University of Mumbai 2009 Diploma Operation Research in Operations Research Management - Question Paper

Diploma in Operations Research Management

15-4-08-Nk-Ex. 121

YMW

BB-8959

[Total Marks : 100

N.B.(1)    Attempt any five questions.

(2)    Figures to the right indicate full marks.

(3)    Necessary explanations at intermediate stages must be given.

(4)    Use of non-programmable Calculator, Statistical Tables and Log tables is permitted.

(5)    Answers should be brief and to the points.

(6)    Assumptions, required in the questions, where necessary must be explained.

1. (a) 3 metal spheres whose radii are 3, 4 and 5 cm. melted and formed into one sphere. If 10 volume of a sphere varies as the cube of its radius, find the radius of the new sphere.

(b) Solve for x, the equation    10

/x + 2 + Vx 1 Vx + 2 - V^x - ¹

2.    (a) Find the square root of

12 + V24 + V56 +V84 (b) Solve for x and y : *

a^x = and a* = x^x.

3.    (a) If a, b and c are any 3 consecutive positive integers, prove that

log ( 1 + ac) = 2 log b (b) Derive the equation of a common tangent to

x² + y² - z and y² = 8x and plot the graph.

4. (a) A contractor builds 3 types of houses. The raw materials needed are given in the form 10 of a matrix A and the cost of buying and tranporting each of the raw materials are also given in the form of a matrix B. Find AB and interpret the entries of (AB)

Cement	Wood	Steel	B= Cement	Purchase	Transport
10	20	5	Wood	15	1
20	30	6	Steel	10	2
25	40	7		20	3

(b) A rectangular area of 1000 square feet is to be enclosed by a fence and then divided in 10 the middle by another fence.

The fence drawn in the middle cost Rs. 0-70 per running foot and the other fence costs Rs. 2-00 per running foot.

Find the minimum cost of fencing.

5. (a) Define (i) Consumers surplus, and (ii) Producers surplus.    . 10

(b) The demand function of the commodity is p = 20 - 3x and the supply function is p = 2x 10 Determine the equilibrium price and show that at this price,

Consumers surplus = 24 and Producers surplus = 16.

Con. 2320-BB-8959-08.

6. (a) Evaluate

7. (a) Find the sum of n terms of the following series :

1

1 1 2-5 ⁺ 5-8

8-11

(ii) 2-5 + 5-8 + 8-11 + 11-14 +.......

(b) A rectangular box with a square base and open top is to be made from 1200 square material. 10 Find volume of the largest box that can be made.

(a)    An Institute of Management uses aptitude test and group discussion for selecting 10 candidates for its management course of those obtaining satisfactory grades in these tests and group discussion. 75% pass the management course examination while only 35% of those whose performance in aptitude test and group discussion was unsatisfactory, pass the management course examination, Can it be said that the aptitude test and group discussion are necessary before admission to the course.

(b)    If    10

then find the sum of the following series :

y +

f?y>v ff~    rthWf<LM&*f' BB-8965

(3 Hours)    *14$^Total Marks : 100

N.B. : (1)    Attempt any five questions.

(2)    All questions carry equal marks.

(3)    Statistical tables and graph papers will be supplied on request.

(4)    Use of non-programmable calculator is allowed.

1. The daily profits in rupees of 100 shops are distributed as follows

Profit per shop	No. of shops
0 - 100	5
100 -	7
200 -	18
300 -	27
400 -	20
500 -	17
600 - 700	6
Total	100

Calculate the arithmatic mean, mode, the lower quartile and the standard deviation (s.d.) of the distribution.

2. (a) Explain with the help of suitable diagrams the concept of positive, negative and lack of correlation between two variables.

(b) A firm believes that its annual profits <y) depend on its expenditure on research (x), in Rs. 1000. The following table presents the information for the preceding 6 years :

Year	2002	2003	2004	2005	2006	2007
X	2	3	5	4	6	10
y	20	25	34	30	31	40

(i)    Calculate the product moment correlation coefficient and comment on its value,

(ii)    Determine the education of the appropriate line of regression and use it to estimate the annual profits when the expenditure on research is Rs. 7,000 in the year.

