Quantitative Techniques for Business: A Comprehensive Guide, Exams of Quantitative Techniques

Download Quantitative Techniques for Business: A Comprehensive Guide and more Exams Quantitative Techniques in PDF only on Docsity!

Quantitative Techniques

MP- 204

Vardhaman Mahaveer Open University, Kota

MP-

Quantitative Techniques

Vardhaman Mahaveer Open University, Kota

Unit-1 : Quantitative Techniques in Business

Unit Structure:

1.0 Objectives

1.1 Introduction

1.2 Meaning

1.3 Classification of Quantitative Techniques

1.4 Role of Quantitative Techniques

1.5 Limitations

1.6 Functions and Their Applications

1.7 Construction of Functions

1.8 Classification of Functions

1.9 Roots of Functions

1.10 Break-Even Analysis

1.11 Sequence and Series (Progression)

1.12 Arithmetic Progression

1.13 Geometric Progression

1.14 Applications of Arithmetic and Geometric Progressions

1.15. Summary

1.16 Key Words

1.17 Self Assessment Test

1.18 References

1.0 Objectives

After reading this unit, you should be able to understand the:

 Meaning of Quantitative Techniques

 Classification of Quantitative Techniques

 Role of Quantitative Techniques

 Limitations of Quantitative Techniques

 Functions and their applications

 Construction and Classification of functions

 Functions related to economics and some special functions

 Roots of a function

 Break Even Analysis

 Sequence and Series

 Arithmetic Progression, Sum of A.P. Series, Arithmetic Mean

 Geometric Progression, Sum of G.P. Series, Geometric Mean

 Applications of A.P. and G.P.

1.1 Introduction

Quantitative technique is a very powerful tool, by using this we can augment 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 alike towards the study of these techniques more and more in the present

times.

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. In this chapter we will see how the 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 behaviour 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 decision-

making – 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. This study material is presented with variety of these techniques with real life problem

areas.

1.2 Meaning

Quantitative techniques are those statistical and operations research programming techniques which help in

the decision making process especially concerning business and industry. These techniques involve the

introduction of the element of quantities i.e., they involve the use of numbers, symbols and other mathematical

expressions. The quantitative techniques are essentially helpful supplement to judgement and intuition. These

techniques evaluate planning factors and alternative as and when they arise rather than prescribe courses of

action. As such, quantitative techniques may be defined as those techniques which provide the decision

maker with a systematic and powerful means of analysis and help, based on quantitative data, in exploring

policies for achieving pre – determined goals. These techniques are particularly relevant to problems of

complex business enterprises.

1.4 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 organisational problems.

(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.

(i) Methods of collecting data

(ii) Classification and tabulation of

collected data

(iii) Probability theory and Sampling

Analysis

(iv) Correlation and regression

(v) Index number

(vi) Time series Analysis

(vii) Interpolation and Extrapolation

(viii) Survey Techniques and

Methodology

(ix) Ratio Analysis

(x) Statistical quality control

(xi) Analysis of Variance

(xii) Statistical Inferences and

Interpretation

(xiii) Theory of Attributes

(i) Linear Programming

(ii) Decision Theory

(iii) Theory of Games

(iv) Simulation:

a. Monte Carlo Techniques

b. System Simulation

(v) Waiting Line (queuing) Theory

(vi) Inventory Planning

(vii) Integrated Production Models

(viii) Network Analysis/ PERT

(ix) Others

a. Non- Linear Programming

b. Dynamic Programming

c. Search Theory

d. Integer Programming

e. Quadratic Programming

f. Parametric Programming

g. The Theory of Replacement etc.

(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.

(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.

(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.

1.5 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 the 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.

1.7 Construction of Functions

Many times in business, we commonly talk about profit functions, loss functions, cost functions and revenue

functions, production function, demand and supply function, consumption function, into function, onto function,

polynomial function, Absolute value function, Step function, Inverse function, Rational and Irrational function,

Algebraic function. The functions are usually set up following the definition and calculation of the functional

values. Now we will take some few examples to illustrate the method of constructing such functions.

Example 1 : A factory has 100 items on hand for shipment to a destination at the cost of Rs 1 a piece

to meet a certain demand d. In case the demand d overshoots the supply. It is necessary to meet the

unsatisfied demand by purchases on the local market at Rs 2 a piece. Construct the cost function if

x is the number shipped from the factory.

Solution: Suppose C(x) denotes the cost function.

