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Quantitative Methods MAT 540 Forecasting 1 Objectives • Upon completion of this lesson, you will be able to: – Apply the most appropriate forecasting method for the properties of the available data – Use technology and information resources to research and communicate issues in Management Science. 2 What Is Forecasting? • Forecasting is the process of predicting the future. It is an integral part of almost all business enterprises. • Examples: – Manufacturing firms forecast demand for their product, to schedule manpower and raw material allocation. – Service organizations forecast customer arrival patterns to maintain adequate customer service. 3 Forecasting Components • Time Frames: – Short-range (one to two months) – Medium-range (two months to one years) – Long-range (more than one or two years) • Patterns: – Trend – Random variations – Cycles – Seasonality 4 Forecasting Methods • Times Series – uses historical data assuming the future will be like the past. • Regression Methods – Regression (or causal ) methods that attempt to develop a mathematical relationship between the item being forecast and factors that cause it to behave the way it does. • Qualitative Methods – Methods using judgment, expertise and opinion to make forecasts. 5 PROPERTIES On passing, 'Finish' button: Goes to Next Slide On failing, 'Finish' button: Goes to Next Slide Allow user to leave quiz: At any time User may view slides after quiz: At any time User may attempt quiz: Unlimited times Time Series Methods • Time Series Methods – Moving average – Weighted moving average – Exponential smoothing – Adjusted exponential smoothing – Seasonal adjustment 7 Moving Average • Moving average – A technique that averages a number of recent actual values, updated as new values become available. • Formula: n Di MAn i1n where: n number of periods in the moving average D data in period i i 8 Moving Average Continued Example: Sales of Comfort brand headache medicine for the past ten weeks at Robert’s Drugs are shown below. If Robert’s Drugs uses a 3-period moving average to forecast sales, what is the forecast for Week 11? Week Sales Week Sales 1 110 6 120 2 115 7 130 3 125 8 115 4 120 9 110 5 125 10 130 9 Moving Average Continued Solution: Forecast 11 = (115 + 110 + 130 ) / 3 ≈ 118.3 Robert’s Drugs Week (t) Salest Forecast t+1 1 110 — 2 115 — 3 125 — 4 120 116.7 5 125 120.0 6 120 123.3 7 130 121.7 8 115 125.0 9 110 121.7 10 130 118.8 11 — 118.3 10 Weighted Moving Average • Weighted Moving Average – More recent values in a series are given more weight in computing the forecast. • Formula n WMAn W D i1 i i where W the weight for period i, between 0% and 100% i Wi 1.00 11 Weighted Moving Average Continued Example: The following data summarizes the historical demand for door sales at Westmore Hardware. Use a weighted moving average method with weights w1 = 0.2, w2 = 0.3, w3 = 0.5 and determine the forecasted demand for August and September. Month Actual Demand March 20 April 25 May 40 June 35 July 30 August 45 12 Weighted Moving Average Continued Solution: Forecast August = 40 • 0.2 + 35 • 0.3 + 30 • 0.5 = 33.5 Forecast September = 35 • 0.2 + 30 • 0.3 + 45 • 0.5 = 38.5 Westmore Hardware Period Week (t) Actual Demand Forecast t+1 1 March 20 — 2 April 25 — 3 May 40 — 4 June 35 31.5 5 July 30 34.5 6 August 45 33.5 7 September — 38.5 13 Exponential Smoothing • Exponential Smoothing – the forecast for the next period is equal to the forecast for the current period plus a proportion () of the forecast error in the current period. • Formula Ft + 1 = Dt + (1 - )Ft where: Ft + 1 = the forecast for the next period Dt = actual demand in the present period Ft = the previously determined forecast for the present period = a weighting factor (smoothing constant). 