Forecasting

Abstract

There are two main reasons why an inventory control system needs to order items some time before customers demand them. First, there is nearly always a lead-time between the ordering time and the delivery time. Second, due to certain ordering costs, it is often necessary to order in batches instead of unit for unit. These two reasons mean that we need to look ahead and forecast the future demand. A demand forecast is an estimated average of the demand size over some future period. But it is not enough to estimate the average demand. We also need to determine how uncertain the forecast is. If the forecast is more uncertain, a larger safety stock is required. Consequently, it is also necessary to estimate the forecast error, which may be represented by the standard deviation or the so-called Mean Absolute Deviation (MAD).

Problems

(*Answer and/or hint in Appendix 1)

• 2.1* Assume the following demands:

  Period	1	2	3	4	5	6	7	8	9	10
  Demand	460	452	458	470	478	480	498	500	490	488

  Determine the moving average with N = 3 after observing the demands in periods 3–10.

• 2.2 Determine ${{\hat{a}}_{5}}$, ${{\hat{a}}_{6}}$, and ${{\hat{a}}_{7}}$ in Example 2.1.

• 2.3 Determine ${{\hat{a}}_{i}}$ and ${{\hat{b}}_{i}}$ for i = 5, 6, 7 in Example 2.2.

• 2.4* Assume that the demand in period t is c + d $\cdot$ t, where c and d are any constants.

  • (a) Apply exponential smoothing with trend starting at t = 1 with ${{\hat{a}}_{0}}=0$ and ${{\hat{b}}_{0}}=0$. Using

    $$ {{\left(\begin{matrix} 1-\alpha& 1-\alpha \\ -\alpha \beta& 1-\alpha \beta \end{matrix} \right)}^{n}}\to 0,\text{ as }n\to \infty$$

    prove that the forecast error will approach 0 in the long run.

  • (b) Apply exponential smoothing starting at t = 1 with ${{\hat{a}}_{0}}=0$. Prove that the forecast error will approach d/α in the long run.

• 2.5

  • (a) Assume that we determine a constant forecast ${{\hat{a}}_{t}}$at time t by minimizing the sum of the quadratic deviations from the demands x _t , x _{t − 1,…,} x _{t − N + 1}. Show that the resulting forecast is a moving average.

  • (b) Assume that we determine a constant forecast ${{\hat{a}}_{t}}$at time t by minimizing the sum of the weighted quadratic deviations for all previous demands x _t , x _{t − 1,}…. The weight for (x _{t − k}  − ${{\hat{a}}_{t}}$)² is w ^k where 0 < w < 1. Show that this is equivalent to exponential smoothing.

• 2.6 Assume that the demand follows a constant model according to (2.1), where a is completely constant and the standard deviation of ε _t is σ.

  • (a) What is the standard deviation of the forecast error in the long run for a moving average?

  • (b) What is the standard deviation of the forecast error in the long run for an exponential smoothing forecast?

  • (c) Show that if we set the standard deviations in (a) and (b) equal we get the relationship (2.10), i.e., α = 2/(N + 1).

• 2.7* A company uses Winters’ trend-seasonal method for an item. The forecast is updated each quarter. Each quarter has a seasonal index. All smoothing constants are equal to 0.2. After the update at the end of 2004 we have:

  $$ {{\hat{a}}_{04.4}}=1000.00 $$
  $$ {{\hat{b}}_{04.4}}=10.00 $$
  $$ {{\hat{x}}_{04.4,05.1}}=808.00 $$
  $$ {{\hat{x}}_{04.4,05.2}}=1020.00 $$
  $$ {{\hat{x}}_{04.4,05.3}}=1648.00 $$
  $$ {{\hat{x}}_{04.4,05.4}}=624.00 $$

  The demand in the first quarter 2005 was 795, and in the second quarter 1023. Update the forecast twice. What is then the forecasts for the coming four quarters?

• 2.8 Assume that the demand is updated each period by Winters’ trend-seasonal method. The smoothing constants are $\alpha =0.2,\quad \beta =0.1,\quad \gamma =0.2$. There are $T=7$ equally long periods during a year. Assume further that the last update took place at the end of period 13 and resulted as follows

  $$ \begin{aligned}& {{{\hat{a}}}_{13}}=5,\ {{{\hat{b}}}_{13}}=1 \\& {{{\hat{F}}}_{7}}={{{\hat{F}}}_{8}}=1.4,\quad {{{\hat{F}}}_{9}}={{{\hat{F}}}_{10}}={{{\hat{F}}}_{11}}=1,\quad {{{\hat{F}}}_{12}}={{{\hat{F}}}_{13}}=0.6 \end{aligned} $$

  The observed demand in period 14 is 3. Carry out the update and determine the forecast for period 17.

• 2.9 Consider the following demand data.

  Period	1	2	3	4	5	6	7	8	9	10
  Demand	430	452	428	470	478	480	498	500	610	488

  Use the trend method in Sect. 2.7.1 to obtain forecasts for periods 11, 12, and 13. Use all data, i.e., the forecast is determined after observing the demand in period 10 and N = 10.

