USA: +1-585-535-1023

UK: +44-208-133-5697

AUS: +61-280-07-5697

DEGENERACY IN THE TRANSPORTATION PROBLEM

We have seen that an initial feasible solution to an m resources/origins and n destination problem consists of (m + n -1) basic variables which is the same as the number of occupied cells. However, if the number of occupied cells: is less than (m + n -1) at any stage of the solution, then the transportation problem is said to have a degenerate solution. Degeneracy as it is called can occur at two stages i.e., at the initial solution or during the testing of the optimal solution. Let us discuss both the cases.

DEGENERACY AT THE INITIAL SOLUTION STAGE

If degeneracy occurs at the initial solution stage, we introduce a very small quantity ε (Greek letter pronounced as epsilon) in one or more of the unoccupied cells to make the number of occupied cells equal to (m+n-1). ε is so small a quantity that its introduction does not change the supply (sources) and demand (destinations) constraints or the rim conditions. E is placed in the unoccupied cell which has the least transportation cost and once e is allotted to it, it is supposed to have been occupied. ε stays in the solution till degeneracy is removed or the final solution is achieved. The value of ε is zero when used in the problems with movement of goods from one cell to another.

The use of ε and degeneracy can be explained with the help of an example.

Example 5.13. A forged parts manufacturing company forges the automobile parts of a particular passenger vehicle at its three plants P-1, P-2 and P-3. The demand of these parts is from distributions located at five location D-I, D-2, D-3, D-4 and D-5. Monthly capacity of the three plants is 100, 60, 80 respectively. Monthly requirement of the auto parts by the five distributors are 50, 50, 60, 40, 80 units respectively. Unit transport costs in Rs. is provided in the following table:

 To

From

D1

D2

D3

D4

D5

P-1

P-2

P-3

6

4

10

10

7

4

8

8

6

8

8

8

4

6

3

REVIEW

  1. 1.   What is a transportation problem? How is it useful in business and industry?
  2. 2.   Explain the use of transportation problem is business and industry giving suitable examples.
  3. 3.   What do you understand by

(a) Feasible solution

(b) North-west solution

(c) Vogel’s approximation method (YAM).

  1. 4.   Discuss various steps involved in finding initial feasible solution of a transportation problem.
  2. 5.   Discuss any two methods of solving a transportation problem. State the advantages and disadvantages of these methods.
  3. 6.   How can an unbalanced transportation problem be balanced? How do you interpret the optimal solution of an unbalanced transportation problem?
  4. 7.   Explain the differences and similarities between the MODI method and stepping stone method used for solving transportation problems.
  5. 8.   What is a transportation method. Explain its objectives. How can we used this model for solving a multiple-site facility beaten problem?
  6. 9.   Which method of solving transportation problems gives a more optimal solution? How will you know when you have achieve the least cost allocation of products between origins and destinations? Explain with examples.
  7. 10.                Formulate a cost minimisation model for the allocation of facilities to locations in problem.

ABC Ltd. is considering the layout of one of its plants divided into three different working areas. There, are three different production facilities and each one has one of them. Assume the data not available.

  1. 11.                An automobile manufacturing company has three factors say F1, F2 and F3 which are feeding 5 different zones north, south, eastern, western and central. The monthly demands of these zones thousand are 25, 40, 30, 25, 20 thousand units respectively. The cost of transporting one unit from the factories F1. F2 and F3 to each of the zones and factory capacity as shown in the table below:

Zone (Transportation costs)

Factory

Capacity (units)

Northern

Southern

Eastern

Western

Central

F1

50,000

8

4

6

10

12

F2

40,000

15

10

6

8

8

F3

30,000

12

8

8

10

10

Perpetrate a transportation model and a LP model.

  1. 1.   XYZ Ltd. has three manufacturing plants P1; P2 and P3 which are stepping the production to three warehouses W1, W2 and W3. The following data is available:

Plant

Production (units)

Warehouse

Requirement (units)

P1

150,000

W1

160,000

P2

120,000

W2

130,000

P3

130,000

W3

80,00

The rate of freight charges/unit is as shown below.

To

W1

W2

W2

P1

1.50

1.60

1.80

From

P2

2.0

1.80

2.50

P3

1.60

1.40

3.00

Determine the initial basic feasible solution using north-west comer method and YAM.

  1. 1.   Plant location of a firm manufacturing a single product has three plants located at A, Band C. Their production during week has been 60, 40 and 50 units respectively. The company has firm commitment orders for 25, 40, 20, 20 and 30 units of the product to customers C-1, C-2, C-3, C-4 and C.5
    respectively. Unit cost of transporting from the three plants to the five customers is given in the table below:

C-1

C-2

C-3

C-4

C-5

Plant location

A

6

1

3

4

6

B

4

4

3

3

2

C

2

6

4

4

6

Use VAM to determine the cost of shipping the product from plant locations to the customers.

