Biz - Quatitative.Managment.Method Chapter.01

8/9/2019 Biz - Quatitative.Managment.Method Chapter.01

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QUANTITATIVE METHODSQUANTITATIVE METHODS

FOR MANAGEMENTFOR MANAGEMENT

(QMM)(QMM)

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Chapter IChapter I

BASIC CONCEPTSBASIC CONCEPTS


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Variables, Constants, and ParametersVariables, Constants, and Parameters

A variable is something whose magnitudecan change, i.e., something that can take ondifferent values.

Variables frequently used are price, profit,revenue, cost, national income, consumption,investment, imports, exports and so on.

Since each variable can assume variousvalues, it must be represented by a symbol(X, Y, P, etc,) instead of a specific number.

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For example, we may represent price by P,profit by , revenue by R, cost by C, nationalincome by Y, and so forth.

A constant is a magnitude that does notchange.

When a constant is joined to a variable, it isoften referred to as the coefficient of that

variable. A coefficient may be symbolic ratherthan numerical.

In order to attain a higher level of generality, wecan use the expression aP in lieu of 7P in amodel (let the symbol a stands for a givenconstant ).


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As a matter of convention, parametric constants(parameters) are normally represented by the

symbols a, b, c, or their counterparts in theGreek alphabet: , , .

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Equations and Inequalities:

An equation is a statement that two expressionsare equal, and an inequality is a statement thattwo expressions are not equal.

An example of an equation is Y = 15.52X

In this equation, X and Y variables. For

example, X may represent the number of bagsof rice (each bag contains 50 kgs) sold by afarmer and Y may represent total income fromrice sales. In this case, 15.52 represents theprice per bag of rice ($15.52). This equationmay be part of a model analyzing rice prices orsome other aspect of the agricultural industry.


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An example of an inequality is 5X < 270

In this case, assume X is a variablerepresenting the number of machine partsmanufactured by a firm. In addition, 5represents the worker-hours required tomanufacture one machine part and 270represents the number of worker-hoursavailable for producing these parts. Thisinequality may be part of a model explaining thefirms production process.

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The RealThe Real--Number SystemNumber System Whole numbers such as 1, 2, 3,..are called

positive integers; these are the numbers mostfrequently used in counting.

The negative counter parts -1, -2, -3,.. arecalled negative integers.

The number 0 (zero) is neither positive nornegative, and is in that sense unique.

The whole set of positive integers, negativeintegers and 0 (zero) is called as the set of allintegers.


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Integers do not exhaust all the possiblenumbers, for we have fractions, such as 2/3,5/4, and 7/3, which if placed on a ruler

would fall between the integers. Also, we havenegative fractions, such as -1/2 and -2/5.Together, these make up the set of all fractions.

The common property of all fractional numbersis that each is expressible as a ratio of twointegers; thus fractions qualify for the

designation rational numbers. But integers arealso rational, because any integer n can beconsidered as the ratio n/1.

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The set of all integers and the set of all fractionstogether form the set of all rational numbers.

The numbers that cannot be expressed asratios of a pair of integers are called asirrational numbers.

Examples of Irrational Numbers:

2 = 1.4142.., which is a nonrepeating,nonterminating decimal.

= 3.1415(representing the ratio of thecircumference of any circle to its diameter),which is also a nonrepeating, nonterminatingdecimal.


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Each irrational number, if placed on a ruler, wouldfall between two rational numbers, so that, just asthe fractions fill in the gaps between the integers

on a ruler, the irrational numbers fill in the gapsbetween rational numbers.

Both rational numbers and irrational numbersconstitute the set of real numbers.

When the set R is displayed on a straight line, werefer to the line as the real line.

The square roots of the negative numbers are theimaginary numbers.

The set of real numbers and the set of imaginarynumbers are mutuall exclusive.

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Integers

Real Numbers

Irrational NumbersRational Numbers

Fraction


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The Concept of SetsThe Concept of Sets

A set is simply a collection of distinct objects.

These objects may be a group of (distinct)numbers, or something else.

The objects in a set are called the elements ofthe set.

A set can be written in two ways, i.e., byenumeration and by description.

