BBA COMMERCEF.Y.B.Com. Mathematics
COMMISSION, BROKERAGE, DISCOUNT
AND PARTNERSHIPUNIT II
3
LINEAR PROGRAMMING PROBLEMSWe find the values of the objective function at these vertices:
z (O) = 800 × 0 + 100 × 0 = 0
z
...
BBA COMMERCEF.Y.B.Com. Mathematics
COMMISSION, BROKERAGE, DISCOUNT
AND PARTNERSHIPUNIT II
3
LINEAR PROGRAMMING PROBLEMSWe find the values of the objective function at these vertices:
z (O) = 800 × 0 + 100 × 0 = 0
z(P) = 800 × 15 + 100 × 10 = 13000
z (C) = 800 (18) + 100 × 0 = 14400 z
(B) = 800 × 0 + 100 × 20 = 2000
z has the maximum value 14400 at the point C (18, 0).
the manufacturer should produce 18 scooters and 0 bicycles, in
order to have maximum profit of Rs. 14400.
Example5:
Minimize z = 40x + 37y
Subject to constraints:10x + 3y 180
2x + 3y 60
x 0, y 0
Solution:
We first draw the line AB and CD whose equation are
10x + 3y = 180 and 2x + 3y = 60.
For the line 10x + 3y = 180 For the line 2x + 3y = 60
The feasible region is shaded in the figure. Its vertices are C (30,
0), B (0, 60) and E, where E is the point of intersection of the lines
AB and CD.
y
60- B (0, 60)
50 -
x y Points
18 0 (18,0)
0 60 (0,60)
x y Points
30 0 (30,0)
0 20 (0,20)Maths_stats_fybcom 62
40-
30-
20-D E(15,10)
10-
(0, 0) 10 20 30 A(30,0)
2x + 3y ≥ 60
10x + 3y ≥ 180
For the point E, we solve the two equations simultaneously.
10x + 3y = 180
2x + 3y = 60
- - -
8x = 120
x =15
2x + 3y = 60 gives 30 + 3y = 60,
i.e., 3y = 30, i.e. y = 10
E is (15, 10)
The values of the objective function z = 40x + 37y at these vertices are:
Vertex (x, y) z = 40x + 37y
C (30, 0) z (C) = 40 × 30 + 37 × 0 = 1200
B (0, 60) z (B) = 40 × 0 + 37 × 60 = 220
E (15, 10) z (E) = 40 × 15 + 37 × 10 = 600 + 370 = 970.
z has minimum value 970 at the point E (15, 10), where x = 15 and
y = 10.
Example6: Minimize z = 4x + 2y
Subject to constraints:x + 3y 3, 2x + y 2, x 0, y 0
Solution:
We first draw the line AB and CD .
For the line x + 3y = 3 For the line 2x + y = 2
The feasible region is shaded in the figure. Its vertices are A (3,
0), D (0, 2) and E, where E is the point of intersection of the lines
AB and CD.
y
3- B (0, 60)
2-D (0,2)
1- B E(0.6,0.8)
x y Point
3 0 A(3,0)
0 1 B(0,1)
x y Point
1 0 C(1,0)
0 2 D(0,2)Maths_stats_fybcom 63
(0, 0) 1 2 3 A(3,0) x
2x + y = 2 x + 3y = 3
To find the point E, we solve the two equations(1) and (2)
simultaneously.
2x + 6y = 6
2x + y = 2
- - -
5y = 4
x = 0.8
4 5
Substituting y = 0.8 in (1), we get,
x + 3 × 0.8 = 3
x + 2.4 = 3
x = 3 – 2.4 = 0.6
Thus E is (0.6, 0.8)
The values of the objective function z = 4x + 2y at the vertices are
calculated below:
Vertex (x, y) z = 4x+ 2y
C (3, 0) z (A) = 4× 3 + 2× 0 = 12
E (0.6, 0.8) z (E) = 4 × 0.6 + 2 × 0.8 = 2.4 + 1.6 = 4
D(0, 2) z (D) = 4 × 0+ 2 × 2= 4
We can see the minimum value of z is 4, at two vertices E(0.6, 0.8) and
D(0, 2). Thus z is minimum at any point on the line segment ED.
