EM 540 Operations Research

1

• EM 540 Operations Research/
• DecS 581 Operations Management

Introduction to Quantitative Analysis (Text
Chapters 12)
2
Todays Class Outline

• Introduction/Overview of class
• Class Operation, Methods and Expectations
• What is Quantitative Analysis (QA) Approach?
• Developing Models, all kinds, new ones each week
• Making some models and playing around with them
• The Role of Computers and Spreadsheet Models in
  the Quantitative Approach
• Fundamentals of Statistics and Variability
• Starting to grasp how to manage it
• A few example problems

3
Results from this Class

• Students will be able to
• Know how this class operates and what is
  expected.
• Understand the use of modeling-Good and Bad
• Introduced to use of computers and spreadsheet
  models to perform QA
• Remember a few of elements statistical
  variability
• Give examples of discrete/ continuous random
  variables
• Explain the difference between discrete and
  continuous probability distributions
• Calculate expected values/variances/use the
  Normal table

4
Details of Class Operation

• Fast Paced-New Tool Each Week-Keep UP!
• You wont need every tool-Learn the best you can
• Mid-Term and Final Exams (55 of class)
• Homework is for you! Keep it in a book! Turn in
  after Exams (10)
• WEEKLY creation of a Homework Problem (15)
• Research Project (10)
• Application Project (10)

Link to Syllabus
5
Class Discussion

• Class Discussion
• Clarity of techniques
• Working simple problems
• Explanation of Application
• How to do this stuff in real life
• What to tell a consultant
• Confidence in selecting tools

6
Introduction

• Mathematical tools have been used for thousands
  of years (Pharaoh's used clay tablets to record
  stores of wheat)
• QA can be applied to a wide variety of problems
  (Production, Operation, Cub Scouts, Home)
• One must understand the specific applicability
  of the technique, its limitations and its
  assumptions (Always examine the assumptions-may
  exclude your world)

7
The Evolution of QA

• 2000
• 1990
• 1980
• 1970
• 1960
• 1950
• 1940
• 1930
• 1920
• 1910
• 1900

• Connectedness-Internet
• Expert Systems and Artificial Intelligence
• Decision Support
• Information System
• Goal Programming
• Decision Theory
• Network Models
• Dynamic Programming
• Game Theory
• Transportation
• Assignment Technique
• Inventory Control
• Queuing Theory
• Markov Analysis

8
The Decision-Making Process
Quantitative Analysis Logic Historic
Data Marketing Research Scientific
Analysis Modeling
Problem
Decision
?
Qualitative Analysis Weather State and federal
legislation New technological
breakthroughs Election outcome
9
Overview of Quantitative Analysis

• Scientific Approach to Managerial Decision Making
• Consider both Quantitative and Qualitative Factors

Quantitative Analysis
Meaningful Information
Raw Data
PG 1.1
10
Operations Conflict
Do the Best Possible
Accept the Optimum Solution
Exceptional OrganizationalPerformance
Continuously Improve
Reject the Optimum Solution
11
Cycle of Continuous Improvement
0.1 - What is the Goal?0.2 - How will I measure
progress toward the Goal? 1. Find the limiting
factor of the system 2. Decide how to Exploit
(optimize) the use of the limiting factor 3.
Subordinate all other factors (make sure the
limiting factor is exploited) 4. If more
capacity is needed, Elevate the limiting
factor 5. If the constraint moves (you improved
or the world changes), start over at Step 1.
12
The Quantitative Analysis Approach

• Define the problem (Systemic View)
• Develop a model (Many Types)
• Acquire data
• Develop a solution
• Test the solution
• Analyze the results and perform sensitivity
  analysis
• Implement the results

13
The QA Approach - Fig 1.1
14
Define the System / Goal / Problems

• Understand relationships
• Clear and concise statement of problem(s)
• May be the toughest part
• Look beyond symptoms to causes
• Problems are related to one another
• Must identify the right problem
• May require specific, measurable objectives

15
Developing the Model of the System

• Model is a representation of a situation
• Models may be physical, logical, scale,
  schematic or mathematical
• Models contain variables (controllable or
  uncontrollable) and parameters
• Controllable variables are called decision
  variables
• Models should be solvable, realistic, easy to
  understand and easy to modify

16
Acquire Relevant Data

• Collect enough to correctly represent the system
• Accurate data is best. But, there are only a few
  elements of the system that must be exact.
  (GIGO)
• Data may come from company reports, company
  documents, interviews, on-site direct
  measurement, and statistical sampling

17
Develop a Solution

• Manipulate the model to arrive at the best
  solution
• Solution must be practical and implementable
• Various methods
• solution of equation(s)
• trial and error
• complete enumeration
• implementation of algorithm

18
Test the Solution

• Check the Validity of the solution
• Must test both input data and model response
• Re-evaluate the accuracy, adequacy and
  completeness of input data
• Collect data from a different source and compare
• Check results for consistency - above all, do the
  make sense?

