Math Problem Statement

Sales of tablet computers at Marika​ Gonzalez's electronics store in​ Washington, D.C., over the past 10 weeks are shown in the table​ below:                                                                                                                    
Week 1 2 3 4 5 6 7 8 9 10 Demand 19 23 29 38 25 29 35 22 25 29

Part 2 ​a) The forecast for weeks 2 through 10 using exponential smoothing with alpha ​= 0.60 and a week 1 initial forecast of 19.0 are ​(round your responses to two decimal​ places): Week 1 2 3 4 5 6 7 8 9 10 Demand 19 23 29 38 25 29 35 22 25 29 Forecast 19.0    19.00    21.40    25.96    33.18    28.27    28.71    32.48    26.19    25.48

Part 3 ​b) For the forecast developed using exponential smoothing ​(alpha ​= 0.60 and initial forecast 19.0​), the MAD​ =    5.40 sales ​(round your response to two decimal​ places). Part 4 ​c) For the forecast developed using exponential smoothing ​(alpha ​= 0.60 and initial forecast 19.0​), the tracking signal​ =    enter your response here ​(round your response to two decimal​ places).

Solution

Let's walk through the process of finding the Tracking Signal step-by-step.

Formula for Tracking Signal:

Tracking Signal=Cumulative Forecast Error (CFE)Mean Absolute Deviation (MAD)\text{Tracking Signal} = \frac{\text{Cumulative Forecast Error (CFE)}}{\text{Mean Absolute Deviation (MAD)}}

Step 1: Compute the Forecast Errors

The forecast error for each week is: Forecast Error=Actual DemandForecast\text{Forecast Error} = \text{Actual Demand} - \text{Forecast}

Here’s the data from the question:

WeekDemand (Actual)ForecastForecast Error
11919.000.00
22319.004.00
32921.407.60
43825.9612.04
52533.18-8.18
62928.270.73
73528.716.29
82232.48-10.48
92526.19-1.19
102925.483.52

Step 2: Compute the Cumulative Forecast Error (CFE)

CFE=(Actual DemandForecast)\text{CFE} = \sum (\text{Actual Demand} - \text{Forecast})

Adding all the forecast errors: CFE=0.00+4.00+7.60+12.048.18+0.73+6.2910.481.19+3.52=14.33\text{CFE} = 0.00 + 4.00 + 7.60 + 12.04 - 8.18 + 0.73 + 6.29 - 10.48 - 1.19 + 3.52 = 14.33

Step 3: Use the Given MAD

The Mean Absolute Deviation (MAD) is provided as: MAD=5.40\text{MAD} = 5.40

Step 4: Calculate the Tracking Signal

Tracking Signal=CFEMAD=14.335.40=2.65\text{Tracking Signal} = \frac{CFE}{MAD} = \frac{14.33}{5.40} = 2.65

Final Answer:

The Tracking Signal is: 2.65\boxed{2.65}

Would you like further explanations or help with any part of this calculation?


Here are 5 related questions:

  1. How does changing the value of alpha affect the exponential smoothing forecast?
  2. What is the significance of a tracking signal outside the range [-4, 4]?
  3. How can you adjust your forecasting model if the tracking signal indicates bias?
  4. How do MAD and MSE compare in measuring forecast accuracy?
  5. What other forecasting techniques could be used instead of exponential smoothing?

Tip: A positive tracking signal suggests the forecast underestimates demand, while a negative signal indicates overestimation.

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Math Problem Analysis

Mathematical Concepts

Forecasting
Exponential Smoothing
Tracking Signal
Mean Absolute Deviation (MAD)

Formulas

Forecast_t = Alpha * Demand_(t-1) + (1 - Alpha) * Forecast_(t-1)
Tracking Signal = Cumulative Forecast Error (CFE) / MAD
Forecast Error = Actual Demand - Forecast

Theorems

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Suitable Grade Level

Undergraduate Business/Statistics