Forecasting Demand in a Supply Chain 4. Forecasting Demand in a Supply Chain

Learning Objectives Understand the role of forecasting for both an enterprise and a supply chain. Identify the components of a demand forecast. Forecast demand in a supply chain given historical demand data using time-series and regression methodologies. Analyze demand forecasts to estimate forecast error.

Overview of Forecasting Methods

Role of Forecasting in a Supply Chain The basis for all planning decisions in a supply chain Used for both push and pull processes Production scheduling, inventory, aggregate planning Sales force allocation, promotions, new production introduction Plant/equipment investment, budgetary planning Workforce planning, hiring, layoffs All of these decisions are interrelated

Role of Forecasting in a Supply Chain – Some Examples Inventory Management Accurate forecasting determines how much inventory a company must keep at various points along its supply chain Quality Management Accurately forecasting customer demand is a key to providing good quality service Strategic Planning Successful strategic planning requires accurate forecasts of future products and markets

Characteristics of Forecasts Forecasts are always inaccurate and should thus include both the expected value of the forecast and a measure of forecast error Long-term forecasts are usually less accurate than short- term forecasts Aggregate forecasts are usually more accurate than disaggregate forecasts In general, the farther up the supply chain a company is, the greater is the distortion of information it receives

Types of Forecasting Methods Depend on Time frame (how far into the future) Short- to mid-range forecast (immediate future - up to two years) Long-range forecast Demand behavior Causes of behavior

Types of Forecasting Methods Demand Behavior Trend a gradual, long-term up or down movement of demand Random variations movements in demand that do not follow a pattern Cycle an up-and-down repetitive movement in demand Seasonal pattern an up-and-down repetitive movement in demand occurring periodically

Types of Forecasting Methods Time (a) Trend (d) Trend with seasonal pattern (c) Seasonal pattern (b) Cycle Demand Random movement Forms of Forecast Movement

Components and Methods Companies must identify the factors that influence future demand and then ascertain the relationship between these factors and future demand, for example: Past demand Lead time of product replenishment Planned advertising or marketing efforts Planned price discounts State of the economy Actions that competitors have taken Notes:

Components and Methods Qualitative Primarily subjective Rely on judgment Time Series Use historical demand only Best with stable demand Causal – Regression Relationship between demand and some other factor Simulation Imitate consumer choices that give rise to demand Notes:

Components of an Observation Observed demand (O) = systematic component (S) + random component (R) Systematic component – expected value of demand Level (current deseasonalized demand) Trend (growth or decline in demand) Seasonality (predictable seasonal fluctuation) Random component – part of forecast that deviates from systematic component (size and variability) Forecast error – difference between forecast and actual demand A good forecast has an error whose size is comparable to the random component

Time-Series Methods Use historical data only Relate the forecast to the time only Linear Trend Line method Static Methods Additive Methods

Time-Series Techniques Best -> Forecast DES: MAPE 8.177

Time-Series Forecasting Methods Assumption: what has occurred in the past will continue to occur in the future Relate the forecast to only one factor - time Linear trend line method Additive methods Static methods Three ways to calculate the systematic component: Multiplicative S = level x trend x seasonal factor Additive S = level + trend + seasonal factor Mixed S = (level + trend) x seasonal factor

Linear Trend Line Method = b a = y - b x where n = number of periods x = = mean of the x values y = = mean of the y values xy - nxy x2 - nx2 x n y S = level + trend y = a + bx where a = intercept coefficient b = slope of the line x = time period (variable coefficient) y = forecast for demand for period x

Linear Trend Line Method x(PERIOD) y(DEMAND) xy x2 1 37 37 1 2 40 80 4 3 41 123 9 4 37 148 16 5 45 225 25 6 50 300 36 7 43 301 49 8 47 376 64 9 56 504 81 10 52 520 100 11 55 605 121 12 54 648 144 78 557 3867 650 Least Squares Example

Linear Trend Line Method Least Squares Example (cont.) x = = 6.5 y = = 46.42 b = = =1.72 a = y - bx = 46.42 - (1.72)(6.5) = 35.2 3867 - (12)(6.5)(46.42) 650 - 12(6.5)2 xy - nxy x2 - nx2 78 12 557