  1. 2.   Solve the following transportation problem. Availability at each plant, requirements at each warehouse and the cost matrix is as shown below:

Warehouse

W1

W2

W3

W4

Availability

P1

200

400

600

200

80

Plant

P2

800

400

400

500

100

P3

400

200

500

400

190

Requirement

60

80

80

120

  1. 3.   There are four supply points P-1, P-2, P-3 and P-4 5/ destination A, B, C, D and E. The following table gives in cost of transportation of materials from supply points to demand statements in rupees.

To

A

B

C

D

E

Plant

P-1

10

12

15

16

18

P-2

12

10

12

10

10

P-3

15

20

6

12

16

P-4

12

18

10

12

12

The present allocation is as follows

P-1 to A 100, P -1 to B -20, P -2 to B-160, P-3 to B-10, P-3 to C-60, P-3 to E-120, P-4 to 0-200, P-4 to E-100. Find an optimal solution for allocations. If we reduce the cost from any supply point to any destination, what do you think will be the impact. Select any case and discuss the outcome.

  1. 1.   Anand Lamps India the (AUL) operates three manufacturing plants and four warehouse. Capacity of the plants and forecast demand of warehouses is as follows :-

Factory

at

Capacity

(tons)

Warehouse

at

Forecast demand

(tons)

P-1

16

A

6

P-2

8

B

7

P-3

8

C

6

D

12

The transportation cost per ton in Rs. is as given below.

From/To

A

B

C

D

P-1

100

120

80

20

P-2

120

200

60

60

P-3

160

200

100

80

ALIL wishes to minimise its transport costs.

  1. 2.   The recruitment policy of the personnel department of a company is such that three types of workers managers, supervisors and workers are required 20, 40, 150 from four placement service centres A, B, C, D and E. The hourly rates in Rs. are given by the matrix below:

Placement service centre

A

B

C

D

Requirement

Managers

16

10

8

12

20

Category

Supervisors

20

24

30

20

40

Workers

200

180

60

160

150

Availability

50

60

200

80

390

Determine the requirement pattern at the lowest cost.

  1. 3.   Find the initial basic feasible solution to the following transportation problem by

(a) Least cost method.

(b) North-west corner rule.

State which of the methods is better

Table

TO

Supply

2

7

4

5

From

3

3

1

8

5

4

7

7

1

6

2

14

Demand

7

9

18

  1. 1.   Solve the following transportation problem by VAM.

Consumers

A

B

C

Available

Suppliers I

6

7

4

14

II

4

3

1

12

III

1

4

7

5

Required

6

10

15

31

Use V AM to find an initial BFS.

  1. 1.   The apex company is the distributor for television receivers. It owns three warehouses with stocking capacity as follows.

Warehouse location

Sets in stock

A

100

B

25

C

75

 

It has the following order for set deliveries:

Market location

Orders

X

80

Y

30

Z

90

 

Delivery costs for warehouses to each customers are largely a function of mileage or distance. The per unit cost have been determined to be:

X

Y

Z

A

B

C

5

3

5

10

7

8

2

5

4

 

The deliveries could be made in many ways but the distributor would like to deliver the T.V. sets in a way that would minimize the delivery cost. Give the distribution schedule. Use VAM only.

  1. 2.   A company has four factories situated in different locations and five warehouses in different cities. The matrix of transportation cost is given below.

Factories

Warehouse

I

II

III

IV

Requiremet

A

4

8

7

6

150

B

9

5

8

8

50

C

6

5

8

7

40

D

5

8

6

3

60

E

7

6

5

8

200

Capacity

(units)

100

80

120

100

 

Find the optimum transportation schedule using VAM only.

  1. 1.   Obtain the initial BFS to the following transportation problem by VAM.

D1

D2

D3

D4

D5

ai

A1

5

7

10

5

3

5

A2

8

6

9

12

14

10

A3

10

9

8

10

15

10

Bj

3

3

10

5

4

25

 

  1. 2.   A steel company has three furnaces and five rolling mills. Transportation cost (rupees per quintal) for sending steel from furnaces to rolling mills are given in the following table :

Furnaces

M1

M2

M3

M4

M5

Availability (Q)

A

B

C

4

5

6

2

4

5

3

5

4

2

2

7

6

1

3

8

12

14

Requirement (Quintal)

4

4

10

8

8

 

How should they meet the requirement? Use YAM.

  1. 3.   A cement factory manager is considering the best way to transport cement from his three and manufacturing centres P, Q and R to depot A, B, C, D and E. The weekly production and demand along with transportation costs per ton are given below:

 

A

B

C

D

E

Tons

P

4

1

3

4

4

60

Q

2

3

2

4

3

35

R

3

5

2

2

4

40

Requirement (Quintal)

22

45

20

18

30

135

 

What should be the distribution programme?

