3. De La Salle University- Dasmariñas
College of Science
Mathematics Department
EDWIN S. BUNAG
Asst. Professor 5
esbunag@dlsud.edu.ph
1997-present
Master of Arts in Mathematics (2001)
Bachelor of Science in Mathematics (1996)
College Algebra, Trigonometry, Math of Investment,
Statistics, Business Calculus, Quantitative Techniques,
Math of Accounting and Finance, Theory of Interest,
Plane and Solid Mensuration

4. Course Description
 This course deals with analysis of decision-making
situations in business environment using probabilities,
inventory, forecasting, linear programming, and
structured linear programming as applied in business
processes. This course will equip the students with the
necessary skills and knowledge of the different
Management Science/Operation Research techniques,
which will develop their decision-making capabilities.
Completion and mastery of the course is a great tool in
decision making for future business executives.

5. Course Outline
 Introduction
 Probability and Probability Distribution
 Decision Analysis
 Utility and Game Theory
 Forecasting
 Linear Programming
 Transportation and Assignment Problems
 Inventory Models
 Waiting Line models
 Simulation

6. Over View (Need for Good Decision)
Managers
Success
A Manager
Should Be a
Good
Decision Maker
Increasing competition
Changing markets
Changing customers requirements
More complex business environment
Complex information needs and system
Increased uncertainty
Larger error costs

7. Problem solving can be defined as the process
of identifying a difference between the actual
and the desired state of affairs and then taking
action to resolve the difference.
Problem Solving and Decision Making

8. Over view (Decision Making Process)
SCIENTIFIC METHOD OF
SOLVING PROBLEM
Observation
Define the Problem
Formulation of Hypothesis
Experimentation
Verification

9. Over view (Decision Making Process)
Decision Making
Structuring the Problem Analyzing the Problem
Define the
problem
Identify the
alternatives
Determine
the criteria
Evaluate
the
alternatives
Choose an
alternative
Evaluate
the results
Implement
the
decision
Qualitative Analysis
Quantitative Analysis

10. MANAGEMENT SCIENCE /
OPERATIONS RESEARCH/DECISION
SCIENCE/QUANTITATIVE TECHNIQUES
 the discipline of using mathematics, and other
analytical methods or quantitative methods, to help
make better business decisions.

11. WHEN DO MANAGERS USE
QUANTITATIVE TECHNIQUES
1. The problem is complex.
2. The problem is especially important (e.g., a great
deal of money is involved).
3. The problem is new.
4. The problem is repetitive.

12. History and Practical Application of
Operations Research / Quantitative
Methods

13. Chapter 2
Introduction to Probability
 Experiments and the Sample Space
 Assigning Probabilities to
Experimental Outcomes
 Events and Their Probability
 Some Basic Relationships
of Probability
 Bayes’ Theorem

14. Probability as a Numerical Measure
of the Likelihood of Occurrence
0 1.5
Increasing Likelihood of Occurrence
Probability:
The event
is very
unlikely
to occur.
The occurrence
of the event is
just as likely as
it is unlikely.
The event
is almost
certain
to occur.

15. REVIEW OF THE BASIC
PROBABILITY CONCEPTS
Experiment -any process that generates outcome.
Sample Space - the set of all possible outcomes of a given experiment.
Event - one or more possible outcomes of an experiment.
Mutually Exclusive Events - two events that can not occur at the same time.
Otherwise not mutually exclusive.

16. THREE TYPES OF PROBABILITY
N
nEP )(
Where:
P(E) refers to the probability of an event will occur.
n refers to the number of elements in the event.
N refers to the number of elements in the sample space.
The Classical Approach
•classical probability defines the probability that an event will occur as

17. THREE TYPES OF PROBABILITY
The Relative Frequency Approach
this method of defining probability uses the relative frequencies
of past occurrences as probabilities.
The Subjective Approach
subjective probabilities are based on the personal belief or
feelings of the person who makes the probability estimate.

18. PROBABILITY RULES
P(A or B) = P(A) + P(B)
If two events
A and B are mutually
exclusive, the
Special Rule of Addition
states that the
Probability of A or B occurring
equals the sum of their
respective probabilities.

19. Addition Rule for Mutually Exclusive Events
Arrival Frequency
Early 100
On Time 800
Late 75
Canceled 25
Total 1000
New England Commuter Airways recently supplied the following
information on their commuter flights from Boston to New York:
What is the probability that a
flight is either early or late?
What is the probability that a
flight is either late or cancelled?

20. PROBABILITY RULES
Person Sex Age
1 Male 31
2 Male 33
3 Female 46
4 Female 29
5 Male 41
•The Addition Rule for Not Mutually Exclusive Events
P(A or B) = P(A) + P(B) – P(A and B)
Example:
The data below refers to the number of persons in the city council.
The members of the council decided to elect a chairperson by random draw.
What is the probability that the chairperson will be either female or over 35?
What is the probability that the chairperson will be either Male or over 32?

