1.
In an Income Tax Office, three assistants
process incoming Income Tax Returns. The first assistant, B_{1}
processes 40% of the returns. The second assistant, B_{2} processes 35%
and the third assistant, B_{3} processes 25% of the returns. The first
assistant has an error rate of 0.04, the second assistant an error rate of 0.06
and the third assistant an error rate of 0.03. A return selected at random from
a days output is found to have an error. Find out the probabilities that it was
processes by the first assistant, the second assistant or third assistant
respectively. (1988)
2.
Consider the all integer
program given below: Max 5 × 1 + 8 × 2 s. t, 6 × 1 + 5 × 2 = 0, 9 × 1 +
4 × 2 = 36, 1 × 1 + 2 × 2 = 10, x1. x2 and integer
1.
Graph the constrains for this problem. Indicate
with heavy dots all the feasible integer solutions.
2.
Find the optimal solution to the L. P.
Relaxation. Round down to find a feasible integer solution.
3.
Find the optional integer solution. Is it the
same as the solution found in part
4.
by rounding down? (1989)
3.
The Alexander Tea Company mixes South Indian
tea, Assamese tea and Ceylon tea to make a new brand of tea. One kg. Of South
Indian tea costs Rs. 8; Assamese tea costs Rs. 10 and Ceylon tea costs Rs. 11.
The company must produce 10.000 kgs. Of a new brand of tea which ought to
include not more than 3, 000 kgs of South Indian tea and 2, 000 kgs of Ceylon
tea and not less than 1, 500 kgs of Assamese tea. Determine the optimum product
mix in order to minimise production cost. (1990)
4.
Describe the hypothesis testing procedure. Also,
explain and illustrate the
1.
testing of population mean, and testing of
difference between means
2.
testing of population proportion, and testing of
difference between proportions.
5.
A TV manufacturer is facing the problem of
selecting a supplier of Cathoderay tube which is the most vital component of
TV. Three foreign suppliers, all equally dependable, have agreed to supply the
tubes. The price per tube and the expected life of a tube for the three
suppliers are as follows:
Supplier

Price

Tube
Expected life per Tube

Supplier
1

Rs. 800
1500 hrs

Supplier
2

Rs. 1000
2000 hrs

Supplier
3

Rs. 1500
4000 hrs

1.
The manufacturer guarantees its customers that it
will replace the TV set if the Tube fails earlier than 1000 hrs. Such a
replacement will cost him Rs. 1000 per tube, over and above the price of the
tube.
2.
Can you help the manufacturer to select a
supplier? Clearly state any assumption you may have made for solving this
problem. (1991)
6.
Estimate the figure of sales for 1993 from the
data given below (1992)
Year

Year
1987

Year
1988

Year
1989

Year
1990

Year
1991

Year
1992

Sales
(ooo units)

Sales
120

Sales
150

Sales
160

Sales
180

Sales
200

Sales
225

7.
A factory can manufacture two products X1 and
X2. Each product is manufactured by a two stage process which involves machines
A and B and the time required is as follows
Item

Machine

Product

A X1

Time 2
Hrs.

Time 3
Hrs.

B X2

Time 1
Hr.

Time 2
Hrs.

1.
Available hours on machine A is 10 Hrs. And
machine B is 16 Hrs.
2.
The contribution of product X1 is Rs. 4 per unit
and X2 is Rs. 3 per unit.
3.
What should be the manufacturing policy for the
factory?
4.
Name the technique you have used in solving this
problem
5.
What are the limitations of this technique? (1993)
8.
Assuming that half of the population is
vegetarian so that the choice of an individual being vegetarian is ½ Assuming
that 100 investigators can take sample of 10 individuals each to see whether
they are vegetarians, how many investigators would you expect to report that
three or less were vegetarians?
9.
Find the optimal transportation schedule from
the following in order to minimise transportation costs: (There exists no
transport facility between C to Z). (1994)
10.
Eight coins are tossed 256 times and the
following: Results are obtained
Number
of heads

Number
of Heads 0

Number
of Heads 1

Number
of Heads 2

Number
of Heads 3

Number
of Heads 4

Number
of Heads 5

Number
of Heads 6

Number
of Heads 7

Frequency

Frequency
2

Frequency
6

Frequency
30

Frequency
52

Frequency
67

Frequency
56

Frequency
32

Frequency
10

1.
Are the coins biased? Use X2 test.
11.
A manufacturer produces five products, P_{1},
P_{2}, P_{3}, P_{4}, and P_{5.} The
requirements of raw materials per ton for each of the products are as indicated
below:
Raw
Product materials