3. (a) State the probability density functions and the properties of the following statistical distribution

(i)    Poisson

(ii)    Normal and

(iii)    Exponential.

Explain the 'forget fullness' property and the distribution for which it is applicable.

(b) In a Restaurant, on a particular morning the amounts spent on breakfast by customers follow a normal distribution with average of Rs. 22.50 and s.d. of Rs. 2.

(i) Estimate the proportion of days on which customers spent between Rs. 20.50 and Rs. 24.50 on breakfast.

. (ii) If on a given morning 540 customers spent Rs. 20 or more on breakfast, what is the total number of customers served ?    

4. (a) The number of articles produced on 55 old and 45 new machines gave the following results

Machines	No. of machines	Mean	S.D.
Old	55	60	10
New	45	50	6

(i)    Find the combined mean and coefficient of variation for both machines together.

(ii)    Determine whether the number of articles produced.on new machines are more uniform than those on old machines.

(b) Two random samples of sizes 10 and 20 are drawn from two normal populations. The sample variances are 25 and 36 respectively. Can we regard the two normal population variances to be equal ?

(a) The probability of an article being defective is given to be 0-2. What is the probability that a random sample of eight articles will have (i) 2 defectives (ii) fewer than 2 defectives?

(b)    The life of a certain type of electronic part is known to follow an exponential distribution with a mean of 3 weeks. What is the probability that a given part will have a life of more than 5 weeks ?

(c)    A chemist claims that his medicine is effective in curing 90% of the patients suffering from flue. To test the claim the medicine was given to 300 patients and 261 were cured. Test his claim at the 1% level of significance. Also determine the 99% confidence limits for the true percentage of the cured patients.

6. (a) The consumer preference for four brands of a product are given below

Brand	A	B	C	D	Total
No. of persons	30	20	40	10	100

(i)    Test the hypothesis that the consumer preferences are equal.

(ii)    Test the hypothesis that the brand C is preferred by as many persons as the other three brands examined.

Use a 1% level of significance.

(b) In a time-study, 10 timings of a particular element were as follows 18, 16, 14, 19, 20, 15, 17, 16, 10, 13.

Calculate 95% confidence limits for the true average time for this element.

(a)    Describe briefly the utility of control charts in industry. Describe the working of the X and R charts.

(b)    The life in hours of three brands of electric bulbs is given below

Brand :

A : 1200 1300 1350 B : 1200 1350 1400 1400 C : 1100 1200 1300 1400 1450 Test at the 5% level whether there is a significant difference among the means of the three brands.

Write short notes on any three

(a)    Measures of Skewness

(b)    Type I and Type II errors

(c)    Components of a time-series

(d)    Paired t-test.

Con. 2449-08.    BB-8968

(3 Hours)    [Total Marks : 100

N.B. : (1)    In Section I, question No. 1 is compulsory. Attempt two questions from the rest.

(2)    In Section II, question No. 6 is compulsory. Attempt two questions from the rest.

(3)    Figures to the right indicate marks to a sub-question.

(4)    Answers to the both the sections are to be written in the same answer-book.

Section I

1.    (a) Define the following    6

(i)    Demand

(ii)    Cross elasticity of demand

(iii)    Demand forecasting

(b)    Distinguish between the following    6

(i)    Average production and marginal production.

(ii)    Past costs and future costs

(iii)    Direct costing and Absorption costing.

(c)    The Total Cost Function y of manufacturing x number of units is given by 6 y = 16,000 + 600 x + 0-2 x².

Calculate -

(i)    Average cost of producing 200 units.

(ii)    If the company doubles the output will it halve its average cost ?

(iii)    what is the average variable cost if no units are produced ?

2.    (a) Explain the functions performed by a managerial economist, in the Indian context. 8 (b) Discuss the characteristics of Managerial Economics.    8

3.    (a) Explain the business applications of price elasticity.    8 (b) Discuss the determinants of demand. 8

4.    (a) Give a summary chart of the methods of demand forecasting. Discuss any two 8

methods of demand forecasting.

