Supply Chain Management Chapter 4 Demand forecasting in
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Supply Chain Management 第四章 供应链的需求预测 Chapter 4 Demand forecasting in a supply chain 7 -2
Time Series Forecasting (Table 7. 1) 例:Natural. Gas. com Forecast demand for the next four quarters. 13
预测需求水平和需求趋势 n n 剔除季节影响后的需求Deseasonalized demand = demand that would have been observed in the absence of seasonal fluctuations 时期数Periodicity (p):在周期内包含的所有时期之 后,季节性周期将重复进行 for demand at Natural. Gas. com(Table 7. 1, Figure 7. 1) p =4 14
Deseasonalizing Demand 15
Deseasonalized demand Period t Demand D 1 8, 000 2 13, 000 3 23, 000 19, 750 4 34, 000 20, 650 5 10, 000 21, 250 6 18, 000 21, 750 7 23, 000 22, 500 8 38, 000 22, 125 9 12, 000 22, 625 10 13, 000 24, 125 11 32, 000 12 41, 000 16
剔除季节性影响后需求以一个固定比率变化,即剔除 季节性影响后的需求与时间t之间存在一个线性关 系 Dt = L + t. T where Dt = deseasonalized demand in period t L = level (deseasonalized demand at period 0) T = trend (rate of growth of deseasonalized demand) In the example, L = 18, 439 and T = 524 17
Time Series of Demand (Figure 7. 3) 18
估计季节性系数 Use the previous equation to calculate deseasonalized demand for each period St = Dt / Dt = seasonal factor for period t In the example, D 2 = 18439 + (524)(2) = 19487 D 2 = 13000 S 2 = 13000/19487 = 0. 67 19
Estimating Seasonal Factors (Fig. 7. 4) 20
预测季节性系数 The overall seasonal factor for a “season” is then obtained by averaging all of the factors for a “season” n 如果数据中存在一个r的季节性循环,对所有pt+i, 1≤i≤p为形式的时期,定义 In the example, there are 3 seasonal cycles in the data and p=4, so S 1 = (0. 42+0. 47+0. 52)/3 = 0. 47 S 2 = (0. 67+0. 83+0. 55)/3 = 0. 68 S 3 = (1. 15+1. 04+1. 32)/3 = 1. 17 S 4 = (1. 66+1. 68+1. 66)/3 = 1. 67 21
预测 Using the original equation, we can forecast the next four periods of demand: F 13 = (L+13 T)S 1 = [18439+(13)(524)](0. 47) = 11868 F 14 = (L+14 T)S 2 = [18439+(14)(524)](0. 68) = 17527 F 15 = (L+15 T)S 3 = [18439+(15)(524)](1. 17) = 30770 F 16 = (L+16 T)S 4 = [18439+(16)(524)](1. 67) = 44794 22
适应性预测法 Ft+l = (Lt + l. Tt )St+l = forecast for period t+l in period t 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 = Forecast error in period t At = Absolute deviation for period t = |Et| MAD = Mean Absolute Deviation = average value of At 23
适应法预测步骤 n n n 初始化: Compute initial estimates of level (L 0), trend (T 0), and seasonal factors (S 1, …, Sp). This is done as in static forecasting. 预测: Forecast demand for period t+1 using the general equation Ft+l = (Lt + l. Tt )St+l. 估计误差: Compute error Et+1 = Ft+1 - Dt+1 修正预测值: Modify the estimates of level (Lt+1), trend (Tt+1), and seasonal factor (St+p+1), given the error Et+1 in the forecast Repeat steps 2, 3, and 4 for each subsequent period 24
Moving Average Example From Natural. Gas. com example (Table 7. 1) At the end of period 4, what is the forecast demand for periods 5 through 8 using a 4 -period moving average? L 4 = (D 4+D 3+D 2+D 1)/4 = (34000+23000+13000+8000)/4 = 19500 F 5 = 19500 = F 6 = F 7 = F 8 Observe demand in period 5 to be D 5 = 10000 Forecast error in period 5, E 5 = F 5 - D 5 = 19500 - 10000 = 9500 Revise estimate of level in period 5: L 5 = (D 5+D 4+D 3+D 2)/4 = (10000+34000+23000+13000)/4 = 20000 F 6 = L 5 = 20000 = F 7 = F 8 26
