Quantitative Techniques in Decision-Making – Management Optional PYQs (1995–2026)

Here are all 138 Quantitative Techniques in Decision-Making questions asked in UPSC Management Optional Paper II from 1995 to 2026, arranged under the 10 topics of the official syllabus. Within each topic the latest questions come first.

Each question ends with a code such as [2024/15m/3b] – the year, the marks, the word limit (where UPSC set one) and the question number in the paper.

1.1] Descriptive statistics

6 questions

1. “Statistics are numerical statements of facts, but all facts stated numerically are not statistics.” Clarify this statement and point out which numerical statements of facts are statistics. [2017/10m/3b]

2. Prior to construction of a dam on a river, the organization entrusted with the task of construction of the dam performed a series of tests to measure the water flow past the proposed location of the dam. The results of the tests were used to construct data shown in Table below. Compute the mean, standard deviation and the coefficient of variation of the river flow in litres per minute. What portion of the water flow occurs at less than 5200 litres per minute? Table: Results of Test.

River Flow (000s of litres/minute)	Frequency
4001–4200	7
4201–4400	21
4401–4600	32
4601–4800	49
4801–5000	58
5001–5200	41
5201–5400	27
5401–5600	11

[2015/10m/1e]

3. In an educational institution, the stipends earned by the students in the first year summer internships of the two year post graduate programme are used to decide the amount of financial aid to be given to the students in the second year of the programme. The stipends earned by the students have the following distribution:

Stipend earned (in ₹)	Number of students
0 – 49,999	231
50,000 – 99,999	304
1,00,000 – 1,49,999	400
1,50,000 – 1,99,999	296
2,00,000 – 2,49,999	123
2,50,000 – 2,99,999	68
3,00,000 or more	23

What is the mode value of the stipend earned? If financial aid is restricted to those students whose stipend earnings are at least 10% lower than the mode value, how many of the applicants qualify? [2012/12m/150w/1a]

4. Using the data given in the table, find: (a) Mean (b) Median (c) Mode (d) Standard deviation (e) Coefficient of Variation. [2011/30m/5i]

5. What observations can you make with your analysis? [2011/5m/5ii]

6. Statistical tools for decision making in business. [2006/20m/200w/1c]

1.2] Probability and probability distributions

23 questions

1. X and Y throw three dice. If X throws 15, then find the chance of Y’s throwing a higher number. [2026/10m/150w/1b]

2. A train is scheduled to arrive at a station at a random time between 09:00 AM and 09:30 AM. The actual arrival time is equally likely to be any moment within this interval. (i) Define a suitable probability density function (PDF). (ii) What is the probability that the train will arrive before 09:13 AM? (iii) Find the expected arrival time and the variance of the arrival time. (iv) Given that the train has not arrived by 09:11 AM, what is the expected arrival time? [2025/10m/1a]

3. A law graduate student randomly guesses at nine multiple-choice questions in All India Bar Examination. There are four possible answers for every question. However there is only one correct answer. Assuming that all questions are independent to each other, find the possibility that the student chooses six correct answers. [2024/15m/2b]

4. A newly constructed house can collapse due to fault in its design. It can also collapse even if it does not have a design fault. The probability that the design of the newly constructed house is faulty is 0.1. The probability that this house collapses if the design is faulty is 0.95, whereas, the probability that the house collapses without any fault in design is 0.45. It is seen that the house has collapsed. What is the probability that it is due to fault in design? [2022/10m/1a]

5. ‘A’ can hit a target 4 times in 5 shots, ‘B’ 3 times in 4 shots and ‘C’ twice in 3 shots. Calculate the probability: (A) ‘A’, ‘B’, ‘C’ all may hit the target. (B) ‘B’ and ‘C’ may hit the target, but ‘A’ may not hit it. (C) ‘C’ and ‘A’ may hit the target, but ‘B’ may not hit it. [2021/6m/2a(i)]

6. A regulatory body of the government has observed that a particular accounting error tends to be associated with an attempt to misstate financial position. Approximately 85% companies attempting to misstate their earnings will commit this accounting error. For a company not trying to misstate its earnings, the likelihood of making this error is approximately 5%. It has been estimated that the percentage of companies attempting to misstate earnings at any given time is only 5%. Suppose that in an audit of ABC company, the accounting error is discovered. What is the probability that it is not an inadvertent mistake (i.e. ABC company has actually been attempting to misstate its earnings)? [2020/10m/1a]

7. A company has 3 plants manufacturing 1000 scooters in a month. Plants ‘A’, ‘B’ and ‘C’ manufacture 500, 300 and 200 scooters per month respectively. 90%, 92% and 95% scooters are rated standard quality in plants ‘A’, ‘B’ and ‘C’ respectively. (i) What is the probability that a scooter selected at random is of standard quality? (ii) What is the probability that a scooter selected at random comes from plant ‘B’ if it is known that the scooter is of standard quality? [2019/10m/1e]

8. (i) Ten accountants were given intensive coaching and four tests were conducted in a month. The scores of tests 1 and 4 are given below:

Sl. No. of Accountants:	1	2	3	4	5	6	7	8	9	10
Marks in 1st Test:	50	42	51	42	60	41	70	55	62	38
Marks in 4th Test:	62	40	61	52	68	51	64	63	72	50

Do the scores from test 1 to test 4 show an improvement? Test at 5% level of significance. [Tables attached] (ii) State the properties of the normal curve. What is a standard normal distribution? Discuss its importance. [2019/15m/3b]

9. The probability that a new advertising campaign will increase sales is 0.80. The probability that the cost of developing the new advertising campaign can be kept within the original budget allocation is 0.40. Assuming that the two events are independent, what is the probability that the cost is not kept within budget or the campaign will not increase sales? [2018/10m/1e]

10. An organisation has been receiving complaints with regard to the work-related stress problems being faced by young employees. In order to address this problem, the organisation has hired a professionally qualified counsellor who has been assigned the task of counselling the young recruits. It is found that the young employees’ arrival at the counsellor’s office fits a Poisson distribution with an average of 5.2 employees every 20 minutes. However, the counsellor is concerned that if too many employees need the services, a crowding problem may develop. (i) The counsellor must determine the probability that four employees will arrive during any 20-minute interval, which could create crowding problem. (ii) The counsellor must also calculate the probability that more than four employees will arrive during any 20-minute period. [Poisson distribution table is attached with this problem]. [2018/15m/3c]

11. 1000 lightbulbs with a mean life of 120 days are installed in a new company. Their length of life is normally distributed with a standard deviation of 20 days. How many bulbs will expire in less than 90 days? If it is decided to replace all the bulbs together, what interval should be allowed between group replacements if not more than 10% should expire before replacement? [2017/10m/1b]

12. A and B decide to meet between 3 p.m. and 4 p.m., but each of them should wait no longer than 10 minutes for the other. Determine the probability that they meet. [2016/2m/1a(i)]

