BDA Problem Set By Prof. Girish Phatak

Second Topic Transportation Model Problems discussed in the class

Q.1 A dairy firm has their plants located in a state. The daily milk production at each plant is

as follows:

PLANT 1 2 3

MILK SUPPLY 6 1 10

Each day, the firm must fulfill the needs of its four distribution centers. Minimum requirement at each center is as follows:

CENTER 1 2 3 4

MILK

SUPPLY

7 5 3 2

Cost in the hundreds of rupees of shipping one million litre from each plant to each

distribution center is given in the following table:

Distribution center

D1 D2 D3 D4

P1 2 3 11 7

P2 1 0 6 1

P3 5 8 15 9

Find initial basic feasible solution for given problem by using:

(a)North – West corner rule

(b) Least cost method and

(c) Vogel’s approximation method

If the objective is to minimize the total transportation cost.

Q.2 A Company has factories at F1, F2, and F3 which supply to warehouses at W1, W2, and W3.

Weekly factory capacities are 200,160,and 90 units, respectively. Weekly warehouse

requirement are 180,120and 150 units respectively. Unit shipping costs (in rupees) are as

follows:

Ware house

W1 W2 W3 SUPPLY

F1 16 20 12 200

F2 14 8 18 160

F3 26 24 16 90

DEMAND 180 120 150 450

Determined the optimal distribution for this company to minimize total shipping cost.

Second Topic Transportation Model Problems for Practice

Q.1 A company has four manufacturing plants and five warehouses. Each plant manufactures

the same product which is sold at different prices in each warehouse area. The cost of

manufacturing and cost of raw material are different in each plant to various factors. The

capacities of the plants are also different. The data are given in the following table:

PLANT ITEM 1 2 3 4

Manufacturing cost (RS) per unit 12 10 08 08

Raw material cost (RS)per unit 08 07 07 05

Capacity per unit time 100 200 120 80

The company has five warehouses. The sales prices, transportation costs and demands are

given in the following table:

ware house transportation cost per unit sale price demand

per unit(Rs.)

1 2 3 4

A 4 7 4 3 30 80

B 8 9 7 8 32 120

C 2 7 6 10 28 150

D 10 7 5 8 34 70

E 2 5 8 9 30 90

(a) Formulate this problem as a transportation problem to maximize profit.

(b) Find the solution using VAM method.

Q.2 Obtain an optimal solution to the transportation problem by UV method. Use VAM method

for obtaining initial feasible solution

D1 D2 D3 D4 CAPACITY

S1 19 30 50 10 7

S2 70 30 40 60 9

S3 40 8 70 20 18

DEMAND 5 8 7 14 34

Q.3 Consider a firm having 2 factories. The firm is to ship its products from the factories to

three- retail stores. The number of units available at factories X and Y are 200 and 300,

respectively while those demanded at retail stores A ,B and C are 100,150 and 250,

respectively. Rather than shipping directly from factories to retail stores, it is asked to

investigate the possibility of Trans- shipment. The transportation cost (in Rupees) per unit is

given in the table .

FACTORY RETAIL STORE

X y A B C

Factory X 0 8 7 8 9

Y 6 0 11 9 10

Retail store A 7 2 0 5 1

B 1 5 1 0 4

C 8 9 7 8 0

Find out the optimal shipping schedule.

Q.4 ABC limited has three production shops supplying a product to five warehouses. The cost of

production varies from shop to shop and cost of transportation from one shop to a warehouse

also varies. Each shop has a specific production capacity and each warehouse has certain amount

of requirement. The cost of transportation are given below:

Ware house

I II III IV V SUPPLY

A 6 4 4 7 5 100

B 5 6 7 4 8 125

C 3 4 6 3 4 175

DEMAN

D

60 80 85 105 70 400

The Cost Of Manufacturing The Product At Different Production Shop Is

shop Variable cost Fixed cost

A 14 7000

B 16 4000

C 15 5000

Find the optimum quantity to be supplied from each shop to different warehouses at minimum

total cost.

Third Topic Assignment Model Problems discussed in the class

Q.1 A job production unit has four jobs A,B,C,D which can be manufactured on each

of the four machines P,Q,R and S. The processing cost of each job on each machine is given in

the table below:

Jobs Machine

P Q R S

Processing cost (Rs.)

A 31 25 33 29

B 25 24 23 21

C 19 21 23 24

D 38 36 34 40

To achieve minimum processing cost, which job will you process on which machine?

Q.2 A workshop has four machines and four tasks for completion. Each of the machines can

perform each of four tasks. Time taken at each of the machines to complete each task is given in

the matrix below:

How should the tasks be assigned to machines requirement of machine hours?

Tasks Machine

A B C D

Processing times (Hrs.)