Exercise
Maximize:
1) z = 5x + 10y, subject to
5x + 8y 40, 3x + y 12, x 0, y 0
2) z = 7x + 6y, subject to
2x + 3y 13, x + y 5, x 0, y 0
3) z = 5x + 3y, subject to
2x + y ≤ 27, 3x + 2y ≤ 48, x ≥ 0, y ≥ 0
4) z = 20x + 30y, subject to
x + y 6, 3x + y 12, x 0, y 0
5) z = 6x + 7y, subject to
2x + 3y 12, 2x + y 8, x 0, y 0
6) z = 30x + 20y, subject to
2x + y 20, x + 3y 15, x 0, y 0
7) z = 20x + 25y, subject to
5x + 2y 50, x + y 12, x 0, y 0
8) z = 90x + 130y, subject to
2x + 3y 18, 2x + y 12, x 0, y 0
Minimize:Maths_stats_fybcom 64
1) z = x + 4y, subject to
x + 3y 3, 2x + y 2, x 0, y 0
2) z = 5x + 2y, subject to
10x + 2y ≥ 20, 5x +5 y 30, x 0, y 0
3) z = 9x + 10y, subject to
x + 2y 30, 3x + y 30, x 0, y 0
4) z = 50x + 55y, subject to
x + 3y 30, 2x + y 20, x 0, y 0
5) z = 80x + 90y, subject to
6x + 5y 300, 2x +3y 120, x 0, y 0
6) z = 13x + 15y, subject to
3x + 4y 360, 2x +y 100, x 0, y 0
7) z = 12x + 20y, subject to
x + y 7, 5x +2y 20, x 0, y 0
Answer:
Maximize:
1) Max. value is 50 at the point (0,5)
y
12-D
10-
8-
6-B
4- P
19
60
,
19
56
2- C A
(0, 0) 1 2 3 4 5 6 7 8 x
5x + 8y = 40
3x + y = 12
2) Max. value is 35 at the point (5,0)
y
6- D
5 - B
4- P(2,3)
3-
2-
6- C A
(0, 0) 1 2 3 4 5 6 x
x + y = 5 2x + 3y = 13
3) Max. value is 75 at the point. (6,15)
y
30- B(0,27)
25 - D (0,24)
20-
15- E(6,15)
10-
5- CMaths_stats_fybcom 65
(0, 0) 5 10 15 20 x
A(13.5,0) 3x + 2y =48
2x + y = 27
4) Max. value is 180 at the point (0,6)
y
12-D(0,12)
10-
8-
6-B
4- E(3,3)
2- C(4,0) A(6,0)
(0, 0) 2 4 6 8 x
x + y = 6
3x + y = 12
5) Max. value is 32 at the point (3,2)
y 8-D(0,8)
7-
6- 2x + y = 8
5 -
4-B(0,4)
3-
2- E(3,2) 2x + 3y = 12
1- C (4,0) A(6,0)
(0, 0) 1 2 3 4 5 6 x
6) Max. value is 310 at the point (9,2)
y
30-
25 -
20-B(0,20)
15-
10-
5- D(0,5) E(9,2) A(10,0)C(15,0)
(0, 0) 5 10 15 20 x x + 3y =15
2x + y =20Maths_stats_fybcom 66
7) Max. value is 300 at the point (0,12)
y
30-
25 - B(0,25)
20- 5x + 2y = 50
15-D(0,12)
10- E
3
10
,
3
26
5- C(12,0)
(0, 0) 5 10 15 20 x
A(10,0) x + y = 12
8) Max. value is 795 the point (4.5, 3)
y
14-
12-D
10- 2x + y = 12
8-
6-B(0,6)
4- E(4.5,3)
2- 2x + 3y = 18
(0, 0) 2 4 6 8 A 10
x
C(6,0)
Minimize:
1) Min. value is 3 at the point (3,0)
y
3-
2- D
1-B P
4 5
3 5
C A
(0, 0) 1 2 3 x + 3y ≥ 3 x
2x + y ≥ 2
2) Min. value is 15 at the point (1,5)
y
10-B
-
8-Maths_stats_fybcom 67
-
6-D
- P(1,5)
4-
- 5x + 5y = 30
2-10x + 2y = 20
-\ A C
(0, 0) 2 4 6 8 A 10
x
3) Min. value is 174 at the point (6,12)
y
30- D(0,30)
25 - 3x + y = 30
20-
15-B(0,15)
10- E(6,12) x + 2y = 30
5- C(10,0) A(30,0)
(0, 0) 5 10 15 20 25 30 x
4) Min. value is 740 at the point (6,8)
y
30-
25 -
20-D(0,20)
15- 2x + y = 20
B(0,10) 10- E(6,8)
5- C(10,0) A(30,0)
(0, 0) 5 10 15 20 25 30 x x + 3y = 30
5) Min. value is 4350 at the point (37.5,15)
y
60- B(0,60)
50 -
40-D(0,40)
30-
20- 2x + 3y = 120 E(37.5,15)
10- A(50,0) C(60,0)
(0, 0) 10 20 30 40 50 60 x
6x + 5y = 300
6) Min. value is 1364 at the point (8,84)
y
120-
100 - D(0,100)
80- E(8,84)
60-
40-
20- C AMaths_stats_fybcom 68
(0, 0) 20 40 60 80 100 120 x
3x + 4y = 360 2x + y = 100
7) Min. value is 84 at the point (7,0)
y
10-D
9 -
8-
7-B
6-
5 - P(2,5)
4-
3 -
2-
1 - C A
(0, 0) 1 2 3 4 5 6 7
x
5x + 2y = 20 x + y = 7
Maths_stats_fybcom 69
UNIT III
4
INTRODUCTION TO STATISTICS AND
DATA COLLECTION
OBJECTIVES:
The objectives of this chapter are to give an overview of:
1. What is statistics and why one should learn it?
2. How did it originate?
3. Scope and limitations.
4. Definition of some basic terms that are used in the subject.
5. Types of data and how they are collected.
6. How should the data be arranged?
4.1 INTRODUCTION
In almost all areas of our daily living in which we make simple