P_{1}

P_{2}

P_{3}

P_{4}

P_{5}

Availability
(tons/day)

G 48% CH
18%

CSS 0.75
0.25 0

Tons/Day
0.34 0.66 0

Tons/Day
0.10 0.90 00

Tons/Day
1.00 0 1.00.5 12.0 18.22.5

Contribution
(Rs. Per ton)

Rs. Per
ton 400

Rs. Per
ton 252

Rs. Per
ton 180

Rs. Per
ton 150

Rs. Per
ton 125

12.
Note: Contribution is
computed as price per unit of the product minus variable cost per unit of the
product.
13.
The marketing constraint for product P_{1}
is such that not more than 10 tons can be sold per day and that of P_{2}
is such that not more than 15 tons can be sold per day.
1.
If X1, X2, X3, X4 and X5 respectively represent
the tons to products P_{1}, P_{2}, P_{3}, P_{4}
and P_{5} to be produced per day test whether the following solutions
are feasible:
1.
X1 = 8.4, X2 = 0, X3 = 12.0. X4 = 0, X5 = 2.5
2.
X1 = 5.0, X2 = 11.0, X = 0, X4 = 19, X = 5.0
3.
X1 = 10, X2 = 12.0, X3 = 4.2, X4 = 4.0, X5 = 5.0
4.
X1 = 6, X2 = 9, X3 = 13.0, X4 = 1.0, X5 = 2.5
2.
Test whether any of the above solutions would
maximize total contribution.
3.
The manufacturer is considering the possibility
of expanding sales of products P_{1} and P_{2} by increasing
promotional expenditure. Which of the following alternatives should be adopted?
1.
Do not promote P_{1} or P_{2}
2.
Promote P_{1} but not P_{2}
3.
Promote P_{2} but not P_{1}
4.
Promote P_{1} and P_{2}
both (1995)
14.
Answer the following questions (1996)
1.
Briefly discuss the significance and limitations
of the study of Correlation analysis.
2.
Compute Karl Pearsons coefficient of
correlations between per capita National income and per capita Consumer
Expenditure from the data given below:
Year

Per
Capita National Income (Rs. )

Per
Capita Consumer Expenditure (Rs. )

Year
1968

Rs. 249

Rs. 237

Year
1969

Rs. 251

Rs. 238

Year
1970

Rs. 248

Rs. 236

Year
1971

Rs. 252

Rs. 240

Year
1972

Rs. 258

Rs. 245

Year
1973

Rs. 269

Rs. 255

Year
1974

Rs. 271

Rs. 254

Year
1975

Rs. 272

Rs. 252

Year
1976

Rs. 280

Rs. 258

Year
1977

Rs. 275

Rs. 251

15.
Answer the following questions
1.
Describe some of the Important Research Design
used in Experimental Hypothesistesting research study.
2.
Why do managers take decisions? Explain decision
making under conditions of uncertainty and risk.
16.
A manufacturer produces three products, A, B and
C. Each product can be produced on either one of two machines I and II. Time
required to produce 1 unit of each product on a machine is
Time to
product 1 unit (hours) Machine

Product
I

Product
II

Product
A

Time 0.5

Time 0.6

Product
B

Time 0.7

Time 0.8

Product
C

Time 0.9

Time
1.05

1.
There are 85 hours available on each machine;
the operating cost is Rs. 5 per hour for machine I and R. 4 per hour for machine
II; and the product requirement are at least 90 units of A, at least 80 units
of B, and at least 60 unit of C. The manufacturer wishes to meet the
requirements at minimum cost.
2.
Solve the given linear programming problem by
the simplex method. (1997)
17.
A panel of two Judges P and Q graded 7 dramatic
performances by independently awarding marks as follows:
Performance

Performance
1

Performance
2

Performance
3

Performance
4

Performance
5

Performance
6

Performance
7

Marks by
P

Marks 46

Marks 42

Marks 44

Marks 40

Marks 43

Marks 41

Marks 45

Marks by
Q

Marks 40

Marks 38

Marks 36

Marks 35

Marks 39

Marks 37

Marks 41

1.
The eighth performance however, which judge Q could
not attend, was awarded 37 marks by judge P. If judge Q had also been present
how many marks would Q have awarded to the eighth performance?
18.
The India Manufacturing Corporation (IMC) has
one plant located on the outskirts of a city. Its production limited to two
produces as naptha (X1) and urea (X2). The unit contribution for each product
has been computed by the firms costing department as Rs. 50 per unit for
product naptha and Rs. 60 per unit for product urea. The time requirements for
each product and total time available in each department (each product passes
through two departments in the plant) are as follows: Department Hours required
Available Hours in a
Month