13. The probabilities of X, Y and Z becoming managers are $\frac{4}{9}$, $\frac{2}{9}$ and $\frac{1}{3}$ respectively. The probabilities that Bonus Scheme will be introduced if X, Y and Z become managers are $\frac{3}{10}$, $\frac{1}{2}$ and $\frac{1}{5}$ respectively. What is the probability that Bonus Scheme will be introduced? And if Bonus Scheme has been introduced, what is the probability that the manager appointed was Y? [2016/4m/1a(ii)]

14. A multiple-choice quiz has 200 questions, each with four possible answers, of which only one is the correct answer. What is the probability that sheer guesswork yields from 25 to 30 correct answers for 80 of the 200 questions about which the student has no knowledge? [2016/5m/1b]

15. If a random variable x follows a Poisson distribution such that $\mathrm{\text{Prob}}(x=1)=\mathrm{\text{Prob}}(x=2)$, find the mean and variance. Also find $\mathrm{\text{Prob}}(x=0)$. [2016/3m/3c(i)]

16. A construction company builds primarily three and four unit apartment buildings (called triplexes and quadruplexes) for investors. It is believed that the total construction time in days follows a normal distribution. The mean time to construct a triplex is 500 days and the standard deviation is 100 days. Recently the President of the company signed a contract to complete a triplex in 625 days. Failure to complete the triplex in 625 days would result in severe penalty fees of Rs. 10 lakhs for every triplex not completed within 625 days. What is the probability that the company will not have to pay penalty fees? Approximately how much penalty fee will it have to pay, if the total number of triplexes to be built is 50? [2015/10m/1a]

17. A study is conducted on 200 residents of a colony who were between the ages of 36 and 50 years, and on 200 residents of the same colony who were over the age of 50 years, to determine whether the source of their daily news is from newspapers or otherwise. Of the 200 residents who were between the ages of 36 and 50 years, 82 got their news primarily from newspapers. Of the 200 residents who were over the age of 50 years, 104 got their news primarily from newspapers. (i) If a resident is selected at random, what is the probability that he or she– (1) got news primarily from newspapers; (2) got news primarily from newspapers and is over 50 years old; (3) got news primarily from newspapers or is over 50 years old? (ii) Suppose that two residents are selected. What is the probability that both got news primarily from newspapers? [2014/10m/1c]

18. A set of parts coming out from a process are measured for their weights. Assuming that the weights are normally distributed with a mean of 73 kg and a standard deviation of 8 kg, what is the probability that a part coming out from this process will weigh between 65 kg and 89 kg? [2014/5m/4b(ii)]

19. In a post office, three clerks were assigned to process incoming mail. The first clerk, $C_1$, processes 40%, the second clerk, $C_2$, processes 35% and the third clerk, $C_3$, processes 25% of the mail. The first clerk has an error rate of 0.04, the second clerk has an error rate of 0.06 and the third clerk has an error rate of 0.03. A mail selected at random from a day’s output is found to have an error. The Postmaster wishes to know the probability that the mail was processed by the first, second or third clerk respectively. Find the probabilities. [2013/6m/1a(i)]

20. The odds that A speaks the truth is 3:2 and the odds that B speaks the truth is 5:3. In what percentage of cases are they likely to contradict each other on an identical point? [2013/4m/1a(ii)]

21. What is the probability of finding a crime-sheeter who is older than 50 years? [2011/15m/5iii]

22. A regimen consisting of a daily dose of vitamin C was tested to determine its effectiveness in preventing the common cold. Ten people who were following the prescribed regimen were observed for a period of one year. Eight survived the winter without a cold. Suppose the probability of surviving the winter without cold is 0.5 when vitamin C regimen is not followed. What is the probability of observing eight or more survivors, given that the regimen is ineffective in increasing resistance to colds? [2009/20m/1a]

23. What are the assumptions behind Binomial, Poisson, Exponential and Normal probability distributions? Give at least one situation for each of the distributions. [2000/20m/200w/5a]

1.3] Sampling distributions and the central limit theorem

7 questions

1. Determine the required sample size for estimating the true weight cereal container from a population of N = 5000. Based on the past records variance of weight is 121 grams and this estimate should be within 2.5 grams of the true average weight with 99% confidence. Will there be a change in the size of the sample if we assume population to be infinite? If so, explain by how much? (Table enclosed) [Figure: Table enclosed with the paper – “AREAS UNDER THE STANDARD NORMAL DISTRIBUTION. The entries in this table are the probabilities that a standard normal variate is between 0 and Z (the shaded area)”: normal curve with the area between 0 and Z shaded; standard table of areas for Z = 0.0 to 3.0 (rows) and second decimal 0.00 to 0.09 (columns)]. [2025/10m/1b]

2. What are the tenets of Central Limit Theorem? Discuss the concept of Standard Error. What is its utility in hypothesis testing? [2021/10m/4b(ii)]

3. The mean selling price of new affordable homes in a metropolitan city is ₹ 22,00,000. The population standard deviation is ₹ 2,00,000. A random sample of 81 new affordable home sales from this city was selected.

(i) What is the probability that the sample mean selling price was more than ₹ 25,00,000? [2020/5m/1b(i)]

(ii) What is the maximum selling price of 10% of affordable home sales in the city whose prices are the lowest? [2020/5m/1b(ii)]

4. The following data gives the marks in Mathematics in a competitive examination in 2018:

37	24	26	31	23	45	35	34	27	25
43	27	21	39	32	24	42	36	48	23
31	46	41	28	34	26	38	40	35	26

Describe the procedure and draw a random sample of size 8 without replacement from the given marks by suitable method. Give an estimate of mean score and its standard error. You may use the following random numbers: 1346 5471 5274 1865 3283 1164 9247 5754 [2019/15m/2c]

5. (i) What is the difference between cluster sampling and stratified sampling? Explain with the help of examples. (ii) A manufacturer of machine tools claims that the mean amount of time required to assemble their top-of-the-line electric saw is 80 minutes with a standard deviation of 40 minutes. Data was collected from a random sample of 64 past purchases of this electric saw. What is the probability that the sample mean will be greater than 88 minutes? [Normal distribution table is placed at the end of the booklet]. [2018/15m/4a]

6. What is ‘central limit theorem’? Give three managerial examples, where ‘central limit theorem’ is applicable. [2014/5m/4b(i)]

7. Sampling Distribution. [2010/10m/1a]

1.4] Hypothesis testing

11 questions

1. In a study to evaluate the effectiveness of two different animal feeds, researchers recorded the weight gains (in kilograms) of animals after fed each type of feed.

Feed A	45	48	47	50	46	49	47	48
Feed B	51	53	52	55	50	52	54	33

(i) If the two samples are considered independent, can we conclude that Feed B is more effective than Feed A at the 0.025 level of significance? (ii) Also examine the case if the same set of eight animals received both feeds, and test at the 0.01 level of significance. (Table enclosed) [Figure: Table enclosed with the paper – “TABLE 6 STUDENT’S t DISTRIBUTION. The first column lists the number of degrees of freedom (ν). The headings of the other columns give probabilities (P) for t to exceed the entry value. Use symmetry for negative t values.”: t-curve with right tail shaded; critical values for ν = 1 to 30, 40, 60, 120 and P = .10, .05, .025, .01, .005] [2025/15m/2b]

2. A random sample of 10 bags of urea are found to have the following weights (kg): 40, 45, 49, 50, 55, 44, 52, 48, 50, 45 Test at 5% level of significance whether the average packing weight can be taken as 50 kg.