Linear trend line y = 35.2 + 1.72x Forecast for period 13 = 57.56 units 70 – 60 – 50 – 40 – 30 – 20 – 10 – 0 – | | | | | | | | | | | | | 1 2 3 4 5 6 7 8 9 10 11 12 13 Actual Demand Period

Adaptive Forecasting Methods The estimates of level, trend, and seasonality are adjusted after each demand observation Estimates incorporate all new data that are observed. Methods: Moving Averages + Weighted Moving Averages Simple Exponential Smoothing Trend-Corrected Exponential Smoothing (Holt’s model) Trend- and Seasonal-Corrected Exponential Smoothing (Winter’s model)

Adaptive Forecasting Methods where Lt = estimate of level at the end of Period t Tt = estimate of trend at the end of Period t St = estimate of seasonal factor for Period t Ft = forecast of demand for Period t (made Period t – 1 or earlier) Dt = actual demand observed in Period t Et = Ft – Dt = forecast error in Period t

Steps in Adaptive Forecasting Initialize Compute initial estimates of level (L0), trend (T0), and seasonal factors (S1,…,Sp) Forecast Forecast demand for period t + 1 Estimate error Compute error Et+1 = Ft+1 – Dt+1 Modify estimates Modify the estimates of level (Lt+1), trend (Tt+1), and seasonal factor (St+p+1), given the error Et+1

Moving Average Assumption: no observable trend or seasonality Systematic component of demand = level The level in period t is the average demand over the last N periods Lt = (Dt + Dt-1 + … + Dt–N+1) / N Ft+1 = Lt and Ft+n = Lt After observing the demand for period t + 1, revise the estimates Lt+1 = (Dt+1 + Dt + … + Dt-N+2) / N, Ft+2 = Lt+1

Moving Average Example I A supermarket has experienced weekly demand of milk of D1 = 120, D2 = 127, D3 = 114, and D4 = 122 gallons over the past four weeks What is the forecast demand for Period 5 using a 4- period moving average? What is the forecast error if demand in Period 5 turns out to be 125 gallons?

Moving Average Example I L4 = (D4 + D3 + D2 + D1)/4 = (122 + 114 + 127 + 120)/4 = 120.75 Forecast demand for Period 5: F5 = L4 = 120.75 gallons Error if demand in Period 5 = 125 gallons E5 = F5 – D5 = 125 – 120.75 = 4.25 Revised demand L5 = (D5 + D4 + D3 + D2)/4 = (125 + 122 + 114 + 127)/4 = 122

Moving Average Example II 3-month Simple Moving Average Jan 120 Feb 90 Mar 100 Apr 75 May 110 June 50 July 75 Aug 130 Sept 110 Oct 90 Nov - ORDERS MONTH PER MONTH – 103.3 88.3 95.0 78.3 85.0 105.0 110.0 MOVING AVERAGE

Moving Average Example II 5-month Simple Moving Average Jan 120 Feb 90 Mar 100 Apr 75 May 110 June 50 July 75 Aug 130 Sept 110 Oct 90 Nov - ORDERS MONTH PER MONTH – 99.0 85.0 82.0 88.0 95.0 91.0 MOVING AVERAGE

Moving Average Example II 150 – 125 – 100 – 75 – 50 – 25 – 0 – | | | | | | | | | | | Jan Feb Mar Apr May June July Aug Sept Oct Nov Actual Orders Month Smoothing Effects 5-month 3-month

Weighted Moving Average Enhancement: Adjusts the moving average method to more closely reflect data fluctuations The level in period t is the weighted average demand over the last N periods Lt = Wt Dt + Wt-1 Dt-1 + … + Wt–N+1 Dt–N+1 Ft+1 = Lt and Ft+n = Lt Wt is the weight for period t, between 0 and 100 percent All weights must add up to 1.