14 Exponential Smoothing Continued • Example Make a prediction, given the following data on Gillmore Hotel check-ins for a 5-month period below. Using the exponential smoothing factor 0.3, how many check-ins can be forecasted for January? Gillmore Hotel Month Room Occupied August 40 September 30 October 45 November 60 December 55 15 Exponential Smoothing Continued • Solution F2 = 0.3 • 40 + 0.7 • 40 = 40 F3 = 0.3 • 30 + 0.7 • 40 = 37 F6 = 0.3 • 55 + 0.7 • 45.6 = 48.4 Gillmore Hotel Period Month Room Occupied Forecast , Ft+1 1 August 40 — 2 September 30 F2 = 40 3 October 45 F3 = 37 4 November 60 F4 = 39.4 5 December 55 F5 = 45.6 6 January — F6 = 48.4 16 Adjusted Exponential Smoothing • Adjusted Exponential Smoothing – exponential smoothing with a trend adjustment factor added. Formula: AFt + 1 = Ft + 1 + Tt+1 where: T = an exponentially smoothed trend factor Tt + 1 + (Ft + 1 - Ft) + (1 - )Tt Tt = the last period trend factor = smoothing constant for trend ( a value between zero and one). 17 Adjusted Exponential Smoothing Continued • Example Given the following data on Gillmore Hotel check-ins for a 6-month period below, with alpha = 0.2 and beta = 0.1, what is the adjusted exponentially smoothed forecast for September? Gillmore Hotel Month Room Occupied July 45 August 36 September 52 October 48 November 56 December 60 18 Adjusted Exponential Smoothing Continued • Solution T3 = 0.1 (43.20 – 45.00) + (1 – 0.1) (0.00) = – 0.18 + 0 = – 0.18 AF3 = 43.20 + (– 0.18) = 43.02 Gillmore Hotel Period Month Room Forecast Trend Adjusted Forecast Occupied (Ft + 1) (Tt+1 ) ( AFt + 1 ) 1 July 45.00 45.00 — — 2 August 36.00 45.00 0.00 45.00 3 September 52.00 43.20 -0.18 43.02 4 October 48.00 44.96 0.01 44.97 5 November 56.00 45.57 0.07 45.64 6 December 60.00 47.65 0.27 47.93 19 Linear Trend Line • Linear Trend Line – A linear regression model that relates demand to time. 20 Seasonal Adjustment • Seasonal Adjustment – An adjustment can be made to the forecast by multiplying the normal forecast by a seasonal factor. • Formula: – A seasonal factor can be determined by dividing the actual demand for each seasonal period by total annual demand: Si =Di/D 21 Seasonal Adjustment Continued • Example: Tony’s Shoe Store sells various types of shoes. Quarterly sales is given by the table below for the past three years, determine the seasonal factors for each quarter. If the forecast for the forth year is 25,000 shoes demanded, determine the seasonally adjusted forecasts. Tony’s Shoe Store Year Winter Spring Summer Fall 1 4800 4500 4100 5500 2 5700 3800 4500 6000 3 6000 4600 4900 6500 22 Seasonal Adjustment Continued • Solution – Swinter = D1/ D = 16500/60900 ≈ 0.2709 – Sspring = D2/ D = 12900/609000 ≈ 0.2118 – Ssummer = D3/ D = 13500/609000 ≈ 0.2217 – Sfall = D4/ D = 18000/609000 ≈ 0.2956 Tony’s Shoe Store Year Winter Spring Summer Fall Total 1 4800 4500 4100 5500 18900 2 5700 3800 4500 6000 20000 3 6000 4600 4900 6500 22000 Total 16500 12900 13500 18000 60900 23 Seasonal Adjustment Continued • Solution Continued: The forecast for 4th year is 25,000 shoes demanded. Seasonally adjusted forecasts: F4 = 25,000 SFwinter = (Swinter)(F4) = (0.2709)(25000) = 6773 SFspring = (Sspring)(F4) = (0.2118)(25000) = 5295 SFsummer = (Ssummer)(F4) = (0.2217)(25000) = 5543 SFfall = (Sfall)(F4) = (0.2956)(25000) = 7390 24 PROPERTIES On passing, 'Finish' button: Goes to Next Slide On failing, 'Finish' button: Goes to Next Slide Allow user to leave quiz: At any time User may view slides after quiz: At any time User may attempt quiz: Unlimited times PROPERTIES On passing, 'Finish' button: Goes to Next Slide On failing, 'Finish' button: Goes to Next Slide Allow user to leave quiz: At any time User may view slides after quiz: