CHAPTER 5                                                             DEMAND FORECASTING

1. The table below shows annual demand (in 100,000 units per year) for Fridgets (they're like Widgets, only cooler). Use this information to calculate a linear trend forecasting model using regression analysis. Use your trend estimate to forecast demand for the years 1995 and 2000.

Year          Demand

1990                 1

1991                 4

1992                 5

1993                 8

Solution:

Year           Demand (S)           Trend (t)           (S − ΣS/n)(t − Σt/n)           (t − Σt/n)²

1990                 1                            1                               5.25                             2.25

1991                 4                            2                               0.25                             0.25

1992                 5                            3                               0.25                             0.25 

1993                 8                            4                               5.25                             2.25

ΣS = 18            Σt = 10         ΣS/n = 18/4 = 4.5           Σt/n = 10/4 = 2.5

Σ(S − ΣS/n)(t − Σt/n) = 11                Σ(t−Σt/n)2 = 5                     b = 11/5 = 2.2

a = 4.5 − (2.2)(2.5) = −1                             Equation: S = −1 + 2.2t

For 1995, t = 6 so S = −1 + (2.2)(6) = 12.2

For 2000, t = 11 so S = −1 + (2.2)(11) = 23.2

2. The table below shows annual demand (in 100,000 units per year) for Smidgets (they're like Widgets, only smaller). Use this information to calculate a linear trend forecasting model using regression analysis. Use your trend estimate to forecast demand for the years 1996 and 2001.

Year          Demand

1990               2

1991               4

1992               5

1993               9

Solution:

Year        Demand (S)        Trend (t)         (S − ΣS/n)(t − Σt/n)          (t − Σt/n)²

1990            2                           1                               4.5                            2.25

1991            4                           2                               0.5                            0.25

1992            5                           3                               0.0                            0.25

1993            9                           4                               6.0                            2.25

ΣS = 20       Σt = 10         ΣS/n = 20/4 = 5         Σt/n = 10/4 = 2.5

Σ(S − ΣS/n)(t − Σt/n) = 11             Σ(t − Σt/n)2 = 5           b = 11/5 = 2.2

a = 5 − (2.2)(2.5) = −0.5                      Equation: S = −0.5 + 2.2t

For 1996, t = 7 so S = −0.5 + (2.2)(7) = 14.9

For 2001, t = 12 so S = −0.5 + (2.2)(12) = 25.9

3. The table below shows annual demand (in 1,000,000 units per year) for Widgets. Use this information to calculate a constant growth forecasting model. Use your growth model to forecast demand for the years 1995 and 2000.

Year          Demand

1990            1.1

1991            1.5

1992            1.5

1993            1.8

Solution:

1.8 = (1.1)(1 + g)⁴ so g = (1.8/1.1)^1/4 − 1 = (1.64)^0.25 − 1 = 0.1310 or 13.10%

For 1995, S = (1.8)(1.1310)² = (1.8)(1.2792) = 2.3

For 2000, S = (1.8)(1.1310)⁷ = (1.8)(2.3672) = 4.3

4. The table below shows annual demand (in 1,000 units per year) for Widjets (they're like Widgets, only faster). Use this information to calculate a constant growth forecasting model. Use your growth model to forecast demand for the years 1996 and 2001.

Year            Demand

1990              1.0

1991              1.4

1992              1.5

1993              1.7

Solution:

1.7 = (1.0)(1 + g)⁴ so g = (1.7/1.0)^1/4 − 1 = (1.7)^0.25 − 1 = 0.1419 or 14.19%

For 1996, S = (1.7)(1.1419)3 = (1.7)(1.4888) = 2.5

For 2001, S = (1.7)(1.1419)8 = (1.7)(2.8900) = 4.9

5. The table below shows semi-annual demand (in thousands) for Shidgets (they're like Widgets, only quieter). A linear trend has been estimated using this data set with t = 1 for 1990.1 and t = 8 for 1993.2. It has an intercept of 0.76 and a slope of 0.20. Use the ratio-to-trend method to calculate seasonal adjustment factors for the first and second half of the year and then forecast the level of demand for 1995.1 and 1995.2. Note: Round all intermediate calculations to two decimal places.

Year              Demand

1990.1              1.0

1990.2              1.1

1991.1              1.4

1991.2              1.5

1992.1              1.8

1992.2              1.9

1993.1              2.3

1993.2              2.3

Solution:

Year           Demand (A)                   Forecast (F)                A/F

1990.1            1.0                                 0.96                        1.04

1990.2            1.1                                 1.16                        0.95

1991.1            1.4                                 1.36                        1.03

1991.2            1.5                                 1.56                        0.96

1992.1            1.8                                 1.76                        1.02

1992.2            1.9                                 1.96                        0.97

1993.1            2.3                                  2.16                       1.06

1993.2            2.3                                  2.36                       0.97

Seasonal factor for Year.1 = (1.04 + 1.03 + 1.02 + 1.06)/4 = 1.04

Seasonal factor for Year.2 = (0.95 + 0.96 + 0.97 + 0.97)/4 = 0.96

Forecast for 1995.1 (t = 11): (1.04)(0.76 + 0.20 * 11) = (1.04)(2.96) = 3.08

Forecast for 1995.2 (t = 12): (0.96)(0.76 + 0.20 * 12) = (0.96)(3.16) = 3.03

6. The table below shows semi-annual demand (in thousands) for Didgets (they're like Widgets, only they're easier to work). A linear trend has been estimated using this data set with t = 1 for 1991.1 and t = 6 for 1993.2. It has an intercept of 1.66 and a slope of 0.24. Use the ratio-to-trend method to calculate seasonal adjustment factors for the first and second half of the year and then forecast the level of demand for 1996.1 and 1996.2.

