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Lecture3

Decision TheoryChapter 5S

2

Certainty - Environment in which relevant parameters have known values

Risk - Environment in which certain future events have probabilistic outcomes

Uncertainty - Environment in which it is impossible to assess the likelihood of various future events

Decision EnvironmentsDecision Environments

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Maximin - Choose the alternative with the best of the worst possible payoffs

Maximax - Choose the alternative with the best possible payoff

Minimax Regret - Choose the alternative that has the least of the worst regrets

Decision Making under UncertaintyDecision Making under Uncertainty

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Payoff Table: An ExamplePayoff Table: An Example

Low Moderate High

Small facility

$10 $10 $10

Medium facility

7 12 12

Large facility

- 4 2 16

Possible Future Demand

Values represent payoffs (profits)

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Maximax SolutionMaximax Solution

Note: choose the “minimize the payoff” option if the numbers in the previous slide represent

costs

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Maximin Solution Maximin Solution

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Minimax Regret SolutionMinimax Regret Solution

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Decision Making Under Risk - Decision Decision Making Under Risk - Decision TreesTrees

State of nature 1

B

Payoff 1

State of nature 2

Payoff 2

Payoff 3

2

Choose A’1

Choose A’2

Payoff 6State of nature 2

2

Payoff 4

Payoff 5

Choose A’3

Choose A’4

State of nature 1

Choose A

’

Choose A’2

1

Decision PointChance Event

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Decision Making with ProbabilitiesDecision Making with Probabilities

Expected Value Approach Useful if probabilistic information regarding the

states of nature is available Expected return for each decision is calculated by

summing the products of the payoff under each state of nature and the probability of the respective state of nature occurring

Decision yielding the best expected return is chosen.

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Example: Burger PrinceExample: Burger Prince Burger Prince Restaurant is considering opening a new restaurant

on Main Street. It has three different models, each with a different seating

capacity. Burger Prince estimates that the average number of customers per

hour will be 80, 100, or 120 with a probability of 0.4, 0.2, and 0.4 respectively

The payoff (profit) table for the three models is as follows.

s1 = 80 s2 = 100 s3 = 120

Model A $10,000 $15,000 $14,000

Model B $ 8,000 $18,000 $12,000

Model C $ 6,000 $16,000 $21,000 • Choose the alternative that maximizes expected payoff

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Decision TreeDecision Tree

1111

.2.2

.4.4

.4.4

.4.4

.2.2

.4.4

.4.4

.2.2

.4.4

dd11

dd22

dd33

ss11

ss11

ss11

ss22

ss33

ss22

ss22

ss33

ss33

PayoffsPayoffs

10,00010,000

15,00015,000

14,00014,0008,0008,000

18,00018,000

12,00012,000

6,0006,000

16,00016,000

21,00021,000

2222

3333

4444

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Management Scientist SolutionsManagement Scientist Solutions

EVPI = Expected payoff under certainty –

Expected payoff under risk

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Lecture2

ForecastingChapter 3

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A statement about the future value of a variable of interest such as demand.

Forecasts affect decisions and activities throughout an organization Accounting, finance Human resources Marketing Operations Product / service design

ForecastForecast

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Accounting Cost/profit estimates

Finance Cash flow and funding

Human Resources Hiring/recruiting/training

Marketing Pricing, promotion, strategy

Operations Schedules, MRP, workloads

Product/service design New products and services

Uses of ForecastsUses of Forecasts

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Elements of a Good ForecastElements of a Good Forecast

Timely

AccurateReliable

Mea

ningfu

l

Written

Easy

to u

se

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Steps in the Forecasting ProcessSteps in the Forecasting Process

Step 1 Determine purpose of forecast

Step 2 Establish a time horizon

Step 3 Select a forecasting technique

Step 4 Gather and analyze data

Step 5 Prepare the forecast

Step 6 Monitor the forecast

“The forecast”

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Types of ForecastsTypes of Forecasts

Judgmental - uses subjective inputs

Time series - uses historical data assuming the future will be like the past

Associative models - uses explanatory variables to predict the future

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Judgmental ForecastsJudgmental Forecasts