[Figure: t-distribution table printed with the question – values of $t_{\alpha }$ for $\alpha$ = 0.05, 0.025, 0.01, 0.005; rows df = 1–30, 40, 60, 120, ∞] [2023/15m/2b]

3. Eleven police personnel were given a test in shooting. Further they were given a month’s training and a second test of equal difficulty was conducted at the end of it. The table below contains the marks awarded in the two tests. Do these marks give evidence that the police personnel benefitted from the training?

Police personnel ID	1	2	3	4	5	6	7	8	9	10	11
Test 1 marks	23	20	19	21	18	20	18	17	23	16	19
Test 2 marks	24	19	22	18	20	22	20	20	23	20	17

[Test at 5% level of significance] (Relevant table is attached at the end of this Paper) [2022/10m/1d]

4. The mean annual income of workers in a company is ₹ 5,000 with a standard deviation of ₹ 1,200. It is suspected that local workers have higher than the average income. A sample of 144 local workers is drawn and their sample mean income is found to be ₹ 5,500. Can it be said that local workers have significantly higher income than the total population? (Use $\alpha$ = 0.05) [2021/10m/1a]

5. (i) Ten accountants were given intensive coaching and four tests were conducted in a month. The scores of tests 1 and 4 are given below:

Sl. No. of Accountants:	1	2	3	4	5	6	7	8	9	10
Marks in 1st Test:	50	42	51	42	60	41	70	55	62	38
Marks in 4th Test:	62	40	61	52	68	51	64	63	72	50

Do the scores from test 1 to test 4 show an improvement? Test at 5% level of significance. [Tables attached] (ii) State the properties of the normal curve. What is a standard normal distribution? Discuss its importance. [2019/15m/3b]

6. The manager of a hotel based in New Delhi reports that the mean number of rooms rented out per night is at least 212 based on his observation for the past several months. One of the corporate officials feels that this number might be somewhat over-estimated. A sample of 150 nights produces a mean of 201.3 rooms and a standard deviation of 45.5 rooms. If these results suggest the manager has inflated his report, he will be given a warning. At 5% level, what is the manager’s fate? [Normal distribution table is placed at the end of the booklet]. [2018/10m/1a]

7. What do you mean by Type–I and Type–II errors? How are they interrelated? In which situations, you would not like to commit Type–I error? Why? [2017/5m/4a(ii)]

8. A vote is to be taken among the residents of a town and the surrounding area to determine whether a proposed chemical plant should be constructed or not. The construction site is within the town limits and for this reason many voters of the surrounding area feel that the proposal will pass because of the large proportion of town voters who favour the construction. To determine if there is a significant difference in the proportion of town voters and surrounding area voters favouring the proposal, a poll is taken. If 120 of the 200 town voters favour the proposal and 240 of 500 surrounding area voters favour the proposal, would you agree that the proportion of town voters favouring the proposal is higher than that of the surrounding area voters? Use a 5% level of significance. [2016/5m/2b(iii)]

9. Technology has led to the rise of workaholics, who are on the job 60 hours or more per week. One of the reasons cited by employees as to why they worked long hours was that they loved their job, because it is stimulating and challenging. A survey of 1564 workaholics included 786 men and 778 women and the results showed that 707 men and 638 women loved their job, because it is stimulating and challenging. At 0.05 level of significance, is the proportion of workaholic men, who loved their job because it is stimulating and challenging, different from the proportion of women? [2014/15m/2b]

10. If a statement is made “Crime-sheeters in the age group of 35 to 50 are more likely than those in the age group of 55 to 70,” how would you verify the statement using Tests of Hypothesis? [2011/10m/5iv]

11. Statistical Tests of Significance. [2001/20m/1c]

1.5] Chi-square and ANOVA

12 questions

1. Explain parametric test and non-parametric test. What are the applications of chi-square (${\chi }^2$) test? Give one example. [2026/10m/150w/1a]

2. State the assumptions made for the study of ANOVA. [2024/10m/1b]

3. State the characteristics of Chi-Square Test. [2023/10m/1d]

4. The following table gives the production in three shifts and the corresponding number of defective goods that turned out in three weeks. Test at 5% level of significance whether weeks and shifts are independent. Table:

Shift	1st week	2nd week	3rd week	Total
I	20	20	20	60
II	5	10	15	30
III	15	20	25	60
Total	40	50	60	150

[Figure: Table of critical values of Chi-Square (${\chi }^2$) printed with the question – curve of $f\left({\chi }^2\right)$ with shaded upper tail; columns $\alpha$ = 0.100, 0.050, 0.025, 0.010, 0.005, 0.001; rows df = 1–30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100] [2023/15m/2a]

5. A city administration conducted a study of the waiting time in the emergency wings of three hospitals. These hospitals are located in three zones of the city far away from each other. The administration is interested in reducing the waiting time at the emergency wings. To study this, a random sample of 10 emergency wing cases at each hospital was selected on a particular day and the waiting time was measured. The results are recorded in the following table. At 0.05 level of significance, is there evidence of a difference in the average waiting times in the three hospitals?

Hospital 1	Hospital 2	Hospital 3
12	7	5
8	3	4
7	7	4
6	5	5
7	5	6
4	6	4
8	8	7
5	6	8
6	4	6
7	7	4

(Relevant table is attached at the end of this Paper) [2022/15m/2b]

6. A certain drug is claimed to be effective in curing cold. In an experiment on 500 persons suffering from cold, half of them were given the drug and the other half were given the sugar pills. Patients’ reactions to the treatment are recorded in the following table:

Treatment	Helped	Harmed	No Effect
Drug	150	30	70
Sugar pills	130	40	80
Total	280	70	150

On the basis of the above data, can it be concluded that there is a significant difference in the effect of drug and sugar pills?

[Figure: Table of critical values of Chi-Square (${\chi }^2$) printed after the question – curve of $f\left({\chi }^2\right)$ with shaded upper tail; columns $\alpha$ = 0.100, 0.050, 0.025, 0.010, 0.005, 0.001; rows df = 1–30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100] [2021/10m/3b(ii)]

7. The pathology laboratory of a hospital has five pathologists. Samples arriving at the laboratory for a particular test are randomly assigned to one of them. An evaluation of the performance during the previous month gave the following counts shown in the table below:

Pathologist	A	B	C	D	E
Number of tests done	850	1000	750	900	1050
Number of reported errors	12	15	11	17	10

The director of the hospital wanted to know whether the proportion of errors in the pathological tests assigned to the five pathologists differ. State appropriate null and alternate hypotheses for the question above. Carry out the test of hypothesis at 5% level of significance and describe the result.

