Quantitative Techniques - Introductions PPT.pptx

5 June 2022 1
Mr. MUSTAQ MULLA, Teaching Assistant, Rani Channamma University Belagavi, P G Centre Jamakhandi

Unit -1
Introduction
• Quantitative technique is a very powerful tool, by using this
we can Judgment our production, maximize profits,
minimize costs, and production methods can be oriented for
the accomplishment of certain pre–determined objectives.
• Quantitative techniques used to solve many of the problems
that arise in a business or industrial area. A large number of
business problems, in the relatively recent past, have been
given a quantitative representation with considerable
degree of success.
• All this has attracted the students, business executives,
public administrators a like towards the study of these
techniques more and more in the present times.
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Meaning And definition:
• Quantitative techniques may be defined as those
techniques which provide the decision makes a
systematic and powerful means of analysis, based
on quantitative data. It is a scientific method
employed for problem solving and decision
making by the management.
• Quantitative techniques are defined as “those
statistical techniques which lead to numerical
analysis of variables, affecting a decision situation
and evaluation of alternative strategies to attain
objectives of organizations.”
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Features
Descriptions of quantitative techniques reveal following characteristics or features.
1. Measurement : Measurement is the basis of quantitative technique. Measurement is
assigning numbers to concepts and phenomena. Measurement generates necessary
data.
2. Numerical analysis : Another basic feature of quantitative techniques is numerical
expression of variables and analysis there-no. Even qualitative characteristics or
phenomenon can be transformed to numbers and symbols using quantitative
techniques.
3. Scientific method : Quantitative techniques for decision making are examples for the
use of scientific methods of management. It offers a systematic and objective
experimentation, observation and evaluation of best strategies.
4. Decision making : It is a support system in decision making process. It provides
decision makers with appropriate tools of evaluation and presentation.
5. Options : Quantitative techniques should evaluate and reveal alternative strategies or
options. There is no scope for decision where there is a single option.
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6 Improvement : Quantitative techniques should replace personal judgment and
intuition. It should lead to improved and quality decisions.
7 Functions of quantitative techniques : Quantitative Techniques are those
methods in which details of a problem or situation are expressed in
numerical terms, so as to support decision making.
8 Quantification : Critical factors affecting a decision situation is transformed
into quantitative or numerical form. It is easy to comprehend, understand
and delegate an issue in numerical form.
9 Analysis : Quantitative techniques enable scientific and systematic study of
any issue. It probes deep into the factors influencing the problem and helps
to express the situation in a comprehensive form.
10 Decision making :Quantitative techniques facilitate the process of decision
making. It sets out all possible alternatives and enables a feasibility study of
each so that the optimal alternative can be chosen.
11 Deployment of resources: Quantitative techniques, if properly applied,
leads to optimal allocation of available limited resources. It avoids wastages
and less efficient usage of resources, and leads to conservation of
resources.
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Scope of Quantitative Techniques
A few specific areas of application are mentioned below.
 Finance and Accounting: Cash flow analysis, Capital budgeting,
Dividend and Portfolio management, Financial planning.
 Marketing Management: Selection of product mix, Sales resources
allocation and Assignments.
 Production Management: Facilities planning, Manufacturing,
Aggregate planning, Inventory control, Quality control, Work
scheduling, Job sequencing, Maintenance and Project planning and
scheduling.
 Personnel Management: Manpower planning, Resource allocation,
Staffing, Scheduling of training programs.
 General Management: Decision Support System and Management
of Information Systems, MIS, Organizational design and control,
Software Process Management and Knowledge Management.
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Classification of Quantitative
Techniques:
There are different types of quantitative techniques. We
can classify them into three categories.