  1. 4.   Solve the following problem and test its optimality

Project

A

Project

B

Project

C

Plant

Pant W

4

8

8

56

Pant X

16

24

16

82

Pant Y

8

16

24

77

Project requirement

72

92

41

215

205

 

  1. 5.   The relevant data on demand, supply and profit per unit of a product manufactured and sold by a company are given below:

Factory

1

2

3

4

5

Supply

P

5

8

14

7

8

100

Q

2

6

7

8

7

20

R

3

4

5

9

8

60

4

10

7

8

6

20

Demand

45

65

70

35

15

135

Given the transportation from A to 3, and D to 2 are not allowed due to certain reasons, Find out using VAM optimal method of transportation from factories to marketing centers.

  1. 1.   Solve the following problem in which cell entries represent unit costs:

D1

D2

D3

Available

Q1

2

7

4

5

Q2

3

3

1

8

Q3

5

4

7

7

Q4

1

6

2

14

Required

7

9

18

44

 

Apply MODI method to test optimality,

  1. 2.   Solve the following transportation problem

To

From

D1

D2

D3

D4

Available

S1

4

3

1

2

80

S2

5

2

3

4

60

S3

3

5

6

3

40

Requirement

50

60

20

50

180 Total

 

Apply MODI method to test its optimality.

  1. 3.   Solve the following transportation problem.

1

2

3

4

5

6

Stock available

F

1

7

5

7

7

5

3

60

A

C

2

9

11

6

11

-

5

20

T

O

3

11

10

6

2

2

8

90

R

Y

4

9

10

9

6

9

12

50

Demand

60

20

40

20

40

40

  1. 4.   A manufacturer of jeans is interested in developing an advertising campaign that will reach four different age groups. Advertising campaigns can be conducted through TV, radio and magazines. The following table gives the estimated cost in paise or exposure for each age group according to the medium employed. In addition, maximum exposure levels possible in each of the media, namely TV, radio and magazines are 40, 30 and 20 millions respectively. Also the minimum desired exposures with each age group, namely 13-18, 19-25, 26-35, 36 and older, are 30, 25, 15 and 10 millions. The objective is to minimize the cost of attaining the minimum exposure level in each age group.

Media

Age groups

13-18

19-25

26-35

36 an older

TV

12

7

10

10

Radio

10

9

12

10

Magazines

14

12

9

12

 

(a) Formulate the above as a transportation problem and find the optimal solution.

(b) Solve the problem if the policy is to provide at least 4 million exposures through TV in the age groups and at least 8 million exposures through TV in the age group 19-25.

  1. 1.   STRONGHOLD Construction Company is interested in taking loans from banks for some of its projects P, Q, R, S, T. The rates of interest and the lending capacity differ from bank to bank. All these projects are to be completed. The relevant details are provided in the following table. Assuming the role of a consultant, advise this company as to how it should take the loans so that the total interest payable may be the least. Are there alternative optimum solutions? If so, indicate one such solution.

Bank

Interest rate in percentage for project

Max. credit

(in thousands)

P

Q

R

S

T

Private Bank

20

18

18

17

17

Any amount

400

250

Nationalized Bank

16

16

16

16

16

Co-operative Bank

15

15

15

14

14

Amount required (in thousands)

200

150

200

75

75

 

  1. 2.   A company has four terminals u, v, w and x. At the start of particular day 10, 4, 6 and 5 trailers respectively are available at these terminals. During the previous night 13, 10, 6 and 6 trailers respectively were loaded at plants A, S, C and D. The company dispatcher has come up with the costs between the terminals and plants as follows:
  1. 1.   The cost conscious company requires for the next month 300, 260 and 180 tonnes of stone chips for its three constructions, C1, C2 and C3 respectively. Stone chips are produced by the company at three, mineral fields taken on short lease. All the available boulders must be curshed into chips. Any excess
    ; chips over the demands at sites C1, C2 and C3 will be sold ex-fields. The fields M1, M2 and M3 will yield 250, 320 and 280 tonnes chips respectively. Transportation costs from mineral fields to construction sites vary according to distances, which are given below in monetary unit (MU).

To

From

C1 C2

C3

M1

M2

M3

85

7

74

5

69

5

 

(a) Determine the optimal economic transportation plan for the company and the overall transportation cost in MU.

(b) What are the quantities to be sold from M1, M2 and M3 respectively?

  1. 2.   A company has 4 different factories in 4 different locations in the country and four sales agencies in four other locations in the country. The cost of productions, the sale price, shipping cost in the cell of matrix, monthly capacities and monthly requirements are given below:

Factory

1

2

3

4

Capacity

Cost of production

A

7

5

6

4

10

10

B

3

5

4

2

15

15

C

4

6

4

5

20

16

D

8

7

6

5

15

15

Monthly requirement

8

12

18

22

Selling price

20

22

25

18

 

Find the monthly production and distribution schedule, which will maximize profits.

