Module 2 – Forecasting 1

Module 2 – Forecasting 1. What is forecasting? Forecasting is defined as estimating the future value that a parameter will take. Most scientific forecasting methods forecast the future value using past data. In Operations Management forecasting is used extensively to estimate future demand of product(s)

2. What is time series forecasting? Time series forecasting uses past data to estimate the future values. Here the performance with respect to time is considered. Time is the independent variable.

3. Mention some simple forecasting models for time series data? Some simple forecasting models using time series data are simple average, moving average and simple exponential smoothing.

4. What is moving average? Moving average is a simple time series forecasting model based on averages of a chosen number of periods. It is used to forecast a constant model or level data.

5. Write the basic equation for simple exponential smoothing? = + (1 ) . This equation can be used when simple exponential smoothing is used as a forecasting model. Here Ft represents the forecast for period 푡+1 푡 푡 퐹t, Dt is the훼퐷 known− demand훼 퐹 for period t and α is the smoothing constant. The basic equation for simple exponential smoothing is = + (1 ) . Here is the smoothed value of the data including Yt the most recent data. 푡 푡 푡−1 푡 푌� 훼푌 − 훼 푌� 푌� 6. Is exponential smoothing a form of weighted average? How? Simple exponential smoothing can be seen as a form of weighted moving average.

Expanding the general equation FDtt+1 =αα +−(1 ) F t, we get 21tt− FDtt+1=α +− αα(1 ) D t −−1 +− αα(1) Dt 2 ++− ... αα( 1 ) DF11 +−(1 α) As t is large and tends to infinity, the term (1- α)t tends to zero. The rest of the terms are all terms involving Dj. It can be seen that the Ft+1 value is a weighted average of 21t− the terms Dt to D1 with weights ααααα,( 1−) ,( 1 −) ,... +− αα( 1 ) . If 0 ≤ α ≤ 1, each weight is smaller than 1 and is decreasing. The highest weight is given to the most recent point and the weights progressively decrease by a factor (1 – α) as the 2 data gets older. As t tends to infinity, the weights are ααααα,( 1−−) ,( 1) ,...This is an infinite geometric series whose first term is α and the common term is (1 – α). α The sum of all the terms of the progression is =1. (1−− (1α ))

7. What are the implications of using small α? A small value of α implies that initial weight given to the recent data is small and the subsequent weights are smaller. This means that more terms contribute to the forecast. This also means that more weight is given to the forecast than to the demand.

8. In the equation Y = a + bt + ε, what does ε represent? What can you say about the mean and variance of ε? The symbol ε represents the error term. It is assumed to be normally distributed with mean = 0 and with small variance. This means that the errors are expected to cancel out each other. The error term is also expected to be small.

9. What do a and b represent in the equation Y = a + bt + ε? In the linear equation Y = a + bt + ε, b is the slope and a is the y intercept – the point in which the line touches the y axis.

10. Write the equations for Holt’s model? The basic equation for Holt’s model is Ft+1 = at + bt. Here at is called the level which represents the smoothed value up to and including the last data. The slope of the line is given by bt and therefore the forecast for the next period Ft+1 = at + bt. The values of at and bt are updated using at = α Dt + (1 – α)(at-1 + bt-1) and bt = β (at – at- 1) + (1 – β)bt-1.

11. How is the Holt’s model different from the linear regression model? Holt’s model is different from linear regression because it computes different values of the slope and intercept at different points using simple exponential smoothing

12. Write the equations for the Winter’s model? Ft+1 = (at + bt)Ct+1 where at and bt are the level and trend as described in the Holt’s model. Ct+1 is the seasonality index for the period that we are forecasting. The equations are at+1 = α(Dt+1/Ct+1) + (1-α)(at + bt) bt+1 = β(at+1 - at ) + (1 - β)bt Ct+p+1 = γ(Dt+1/at+1) + (1 - γ)Ct+1

13. What is seasonality index and how is it calculated? Seasonality index captures the effect of the season on the data. It can be defined as Si = Di/Average. For example, if there are 4 seasons, we can compute the average of the demands of four seasons. The demand in a period divided by the average gives the seasonality index for the period. It is important to know the number of periods that constitute a season.

