Quantitative Techniques

Quantitative Techniques Barhate Mangesh Roll No- PG/509/MBA (I)/2009J

Name of Student

Mr. Barhate Mangesh Tukaram

Roll No

PG/509/MBA(I)/2009J

Institute

Silver Bright Institute of Management (SBIM), Pune

Subject

Quantitative Techniques

Date

10 Jan 2010

QUANTITATIVE TECHNIQUES

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INDEX

- Preamble - Methodology of Quantitative Techniques - Set Theory -

Logarithms Linear Programming Matrix Data Analysis Mean Median Mode Dispertipn Skuewness Deviation

-

Measures of Center and Location Measures of Variation Methods of Time Series Wrapping Up

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Preamble Scientific methods have been man’s outstanding asset to pursue an ample number of activities. It is analyzed that whenever some national crisis, emerges due to the impact of political, social, economic or cultural factors the talents from all walks of life amalgamate together to overcome the situation and rectify the problem. Quantitative techniques had facilitated the organization in solving complex problems on time with greater accuracy. The historical development will facilitate in managerial decision-making & resource allocation, The methodology helps us in studying the scientific methods with respect to phenomenon connected with human behavior like formulating the problem, defining decision variable and constraints, developing a suitable model, acquiring the input data, solving the model, validating the model, implementing the results. The major advantage of mathematical model is that its facilitates in taking decision faster and more accurately. Managerial activities have become complex and it is necessary to make right decisions to avoid heavy losses. Whether it is a manufacturing unit, or a service organization, the resources have to be utilized to its maximum in an efficient manner. The future is clouded with uncertainty and fast changing, and decisionmaking – a crucial activity – cannot be made on a trial-and-error basis or by using a thumb rule approach. In such situations, there is a greater need for applying scientific methods to decision-making to increase the probability of coming up with good decisions. Quantitative Technique is a scientific approach to managerial decision-making. The successful use of Quantitative Technique for management would help the organization in solving complex problems on time, with greater accuracy and in the most economical way. Today, several scientific management techniques are available to solve managerial problems and use of these techniques helps managers become explicit about their objectives and provides additional information to select an optimal decision.

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Methodology of Quantitative Techniques

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Scope of Quantitative Technique The scope and areas of application of scientific management are very wide in engineering and management studies. Today, there are a number at quantitative software packages available to solve the problems using computers. This helps the analysts and researchers to take accurate and timely decisions. This book is brought out with computer based problem solving. A few specific areas are mentioned below. 1. Finance and Accounting: Cash flow analysis, Capital budgeting, Dividend and Portfolio management, Financial planning. 2. Marketing Management: Selection of product mix, Sales resources allocation and Assignments. 3. Production Management: Facilities planning, Manufacturing, Aggregate planning, Inventory control, Quality control, Work scheduling, Job sequencing, Maintenance and Project planning and scheduling. 4. Personnel Management: Manpower planning, Resource allocation, Staffing, Scheduling of training programmers. 5. General Management: Decision Support System and Management of Information Systems, MIS, Organizational design and control, Software Process Management and Knowledge Management. From the various definitions of Quantitative Technique it is clear that scientific management hen got wide scope. In general, whenever there is any problem simple or complicated the scientific management technique can be applied to find the best solutions. In this head we shall try to find the scope of M.S. by seeing its application in various fields of everyday lift this include define operation too.