statements we are actually doing what one may call statistical thinking.
What may seem like a simple ―I walk on an average 4 kms per day‖ is a
statement which involves statistics. ‗There are 60 percent chances that a
particular political party will get reelected at the next election‖, ―This year
it was much hotter than the same month previous year‖, are all examples
of the common man doing statistics without realizing it. Due to the simple
capacity to observe people, things and event in our environment we notice
similarities and differences and look for patterns and regularities as part of
a survival mechanism to sense danger or opportunity in an effort to grow
wiser and understand our environment better. The simple act of
observation results in us getting data. The word data means fact, and is
plural of the word datum. This in turn makes us look for connections
among the data that we have noticed. While most of us make connections
and make conclusions which affects us and others around us the question
that we must ask is ‗How mature or informed our conclusions are?‖, ―How
can we be sure that our conclusions are factually correct?‖ These
questions also lead us to ask ―Is there a scientific, rational way of
determining whether our conclusions are valid? Can we use structured,
methodical, time tested ways to make these conclusions?‖ The remaining
part of this chapter is devoted to answer these questions.
4.1.1 What is statistics?
In its simplest sense statistics refers to the science of collecting
data, assortment of data and analyzing them to reach a definite conclusion.
The methods by which the data are analysed are called as StatisticalMaths_stats_fybcom 70
Methods.
The word statistics seems to have originated from the German
word ―statistik‘, the Italian word ‗statistica‘ or the Latin word ‗status‘ all
meaning ‗state‘ in the political sense of the word. This in turn seems to
have come because statistics was widely used initially to collect data for
the state ( Government) so that officials could use the data for better
planning in the future. Statistical thinking however has a long history as
from very early times kings and rulers have been collecting data about
their populations and resources.
4.1.2 Functions of Statistics
Following are the important functions of statistics
1. It represents data in a definite form
One of the important function of statistics is that it enables one to
make statements which are precise and in quantitative terms. To say
that India is a overpopulated country, or that a country has high level
of poverty are all general statements but do not convey any precise
meaning. Words such as ―high‖, ―low‖, ―good‖, ―bad‖, are very vague
and subject to interpretation by different people in different manner.
Statements of facts made in exact quantitative terms are more
convincing.
2. It simplifies complex data or a mass of figures.
Statistics not only helps one to express data in a concrete definite
form but also reduces the data to a few significant figures which give
the essence of the issue under consideration.
3. It facilitates comparison.
Generally plain facts by themselves have little value unless they are
seen in the correct context in which they occur. This requires that we
place the facts in relation to facts of similar type but in other
situations. For example the fact that the population of a country is 12
million by itself would have a restricted meaning but in the context of
how big the geographical area is and what are the populations of other
similarly placed countries would throw light on some very important
aspects of population density and other issues such as crowding and
pollution.