If d (^)  x,

C(x) = x + 2(d-x)

If d < x, C(x) = x

It can be combined into a single representation by writing

C(x) = x + 2 max (0, d-x)

This definition of C(x) is seen to be equivalent to the earlier one by considering separately the cases

d (^)  x and d < x.

Example 2: If f(x) = 2x + 1, find the range if domain is {-1,2,3} and hence find the function.

Solution: Here f(x) = 2x + 1

= f (-1) = -1 × 2 + 1 = -1,

= f (2) = 2 × 2 + 1 = 5,

= f (3) = 3 × 2 + 1 = 7

So, Range = {-1, 5, 7}

And f = {(-1, -1), (2, 5), (3, 7)}

1.8 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.

1.8.1 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

for example: q = a - bp or q = (10/p) or q = - 10p

2

• p + 5

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).

3. Cost Function:

3.1. Total Cost Function 3. 2. Average Cost Function

3.1 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).

P

Price

Demand

X

Demand Function

P

Price

Supply

X

Supply Function

30x = 18,000x = (18,000 / 30) = 600

600 tins per day.

1.8.2 Some Special Functions:

In this case, we have some special functions which can be used in economics as well as computer science

area too. These functions are defined as follows:

1. Polynomial function:

The general form of the function

f (x) or y = f (x 1,

x 2,

x 3,……..,

x n

) = a n

x

n

• a n-

x

n-

• a n-

x

n-

• ……+ a 0

……(i)

Where a i

’s are real numbers, a 1  0, and n is positive integer is called a polynomial of degree n. The real

constants a n,

a n-1,

a n-2,

a 0

are called coefficients. The exponent n is called the degree of the polynomial. For

n = 1, the equation (a) can be written as

y = a 1

x + a 0

(a (^)  0)

This is usually written as

y = a + bx (b (^)  0)

Where a = a 0

and b = a

Such Polynomial functions are called linear function

(i. e. degree 1) and has the domain {x: x (^) R}.

(b) Now for n = 2, the equation (i) can be written as:

y = a 2

x

2

• a 1

x

1 +a 0

(a 2  0)^ or^ y = ax

2

• bx + c, (a (^)  0)

Where a = a 2,

b = a 1

and c = a 0

. Such polynomial functions are called second degree functions or quadratic

functions.

The x -intercept of a quadratic function is very important. The x- intercept is that point (or points) at which

the graphs crosses the x-axis; that is, the point at which y = 0. If such points exist, they satisfy the equation:

ax

2

• bx + c = 0.

The graph of the linear function is the straight line y = a + bx which cuts the x-axis at a distance -b/a units

from the origin and y-axis at a distance of a units from the origin as shown.

Functions like f(x) = x

3

• (^2) x

g(x) = 2 + x - x

4

are polynomial function but 2x + x

2/ is not a polynomial function.

O

Y

B(- b / a,0)

X

A(0,b)

y = a + bx

Inverse Mapping

Example 1: The demand for certain item is given by q = 150 - 3p Where q denotes the amount

demanded and p the price per unit. It costs Rs 4 to produce each unit. What is the profit function

of the firm for this item?

Assume Y = total profit

R = total revenue

C = total cost

Since R = pq, C = 4q,

Profit = Total Revenue – Total cost

Y= R - C

= pq - 4q

= q (p - 4)

= (150 – 3p) (p – 4) [Substituting q = (150 – 3p) given]

= -3p

2

• 162p -600 which is a quadratic function in p.

2. Absolute Value Function:

The functional relationship of the form f (x) or y = |x| is known as an absolute value function, where x

represents the absolute value of x and is defined as given below, and the graph of this function is also shown

3. Step Function:

If a function is defined on a closed interval [a, b] and assume a constant value in the interior of each sub-

interval say [a, x 1

], [x 2,

x 3

], …………. , [x n,

b] of [a, b] where a < x 1

< x 2

< …… < x n

< b, then such function

is called a step function. Symbolically it may be expressed as:

y or f (x) = k i

for all values of x in the ith sub-interval.

The graph of this function is given below:

x, for x = 0

y = |x| = - x, x < 0

0, x = 0

O (^) X

Y

y (^3)

y 2

y 1

Step Function

An expression which involves root extraction on terms involving x is called irrational function. The function

such as x , 2 7 6

2 x  x  are examples of irrational functions.

