Demand Forecasting: Time Series Models Professor Stephen R Demand Forecasting: Time Series Models Professor Stephen R. Lawrence College of Business and Administration University of Colorado Boulder, CO 80309-0419

Forecasting Horizons Long Term Medium Term Short Term 5+ years into the future R&D, plant location, product planning Principally judgement-based Medium Term 1 season to 2 years Aggregate planning, capacity planning, sales forecasts Mixture of quantitative methods and judgement 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 Scheduling existing resources How many employees do we need and when? How much product should we make in anticipation of demand? 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? 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 Types of Forecasts Qualitative --- based on experience, judgement, knowledge; Quantitative --- based on data, statistics; 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 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 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. 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 Forecast Ft is average of n previous observations or actuals Dt : Note that the n past observations are equally weighted. 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 ... Include n most recent observations Weight equally Ignore older observations weight 1/n ... n 3 2 1 today

Moving Average n = 3

Example: Moving Average Forecasting

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

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

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

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

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

Exponential Smoothing: Math

Exponential Smoothing: Math

Exponential Smoothing: Math Thus, new forecast is weighted sum of old forecast and actual demand Notes: Only 2 values (Dt and Ft-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 Forecast for k periods into future is:

Exponential Smoothing

Example: Exponential Smoothing

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

Holt’s Method: Double Exponential Smoothing What happens when there is a definite trend? A trendy clothing boutique has had the following sales over the past 6 months: 1 2 3 4 5 6 510 512 528 530 542 552 Actual Demand Forecast Month

Holt’s Method: Double Exponential Smoothing 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  Smooth the base forecast Bt Smooth the trend forecast Tt Forecast k periods into future Ft+k with base and trend

ES with Trend a = 0.2, b = 0.4

Example: Exponential Smoothing with Trend

Winter’s Method: Exponential Smoothing w/ Trend and Seasonality 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 g 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 Smooth the base forecast Bt Smooth the trend forecast Tt Smooth the seasonality forecast St

Winter’s Method: Exponential Smoothing w/ Trend and Seasonality Forecast Ft with trend and seasonality Smooth the trend forecast Tt Smooth the seasonality forecast St

ES with Trend and Seasonality a = 0.2, b = 0.4, g = 0.6

Example: Exponential Smoothing with Trend and Seasonality

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

Forecasting Performance Measures

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

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

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

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

Fortunately, there is software...