Tuesday, July 5, 2011

7. Quantitative Techniques in Decision Making - Topical Analysis

1.      In an Income Tax Office, three assistants process incoming Income Tax Returns. The first assistant, B1 processes 40% of the returns. The second assistant, B2 processes 35% and the third assistant, B3 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, ex-plain 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 Cathode-ray 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:
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 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
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 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, P1, P2, P3, P4, and P5. The requirements of raw materials per ton for each of the products are as indicated below:
Raw Product materials
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 P1 is such that not more than 10 tons can be sold per day and that of P2 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 P1, P2, P3, P4 and P5 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 P1 and P2 by increasing promotional expenditure. Which of the following alternatives should be adopted?
1.      Do not promote P1 or P2
2.      Promote P1 but not P2
3.      Promote P2 but not P1
4.      Promote P1 and P2 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:
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 Hypothesis-testing 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 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
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 91
Sales 97
Sales 108
Sales 121
Sales 67
Sales 124
Sales 51
Sales 73
Sales 111
Sales 57
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 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 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 two-variable example Also highlight the areas of its successful and non-successful 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 machine-minute 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 minimum-cost 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)

1 comment:

  1. hello!! any suggestions for QT coaching centers in UPSC in Delhi