Weighted Moving Average Example MONTH WEIGHT DATA August 17% 130 September 33% 110 October 50% 90 = (0.50)(90) + (0.33)(110) + (0.17)(130) = 103.4 orders November Forecast

Simple Exponential Smoothing Assumption: No observable trend or seasonality Systematic component of demand = level Initial estimate of level L0 is assumed to be the average of all historical data (this is optional)

Simple Exponential Smoothing Given data for Periods 1 to n (this step is optional) Current forecast Revised forecast using smoothing constant 0 < a < 1 Thus

Simple Exponential Smoothing Use supermarket data to forecast period 5 using a=0.1 E1 = F1 – D1 = 120.75 –120 = 0.75

Simple Exponential Smoothing If we continue the calculations for the next periods (2, 3 & 4), we obtain F5 = 120.72 a 0.1 Period Dt Lt Ft Et   120.75 1 120 120.68 0.75 2 127 121.31 -6.33 3 114 120.58 7.31 4 122 120.72 -1.42 5

Simple Exponential Smoothing Effect of Smoothing Constant 0.0  1.0 If = 0.20, then Lt +1 = 0.20Dt + 0.80 Lt If = 0, then Lt +1 = 0Dt + 1 Lt = Lt Forecast does not reflect recent data If = 1, then Lt +1 = 1Dt + 0 Lt =Dt Forecast based only on most recent data

Simple Exponential Smoothing Period Month Dt Lt Ft Et   46.42 1 Jan 37 43.59 9.42 2 Feb 40 42.51 3.59 3 Mar 41 42.06 1.51 4 Apr 40.54 5.06 5 May 45 41.88 -4.46 6 Jun 50 44.32 -8.12 7 Jul 43 43.92 1.32 8 Aug 47 44.84 -3.08 9 Sep 56 48.19 -11.16 10 Oct 52 49.33 -3.81 11 Nov 55 51.03 -5.67 12 Dec 54 51.92 -2.97 13

Simple Exponential Smoothing Period Month Dt Lt Ft Et   46.42 1 Jan 37 41.71 9.42 2 Feb 40 40.85 1.71 3 Mar 41 40.93 -0.15 4 Apr 38.96 3.93 5 May 45 41.98 -6.04 6 Jun 50 45.99 -8.02 7 Jul 43 44.50 2.99 8 Aug 47 45.75 -2.50 9 Sep 56 50.87 -10.25 10 Oct 52 51.44 -1.13 11 Nov 55 53.22 -3.56 12 Dec 54 53.61 -0.78 13

Simple Exponential Smoothing

Trend-Corrected Exponential Smoothing (Holt’s Model) Appropriate when the demand is assumed to have a level and trend in the systematic component of demand but no seasonality Systematic component of demand = level + trend

Trend-Corrected Exponential Smoothing (Holt’s Model) Obtain initial estimate of level and trend by running a linear regression Dt = at + b T0 = a, L0 = b In Period t, the forecast for future periods is Ft+1 = Lt + Tt and Ft+n = Lt + nTt Revised estimates for Period t (β is a smoothing constant factor) Lt+1 = aDt+1 + (1 – a)(Lt + Tt) Tt+1 = b(Lt+1 – Lt) + (1 – b)Tt

Trend-Corrected Exponential Smoothing (Holt’s Model) MP3 player demand D1 = 8,415, D2 = 8,732, D3 = 9,014, D4 = 9,808, D5 = 10,413, D6 = 11,961 a = 0.1, b = 0.2 Using regression analysis (line trend method) L0 = 7,367 and T0 = 673 Forecast for Period 1 F1 = L0 + T0 = 7,367 + 673 = 8,040

Trend-Corrected Exponential Smoothing (Holt’s Model) Revised estimate L1 = aD1 + (1 – a)(L0 + T0) = 0.1 x 8,415 + 0.9 x 8,040 = 8,078 T1 = b(L1 – L0) + (1 – b)T0 = 0.2 x (8,078 – 7,367) + 0.8 x 673 = 681 With new L1 F2 = L1 + T1 = 8,078 + 681 = 8,759 Continuing F7 = L6 + T6 = 11,399 + 673 = 12,072