  1. 1.   A company wishes to determine an investment strategy for each of the next four years. Five investment types have been selected, investment capital levels have been established for each investment type. An assumption is that amounts invested in ‘any year will remain invested until the end of the planning horizon of four years. The following table summarises the data for this problem. The values in the body of the table represent net return on investment of one rupee upto the end of the planning horizon. For
    example, a rupee invested in type B at the beginning of year 1 will grow to Rs. 1.90 by the end of the fourth year, yielding a net return of 0.90.

Investment made at the beginning of year

Investment type

Rupees available in (000’s)

A

B

C

D

E

NET RETURN DATA

1

0.90

0.90

0.60

0.75

1.00

500

2

0.55

0.65

0.40

0.60

0.50

600

3

0.30

0.25

0.30

0.50

0.20

750

4

0.15

0.12

0.25

0.35

0.10

800

Maximum rupees investment (in 000’s)

750

600

500

1,000

 

The object in this problem is to determine the amount to be invested at the beginning of each year in an investment type so as to maximize the net rupee return for the four-year period.

Solve the above transportation problem and get an optimal solution. Also calculate the net return on investment for the planning horizon of four-year period.

  1. 1.  
Investment made

Net return rate

Available

P

Q

R

S

1

95

80

70

60

70

2

75

65

60

50

40

3

70

45

50

40

90

4

60

40

40

30

30

4

60

40

40

30

30

Maximum Investment (in lacs)

40

50

60

60

 

Solve the given problem so as to give maximum return.

  1. 2.   Following is the profit matrix based on four factories and three sales depots of the company.

Sales depots

S1

S2

S3

S4

F1

6

6

1

10

Factories     F2

-2

-2

-4

150

F3

3

2

2

50

F4

8

5

3

100

Requirements

80

120

150

 

Determine the most profitable distribution schedule and the corresponding profit, assuming no profit in case of surplus production.

  1. 1.   Determine the optimal solution to the problem given below and find the minimum cost of transportation.

To

From

E

F

G

H

I

ai

A

4

7

3

8

2

4

B

1

4

7

3

8

7

C

7

2

4

7

7

9

D

4

8

2

4

2

2

bi

8

3

7

2

2

 

  1. 2.   A transport corporation has trucks at 3 garages A, Band C in a city. The number of trucks available in each garage, the number of trucks required by each customer and the distance (Km) from garage to customer’s locations are given below:

1

2

3

4

Availability

Garages

A

7

6

9

12

12

B

8

6

7

10

8

C

10

7

8

12

10

Requirements

4

3

5

8

 

Just before sending the trucks, it is known that road from B to customers 2 is closed for traffic due to road block. How should the trucks be assigned to the customer is order to minimize the total distance to be covered to the road block.

  1. 3.   A leading firm has three auditors. Each auditor can work up to 160 hours during the next month, during which time three projects must be completed. Project 1 will take 130 hours.
  2. 4.   Determine an initial basic feasible solution to the following transportation problem using north-corner rule:

To

Available

From

3

4

6

8

9

20

2

10

1

5

8

30

7

11

20

40

3

15

2

1

9

14

16

13

Demand

40

6

8

18

6

 

  1. 5.   Determine an initial basic feasible solution to the following transportation problem using row minima method:

To

Available

From

5

2

4

2

22

4

8

1

6

15

4

6

7

5

8

Demand

7

12

17

9

 

  1. 6.   Find the initial basic feasible solution of the following transportation problem by approximate method :

W1

W2

W3

W4

Capacity

F1

19

30

50

10

7

Factory

F2

70

30

40

60

9

F3

40

8

70

20

18

Requirement

5

8

7

14

34 (Total)

  1. 7.   Determine an initial basic feasible solution to the following L.P. using:

(a) North-west comer rule,

(b)  Vogel’s approximation method.

A1

B1

C1

D1

E1

Supply

A

2

11

10

3

7

4

Origin

B

1

4

7

2

1

8

C

3

9

4

8

12

9

Demand

3

3

4

5

6

  1. 8.   Solve the transportation problem for which the cost, origin availabilities and destination requirements are given below:

D1

D2

D3

D4

D5

D6

a1

O1

1

2

1

4

5

2

30

O2

3

3

2

1

4

3

50

O3

4

2

5

9

6

2

75

O4

3

1

7

3

4

6

20

b1

20

40

30

10

50

25

175 (Total)

 

  1. 9.   Give a mathematical formulation of the transportation and simplex methods. What are the differences in the nature of problems that can be solved by these methods.
  1. 1.   Given below is the unit costs array with supplies ai = 1,2,3 and demands b1j= 1,2,3 and 4.

1

2

3

4

a1

1

8

10

7

6

50

Source

2

12

9

4

7

40

3

9

11

10

8

30

b1

25

32

40

23

120 (Total)

Find the optimal solution to the above Hitchcock problem.