(b) Explain the recent trends in demand forecasting.    8

5.    (a) Discuss the diseconomies of scale.    8

(b) Distinguish between cost control and.cost reduction. Discuss the factors essential 8 for the success of a cost reduction program.

f'M* "\yipioKUA IA oPenrciJ-ronS    Fcv \~ncmuy

PctfW'T ~ fvt jnc.i fJCo-i t-Ai coy\on\)tS rthd f)cc>oLm-ffW

P4/RT-Ex-08-6C0    _fQ /T* ,ZOO$'

Con. 2449BB896808.    2 Section II

6. (a) Define the following :-    6

(i)    Monopoly

(ii)    Cost of retained earnings

(iii)    Net present value.

(b) Find using straight line method    6

(i)    depreciation during the year

(ii)    depreciation at the end of the year

(iii)    value of the asset at the end of the year given that

(c) A company is manufacturing five products. The accounts department has supplied 6 the following data

Product	Price Rs	Variable Cost Per Unit	% Share of each Product to total sales
A	5	4	10
B	5	3	15
C	10	6	20
D	10	7	25
E	10	8	30

The capacity of the firm in terms of rupees is Rs. 10,00,000 and the fixed costs per year are Rs. 2,00,000. Calculate the BER

7. (a)    Explain the Rent Theory of Profits.    8

(b)    Explain the features, causes and disadvantages of monopoly.    8

, 8. (a)    Distinguish between perfect competition and monopolistic competition.    4

(c)    Discuss the general principles of Oligopoly pricing.    8

9. (a)    Explain the advantages of price leadership situation.    8

(b)    What are economic indicators ? Discuss external economic indicators.    8

10 (a)    What is a Master Budget ? What are its components ?    8

(b)    What are the components of the feasibility report in project planning ?    8

Mcrr Tptanna '\r> OP<msJh'or>s cseonrcb W?r

P4/RT-EX-08-598    I    '    \

~    na-naacnnerM    .

Con. 2439-08.|?Jl|\JU2, \U -'BoVCS    OPtfYClVfoDS ^<Re<5CTf7yCh'BB-9343

(3 Hours)    [Total Marks : 100

N.B. : (1)    Attempt any three questions from each section.    

(2)    Answers to each section should be written in separate books.

(3)    Figures to the right indicate full marks.

(4)    Necessary explanations at intermediate stages must be given.

(5)    Assumptions, wherever necessary must be stated clearly.

(6)    Use of ordinary calculator and statistical table is allowed.

Section I

1.    (a) Solve the following LPP    8

Maximise Z = 30x., + 16x₂ + 25x₃

Subject to 8x., + 4x₂ + 5x₃ < 1000 5x₁ + 3x₂ + 3x₃ < 650 3x., + 2x₂ + 3x₃ < 420 x_v x₂ x₃ > 0

(b) Write down the Dual.problem of the following LPP and find out the optimum solutions 8 of both primal and dual problems.

2.    Solve the following problem- graphically -    20

Maximise Z = 20x₁ + 10x₂

Subject to x₁ + 2x₂ < 40 3x.,+ x₂ < 30 4x₁ + 3x₂ >60 x_1f x₂ > 0

(a)    if the objective function coefficients (20 ,10) change to (10, 20), what will be the optimal solution.

(b)    If the RHS coefficient change from (40, 30, 60) to (50, 50, 50) respectively find the new optimal solution.

(c)    A new constraint x-, + x₂ < 45, be added find the new optimal solution if the present optimal solution is affected.

(d)    If product x₃ with cost 5 and resource requirements (3, 2, 5) respectively be introduced, find the Optimal solution.

