Lecture 3 Decision Theory Chapter 5S 1 Decision Environments 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 2

Decision Making under Uncertainty 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 3 Payoff Table: An Example Possible Future Demand Low Moderate High Small facility $10 $10 $10

Medium facility 7 12 12 Large facility -4 2 16 Values represent payoffs (profits) 4 Maximax Solution Note: choose the minimize the payoff option if the numbers in the previous

slide represent costs 5 Maximin Solution 6 Minimax Regret Solution 7 Decision Making Under Risk - Decision Trees Decision Point Chance Event A e s o o h C State

re 1 u t a of n State of C ho o State os natur 2 e2 ure t a n f 1

Choose Payoff 2 A2 Payoff 3 3 Payoff 4 A4 Payoff 5 A e s o o Ch 2 Choose e

A 2 State of 1 eA Choos B 1 Payoff 1 natur e2 Payoff 6 8 Decision 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. 9 Example: 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 10 Decision Tree d1 1 d2 d3 2 3 4

s1 s2 s3 s1 s2 s3 s1 s2 s3 .4 .2 .4 .4 .2 .4 .4 .2 .4 Payoffs 10,000 15,000 14,000 8,000 18,000 12,000

6,000 16,000 21,000 11 Management Scientist Solutions EVPI = Expected payoff under certainty Expected payoff under risk 12 Lecture 2 Forecasting Chapter 3 13 Forecast 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 14 Uses of Forecasts 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 15 Elements of a Good Forecast Timely Reliable n i n a e M g l fu Accurate Written

y s Ea to e s u 16 Steps in the Forecasting Process The forecast Step 6 Monitor the forecast Step 5 Prepare the forecast Step 4 Gather and analyze data Step 3 Select a forecasting technique Step 2 Establish a time horizon Step 1 Determine purpose of forecast 17 Types 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 18 Judgmental Forecasts Executive opinions Sales force opinions Consumer surveys

Outside opinion Delphi method Opinions of managers and staff Achieves a consensus forecast 19 Time Series Forecasts Trend - long-term movement in data Seasonality - short-term regular variations in data Cycle wavelike variations of more than one years duration Irregular variations - caused by unusual circumstances 20 Forecast Variations Figure 3.1

Irregular variation Trend Cycles 90 89 88 Seasonal variations 21 Smoothing/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 22

Example of Moving Average Sales of gasoline for the past 12 weeks at your local Chevron (in 000 gallons). If the dealer uses a 3-period moving average to forecast sales, what is the forecast for Week 13? Past Sales Week 1 2 3 4 5 6 Sales 17 21 19 23 18 16 12

Week 7 8 9 10 11 Sales 20 18 22 20 15 22 23 Management Scientist Solutions MA(3) for period 4 = (17+21+19)/3 = 19 Forecast error for period 3 = Actual Forecast = 23 19 =4 24

MA(5) versus MA(3) Actual 1 2 3 4 5 6 7 8 9 10 11 12 MA(3) 17 21 19 23 18 16 20 18 22 20 15

22 MA(5) 19 21 20 19 18 18 20 20 19 MA Forecast Graph 25 20 19.6 19.4 19.2 19 18.8 19.2 19 Actual/MAForecast sale values

Week Actual 15 MA(3) 10 MA(5) 5 0 1 2 3 4 5 6 7 8 9 10 11 12

Week 25 Exponential 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. 26 Exponential Smoothing Ft+1 = Ft + (At - Ft) Weighted averaging method based on previous forecast plus a percentage of the forecast error A-F is the error term, is the % feedback 0 1

27 Picking a Smoothing Constant Actual 50 Demand .4 .1 45 40 35 1 2 3 4 5 6 7

8 9 10 11 12 Period 28 Linear Trend Equation Suitable for time series data that exhibit a long term linear trend Ft Ft = a + bt a Ft = Forecast for period t t = Specified number of time periods a = Value of Ft at t = 0 b = Slope of the line 0 1 2 3 4 5 t

29 Linear Trend Example Linear trend equation F11 = 20.4 + 1.1(11) = 32.5 Sale increases every time period @ 1.1 30 Actual vs Forecast Actual/Forecasted sales Linear Trend Example 35 30 25 20 Actual 15 Forecast

10 5 0 1 2 3 4 5 6 7 8 9 10 Week F(t) = 20.4 + 1.1t 31

Forecasting with Trends and Seasonal Components 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 seasons during 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

32 Management Scientist Solutions 33 Interpretation of Seasonal Indices Seasonal index for season 2 (Fathers 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 34 Forecast Accuracy

Error - difference between actual value and predicted value Mean Absolute Deviation (MAD) Average absolute error Mean Squared Error (MSE) Average of squared error 35 Associative 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 36 Regression Analysis An Example 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 $350,000 $300,000 $250,000 $200,000 Series2 $150,000 $100,000 $50,000 $0 10 00 10 08 19 50 17 83 10

50 17 50 14 03 15 00 18 00 30 00 19 30 20 80 16 80 Price 60 0 Home-Size (Square feet) Linear model seems reasonable A straight line is fitted to a set of sample points 37

Regression 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 38 Forecast Accuracy Error - difference between actual value and predicted value Mean Absolute Deviation (MAD) Average absolute error

Mean Squared Error (MSE) Average of squared error 39 MAD and MSE MAD = Actual forecast n MSE = ( Actual forecast) 2

n 40 Measure 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 0.01 2 22.9 22.6 0.3 0.09 3 25.5 23.7 1.8 3.24 4 21.9 24.8 -2.9

8.41 5 23.9 25.9 -2 4 6 27.5 27 0.5 0.25 7 31.5 28.1 3.4 11.56 8 29.7 29.2 0.5 0.25 9 28.6 30.3 -1.7 2.89 10 31.4

31.4 0 0 Sum of E(squared) 30.7 41 Forecasting Accuracy Estimates Example 10 of textbook Period 1 2 3 4 5 6 7 8 MAD= MSE= Actual 217 213 216 210

213 219 216 212 Forecast 215 216 215 214 211 214 217 216 (A-F) 2 -3 1 -4 2 5 -1 -4 -2 |A-F| 2

3 1 4 2 5 1 4 22 (A-F)^2 4 9 1 16 4 25 1 16 76 2.75 9.50 42 Sources of Forecast errors Model may be inadequate Irregular variations

Incorrect use of forecasting technique 43 Characteristics 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 44 Choosing 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 45