  1. 2.   Consider four bases of operations B and three tartets T. The tons of bombs per air-craft from any base that can be delivered to any target are given in the following table:

1

2

3

1

8

6

5

Base (B)

2

6

6

6

3

10

8

4

4

8

6

4

 

The daily sortie capability of each of the four bases is ISO sorties per day. The daily requirement in sorties over each individual target is 200. Find the allocation of sorties from each base to each target which maximizes the total tonnage over all the three targets explaining each step.

  1. 3.   General Electrodes is a big electrode manufacturing company. It has two factories and three main distribution centers in three cities. The supply and demand transportation. How should the trips be scheduled so that the cost of transportation is minimized?

The present cost of transportation is around Rs. 3,100 per month. What can be the maximum savings by proper scheduling?

Centers A B C
Requirement 50 50 150
Cost per trip from X plant 25 35 10
Cost per trip from Y plant 20 5 80
Capacity of plant X 15 units of electrodes
  1. 1.   A company has decided to manufacture some or all of five new products at three of its plants. production capacity of each of these three plants is as follows:
Plant No. Production capacity in total number of units
1 40
2 60
3 90

Sales potential of the five products is as follows:

Product No. 1 2 3 4 5
Market potential in units 30 40 70 40 60

Plant No. cannot produce product No.5. The variable cost per unit for the respective plant and product combination is given below:

Product No. 1 2 3 4 5
Plant No.1 20 19 14 21 16
Plant No.2 15 20 13 9 16
Plant No.3 18 15 18 20

Based on above data, determine the optimum product to plant combination by programming.

  1. 1.   A fertilizer company has three plants A, Band C which supply to six major distribution centers 1, 2, 3, 4, 5 and 6. The table below gives the transportation costs per case, the plant annual capacities, and predicted annual demands at different centers in terms of thousands of cases. The variable production costs per case Rs. 8.50, Rs. 9.40 and Rs. 7 respectively at plants A, Band C. Determine the minimum cost production and transportation allocation.

Transportation cost Rs. per case

Major distribution centers

1

2

3

4

5

6

Annual production in thousands of cases

A

2.50

3.50

5.50

4.50

1.50

4.00

2,200

Plants

B

4.60

3.60

2.60

5.10

3.10

4.10

3,400

C

5.30

4.30

4.80

2.30

3.30

2.80

1,800

Annual demand in thousand of cases

850

750

420

580

1,020920

 

Prove that if the variable production costs are the same at every plant, once can obtain an optimal allocation by using transportations costs only.

  1. 2.   Describe the transportation problem. Give method of finding an initial feasible solution. Explain what is meant by -an optimality test. Give the method of improving over the initial solution to reach the optimal feasible solution.
  2. 3.   The unit costs of transportation from site i to site j are given below. At site i= 1, 2, 3 stocks of 15O, 200 and 170 units respectively are available. 300 units are to be sent to site 4 and the rest to site 5. Find the cheapest way of doing this.

To

1

2

3

4

5

1

-

3

4

10

7

2

1

-

2

16

6

From

3

7

4

-

12

13

4

8

3

9

-

5

5

2

1

7

5

-

 

(Hint. In accordance with the restriction of supply and demand, table 3.142reduces to the following table:

To

4

5

Available

1

10

7

150

From

2

16

6

200

3

12

13

170

Required

300

220

 

  1. 1.   Consider the following unbalanced problem:

To

From

1

2

3

Supply

1

5

1

7

10

2

6

4

6

80

3

3

2

5

15

Demand

75

20

50

Since there is not enough supply, some of the demands at these destinations may not be satisfied. Suppose that there are penalty costs for every unsatisfied demand unit which are given by 5, 3 and 2 for destinations I, 2 and 3 respectively. Find the optimal solution.

[Hints. The balanced transportation table with dummy source and associated penalty costs in shown below.

To

From

1

2

3

Supply

1

5

1

7

10

2

6

4

6

80

3

3

2

5

15

Dummy

5

3

2

40

Demand

75

20

50

This table can now be solved by the usual MODI method.]

  1. 2.   A production control superintendent finds the following information on his desk. In departments A, B and C the number of surplus pallets is 18, 27 and 21 respectively. In departments G, H, I and J number of pallets required is 14, 12, 23 and 17 respectively. The time in minutes to move a pallet from
    one department to another is given below.

To

From

G

H

I

J

A

13

25

12

21

B

18

23

14

9

C

23

15

12

16

 

What is the optimal distribution plan to minimize the moving time?

  1. 1.   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.

To

From

E

F

G

H

J

A

8

10

12

17

15

B

15

13

18

11

C

14

20

6

10

13

D

13

19

7

6

12

 

The present allocation is as follows.

A to E 90, A to F 10, C to F 10, C to G 50, C to J 120, D to H 210, D to 170.

(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?

  1. 2.   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 in rupees from each warehouse to each market.