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Set Theory Set Theory is the mathematical science of the infinite. It studies properties of sets, abstract objects that pervade the whole of modern mathematics. The language of set theory, in its simplicity, is sufficiently universal to formalize all mathematical concepts and thus set theory, along with Predicate Calculus, constitutes the true Foundations of Mathematics. As a mathematical theory, Set Theory possesses a rich internal structure, and its methods serve as a powerful tool for applications in many other fields of Mathematics. Set Theory, with its emphasis on consistency and independence proofs, provides a gauge for measuring the consistency strength of various mathematical statements. There are four main directions of current research in set theory, all intertwined and all aiming at the ultimate goal of the theory: to describe the structure of the mathematical universe. They are: inner models, independence proofs, large cardinals, and descriptive set theory. The following basic facts are excerpted from “Introduction to Set Theory,” Third Edition, by Karel Hrbacek and Thomas Jech, published by Marcel Dekker, Inc., New York 1999. 1. Ordered Pairs 2. Relations 3. Functions 4. Natural Numbers 5. Cardinalities of Sets 6. Finite Sets 7. Countable Sets 8. Real Numbers 9. Uncountable Sets Applications Nearly all mathematical concepts are now defined formally in terms of sets and set theoretic concepts. For example, mathematical structures as diverse as graphs, manifolds, rings, and vector spaces are all defined as sets having various (axiomatic) properties. Equivalence and order relations are ubiquitous in mathematics, and the theory of relations is entirely grounded in set theory. Set theory is also a promising foundational system for much of mathematics. Since the publication of the first volume of Principia Mathematica, it has been claimed that most or even all mathematical theorems can be derived using an aptly designed set of axioms for set theory, augmented with many definitions, using first or second order logic. For example, properties of the natural and real numbers can be derived within set theory, as each number system can be identified with a set of equivalence classes under a suitable equivalence relation whose field is some infinite set. Set theory as a foundation for mathematical analysis, topology, abstract algebra, and discrete mathematics is likewise uncontroversial; mathematicians accept that (in principle) theorems in

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these areas can be derived from the relevant definitions and the axioms of set theory. Few full derivations of complex mathematical theorems from set theory have been formally verified, however, because such formal derivations are often much longer than the natural language proofs mathematicians commonly present. One verification project, Metamath, includes derivations of more than 10,000 theorems starting from the ZFC axioms and using first order logic.

Logarithms The logarithm is perhaps the single, most useful arithmetic concept in all the sciences; and an understanding of them is essential to an understanding of many scientific ideas. Logarithms may be defined and introduced in several different ways. But for our purposes, let's adopt a simple approach. This approach originally arose out of a desire to simplify multiplication and division to the level of addition and subtraction. Of course, in this era of the cheap hand calculator, this is not necessary anymore but it still serves as a useful way to introduce logarithms.

Important Facts and Formulae Logarithm: If a is a positive real number, other than 1 and am = X, then we write: m = loga x and we say that the value of log x to the base a is m.

Example: (i) 103 = 1000 => log10 1000 = 3 (ii) 2-3 = 1/8 => log2 1/8 = - 3 (iii) 34 = 81 => log3 81=4 (iiii) (.1)2 = .01 => log(.l) .01 = 2.

Properties of Logarithms:

1. loga(xy) = loga x + loga y 2. loga (x/y) = loga x - loga y

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3.logx x=1 4. loga 1 = 0 5.loga(xp)=p(logax)

1

6. logax =1/logx a 7. logax = logb x/logb a=log x/log a.

Remember: When base is not mentioned, it is taken as 10. Common Logarithms: Logarithms to the base 10 are known as common logarithms. The logarithm of a number contains two parts, namely characteristic and mantissa. Characteristic: The integral part of the logarithm of a number is called its characteristic. Case I: When the number is greater than 1. In this case, the characteristic is one less than the number of digits in the left of the decimal point in the given number. Case II: When the number is less than 1. In this case, the characteristic is one more than the number of zeros between the decimal point and the first significant digit of the number and it is negative. Instead of - 1, - 2, etc. we write, 1 (one bar), 2 (two bar), etc. Example: Number

Characteristic

Number

Characteristic

348.25

2

0.6173

1

46.583

1

0.03125

2

9.2193

0

0.00125

3

Mantissa: The decimal part of the logarithm of a number is known is its mantissa. For mantissa, we look through log table.

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Linear Programming Linear programming is a widely used mathematical modeling technique to determine the optimum allocation of scarce resources among competing demands. Resources typically include raw materials, manpower, machinery, time, money and space. The technique is very powerful and found especially useful because of its application to many different types of real business problems in areas like finance, production, sales and distribution, personnel, marketing and many more areas of management. As its name implies, the linear programming model consists of linear objectives and linear constraints, which means that the variables in a model have a proportionate relationship. For example, an increase in manpower resource will result in an increase in work output. Essentials of Linear Programming Model For a given problem situation, there are certain essential conditions that need to be solved by using linear programming. 1. Limited resources : limited number of labour, material equipment and finance 2. Objective : refers to the aim to optimize (maximize the profits or minimize the costs). 3. Linearity : increase in labour input will have a proportionate increase in output. 4. Homogeneity : the products, workers' efficiency, and machines are assumed to be identical. 5. Divisibility : it is assumed that resources and products can be divided into fractions. (in case the fractions are not possible, like production of one-third of a computer, a modification of linear programming called integer programming can be used).

