11/16/12 1 Introduction to Quantitative Analysis Bus-221-QM Lecture 14 Course review To accompany Quantitative Analysis for Management, Eleventh Edition, by Render, Stair, and Hanna Course Overview 2-2 Lecture TITLE 1 Break Even Analysis (Chapter 1) 2 Bayes Theorem Regression Analysis (Chapter 2, not English) 3, 4, 5, 6 Decision Analysis (Chapter 3) 7, 8 Forecasting (Chapter 5) 9 Inventory Control (Chapter 6) 10, 11 Linear Programming (Chapter 7) 12 Project Management (Chapter 13) 13 Waiting Lines (Chapter 14) The Quantitative Analysis Approach 1-3 Implementing the Results Analyzing the Results Testing the Solution Developing a Solution Acquiring Input Data Developing a Model Defining the Problem Figure 1.1 Developing a Model Quantitative analysis models are realistic, solvable, and understandable mathematical representations of a situation. 1-4 There are different types of models: $ Advertising $ Sales Y = b 0 + b 1 X Schematic models Scale models Acquiring Input Data Input data must be accurate 1-5 Data may come from a variety of sources such as company reports, company documents, interviews, on-site direct measurement, or statistical sampling. Garbage In Process Garbage Out How To Develop a Quantitative Analysis Model 1-6 A mathematical model of profit: Profit = Revenue – Expenses
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11/16/12
1
Introduction to Quantitative Analysis
Bus-221-QM Lecture 14
Course review
To accompany Quantitative Analysis for Management, Eleventh Edition, by Render, Stair, and Hanna
Course Overview
2-2
Lecture TITLE 1 Break Even Analysis (Chapter 1) 2 Bayes Theorem
Three Types of Probabilities ! Marginal)(or)simple))probability)is)just)the)probability)of)a)
single)event)occurring.)P)(A))
2-14
! Joint probability is the probability of two or more events occurring and is equal to the product of their marginal probabilities for independent events.
P (AB) = P (A) x P (B) ! Conditional probability is the probability of event
B given that event A has occurred. P (B | A) = P (B)
! Or the probability of event A given that event B has occurred
P (A | B) = P (A)
Revising Probabilities with Bayes� Theorem
2-15
Posterior Probabilities
Bayes� Process
Bayes� theorem is used to incorporate additional information and help create posterior probabilities.
Prior Probabilities
New Information
Figure 2.4
General Form of Bayes’ Theorem
2-16
)()|()()|()()|()|(
APABPAPABPAPABPBAP
!!+=
We can compute revised probabilities more directly by using:
where the complement of the event ; for example, if is the event “fair die”, then is “loaded die”.
A random variable assigns a real number to every possible outcome or event in an experiment.
X = number of refrigerators sold during the day
Probability Distribution of a Discrete Random Variable
2-18
The students in Pat Shannon’s statistics class have just completed a quiz of five algebra problems. The distribution of correct scores is given in the following table:
For discrete random variables a probability is assigned to each event.
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Probability Distribution of a Discrete Random Variable
The Probability Distribution follows all three rules: 1. Events are mutually exclusive and collectively exhaustive. 2. Individual probability values are between 0 and 1. 3. Total of all probability values equals 1.
Table 2.6
Expected Value of a Discrete Probability Distribution
2-20
( ) ( )∑=
=n
iii XPXXE
1
( ) )(...)( 2211 nn XPXXPXXPX +++=
The expected value is a measure of the central tendency of the distribution and is a weighted average of the values of the random variable.
where iX)( iXP
∑=
n
i 1)(XE
= random variable’s possible values = probability of each of the random variable’s
possible values = summation sign indicating we are adding all n
possible values = expected value or mean of the random sample
Variance of a Discrete Probability Distribution
2-21
For a discrete probability distribution the variance can be computed by
)()]([∑=
−==n
iii XPXEX
1
22 Varianceσ
where iX)(XE
)( iXP
= random variable’s possible values = expected value of the random variable = difference between each value of the random
variable and the expected mean = probability of each possible value of the
random variable
)]([ XEXi −
Variance of a Discrete Probability Distribution
2-22
A related measure of dispersion is the standard deviation.
2σVarianceσ ==where
σ= square root = standard deviation
Probability Distribution of a Continuous Random Variable
2-23
Since random variables can take on an infinite number of values, the fundamental rules for continuous random variables must be modified.
! The sum of the probability values must still equal 1.
! The probability of each individual value of the random variable occurring must equal 0 or the sum would be infinitely large.
The probability distribution is defined by a continuous mathematical function called the probability density function or just the probability function.
! This is represented by f (X).
Probability Distribution of a Continuous Random Variable
! The normal distribution is specified completely when we know the mean, µ, and the standard deviation, σ .
The Normal Distribution
" The normal distribution is symmetrical, with the midpoint representing the mean.
" Shifting the mean does not change the shape of the distribution.
" Values on the X axis are measured in the number of standard deviations away from the mean.
" As the standard deviation becomes larger, the curve flattens.
" As the standard deviation becomes smaller, the curve becomes steeper.