If we let S represent the set of three numbers 2,3, and 4, we can write, by enumeration of theelements, S = {2, 3, 4}

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If we let I denote the set of all positive integers,enumeration becomes difficult, and we mayinstead simply describe the elements and write

I = {x | x a positive integer}

This is read as follows: I is the set of all(numbers) x, such that x is a positive integer.

Example: The set of all real numbers greaterthan 2 but less than 5 (call it J) can beexpressed symbolically as J = {x | 2 < x


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A set with a finite number of elements is calleda finite set.

Set I and set J, each with an infinite number ofelements, are, on the other hand, examples ofan infinite set.

Finite sets are always denumerable (orcountable), i.e., their elements can be countedone by one in the sequence 1, 2, 3, Infinitesets may be either denumerable (set I above),

or nondenumerable (set J above).

The membership in a set is indicated bysymbol, which is read : is an element of.

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Relationships between SetsRelationships between Sets If two sets S1 and S2 happen to contain identical

elements,

S1 = {2, 7, a, f} and S2 = {2, a, 7, f}

Then S1 and S2 are said to be equal (S1= S2)

The order of appearance of the elements in aset is immaterial.

If one element is different, two sets are notequal.

One set may be a subset of another set.


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T is a subset of S if and only if x T implies

x S.

Using the set inclusion symbols (is containedin) and (includes), we may then write

T S or S T

It is possible that two given sets happen to besubsets of each other. When this occurs, we

can be sure that these two sets are equal. Tostate this formally: we can have S1S2 andS2S1

if and only if S1 = S2.

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The set of all integers is a subset of the set ofall rational numbers.

The set of all rational numbers is a subset ofthe set of all real numbers.

2n numbers of subsets can be formed from theset having n numbers of elements.

The set itself can be considered as one of thesubsets.

A null set is considered a subset of any set.

A set containing no element is called null set orempty set, and is denoted by or { }.


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The two sets are said to be disjoint if they haveno elements in common.

For example, the set of all positive integers andthe set of all negative integers are disjoint sets.

The two sets are neither equal nor disjoint; also,neither set is a subset of the other, when theyhave some elements in common but someelements peculiar to each.

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Operation on SetsOperation on Sets

1. Intersection: A B = {x | x A and x B}

2. Union: A B = { x | x A or x B}

3. = {x | x U and x A} where U is theuniversal set.

Example 1: If A = {3, 5, 7} and B = {2, 3, 4, 8}

A B = {2, 3, 4, 5, 7, 8}

A B = {3}


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Example 2:

The union of the set of all integers and the set

of all fractions is the set of all rationalnumbers.

The union of the rational-number set and theirrational-number set yields the set of all realnumbers.

Example 3:

If A = {-3, 6, 10} and B = {9, 2, 7, 4},

Then A B = . Set A and B are disjoint.

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Example 4:If U = {5, 6, 7, 8, 9} and A = {5, 6},

then = {7, 8, 9}

Example 5:

The complement of U is a null set. ( = ).


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A B

A B

AA

A~

A~

Compliment

Intersection

Union

A B

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Example 6:

The set A includes all people who have seen arecent television advertisement for a product.

The set B includes all the people who haveheard a recent radio advertisement for thesame product.

The intersection of these two sets forms a setC, where C = A B.

The set C includes all the people who havebeen exposed to both advertisements.


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Example 7:

A restaurant chain operates two types of

restaurants. Set S includes all the people who have been

customers at the chains cafeteria-stylerestaurants and set T includes all the customersat the table-service restaurants.

The union of S and T, designated as set R

where R = S U T, includes all the people whohave been to either type of restaurant or toboth.

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Laws of Set Operations:

1. A U B = B U A (Commutative law of unions)

2. A B = B A (Commutative law of intersections)

3. A U (B U C) = (A U B) U C

(Associative law of unions)

4. A (B C) = (A B) C

(Associative law of intersections)5. A U (B C) = (A U B) (A U C)

(Distributive law of unions)

6. A (B U C) = (A B) U (A C)

Distributive law of intersections


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Example 8: Verify the distributive law, given

A = {4, 5}, B = {3, 6, 7}, and C = {2, 3}

I. Left: A U (B C) = {4, 5} U {3} = {3, 4, 5}

Right: (A U B) (A U C) = {3, 4, 5, 6, 7} {2, 3, 4, 5} = {3, 4, 5}

II. Left: A (B U C) = {4, 5} {2, 3, 6, 7} =

Right: (A B) U (A C) = U =

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Class Assignment 1:

If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Set A = {1, 2, 3, 4, 5, 6}

Set B = {2, 4, 6, 8, 10}

Find 1. A U B 2. A B 3. A U

4. A U 5. A U


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Example 9:

Set S includes all people who have purchased ahousehold cleaning product at a supermarket.