21. PROBABILITY RULES
•Multiplication Rule with Independent Events
•Multiplication Rule with Dependent Events
P(A and B) = P(A) P(B)
Example:
If two coins are flipped once, what is the probability that both
coins will turn up heads?
A nationwide survey found that 72% of people in the United
States like pizza. If 3 people are selected at random, what is the
probability that all three like pizza?

22. PROBABILITY RULES
Example:
A bag of fruits contains six mangoes, four atis and five guavas.
If you are sampling without replacement, what is the probability
of getting a mango and an atis in that order?

23. PROBABILITY RULES
The probability of event A
occurring given that the
event B has occurred is
written P(A|B)
A Conditional Probability is the probability of a
particular event occurring, given that another
event has occurred.
P(A|B) = P(A and B)/P(B)
P(B|A) = P(A and B)/P(A)

24. PROBABILITY RULES
Major Male Female Total
Accounting 170 110 280
Finance 120 100 220
Marketing 160 70 230
Management 150 120 270
Total 600 400 1000
The Dean of the School of Business at Owens
University collected the following information about
undergraduate students in her college:
Given that the student is a
female, what is the
probability that she is an
accounting major?
Given that the student is a
male, what is the
probability that he is a
Marketing major?

25. Bayes’ Theorem
New
Information
Application
of Bayes’
Theorem
Posterior
Probabilities
Prior
Probabilities
 Often we begin probability analysis with initial or
prior probabilities.
 Then, from a sample, special report, or a product
test we obtain some additional information.
 Given this information, we calculate revised or
posterior probabilities.
 Bayes’ theorem provides the means for revising the
prior probabilities.

26. A manufacturing firm receives shipments of parts from
different suppliers. There is a 65% chance that a part is
from supplier 1 and 35% that a part is from supplier 2.
Additional information is given on the conditional
probability of receiving good and bad parts from two
suppliers:
Good Parts Bad Parts
Supplier 1 0.98 0.02
Supplier2 0.95 0.05
Find the probability that a bad part is from supplier 1?
Find the probability that a bad part is from supplier 2?
Example (page 43)

27. Bayes’ Theorem
1 1 2 2
( ) ( | )
( | )
( ) ( | ) ( ) ( | ) ... ( ) ( | )
i i
i
n n
P A P B A
P A B
P A P B A P A P B A P A P B A
 To find the posterior probability that event Ai will
occur given that event B has occurred, we apply
Bayes’ theorem.
 Bayes’ theorem is applicable when the events for
which we want to compute posterior probabilities
are mutually exclusive and their union is the entire
sample space.

28. Tabular Approach
 Step 1
Prepare the following three columns:
Column 1 The mutually exclusive events for which
posterior probabilities are desired.
Column 2 The prior probabilities for the events.
Column 3 The conditional probabilities of the new
information given each event.

29. Tabular Approach
 Step 2
Column 4
Compute the joint probabilities for each event
and the new information B by using the
multiplication law.
Multiply the prior probabilities in column 2 by
the corresponding conditional probabilities in
column 3. That is, P(Ai IB) = P(Ai) P(B|Ai).

30. Tabular Approach
 Step 3
Column 4
Sum the joint probabilities. The sum is the
probability of the new information, P(B).

31.  Step 4
Column 5
Compute the posterior probabilities using the
basic relationship of conditional probability.
The joint probabilities P(Ai IB) are in column 4
and the probability P(B) is the sum of column 4.
Tabular Approach
)(
)(
)|(
BP
BAP
BAP i
i

32. Assignment
1. Give 2 example for each probability rules.
2. Examples should be business related problems.
3. Provide complete solution.
4. Discuss how it can be used in decision making.

33. Random Variable
Binomial Probability Distribution
Poisson Probability Distribution
Exponential Probability Distribution
Normal Probability Distribution

34. Random Variable
Is a function whose value is a real number
determined by each element in the sample space.
Example
A coin is tossed three times. List down the elements of the
sample space. List down the possible values of the following
random variables:
X: the number of heads that fall
Y: the number of tails that fall
W: the number of heads minus the number of tails
A random variable is a numerical description of the
outcome of an experiment.