19
Analyze the Results

• Understand what action is implied by the solution
• Determine the implications of this action
• Conduct sensitivity analysis - change input value
  or model parameter and see what happens
• Use sensitivity analysis to help gain
  understanding of problem (as well as for answers)

20
Implement the Results

• Incorporate the solution into the company
• Monitor the results
• Use the results of the model and sensitivity
  analysis to help you sell the solution to
  management

21
Modeling in the Real World

• Models are may be complex
• Models can be expensive
• Models can be difficult to understand / sell
• Models are used in the real world by real
  organizations to solve real problems

22
How to Develop a Numerical Model
Profit Example

• Profits Revenue - Expenses

Profits (Price per Unit)(Number of Units
Sold) - Fixed Cost - (Variable
Costs per Unit)(Number of Units Sold)
Profits 10x - 1,000 - 5x
23
Financial Model Graphic View

Sales
Unit Price
Quantity Sold --gt
24
Finding Important Data Points
Breakeven Point
0 (Unit Price)(Number Sold) - Fixed Cost -
(Variable Cost/Unit)(Number Sold)
Then (Unit Price)(Number of Sold) -
(Variable Cost/Unit)(Number Sold) Fixed Cost
And (Unit Price - Variable Cost/Unit)(Number
Sold) Fixed Cost
25
Math Manipulation of Basic Equations
Breakeven Point - continued
Dividing both sides by (Unit Price - Variable
Cost/Unit)
We have BEP(Units) Fixed Cost/(Unit Price -
Variable Cost/Unit)
26
Even Simple Models Can Help Managers

• Gain deeper insight into the nature of their
  system and business relationships
• Find better ways to assess value in such
  relationships and
• See a way of managing / reducing (or at least
  understanding) uncertainty that surrounds
  business plans and actions

27
Financial Model ABC Accounting View

Sales
Quantity Sold --gt
28
Financial Model Fixed Cost View

Sales
Quantity Sold --gt
29
Financial Model Fixed Cost View

Sales
Quantity Sold --gt
30
Financial Model ABC Accounting View

Sales
Quantity Sold --gt
31
Models

• Are less expensive and disruptive than
  experimenting with real world systems
• Allow What if questions to be asked
• Encourage management input
• Build decision maker intuition
• Help communicate problems and solutions to others
• May provide the only way to solve large or
  complex problems in a timely fashion

32
The Downside Models

• Complex model are expensive and time-consuming to
  develop and test
• Models can be misused, misunderstood and feared
  because of their mathematical complexity
• Tend to downplay the role and value of
  non-quantifiable information
• Simplifying assumptions can distort the variables
  of the real world

33
Some Suggestions

• Use descriptive models
• Use simple models and let the mind extrapolate
  whenever possible
• Try to understand why the managers involved
  decide things the way they do
• Include managerial and organizational changes
  with other recommendations from the model
• Be a system-wide thinker

34
Mathematical Models Characterized by Risk

• Deterministic models - we know all values used in
  the model with certainty
• Probabilistic models - we know the probability
  that parameters in the model will take on a
  specific value

35
Simulation (Stochastic) Model
What is the Expected Value a Single Die when
Rolled in Combination with Other Fair Dice?
Range of a 90 confidence interval?
Mean?
3.5 7.0 10.5 14 17.5
1.6 1.1 0.9 0.8 0.7
0.7ltgt6.3 3.0ltgt11. 5.7ltgt15. 8.5ltgt19. 11ltgt23.
1 Die 2 Dice 3 Dice 4 Dice 5 Dice
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Physical Factory Model

• Work flows from left to right through processes
  with capacity shown.