14. Mention some measures of goodness of forecasts? Some measures of goodness of forecasts include, mean squared deviation, mean absolute deviation, mean percentage deviation etc.

15. What is a causal model? In a causal model, there is an independent variable or a causal variable that impacts the dependent variable (demand).

16. Write equations for causal model? The equation for a causal model is Y = a + bX where X is the independent (causal) variable and Y is the dependent variable. This is a linear model. Other models exist.

Problems

1. Given the data 92, 93, 92, 91, 93, 94, 92 find the forecast for the eighth period using simple average, weighted average (weight of 1 for the first four periods and 2 for the remaining three), 3 period moving average? Simple average = (92 + 93 + 92 + 91 + 93 + 94 + 92)/7 = 92.28 Weighted moving average = [ (92+ 93 + 92 + 91) + 2(93 + 94 + 92)]/10 = 92.6 Three period moving average = (93 + 94 + 92)/3 = 93

2. Given the data 92, 93, 92, 91, 93, 94, 92 find the forecast for the eighth period using simple exponential smoothing? Use α = 0.3 and initial forecast using simple average? Simple average = 92.28; F1 = 92.28, α = 0.3 Ft+1 = αDt + (1- α)Ft; F2 = 92.196, F3 = 92.44, F4 = 92.31, F5 = 91.91, F6 = 92.24, F7 = 92.77, F8 = 92.54

3. Given the data 63, 64, 66, 67, 67, 69, 71, 72 find the forecast for the eighth period using simple average, and 3 period moving average? Is it a good forecast? Why or why not? Simple average = 67.375, three period moving average forecast = (69 + 71 + 72)/3 = 70.666. Both are not good forecasts because the data shows increasing trend while the forecasting models used are for constant (level) data and indicate a central value (average).

4. Given the data 63, 64, 66, 67, 67, 69, 71, 72 find the forecast for the ninth period using simple exponential smoothing? Use α = 0.3 and initial forecast using simple average. Is it a good forecast? Why or why not? F1 = average = 67.375, Ft+1 = αDt + (1- α)Ft; F2 = 66.06, F3 = 65.44, F4 = 65.61, F5 = 66.03, F6 = 66.32, F7 = 67.12, F8 = 68.27, F9 = 69.40 . The forecast is not good because the data shows increasing trend while simple exponential smoothing is to be used for constant (level) data and indicates a central value (average).

5. Derive the expression for a and b in the equation Y = a + bt? n =++ε 2 We wish to derive yt a bt and find a and b such that ∑( yt −− a bt) is minimized. t=1 Partially differentiating the residue with respect to a and b and setting the first derivative to

2 zero, we get the equations ∑∑yt = na + b t and ∑tyt = a ∑∑ t + b t . Here a and b are unknowns and the other terms can be computed. Solving these equations we get the values of a and b.

6. Given the data 63, 64, 66, 67, 67, 69, 71, 72 find the forecast for the ninth period using linear regression? We compute ΣY = 539, Σt = 36, Σt2 = 204 and ΣYt = 2479. The equations are 539 = 8a + 36b and 2479 = 36a + 204b. Solving, we get b = 1.274 and a = 61.64. F9 = a + 8b = 73.11

7. Given the data 63, 64, 66, 67, 67, 69, 71, 72 find the forecast for the ninth period using Holt’s model? Use α = β = 0.2. The equations for Holt’s model have been given earlier. Using these equations, we get F2 = 64.29, F3 = 65.51, F4 = 66.91, F5 = 68.23, F6 = 69.23, F7 = 70.42, F8 = 71.80 and F9 = 73.1

8. Data for four quarters for three years is 81, 62, 76, 55, 85, 65, 79, 60, 90, 69, 84, 64. Find the forecast for the next four periods using a simple seasonality model computing seasonality indices? We assume that there are 4 seasons and data for three years (say). The total demands for 3 years are 274, 289 and 307. The forecasted total demand is 323 and per season it is 80.75. The seasonality indices (average) are 1.18, 0.9, 1.09 and 0.82. The forecasted values are 95.3, 72.68, 88, 67.