Table: ${\chi }^2$ (Chi-Squared) Distribution: Critical Values of ${\chi }^2$

Degrees of freedom	Significance level 5%	Significance level 1%	Significance level 0.1%
1	3.841	6.635	10.828
2	5.991	9.210	13.816
3	7.815	11.345	16.266
4	9.488	13.277	18.467
5	11.070	15.086	20.515
6	12.592	16.812	22.458
7	14.067	18.475	24.322
8	15.507	20.090	26.124
9	16.919	21.666	27.877
10	18.307	23.209	29.588

[2020/15m/4b]

8. The following represent the number of units produced by four different workers using five different types of machines:

Worker	A	B	C	D	E
I	40	50	30	70	60
II	50	70	70	40	50
III	70	60	70	80	80
IV	30	50	40	80	20

On the basis of this information, can it be concluded that (i) workers do not differ with regard to mean productivity (ii) mean production is same for different machines? [Test at 5% level of significance] [Tables attached] [2019/20m/2b]

9. A movie producer is bringing out a new movie. In order to map out his advertising campaign, he wants to determine whether the movie will appeal most to a particular age group or whether it will appeal equally to all age groups. The producer takes a random sample from persons attending the preview of the movie, and obtains the following results. Use ${\chi }^2$ test to derive the conclusion (for 6 degrees of freedom and 5% significance level, ${\chi }^2$ critical value is 12.592):

Age groups	Under 20	20–39	40–59	60 and above
Liked the movie	320	80	110	200
Disliked the movie	50	15	70	60
Indifferent	30	5	20	40

[2017/10m/1a]

10. Out of 200 boys, 75 are intelligent of which 40 have skilled fathers, while 85 of the unintelligent boys have unskilled fathers. Do these figures support the hypothesis that skilled fathers have intelligent boys? (Use Chi-square values 3.841, 5.991, 7.815 and 9.488 for degrees of freedom 1, 2, 3 and 4 respectively.) [2016/5m/2c(i)]

11. The determination of yields of three different varieties (say X, Y and Z) was subject of a recent experiment. Different blocks chosen at random from a larger group were used for this purpose. The data recorded were as follows:

Blocks / Varieties	X	Y	Z
1	12	16	30
2	5	10	18
3	7	28	35
4	10	26	51

(i) Analyze the above two-way classified data. (ii) Test for the effects of blocks and varieties at the 5% level of significance. (iii) Determine which varieties differ, if any. Is it possible to do the same with the blocks? [2013/20m/2b]

12. The following table gives the number of aircraft accidents that occurs during the various days of the week. Find whether the accidents are uniformly distributed over the week.

Days	Sun	Mon	Tue	Wed	Thurs	Fri	Sat
No. of accidents	14	16	8	12	11	9	14

(Given: the values of chi-square significant at 5, 6, 7 d.f. are respectively 11.07, 12.59, 14.07 at 5% level of significance). [2009/20m/1d]

1.6] Simple correlation and regression

19 questions

1. (i) What is the meaning of positive and negative correlation? Explain with suitable example. (ii) Define regression. What are coefficients of regression and what are their properties? Give examples. [2026/15m/3a]

2. A company gives on-the-job training to its salesmen, which is followed by a test. The company is considering whether there is any relationship of test scores and sales performance made by the ten salesmen during last one year, and also whether the company should terminate the services of any salesman who does not perform well in the test:

Test Score	50	60	58	47	50	33	65	43	46	68
Sales (’000)	48	65	50	48	55	58	63	48	50	70

Give your comment whether the company should terminate the services of any salesman who does not perform well in the test. [2024/15m/2a]

3. The area sales manager of a company has compiled the following data for the eight territories under his/her supervision. All the territories have similar size and consumer characteristics. The sales manager believes that the number of selling agents assigned to a territory has an impact on its sales revenue:

Sales territory	Sales (crores of ₹)	Number of selling agents
1	100	10
2	80	10
3	60	7
4	120	15
5	150	20
6	90	12
7	70	8
8	130	14

(i) Find the Pearson’s correlation coefficient between the two variables mentioned above. Is the sales manager correct in his/her belief? (ii) Develop a linear regression model and using it, predict the sales in a territory if 16 selling agents are assigned to it. [2022/15m/2a]

4. Twelve salesmen are ranked for efficiency and their length of service. Find the rank correlation coefficient.

Salesmen	Efficiency	Length of Service
A	1	2
B	2	1
C	3	5
D	5	3
E	5	9
F	5	7
G	7	7
H	8	6
I	9	4
J	10	11
K	11	10
L	12	11

[2021/5m/4b(i)]

5. Entrepreneurs who want to start new businesses in a city can seek financial support from its City Investment Promotion Board (CIPB). The officers in CIPB evaluate applications for financial support and give their recommendations. This evaluation process takes some time. The time spent by the ten officers in CIPB and the number of applications they evaluated are given in the following table:

Time spent (in days)	Applications evaluated (Number)
127	20
59	12
87	21
21	4
60	15
121	31
15	3
18	5
27	3
75	17

(i) Based on the information, determine the least squared linear regression model. [2020/10m/3a(i)]

(ii) What does the slope of the linear regression model represent? [2020/5m/3a(ii)]

(iii) Based on the above data, what is the approximate predicted number of applications for financial support that an officer can evaluate with 100 days of work? [2020/5m/3a(iii)]

6. You are the manager of a small steel mill and all your inputs except labour are fixed in the short-run. You suspect that as the input of labour increases, the output of steel will also increase. But you would like to know more about the nature of this relationship so that you can predict the output of steel associated with any level of labour input and use this information to determine the optimal level of labour input. To achieve this goal, you conducted an experiment in which the level of labour input is varied for 8 working days. This is shown in the following table:

Observation	Number of workers	Tons of steel production
1	1	4
2	2	6
3	3	10
4	4	10
5	5	15
6	6	15
7	7	16
8	8	20

(i) Draw a scatter diagram. (ii) Develop the predictive regression model after finding out the values of intercept and slope. (iii) If the number of workers is increased up to 10, what would be the output of steel? [2018/20m/2a]

7. The following are the monthly figures of advertising expenditure and sales of a company. It is found that advertising expenditure has impact on sales generally after 2 months. Allowing for this time lag, calculate the coefficient of correlation:

Month	Advertising expenditure (in thousand ₹)	Sales (in units)
January	50	1200
February	60	1500
March	70	1600
April	90	2000
May	120	2200
June	150	2500
July	140	2400
August	160	2600
September	170	2800
October	190	2900
November	200	3100
December	250	3900

[2017/10m/4a(i)]

8. The two regression lines obtained from a certain data were $Y=X+5$ and $16X=9Y-94$. Find the variance of X, if the variance of Y is 16. Also find the covariance between X and Y. [2016/4m/1a(iii)]

9. There are 50 students in a class. The regression equation of marks in Economics (X) on marks in Management (Y) is $3Y-5X+180=0$. The mean marks in Management is 44 and the variance of marks in Economics is $\frac{9}{16}$th of the variance of marks in Management. Find the mean marks in Economics and the coefficient of correlation between marks in the two subjects. [2013/10m/1b]

10. Comment briefly on the Standard Error of Estimate for any given set of observations. [2010/10m/2b]

11. Assuming a linear regression model of the type $Y_i=a+b{X}_i+U_i$, $i=1,2,\dots ,29$ the following data have been collected $\sum X_i^2=100$, $\sum X_i\prime Y_i\prime =72$, $\sum X_i=2610$, $\sum Y_i=1740$ where $X_i\prime$ and $Y_i\prime$ are deviations of $X_i$ and $Y_i$ from their respective sample means $\overline{X}$ and $\overline{Y}$. What is the least square estimate of $a$ and $b$? [2009/20m/1c]