Simple Exponential Smoothing Example From Natural. Gas. com example, forecast demand for period 1 using exponential smoothing L 0 = average of all 12 periods of data = Sum(i=1 to 12)[Di]/12 = 22083 F 1 = L 0 = 22083 Observed demand for period 1 = D 1 = 8000 Forecast error for period 1, E 1, is as follows: E 1 = F 1 - D 1 = 22083 - 8000 = 14083 Assuming α = 0. 1, revised estimate of level for period 1: L 1 = αD 1 + (1 -α)L 0 = (0. 1)(8000) + (0. 9)(22083) = 20675 F 2 = L 1 = 20675 Note that the estimate of level for period 1 is lower than in period 0 28
需求趋势修正后的指数平滑法Trend-Corrected Exponential Smoothing (Holt’s Model) 系统需求有需求水平和需求趋势没有季节性变动 系统成分=需求水平+需求趋势 n Obtain initial estimate of level and trend by running a linear regression of the following form: Dt = at + b T 0 = a L 0 = b In period t, the forecast for future periods is expressed as follows: Ft+1 = Lt + Tt Ft+n = Lt + n. Tt 观测到t+1期需求后,修正 Lt+1 = Dt+1 + (1 - )(Lt + Tt) Tt+1 = b(Lt+1 - Lt) + (1 -b)Tt α为需求水平的平滑系数,0< α<1 β为需求趋势的平滑系数,0< β<1。 n 29
Trend-Corrected Exponential Smoothing Example: Tahoe Salt demand data. Forecast demand for period 1 using Holt’s model (trend corrected exponential smoothing) Using linear regression, L 0 = 12015 (linear intercept) T 0 = 1549 (linear slope) Forecast for period 1: F 1 = L 0 + T 0 = 12015 + 1549 = 13564 Observed demand for period 1 = D 1 = 8000 E 1 = F 1 - D 1 = 13564 - 8000 = 5564 Assume = 0. 1, = 0. 2 L 1 = D 1 + (1 - )(L 0+T 0) = (0. 1)(8000) + (0. 9)(13564) = 13008 T 1 = (L 1 - L 0) + (1 - )T 0 = (0. 2)(13008 - 12015) + (0. 8)(1549) = 1438 F 2 = L 1 + T 1 = 13008 + 1438 = 14446 F 5 = L 1 + 4 T 1 = 13008 + (4)(1438) = 18760 30
需求趋势和季节性需求修正后的指数平滑法 Trend- and Seasonality-Corrected Exponential Smoothing (Winter Model) n 系统需求有需求水平、需求趋势和季节性变动 系统需求=(需求水平+需求趋势)×季节性需 求 n Assume periodicity p n Obtain initial estimates of level (L 0), trend (T 0), seasonal factors (S 1, …, Sp) using procedure for static forecasting n In period t, the forecast for future periods is given by: Ft+1 = (Lt+Tt)(St+1) and Ft+n = (Lt + n. Tt)St+n 31
Trend- and Seasonality-Corrected Exponential Smoothing (continued) After observing demand for period t+1, revise estimates for level, trend, and seasonal factors as follows: Lt+1 = (Dt+1/St+1) + (1 - )(Lt+Tt) Tt+1 = (Lt+1 - Lt) + (1 - )Tt St+p+1 = (Dt+1/Lt+1) + (1 - )St+1 a 为需求水平的平滑系数,0< <1 b 为需求趋势的平滑系数,0< <1 g为季节性需求的平滑系数,0< <1 32
Trend- and Seasonality-Corrected Exponential Smoothing Example: Tahoe Salt data. Forecast demand for period 1 using Winter’s model. Initial estimates of level, trend, and seasonal factors are obtained as in the static forecasting case L 0 = 18439 T 0 = 524 S 1=0. 47, S 2=0. 68, S 3=1. 17, S 4=1. 67 F 1 = (L 0 + T 0)S 1 = (18439+524)(0. 47) = 8913 The observed demand for period 1 = D 1 = 8000 Forecast error for period 1 = E 1 = F 1 -D 1 = 8913 - 8000 = 913 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 L 1 = a(D 1/S 1)+(1 -a)(L 0+T 0) = 0. 1)(8000/0. 47)+(0. 9)(18439+524)=18769 T 1 = b(L 1 -L 0)+(1 -b)T 0 = (0. 2)(18769 -18439)+(0. 8)(524) = 485 S 5 = g(D 1/L 1)+(1 -g)S 1 = (0. 1)(8000/18769)+(0. 9)(0. 47) = 0. 47 F 2 = (L 1+T 1)S 2 = (18769 + 485)(0. 68) = 13093 33
7. 5 预测误差的度量 n 一个好的预测方法应该反映系统需求部分而不是随机需求部分。 随机需求部分会以预测误差的形式表现出来。 t期的预测误差Forecast error = Et = Ft - Dt n 平均方差Mean squared error (MSE) n 绝对离差Absolute deviation = At = |Et| n 平均绝对离差Mean absolute deviation (MAD) When the random component is normally distributed = 1. 25 MAD 34
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