Trend-Corrected Exponential Smoothing (Holt’s Model) Period   Demand L T Forecast |Et| Et2 (|Et|/Dt)% 35.212 1.7238 January 1 37 36.97 1.73 36.94 0.06 0.00 0.17 February 2 40 39.35 1.93 38.70 1.30 1.69 3.25 March 3 41 41.14 1.89 41.28 0.28 0.08 0.68 April 4 40.01 0.98 43.03 6.03 36.31 16.29 May 5 45 43.00 1.58 41.00 4.00 16.04 8.90 June 6 50 47.29 2.40 44.58 5.42 29.37 10.84 July 7 43 46.34 1.39 49.69 6.69 44.71 15.55 August 8 47 47.37 1.28 47.74 0.74 0.54 1.57 September 9 56 52.33 2.39 48.65 7.35 54.01 13.12 October 10 52 53.36 1.98 54.71 2.71 5.21 November 11 55 55.17 55.33 0.33 0.11 0.61 December 12 54 55.55 1.46 57.10 3.10 9.58 5.73 57.01 b 0.300 MAD 3.167 a 0.500 MSE 16.649 MAPE 6.826

Trend-Corrected Exponential Smoothing (Holt’s Model)

Trend- and Seasonality-Corrected Exponential Smoothing Appropriate when the systematic component of demand is assumed to have a level, trend, and seasonal factor Systematic component = (level + trend) x seasonal factor Ft+1 = (Lt + Tt)St+1 and Ft+l = (Lt + lTt)St+l

Trend- and Seasonality-Corrected Exponential Smoothing After observing demand for period t + 1, revise estimates for level, trend, and seasonal factors Lt+1 = a(Dt+1/St+1) + (1 – a)(Lt + Tt) Tt+1 = b(Lt+1 – Lt) + (1 – b)Tt St+p+1 = g(Dt+1/Lt+1) + (1 – g)St+1 a = smoothing constant for level b = smoothing constant for trend g = smoothing constant for seasonal factor

Winter’s Model Tahoe Salt Example L0 = 18,439 T0 = 524 S1= 0.47, S2 = 0.68, S3 = 1.17, S4 = 1.67 F1 = (L0 + T0)S1 = (18,439 + 524)(0.47) = 8,913 The observed demand for Period 1 = D1 = 8,000 Forecast error for Period 1 = E1 = F1 – D1 = 8,913 – 8,000 = 913 Year, Qtr Period t Demand Dt 1,2 1 8,000 1,3 2 13,000 1,4 3 23,000 2,1 4 34,000 2,2 5 10,000 2,3 6 18,000 2,4 7 3,1 8 38,000 3,2 9 12,000 3,3 10 3,4 11 32,000 4,1 12 41,000

Winter’s Model Assume a = 0.1, b = 0.2, g = 0.1; revise estimates for level and trend for period 1 and for seasonal factor for Period 5 L1 = a(D1/S1) + (1 – a)(L0 + T0) = 0.1 x (8,000/0.47) + 0.9 x (18,439 + 524) = 18,769 T1 = b(L1 – L0) + (1 – b)T0 = 0.2 x (18,769 – 18,439) + 0.8 x 524 = 485 S5 = g(D1/L1) + (1 – g)S1 = 0.1 x (8,000/18,769) + 0.9 x 0.47 = 0.47 F2 = (L1 + T1)S2 = (18,769 + 485)0.68 = 13,093

Mean Squared Error MSEt Winter’s Model Period t Demand Dt Level Lt Trend Tt Seasonal Factor St Forecast Ft Error Et Absolute Error At Mean Squared Error MSEt MADt % Error MAPEt TSt   18,439 524 1 8,000 18,769 485 0.47 8,913 913 832,857 11 11.41 1.00 2 13,000 19,240 482 0.68 13,093 93 420,727 503 6.06 2.00 3 23,000 19,716 481 1.17 23,076 76 282,393 360 4.15 3.00 4 34,000 20,214 484 1.67 33,730 -270 270 230,049 338 3.31 2.40 5 10,000 20,776 500 9,637 -363 363 210,321 343 3.37 1.31 6 18,000 21,797 604 14,458 -3,542 3,542 2,265,821 876 20 6.09 -3.53 7 22,127 549 26,202 3,202 3,406,502 1,208 14 7.21 0.09 8 38,000 22,683 551 37,898 -102 102 2,981,997 1,070 6.34 0.01 9 12,000 23,479 600 10,855 -1,145 1,145 2,796,424 1,078 10 6.70 -1.06 23,543 493 0.69 16,715 3,715 3,896,961 1,342 29 8.89 1.92 32,000 24,399 565 1.16 27,801 -4,199 4,199 5,145,678 1,602 13 9.27 -1.01 12 41,000 24,921 557 41,731 731 4,761,357 1,529 8.65 -0.58 12,015 17,703 15 31,167 16 45,307 alpha 0.1 Beta 0.2 Gamma