Executive opinions Sales force opinions Consumer surveys Outside opinion Delphi method

Opinions of managers and staff Achieves a consensus forecast

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Time Series ForecastsTime Series Forecasts

Trend - long-term movement in data Seasonality - short-term regular variations in

data Cycle – wavelike variations of more than one

year’s duration Irregular variations - caused by unusual

circumstances

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Forecast VariationsForecast Variations

Trend

Irregularvariation

Seasonal variations

908988

Figure 3.1

Cycles

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Smoothing/Averaging MethodsSmoothing/Averaging Methods

Used in cases in which the time series is fairly stable and has no significant trend, seasonal, or cyclical effects

Purpose of averaging - to smooth out the irregular components of the time series.

Four common smoothing/averaging methods are: Moving averages Weighted moving averages Exponential smoothing

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Sales of gasoline for the past 12 weeks at your Sales of gasoline for the past 12 weeks at your local Chevron (in ‘000 gallons). If the dealer local Chevron (in ‘000 gallons). If the dealer uses a 3-period moving average to forecast uses a 3-period moving average to forecast sales, what is the forecast for Week 13?sales, what is the forecast for Week 13?

Example of Moving AverageExample of Moving Average

Past Sales

WeekWeek SalesSales WeekWeek SalesSales 1 17 7 201 17 7 20 2 21 8 182 21 8 18 3 19 9 223 19 9 22 4 23 10 204 23 10 20 5 18 11 155 18 11 15

6 166 16 12 12 22 22

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Management Scientist SolutionsManagement Scientist Solutions

MA(3) for period 4

= (17+21+19)/3 = 19

Forecast error for period 3 = Actual – Forecast = 23 – 19

= 4

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MA(5) versus MA(3)MA(5) versus MA(3)

Week Actual MA(3) MA(5)1 172 213 194 23 195 18 216 16 20 19.67 20 19 19.48 18 18 19.29 22 18 19

10 20 20 18.811 15 20 19.212 22 19 19

MA Forecast Graph

0

5

10

15

20

25

1 2 3 4 5 6 7 8 9 10 11 12

Week

Actu

al/M

A Fo

reca

st s

ale

valu

es

Actual

MA(3)

MA(5)

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Exponential SmoothingExponential Smoothing

• Premise - The most recent observations might have the highest predictive value. Therefore, we should give more weight to the more recent time periods

when forecasting.

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Exponential SmoothingExponential Smoothing

Weighted averaging method based on previous forecast plus a percentage of the forecast error

A-F is the error term, is the % feedback

Ft+1 = Ft + (At - Ft)

10

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Picking a Smoothing ConstantPicking a Smoothing Constant

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40

45

50

1 2 3 4 5 6 7 8 9 10 11 12

Period

Dem

and .1

.4

Actual

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Linear Trend EquationLinear Trend Equation

Ft = Forecast for period t t = Specified number of time periods a = Value of Ft at t = 0 b = Slope of the line

Ft = a + bt

0 1 2 3 4 5 t

Ft

a

Suitable for time series data that exhibit a long term linear trend

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Linear Trend ExampleLinear Trend Example

F11 = 20.4 + 1.1(11) = 32.5

Linear trend equation

Sale increases every time period @ 1.1

units

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Actual Actual vsvs Forecast Forecast

Linear Trend Example

0

5

10

15

20

25

30

35

1 2 3 4 5 6 7 8 9 10

Week

Act

ual

/Fo

reca

sted

sal

es

Actual

Forecast

F(t) = 20.4 + 1.1t

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Forecasting with Trends and Seasonal Forecasting with Trends and Seasonal Components – An ExampleComponents – An Example

Business at Terry's Tie Shop can be viewed as falling into three distinct seasons: (1) Christmas (November-December); (2) Father's Day (late May - mid-June); and (3) all other times.

Average weekly sales ($) during each of the three seasonsduring the past four years are known and given below.

Determine a forecast for the average weekly sales in year 5 for each of the three seasons.