12. Explain the difference between the product moment correlation coefficient and rank correlation coefficient. The rankings of ten students in two subjects A and B are as follows:

A	3	5	8	4	7	10	2	1	6	9
B	6	4	9	8	1	2	3	10	5	7

Find the correlation coefficient and comment on the result. [2009/60m/2]

13. scatter diagrams to visualize the relationship between two variables. [2000/20m/8a]

14. regression analysis to estimate the relationship between two variables. [2000/20m/8b]

15. 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? [2000/20m/8c]

16. From the following data obtain the two regression equations.

Sales	91	97	108	121	67	124	51	73	111	57
Purchase	71	75	69	97	70	91	39	61	80	47

[1998/60m/2]

17. A panel of two Judges P and Q graded 7 dramatic performances by independently awarding marks as follows:

Performance	1	2	3	4	5	6	7
Marks by P	46	42	44	40	43	41	45
Marks by Q	40	38	36	35	39	37	41

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? [1997/60m/7]

18. Briefly discuss the significance and limitations of the study of Correlation analysis. [1996/30m/7a]

19. Compute Karl Pearson’s 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.)
1968	249	237
1969	251	238
1970	248	236
1971	252	240
1972	258	245
1973	269	255
1974	271	254
1975	272	252
1976	280	258
1977	275	251

[1996/30m/7b]

1.7] Time series and forecasting

11 questions

1. ABC computer company is engaged in repairs and service of computers. It is trying to forecast the demand for next 12 months, based on the accumulated demand data as given below. The company wants to follow exponential smoothing forecasts using smoothing constant (α) equal to 0.30 and 0.50: Demand for Repair.

Period	Month	Demand	Period	Month	Demand
1	January	40	7	July	45
2	February	43	8	August	48
3	March	44	9	September	57
4	April	40	10	October	55
5	May	45	11	November	56
6	June	50	12	December	55

The forecast for period one may be taken as 40. The present your result in a table. [2026/20m/2a]

2. The electrical power demand of TATA Power Corporation from 2018 to 2024, measured in megawatts, is provided below. Using the data, forecast the demand for the year 2025 by fitting and plotting the linear trend line.

Year	Electrical Power Demand (Megawatts)
2018	94
2019	99
2020	100
2021	110
2022	125
2023	162
2024	142

[2025/15m/2a]

3. A department store had forecast the sales of ₹ 1,10,000 for last week. The actual sales turned out to be ₹ 1,25,000. Given $\alpha$ = 0.1, what is the forecast for this week? If sales this week turn out to be ₹ 1,20,000, what is the forecast for the next week? [2016/5m/2b(i)]

4. (i) Forecast of demand for cakes made by a baking company needs to be made with accuracy. The bakery markets cakes through a chain of food stores. It has been experiencing over- and under-production due to forecasting errors. The demand in dozens of cakes for the past four weeks is shown in the table below. Cakes are made for sale on the following day, for example, Monday’s cake production is for Tuesday’s sales and so on. The bakery is closed on Saturday. So Friday’s production must satisfy demand for both Saturday and Sunday:

Day	4 weeks ago	3 weeks ago	2 weeks ago	1 week ago
Monday	2200	2400	2300	2400
Tuesday	2000	2100	2200	2200
Wednesday	2300	2400	2300	2500
Thursday	1800	1900	1800	2000
Friday	1900	1800	2100	2000
Saturday + Sunday	2800	2700	3000	2900

Using a weighted average of 0.40, 0.30, 0.20 and 0.10 for week 1, week 2, week 3 and week 4 ago respectively, what is the weighted average forecast for Monday of the current week? (ii) The bakery also produces bread and is planning its purchases of ingredients for bread production. If bread demand forecast for last week had been 22000 loaves, and only 21000 loaves were actually demanded, what would the forecast be for this week using exponential smoothing method with smoothing coefficient $\alpha =0.10$? (iii) In part (ii) above, if actual demand in current week turns out to be 21500, what will be the forecast for the next week with exponential smoothing method, using $\alpha =0.10$? [2013/15m/3b]

5. What are your observations on the sales data given in the table? [2011/5m/4i]

6. Estimate the sales likely to occur in quarters 17, 18, 19 and 20. [2011/20m/4ii]

7. Discuss the various methods of forecasting. What are the precautions that you would like to take to ensure that the forecast is realistically achievable? What approach would you like to recommend for resource allocation to support the activities to achieve the performance target? [2007/60m/6]

8. Forecasting techniques. [2003/20m/200w/1b]

9. Time Series Analysis. [2002/20m/200w/1d]

10. Use of National Income in business forecasting. [1997/30m/8a]

11. What are the major forecasting methods? Give a brief description of these methods. How would you forecast the demand for television in the year 2000? What factors would influence the choice of a forecasting method? [1995/60m/5]

1.8] Decision theory

10 questions

1. A decision problem has been expressed in the following payoff table:

States of Nature	Alternatives
	$A_1$	$A_2$	$A_3$
$E_1$	190	186	182
$E_2$	164	162	166
$E_3$	142	144	174

Which alternative will be selected if the following criterion is used? (A) Maximax criterion (B) Maximin criterion (C) Hurwicz’s criterion (Assume degree of optimism is 0.7) [2021/9m/2a(ii)]

2. E-Education is a new start-up that develops and markets MBA courses offered over the Internet. The company is currently located in Bangalore and employs 150 people. Due to strong growth, the company needs additional office space. The company has the option of leasing additional space at its current location in Bangalore for the next two years, but after that will need to move to a new building. Another option the company is considering is moving the entire operation to a new building. Another option the company is considering is moving the entire operation into a small Central India town immediately. A fourth option is for the company to lease a new building in Bangalore immediately. If the company chooses the first option and leases new space at its current location, it can, at the end of two years, either lease a new building in Bangalore or move to a small town in Central India. The following are some additional facts about the alternatives and current situation: 1. The company has a 75% chance of surviving the next two years. 2. Leasing the new space for two years at the current location in Bangalore would cost ₹ 7,50,000 per year. 3. Moving the entire location to Central India would cost ₹ 10 lakhs. Leasing a new building would cost only ₹ 5,00,000 per year. 4. Moving to a new building in Bangalore would cost ₹ 2,00,000 and leasing the new building would cost ₹ 6,50,000. 5. The company can cancel the lease at any time. 6. The company would build its own building in five years, if it survives. 7. Assume all other costs and revenues are the same no matter where the company is located. What should E-Education do? [2019/20m/3a]

3. A firm is planning to develop and market a new drug. The cost of extensive research to develop the drug has been estimated at ₹ 1,00,000. The manager of the research program has found that there is a 60% chance that the drug will be developed successfully. The market potential has been assessed as follows:

Market Potential	Probability	PV of Profit (₹)
Large Market	0.1	50,000
Moderate Market	0.6	25,000
Low Market	0.3	10,000