Product
Neptha

Product
Urea

Month 1

Cost 3,
000

Cost 1,
500

Month 2

Cost 2,
000

Cost 1,
500

In addition the demand for the products restricts
the production to a maximum of 400 units of each of these products. The IMC
wants to maximise is profit.
1.
Make a Linear Programming Model for this
problem.
2.
Solve this problem graphically and state how
many units of each product should be producted and how much will be the maximum
profit for this company.
19.
From the following data obtain the two
regression equations
Sales

Sales 91

Sales 97

Sales
108

Sales
121

Sales 67

Sales
124

Sales 51

Sales 73

Sales
111

Sales 57

Purchase

Purchase
71

Purchase
75

Purchase
69

Purchase
97

Purchase
70

Purchase
91

Purchase
39

Purchase
61

Purchase
80

Purchase
47

20.
Demonstrate the use of probability theory and
decision trees in risk analysis in microeconomic decision making. (1999)
21.
A departmental store wishes to procure the
following readymade garments:
Type of
Garment

Type A

Type B

Type C

Type D

Type E

Quantity

Quantity
300

Quantity
200

Quantity
150

Quantity
500

Quantity
400

1.
Tenders are submitted by four different
manufacturers who undertake to supply not more than quantities below (all types
of garments combined):
Manufacturer

Manufacturer
I

Manufacturer
II

Manufacturer
III

Manufacturer
IV

Total
Quantity

Quantity
600

Quantity
500

Quantity
300

Quantity
400

2.
The departmental store estimates that its profit
per garment will vary with the manufacturer as shown in the matrix below:
3.
How should orders be placed?
22.
What are the assumptions behind Binomial
Poisson? Exponential and Normal probability distributions? Give at least one
situation for each of the distributions. (2000)
23.
Describe a situation which you would consider as
decision making under risk. Justify your answer.
24.
Describe two situations which involve
optimization under constraints. What is the difference in the natures of
solution procedure of problems without constraints and with constraints?
25.
An optimal policy has the property that whatever
the initial state and initial decision are, the remaining decisions must
constitute an optimal policy with regard to the state resulting from the first
decision (R. E. Bellman). Illustrate an applications of the above principle.
26.
Comment and illustrate how modeling enables
quick and economical experimentation for finding an optimum solution for a
given problem.
27.
Explain with the help of examples as to how one should
see
1.
scatter diagrams to visualize the relationship
between two variables
2.
regression analysis to estimate the relationship
between two variables
3.
regression equation to predict future values of
the dependent variable. Also briefly highlight the limitations of regression
and correlation analysis. What are the assumptions behind regression analysis?
28.
Statistical Tests of Significance (2001)
29.
Time Series Analysis (2002)
30.
Forecasting techniques (2003)
31.
Comment and illustrate how statistical models
enable quick and economical experimentation for finding an optimum solution for
a given problem (60).
32.
Linear Programming [LP] is a mathematical
modeling technique designed to optimize the usage of limited resources. [Hamdy
A. Taha] Illustrate the basic elements of an L. P. Model by using a simple
twovariable example Also highlight the areas of its successful and
nonsuccessful applications (60). (2004)
33.
Linear Programming as a problem solving
technique
34.
Statistical tools for decision making in
business (2006)
35.
A company manufactures three products X, Y and
Z. Their profits per unit are Rs. 300, Rs. 200, and Rs. 400 respectively. The
company has two machines and the required processing time in minutes on each
machine for each product is given below: Machines 1 and 2 have 2, 000 and 2,
500 machineminute respectively. The upper limit for the production volumes of
X, Y and Z are 100 units, 200 units and 50 units respectively. But the company
must produce a minimum of 50 units of X to meet contractual obligations. Determine
the optimum production policy for the company. Which technique did you employ
for this? Explain the various underlying assumptions of the technique (60)
36.
Define the dual of a linear programming problem.
State the functional properties of duality. Explain the advantages. Three food
products are available at costs of Rs. 10, Rs. 36 and Rs. 24 per unit
respectively. They contain 1, 000, 4, 000 and 2, 000 calories per unit,
respectively and 200, 900 and 500 protein units per unit, respectively. It is
required to find the minimumcost diet containing at least 20, 000 calories and
3, 000 units of protein. Formulate and solve the given problem as a LP problem.
Write the dual and use it to check the optimal solution of the given problem
(60). (2007)
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