Winter’s Model

Time Series Models Forecasting Method Applicability Moving average No trend or seasonality Simple exponential smoothing Holt’s model Trend but no seasonality Winter’s model Trend and seasonality

Static Forecasting Methods Static methods assumes that the estimates of level, trend, and seasonality within the systematic component do not vary as new demand is observed where L = estimate of level at t = 0 T = estimate of trend St = estimate of seasonal factor for Period t Dt = actual demand observed in Period t Ft = forecast of demand for Period t

Static Forecasting Methods - Tahoe Salt Example Year Quarter Period, t Demand, Dt 1 2 8,000 3 13,000 4 23,000 34,000 5 10,000 6 18,000 7 8 38,000 9 12,000 10 11 32,000 12 41,000

Static Forecasting Methods - Tahoe Salt Example Observe that the demand is seasonal, the lowest occurs every second quarter of each year, and the overall demand pattern repeats every year.

Static Forecasting Methods - Tahoe Salt Example How to estimate level, trend, and seasonality factors? Basic Steps: Deseasonalize demand and run the linear trend line method to estimate level and trend To ensure that each season is given equal weight when deseasonalizing the demand, we must take the average of p consecutive periods of demand. Estimate seasonal factors The seasonal factor of a period t is the ratio between the actual demand and the deseasonalized demand

Static Forecasting Methods - Tahoe Salt Example Estimate Level and Trend Deseasonalized Demand formulas: Periodicity p = 4, t = 3

Static Forecasting Methods - Tahoe Salt Example Estimate Level and Trend

Static Forecasting Methods - Tahoe Salt Example Estimate Level and Trend A linear relationship exists between the deseasonalized demand and time based on the change in demand over time

Static Forecasting Methods - Tahoe Salt Example The values of L and T for the deseasonalized demand can be estimated using the linear trend line method: See also later slides on Linear Regression Estimate Level and Trend

Static Forecasting Methods - Tahoe Salt Example Estimating Seasonal Factors

Static Forecasting Methods - Tahoe Salt Example Estimating Seasonal Factors Given r seasonal cycles in the data, for all periods of the form pt+1,1<=i<=p, we should obtain the seasonal factor for each period: For r = 3 (seasonal cycles) and p=4 (periods) we get:

Static Forecasting Methods - Tahoe Salt Example Forecast for the next 4 periods Given the level, trend and all seasonal factors calculated earlier, using the following formula we obtain the forecast for the next 4 quarters:

Static Forecasting Methods - Tahoe Salt Example

Forecast Errors

Measures of Forecast Error Declining alpha

Forecast Accuracy Forecast error difference between forecast and actual demand MAD mean absolute deviation MAPD mean absolute percent deviation Cumulative error Average error or bias

Mean Absolute Deviation (MAD)  Dt - Ft  n MAD = where t = period number Dt = demand in period t Ft = forecast for period t n = total number of periods  = absolute value

PERIOD DEMAND, Dt Ft ( =0.3) (Dt - Ft) |Dt - Ft| MAD Example 1 37 37.00 – – 2 40 37.00 3.00 3.00 3 41 37.90 3.10 3.10 4 37 38.83 -1.83 1.83 5 45 38.28 6.72 6.72 6 50 40.29 9.69 9.69 7 43 43.20 -0.20 0.20 8 47 43.14 3.86 3.86 9 56 44.30 11.70 11.70 10 52 47.81 4.19 4.19 11 55 49.06 5.94 5.94 12 54 50.84 3.15 3.15 557 49.31 53.39 PERIOD DEMAND, Dt Ft ( =0.3) (Dt - Ft) |Dt - Ft| MAD = 53.39/11 = 4.85