7. Algebraic Function:

A function consisting a finite number of terms involving powers and roots of the variable x and the four basic

mathematical operations (addition, subtraction, multiplication and division) is called an algebraic function. In

general, it can be expressed as

y

n

• A 1

y

n-

• …………. + A n

= 0 where A 1,

A

2, …………..

, A

n

are rational functions of x.

There are two categories of algebraic functions namely: explicit and implicit algebraic functions. For example,

y = 3 x  2 x is and enplicat algebraic function, where as xy

2

• xy + x

2 = 0 is an implicit function.

8. Transcendental Function:

All functions which are not algebraic are called transcendental functions. These functions include:

1. Trigonometric Functions:

The trigonometric functions of an angle θ (θ be any real number) are given by:

sin θ = sin θ

c , cos θ = cos θ

c , tan θ = tan θ

c

cosec θ = cosec θ

c , sec θ = sec θ

c , cot θ = cot θ

c

where θ denote the angle whose radian measure is in θ.

The sin and cosec are said to be co - functions, as are the cos and sec and tan and cot.

The trigonometry functions are defined similarly for negative angles. An angle θ may be measured in degrees

or radians. However, in calculus and its applications to business and economics radian measure is usually

more convenient.

The trigonometric functions are very useful in the study of business cycles, seasonal or other cyclic variations

are described by sine or cosine functions.

2. Exponential Functions:

A function having a constant base and a variable exponent is called an exponent function, such as

(i) y = a

x , a (^)  1, a > 0 (ii) y = k a

x a (^)  1, a > 0

(iii) y = k a

bx a (^)  1, a > 0 (iv) y = k e

x

where a, b, e, and k are constants and x is exponent.

In calculus and its applications to business and economics, such functions are useful for describing sharp

increase and decrease in the value of dependent variable. For example the graph of exponential function y

= ka

x indicates rise to the right in the value of y for a > 1 and k > 0 whereas indicates fall to the left for a <

1 and k >0, as shown in the fig.

O

(a)

K

Y

y = ka

x

(a> 1, k>0)

X

The rules governing the exponents are as under:

(i) a

x

a

x 2

= a

x 1

• x 2

(ii) a

x 1

/a

x 2

= a

x 1

• x 2

(iii) (a

x 1

x 2

= a

x 1

x 2

(iv) (a.b)

x 1

= a

x

b

x 1

(v) (a /b)

x 1

= a

x

b

• x 1

(vi) a

0 = 1.

3. Logarithmic Functions:

A logarithmic function is expressed as log a

x where a > 0, a (^)  1 is the base. It is read as “ y is the log to the

base a of x”. This relationship may also be expressed by the equation x = a

y

. It is an exponential function.

Thus, logarithmic and exponential functions are inverse functions, i.e. if x is an exponential function of y, then

y is a logarithmic function of x.

Although the base of logarithm can be any positive number other than 1, but most widely used bases are

either 10 ( common or Briggsian logarithms) or e = 2.718 (natural or Naperian logarithms). By convention,

log x denotes the common logarithm of x in x denotes natural logarithm of x. If any other base is meant, it is

specified.

Some important properties of the logarithms are as follows. If x and y are positive real number, then

(i) log a

xy = log a

x + log a

y (ii) log a

(xy) = log a

x - log a

y (iii) log a

x

n = n log a

x

(iv) log a

x

1/n =(1 / n) log a

x (v) log a

x = log a

b × log b

x

(vi) Logarithm of zero and negative number is not defined.

Since exponential function x = a

y and logarithmic function y = log a

x are universe functions, therefore graph of

these curves for a particular value of ‘a’ can be obtained from the graph of each other by taking reflection

about the line y = x, as shown in the fig

Incommensurable Power Functions: A function having a variable base and a constant exponent is called and

incommensurable power function, such as y = 5 x or x

3/ or x

a etc.

9. Even and Odd Functions:

If a function not change its sign of its independent variable changed, then it is said to be an even function, i.e.

f (-x) = f(x). The examples of even functions are x

6 , cos x, etc. It follows that the graph of an even function

is symmetrical about the y - axis.

On the other hand, f(x) is said to be odd if f(-x) = -f(x). The examples of odd functions are: x

7 , 10. Periodic

Function: If f(x + T) = f(x), where T is a real number, then f(x) is called a periodic function. The real number

T is called a period of f(x).

The least positive period of a periodic function is called the principal period of that function. Since for all real

numbers x, sin(2π + x) = sin x and cos (2π + x ) = cos x, therefore the function sin x and cos x are periodic

functions with period 2π. Other than this 4π, 2π, 6π etc. are a lso periods of sin x and cos x.