• 2.10 Consider the example described in Sect. 2.7.3 dealing with a spare part that is exclusively used as a replacement when maintaining a certain machine. The maintenance activities are carried out by the company selling the spare part, and the customers have to order the maintenance two months in advance. Consider the following data showing demand and the known number of machines undergoing maintenance.

  Month	1	2	3	4	5	6	7	8
  Demand	66	45	77	78	64	79
  No of machines	110	97	150	143	125	160	151	172

  Use linear regression to forecast demand in months 7 and 8.

• 2.11 Consider the following demand data.

  Period	1	2	3	4	5	6	7	8	9	10
  Demand	0	0	0	0	22	0	0	0	27	0

  Use Croston’s method to update a demand forecast as described in Sect. 2.8. At the end of period 0 after a positive demand in this period we have ${{\hat{k}}_{0}}=6$ and ${{\hat{d}}_{0}}=31$. Use $\alpha =\beta =0.2$.

• 2.12* Consider the ARIMA(0, 2, 2) model

  $$ {{x}_{t}}=2{{x}_{t-1}}-{{x}_{t-2}}+{{\varepsilon }_{t}}-(2-\alpha -\alpha \beta){{\varepsilon }_{t-1}}+(1-\alpha){{\varepsilon }_{t-2}}$$

  .

  Show that this model is equivalent to exponential smoothing with trend.

• 2.13* Consider the following monthly demands:

  Month	1	2	3	4	5	6
  Demand	718	745	767	728	788	793

  • (a) Determine forecasts by exponential smoothing. Update also MAD. At the end of month 0 the forecast was 730 and MAD was 20. Use α = 0.2 for both updates.

  • (b) Determine instead forecasts by exponential smoothing with trend. The initial forecast is ${{\hat{a}}_{0}}=730$and ${{\hat{b}}_{0}}=0$. Use α =  β  = 0.2. What is the forecast for month 5 at the end of month 2?

• 2.14* A company is using exponential smoothing with trend to forecast the demand for a certain product. The smoothing constants are α = 0.2 and β = 0.1. MAD is updated with exponential smoothing with α = 0.1. At the end of month 4 the mean and trend were 220 and 10, respectively, and MAD was 35. The demand in month 5 is 250. Update forecast and MAD and estimate mean and standard deviation for the total demand during the three months 6–8. The forecast errors are independent over time.

• 2.15 The demand for a product has a clear trend. The company producing the product is therefore determining its forecasts by applying exponential smoothing with trend. At the end of March the updated mean was 50 and the trend was 13. MAD was 2. The smoothing constants for exponential smoothing with trend are $\alpha =0.2$ and $\beta =0.05$. MAD is updated by exponential smoothing with $\alpha =0.1$. The demands during April, May, and June are 57, 64, and 73 respectively. Update forecast and MAD during these months. Determine also a forecast for the total demand during July and August and the standard deviation for this period. (Forecast errors during different periods are assumed to be independent.)

• 2.16 The forecast for an item is determined by exponential smoothing with α = 0.2. When updating MAD, α = 0.1. After the update at the end of period 6 the forecast is 320 and MAD = 82. We have just observed the demand in period 7, which is 290.

  • (a) Update forecast and MAD.

  • (b) Consider the first half of period 10. After the update in (a), what is the forecast and the standard deviation for the demand during this half period. Forecast errors during different periods are assumed to be independent.

  • (c) Assume that you wish to change forecasting method to a moving average, so that the forecast is still based on data with the same average age. How would you choose N?

• 2.17 When estimating the forecast error it is common to use that MAD is coupled to the standard deviation as

  $$ \sqrt{\pi /2}\cdot MAD=\sigma$$

  under the assumption of Normal distribution. Prove this.

• 2.18 Both the forecast and MAD are updated monthly by exponential smoothing. Both smoothing constants are equal to 0.1. After the update that took place at the end of March the forecast was 132 and MAD was 42.

  (a) In April the demand is 92. Update forecast and MAD.

  (b) In this forecast, what is the weight for the demand in January?

• 2.19 Consider the following monthly demands:

  Month	1	2	3	4	5	6
  Demand	668	745	778	728	634	789

  • (a) Determine forecasts by exponential smoothing. At the end of month 0 the forecast was 730. Use α  = 0.2.

  • (b) Update also MAD. At the end of month 0 MAD was 30. Use again α  = 0.2. What is the estimate for σ at the end of period 6?

  • (c) For comparison, use also the updating procedure (2.51). Set the initial  $\sigma =\sqrt{\pi /2}\cdot MAD=\sqrt{\pi /2}\cdot 30$. Use again α = 0.2. What is now the estimate for σ at the end of period 6?

• 2.20 Consider (2.49) and (2.58). Assume that we use the same α and that $\left| {{z}_{0}}\right|\le MA{{D}_{0}}$. Prove that this implies $\left| {{z_t}} \right| \le MA{D_t}$ for all t > 0.