Other Accuracy Measures Mean absolute percent deviation (MAPD) MAPD = |Dt - Ft| Dt Cumulative error E = et Average error E = et n

Forecast Control Tracking signal (Dt - Ft) E MAD Tracking signal = = monitors the forecast to see if it is biased high or low 1 MAD ≈ 0.8 б Control limits of 2 to 5 MADs are used most frequently Tracking signal = = (Dt - Ft) MAD E

Tracking Signal Values 1 37 37.00 – – – 2 40 37.00 3.00 3.00 3.00 3 41 37.90 3.10 6.10 3.05 4 37 38.83 -1.83 4.27 2.64 5 45 38.28 6.72 10.99 3.66 6 50 40.29 9.69 20.68 4.87 7 43 43.20 -0.20 20.48 4.09 8 47 43.14 3.86 24.34 4.06 9 56 44.30 11.70 36.04 5.01 10 52 47.81 4.19 40.23 4.92 11 55 49.06 5.94 46.17 5.02 12 54 50.84 3.15 49.32 4.85 DEMAND FORECAST, ERROR E = PERIOD Dt Ft Dt - Ft (Dt - Ft) MAD – 1.00 2.00 1.62 3.00 4.25 5.01 6.00 7.19 8.18 9.20 10.17 TRACKING SIGNAL TS3 = = 2.00 6.10 3.05 Tracking signal for period 3

Statistical Control Charts  = (Dt - Ft)2 n - 1 Using  we can calculate statistical control limits for the forecast error Control limits are typically set at  3

Statistical Control Charts Errors 18.39 – 12.24 – 6.12 – 0 – -6.12 – -12.24 – -18.39 – | | | | | | | | | | | | | 0 1 2 3 4 5 6 7 8 9 10 11 12 Period UCL = +3 LCL = -3

Selecting the Best Smoothing Constants

Selecting the Best Smoothing Constant

Selecting the Best Smoothing Constant

Application at Tahoe Salt

Forecasting Demand at Tahoe Salt Moving average Simple exponential smoothing Trend-corrected exponential smoothing Trend- and seasonality-corrected exponential smoothing

Forecasting Demand at Tahoe Salt

Forecasting Demand at Tahoe Salt Moving average L12 = 24,500 F13 = F14 = F15 = F16 = L12 = 24,500 s = 1.25 x 9,719 = 12,148

Forecasting Demand at Tahoe Salt

Forecasting Demand at Tahoe Salt Single exponential smoothing L0 = 22,083 L12 = 23,490 F13 = F14 = F15 = F16 = L12 = 23,490 s = 1.25 x 10,208 = 12,761

Forecasting Demand at Tahoe Salt

Forecasting Demand at Tahoe Salt Trend-Corrected Exponential Smoothing L0 = 12,015 and T0 = 1,549 L12 = 30,443 and T12 = 1,541 F13 = L12 + T12 = 30,443 + 1,541 = 31,984 F14 = L12 + 2T12 = 30,443 + 2 x 1,541 = 33,525 F15 = L12 + 3T12 = 30,443 + 3 x 1,541 = 35,066 F16 = L12 + 4T12 = 30,443 + 4 x 1,541 = 36,607 s = 1.25 x 8,836 = 11,045

Forecasting Demand at Tahoe Salt

Forecasting Demand at Tahoe Salt Trend- and Seasonality-Corrected L0 = 18,439 T0 =524 S1 = 0.47 S2 = 0.68 S3 = 1.17 S4 = 1.67 L12 = 24,791 T12 = 532 F13 = (L12 + T12)S13 = (24,791 + 532)0.47 = 11,940 F14 = (L12 + 2T12)S13 = (24,791 + 2 x 532)0.68 = 17,579 F15 = (L12 + 3T12)S13 = (24,791 + 3 x 532)1.17 = 30,930 F16 = (L12 + 4T12)S13 = (24,791 + 4 x 532)1.67 = 44,928 s = 1.25 x 1,469 = 1,836