Set T includes those who have bought the productat a hardware store.

Set R includes those who have purchased it at adrug store.

In order to understand its customers and develop amarketing strategy, the products manufacturerwants to know the set of all people who have bothbought the product in a supermarket and eitherone or both of the other two types of outlets.

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S

T

R

S (T U R) = (S T) U (S R)


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The relationship is shown in the Venn diagramgiven above.

The set T U R includes all of circles T and R

including their intersection. The firm needs to know the set, which includes

the intersection of set S with the union of T and R(see the shaded area of the Venn diagram).

The shaded area demonstrates the distributivelaw, i.e., S (T U R) = (S T) U (S R).

The shaded area includes those people who havepurchased the product at (1) both a super marketand a hardware store, (2) both a super marketand a drug store, or (3) a super market, hardwarestore, and drug store.

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Class Assignment 2:

A food manufacturer conducts a survey of consumerusage of three similar breakfast cereals (A, B, and C)which it produces. A total of 5000 consumers issampled and the following results are obtained:

i. 1500 have used cereal A.

ii. 600 have used cereal B.

iii.700 have used cereal C.

iv.300 have used A and B.

v. 200 have used A and C.

vi.50 have used B and C.

vii.None have used all three.


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1.Draw a Venn diagram showing these surveyresults.

2.With these results, demonstrate the distributivelaw of sets of the form, A (B U C) = (A B) U(A C).

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Exponents:Exponents:Rule I: xm X xn = xm+n

Proof: xm X xn=(x X x XX x)(x X xX x)

m terms n terms

Rule II: xm/ xn = xm-n (x 0)

Proof: xm/ xn = (xXxXXx) / (xXxXXx)

= xXxXXx = xm-n

m-n terms

Rule III: X-n = 1 / xn (x 0)


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Rule IV: x0 = 1 (x 0)

Rule V: x1/n = x

Rule VI: Xm/n = Xm

Rule VII: (xm)n = xmn

Rule VIII: xm X ym = (xy)m

Rule IX: Xm/Ym = (X/Y)m

n

n

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Class Assignment 3:Evaluate each of the following expressions.

1.(27)-1/3

2.(82)1/3

3.(125)1/3 / 52

4.[(7)0(8)1/3]-5

5.(271/3)(272/3)


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Simplify the following expressions:

1.(X4) (X1/2) (X-3)

2.(1 / X3) (X2 / Y1/3)

3.(X6 / Y2) / (6 / Y)

4.(XYW)2(W2X3)1/2

5.(1 / W2) (3XYW)4

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Factoring

Example 1:

4Y3 5XY2 + 6Y

= Y(4Y2 5XY + 6)

Example 2:

Y = (X + a) (X + b) = X2 + (a +b) X + abY = X2 7X + 12 = (X 4)(X 3)


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Example 3:

Y = abX2 + (ad + cb)X + cd = (aX + c)(bX + d)

Y = 8X2 + 26X + 15 = (4X + 3)(2X + 5)

Example 4:

Y = X2 25 = (X + 5)(X 5)

Example 5:

X3 + a3 = (X + a)(X2 aX + a2)

X3 + 27 = (X + 3)(X2 3X + 9)

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Example 6:

X3 a3 = (X a)(X2 + aX + a2)

X3 125 = (X 5)(X2 + 5X + 25)


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Class Assignment 4:

Completely factor each of the following

expressions.

1. 6X3 4X2 + 12XY

2. Y2 + 6Y 16

3. 2X2 6X 8

4. X2 225

5. X3 + 8000

6. X3 216

7. 5Y2 17Y + 14

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Biz - Quatitative.Managment.Method Chapter.01