MarketRequest50
Process A B C D E
RM
FG
Capability Parts 3.5 3.5 3.5 3.5 3.5per Day
How many can we promise to produce in ten days?
Production capacity of each process is
determined by a fair die (namely 1,2,3,4,5, or 6
parts per day)
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QM For Windows
38
QM For Windows
39
Excel QM
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Excel QMs Main Menu of Models
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Summary of Possible Problems in Using Models

• Define the Problem
• Conflicting viewpoints
• Departmental impacts
• Assumptions
• Develop a Model
• Fitting the Model
• Understanding the Model

• Acquire Input Data
• Accounting Data
• Validity of Data
• Develop a Solution
• Complex Mathematics
• Only One Answer is Limiting
• Solutions become quickly outdated

42
Summary of Possible Problems in Using Models

• Test the Solution
• Identifying appropriate test procedures
• Analyze the Results
• Holding all other conditions constant
• Identifying cause and effect

• Implement the Solution
• Selling the solution to others

43
Take a Break!
44
Variation!

• Life is uncertain!
• We must deal with Risk!
• A probability is a numerical statement about the
  likelihood that an event will occur

45
Basic Statements About Probability

• 1. The probability, P, of any event or state of
  nature occurring is greater than or equal to 0
  and less than or equal to 1. That is
• 0 ? P(event) ? 1
• 2. The sum of the simple probabilities for all
  possible outcomes of an activity must equal 1

46
Example 2.1

• Demand for white latex paint at Diversey Paint
  and Supply has always been 0, 1, 2, 3, or 4
  gallons per day. (There are no other possible
  outcomes when one outcome occurs, no other can.)
  Over the past 200 days, the frequencies of
  demand are represented in the following table

47
Example 2.1 - continuedFrequencies of Demand

• Number of Days
• 40
• 80
• 50
• 20
• 10
• Total 200

• Quantity Demanded (Gallons)
• 0
• 1
• 2
• 3
• 4

48
Example 2.1 - continuedProbabilities of Demand

• Quantity Frequency
• Demanded (days)
• 0 40
• 1 80
• 2 50
• 3 20
• 4 10
• Total days 200

• Probability
• (40/200) 0.20
• (80/200) 0.40
• (50/200) 0.25
• (20/200) 0.10
• (10/200) 0.05
• Total
• probability 1.00

49
Types of Probability

• Objective probability
• Can be determined by experiment or observation
• Probability of heads on coin flip
• Probably of spades on drawing card from deck

50
Introduction - continuedTypes of Probability

• Subjective probability
• Based upon judgement
• Can be determined by
• judgement of expert
• opinion polls
• Delphi method
• etc.

51
Mutually Exclusive / Collectively Exhaustive

• Events are said to be mutually exclusive if only
  one of the events can occur on any one trial
• Events are said to be collectively exhaustive if
  the list of outcomes includes every possible
  outcome heads and tails as possible outcomes of
  coin flip

52
Example 2

• Outcome
• of Roll
• 1
• 2
• 3
• 4
• 5
• 6

• Probability
• 1/6
• 1/6
• 1/6
• 1/6
• 1/6
• 1/6
• Total 1

Rolling a die has six possible outcomes
53
Example 2a

• Outcome
• of Roll 5
• Die 1 Die 2
• 1 4
• 2 3
• 3 2
• 4 1

• Probability
• 1/36
• 1/36
• 1/36
• 1/36

Rolling two dice resulting in a total of five
spots showing. There are a total of 36 possible
outcomes.
Probability of rolling 5 on two dice
4/36 or 1/9
54
Example 3
Draws Mutually
Collectively
Exclusive Exhaustive

• Draw a spade and a club
• Draw a face card and a number card
• Draw an ace and a 3
• Draw a club and a non-club
• Draw a 5 and a diamond
• Draw a red card and a diamond

• Yes No
• Yes Yes
• Yes No
• Yes Yes
• No No
• No No

One card cantbe both
Draw represents all possible options
55
Probability of Mutually Exclusive Events

• P(event A or event B)
• P(event A) P(event B)
• or
• P(A or B) P(A) P(B)
• i.e.,
• P(spade or club) P(spade) P(club)
• 13/52 13/52
• 26/52 1/2 50

P(B)

56
Probability(A and B)(Venn Diagram)
P(A)
57
Probability (A or B)
-

P(A)
P(B)
P(A and B)

P(A or B)
58
Probability of Events Not Mutually Exclusive
Event A Red hair Event B Female

• P(event A or event B)
• P(event A) P(event B) -
• P(event A and event B both occurring)
• or
• P(A or B) P(A) P(B) - P(A and B)

M
F
P(Red hair).1 P(Female).5 P(Red hair
and Female).08 P(Red hair and Male).02
P(Red hair OR Female) .1 .5 - .08 .52
59
Statistical Dependence

