Demand Forecasting: Time Series Models Professor Stephen R. Lawrence College of Business and Administration University of Colorado Boulder, CO

Forecasting Horizons o Long Term  5+ years into the future  R&D, plant location, product planning  Principally judgement-based o Medium Term  1 season to 2 years  Aggregate planning, capacity planning, sales forecasts  Mixture of quantitative methods and judgement o Short Term  1 day to 1 year, less than 1 season  Demand forecasting, staffing levels, purchasing, inventory levels  Quantitative methods

Short Term Forecasting: Needs and Uses o Scheduling existing resources  How many employees do we need and when?  How much product should we make in anticipation of demand? o Acquiring additional resources  When are we going to run out of capacity?  How many more people will we need?  How large will our back-orders be? o Determining what resources are needed  What kind of machines will we require?  Which services are growing in demand? declining?  What kind of people should we be hiring?

Types of Forecasting Models o Types of Forecasts  Qualitative --- based on experience, judgement, knowledge;  Quantitative --- based on data, statistics; o Methods of Forecasting  Naive Methods --- eye-balling the numbers;  Formal Methods --- systematically reduce forecasting errors; –time series models (e.g. exponential smoothing); –causal models (e.g. regression).  Focus here on Time Series Models o Assumptions of Time Series Models  There is information about the past;  This information can be quantified in the form of data;  The pattern of the past will continue into the future.

Forecasting Examples o Examples from student projects:  Demand for tellers in a bank;  Traffic on major communication switch;  Demand for liquor in bar;  Demand for frozen foods in local grocery warehouse. o Example from Industry: American Hospital Supply Corp.  70,000 items;  25 stocking locations;  Store 3 years of data (63 million data points);  Update forecasts monthly;  21 million forecast updates per year.

Simple Moving Average o Forecast F t is average of n previous observations or actuals D t : o Note that the n past observations are equally weighted. o Issues with moving average forecasts:  All n past observations treated equally;  Observations older than n are not included at all;  Requires that n past observations be retained;  Problem when 1000's of items are being forecast.

Simple Moving Average o Include n most recent observations o Weight equally o Ignore older observations weight today n 1/n

Moving Average n = 3

Example: Moving Average Forecasting

Exponential Smoothing I o Include all past observations o Weight recent observations much more heavily than very old observations: weight today Decreasing weight given to older observations

Exponential Smoothing I o Include all past observations o Weight recent observations much more heavily than very old observations: weight today Decreasing weight given to older observations

Exponential Smoothing I o Include all past observations o Weight recent observations much more heavily than very old observations: weight today Decreasing weight given to older observations

Exponential Smoothing I o Include all past observations o Weight recent observations much more heavily than very old observations: weight today Decreasing weight given to older observations

Exponential Smoothing: Concept o Include all past observations o Weight recent observations much more heavily than very old observations: weight today Decreasing weight given to older observations

Exponential Smoothing: Math

o Thus, new forecast is weighted sum of old forecast and actual demand o Notes:  Only 2 values (D t and F t-1 ) are required, compared with n for moving average  Parameter a determined empirically (whatever works best)  Rule of thumb:  < 0.5  Typically,  = 0.2 or  = 0.3 work well o Forecast for k periods into future is:

Exponential Smoothing  = 0.2

Example: Exponential Smoothing

Complicating Factors o Simple Exponential Smoothing works well with data that is “moving sideways” (stationary) o Must be adapted for data series which exhibit a definite trend o Must be further adapted for data series which exhibit seasonal patterns

Holt’s Method: Double Exponential Smoothing o What happens when there is a definite trend? A trendy clothing boutique has had the following sales over the past 6 months: Month Demand Actual Forecast

Holt’s Method: Double Exponential Smoothing o Ideas behind smoothing with trend:  ``De-trend'' time-series by separating base from trend effects  Smooth base in usual manner using   Smooth trend forecasts in usual manner using  o Smooth the base forecast B t o Smooth the trend forecast T t o Forecast k periods into future F t+k with base and trend

ES with Trend  = 0.2,  = 0.4

Example: Exponential Smoothing with Trend

Winter’s Method: Exponential Smoothing w/ Trend and Seasonality o Ideas behind smoothing with trend and seasonality:  “De-trend’: and “de-seasonalize”time-series by separating base from trend and seasonality effects  Smooth base in usual manner using   Smooth trend forecasts in usual manner using   Smooth seasonality forecasts using  o Assume m seasons in a cycle  12 months in a year  4 quarters in a month  3 months in a quarter  et cetera

Winter’s Method: Exponential Smoothing w/ Trend and Seasonality o Smooth the base forecast B t o Smooth the trend forecast T t o Smooth the seasonality forecast S t

Winter’s Method: Exponential Smoothing w/ Trend and Seasonality o Forecast F t with trend and seasonality o Smooth the trend forecast T t o Smooth the seasonality forecast S t

ES with Trend and Seasonality  = 0.2,  = 0.4,  = 0.6

Example: Exponential Smoothing with Trend and Seasonality

Forecasting Performance o Mean Forecast Error (MFE or Bias): Measures average deviation of forecast from actuals. o Mean Absolute Deviation (MAD): Measures average absolute deviation of forecast from actuals. o Mean Absolute Percentage Error (MAPE): Measures absolute error as a percentage of the forecast. o Standard Squared Error (MSE): Measures variance of forecast error How good is the forecast?

Forecasting Performance Measures

o Want MFE to be as close to zero as possible -- minimum bias o A large positive (negative) MFE means that the forecast is undershooting (overshooting) the actual observations o Note that zero MFE does not imply that forecasts are perfect (no error) -- only that mean is “on target” o Also called forecast BIAS Mean Forecast Error (MFE or Bias)

Mean Absolute Deviation (MAD) o Measures absolute error o Positive and negative errors thus do not cancel out (as with MFE) o Want MAD to be as small as possible o No way to know if MAD error is large or small in relation to the actual data

Mean Absolute Percentage Error (MAPE) o Same as MAD, except... o Measures deviation as a percentage of actual data

Mean Squared Error (MSE) o Measures squared forecast error -- error variance o Recognizes that large errors are disproportionately more “expensive” than small errors o But is not as easily interpreted as MAD, MAPE -- not as intuitive

Fortunately, there is software...