Properties of Linear Programming Model The following properties form the linear programming model: 1. Relationship among decision variables must be linear in nature. 2. A model must have an objective function. 3. Resource constraints are essential. 4. A model must have a non-negativity constraint.

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Formulation of Linear Programming Formulation of linear programming is the representation of problem situation in a mathematical form. It involves well defined decision variables, with an objective function and set of constraints. Objective function The objective of the problem is identified and converted into a suitable objective function. The objective function represents the aim or goal of the system (i.e., decision variables) which has to be determined from the problem. Generally, the objective in most cases will be either to maximize resources or profits or, to minimize the cost or time. For example, assume that a furniture manufacturer produces tables and chairs. If the manufacturer wants to maximize his profits, he has to determine the optimal quantity of tables and chairs to be produced. Let x1 = Optimal production of tables p1 = Profit from each table sold x2 = Optimal production of chairs p2 = Profit from each chair sold. Hence, Total profit from tables = p1 x1 Total profit from chairs = p2 x2 The objective function is formulated as below, Maximize Z or Zmax = p1 x1 + p2 x2 Constraints When the availability of resources are in surplus, there will be no problem in making decisions. But in real life, organizations normally have scarce resources within which the job has to be performed in the most effective way. Therefore, problem situations are within confined limits in which the optimal solution to the problem must be found. Considering the previous example of furniture manufacturer, let w be the amount of wood available to produce tables and chairs. Each unit of table consumes w1 unit of wood and each unit of chair consumes w2 units of wood. For the constraint of raw material availability, the mathematical expression is, w1 x1 + w2 x2 £ w

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In addition to raw material, if other resources such as labour, machinery and time are also considered as constraint equations. Non-negativity constraint Negative values of physical quantities are impossible, like producing negative number of chairs, tables, etc., so it is necessary to include the element of non-negativity as a constraint i.e., x1, x2 ³ 0 Example for Linear Programming by Graphical Method

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Matrix A matrix is an ordered set of numbers listed rectangular form. Example. Let A denote the matrix [2 [5 [3 This matrix A

5 7 8] 6 8 9] 9 0 1] has three rows and four columns. We say it is a 3 x 4 matrix.

We denote the element on the second row and fourth column with a2,4.

Properties of Addition The basic properties of addition for real numbers also hold true for matrices. Let A, B and C be m x n matrices 1. A + B = B + A commutative 2. A + (B + C) = (A + B) + C associative 3. There is a unique m x n matrix O with A+O = A

additive identity

4. For any m x n matrix A there is an m x n matrix B (called -A) with A+B = O

additive inverse

Properties of Matrix Multiplication Unlike matrix addition, the properties of multiplication of real numbers do not all generalize to matrices. Matrices rarely commute even if AB and BA are both defined. There often is no multiplicative inverse of a matrix, even if the matrix is a square matrix. There are a few properties of multiplication of real numbers that generalize to matrices. We state them now. Let A, B and C be matrices of dimensions such that the following are defined. Then 1. A(BC) = (AB)C

associative

2. A(B + C) = AB + AC

distributive

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3. (A + B)C = AC + BC

distributive

4. There are unique matrices Im and In with Im A = A In = A

multiplicative identity

Solving Linear Systems of Equations as Application Reduced Row Echelon Form When solving linear systems, we first transform the system into an augmented matrix. At that point our goal is to transform the matrix into an "easier" matrix whose corresponding linear system has the same solution set. We now defined what it means for a matrix to be "easier". Definition An m x n matrix is in reduced row echelon form if it satisfies the following properties: 1. All zero rows, if any, are at the bottom of the matrix 2. The first nonzero entry of each row is a one. This one is called the leading one or the corner. 3. Each corner is to the right and below the preceding corner. 4. The columns containing a leading one have zeros in all other entries. If only 1, 2, and 3 are satisfied, then the matrix is in row echelon form.