35. Random Variable
Sample
space
HHH
HHT
HTH
THH
HTT
THT
TTH
TTT
Random
Variable
X: no. of heads
3
2
2
2
1
1
1
0
Random
Variable
Y: no. of tails
0
1
1
1
2
2
2
3
Random
Variable
W: X – Y
3
1
1
1
-1
-1
-1
-3

36. Probability Distribution
Probability distribution is a list of all possible outcome
of a random variable with their corresponding probabilities.
Example
Probability distribution of the following random variable:
Random Variable
X
P(X=0) = 1/8
P(X=1) = 3/8
P(X=2) = 3/8
P(X=3) = 1/8
Random Variable
Y
P(Y=0) = 1/8
P(Y=1) = 3/8
P(Y=2) = 3/8
P(Y=3) = 1/8
Random Variable
W
P(W=-3) = 1/8
P(W=-1) = 3/8
P(W=1) = 3/8
P(W=3) = 1/8
Note:
Discrete Random Variable : defined over a discrete sample space.
Continuous Random Variable: defined over a continuous sample space.

37. Expected Value and Variance
The expected value, or mean, of a random variable
is a measure of its central location.
The variance summarizes the variability in the
values of a random variable.
The standard deviation, , is defined as the positive
square root of the variance.
Var(x) = 2 = (x - )2f(x)
E(x) = = xf(x)

38.  a tabular representation of the probability
distribution for TV sales was developed.
 Using past data on TV sales, …
Example: JSL Appliances
Number
Units Sold of Days
0 80
1 50
2 40
3 10
4 20
200
x f(x)
0 .40
1 .25
2 .20
3 .05
4 .10
1.00

39. Types of Probability Distribution
Discrete Probability Distribution
1. Binomial Probability Distribution
2. Poisson Probability Distribution
Continuous Probability Distribution
3. Exponential Probability Distribution
4. Normal Probability Distribution

40. The probability distribution is defined by a
probability function, denoted by f(x), which provides
the probability for each value of the random variable.
The required conditions for a discrete probability
function are:
Discrete Probability Distributions
f(x) > 0
f(x) = 1

41. The Binomial Probability Distribution
It describes discrete data resulting from an experiment
called a Bernoulli process.
Properties of the Binomial Distribution
1. The sample consists of a fixed number of observations, n.
2. Each trial has only two possible outcomes.
3. The probability of a success and failure on any trial
remains fixed over time.
4. The trials are statistically independent.

42. The Binomial Probability Distribution
Where:
P(r) is the probability of r successes in n trials.
n is the total number of trials.
r represents a certain number of successes.
p represents the probability of success.
q represents the probability of failure.
rnr
rn qpCP(r)
Binomial Formula
( )!
( ) (1 )
!( )!
x n xn
f x p p
x n x

43. Binomial Probability
Distribution
(1 )np p
 Expected Value
 Variance
 Standard Deviation
E(x) = = np
Var(x) = 2 = np(1 p)

44. Example: Evans Electronics
 Binomial Probability Distribution
Evans is concerned about a low retention rate for
employees. In recent years, management has seen a
turnover of 10% of the hourly employees annually.
Thus, for any hourly employee chosen at random,
management estimates a probability of 0.1 that the
person will not be with the company next year.
Choosing 3 hourly employees at random, what is the
probability that 1 of them will leave the company this
year?

45. The Binomial Probability Distribution
Sample Problem 2
5 employees are required to operate a chemical process; the
process cannot be started until all 5 workstations are manned.
Employee records indicate there is a 0.4 chance of any one
employee being late, and we know that they all come to work
independently of each other. Management is interested in
knowing the probabilities of 0, 1, 2, 3, 4, or 5 employees being
late, so that a decision concerning the number of backup
personnel can be made.

46. The Poisson Probability Distribution
Named after the mathematician and physicist
Siméon Poisson (1781-1840).
It describes the distribution of arrivals per unit time at a
service facility.
A Poisson distributed random variable is often useful in
estimating the number of occurrences over a specified interval
of time or space.
It is used to describe a number managerial situations including
the demand (arrivals) of patience at a health clinic, the
distribution of telephone calls going through a central switching
system, the arrivals of vehicles at a toll booth, the number
of accidents at an intersection, and the number of looms
waiting for service in a textile mill.

47. The Poisson Probability Distribution
THE POISSON FORMULA
x
e
–
P(x) =
x
Where:
P(x) the probability of exactly x occurrences.
x
Lambda (the average number of occurrences per
interval of time) raised to the x power.
e–
e (2.71828…) raised to the negative lambda power.
x x factorial
!
)(
x
e
xf
x

48. The Poisson Probability Distribution
Sample Problem 1
The manager of DWEIN BANK records the arrival of
customers and on the average; three costumers arrive
per minute at the bank during the noon to 1 P.M. hour.
What is the probability that in a given minute exactly
two customers will arrive? What is the probability that
more than two customers will arrive in a given minute?

49. Example: Mercy Hospital
Patients arrive at the
emergency room of Mercy
Hospital at the average
rate of 6 per hour on
weekend evenings.
What is the
probability of 4 arrivals in
30 minutes on a weekend evening?
MERCY

50. The Exponential Probability Distribution
It describes a continuous random variable of the interarrival time.
It describes the distribution of service time at a service facility.
Some Applications
Used in waiting line theory to model the length of time between
arrivals in processes such as customers at a bank’s ATM, clients
in a fast-food restaurant and patients entering a hospital
emergency room.