Year Season 1 2 3 4 1 1856 1995 2241 2280 2 2012 2168 2306 2408 3 985 1072 1105 1120

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Management Scientist SolutionsManagement Scientist Solutions

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Interpretation of Seasonal IndicesInterpretation of Seasonal Indices Seasonal index for season 2 (Father’s Day) = 1.236

Means that the sale value of ties during season 2 is 23.6% higher than the average sale value over the year

Seasonal index for season 3 (all other times) = 0.586 Means that the sale value of ties during season 3 is 41.4%

lower than the average sale value over the year

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Forecast AccuracyForecast Accuracy

Error - difference between actual value and predicted value

Mean Absolute Deviation (MAD)

Average absolute error

Mean Squared Error (MSE)

Average of squared error

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Associative ForecastingAssociative Forecasting

Predictor variables - used to predict values of variable interest

Regression - technique for fitting a line to a set of points

Least squares line - minimizes sum of squared deviations around the line

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Regression Analysis – An ExampleRegression Analysis – An ExampleHome-Size (Square feet) Price

600 $72,000

1050 $116,300

1800 $152,000

922 $80,500

1950 $141,900

1783 $124,000

1008 $117,000

1840 $165,900

3700 $153,500

1092 $126,500

1950 $122,000

1403 $140,000

1680 $223,000

1000 $99,500

2310 $211,900

1300 $121,900

1930 $169,000

3000 $156,000

1362 $123,500

1750 $136,000

2080 $194,900

1344 $128,500

2130 $302,000

1500 $142,000

2400 $146,000

2272 $180,000

1050 $126,500

1610 $139,500

$0

$50,000

$100,000

$150,000

$200,000

$250,000

$300,000

$350,000

600

1000

1008

1950

1783

1050

1750

1403

1500

1800

3000

1930

2080

1680

Series2

• Linear model seems reasonable• A straight line is fitted to a set of sample points

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Regression ResultsRegression Results Use MS-Excel macro Template posted at class website

y = 85972.78 + 35.65x

Price = 85972.87 + 35.65(Square footage)

Forecast price of a 2000 square feet house

y = 85972.78 + 35.65(2000) = $157,272.78

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Forecast AccuracyForecast Accuracy

Error - difference between actual value and predicted value

Mean Absolute Deviation (MAD)

Average absolute error

Mean Squared Error (MSE)

Average of squared error

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MAD and MSEMAD and MSE

MAD = Actual forecast

n

MSE = Actual forecast)

2

n

(

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Measure of Forecast AccuracyMeasure of Forecast Accuracy MSE = Mean Squared Error

Week # Actual (A) Forecast(F) Error =E =A-F E(squared)1 21.6 21.5 0.1 0.012 22.9 22.6 0.3 0.093 25.5 23.7 1.8 3.244 21.9 24.8 -2.9 8.415 23.9 25.9 -2 46 27.5 27 0.5 0.257 31.5 28.1 3.4 11.568 29.7 29.2 0.5 0.259 28.6 30.3 -1.7 2.89

10 31.4 31.4 0 0

Sum of E(squared) 30.7

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Forecasting Accuracy Estimates Forecasting Accuracy Estimates Example 10 of textbookExample 10 of textbook

Period Actual Forecast (A-F) |A-F| (A-F)^21 217 215 2 2 42 213 216 -3 3 93 216 215 1 1 14 210 214 -4 4 165 213 211 2 2 46 219 214 5 5 257 216 217 -1 1 18 212 216 -4 4 16

-2 22 76

MAD= 2.75MSE= 9.50

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Sources of Forecast errorsSources of Forecast errors

Model may be inadequate Irregular variations Incorrect use of forecasting technique

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Characteristics of ForecastsCharacteristics of Forecasts

They are usually wrong A good forecast is more than a single number Aggregate forecasts are more accurate The longer the forecast horizon, the less accurate

the forecast will be Forecasts should not be used to the exclusion of

known information

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Choosing a Forecasting TechniqueChoosing a Forecasting Technique

No single technique works in every situation Two most important factors

Cost Accuracy

Other factors include the availability of: Historical data Computers Time needed to gather and analyze the data Forecast horizon