Example Of the following three matrices,

The A and B are in rref, while C is not.

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The main purpose of putting a matrix in rref is that this form makes the solution of the linear system easy to identify. For example A corresponds to the system x1 = 4

x2 = 2

x3 = x3

or in parametric form we get the line x1 = 4

x2 = 2

x3 = t

B corresponds to the system x1 + 3x3 = 5

x2 - x3 = 0

x3 = x3

This also gives us a line. In parametric form it is x1 = 5 - 3t

x2 = t

x3 = t

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Data Analysis Data and Statistical data: Statistics is a branch of applied mathematics concerned with the collection and interpretation of quantitative data and the use of probability theory to estimate population parametersStatistical methods can be used to summarize or describe a collection of data; this is called descriptive statistics. Data: A collection of values to be used for statistical analysis. A dictionary defines data as facts or figures from which conclusions may be drawn. Data may consist of numbers, words, or images, particularly as measurements or observations of a set of variables. Data are often viewed as a lowest level of abstraction from which information and knowledge are derived. Thus, technically, it is a collective or plural noun. Datum is the singular form of the noun data. Data can be classified as either numeric or nonnumeric. Specific terms are used as follows: Types of Data Qualitative data are nonnumeric. 1. {Poor, Fair, Good, Better, Best}, colors (ignoring any physical causes), and types of material {straw, sticks, bricks} are examples of qualitative data. 2. Qualitative data are often termed categorical data. Some books use the terms individual and variable to reference the objects and characteristics described by a set of data. They also stress the importance of exact definitions of these variables, including what units they are recorded in. The reason the data were collected is also important. Quantitative data are numeric. Quantitative data are further classified as either discrete or continuous. Discrete data are numeric data that have a finite number of possible values.

o

A classic example of discrete data is a finite subset of the counting numbers, {1,2,3,4,5} perhaps corresponding to {Strongly Disagree... Strongly Agree}. When data represent counts, they are discrete. An example might be how many students were absent on a given day. Counts are usually considered exact and integer.

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Continuous data have infinite possibilities: 1.4, 1.41, 1.414, 1.4142, 1.141421... The real numbers are continuous with no gaps or interruptions. Physically measureable quantities of length, volume, time, mass, etc. are generally considered continuous. At the physical level (microscopically), especially for mass, this may not be true, but for normal life situations is a valid assumption. Data analysis is a process of gathering, modeling, and transforming data with the goal of highlighting useful information, suggesting conclusions, and supporting decision making. Data analysis has multiple facets and approaches, encompassing diverse techniques under a variety of names, in different business, science, and social science domains. Frequency Distribution: The distribution of empirical data is called a frequency distribution and consists of a count of the number of occurrences of each value. If the data are continuous, then a grouped frequency distribution is used. Typically, a distribution is portrayed using a frequency polygon or a histogram. Mathematical distributions are often used to define distributions. The normal distribution is, perhaps, the best known example. Many empirical distributions are approximated well by mathematical distributions such as the normal distribution. Grouped Frequency Distribution A grouped frequency distribution is a frequency distribution in which frequencies are displayed for ranges of data rather than for individual values. For example, the distribution of heights might be calculated by defining one-inch ranges. The frequency of individuals with various heights rounded off to the nearest inch would be then be tabulated. Graphical presentation of Frequency distribution: Histogram: A histogram is a graphical display of tabulated frequencies. A histogram is the graphical version of a table that shows what proportion of cases fall into each of several or many specified categories.

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Example of a histogram of 100 values Advantages • • •

Visually strong Can compare to normal curve Usually vertical axis is a frequency count of items falling into each category

Disadvantages • • •

Cannot read exact values because data is grouped into categories More difficult to compare two data sets Use only with continuous data

Frequency Polygons: Frequency polygons are a graphical device for understanding the shapes of distributions. They serve the same purpose as histograms, but are especially helpful in comparing sets of data. Frequency polygons are also a good choice for displaying cumulative frequency distributions. To create a frequency polygon, start just as for histograms, by choosing a class interval. Then draw an X-axis representing the values of the scores in your data. Mark the middle of each class interval with a tick mark, and label it with the middle value represented by the class. Draw the Yaxis to indicate the frequency of each class. Place a point in the middle of each class interval at the height corresponding to its frequency. Finally, connect the points. You should include one class interval below the lowest value in your data and one above the highest value. The graph will then touch the X-axis on both sides.