They are:
1. Mathematical Quantitative Techniques
2. Statistical Quantitative Techniques
3. Programming Quantitative Techniques
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1. Mathematical Quantitative Techniques
Mathematical Quantitative Techniques:
A technique in which quantitative data are used along with the principles of
mathematics is known as mathematical quantitative techniques. Mathematical
quantitative techniques involve:
1. Permutations and Combinations:
Permutation means arrangement of objects in a definite order. The number of
arrangements depends upon the total number of objects and the number of objects
taken at a time for arrangement. The number of permutations or arrangements is
calculated by using the
following formula:- nPr = n !
(n-r)!
Combination means selection or grouping objects without considering their order.
The number of combinations is calculated by using the following formula:-
nCr = n!
(n-r) !
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2. Set Theory:- Set theory is a modern mathematical device which solves
various types of critical problems.
3. Matrix Algebra: Matrix is an orderly arrangement of certain given numbers
or symbols in rows and columns. It is a mathematical device of finding out
the results of different types of algebraic operations on the basis of the
relevant matrices.
4. Determinants: It is a powerful device developed over the matrix algebra.
This device is used for finding out values of different variables connected
with a number of simultaneous equations.
5. Differentiation: It is a mathematical process of finding out changes in the
dependent variable with reference to a small change in the independent
variable.
6. Integration: Integration is the reverse process of differentiation.
7. Differential Equation: It is a mathematical equation which involves the differential
coefficients of the dependent variables.
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Mr. MUSTAQ MULLA, Teaching Assistant,
Rani Channamma University Belagavi, P G
Centre Jamakhandi

2. Statistical Quantitative Techniques
Statistical techniques are those techniques which are used in conducting
the statistical enquiry concerning to certain Phenomenon. They include all
the statistical methods beginning from the collection of data till
interpretation of those collected data.
Statistical techniques involve:
1. Collection of data:
One of the important statistical methods is collection of data. There are
different methods for collecting primary and secondary data.
2. Measures of Central tendency, dispersion, skewness and Kurtosis :
Measures of Central tendency is a method used for finding the average of a
series while measures of dispersion used for finding out the variability in a
series. Measures of Skewness measures asymmetry of a distribution while
measures of Kurtosis measures the flatness of peakedness in a distribution.
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3. Correlation and Regression Analysis:
Correlation is used to study the degree of relationship among two or more variables. On the other
hand, regression technique is used to estimate the value of one variable for a given value of
another.
4. Index Numbers:
Index numbers measure the fluctuations in various Phenomena like price, production etc
over a period of time, They are described as economic barometers.
5. Time series Analysis:
Analysis of time series helps us to know the effect of factors which are responsible for changes:
6. Interpolation and Extrapolation:
Interpolation is the statistical technique of estimating under certain assumptions, the
missing figures which may fall within the range of given figures. Extrapolation provides
estimated figures outside the range of given data.
7. Statistical Quality Control
Statistical quality control is used for ensuring the quality of items manufactured. The
variations in quality because of assignable causes and chance causes can be known with the help
of this tool. Different control charts are used in controlling the quality of products.
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8. Ratio Analysis:
Ratio analysis is used for analyzing financial
statements of any business or industrial concerns
which help to take appropriate decisions.
9. Probability Theory:
Theory of probability provides numerical values of
the likely hood of the occurrence of events.
10. Testing of Hypothesis
Testing of hypothesis is an important statistical tool
to judge the reliability of inferences drawn on the
basis of sample studies.
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3. Programming Quantitative Techniques
Programming techniques are also called operations
research techniques. Programming techniques are model
building techniques used by decision makers in modern
times.
Programming techniques involve:
1. Linear Programming:
Linear programming technique is used in finding a solution
for optimizing a given objective under certain constraints.
2. Queuing Theory:
Queuing theory deals with mathematical study of queues.
It aims at minimizing cost of both servicing and waiting.
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3. Game Theory:
Game theory is used to determine the optimum strategy in a competitive situation.
4. Decision Theory:
This is concerned with making sound decisions under conditions of certainty, risk and
uncertainty.
5. Inventory Theory:
Inventory theory helps for optimizing the inventory levels. It focuses on minimizing
cost associated with holding of inventories.