51. The Exponential Probability Distribution
THE EXPONENTIAL FORMULA
P(T t) = 1 – e– t
Where:
P(T t) the probability that the
service time T will be less
than or equal to t.
the mean service rate
e 2.71828…

52. The Exponential Probability Distribution
Sample Problem 1
Suppose the distribution in the
figure represents the time it takes for a
microcomputer repair facility to repair
1 unit, and suppose the mean repair
time has been found to be 3 hours.
What is the probability that the service
on a faulty microcomputer will be
completed in 2 or fewer hours?
0
p(t)
t

53. The Exponential Probability Distribution
Sample Problem 2
Suppose that customers arrive at a bank’s ATM at the rate
of 20 per hour. If a customer has just arrived, what is the
probability that the next customer will arrive within 6 minutes?

54. The Normal Probability Distribution
The normal probability distribution is frequently
referred to as the Gaussian distribution
(named after the mathematician-astronomer
Karl Gauss 1777-1855).
x
z
Standard Score
It is characterized by a normal curve.
* it is bell shaped
* it has a single highest peak
* it is symmetrical about the center
* the curve is asymptotic to the horizontal
line
* the area under the curve is 100% or 1

55. The Normal Probability Distribution
Applications:
1. Lifetimes of batteries in a certain application are
normally distributed with mean 50 hours and
standard deviation 5 hours.
a. Find the probability that a randomly selected
battery lasts between 42 and 52 hours.
b. Find the 40th percentile of battery lifetimes

56. The Normal Probability Distribution
2. A process manufactures ball bearings whose diameters are
normally distributed with mean 2.505 cm and standard deviation
0.008 cm. Specifications call for the diameter to be in the interval
2.5 0.01 cm.
a. What proportion of the ball bearings will meet the
specification?
b. The process can be recalibrated so that the mean will
be equal to 2.5 cm, the center of the specification
interval. The standard deviation of the process
remains 0.008 cm. What proportion of the diameters
will meet the specification?

57. The Normal Probability Distribution
c. The process has been recalibrated so that the mean
diameter is now 2.5 cm. To what value must the standard
deviation be lowered so that 95% of the diameters will
meet the specification?

58. The Normal Probability Distribution
Shaft manufactured for use in optical storage devices have
diameters that are normally distributed with mean 0.652 cm
and standard deviation 0.003 cm. The specification for the
shaft diameter is 0.650 0.005 cm.
a. What proportion of the shafts manufactured by this process
meet the specifications?
b. The process mean can be adjusted through calibration. If the
mean is set to 0.650 cm, what proportion of the shafts will
meet the specifications?
c. From part b, how many shaft will be rejected if there are
100,000 shaft produced.

59. Qualitative analysis is based primarily on the
manager’s judgment and experience; it includes
the manager’s intuitive “feel” for the problem
and is more an art than a science.

60. Quantitative analysis will concentrate on the
quantitative facts or data associated with the
problem and develop models or mathematical
expressions that describe the objectives, constraints,
and other relationships that exist in the problem.
Quantitative Analysis Process
 Model Development
 Data Preparation
 Model Solution
 Report Generation

61. Types of models:
1. Iconic models are physical replicas of real
objects.
Ex.
Model Development

62. Types of models:
2. Analog models are physical in form but do not have
the same physical appearance as the object being
modeled.
Ex.
The position of the needle on the
dial of a speedometer represents
the speed of the automobile.
Model Development

63. Types of models:
3. Mathematical models are representations of a
problem by a system of symbols and mathematical
relationships or expressions.
Model Development

64. Models Used in Economics
Cost function C(x)
is the cost of producing x units of the
commodity.
Revenue function R(x)
is the revenue obtained from selling x
units of the commodity R(x)=xp(x).
Profit function P(x)
is the profit obtain from selling x units of
the commodity P(x)=R(x) – C(x).

65. Breakeven Analysis
Market research indicates that consumers will buy
x thousand units of a particular kind of coffee maker
when the unit price is dollars.
The cost of producing the x thousand units is
thousand dollars.
a. What are the revenue and profit functions
for this production process?
b. For what values of x is production of coffee
makers profitable?

66. Linear Programming Model
Objective Function: Maximize Profit: Z = 8x + 6y
Subject to:
(Assembly constraint) x + 2y 60
(Finishing constraint) x + 4y
(Implicit constraint) x, y

67. Types of criteria:
1. Single-criterion decision problems are
those in which the objective is to find the
best solution with respect to one criteria.
2. Multicriteria decision problems are
those that involve more than one criterion.