The break - even analysis seeks to determine the output at which the producers will break-even, i.e. the

level of output where his costs will exactly meet his revenue.

The total profit associated with an output of q units is given by

Profit (P) = Total revenue – Total cost

= (Price) (Quantity) – {Fixed cost + (Variable cost) (Quantity)}

= p. q – ( k + v. q)

Solved Examples

Example 1. A firm produces an item whose production cost function is C = 80 + 4x, where x is the

number of items produced. If entire stock is sold at the rate of Rs. 8 then determine the revenue

function. Also obtain the ‘break – even’ point.

Solution. The revenue function is given by R = 8x. Also given that , C = 80 + 4x. Therefore

Profit, P = R – C = 8x – (80 + 4x) = 4x – 80

The break- even points occur when R – C = 0 or R = C, i.e. 8x = 80 + 4x or x = 20 (units).

Example 1. A company producing dry cells introduces production bonus for its employees which

increases the cost of production. The daily cost of production C(x) for x number of cells is Rs. (3.5x

(a) If each cell is sold for Rs. 6 , determine the number of cells that should be produced to ensure no

loss.

(b) If the selling price is increased by 50 paise, what would be the break – even point?

(c) If at least 6000 cells can be sold daily, what price the company should charge per cell to guarantee

no loss?

Solution. Let R(x) be the revenue due to the sales of x number of cells.

(a) Given that, cost of each cell is Rs. 6. Then R(x) = 6x. For no loss, we must have

R(x) = C(x) or 6x = 3.5x + 12,000 or 12,000/2.5 = 4,800 cells.

(b) Increased selling price is, Rs.(6 + 0.50) = Rs. 6.5. Thus, R(x) = 6.5x. Now, for break- even point, we

must have

R(x) = C(x) or 6.5x = 3.5x + 12,000 or x = 12,000/3 = 4000 cells.

(c) Let p be the unit selling price. Then revenue from the sale of 6000 cells will be, R(p) = 6000 p.

Thus, for no loss we must have

R(p) = C(p) or 6000p = 3.5 × 6000 + 12,000 or p = 33,000/6,000 = Rs. 5.5.

By definition, break- even analysis determines the optimum value of q for which profit P equals

zero, i.e. Total revenue = Total cost

or p. q – ( k + v. q) = 0

q* (optimum)

k Fixed Cost

p – v Selling Price – Variable Cost

1.11 Sequence and Series (Progression)

1.11.1 Sequences

A sequence is a function, whose domain is the set N of natural numbers or rule set of N. We may also define

a sequence as an arrangement of numbers according to a certain fixed rule. The different numbers in

sequences are called its terms and are generally denoted by a 1,

a 2,

a 3, …..

or T 1,

T

2,

T

3.....

etc. Some examples of

sequences are:

1, 3, 5, 7, 9, …… …(i)

1, 3, 9, 27, ….. …(ii)

In (i), the terms are arranged such that the difference between two consecutive terms is the same i.e. , 2.

In (ii), the terms are arranged in such a way that the ratio of two consecutive terms is the same i.e., 3.

Note. It is obvious that if the rule (which the terms of a sequence obey) is known, then all or any

term can be found.

General Term

The nth term of a sequence is called the general term and is denoted by a n

or T n

Finite and Infinite Sequences

Finite Sequence : A sequence containing finite number of terms is called a finite sequence.

For example, 2, 4, 6, 8, …, 40

24, 21, 18, … upto 6 terms.

Infinite Sequence : A sequence is called infinite if it is not a finite sequence or in other words, a

sequence having infinite terms is called an infinite sequence.

For example, 1, 3, 5, 7, …, to infinity

Note. If the general term (= T n

) of a sequence is given, we can find T 1,

T

2,

T

3

..... by putting n = 1, 2,

3, …. in T n

1.11.2 Series

The terms of a sequence connected by +ve or - ve sign from a series.

For example, 1+ 3+ 5+ 7+ …

The different terms of the series are denoted by or T 1,

T

2,

T

3.....

Solved Examples

Example 1. Write the first three terms of the sequences whose nth terms are

( i) 3n - 5 (ii) 2

n (iii) n

2

Solution. (i) Here T n

= 3n - 5

Putting n = 1, 2, 3 in T n

, we get

T

1

= 3.1 - 5 = - 2 T

2

= 3.2 - 5 = 1 T

3

(ii) Here T n

n