At any time User may attempt quiz: Unlimited times Forecasting Accuracy Forecast error • Mean Absolute Deviation (MAD) – the average, absolute difference between the forecast and the demand – Formula: D F MAD t t n where: t the period number D demand in period t t F the forecast for period t t n the total number of periods 27 Forecasting Accuracy Continued • Mean Absolute Percent Deviation (MAPD) – Absolute error as a percentage of demand – Formula: Dt Ft MAPD Dt • Cumulative Error (E) – the sum of the forecast errors – Formula: E =et • Average Error ( E ) – The per-period average of cumulative error – Formula: et E n 28 Forecasting Accuracy Continued • Example: Freeman’s Electronics recorded recent past demand for large screen LCD TVs in the following table. Based in the three month Freeman’s Electronics moving average, forecasted Month Actual Demand February 20 demand for May, June, July, March 22 August and September are April 33 May 35 25, 30, 33, 38, 40 respectively, June 31 Find the MAD, MAPD, E, and E . July 48 August 41 29 Forecasting Accuracy Continued • Solution: MAD = 29/4 = 7.25; MAPD = 29/155 = 0.187 = 18.7% E = 29; E = 29/4 = 7.25 Freeman’s Electronics Month Demand, Dt Forecast, Ft Error, (Dt – Ft ) │Dt – Ft │ February 20 — — — March 22 — — — April 33 — — — May 35 25 10 10 June 31 30 1 1 July 48 33 15 15 August 41 38 3 3 September — 40 — — Total 155 29 29 30 PROPERTIES On passing, 'Finish' button: Goes to Next Slide On failing, 'Finish' button: Goes to Next Slide Allow user to leave quiz: At any time User may view slides after quiz: At any time User may attempt quiz: Unlimited times Time Series Forecasting Using Excel 32 Time Series Forecasting Using QM 33 Regression Methods • Linear regression – relates demand (dependent variable) to an independent variable. • Regression method – technique for fitting a line to a set of points • Correlation – A measure of the strength of the relationship between independent and dependent variables • Coefficient of determination – the percentage of the variation in the dependent variable that results from independent variable 34 Regression Methods Continued • Least Square Method: • Formula for the Correlation y a bx Coefficient: a y bx r n xy x y x2 x2 n y2 y 2 b xy n x y n 2 x2 n x where : x x mean of the x data n • Coefficient of Determination is y y mean of the y data n computed by r squared, r2. 35 Regression Methods Continued • Example: Infoworks is a large computer discount store that sells computers and ancillary equipment. Infoworks has collected historical data on computer sales and printer sales for the past 10 years . a. Develop a linear regression model relating printer sales to computer sales in order to forecast printer demand in year 11 if 1,300 computers are sold. b. Determine the correlation coefficient and the coefficient of determination. 36 Regression Methods Continued • Example: Infoworks Year PCs sold Printers Year PCs sold Printers Sold Sold 1 1,045 326 6 1,117 506 2 1,610 510 7 1,066 612 3 860 296 8 1,310 560 4 1,211 478 9 1,517 590 5 975 305 10 1,246 676 37 Regression Methods Continued • Solution: (a) a = 74.77 PC Printers b = 0.34 Year Sales Sold y = 74.44 + 0.34x 1 1,045 326 2 1,610 510 x= 1300 3 860 296 y = 74.44 + 0.34x 4 1,211 478 y = 74.44 + 0.34 (1300) 5 975 305 y = 516.77 6 1,117 506 7 1,066 612 (b) 8 1,310 560 r = 0.599 9 1,517 590 r2 = 0.359 10 1,246 676 11 1,300 517 38 PROPERTIES On passing, 'Finish' button: Goes to Next Slide On failing, 'Finish' button: Goes to Next Slide Allow user to leave quiz: At any time User may view slides after quiz: At any time User may attempt quiz: Unlimited times Summary • Forecasting components • Time series methods • Forecast accuracy • Time series forecasting using Excel & QM • Regression methods 40