2-26
The Normal Distribution
2-27
| | |
40 µ = 50 60
| | |
µ = 40 50 60
Smaller µ, same σ
| | |
40 50 µ = 60
Larger µ, same σ
Figure 2.8
The Normal Distribution
2-28
µ Figure 2.9
Same µ, smaller σ
Same µ, larger σ
Using the Standard Normal Table
2-29
Step 1 Convert the normal distribution into a standard normal distribution.
! A standard normal distribution has a mean of 0 and a standard deviation of 1
! The new standard random variable is Z
σµ−
=XZ
where X = value of the random variable we want to measure µ = mean of the distribution σ = standard deviation of the distribution Z = number of standard deviations from X to the mean, µ
Using the Standard Normal Table
2-30
For example, µ = 100, σ = 15, and we want to find the probability that X is less than 130.
EMV (alternative i) = (payoff of first state of nature) x (probability of first state of nature) + (payoff of second state of nature) x (probability of second state of nature) + … + (payoff of last state of nature) x (probability of last state of nature)
3-41
Expected Value of Perfect Information (EVPI)
• EVPI places an upper bound on what you should pay for additional information.
EVPI = EVwPI – Maximum EMV
• EVwPI is the long run average return if we have perfect information before a decision is made.
EVwPI = (best payoff for first state of nature) x (probability of first state of nature) + (best payoff for second state of nature) x (probability of second state of nature) + … + (best payoff for last state of nature) x (probability of last state of nature)
• Delphi Method – This is an iterative group process where (possibly geographically dispersed) respondents provide input to decision makers.
• Jury of Executive Opinion – This method collects opinions of a small group of high-level managers, possibly using statistical models for analysis.
• Sales Force Composite – This allows individual salespersons estimate the sales in their region and the data is compiled at a district or national level.
• Consumer Market Survey – Input is solicited from customers or potential customers regarding their purchasing plans.
5-52
Time-Series Models
• Time-series models attempt to predict the future based on the past.
• Five uses of inventory: – The decoupling function – Storing resources – Irregular supply and demand – Quantity discounts – Avoiding stockouts and shortages
• Decouple manufacturing processes. – Inventory is used as a buffer between stages in a
manufacturing process. – This reduces delays and improves efficiency.
6-74
Importance of Inventory Control
• Storing resources. – Seasonal products may be stored to satisfy off-
season demand. – Materials can be stored as raw materials, work-in-
process, or finished goods. – Labor can be stored as a component of partially
completed subassemblies. • Compensate for irregular supply and demand.
– Demand and supply may not be constant over time.
– Inventory can be used to buffer the variability.
Q = number of pieces to order EOQ = Q* = optimal number of pieces to order
D = annual demand in units for the inventory item Co = ordering cost of each order Ch = holding or carrying cost per unit per year
Annual ordering cost = × Number of
orders placed per year
Ordering cost per
order
oCQD
=
6-78
Inventory Costs in the EOQ Situation
Mathematical equations can be developed using:
Q = number of pieces to order EOQ = Q* = optimal number of pieces to order
D = annual demand in units for the inventory item Co = ordering cost of each order Ch = holding or carrying cost per unit per year
Annual holding cost = × Average inventory
Carrying cost per unit
per year
hCQ2
=
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6-79
Economic Order Quantity (EOQ) Model
hCQ2
cost holding Annual =
oCQD
=cost ordering Annual
h
o
CDC
Q2
== *EOQ
Summary of equations:
6-80
Purchase Cost of Inventory Items
• Total inventory cost can be written to include the cost of purchased items.
• Given the EOQ assumptions, the annual purchase cost is constant at D × C no matter the order policy, where – C is the purchase cost per unit. – D is the annual demand in units.
• At times it may be useful to know the average dollar level of inventory:
2level dollar Average
)(CQ=
6-81
Purchase Cost of Inventory Items
• Inventory carrying cost is often expressed as an annual percentage of the unit cost or price of the inventory.
• This requires a new variable. Annual inventory holding charge as
a percentage of unit price or cost I =
! The cost of storing inventory for one year is then
Cs = setup cost Ch = holding or carrying cost per unit per
year p = daily production rate d = daily demand rate t = length of production run in days
6-86
Annual Carrying Cost for Production Run Model
Maximum inventory level = (Total produced during the production run) – (Total used during the production run) = (Daily production rate)(Number of days production)
– (Daily demand)(Number of days production) = (pt) – (dt)
since Total produced = Q = pt
we know pQt =
Maximum inventory
level !"
#$%
&−=−=−=pdQ
pQd
pQpdtpt 1
6-87
Annual Carrying Cost for Production Run Model
Since the average inventory is one-half the maximum:
!"
#$%
&−=pdQ
12
inventory Average
and
hCpdQ!"
#$%
&−= 1
2cost holding Annual
6-88
Annual Setup Cost for Production Run Model
sCQD
=cost setup Annual
Setup cost replaces ordering cost when a product is produced over time.
replaces
oCQD
=cost ordering Annual
6-89
Determining the Optimal Production Quantity
By setting setup costs equal to holding costs, we can solve for the optimal order quantity