6. Net work programming:
It is a technique of planning, scheduling, controlling, monitoring and co-ordinating
large and complex projects comprising of a number of activities and events. It
serves as an instrument in resource allocation and adjustment of time and cost up
to the optimum level. It includes CPM, PERT etc.
7. Simulation:
It is a technique of testing a model which resembles a real life situations
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8. Replacement Theory:
It is concerned with the problems of replacement of machines, etc due to their
deteriorating efficiency or breakdown. It helps to determine the most economic replacement
policy.
9. Non Linear Programming:
It is a programming technique which involves finding an optimum solution to a problem in
which some or all variables are non-linear.
10. Sequencing:
Sequencing tool is used to determine a sequence in which given jobs should be performed
by minimizing the total efforts.
11. Quadratic Programming:
Quadratic programming technique is designed to solve certain problems, the objective
function of which takes the form of a quadratic equation.
12. Branch and Bound Technique
It is a recently developed technique. This is designed to solve the combinational
problems of decision making where there are large number of feasible solutions. Problems of
plant location, problems of determining minimum cost of production etc. are examples of
combinational problems.
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Role of Quantitative Techniques
These techniques are especially increasing since World War II in the technology of business
administration.
These techniques help in solving complex and intricate problems of business and industry.
Quantitative
techniques for decision making are, in fact, examples of the use of scientific method of
management. Their
role can be well understood under the following heads:
(i) Provide a tool for scientific analysis:
These techniques provides executives with a more precise description of
the cause and effect relationship and risks underlying the business operations in
measurable terms and this eliminates the conventional intuitive and subjective
basis on which managements used to formulate their decisions decades ago. In fact,
these techniques replace the intuitive and subjective approach of decision making
by an analytical and objective approach. The use of these techniques has
transformed the conventional techniques of operational and investment problems in
business and industry. Quantitative techniques thus encourage and enforce
disciplined thinking about organizational problems.
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(ii) Provide solution for various business problems:
These techniques are being used in the field of production, procurement, marketing,
finance and allied fields. Problems like, how best can the managers and executives
allocate the available resources to various products so that in a given time the profits
are maximum or the cost is minimum? Is it possible for an industrial enterprise to
arrange the time and quantity of orders of its stock such that the overall profit with
given resources is maximum? How far is it within the competence of a business
manager to determine the number of men and machines to be employed and used in
such a manner that neither remains idle and at the same time the customer or the
public has not to wait unduly long for service? And similar other problems can be solved
with the help of quantitative techniques.
(iii) Enable proper deployment of resources:
It render valuable help in proper deployment of resources. For example, PERT enables
us to determine the earliest and the latest times for each of the events and activities
and thereby helps in identification of the critical path. All this helps in the deployment
of the resources from one activity to another to enable the project completion on time.
This techniques, thus, provides for determining the probability of completing an event
or project itself by a specified date.
(iv) Helps in minimizing waiting and servicing costs:
This theory helps the management men in minimizing the total waiting and servicing
costs. This technique also analyses the feasibility of adding facilities and thereby helps
the business people to take a correct and profitable decision.
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(v) Assists in choosing an optimum strategy:
Game theory is especially used to determine the optimum strategy in a
competitive situation and enables the businessmen to maximise profits or
minimize losses by adopting the optimum strategy.
(vi) They render great help in optimum resource allocation
Linear programming technique is used to allocate scarce resources in an
optimum manner in problem of scheduling, product – mix and so on.
(vii) Enable the management to decide when to buy and how much to buy:
The techniques of inventory planning enables the management to decide
when to buy and how much to buy.
(viii) They facilitate the process of decision making:
Decision theory enables the businessmen to select the best course of
action when information is given to probabilistic form. Through decision
tree techniques executive’s judgement can systematically be brought into
the analysis of the problems. Simulation is an other important technique
used to imitate an operation or process prior to actual performance. The
significance of simulation lies in the fact that it enables in finding out the
effect of alternative courses of action in situation involving uncertainty
where mathematical formulation is not possible. Even complex groups of
variables can be handled through this technique.
