Mth426 group13 final_report

1. MTH426A: Introduction to Mathematical Modeling
Project Report: Indian Education System
Group: 13
Instructor: Prof. Prawal Sinha
Group members:
1. Dhruv Roosia (13252)
2. Jitendra Gawariya (13327)
3. Keshaw Singh (13347)
4. Kuldeep Soni(13362)
5. Nakul Surana (13418)
Aim
• Making Model of Indian Education System.
• Analysis of female to male ratio in skilled labor.
• Analysis of total skilled workforce in India to meet future needs.
Introduction
Quality education is the major driving force behind a nation’s development. India currently faces
a severe shortage of well-trained, skilled workers. Even among the educated workforce, there are
sections with little or no job skills. This makes them largely unemployable. Only 2.3% of Indians
are employable as compared to more than 50% in several other countries. Therefore, India must
focus on scaling up education quality and removing gender bias to meet the demands of
employers and drive economic growth. Our nation, with more than 50% of its population under
25 years, has a lot of potential to create an impact on the world map. A well-trained workforce
needs to be the driving force behind that.
System Characterization
Objects:
Residents of India
- Since we are aiming to predict the strength of workforce in India.
System:

2. (i) Open
- There is interaction within the residents as well as with other elements of the society
(ii) Deterministic
- The values assumed by the variables and the changes in the variables are predictable with
certainty
(iii) Dynamic
- The variables being considered are time-dependent since population and the related
variables change with time
(iv) Discrete
- Time element is treated as discrete since data related to all the variables is known at
discrete time instants only
(v) White box
- We explicitly express the relation between the variables and the objects of the system
Environment:
Society
Variables:
Symbol Description
Ni, y Number of live births in the year ‘y’
di, e, y Dropout rate at elementary level (classes 1-8) for year ‘y’
di, s, y Dropout rate at secondary level (classes 9-12) for year ‘y’
pfy College performance factor for year ‘y’
gi, y (= GERi, y) Gross enrollment ratio for colleges in the year ‘y’
GBy General budget for year ‘y’

3. SLf, y Strength of skilled female labor in the year ‘y’
SLm, y Strength of skilled male labor in the year ‘y’
Vk, y Vocational training factor for the year ‘y’
i = m (male), f (female)
Dropout rates at the elementary and secondary level are directly obtainable from authentic
sources. Similarly, the gross enrollment ratio and the data regarding the General Budget and its
distribution can be obtained. For calculating the values of variables like performance factor and
social bias (in case of women), we construct some appropriate self-defined metrics which are
included in the discussion that follows.
Problem Formulation
Assumptions
• For elementary education, average of primary and upper primary gives the dropout.
• Similarly, those of high and senior secondary for secondary education.
• Social bias, performance factor and vocational training factor have been approximated by
self-constructed metrics.
e.g. – social bias based on literacy rate.
In order to calculate number of skilled female and male labor in India. We designed the
illustrated flow diagram –

4. So the final formula for the strength of male and female labor, respectively, are –
(i) SLm, y = Ny-22 (1 - d1, e, y) (1 - d1, s, y) [gm, y-4 * pfy + (1 - gm, y-4) GBy-4 / (1 - (d1, e, y-12) (d1, s, y-4))]
(i) SLf, y = Ny-22 (1 - d2, e, y) (1 - d2, s, y) [gf, y-4 * pfy + (1 - gf, y-4) GBy-4 / (1 - (d2, e, y-12) (d2, s, y-4))]
Logic – Let us understand this while taking a particular year say 2000. So number of skilled
workers in the year 2000 is considered as a sum of students graduated from colleges and students
graduated from vocational training institutes in the same year.
Calculate number of students graduating from colleges in the year ‘Y’ –
Nm, Y-22 * (1-dm, e , (Y-10)) * (1-dm, s, (Y-4)) * GERm, (Y-4) * pfY
where -
Nm, Y-22 - number of male births in the year Y-22 (2000-22 =1978), .
year when the person is born
dm, e, (Y-10) - dropout rate for male in the year Y-10 (2000-10 =1990) i.e. in 5th standard
the year when the person is of 10 years (5
th
Standard)
dm, e, (Y-4) - dropout rate for male in the year Y-4 (2000-4 =1996) i.e. in 12th standard
A year when a labor in of 18 years (12
th
Standard)
GERm,(Y-4) - gross enrollment ratio of Indian colleges in the year Y-4 (2000-4 = 1996)
A year when a labor entered in college
pfY - performance factor of colleges, a indicator which reflects the number of polished
individuals prepared for market.
Calculate number of students given vocational training in the year ‘Y’ –
Nm, Y-22 * (1-dm, e, (Y-10))(1-dm, s, (Y-4)) * (1- GERm, (Y-4)) * Vk
where -
Vk, Y = gain by vocational training formulated as -
GBY-4
(1-dm, e, (Y-10))(1-dm, s, (Y-4))
GBY-4 = General Budget in the year Y-4 (2000-4 = 1996)