Market

I

II

III

IV

Supply

A

5

2

4

3

22

Warehouse

B

4

8

1

6

15

C

4

6

7

5

8

Requirement

7

12

17

9

The shipping clerk has working 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 to 1 unit from C to III.

(a) Check and see if the clerk has made the optimal schedule.

(b) Find the optimum 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?

  1. 3.   A company has factories at A, Band C which supply warehouses at D, E, F and G. Monthly factory capacities are 250, 300 and 400 units respectively for regular production. If overtime production is utilised, factories A and B can produce 50 and 75 additional units respectively at overtime incremental costs of Rs. 4 and Rs. 5 respectively. The current warehouse requirements are 200, 225, 275 and 300 units respectively. Unit transportation costs in rupees from factories to warehouses are as follows:

To

From

D

E

F

G

A

11

13

17

14

B

16

18

14

10

C

21

24

13

10

 

Determine the optimum distribution for this company to minimize costs.

[Hint. First table is made which takes into account the overtime production and the corresponding production costs.

D

E

F

G

Supply

A

11

15

13

17

17

21

14

18

250

50

B

16

21

18

23

14

19

10

15

300

75

C

21

24

13

10

400

Demand

200

225

275

300

In the above table, total supply = 1,075 units

total demand =1,000 units

Therefore, we add a dummy warehouse with demand of 75 units and cost coefficients zero in each of its cell.

D

E

F

G

Dummy

Supply

A

11

15

13

17

17

21

14

18

0

0

250

50

B

16

21

18

23

14

19

10

15

0

0

300

75

C

21

24

13

10

0

400

Demand

200

225

275

300

75

The initial feasible solution can now be obtained and can be optimized using MODI method].

  1. 1.   A company has plants at A, B and C which have capacities to produce 300 kg, 200 kg and 500 kg respectively of a particular chemical per day. The production costs per kg in these plants are Re. 0.70, Re. 0.60 and Re. 66 respectively. Four buk consumers have placed orders for the product on the following basis:

kg required per day

Price offered Rs. 1 kg

I

400

1.00

II

250

1.00

Customer

III

350

1.02

IV

150

1.03

Shipping costs (in paise per kg) from plants to consumers are given in the table below.

I

II

III

IV

A

3

5

4

6

From

B

8

11

9

12

C

4

6

2

8

Work out an optimal schedule for the above situation. Under what conditions would you change the schedule?

  1. 1.   ABC manufacturing company wishes to develop a monthly production schedule for the next months. Depending upon the sales commitments, the company can either keep the production constant, allowing fluctuations in inventory or inventories can be maintained at a constant level, with fluctuating production. Fluctuating production in working overtime, the cost of which is estimated to be double the normal production cost of Rs. 12 per unit. Fluctuating inventories result in inventory carrying cost of Rs. per unit. If the company fails to fulfill its sales commitment, it incurs a shortage cost of Rs. 4 per unit per month. The production capacities for the next three months are shown below.

Production capacity

Month

Regular

Overtime

Sales

1

50

30

60

2

50

0

120

3

60

50

40

Determine the optimal production schedule.

(Hint. Here regular and overtime production capacity is the source and the sales is the destination. The costs for different cells may be computed as follows:

(a) For items produced and sold in the same month, there will be no inventory carrying cost. Thus the costs for cells (1, 1), (2, 2), (3, 3) are Rs. 12 each and for cells (1, 1), (3, 3) are Rs. 24 each.

(b)  For items produced in a particular month and sold in subsequent months, additional inventory cost of Rs, 2 per month will be incurred. Thus cells (1, 2), (1, 3), (2, 3), (1, 2) and (1, 3) will have costs of Rs. 14, Rs. 16, Rs. 14, Rs. 26 and Rs. 28 respectively.

(c) For items produced in a particular month to meet the backlog of sales during previous months, In addition to the production costs (normal or overtime), shortage costs of Rs. 4 per month will be incurred. Therefore, for cells (2, 1), (3, 2), (3,1), (31, 2) and (31, 1) the costs will be Rs. 16, Rs. 16, Rs. 20, Rs. 28 and Rs. 32 respectively. Thus the equivalent transportation table for the given problem will be one shown below:

1

2

3

Dummy

Production capacity

1

12

14

16

0

50

2

16

12

14

0

50

Month

3

20

16

12

0

60

11

24

26

28

0

30

31

32

28

24

0

50

Sales

60

120

40

20

Review

  1. 2.   (i) Initial basic feasible solution

(ii) Northwest corner rule

(iii) Unbalanced transportation problem.

  1. 3.   (i) Initial basic feasible solution

(ii) Transportation problem

(iii) North-west corner rule.

  1. 4.   (i) Degenerate solution

(ii) Prohibited routes in transportation problem,

  1. 5.   (i) Balanced transportation problem.

(ii) Initial Basic Feasible Solution of a transportation problem.