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(ix) Through various quantitative techniques management can
know the reactions of the integrated business systems:
The Integrated Production Models techniques are used to
minimise cost with respect to work force, production and
inventory. This technique is quite complex and is usually
used by companies having detailed information concerning
their sales and costs statistics over a long period. Besides,
various other O.R. techniques also help in management
people taking decisions concerning various problems of
business and industry. The techniques are designed to
investigate how the integrated business system would react
to variations in its component elements and/or external
factors.
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Functions of Quantitative Techniques:
The following are the important functions of quantitative techniques:
1. To facilitate the decision-making process
2. To provide tools for scientific research
3. To help in choosing an optimal strategy
4. To enable in proper deployment of resources
5. To help in minimizing costs
6. To help in minimizing the total processing time
required for performing a set of jobs
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Functions and Their Applications
A function is a technical term used to symbolise relationship between
variables. When one or more independent variables are related to
dependent variables then the dependent variable is said to be the
function of independent variables. A function explains the nature of
relationship between dependent and independent variables. A
relationship may be a formula or a graph or a mathematical equation,
Classification of Functions
We have already learnt that for a function f : A B, f associates all elements
of set A to set B and each element of set A is associated to a unique
element of a set B. Thus, we may associate different element of set B or
we may associate more than one element of set A to same element of set
B (but same element of set A cannot be associated with more than one
element of set B). Also, all elements of set B may or may not have their
pre-images in A.
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Functions Related To Economics:
In the case of functions in economics, the variables are hypothetical
quantities and not actual observable quantities as in physical science.The
range and the domain of economics functions are made up of nonnegative
quantities so that the graphs of these functions are in first quadrant only.
In other words, for the purpose of economic analysis, only that part of a
curve is relevant which lies in the first quadrant.
1. Demand Function:
As we know that the quantity demanded of a particular commodity by the
buyers in the market is depending on the price. As the prices increases,
the demand is decreases shown as figure. If q is the quantity of a
commodity demanded and p is the price then the demand function is
given by : q = f (p) shows q depends on p price
p = g (q) shows p depends on q
2. Supply function:
Price of any particular commodity in the market depends on the quantity
of supply. As the quantity of supply increases the price is also increases. If
x is the quantity of supply and p is its price then the supply function is
given by x = f (p) or p = f (x).
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Functions Related To Cost:
1 Total Cost Function. 2 Average Cost Function.
1 Total cost function:
The total cost (C) which is equal to sum of fixed cost and variable cost, of
production in a firm, is depending on the quantity produced of a particular
commodity. As the production increases the total cost also increases. If x is
the quantity produced at total cost T then the total cost function is given
by T = f (x).
2 Average Cost Function:
Average cost function or cost per unit is obtained by dividing the total cost
function by the quantity produced. Average cost function ,
A = (T / x) = (f (x) / x).
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3 Total Revenue Function:
The total revenue in a particular commodity of a firm is depending on the
number units sold. If p is the price of any commodity and q is the quantity sold
then the total revenue function is given by R = p.q, and the average revenue
function or revenue per unit is obtained by dividing the total revenue function
by the units of quantity sold, it is equal to price (p) of the commodity. AR =
(R / q) = (Pq / q) = p = price
5. Profit Function:
Profit on a particular commodity of a firm is equal to the difference or
total revenue and the total cost the profit function is given by z = R – T.
6. Production Function:
The output product is depending in the various inputs like capital,
labour, raw material etc. If q is the output quantity of a particular
commodity and x, y and z are the input variables in a firm then the
production function as sin y. q = f (x, y, z)
For example: If labour (1) and capital (K) are inputs variables then the
output is given by Q = f (L, K) =
ALα - Kβ, α + β = 1 where A, α and β are constants is known as Cobb-
Douglas production function.