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Advantages • • •

Visually appealing Can compare to normal curve Can compare two data sets

Disadvantages • •

Anchors at both ends may imply zero as data points Use only with continuous data

Frequency Curve:

A smooth curve which corresponds to the limiting case of a histogram computed for a frequency distribution of a co the number of data points becomes very large.

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Advantages •

Visually appealing

Disadvantages • •

Anchors at both ends may imply zero as data points Use only with continuous data

Describing Data Numerically Center and Location

Other Measures of Location

Variation Range

Mean

Percentiles Variance

Median Quartiles Mode

Standard Deviation Interquartile Range

Weighted Mean

Coefficient of Variation

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Measures of Center and Location

Center and Location

Mean

Mode

Median

Weighted Mean

n

∑x x=

i=1

n

∑x

µW

i

i=1

The Mean is the arithmetic average of data values Sample mean n = Sample Size

n

∑ =

x

i

x

=

i= 1

n

+

1

x

+ L n

2

+

x

n

N = Population Size

N

∑ µ

=

x

i= 1

N

i

i

i

Mean (Arithmetic Average) N

x

i

i

N

µ=

∑w x ∑w wx ∑ = ∑w

XW =

i

i

=

x

1

+

x

2

+ L N

+

Population mean The most common measure of central tendency Mean = sum of values divided by the number of values Affected by extreme values (outliers)

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x

N

i

Quantitative Techniques Barhate Mangesh Roll No- PG/509/MBA (I)/2009J Median In an ordered array, the median is the “middle” number If n or N is odd, the median is the middle number If n or N is even, the median is the average of the two middle numbers

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

Median = 3

Median = 3

Mode

A measure of central tendency Value that occurs most often Not affected by extreme values Used for either numerical or categorical data There may may be no mode There may be several modes

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Mode = 5

No Mode

0 1 2 3 4 5 6 7 8 9 10

Mean = 3

Mean = 4

1 + 2 + 3 + 4 + 5 15 = =3 5 5

1 + 2 + 3 + 4 + 10 20 = =4 5 5

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Quantitative Techniques Barhate Mangesh Roll No- PG/509/MBA (I)/2009J Weighted Mean Used when values are grouped by frequency or relative importance

Example: Sample of 26 Repair Projects Days to Complete

Frequency

5

4

6

12

7

8

8

2

Weighted Mean Days to Complete: XW =

∑w x ∑w

i i

=

(4× 5) + (12× 6) + (8× 7) + (2× 8) 4 + 12 + 8 + 2

=

164 = 6.31 days 26

i

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Shape of a Distribution Describes how data is distributed Symmetric or skewed

Symmetric

Left-Skewed

Right-Skewed

Mean < Median < Mode Mean = Median = Mode Mode < Median < Mean (Longer tail extends to left)

(Longer tail extends to right)

Other Location Measures

Other Measures of Location Percentiles

Quartiles

The pth percentile in a data array:

p% are less than or equal to this value (100 – p)% are greater than or equal to this value

(where 0 ≤ p ≤ 100)

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1st quartile = 25th percentile 2nd quartile = 50th percentile = median 3rd quartile = 75th percentile

Quantitative Techniques Barhate Mangesh Roll No- PG/509/MBA (I)/2009J Quartiles

Quartiles split the ranked data into 4 equal groups

25%

25%

25%

25%

A Graphical display of data using 5-number summary: Minimum -- Q1 -- Median -- Q3 -- Maximum Shape of Box and Whisker Plots The Box and central line are centered between the endpoints if data is symmetric around the median

A Box and Whisker plot can be shown in either vertical or horizontal format Distribution Shape and Box and Whisker Plot

Left-Skewed

Q1

Q2 Q3

Symmetric

Q1 Q2 Q3

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Right-Skewed

Q1 Q2 Q3


Measures of Variation

Variation Range Interquartile Range

Variance

Standard Deviation

Population Variance

Population Standard Deviation

Sample Variance

Sample Standard Deviation


Variation Measures of variation give information on the spread or variability of the data values.