Forecasting Demand at Tahoe Salt

Forecasting Demand at Tahoe Salt Forecasting Method MAD MAPE (%) TS Range Four-period moving average 9,719 49 –1.52 to 2.21 Simple exponential smoothing 10,208 59 –1.38 to 2.15 Holt’s model 8,836 52 –2.15 to 2.00 Winter’s model 1,469 8 –2.74 to 4.00

Regression Methods

Regression Methods Linear regression Correlation a mathematical technique that relates a dependent variable to an independent variable in the form of a linear equation Correlation a measure of the strength of the relationship between independent and dependent variables

Linear Regression Example (WINS) x (ATTENDANCE) y xy x2 4 36.3 145.2 16 6 40.1 240.6 36 6 41.2 247.2 36 8 53.0 424.0 64 6 44.0 264.0 36 7 45.6 319.2 49 5 39.0 195.0 25 7 47.5 332.5 49 49 346.7 2167.7 311

Linear Regression Example (cont.) y = = 43.36 b = = = 4.06 a = y - bx = 43.36 - (4.06)(6.125) = 18.46 49 8 346.9 xy - nxy2 x2 - nx2 (2,167.7) - (8)(6.125)(43.36) (311) - (8)(6.125)2

Linear Regression Example (cont.) | | | | | | | | | | | 0 1 2 3 4 5 6 7 8 9 10 60,000 – 50,000 – 40,000 – 30,000 – 20,000 – 10,000 – Linear regression line, y = 18.46 + 4.06x Wins, x Attendance, y y = 18.46 + 4.06(7) = 46.88, or 46,880 Attendance forecast for 7 wins

Correlation and Coefficient of Determination Correlation, r Measure of strength of relationship Varies between -1.00 and +1.00 Coefficient of determination, r2 Percentage of variation in dependent variable resulting from changes in the independent variable

Computing Correlation n xy -  x y [n x2 - ( x)2] [n y2 - ( y)2] r = Coefficient of determination r2 = (0.947)2 = 0.897 r = (8)(2,167.7) - (49)(346.9) [(8)(311) - (49)2] [(8)(15,224.7) - (346.9)2] r = 0.947

Correlation Coefficient y x (a) Perfect negative correlation y x (e) Perfect positive correlation y x (b) Negative correlation y x (d) Positive correlation High Moderate Low Correlation coefficient values | | | | | | | | | –1.0 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1.0 y x (c) No correlation

Standard Error of the Estimate A forecast is just a point estimate of a future value This point is actually the mean of a probability distribution 4.0 – 3.0 – 2.0 – 1.0 – | | | | | | | 0 1 2 3 4 5 6 7 Area payroll (in $ billions) Nodel’s sales (in$ millions) 3.25 Regression line,

Standard Error of the Estimate where y = y-value of each data point yc = computed value of the dependent variable, from the regression equation n = number of data points

Standard Error of the Estimate Computationally, this equation is considerably easier to use We use the standard error to set up prediction intervals around the point estimate

Standard Error of the Estimate 4.0 – 3.0 – 2.0 – 1.0 – | | | | | | | 0 1 2 3 4 5 6 7 Area payroll (in $ billions) Nodel’s sales (in$ millions) 3.25 The standard error of the estimate is $306,000 in sales

Regression Analysis with Excel

Regression Analysis with Excel Maosplace.xls – Linear Regression

Study the relationship of demand to two or more independent variables Multiple Regression Study the relationship of demand to two or more independent variables y = 0 + 1x1 + 2x2 … + kxk where 0 = the intercept 1, … , k = parameters for the independent variables x1, … , xk = independent variables

Multiple Regression with Excel

Example of Multi-Regression for Demand Forecasting Day Temperature Flyer Sales 1 63 ✓ 152 2 70 168 3 73 180 4 75 235 5 80 236 6 82 225 7 85 268 8 88 330 9 90 314 10 91 306 11 92 374 12 192 13 98 340 14 100 388 15 317 16 87 283 17 84 258 18 310 19 226 20 214 21 76 198 Can we predict customer reactions or the demand for a product (ice cream)? : Flyer was included in the local daily newspaper