5. *Same calculation can be done for females as well.
Objective in mathematical terms -
(i) max. f1 = SLm, Y/SLf, Y , i.e., maximizing female by male labor ratio
(ii) max. f2 = SLm, Y + SLf, Y , i.e., maximizing the strength of total skilled workforce
With respect to given restrictions.
Restrictions:
0 <= pf <= 1 i = 1 for Male
0 <= d <= 1 i = 2 for Female
0 <= g <= 1
0 <= Vk <= 1
GB >= Infrastructure +Incentive + Training
0 <= GB <= 100
Analysis
Here the aim boils down to maximizing two required functions. For this here we are using
Lagrangian function, it works as follows -

6. Data Point table:-
Year Male Labor Female Labor Ratio Total Labor
1972 5711.2868 388.4313385 0.068011177 6099.718139
1973 6242.755288 586.308024 0.093918149 6829.063312
1974 6809.272535 452.5713805 0.066463984 7261.843915
1975 7417.902743 834.896856 0.112551605 8252.799599
1976 8063.086759 522.7789065 0.064836076 8585.865665
1977 8745.938915 1082.292878 0.123748049 9828.231793
1978 9467.582299 598.7699462 0.063244229 10066.35225
1979 10229.19723 1274.255366 0.124570417 11503.4526
1980 11030.27687 1083.176275 0.09820028 12113.45315
1981 12481.73402 1239.059768 0.099269842 13720.79379
1982 14013.59095 1287.475826 0.09187337 15301.06678
1983 15673.00866 1836.324474 0.117164771 17509.33313
1984 17462.43024 1507.046607 0.086302226 18969.47685
1985 19753.12774 2475.180659 0.125305759 22228.3084
1986 22255.94616 1821.939811 0.081863058 24077.88598
1987 24986.4374 3203.269205 0.128200317 28189.7066
1988 27966.1022 2175.870773 0.077803863 30141.97297
1989 31217.38512 3905.650034 0.125111377 35123.03516
1990 34768.66489 3754.63775 0.107989126 38523.30264
1991 38386.00359 4450.454836 0.11593952 42836.45843
1992 42295.11851 4241.994971 0.100295143 46537.11349
1993 46500.41914 5638.041745 0.121247117 52138.46089
1994 51004.157 4772.68913 0.093574513 55776.84613
1995 55528.5473 6827.157529 0.122948607 62355.70483
1996 60326.27765 5375.20781 0.089102262 65701.48546
1997 65432.71203 7817.719488 0.119477235 73250.43152
1998 70898.67113 6021.952823 0.084937457 76920.62395
1999 75393.27376 9067.024425 0.120263042 84460.29818
2000 83048.77541 14500.65281 0.174604053 97549.42822
2001 77068.99823 13601.04471 0.176478805 90670.04294
2002 81479.90237 13681.05804 0.167907148 95160.96041
2003 79320.69909 14208.31399 0.179124921 93529.01308
2004 79946.74569 12867.14301 0.160946426 92813.88871
2005 85728.21433 15615.80445 0.182154785 101344.0188

7. 2006 93362.4356 15134.70243 0.162106979 108497.138
2007 94016.91254 17368.41834 0.18473717 111385.3309
2008 102212.6128 16594.14662 0.162349305 118806.7594
2009 118138.7439 22838.10765 0.193315985 140976.8516
2010 118295.4829 28478.3322 0.240738966 146773.8151
2011 119406.6363 29423.25355 0.246412213 148829.8899
2012 150928.1917 32152.54871 0.213032094 183080.7404
2013 161000.178 38195.11571 0.237236481 199195.2937
2014 165126.637 39046.84354 0.23646605 204173.4806
Differentiating the former L1 and L1 function w.r.t. corresponding variables and substituting
other variables values for different years, we get the following result -
Plots: -