  1. 6.   Describe transportation problem with its general mathematical formation.
  2. 7.   Explain the various steps involved in solving a transportation problem by applying the N. West Corner Method.
  3. 8.   Describe transportation problem with its general mathematical formulation.
  4. 9.   Explain various steps in Vogel’s approximation method for finding initial
    feasible solution of the transportation problem.
  5. 10.                Explain various steps involved in solving a transportation problem by anyone of the Transportation Problem
  6. 11.                Write a brief note on Vogel’s approximation method to solving problem.
  7. 12.                Explain the various steps involved in solving a transportation problem by applying the North West Corner Method.
  8. 13.                Describe sequences of steps in MODI Method of solving a transportation problem.
  9. 14.                Explain the transportation problem giving examples.
  10. 15.                Explain

(a) NWCM,

(b)  LCEM,

(c)  VAM and Test of optimality by

(i)    stepping stone method and

(ii)   MODI method. Take suitable examples.

  1. 1.   Find the initial basic feasible solution of the following transportation problem with the help of North-west corner method.

A

B

C

X

11

21

16

14

Plant

Y

07

17

13

26

Available at plant

Z

11

23

21

36

18

28

28

Market requirement

  1. 1.   Find the initial basic feasible solution of the following transportation problem using least cost method:

W1

W2

W3

W4

Factory Capacity

F1

30

25

40

20

100

F2

29

26

35

40

250

F3

31

33

37

30

150

Warehouse requirement

90

160

200

50

 

  1. 2.   A construction company needs 3, 3, 4 and 5 million cubic feet of fill at 4 dam sites. It can transfer the fill from three mounds A, Band C where 2, 6 and 7 million cubic feet of fill is available. Costs (in lakhs of Rs.) of transporting one million cubic feet of fill from the mounds to 4 dam sites are:

From/To

I

II

III

IV

A

15

10

17

18

B

16

13

12

13

C

12

17

20

11

Determine the optimum distribution for this company to minimize the total cost.

  1. 3.   Suggest an optimal transportation plan with a view to minimize cost from form the following information.

Source Destination

 

Cost of shipping per unit

Unit Demand

F1

F2

F3

W1

Rs.

0.9

1

1

5

W2

Rs.

1

1.4

0.8

20

W3

Rs.

1.3

1

0.8

20

Units Available

20

15

10

45

 

  1. 1.   A distribution system has the following constraints:

Factory

Capacity (units)

Warehouse

Demand (units)

A

45

I

25

B

15

II

55

C

40

III

20

The transportation cost per unit (Rs.) associated with each route is as follows:

From

I

II

III

A

10

7

8

B

15

12

9

C

7

8

12

 

Find the optimum transportation schedule and the minimum total cost of transportation.

  1. 2.   Given the following information, compute optimal transportation cost using any method.

Project           Requirement per week, Truck Available Per week.

Loads

Plants

A

45

X

35

B

50

Y

40

C

20

Z

40

 

Cost information

To Project

From

A

B

C

Plant X

5

10

10

Plant Y

20

30

20

Plant Z

5

8

12

 

  1. 3.   Determine the optimum solution to the following problem:

Cost Matrix

To

 

 

T1

T2

T3

T4

Availability

F1

10

20

5

7

10

F2

13

9

12

8

20

From

F3

4

15

7

9

30

F4

14

7

9

10

40

F5

3

12

6

19

40

Demand

60

60

20

10

 

  1. 1.    Solve the following transportation problem as

(a) maximization problem and

(b) minimization problem

 

D1

D2

D3

D4

D5

D6

Available

Q1

Q2

Q3

Q4

2

3

3

4

1

2

5

2

3

2

4

2

3

4

2

1

2

3

4

2

5

4

1

2

60

40

60

30

Required

30

50

20

40

30

10

 

  1. 2.   Solve the following transportation cost problem:

 

A

B

C

D

E

F

Available

Q

R

S

T

5

5

2

6

3

6

1

10

7

12

2

9

3

5

4

5

8

7

8

10

5

11

2

9

3

4

2

8

Required

3

3

6

2

1

2

 

  1. 3.   Find the initial basic feasible solution by at least three different methods for the following transportation problem:

From/To

D1

D2

D3

D4

Available

F1

F2

F3

10

1

7

7

6

4

3

7

5

6

3

6

3

5

7

Demand

3

2

6

4

 

 

  1. 4.   Solve the following cost minimising transportation problem.

 

Warehouses

Capacity

Factory

95

115

195

105

180

180

80

40

95

15

30

70

12

7

1

Demand

5

4

4

11

 

  1. 5.   Describe transportation problem with its general mathematical formulation.
  2. 6.   Explain various steps in Vogel’s approximation method for finding initial basic feasible solution of the transportation problem.
  3. 7.   Explain various steps involved in solving a transportation problem by anyone of the method to solve it.
  4. 8.   Write a brief note on Vogel’s approximation method to solving transportation
    problem.
  5. 9.   Explain the various steps involved in solving a transportation problem by applying the North West Comer Method.
  6. 10.                Describe sequence of steps in MODI Method of solving a transportation problem.
  7. 11.                Discuss various methods of getting basic feasible solution of Transportation
    problem. Which one would you prefer and why?
  8. 12.                Describe transportation problem with its general mathematical formulation.
  9. 13.                Explain the various steps involved in solving a transportation problem by applying the North West Comer method.
  1. 1.   Find the initial basic feasible solution of the following transportation problem with the help of North-West Comer Method.