Do you see any relationships in these data? Day Temperature Flyer Sales 1 63 152 2 70 168 3 73 180 4 75 235 5 80 236 6 82 225 7 85 268 8 88 330 9 90 314 10 91 306 11 92 374 12 192 13 98 340 14 100 388 15 317 16 87 283 17 84 258 18 310 19 226 20 214 21 76 198 Two parameters possibly affecting sales: temperature and flyer advertisement in the local daily newspaper Flyer = 0: No flyer included in the local daily newspaper Flyer = 1: Otherwise

Sales Forecast – Single Regression Use as predictor variable the temperature, assuming a linear relationship between sales and temperature Can I use this model to forecast the demand for a day in 2 months from now? Can I make forecasts outside the above temperature range? Should I cleanup the historical data (e.g. days with sold outs)?

Sales Forecast - Multiple Regression Can I increase predictability by considering flyer promotions on particular days? The relationship with TWO explanatory (predictor) variables can be formalized as follows: Sales(t) = B0 + B1*Temperature(t)+ B2*Flyer(t)

Sales Forecast - Multiple Regression Better predictability than the original regression (R Square was 0.8928) SUMMARY OUTPUT Regression Statistics Multiple R 0.969768733 R Square 0.940451395 Adjusted R Square 0.933834884 Standard Error 14.95415498 Observations 21 ANOVA   df SS MS F Significance F Regression 2 63571.28991 31785.64 142.1370421 9.41552E-12 Residual 18 4025.281521 223.6268 Total 20 67596.57143 Coefficients t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept -250.4242164 31.33367525 -7.99218 2.48511E-07 -316.2538254 -184.5946075 Temperature 6.027360446 0.380548578 15.83861 5.16976E-12 5.227857551 6.82686334 Flyer 3.000080948 8.093034548 0.370699 0.715188186 -14.00275371 20.0029156 Very small chance (<0.05) the independent values are not useful B0; B1; B2 Significance

Sales Forecast - Multiple Regression

Application in Practice

Demand Forecasting Extrapolate past demand data into the future using time-series forecasting methods: Select a preferred forecasting method from a set of techniques Quantify the forecasting errors to be experienced when forecasting the future Use regression analysis to build predictive models that will relate targeted customer segments with casual variable such as price, inventory, and so on.

The Role of IT in Forecasting Forecasting module is core supply chain software Can be used to best determine forecasting methods for the firm and by product categories and markets Real time updates help firms respond quickly to changes in marketplace Facilitate demand planning

Risk Management Errors in forecasting can cause significant misallocation of resources in inventory, facilities, transportation, sourcing, pricing, and information management Common factors are long lead times, seasonality, short product life cycles, few customers and lumpy demand, and when orders placed by intermediaries in a supply chain Mitigation strategies – increasing the responsiveness of the supply chain and utilizing opportunities for pooling of demand

Forecasting In Practice Collaborate in building forecasts Share only the data that truly provide value Be sure to distinguish between demand and sales

Basic Approach Understand the objective of forecasting. Integrate demand planning and forecasting throughout the supply chain. Identify the major factors that influence the demand forecast. Forecast at the appropriate level of aggregation. Establish performance and error measures for the forecast. Notes:

Forecasting In Practice 1. Identify the purpose of forecast 2. Collect historical data 3. Plot data and identify patterns 6. Check forecast accuracy with one or more measures 5. Develop/compute forecast for period of historical data 4. Select a forecast model that seems appropriate for data 7. Is accuracy of forecast acceptable? No 8b. Select new forecast model or adjust parameters of existing model Yes 9. Adjust forecast based on additional qualitative information and insight 10. Monitor results and measure forecast accuracy 8a. Forecast over planning horizon

Summary of Learning Objectives Understand the role of forecasting for both an enterprise and a supply chain Identify the components of a demand forecast Forecast demand in a supply chain given historical demand data using time-series methodologies Analyze demand forecasts to estimate forecast error