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7. Consumption Function:
The total consumption function of a firm is depending on the income as income increases
the consumption expenditure also increases. If C is the consumption and I is the income then
the consumption function given by C = f (5) = a + bI where a, b are constants.
Example1:
A company sells x tins of chicken each day at Rs. 80 per tin. the cost of production and selling
price of these tins is Rs. 50 per tin plus a fixed daily overhead cost of Rs. 18,000. Determine
the profit function. What is the profit if 2,000 tins are produced and sold a day? Find out the
number of tins produced in a day with no profit and no loss.
Solution:
a). Per day x tins of chicken are produced with cost Rs. 80 per tin the revenue received per day
is Revenue function
R (x) = 80. x
The cost of production per day cost function is c(x) = 1,800 + 50x
If P (x) is the profit function is given by
P (x) = R(x) - c(x)
= 80x - (50x +1,800) = 30x - 1,800
If 2,000 tins are produced and sold in a day then the profit is given by
P (2,000) = 30 × 2,000 - 18,000
= 60,000 - 18,000 = Rs. 42,000
b). for no profit and no loss then profit function is zero
P (x) = 0 = 30x - 18,000 = 0
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USES OF QUANTITATE TECHNIQUES
Business and Industry
Quantitative techniques render valuable services in the
field of business and industry. Today, all decisions in
business and industry are made with the help of
quantitative techniques. Some important uses of
quantitative techniques in the field of business and
industry are given below:
1. Quantitative techniques of linear programming is
used for optimal allocation of scarce resources in the
problem of determining product mix.
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2. Inventory control techniques are useful in dividing when and how much
items are to be purchase so as to maintain a balance between the cost of
holding and cost of ordering the inventory
3. Quantitative techniques of CPM, and PERT helps in determining the
earliest and the latest times for the events and activities of a project.
This helps the management in proper deployment of resources.
4. Decision tree analysis and simulation technique help the management in
taking the best possible course of action under the conditions of risks
and uncertainty.
5. Queuing theory is used to minimize the cost of waiting and servicing of
the customers in queues.
6. Replacement theory helps the management in determining the most
economic replacement policy regarding replacement of an equipment.
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Methodology of Q.T.
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1. Formulating the Problem :
As a first step, it is necessary to clearly understand the problem situations. It is
important to know how it is characterized and what is required to be determined.
Initially it is important to identify the key decision and the objective from the problem.
Then, it involves determination of the number of decision variables and the relationship
between variables. The measurable guaranties that are represented through these
variables are notified. Finally the practical limitations or constraints are also inferred
from the problem.
2. Defining the Decision Variables and Constraints :
In a given problem situation, defining the key decision variables are important.
Identifying these variables helps us to develop the model. For example, consider
a manufacturer who is manufacturing three products A, B and C using two
machines, I and II. Each unit of product A takes 2 minutes on machine I and 5
minutes on machine II. `Product B takes 1 minute on machine I and 3 minutes
on machine II.
Similarly, product C takes 4 minutes and 6 minutes on machine I and machine
II, respectively. The total available time on machine I and machine II are 100
hours and 120 hours, respectively. Each unit of A yields a profit of Rs. 3.00, B
yields Rs. 4.00 and C yields Rs. 5.00. What should be level of production of
products A, B and C that should be manufactured by the company so as to
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The decision variables, objective and constraints are identified
from the problem.
The company is manufacturing three products A, B and C. Let A
be x1, B be x2 and C be x3. x1, x2 and x3 are the three decision
variables in the problem. The objective is to maximize the
profits. Therefore, the problem is to maximize the profit, i.e., to
know how many units of x1, x2 and x3 are to be manufactured.
There are two machines available, machine I and machine II
with total machine hours available as 100 hours and 120 hours.
The machine hours are the resource constraints, i.e., the
machines cannot be used more than the given number of hours.
To summarize,
1 Key decision : How many units of x1, x2 and x3 are to be
manufactured?