I 51 77 49 55

II 32 34 59 68

III 37 44 70 54

IV 55 55 58 55

Q.3 A pharmaceutical company has four branches, one each at city A, B, C and D. A branch

manager is to be appointed one at each city, out of four candidates P, Q, R and S. The monthly

business depends upon the city and the effectiveness of the branch manager in that city.

Branch

manager

City

A B C D

Monthly business (Rs. lakhs)

P 11 11 9 9

Q 13 16 11 10

R 12 17 13 8

S 16 14 16 12

Which manager should be appointed at which city so as to get maximum total monthly business?

Q.4 The production cost or products P1, P2, P3, P4 and P5 per unit made on machines M1, M2,

M3, M4 and M5 are tabulated below:

P1 P2 P3 P4 P5

M1 50 80 30 40 45

M2 60 30 40 40 50

M3 40 40 50 45 35

M4 35 40 30 35 50

M5 40 45 50 45 45

Selling prices per unit are as follows:

P1= Rs. 80, P2=Rs.90, P3=Rs.105, P4= Rs.70 and P5 = Rs. 65

Decide which product should be made on which machine to realize maximum profit.

Q.5 Darda oil mills have four plants each of which can manufacture anyone of the four

products. The manufacturing costs differ from plant to plant and so do the sales revenues. The

revenue and cost details are as given below:

Sales revenue (Rs. Lakhs)

Plants Products

I II III IV

A

B

C

D

70

80

75

78

88

90

87

85

69

71

73

74

82

94

80

89

Manufacturing cost (Rs. Lakhs)

Plants Products

I II III IV

A

B

C

D

59

65

62

65

70

73

72

74

55

55

59

58

71

79

68

76

Suggest which plant should produce which product to maximize profit?

Q.6 A company has four territories open, and four salesman available for an assignment. The

territories are not equally rich in its sales potential. It is estimated that, a typical salesman,

operating in each territory would bring in the following annual sales.

Territory I II III IV

Annual

Sales(Rs.)

126000 105000 84000 63000

The four salesmen also differ in their ability. It is estimated that, working under the same

conditions, their yearly sales would be proportionately as follows:

Salesman A B C D

Proportion 7 5 5 4

Assign the salesmen to each territory if the criterion is maximum expected total sales.

Third Topic Assignment Model Problems for Practice

Q.1 A departmental head has four subordinates and four tasks for completion. The subordinates

differ in their capabilities and tasks differ in their work contents and intrinsic difficulties. His

estimate of time for each subordinate and each task is given in matrix below:

Tasks Subordinates

I II III IV

Processing cost (Rs.)

A 17 25 26 20

B 28 27 23 25

C 20 18 17 14

D 28 25 23 19

How should the tasks be assigned to minimize requirements of man-hours?

Q.2 A departmental head has three subordinates and four tasks for completion. The employees

differ in their capabilities and the tasks differ in their work contents. With the performance

matrix given below, which three of four tasks should be assigned to subordinates?

Tasks Subordinates

I II III

A

B

C

D

9

8

20

21

12

13

12

15

11

17

13

17

Q.3 A gear manufacturer requires 2000 numbers per month of each of the six types of gears. Six

hobbing machines are available to process these gears. The gears differ in their work contents-

gear with, number of teeth, module etc-and machine differ in their capabilities-speeds, feeds and

ability to take depth of cut. The production control department has prepared the machine wise

cost matrix as shown in the matrix below:

Gear Hobbing machines

M1 M2 M3 M4 M5 M6

I

II

III

IV

V

VI

15

20

19

30

6

13

18

16

16

-

8

12

13

12

15

42

10

16

10

14

-

38

12

14

-

18

19

35

9

15

14

15

20

36

10

18

Gear I can be assigned to machine M5 because of steep helix angle. Gear III can not be assigned

to machine M4 as it is not within the capacity of this machine. And gear IV can not be loaded on

machine M2 because of limitations of process capability of the machine. Find the optimum

assignment schedule.

Q.4 A salesman has to visit five cities. He wishes to start from a particular city, visit each city

once and return to starting city. The cost of going from a city to another in Rs. is given below:

From

City

To city

A B C D E

A

B

C

D

E

0

16

18

21

11

12

0

17

14

13

15

13

0

18

12

17

18

14

0

18

11

12

17

16

0

Determine the least cost route.

Q.1 A departmental store purchases Christmas trees, which can be ordered

only in lots of 100. Each tree selling price Rs. 40 each. Unsold trees, however,

have no salvage value. The purchase price of the trees is Rs. 25 each The

probability distribution obtained from analysis of past data is given below:

Trees sold probabilities

100 0.20

200 0.35

300 0.25

400 0.15

500 0.05

(a) Setup a payoff table

(b) How much quantity should the departmental store buy to maximize its

profit?

Q.2 A manufacturer of sewing machines is faced with the problem of selecting one of

the two models for manufacturing. The profit depends on the market acceptability

of the model which are present is uncertain but is had been broadly classified into

four categories: excellent, good, fair and poor.