4. It helps in forecasting.
Statistics helps in predicting what may happen in future on the basis of
past data and analysis. It is useful for anybody in business to have an
idea of what might be the possibility of selling goods or services so
that the person can plan for any future changes in demands. Production
Planning, inventories are all based on statistical modes which help by
predicting excess or shortfalls in demands.
5. It helps in formulating policies
Due to its potential to do forecasting statistics helps policy makers inMaths_stats_fybcom 71
drafting policies which help in better governance. The annual finance
budget relies heavily on statistical methods to decide issues of direct
and indirect taxation and budgetary allocations based on past data and
its analysis.
4.1.3 Scope of Statistics
There has been hardly any area whether from the exact
sciences or social sciences where statistics has not been applied, whether it
be trade, industry, commerce, economics, life sciences, education where
statistical methods have not been applied. However certain fields have
used statistics very frequently and effectively. We list some of the
important fields.
1. State : As we have stated earlier from very early times statistics have
been used by governments in framing policies on the basis of data
about population, military, crimes, education etc The present day
governments have special departments which maintain a variety of
data of significance to the state. The extensive proliferation of
technologies such as the internet has made huge amount of data
available making statistical tools absolutely indispensible.
2. Business : With the advent of globalization and the extensive range of
operations Business has grown almost exponentially in many areas.
Business now extends across geographical and political boundaries.
The large amount of data that it generates is used for analysis and
forecasting of consumption patterns and has given rise to advance
techniques such as Data Mining.
4.1.4 Limitations of statistics
Although there are many areas where statistics is useful there are certain
limitations also as there are areas where it is not applicable and there are
possibilities of its misuse. We indicate some of them as under.
1. Statistics does not deal with individual measurement.
Statistical measurements are generally mass measurements and it deals
with a collection of data and not any particular measurement. In fact a
particular measurement can be very different form an average.
2. Statistics deals with quantitative characteristics.
Statistics are numerical statement of facts. There are many aspects of
life which cannot be quantified and are therefore not within the
purview of statistics. For e.g. The honesty or integrity of person,
whether a person is affectionate or not, whether or not a person is
intelligent are some qualities which are not easily quantifiable and not
available for statistical analysis.
3. Statistical results are true on an average.
Unlike exact sciences in which there is a clear cause effect relationship
in the phenomena which is being studied, in statistics the results areMaths_stats_fybcom 72
true only on an average. Any individual element in the data collected
may not have any resemblance with the average.
4. Statistics is one of the many methods of studying phenomena.
A statistical study is not complete in itself. There are many ways in
which phenomena can be studied and one of it is statistics. Generally
statistics complements other methods to substantiate the understanding
of phenomena.
5. Statistics can be misused.
Perhaps the greatest limitations of statistics are that it can be misused.
Many people use statistics to prove a point to which they have already
subscribed. For e.g if a person wants to prove that malnutrition deaths
have been high in an area then the researcher can deliberately collect
data that will favour that conclusion and ignore the data that will not
be favorable. Although there are ways in which this bias can be
detected an intelligent researcher can easily mislead people. One of the
reasons why one should learn statistics to understand how one can be
mislead by such people and be cautious.
4.2 BASIC STATISTICAL CONCEPTS.
In using statistics it is necessary that we have a clear understanding of the
words that are used. We give some definitions of some of the very basic
concepts in statistics.
4.2.1 Data
The word data is plural of the word datum which in Greek means fact. It
is a collection of observations expressed in numerical quantities. Data is
always used in the collective sense and not in singular.
4.2.2 Population
The word population in statistics means the totality of the set of objects
under study. It should not be understood in the limited sense in which it is
generally used to mean people in a certain city or country.
4.2.3 Sample
A sample is a selected number of entities or individuals which form a part
of the population under study. The study of a sample is more practical and
economical in most situations where the population is large and is used to
make conclusions about the entire population.
4.2.4 Characteristic
The word characteristic means an aspect possessed by an individual entity.
We may study the rainfall of a certain region, or the marks scored byMaths_stats_fybcom 73
students in a certain school. These are referred to as characteristics.