Range Simplest measure of variation Difference between the largest and the smallest observations:

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Range = xmaximum – xminimum Example: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Range = 14 - 1 = 13 Interquartile Range Can eliminate some outlier problems by using the interquartile range Eliminate some high-and low-valued observations and calculate the range from the remaining alues. Interquartile range = 3rd quartile – 1st quartile Interquartile Range

Example: X

minimum 25%

12

Median (Q2)

Q1 25%

30

25%

45

maximum

25%

57

Interquartile range = 57 – 30 = 27

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X

Q3

70


Variance Average of squared deviations of values from the mean

Sample variance: n

∑ =

2

s

(x

− x )

i

i= 1

n - 1 N

∑ =

2

σ

2

(x

− µ)

i

2

i= 1

N

Population variance: n

∑ =

s

(x

−

i

2

i= 1

n - 1 N

∑ =

σ

x )

(x

i

− µ)

2

i= 1

N

Standard Deviation Most commonly used measure of variation Shows variation about the mean Has the same units as the original data Sample standard deviation:

∑ s

=

n

(x

i =

−

i

n

-

1

i

−

N

∑ σ

=

x

)

2

1

i =

(x 1

N

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µ)

2

Quantitative Techniques Barhate Mangesh Roll No- PG/509/MBA (I)/2009J Population standard deviation:

Comparing Standard Deviations

Data A 11

12

13

14

15

16

17

18

19

20 21

Mean = 15.5 s = 3.338

Data B 11

12

13

14

15

16

17

18

19

20 21

Mean = 15.5 s = .9258

Data C 11

12

13

14

15

16

17

18

19

20 21

Mean = 15.5 s = 4.57


Measures relative variation Always in percentage (%) Shows variation relative to mean Is used to compare two or more sets of data measured in different units

Population

Sample

s CV =  x

 σ  CV =   ⋅ 100% µ 

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  ⋅100%  


The Empirical Rule If the data distribution is bell-shaped, then the interval: contains about 68% of the values in the population or the sample

µ ± 1σ

68%

µ

µ ± 1σ

µ ± 2σ contains about 95% of the values in the population or the sample

µ ± 3σcontains about 99.7% of the values in the population or the sample

95%

99.7%

Tchebysheff’s Theorem

µ ± 2σ

µ ± 3σ

Regardless of how the data are distributed, at least (1 - 1/k2) of the values will fall within k standard deviations of the mean Examples: (1 - 1/12) = 0% ……..... k=1 (µ ± 1σ) (1 - 1/22) = 75% …........k=2 (µ ± 2σ) (1 - 1/32) = 89% ………. k=3 (µ ± 3σ)

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Methods of Time Series Time Series Time series is set of data collected and arranged in accordance of time. According to Croxton and Cowdon,”A Time series consists of data arranged chronologically.” Such data may be series of temperature of patients, series showing number of suicides in different months of year etc. The analysis of time series means separating out different components which influences values of series. The variations in the time series can be divided into two parts: long term variations and short term variations. Long term variations can be divided into two parts: Trend or Secular Trend and Cyclical variations. Short term variations can be divided into two parts: Seasonal variations and Irregular Variations.

VARIATIONS IN TIME SERIES

LONG TERM VARIATIONS

TREND

SHORT TERM VARIATIONS

SEASONAL VARIATIONS

CYLICAL VARIATIONS

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IRREGULAR VARIATIONS


Wrapping Up Quantitative Techniques adopt a scientific approach to decision-making. In this approach, past data is used in determining decisions that would prove most valuable in the future. The use of past data in a systematic manner and constructing it into a suitable model for future use comprises a major part of scientific management. For example, consider a person investing in fixed deposit in a bank, or in shares of a company, or mutual funds, or in Life Insurance Corporation. The expected return on investments will vary depending upon the interest and time period. We can use the scientific management analysis to find out how much the investments made will be worth in the future. There are many scientific method software packages that have been developed to determine and analyze the problems. Quantitative Technique is the scientific way to managerial decision-making, while emotion and guess work are not part of the scientific management approach. This approach starts with data. Like raw material for a factory, this data is manipulated or processed into information that is valuable to people making decision. This processing and manipulating of raw data into meaningful information is the heart of scientific management analysis.

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Quantitative Techniques

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