5 5 6 5 9 11 3 116 8 11 2 2 10

5629

4 4 6 2 4 2 22

Q.20.The following table gives the cost of transporting material from supply points A, B, C and D to demand points E, F, G, H and J:

From

To

E F G H J

A

B

C

D

8

15

14

13

10

13

20

19

12

18

6

7

17

11

10

6

15

9

13

12

The present allocation is as follows: A to E---90, A to F---10, B to F---150, C to F---10, C to G---50, C to J---120, D to H---210, D to J---70;(a) Check if this allocation is optimum. If not, find an optimum schedule.(b) If in the above problem the transportation cost from A to G is reduced to 10, what will be the new optimum schedule?

Q21. Determine an IBFS to the following transportation problem using northwest-cell method.

From/To 1 2 3 4 5 Available

A 3 4 6 8 9 20

B 2 10 1 5 8 30

C 7 11 20 40 3 15

D 2 1 9 14 16 13

Demand 40 6 8 18 6

Group No. – 8

Q22. A company has three factories, say F1, F2, F3 that are feeding 5 zones-north, south, eastern, western and central whose monthly demands are 20, 30, 25, 25 and 20 thousands units respectively. The transportation Cost per unit is given below:

If the factories capacities are 40, 30, & 50 thousands units respectively, then

1. Make a transportation model

2. Make a LP model

Factory

Zones

North South East West Central

F1

F2

F3

5 4 7 10 12

13 10 7 5 6

12 9 8 7 10

Q23. Determine an IBFS to the following Transportation Problem using:

(a) North-west corner cell method

(b) VAM

Origin

Destination

A1 B1 C1 D1 E1 Supply

A

B

C

2 11 10 3 7

1 4 7 2 1

3 9 4 8 12

4

8

9

Demand 3 3 4 5 6

Q24. A firm manufacturing a product has three plants has locations X, Y & Z. The three plants have produced 60, 35 & 40 units respectively during this week. The firm has made commitments to cell 22, 45, 20, 18 & 30 units of the product to customers A, B, C, D & E respectively. The Transportation Cost per unit is given below.

Plant Location

Customer

A B C D E

X

Y

Z

4 1 3 4 4

2 3 2 2 3

3 5 2 4 4

Use VAM to determine cost of shifting the product from plant location to customer. Does your solution provide least cost transportation schedule?

Group No. – 9

Q25. A company has factories at 4 different places 1, 2, 3, & 4 which supply items to warehouses A, B, C, D & E. Monthly factory capacities are 200,175,150 and 325 respectively.

Monthly warehouse requirements are 110, 90, 120, 230 & 160 respectively. Unit shipping costs in Rs. Are given in the table:

FROM

TO

A B C D E

1

2

3

4

13 - 31 8 20

14 9 17 26 10

25 11 12 17 15

10 21 13 - 17

Shipments from 1 to B and from 4 to D are not possible. Determine the optimum distribution plan to minimize the shipping cost.

Q26. The personnel department of a construction company recruits 3 categories of workers A, B, & C of 100, 70 & 85 numbers respectively from centers P, Q, R & S. There hourly rates in rupees are given by the matrix below. Find the recruitment pattern at the lowest cost.

CATEGORY CENTRE

P Q R S

REQUIREMENT

A

B

C

12 13 15 14

15 14 13 12

10 12 14 13

100

70

85

AVAILABILITY 35 70 100 50 255

Q27. Consider the following transshipment problem involving 4 sources and 2 destinations. The supply values of the sources S1, S2, S3 & S4 are 200, 250, 200 & 450 units respectively. The demand values of the destinations D1 & D2 are 550 and 550 units respectively. The transportation cost per unit b/w different sources and destinations are summarized in the following table:

Group No. – 10

Q28. The ABC tool company has a sales force of 25 men who work out of 3 regional offices. The company produces 4 basic lines of hand tools. The sales manager feels that 6 sales-men are needed to distribute product line 1; 10 sales-men are needed to distribute product line 2; 4 sales-men are needed to distribute product line 3; & 5 sales-men are needed to distribute product line 4. The cost in Rs. Per day of assigning salesman from each of the offices for selling each of the product lines are as follows:

REGIONAL

OFFICE

PRODUCT LINES

1 2 3 4

A

B

C

20 21 16 18

17 28 14 16

29 23 19 20

At the present time the 10 salesman are allocated to office A; 9 salesmen are allocated to office B and 7 salesmen are allocated to office C. How many salesmen should be assigned from each office to selling each product line in order to minimize cost?