Note: Round all intermediate calculations to two decimal places.

Year          Demand

1991.1         1.8

1991.2         2.4

1992.1         2.2

1992.2         2.8

1993.1         2.5

1993.2         3.3

Solution:

Year            Demand (A)             Forecast (F)              A/F

1991.1             1.8                            1.90                      0.95

1991.2             2.4                            2.14                      1.12

1992.1             2.2                            2.38                      0.92

1992.2             2.8                            2.62                      1.07

1993.1             2.5                            2.86                      0.87

1993.2             3.3                            3.10                      1.06

Seasonal factor for Year.1 = (0.95 + 0.92 + 0.87)/3 = 0.91

Seasonal factor for Year.2 = (1.12 + 1.07 + 1.06)/3 = 1.08

Forecast for 1996.1 (t = 11): (0.91)(1.66 + 0.24 * 11) = (0.91)(4.30) = 3.91

Forecast for 1996.2 (t = 12): (1.08)(1.66 + 0.24 * 12) = (1.08)(4.54) = 4.90

7. The table below shows the demand for Fidgets (they're like Widgets, only they're more active) over an eight-month period. Calculate a four-period moving average forecast for September. Also evaluate the quality of the four-period moving average forecasting model by calculating the root-mean-square error for the data set. Note: Round all intermediate calculations to two decimal places.

Month               Demand

Jan                           10

Feb                          11

Mar                           5

Apr                            8

May                          6

Jun                          11

Jul                              5

Aug                          11

Solution:

Month                   Demand (A)                  Forecast (F)              (A − F)²

Jan                               10

Feb                              11

Mar                               5

Apr                                8

May                               6                                   8.5                           6.25

Jun                               11                                   7.5                           12.25

Jul                                   5                                   7.5                             6.25

Aug                               11                                   7.5                           12.25

Forecast for September: (6 + 11 + 5 + 11)/4 = 33/4 = 8.25

RMSE = [(6.25 + 12.25 + 6.25 + 12.25)/4)]^0.5 = (37/4)^0.5 = 9.25^0.5 = 3.04

8. The table below shows the demand for Gadgets (they're like Widgets, only they're more mechanical) over a five-month period. Calculate exponential smoothing forecasts for each month and for July. Use a coefficient of 0.5 and assume that the forecast for January was 8. Also evaluate the quality of the exponential smoothing model by calculating the root-mean-square error for the data set. Note: Round all intermediate calculations to two decimal places.

Month           Demand

Jan                     10

Feb                      5

Mar                   10

Apr                      8

May                     5

Jun                     10

Solution:

Month              Demand (A)         Forecast (F)           (A − F)²

Jan                           10                          8.00                    4.00

Feb                             5                          9.00                   16.00

Mar                          10                          7.00                     9.00

Apr                             8                           8.50                     0.25

May                            5                           8.25                   10.56

Jun                            10                           6.63                   11.36

Forecast for July: (6.63)(0.50) + (10)(1 − 0.50) = 8.32

RMSE = [(4 + 16 + 9 + 0.25 + 10.56 + 11.36)/6]^0.5 = (51.17/6)^0.5 = 8.53^0.5 = 2.92

9. A firm has determined that its average level of sales (S_t) per week in thousands of dollars during a given year depends on the previous year's level of sales (S_t−1), the previous year's level of advertising (A_t−1) per month in thousands of dollars, and the current year's rate of annual industry growth (G_t) in percentage terms. The firm has also determined that the level of industry growth in the current period depends on the previous period's rate of

industry growth (G_t−1) and on current period sales by the firm. During the current period, the firm's level of sales was $100,000, advertising was $40,000, and the rate of growth in the industry was 4 percent. The firm estimated the following two-equation econometric model:

S_t = 4 + 0.40S_t−1 + 0.10A_t−1 + G_t and G_t = 1 + 0.5G_t−1 + 0.5S_t

(i) Formulate a single-equation forecasting equation from this model.

(ii) Forecast the level of sales in the next period.

Solution:

Substitution yields: S_t = 4 + 0.40S_t−1 + 0.10A_t−1 + 1 + 0.5G_t−1 + 0.5S_t

Solving for S_t+1 yields: S_t+1 = 10 + 0.80S_t + 0.20A_t + G_t

The forecast is: S_t+1 = 10 + (0.80)(100) + (0.20)(40) + 4 = 102

10. A firm has determined that its average level of sales (S_t) per week in thousands of dollars during a given year depends on the previous year's level of sales (S_t−1), the previous year's level of advertising (A_t−1) per month in thousands of dollars, and the current year's rate of annual industry growth (G_t) in percentage terms. The firm has also determined that the level of industry growth in the current period depends on the previous period's rate of

industry growth (G_t−1) and on current period sales by the firm. During the current period, the firm's average level of sales was $100,000, advertising was $10,000, and the rate of growth in the industry was 2 percent. The firm estimated the following two-equation econometric model:

S_t = 5 + 0.50S_t−1 + 0.10A_t−1 + 2G_t and G_t = 0.5G_t−1 + 0.25S_t

(i) Formulate a single-equation forecasting equation from this model.

(ii) Forecast the level of sales in the next period.

Solution:

Substitution yields: S_t = 5 + 0.50S_t−1 + 0.10A_t−1 + G_t−1 + 0.5S_t

Solving for S_t+1 yields: S_t+1 = 10 + S_t + 0.20A_t + 2G_t

The forecast is: S_t+1 = 10 + 100 + (0.20)(10) + (2)(2) = 116