2 Decision variables : x1, x2 and x3
3 Objective : To maximize profit
4 Constraint : Machine hours
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3. Developing a Suitable Model.
A model is a mathematical representation of a problem situation.
The mathematical model is in the form of expressions and
equations that replicate the problem.
For example, the total profit from a given number of products
sold can be determined by subtracting selling price and cost
price and multiplying the number of units sold. Assuming selling
price, sp as Rs. 40 and cost price, cp as Rs. 20, the following
mathematical model expresses the total profit, tp earned by
selling number of unit x.
T P = (SP – CP) x
T.P= (40 – 20) x
T P = 20 x
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4. Acquiring the Input Data :
Acquiring accurate data for input values are essential.
Even though the model is a perfect representation of
reality, improper data will result in misleading results.
Collecting accurate data can be one of the most difficult
steps in performing quantitative analysis.
5. Solving the Model :
Solving the model involves manipulating the model to
arrive at the optimal solution. In some cases, this
requires that an equation be solved for the best
decision. In other cases, you can use a trial and error
method, trying various approaches and picking the one
that results in the best solution
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5. Validating the Model
A validation is a complete test of the model to confirm that it provides
an accurate representation of the real problem. This helps us in
determining how good and realistic the solution is. During the model
validation process, inaccuracies can be rectified by taking corrective
actions, until the model is found to be fit.
6. Implementing the Results
The final step is to implement the results. This is the process of
incorporating the solution into the company. Even if the solution is
optimal and will result in millions of dollars in additional profits, if
managers resist the new solution, all of the efforts of the analysis are of
no value.
After the solution has been implemented, it should be closely
monitored. Over time, there may be numerous changes that call for
modifications of the original solution. A changing economy, fluctuating
demand and model enhancements requested by managers and decision
makers are only a few examples of changes that might require the
analysis to be modified.
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Limitations
Quantitative techniques though are a great aid to management but still they
cannot be substitute for decision making. The choice of criterion as to what is
actually best for the business enterprise is still that of an executive who has to fall
back upon his experience and judgement. This is so because of the several
limitations of quantitative techniques. Important limitations of these techniques
are as given below:
(i) The inherent limitation concerning mathematical expressions:
Quantitative techniques involve the use of mathematical models, equations and
similar other mathematical expressions. Assumptions are always incorporated in
the derivation of an equation and such an equation may be correctly used for the
solution of the business problems when the underlying assumptions and variables
in t he model are present in the concerning problem. IF this caution is not given
due care then there always remains the possibility of wrong application of the
quantitative techniques. Quite often the operations researchers have been
accused of having many solutions without being able to find problems that fit.
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(ii) High costs are involved in the use of quantitative techniques:
Quantitative techniques usually prove very expensive. Services of
specialised persons are invariably called for while using quantitative
techniques. Even in big business organisations we can expect that
quantitative techniques will continue to be of limited use simply
because they are not in many cases worth their cost. As opposed to
this a typical manager, exercising intuition and judgement, may be
able to make a decision very inexpensively. Thus, the use of
quantitative techniques is a costlier affair and this in fact constitutes
a big and important limitation of such techniques.
(iii) Quantitative techniques do not take into consideration the
intangible factors i.e., non measurable human factors:
Quantitative techniques make no allowances for intangible factors
such as skill, attitude, vigour of the management people in taking
decisions but in many instances success or failure hinges upon the
consideration of such non-measurable intangible factors. There
cannot be any magic formula for getting an answer to management
problems; much depends upon proper managerial attitudes and
policies.
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(iv) Quantitative techniques are just the tools of analysis
and not the complete decision making process:
It should always be kept in mind that
quantitative techniques, whatsoever it may
be, alone cannot make the final decision. They
are just tools and simply suggest best
alternatives but in final analysis many
business decisions will involve human
element. Thus, quantitative analysis is at best
a supplement rather than, a substitute for
management; subjective judgement is likely to
remain a principal approach to decision
making.
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Thank You
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