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:

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, P₁, P₂, P₃, P₄, and P_5. The requirements of raw materials per ton for each of the products are as indicated below:

Raw Product materials	P1	P2	P3	P4	P5
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:

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 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	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 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)