Market

A

B

C

X

11

21

16

14

Plant

Y

07

17

13

26

Available at plant

Z

11

23

21

36

18

28

25

Market requirement

  1. 2.   Find the initial basic feasible solution of the following transportation problem using least cost method;

W1

W2

W3

W4

Factory capacity

F1

30

25

40

20

100

F2

29

26

35

40

250

F3

31

33

37

30

150

Warehouse requirement

90

160

200

50

 

  1. 3.   Suggest an optimal transportation plan with view to minimize following information.

Cost of shipping per unit

Source Destination

F1

F2

F3

Units Demanded

W1

W2

W3

0.9

1

1.3

1

1.4

1

1

0.8

0.8

5

20

20

Units Available

20

15

10

45

 

  1. 1.   A construction company needs 3, 3, 4 and 5 million cubic feet of fill at 4 dam sites. It can transfer the fill from three mounds A, B and C where 2, 6 and 7 million cubic feet of fill is available. Costs (in lakhs of Rs.) of transporting one million cubic of fill from the mounds to 4 dam sites are:

From/To

I

II

III

IV

A

15

10

17

18

B

16

13

12

13

C

12

17

20

11

 

Determine the optimum distribution for this company to minimize the total cost.

  1. 2.   A distribution system has the following constraints:

Factory

Capacity (units)

Warehouse

Demand (units)

B

B

C

45

15

40

I

II

III

25

55

20

 

The transportation cost per unit (Rs.) associated with each route are as follows:

From

I

II

III

A

B

C

10

15

7

7

12

8

8

9

12

 

Find the optimum transportation schedule and the minimum total cost of
transportation.

  1. 3.   Given the following information, compute optimal transportation cost using any method.

Project

Requirement per week, truck loads

Plants

Available per week, truck loads

A

B

C

45

50

20

X

Y

Z

35

40

40

 

Cost information From

To project

A

B

C

Plant X

Plant Y

Plant Z

5

20

5

10

30

8

10

20

12

 

  1. 4.   Solve the following transportation problem:

S1

S2

S3

S4

D1

D2

D3

D4

5

10

12

5

7

12

10

7

8

15

7

6

9

18

9

9

125

125

125

125

25

15

30

10

 

  1. 5.   Solve the following transportation problem. The cost matrix is given below:

Requirement

A

B

C

Source

1

2

3

4

7

10

12

10

40

9

12

10

10

30

12

9

14

12

20

25

15

30

10

  1. 6.   Find the optimum cost of transportation for the following problem:

W

X

Y

Z

Availability

I

II

III

IV

30

40

40

50

30

50

40

20

30

30

40

30

60

50

60

70

200

400

300

200

Requirement

350

450

200

100

 

  1. 104.             Find the optimum sol. to the following transportation problem in which the cells contain the transportation cost in rupees:

W1

W2

W3

W4

W5

Available

F1

F2

F3

F4

7

8

6

5

6

5

8

7

4

6

9

7

5

7

6

8

9

8

5

6

40

30

20

10

Required

30

30

15

20

5

100

 

  1. 105.             The table below gives the information regarding the quantity required by 4 markets and supply capacity of 3 warehouses. The unit transportation cost from warehouse market is also given below. Find the optimum allocation that minimizes the total shipping cost.

Market

Warehouse

1

2

3

4

Supply

A

B

C

Requirement

5

4

4

7

2

8

6

12

4

1

7

17

3

6

5

9

22

15

8

 

  1. 106.             Find the basic feasible sol. of the following transportation problem by VAM. Also find the optimal transportation plan

1

2

3

4

5

Available

A

B

C

D

4

5

3

2

3

2

5

4

1

3

6

4

2

4

3

5

6

5

2

3

80

60

40

20

Required

60

60

30

40

10

200 (Total)

 

  1. 107.             Solve the following transportation problem

To

From

A

B

C

Available

P1

P2

P3

4

16

8

8

24

16

8

16

24

56

82

77

Required

72

102

41

 

  1. 108.             (23) Solve the following transportation problem for maximum profit

Market

Warehouse

A

B

C

D

X

Y

Z

12

8

14

18

7

3

6

10

11

25

18

20

 

Availability at Warehouse Demand in Markets
X : 200 Units

Y : 500 Units

Z : 300 Units

A : 180 Units

B : 320 Units

C : 100 Units

D : 400 Units