Identify alternate optimal solutions, if any.

29. Find the initial basic feasible solution of the following transportation problem by Vogel’s approximation method.

Warehouses

W1 W2 W3 W4 CAPACITY

F1 10 30 50 10 7

Factory F2 70 30 40 60 9

F3 40 8 70 20 18


Q30. If the matrix element represents the unit transportation time solve the following transportation problem

To

From Available

D1 D2 D3 D4

O1 10 0 20 11 15

O2 1 7 9 20 25

O3 12 14 16 18 5


(Ans 12)

Group No. – 11

Q31. Use least cost method to find IBFS and then find the minimum transportation cost for the following problem (Ans 180)

D1 D2 D3 D4

O1 1

(20)

2 1

(10)

4

O2 3

3

(20)

2

(20)

1

(10)

O3 4 2

(20)

5 9

Q32. An enterprise has three factories at locations A, B and C, which supplies three shops, located at D, E and F. Monthly factory capacities are 10,80 and 15 units respectively. Monthly shop requirements are 75,20 and 50 units respectively. Unit shipping costs (in Rs ) are given below

Factory Shops

D E F

A 5 1 7

B 6 4 6

C 3 2 5

The penalty costs are not satisfying demand at the shops D, E and F are Rs. 5 , Rs 3 and Rs. 2 .respectively. Determine the optimal distribution.

(Ans (A,E) =10 units, (B,D) =60 units, (B,E) =10 units, (B,F) =10 units, (C,D) =15 units, Penalty Rs 80)

Q33. Consider a transportation problem with m=3 and n=4 where

C11=2 C12=3 C13=11 C14=7

C21=1 C22=0 C23=6 C24=1

C31=5 C32=8 C33=15 C34=9

Suppose S1 =6 ,S2=1 and S3=10 whereas D1=7 ,D2=5 and D4=2.Apply the transportation simplex method to find out an optimal solution. (Ans minimum cost=100).

Group No. – 12

Q34. Using VAM, solve the following transportation problem for maximum profit: (ANS: 15300 )

A B C D SUPPLY

12 18 06 25 200

8 7 10 18 500

14 03 11 20 300

DEMAND 180 320 100 400

Q35. Find the optimal transportation cost for the following:

A B C D SUPPLY

03 07 09 10 120

04 05 07 12 170

09 12 05 06 210

DEMAND 290 150 70 90

Q36. Consider the following transportation problem

Factory Godowns Stock available

1 2 3 4 5 6

A 7 5 7 7 5 3 60

B 9 11 6 11 - 5 20

C 11 10 6 2 2 8 90

D 9 10 9 6 9 12 50

DEMAND 60 20 40 20 40 40

It is not possible to transport any quantity from factory B to godown5. Determine

(1) Initial solution by VAM

(2) Optimal basic solution

(3) Is the optimum solution unique? If not , find the alternative optimum solution.

(Ans degeneracy,1120Rs, no,min cost 1120/-)

( ANS Rs. 2430/- )

Group No. – 13Q. 37. A company has three plants A, Band C and three warehouses X, Y and Z. Number of units available at the plants is 60, 70 and 80, respectively. Demands at X, Y and Z are 50, 80 and 80, respectively. Unit Costs of transportation are as follows:

Q. 38. The cost-requirement table for the transportation problem is given as below:

Find optimum solution.

Q. 39. Solve the following transportation problem

Q. 40. Optimize the following profit matrix for transportation using VAM. Test the optimality using stepping stone method.

Availability

700

700

700

Demand of each market is 600.

Markets

To Warehouse

W1 W2 W3

Agra

Lucknow

Allahabad

32

30

40

20

16

12

17

13

18

tutorial-dcdcd3_ or_transportation (1)

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