The present value (PV) figures do not include the cost of research. While the firm is considering this proposal, a second proposal, almost similar, comes up for consideration. The second one also requires an investment of ₹ 1,00,000, but the PV of all profit is ₹ 12,000. Of course the return on investment in the second proposal is certain. Recommend, by applying appropriate decision criteria, what decision should the firm take regarding the investment of ₹ 1,00,000? [2016/15m/3a]

4. XYZ, a real estate development firm, is considering several alternative development projects. These include building and leasing an office space, purchasing a parcel of land and constructing an office building to rent, buying and leasing a warehouse, building a shopping center, and building and selling condominiums. The financial success of these projects depends on interest rate movements in the next five years. The various development projects and their five year financial return (Rs. Crores), given that the interest rates will decline, remain stable, or increase are shown in the payoff table below. Table: Payoffs from Development Projects (Rs. Crores)

Project	Interest Rate: Decline	Interest Rate: Stable	Interest Rate: Increase
Office Park	5	17	45
Office Building	15	19	24
Warehouse	17	14	10
Shopping Center	7	24	36
Condominiums	32	15	6

Determine the best investment option using the following criteria: (i) Maximin criterion. (ii) Equal likelihood criterion. (iii) Hurwitz criterion with a coefficient of optimism = 0.3. [2015/15m/2a]

5. A pharmaceuticals company is reviewing its employee healthcare program. Currently the company pays a fixed fee of Rs. 1500 per month for each employee, regardless of the number of rupee amount of medical claims filed to healthcare insurer A. Another healthcare provider B has offered to charge the company a fixed fee of Rs. 500 per month per employee and Rs. 150 per claim filed. A third health insurer C charges Rs. 1000 per month per employee and Rs. 50 per claim filed. (i) How will the average number of claims per employee per month affect the decision to choose the healthcare program? (ii) When will it be best to go for option provided by C? (iii) When will it be best to go for option provided by B? [2015/15m/4c]

6. An oil company is considering a bid for a shale oil development contract to be awarded by the Central Government. The company has decided to bid ₹ 660 crores. The company estimates that it has a 60% chance of winning the contract with this bid. If the firm wins the contract, it can choose one of the three methods for getting the oil from shale. It can develop a new method for oil extraction, use an existing (inefficient process), or subcontract the processing out to a number of smaller companies once the shale has been excavated. The results from these alternatives are given in the table below:

Alternative	Outcomes	Probability	Profit (₹ in crores)
Develop new process	Great success	0.30	3,600
Develop new process	Moderate success	0.60	1,800
Develop new process	Failure	0.10	-600
Use present process	Great success	0.50	1,800
Use present process	Moderate success	0.30	1,200
Use present process	Failure	0.20	-240
Subcontract	Moderate success	1.00	1,500

The cost of preparing the contract proposal is ₹ 12 crores. If the company does not make a bid, it will invest in an alternative venture with a guaranteed profit of ₹ 180 crores. (i) Should the company make the bid? (ii) If yes, what method for oil extraction should it adopt? [2013/15m/2c]

7. The Black Gold Oilfield Company has recently acquired rights to a new potential source of natural oil in Bay of Bengal. The current market value of these rights is ₹ 900 crore. However if there is natural oil at the site, it is estimated to be worth ₹ 8000 crore. In this case, the company would have to pay ₹ 1000 crore in drilling costs to extract oil. The company believes there is a 0.25 probability that the proposed drilling site actually will hit the natural oil reserve. Alternately, the company can pay ₹ 300 crore to first carry out a seismic survey at the proposed drilling site. Historically if the seismic survey produces a favourable result, there is a 0.5 probability of hitting oil at the drilling site. However, if the seismic survey produces an unfavourable result, there is only a 0.14 probability of hitting oil. The probability of an unfavourable survey when no oil is present is 0.8. The probability of a favourable seismic survey when oil is present at the drilling site is 0.6. What is the optimal decision strategy, using the Expected Monetary Value (EMV) criterion? [2012/30m/300w/8c]

8. The company is presented with three options: make in house, buy locally, import. There are three market scenarios: demand may increase, demand may decrease and demand may remain at the present level. The table below gives the expected profit the company will make for each combination. Which option should the company choose if the probability of demand increasing is 0.3, probability of demand decreasing is 0.45 and probability of demand remaining as at present level is 0.25?

Expected Profits for Each Combination

Options	Demand Scenarios: May increase	Demand Scenarios: May decrease	Demand Scenarios: May remain at present level
Make in house	₹ 100	₹ 20	₹ 40
Buy locally	₹ 50	₹ 70	₹ 30
Import	₹ 65	₹ 80	₹ 35

[2011/10m/4iv]

9. In question (iv), all other data remaining same, which option will you choose if the probabilities of all market conditions are unknown? Why? [2011/15m/4v]

10. Demonstrate the use of probability theory and decision trees in risk analysis in microeconomic decision making. [1999/60m/7]

1.9] Index numbers

2 questions

1. State the chief characteristics of Index Numbers. [2023/10m/1a]

2. What is Fisher’s ideal index? Why is it called ideal? Describe the steps involved in the computation of Fisher’s ideal index number. [2021/10m/1e]

1.10] Linear programming

37 questions

1. (i) Explain the steps required for formulating a linear programming model. (ii) Find out the solution for a store from the following data: $\begin{array}{cc}\hfill \hfill & \hfill \mathrm{\text{Maximize}}Z=50x_1+60x_2\hfill \\ \hfill \hfill & \hfill \mathrm{\text{subject to}}\hfill \\ \hfill \hfill & \hfill x_1+1.5x_2\le 40\hfill \\ \hfill \hfill & \hfill 4x_1+4.5x_2\le 150\hfill \\ \hfill \hfill & \hfill x_1,x_2\ge 0\hfill \end{array}$ Profit is in ₹. Draw graph to arrive at result. (Graph paper is enclosed) [2026/15m/2b]

2. Techline Pvt. Ltd. is a consumer electrics manufacturing concern producing three models of Smart Devices: X, Y and Z. The company operates under the constraints of two key resources, viz. raw materials and labor hours. Each week, the availability of raw materials is restricted to 800 kg, while labor availability is limited to 600 hours. To maximize profit, the company needs to decide, how many units of each product to manufacture weekly without exceeding the available resources. The relevant details for each product are provided below:

Product	Raw Material per unit (Kg)	Labor Hours per unit (Hours)	Profit contribution per unit (₹)
X	6	5	80
Y	4	4	60
Z	2	3	40

(i) Formulate this scenario as a Linear Programming Problem (LPP), and solve it for optimal product mix. (ii) Based on the optimal solution, analyze the situation if the company wants to introduce a New product W, requiring 2 kg of raw materials and 2 hours of labor hours per unit. Its estimated profit contribution is ₹35/- per unit. Should the company include this new product in its production mix? [2025/20m/2c]

3. Explain the necessary conditions for application of simplex method to be applied to linear programming problems. [2024/10m/1a]

4. (i) For conducting Third Semester Practical Examination, the Chemistry department of a university affliated college requires 10, 12 and 7 units of three chemicals X, Y and Z respectively. The chemicals are available in two boxes—box A and box B. Box A contains 3, 2 and 1 unit of X, Y and Z respectively and costs ₹300. Box B contains 1, 2 and 2 units of X, Y and Z respectively and costs ₹200. Find how many boxes of each type should be purchased by the department so that the total cost is minimal by formulating the problem in linear programming problem and solve it through graphical method. (ii) Comment on the sensitivity analysis. [2024/20m/2c]

5. A firm produces three products A, B and C which pass through three departments: Milling, Lathe and Grinder. Each unit of Product A requires 8, 4 and 2 hours respectively in the three departments. Each unit of Product B requires 2, 3 and 0 hours respectively in the three departments. Each unit of Product C requires 3, 0 and 1 hours respectively in the three departments. The available capacity on the machines which might limit output is given below. Table: Capacity of the Machines for Production.