• Events are either
• statistically independent (the occurrence of one
  event has no effect on the probability of
  occurrence of the other) or
• statistically dependent (the occurrence of one
  event gives information about the occurrence of
  the other)

60
Which Are Independent?

• (a) Your education
• (b) Your income level
• (a) Draw a Jack of Hearts from a full 52 card
  deck
• (b) Draw a Jack of Clubs from a full 52 card
  deck
• (a) Chicago Cubs win the National League pennant
• (b) Chicago Cubs win the World Series

61
Probabilities - Independent Events

• Marginal probability the probability of an
  event occurring P(A)
• Joint probability the probability of multiple,
  independent events, occurring at the same time
  P(AB) P(A)P(B)
• Conditional probability (for independent events)
  
• the probability of event B given that event A has
  occurred P(BA) P(B)
• or the probability of event A given that event B
  has occurred P(AB) P(A)

That is, No Change
62
Probability(AB) Independent Events
Conditional Probability makes no difference if
the events are independent
63
Statistically Independent Events

• 1. P(black ball drawn on first draw)
• P(B) 0.30 (marginal probability)
• 2. P(two white balls drawn)
• P(WW) P(W)P(W) 0.700.70 0.49 (joint
  probability for two independent events)

• A bucket contains 3 black balls, and 7 white
  balls.
• We draw a ball from the bucket, replace it, stir
  and draw a second ball.

64
Statistically Independent Events - continued

• 1. P(black ball drawn on second draw, first draw
  was white)
• P(BW) P(B) 0.30
• (conditional probability)
• 2. P(white ball drawn on second draw, first draw
  was white)
• P(WW) 0.70
• (conditional probability)

• A bucket contains 3 black balls, and 7 white
  balls.
• We draw a ball from the bucket, replace it, stir
  and draw a second ball.

65
Probabilities - Dependent Events

• Marginal probability probability of an event
  occurring P(A)
• Conditional probability (for dependent! events)
• the probability of event B given that event A has
  occurred P(BA) P(AB)/P(B)
• the probability of event A given that event B
  has occurred P(AB) P(AB)/P(A)

66
Probability (AB)
AB
A
B
/

P(AB)
P(B)
P(AB)
P(AB) P(AB)/P(B)
67
Probability (BA)
AB
A
B

P(BA)
P(A)
P(BA) P(AB)/P(A)
68
Statistical Events

• Assume that we have an urn containing 10 balls of
  the following descriptions
• 4 are white (W) and lettered (L)
• 2 are white (W) and numbered (N)
• 3 are black (B) and lettered (L)
• 1 is black (B) and numbered (N)

• Then
• P(WL) 4/10 0.40
• P(WN) 2/10 0.20
• P(W) 6/10 0.60
• P(BL) 3/10 0.3
• P(BN) 1/10 0.1
• P(B) 4/10 0.4

69
Statistically Dependent Events

• Then
• P(LB) P(BL)/P(B)
• 0.3/0.4 0.75
• P(BL) P(BL)/P(L)
• 0.3/0.7 0.43
• P(WL) P(WL)/P(L)
• 0.4/0.7 0.57

If we draw a Black (B)
If we draw an L (L)
If we draw an L (L)
70
Joint Probabilities, Dependent Events

• Your stockbroker informs you that if the stock
  market reaches the 10,500 point level by January,
  there is a 70 probability the Tubeless
  Electronics will go up in value. Your own
  feeling is that there is only a 40 chance of the
  market reaching 10,500 by January.
• What is the probability that both the stock
  market will reach 10,500 points, and the price of
  Tubeless will go up in value?

71
Joint Probabilities, Dependent Events (more)

• Then
• P(MT) P(TM)P(M)
• (0.70)(0.40)
• 0.28

• Let M represent the event of the stock market
  reaching the 10,500 point level, and T represent
  the event that Tubeless goes up.

72
Revising Probabilities Bayes Theorem

• Bayes theorem can be used to calculate revised
  or posterior probabilities

73
Posterior Probabilities

• A cup contains two dice identical in appearance.
  One, however, is fair (unbiased), the other is
  loaded (biased). The probability of rolling a 3
  on the fair die is 1/6 or 0.166. The probability
  of tossing a 3 on the loaded die is 0.60.
• You have no idea which die is which, but you
  select one by chance. The probability of picking
  the fair die is 0.5.
• Then you toss the die. The result is a 3.
• What is the probability that the die you selected
  (and rolled) was fair?