4.2.5 Variables and attributes.
In statistics characteristics are of two types. Measurable and nonmeasurable. Measurable characteristics are those that can be quantified as
expressed in numerical terms. The measurable characteristics are known
as variables. A non-measurable characteristic is qualitative in nature and
cannot be quantified. Such a characteristic e.g nationality, religion, etc are
called as attributes.
4.3.1 Collection of Data
To apply statistical methods or to study a problem we must collect data.
Collection of data is very important as it forms the basis of the analysis. It
is important to understand the techniques of collection of data because if
the data is not collected properly its reliability itself becomes questionable
and our entire analysis will be on weak foundations.
Based on whether the investigator collects data himself/herself or uses
data collected by some other person or agency data is classified into two
types. Primary data and Secondary data.
1. Primary Data
Primary data are those which are collected directly from the field of
enquiry for a specific purpose. This is raw data original in nature and
directly collected from the population. The collection of the data can
be made through two methods. a) Complete enumeration or census
method or b) Sampling survey methods.
The complete enumeration or census method is a study of the entire
population and data are collected about each individual of the
population. Generally this is a laborious, time consuming and
expensive method. Large organizations and Government semi -
government organizations, Research institutions and Public sector
bodies such as the RBI ( Reserve Bank Of India ) can afford such
methods of collecting data.
2. Secondary Data
Secondary data are such information which has already been collected
by some agency for a specific purpose and is subsequently compiled
by the investigator from that source for application in a different area.
Data used by any other person or agency other than the one which
collected it constitutes secondary data. The same data is primary when
collected by the source agency and becomes secondary when used by
any other agency. Data after analysis are also termed as secondary
data.
4.3.2 Collection of primary data
Primary data can be collected by the investigator in following ways.Maths_stats_fybcom 74
i) By direct personal observation
The investigator may collect data by direct observation. This can
be done by meeting and interrogating people who may supply the desired
information. Such data is directly obtained and can be very reliable but it
is a time consuming and costly process.
ii) By indirect oral investigation
Here the information is collected not by questioning the concerned
people but by asking people connected with the concerned people. These
people can be called as witnesses who have knowledge about the persons
concerned or situation involved. Here the investigator has the added
responsibility of ensuring that the witnesses are not biased and therefore
the reliability of data cannot be questioned. Sometimes this can be the
only way to get the desired information as the people themselves are either
not accessible or not willing to give the information.
iii) By sending questionnaires by mail or email
A questionnaire is a proforma containing a set of questions. A collection
of questions relevant to the area of study is created and sent by post or
email to selected people with a request to fill up and returned by post or
email. This method can cover a large population economically but does
not necessarily result in good response. The questionnaire has also to be
carefully drafted so as to not have leading questions which direct the
person to an answer desired by the investigator.
iv) By sending schedules through paid investigation
This method is used quite widely particularly by market researchers. Here
schedules are prepared and the investigators are trained to meet people
concerned with the schedules. A schedule is a form where information is
to be noted by an enumerator who questions people. The investigators are
to fill up the schedules on the basis of answers given by the respondents.
The success of this method largely depends on how efficient the
investigators are and how tactfully they collect the needed data.
4.3.3 Collection of Secondary Data
Secondary data are those which are collected by some other agency and
are used for further investigation. The sources of secondary data can be
classified into two:
a) Published sources b) Unpublished sources
Published Sources
Some of the published sources providing secondary data are:
a. Government Publications: Government, semi-government and private
organizations collect data related to business, trade, prices, consumption;
production, industries, income, health, etc. These publications are very
powerful source of secondary data. Central Statistical Organization
(C.S.O.), National Sample Survey Organization (N.S.S.O.), office of theMaths_stats_fybcom 75
Registrar, and Census Commissioner of India, Directorate of Economics
and Statistics and Labour Bureau-Ministry of labour are a few government
publications
b. International Publications : Various governments in the world and
international agencies regularly publish reports on data collected by them
on various aspects. For example, U.N.O.‘s Statistical Year book,
Demography Year Book.
c. Semi-official Publications : Local bodies like District Boards,
Municipal Corporations publish periodicals providing information about
vital factors like health, births, deaths etc.
d. Reports of Committees and Commissions : At times state and central
governments appoint committees and commissions with a specific
reference to study a phenomenon. The reports of these committees and
commissions provide important secondary data. For example, Kothari
commission report on education reforms, Report of National Agricultural
Commission.
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