Machine Type	Available time (in machine hours per week)
Milling Machine	250
Lathe Machine	150
Grinder Machine	50

The unit profit would be: ₹ 20, ₹ 6 and ₹ 8 for products A, B and C respectively. (i) Formulate the problem in LPP. (ii) How much of each product should the firm produce in order to maximize profit? [2023/20m/2c]

6. Agrofarms Ltd. is a company engaged in large-scale cultivation of organic vegetables, grains and cereals. In 200 acres of land, Agrofarms Ltd. grows only tomatoes and onions. For the upcoming season, it estimates that it can make a profit of ₹ 7,000 per acre of tomatoes and ₹ 2,000 per acre of onions. During this growing season, each acre of tomatoes will require 4 tons of fertilizers and 3 tons of pesticides, whereas each acre of onions will require 2 tons of fertilizers and 1 ton of pesticides. Agrofarms Ltd. has contracted for at most 600 tons of fertilizers and 330 tons of pesticides.

(i) How many acres of land should be devoted to each crop to maximise its profit for the season? Is there any land that remains unfarmed? [2022/10m/2c(i)]

(ii) What is the minimum profit per acre of onions that would make it economically feasible for Agrofarms Ltd. to grow onions? [2022/6m/2c(ii)]

(iii) If the profit per acre of onions were ₹ 2,500, how many acres of land should Agrofarms Ltd. plant of each crop to maximize the profit for the season? [2022/4m/2c(iii)]

7. Solve the following Linear Programming Problem by Graphical Method: $\begin{array}{cc}\hfill \hfill & \hfill \mathrm{\text{Minimize cost:}}C=5x_1+6x_2\hfill \\ \hfill \hfill & \hfill \mathrm{\text{subject to:}}\hfill \\ \hfill \hfill & \hfill x_1+x_2=1000\hfill \\ \hfill \hfill & \hfill x_1\le 300\hfill \\ \hfill \hfill & \hfill x_2\ge 150\hfill \\ \hfill \hfill & \hfill x_1,x_2\ge 0\hfill \end{array}$ [2021/10m/3b(i)]

8. A State Public Health Department has allocated ₹ 40,00,000 for advertising information on Coronavirus pandemic. Two types of advertisements will be used: Radio and Newspaper. Each radio advertisement costs ₹ 20,000 and reaches an estimated 3000 people. Each newspaper advertisement costs ₹ 50,000 and reaches an estimated 7000 people. In planning the advertising campaign, the Public Health Department would like to reach as many people as possible, but they have decided that at least 10 advertisements of each type must be used. Also, the number of radio advertisements must be at least as many as the number of newspaper advertisements. How many advertisements of each type should be used? How many people will this reach? Find the optimal solution that best satisfies all requirements. [2020/15m/2c]

9. The table given below gives the cost per unit (in ₹) for transporting goods from each of three depots 1, 2 and 3 to markets A, B and C. It also depicts stock available at each depot and demand at each market.

	Market A	Market B	Market C	Stock
Depot 1	23	21	27	14
Depot 2	26	25	15	12
Depot 3	20	21	21	13
Demand	11	14	14

(i) Get initial feasible solution using North West Corner method. [2020/4m/4c(i)]

(ii) Calculate initial cost. [2020/2m/4c(ii)]

(iii) Find out optimum cost and allocations from each depot to each market. [2020/9m/4c(iii)]

10. A chemical company produces two chemical products: ‘A’ and ‘B’. Production of both products requires the same process, ‘I’ and ‘II’. The production of ‘B’ results also in a by-product ‘C’ at no extra cost. The product ‘A’ can be sold at a profit of ₹ 3 per unit and ‘B’ at a profit of ₹ 8 per unit. Some of this by-product can be sold at a unit profit of ₹ 2, the remainder has to be destroyed and the destruction cost is ₹ 1 per unit. Forecasts show that up to 5 units of ‘C’ can be sold. The company gets 3 units of ‘C’ for each unit of ‘B’ produced. The manufacturing times are 3 hours each per unit for ‘A’ on process ‘I’ and ‘II’, and 4 hours and 5 hours per unit for ‘B’ on process ‘I’ and ‘II’, respectively. Because the product ‘C’ results from producing ‘B’, no time is used in producing ‘C’. The available times are 18 and 21 hours of process ‘I’ and ‘II’, respectively. Formulate this problem as an LP model to determine the quantity of ‘A’ and ‘B’ which should be produced, keeping ‘C’ in mind, to make the highest profit for the company. [2019/10m/1a]

11. A company manufactures two products ‘Alpha’ and ‘Beta’. Each unit of Alpha requires 1 hour of machining and 2 hours of skilled labour, whereas each unit of Beta uses 2 hours of machining and 1 hour of skilled labour. For the forthcoming month, the machining capacity is limited to 720 hours and the skilled labour is limited to 780 hours. Not more than 320 units of Alpha can be sold during a month. For the maximisation of profits, determine the optimal product mix and maximum profit. Unit contribution from Alpha is ₹ 6 and from Beta is ₹ 4. Use Simplex method. [2017/20m/2a]

12. A company manufactures two kinds of machines, each requiring different manufacturing techniques. The deluxe machine requires 18 hours of labour, 8 hours of testing and yields a profit of ₹ 450. The standard machine requires 3 hours of labour, 4 hours of testing and yields a profit of ₹ 250. There are 800 hours of labour and 600 hours of testing available each month. A marketing forecast has shown that the monthly demand for the standard machine is to be more than 150. Management wants to know the number of each model to be produced monthly that will maximize total profit. Formulate and solve this as a LPP. [2016/10m/1c]

13. XYZ Grocery sells two types of soft drinks: the brand name A1 Cola and the cheaper store brand XYZ Cola. The margin of profit on the A1 Cola is about 50 paise per bottle, whereas the XYZ Cola grosses 70 paise per bottle. On the average the store sells 500 bottles of both the Colas a day. Although A1 is a better recognized name, customers tend to buy more of the XYZ brand because it is considerably cheaper. It is estimated that the XYZ brand outsells the A1 brand by a ratio of at least 2 : 1. However XYZ Grocery sells at least 100 bottles of A1 a day. (i) How many bottles of each brand should the store carry daily to maximize the total profit? (ii) Determine the ratio of the profits per bottle of A1 and XYZ Colas that will keep the solution in (i) unchanged? [2015/14m/3c]