9. Data for four quarters for three years is 81, 62, 76, 55, 85, 65, 79, 60, 90, 69, 84, 64. Find the forecast for the next four periods using Winter’s model. α = β = 0.2, γ = 0.3.

The equations for Winters model are given: Ft+1 = (at + bt)Ct+1 where at and bt are the level and trend as described in the Holt’s model. Ct+1 is the seasonality index for the period

that we are forecasting. The equations are at+1 = α(Dt+1/Ct+1) + (1-α)(at + bt); bt+1 = β(at+1 - at )

+ (1 - β)bt; Ct+p+1 = γ(Dt+1/at+1) + (1 - γ)Ct+1. The computations give C13 = 0.3, C14 = 0.225, C15 = 0.275, c16 = 0.2, a12 = 313.9, b12 = 4.22, F13 = 95.4. Using F13 = D13, we get F14 =71, F15 = 89.5, F16 = 66

10. Students believe that the salary they can expect during a placement process is related to their academic performance. The CGPA (indicator of performance) and the salary obtained by six students are (7, 6), (6.8, 5.8), (7.5, 6.5), (8, 7), (8.2, 7.5) and (8.6, 8). Find the salary that a student with CGPA 8.7 can expect? We build a causal model of the form Y = a + bX where Y is the salary and X is the CGPA. The equations are ΣY = na + bΣX; ΣXY = aΣX + bΣX2. Computing, we get ΣX = 46.1, ΣY = 40.8, ΣXY = 316.49 and ΣX2 = 281.14. Solving, we get a = 7.11 and b = -0.04 For X = 8.7, Y = 6.76.

11. Given the data 92, 93, 92, 91, 93, 94, 92 find the forecast for the eighth period using simple average. Compute the mean absolute deviation?

∑ DFii− n F8 = 92.43. Mean absolute deviation = MAD = . Computing, we get MAD = n (.43 + 0.57 + 0.43 + 1.43 + 0.57 + 1.57 + 0.43)/7 = 0.776

12. Given the data 83, 87, 90, 92, 96, 99, find the forecast for the eighth period using linear regression? We compute ΣY = 547, Σt = 21, Σt2 = 91 and ΣYt = 1969. The equations are 547 = 6a + 21b and 1969 = 21a + 91b. Solving, we get b = 3.11 and a = 80.27. F8 = a + 7b = 102

13. Given the data 83, 87, 90, 92, 96, 99 find the forecast for the eighth period using Holt’s model? Use α = β = 0.2. The equations for Holt’s model are given earlier. Performing the computations, we get F7 = a6 + b6 = 99.14 + 3.21 = 102.35

14. Given the data 92, 93, 92, 91, 93, 94, 92 find the forecast for the eighth period using simple exponential smoothing? Use α = 0.2 α = 0.7 and initial forecast using simple average? Explain the effect of α on the contribution of the data for the various periods At α = 0.2, F7 = 92.63. Since we are building a constant model, F8 = 92.63. At α = 0.7, F7 = 93.55. Since we are building a constant model, F8 = 93.55. Lower α gives more weight to forecast while high α gives more weight to demand.

15. Consider the data 92, 93, 92, 91, 93, 94, 92. Find the mean absolute deviation for the forecast using simple exponential smoothing α = 0.2. The forecasts are F1 = 92.43, F2 = 92.34, F3 = 92.47, F4 = 92.38, F5 = 92.43, F6 = 92.43, F7 = 92.43. MAD = (0.43 + 0.66 + 0.47 + 1.38 + 0.9 + 1.71 + 0.63)/7 = 0.883.