3.    (a) Goods are transported from factories F_1? F₂ and F₃ to the warehouse W_v W₂, W₃ and 10

W₄ cost of transportation, in Rs. from each factory to each warehouse, in Rs. given in the table below. Also number of demand units and supply units are given-

	w1	w2	w3	w4	Supply (units)
Fl	3	5	2	4	.100
F2	6	3	7	2	80
f3	9	4	2	5	40
Demand	70	50	40	60

Determine how many units from each factory to each warehouse should be transported

so as to minimize the to transportation cost.    ,,

[ TURN OVER

B

	4	3	3
A	1	7	1
	-2	-2	12

4. The following table gives completion time, in hours for each worker for each job.    16

		J2	3
W,	3	5	2	4
w2	6	3	7	2
W3	9	4	2	5
W4	8	3	2	5

(a)    Determine the optimum assign on solution.

(b)    Suppose the completion time of J₄ by W₂ change from 9 hours to 5 hours, obtain the optimum assignment solution.

Section II

5.    A project consists of 8 activities.    16 A(3), B(4), C(2), D(3), E(5), F(7), G(8), H(2).

Figures, in brackets, denote durations in days of the activities. The following relationship amongst the activities hold.

(a)    A, B and C are the starting activities of the project.

(b)    A precedes D, B precedes E and F and C precedes G.

(c)    D and F precedes H and

(d)    C and F control G

(e)    G and H are ending activities.

Draw a network diagram. Find EST, LFT, LST for each activity and determine the critical path and project duration. For each activity find the total float, Free float, Interference float and Independent Float.

6.    (a) The following table gives, for each activity of a project. Normal Duration (ND), Crash 8

Duration (CD) in days, Normal Cost (NC), Cost Cash (CC), in Rs. Indirect Cost is Rs. 50 per days :-

Activity	1-2	1-3	2-4	2-5	3-4	4-5
ND	7	3	2	9	6	3
CD	5	1	1	4	2	2
NC	100	150	50	100	100	80
CC	200	350	90	400	200	100

(i)    Determine the minimum project duration and the corresponding Project Cost,

(ii)    Determine the minimum project cost and the corresponding project duration.

(b) In a municipal hospital, patients arrivals are to be considered as Poisson with an 8 average of interarrival time 10 minutes. The doctor's time for examination plus time of dispensing medicine is distributed negative exponentially with an average of 6 minutes.

(i)    What are the chances that a new patient will see the doctor without having to wait ?

(ii)    For what percentage of time, the doctor will remain idle ?

(iii)    Find the average queue length, average number of patients in the system, average waiting time and average time spent in the system.

7. (a) The following table gives the optimistic, most likely and pessimistic project activity 10 duration, in days.

Find the mean time and variance for each activity of the project.

What is the probability that, the project will be completed in 4 days later than expected duration ?

Activity	1-2 1-3 2-4	2-5 3-4 4-5
Optimistic Most Likely Pessimistic	3    5 1 4    6 3 5    7 5	1 4 4 4 8 5 7 12 6

(b) A 2 x 2 pay-off matrix for player A is given below. Then will be a riddle point only if- 6

(i)    p < q, p > 5    ;

5 6 P q

(ii)    P > q, P < 5

(iii)    neither of the outer (i) and (ii)

8. (a) For an LPP, the optimum simplex table is as follows    10

Basic	c	*1	x2	*3	Si	S2	s3	b
x2		1/2	1	0	1/3	-1/3		-1
X3		5/6	0	1	-1/6	2/3		21
S3		-5/3	0	0	-2/3	-1/6		15
	A	25/2	0	0	15/2	10	0	105

(i)     Find the missing numbers.

Find the missing numbers. Find the original LPP.

(ii)    Find the original LPP.

(b) Write short notes on    6

(i)    Alternative optima in a LPP.

(ii)    Saddle Point in a game.

(iii)    unbounded solution.

• «
•  Start 
•  Prev 
•  1 
•  2 
•  Next 
•  End 
• »

Unable to read this question paper?

Attachment: university-of-mumbai-2009-diploma-operation-research-51669-1.pdf

• 1
• 2
• 3
• 4
• 5

( 0 Votes )

Add comment

Name (required)

E-mail (required, but will not display)

Notify me of follow-up comments

Refresh

Send

Cancel

JComments

Earning:   Approval pending.