14. A small manufacturing firm wants to produce two types of products, ‘Normal’ and ‘Premium’. The profit from selling either of these products is same and equal to ₹ 100 per piece. Each of these products requires processing in three departments–Casting, Welding and Finishing. ‘Normal’ needs 9 hours, 1 hour and 4 hours in Casting, Welding and Finishing, respectively. ‘Premium’ needs 4 hours, 1 hour and 7 hours, respectively in these departments. The capacities of these departments are 36 hours, 5 hours and 28 hours, respectively. (i) Formulate the problem as a typical Linear Programming Problem (LPP) so as to find the optimal number of ‘Normal’ and ‘Premium’ to be produced. (ii) Find the maximum achievable profit. (iii) What is the right product mix? [2014/20m/3a]

15. Explain the concept of shadow price in a linear programming problem. How can this concept be used for managerial decision? [2014/5m/4b(iii)]

16. A light metal company manufactures two products A and B. Each product must pass through two processing sections L and M. A good number of machines are available in both the sections. One unit of product A requires 2 hours of processing time in L and 1 hour in M. One unit of product B requires 1 hour of processing time in L and 4 hours in M. Total time available in section L is 6000 hours whereas in section M, it is 10000 hours. The net profit for product A is ₹ 3.50 per unit and for product B is ₹ 5.00 per unit. The company wishes to maximize the total net profit. (i) Formulate the problem as a linear programming problem. (ii) Find the optimum production schedule that will maximize the total net profit per week by graphical method or otherwise. [2013/10m/1c]

17. MNP Manufacturing produces ornate, decorative wood frame doors and windows. Each item produced goes through three manufacturing processes: cutting, sanding and finishing. Each door produced requires 1 hour in cutting, 30 minutes in sanding and 30 minutes in finishing. Each window requires 30 minutes in cutting, 45 minutes in sanding and 1 hour in finishing. In the coming week, MNP has 40 hours of cutting capacity, 40 hours of sanding capacity and 60 hours of finishing capacity available. Assume that all doors produced can be sold for a profit of ₹ 500 each and all windows can be sold for a profit of ₹ 400 each. Formulate a linear programming model to solve the problem. Use the graphical method to find the optimal solution. [2012/12m/150w/5a]

18. Formulate the following problem as Linear Programming Problem: How much of products A and B should be produced to minimise the weekly total cost of machines, men and material? [2011/10m/1i]

19. Formulate the following problem as Linear Programming Problem: How much of products A and B should be produced to maximise the weekly total profit of the company? [2011/10m/1ii]

20. Solve question (i) using graphical method, to minimise the weekly total cost. [2011/20m/1iii]

21. Do you think it is possible to have multiple solutions resulting in the same minimum weekly total cost in case of minimisation problem and resulting in the same maximum weekly total profit in the maximisation problem? If yes, how should the company identify such situations? [2011/10m/1iv]

22. Prepare a typical Tableau to solve the minimisation problem by using Simplex Method. [2011/10m/1v]

23. The Sea Wharf Restaurant would like to determine the best way to allocate a monthly advertising budget of Rs. 50,000 between newspaper advertising and radio advertising. Management has decided that at least 25% of the budget must be spent on each type of media, and that the amount of money spent on newspaper advertising must be at least twice the amount spent on radio advertising. A marketing consultant has developed an index that measures audience exposure per rupee of advertising on a scale from 0 to 100, with higher values implying greater audience exposure. If the value of the index for newspaper advertising is 50 and that for radio advertising is 80, how should the restaurant allocate its advertising budget in order to maximise the value of the total audience exposure? (i) Formulate the problem as a linear programming problem. (ii) Find the optimal solution to the problem using the graphical solution procedure. (iii) Which of the constraints are binding? (iv) How sensitive is your solution to the estimation of index values by the marketing consultant? [2010/40m/2a]

24. Discuss the managerial utility of the sensitivity analysis in linear programming. [2009/20m/1b]

25. A small firm makes three similar products, which all follow the same three-step process, consisting of milling, inspection and drilling. Product A requires 12 minutes of milling, 5 minutes for inspection, and 10 minutes of drilling per unit; product B requires 10 minutes of milling, 4 minutes for inspection, and 8 minutes of drilling per unit; product C requires 8 minutes of milling, 4 minutes for inspection, and 16 minutes of drilling. The department has 20 hours available during the next period for milling, 15 hours for inspection, and 24 hours for drilling. Product A contributes Rs. 2.40 per unit to profit, product B contributes Rs. 2.50 per unit, and product C contributes Rs. 3.00 per unit. Determine the optimal mix of products in terms of maximizing contribution to profits for the period. [2009/60m/3]

26. 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 ₹ 10, ₹ 36 and ₹ 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. [2007/60m/4]

27. A company manufactures three products X, Y and Z. Their profits per unit are ₹ 300, ₹ 200, and ₹ 400 respectively. The company has two machines and the required processing time in minutes on each machine for each product is given below: [Table: processing time in minutes of products X, Y and Z on Machines 1 and 2 – not reproduced in any available source] [?] 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. [2006/60m/4]

28. Linear Programming as a problem solving technique. [2005/20m/200w/1d]

29. 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. [2004/60m/4]

30. 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? [2000/20m/200w/5c]

31. A departmental store wishes to procure the following readymade garments:

Type of Garment	A	B	C	D	E
Quantity	300	200	150	500	400

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

Manufacturer	I	II	III	IV
Total Quantity	600	500	300	400

The departmental store estimates that its profit per garment will vary with the manufacturer as shown in the matrix below: [Profit-per-garment matrix (manufacturers I–IV × garment types A–E) missing in every available source] [?] How should orders be placed? [1999/60m/8]

32. Make a Linear Programming Model for this problem. [1998/30m/1a]

33. 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. [1998/30m/1b]

34. 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.

Product	Machine I	Machine II
A	0.5	0.6
B	0.7	0.8
C	0.9	1.05

There are 85 hours available on each machine; the operating cost is Rs. 5 per hour for machine I and Rs. 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. Solve the given linear programming problem by the simplex method. [1997/60m/6]

35. If $X_1,X_2,X_3,X_4$ and $X_5$ respectively represent the tons to products P1, P2, P3, P4 and P5 to be produced per day test whether the following solutions are feasible: (i) $X_1=8.4,X_2=0,X_3=12.0,X_4=0,X_5=2.5$ (ii) $X_1=5.0,X_2=11.0,X_3=0,X_4=19,X_5=5.0$ [?] (iii) $X_1=10,X_2=12.0,X_3=4.2,X_4=4.0,X_5=5.0$ (iv) $X_1=6,X_2=9,X_3=13.0,X_4=1.0,X_5=2.5$ [1995/20m/3a]

36. Test whether any of the above solutions would maximize total contribution. [1995/20m/3b]

37. 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? (i) Do not promote P1 or P2 (ii) Promote P1 but not P2 (iii) Promote P2 but not P1 (iv) Promote P1 and P2 both. [1995/20m/3c]

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Last updated: 30 October 2026