Math Problem Statement

Assume an initial starting Ft of 291 units, a trend ( Tt ) of eight units, an alpha of 0.40, and a delta of 0.40. If actual demand turned out to be 279, calculate the forecast for the next period.

Note: Round your answer to 1 decimal place.

Solution

To calculate the forecast for the next period using a trend-adjusted exponential smoothing model, we use the following formulas:

  1. Updated Forecast: Ft+1=Ft+TtF_{t+1} = F_t + T_t Where:

    • FtF_t is the forecast for the current period (291 units)
    • TtT_t is the trend for the current period (8 units)
  2. Updated Trend: Tt+1=δ(FtFt1)+(1δ)TtT_{t+1} = \delta \cdot (F_t - F_{t-1}) + (1 - \delta) \cdot T_t Where:

    • δ\delta is the smoothing constant for the trend (0.40)
    • FtF_t is the forecast for the current period (291 units)
    • Ft1F_{t-1} is the forecast for the previous period (since we're given the starting forecast for tt, we assume the previous period's forecast Ft1F_{t-1} is also 291)
    • TtT_t is the trend for the current period (8 units)
  3. Forecast for the next period: Ft+1=αAt+(1α)(Ft+Tt)F_{t+1} = \alpha \cdot A_t + (1 - \alpha) \cdot (F_t + T_t) Where:

    • α\alpha is the smoothing constant for the level (0.40)
    • AtA_t is the actual demand for the current period (279 units)
    • FtF_t is the forecast for the current period (291 units)
    • TtT_t is the trend for the current period (8 units)

Step-by-step calculation:

  1. Update Trend: Using the formula: Tt+1=0.40(291291)+(10.40)8T_{t+1} = 0.40 \cdot (291 - 291) + (1 - 0.40) \cdot 8 Since Ft=Ft1F_t = F_{t-1}, the term (FtFt1)=0(F_t - F_{t-1}) = 0, so: Tt+1=0+0.608=4.8T_{t+1} = 0 + 0.60 \cdot 8 = 4.8

  2. Updated Forecast: Using the formula: Ft+1=0.40279+(10.40)(291+8)F_{t+1} = 0.40 \cdot 279 + (1 - 0.40) \cdot (291 + 8) Ft+1=0.40279+0.60299F_{t+1} = 0.40 \cdot 279 + 0.60 \cdot 299 Ft+1=111.6+179.4=291.0F_{t+1} = 111.6 + 179.4 = 291.0

So, the forecast for the next period is 291.0 units.


Would you like more details on any part of this process? Here are some related questions for further exploration:

  1. How do different values of α\alpha and δ\delta affect the forecast?
  2. What is the difference between simple exponential smoothing and trend-adjusted exponential smoothing?
  3. How would we adjust this method if we had multiple periods of actual demand data?
  4. How can we use this model to forecast for multiple periods ahead?
  5. What other forecasting models can be used for trend and seasonality?

Tip: In trend-adjusted exponential smoothing, the trend component is especially useful when there's consistent movement in the data (either upward or downward), which makes the trend adjustment more accurate.

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Math Problem Analysis

Mathematical Concepts

Exponential Smoothing
Time Series Forecasting
Trend Adjustment

Formulas

F_{t+1} = F_t + T_t
T_{t+1} = \delta \cdot (F_t - F_{t-1}) + (1 - \delta) \cdot T_t
F_{t+1} = \alpha \cdot A_t + (1 - \alpha) \cdot (F_t + T_t)

Theorems

Trend-Adjusted Exponential Smoothing

Suitable Grade Level

Grades 10-12