8. Regression:-
As we can see from the above data, the two most influencing factors are – Dropout rate at
elementary level and College Performance index.
We believe that in order to uphold the Indian society we have to put tremendous workforce on
improving primary education, i.e. to decrease the dropout rate. This is is also well supported by
the corresponding data points.
With the help of regression analysis done on the dropout rates (di,e,y) with respect to incentives,
infrastructure, teacher training, pupil teacher ratio and social bias, we observe the following as
highly significant:
- Teacher training
- Social bias (especially against girl child)
ANOVA Table and accuracy of model
Regression Statistics
Multiple R 0.990681
R Square 0.981449
Adjusted R Square 0.979877
Standard Error 2.532314
Observations 65
ANOVA
df SS MS F
Significan
ce F
Regression 5
2001
7
400
3
624.28
9
1.0305E-4
9
Residual 59 378.3 6.41
Total 64
2039
5
Coefficient SE t Stat P-value
Intercept 69.26921 8.359 8.29 1.8E-11
X Variable 1 (Incentive) -0.00183 2E-04 -7.46 4.6E-10
X Variable 2 (Teacher pupil
ratio) -0.00201 3E-04 -6.13 8E-08

9. X Variable 3 (Infrastructure) -0.04029 0.007 6.01 1.2E-07
X Variable 4 (Teacher
Training) 0.110598 0.157 0.71 0.48307
X Variable 5 (Social bias) 0.328256 0.058 5.68 4.4E-07
Plot of Residuals against the Fitted Values yˆi :
Plot of the (preferably the externally studentized residuals i ) versus the corresponding fitted
values yˆi is useful for detecting several common types of model inadequacies -
Plot
Seems
Fitted
In the
Horizontal Band
Plot between calculated and given strength of skilled labor (Checking Model Accuracy)

10. Our formulation provides results in phase with the actual data points and proportional in most of
the areas, we could also precisely do the same with the help of dummy variable.
Results and Discussion
Differentiating and studying our objective functions f1 and f2 with respect to the variables, we
obtain a priority order for the variables as follows:
1) di, e, y = Dropout rate at elementary level for the year ‘y’
2) pfy = College performance factor for the year ‘y’
3) di, s, y = Dropout rate at secondary level for the year ‘y’
4) GBy = General budget of the year ‘y’
5) gi, y = Gross enrollment ratio for colleges in the year ‘y’
With the help of regression analysis done on the dropout rates (di, e, y) with respect to amount
spent on incentives, infrastructure, teacher training, pupil teacher ratio and social bias, we
observe the following factors as being highly significant:
• Teacher training
• Social bias (specially against a girl child)
In order to attain our goal of making India skilled, we have to focus on these factors.
Conclusion
This project gives us an insight into the future employment needs of the nation for skilled
workforce, both female and male. The estimates given here are evidently way off the true
figures.
• Performance factor could not be quantified properly due to lack of data.
• Vocational training gain factor is taken as one which is difficult to achieve in real world.
But with more in-depth analysis and more informative data, more useful inferences about the
future could be made.
• Analysis can be done by considering each sector like service, manufacturing, etc.
• Cross validation of the model with sufficient data may give further insights.

11. References
[1] Skill Development in India. Available at:
http://www.kas.de/wf/doc/kas_42848-1522-2-30.pdf?151016072126
[2] India Fertility Crude birth rate, 1950-2015. Retrieved from:
https://knoema.com/atlas/India/topics/Demographics/Fertility/Crude-birth-rate
[3] Eckner, Andreas (2012). "A framework for the analysis of unevenly spaced time series data."
Preprint. Available at: http://www.eckner.com/papers/unevenly_spaced_time_series_analysis .
[4] National Level Educational Statistics, MHRD, Govt. of India. Retrieved from:
http://mhrd.gov.in/statist
[5] World Bank Open Data, The World Bank. Retrieved from:
http://data.worldbank.org/