The profits or losses (losses are indicated by negative sign) expected by the management

from the different levels of market acceptability of the models are as follows:

__________________________________________________________________

Market Profit (Rs.) for the model for the

Indicated market acceptability

__________________________

Deluxe Janata

__________________________________________________________________

Excellent 60,000 78,000

Good 28,000 38,000

Fair 18,000 8,000

Poor 8,000 -12,000

__________________________________________________________________

Which product should the company select from the standpoint of maximin (gain)

criterion?

Q.3 A company is making a large boiler installation. A certain automatic monitoring unit is

critical for the operation of the whole system. At the time of original order, the spares for this

unit can be purchased for Rs. 2,000 per unit. The probability distribution for the failure of the

unit during the life time of installation is given as :

__________________________________________________________

_______Failure_________________________Probability____________

0 0.35

1 0.25

2 0.20

3 0.15

4 0.05

___________________________________________________________

If a spare is needed and is not available, the total cost of idle time and replacement cost

will be Rs. 15,000. Unused spares have no salvage value.

Determine the optimal no. of spares to be ordered.

Q.4 A newspaper boy is thinking of selling a special one time edition of a sports

magazine to his regular newspaper customers. Based on his knowledge of his customers,

he believes that he can sell between 9 to 12 copies.

The magazines can be purchased at Rs. 8 each and can be sold for Rs. 12

each. Magazine that are not sold can be returned to the publisher for a

refund of 50%.

(a) Construct the decision matrix for the above inventory problem

indicating possible monetary consequences.

(b) Determine the best decision from the stand point of

(i) Maximin criteria (ii) Maximax criteria

(iii)Hurwiez a-criterion assuming a=0.40

(iv) Minimax regret criteria (v) Laplace criteria

Q.5 Agent Corner, an authorized dealer in domestic appliances find that the cost of

holding refrigerator in stock for a week is Rs. 50.

Customers who cannot get a new refrigerator immediately wanted to go to another dealer

for which expected profit is Rs. 350 per customer.

Probability distribution of demand is as follows:

No. of refrigerator: 0 1 2 3 4 5 6

Probability : 0.05 0.10 0.20 0.30 0.20 0.10 0.05

Assuming that there is no time lag between ordering and delivery, how many

refrigerators should we order per week?

Q.6 A departmental store buys Christmas tree at a landing cost of Rs 25 each and sells them

at an average of Rs 40. Any tree left over after the selling season has no resale value.

The productivity distribution of sale of trees derived from analysis of pas t sales data is

under:

Tree (sale) Probability

100 0.10

200 0.15

300 0.35

400 0.20

500 0.10

600 0.05

700 0.05

a) How many trees should be department store buy to maximize its profit?

b) If trees left after the selling season cost Rs 5 each to remove ,does it affect the

inventory decision?

Q.7 A newspaper boy is thinking of selling a special one time edition of a sports magazine to his

regular newspaper customers. Based on his knowledge of his customer the copies of the

magazine with probabilities estimated as under:

No of copies probability

6 0.10

7 0.15

8 0.35

9 0.20

10 0.10

11 0.05

12 0.05

The magazine can be purchased at Rs 8 each and can be sold for Rs 12 Each.

a) Magazines that are not sold can be required to the publisher for a refund of

50%.Determine optimum quantity to be purchased?

b) If the publisher does not take back the unsold magazines and the boy is forced to sell

them as scrap at rs 1.50, what should be the order quantity.

c) And if the boy gets magazines “on sale basis”,what quantity should be ordered?

Q.8 A Ship building company has launched a program for the construction of new class of

ships. Certain spare units like prime over, each costing 200000 have to be purchased. If these

units are not available when needed, a serious loss is incurred which is in order of Rs 10000000

each instance requirements of spares with the corresponding probabilities are given below.

Nos of spares: 0 1 2 3 4 5

Probability of 0.876 0.062 0.041 0.015 0.005 0.001

requirement

How many spares should the company buy in order to optimize inventory decision?

Fifth Topic Decision Analysis Problems for Practice

Q.1 A perishable item is ordered only once each demand period. Acquisition cost is $3,

selling price is $5, and salvage value is $1.50. What is optimal order quantity? Given:

Demand Probability

100 0.1

110 0.2

120 0.2

130 0.3

140 0.1

150 0.1

Q.2 A newspaper boy buys papers for Rs 1.30 each and sells them for Rs 1.40 each. He

cannot return unsold newspapers. Daily demand has the following distribution.

No. of customers: 23 24 25 26 27 28 30 31 32

Probability: 0.01 0.03 0.06 0.10 0.20 0.25 0.10 0.05 0.05

If each day's demand is independent of the previous day’s, how many papers he should order

each day?