74
Posterior Probabilities Continued

• We know that
• P(fair) 0.50 P(loaded) 0.50
• And
• P(3fair) 0.166 P(3loaded) 0.60
• Then
• P(3 and fair) P(3fair)P(fair)
  (0.166)(0.50) 0.083
• P(3 and loaded) P(3loaded)P(loaded)
• (0.60)(0.50) 0.300

75
Posterior Probabilities Continued

• A 3 can occur in combination with the state fair
  die or in combination with the state loaded
  die. The sum of their probabilities gives the
  unconditional or marginal probability of a 3 on a
  toss
• P(3) 0.083 0.300 0.383.
• Then, the probability that the die rolled was the
  fair one is given by

Or, a 78 chance you selected the loaded die
76
General Form of Bayes Theorem
P(AB)

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77
Further Probability Revisions

• To obtain further information as to whether the
  die just rolled is fair or loaded, lets roll it
  again.
• Again we get a 3.
• Given that we have now rolled two 3s with your
  selected die, what is the probability that the
  die you rolled is fair?

78
Further Probability Revisions - continued

• P(fair) 0.50, P(loaded) 0.50 as before
• P(3,3fair) (0.166)(0.166) 0.027
• P(3,3loaded) (0.60)(0.60) 0.36
• P(3,3 and fair) P(3,3fair)P(fair)
• (0.027)(0.5) 0.013
• P(3,3 and loaded) P(3,3loaded)P(loaded)
• (0.36)(0.5) 0.18
• P(3,3) 0.013 0.18 0.193

79
Further Probability Revisions - continued
So, 93 sure it was a loaded die
80
Further Probability Revisions - continued

• To give the final comparison
• P(fair3) 0.22
• P(loaded3) 0.78
• P(fair3,3) 0.067
• P(loaded3,3) 0.933

Increased confidence resulted from additional
knowledge.
81
Random Variables

• Discrete random variable - can assume only a
  finite or limited set of values- I.e., the number
  of automobiles sold in a year
• Continuous random variable - can assume any one
  of an infinite set of values - I.e., temperature,
  product lifetime

82
Variables (Numeric)
83
Variables (Non-numeric)
84
Probability Distributions
Figure 2.5 Probability Function
85
Expected Value Discrete Probability Distribution
Center of Gravity type equation.
86
Variance of a Discrete Probability Distribution
Moment of Inertia type equation.
87
Binomial Distribution

• Assumptions
• 1. Trials follow Bernoulli process only two
  possible outcomes
• 2. Probabilities stay the same from one trial to
  the next
• 3. Trials are statistically independent
• 4. Number of trials is some positive integer

88
Binomial Distribution
n number of trialsr number of successesp
probability of successq probability of failure
Probability of r successes in n trials
89
Binomial Distribution
Mean
Variance
90
Binomial Distribution
N 5, p 0.50
91
Probability Distribution Continuous Random
Variable

• Probability density function - f(X)

Normal Distribution
2
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-
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/
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92
Normal Distribution for Different Values of ?
Fig. 2.7
?50
?40
?60
? 2
93
Normal Distribution for Different Values of ??
Fig. 2.8
?0.1
? 1
?0.2
?0.3
94
Three Common Areas Under the Curve

• Three Normal distributions with different areas

95
The Relationship Between Z and X
?100 ?15
XZ
96
Haynes Construction Example - Fig. 2.11
What is the probability Haynes will complete
before penalties kick in at 125 days?
97
Haynes Construction Example - Fig. 2.12
What is the probability Haynes will complete in
time to receive Bonus (Less than 75 days)?
98
Haynes Construction Example - Fig. 2.13
What is the probability Haynes will complete
between 110 and 125 days?
99
Negative Exponential Distribution
Expected value 1/? Variance 1/?2
?5 On average, 5 units can be serviced per
period So Expected Service Time0.2 period Area
under the curve shows probability of being less
than that Service Time
Expected Service Time
100
The Poisson Distribution
Expected value ? Variance ?
?2
101
Congratulation! You made it to the end!

• Things to do this Week
• Zip through the first 80 Pages of a heavily
  mathematical text! You will have to learn to be
  quick!
• Work the assigned homework for your Homework Book
  (not to turn-in yet -- the book is due at mid
  term).
• Invent a Homework problem from Chapter 2
  material. Work your problem, turn-in the problem
  and solution (jholt_at_wsu.edu).
• Read Chapter 3
• Overload?? Never! The important part is to
  remember, we will be making models of all kinds.
  And, variability is an issue. Dr Holt