**The Fresh Detergent Case**

Enterprise Industries produces Fresh, a brand of liquid detergent. In order to more effectively manage its inventory, the company would like to better predict demand for Fresh. To develop a prediction model, the company has gathered data concerning demand for Fresh over the last 33 sales periods. Each sales period is defined as one month. The variables are as follows:

**Period =**Time period in month**Demand**= Y = demand for a large size bottle of Fresh (in 100,000)**Price**= the price of Fresh as offered by Ent. Industries**AIP**= the Average Industry Price**ADV**= Enterprise Industries Advertising Expenditure (in $100,000) to Promote Fresh in the sales period.**DIFF**= AIP – Price = the “price difference” in the sales period

Only the trend of PRICE is negative. Other four variables have positive trends. However, the R^{2} values suggest that for ADV and DEMAND only the linear model is explained by the data points moderately (66% and 51% respectively). For all the other three variables, the R^{2} values are too poor to accept the models as adequates because very few percent of data points actually represents the linear model.

As expected, the Demand is negatively correlated with Price. But the regression line equation cannot be relied upon due to poor R^{2} value. For other three variables, there is a positive correlation. Out of these, for the ADV variable, the regression line can be adequate for the R^{2} value is moderately higher.

__Interpretation__

Strong positive correlation is found between

- PERIOD and ADV
- PERIOD and DEMAND
- AIP and DIFF
- DIFF and ADV
- DIFF and DEMAND
- ADV and DEMAND

Strong negative correlation exists between

- PRICE and DIFF
- PRICE and ADV
- PRICE and DEMAND

PERIOD | DEMAND | Forecast MA(3) |
Forecast MA(6) |
Absotute Error – MA(3) | Absotute Error – MA(6) | |

1 | 9.4 | |||||

2 | 10.3 | |||||

3 | 11.5 | |||||

4 | 11.1 | 10.4 | 0.7 | |||

5 | 11 | 11.0 | 0.0 | |||

6 | 10.5 | 11.2 | 0.7 | |||

7 | 10.2 | 10.9 | 10.6 | 0.7 | 0.4 | |

8 | 8.9 | 10.6 | 10.8 | 1.7 | 1.9 | |

9 | 8.3 | 9.9 | 10.5 | 1.6 | 2.2 | |

10 | 8.12 | 9.1 | 10.0 | 1.0 | 1.9 | |

11 | 8.8 | 8.4 | 9.5 | 0.4 | 0.7 | |

12 | 9.8 | 8.4 | 9.1 | 1.4 | 0.7 | |

13 | 10.1 | 8.9 | 9.0 | 1.2 | 1.1 | |

14 | 11.3 | 9.6 | 9.0 | 1.7 | 2.3 | |

15 | 12.5 | 10.4 | 9.4 | 2.1 | 3.1 | |

16 | 12.4 | 11.3 | 10.1 | 1.1 | 2.3 | |

17 | 12.1 | 12.1 | 10.8 | 0.0 | 1.3 | |

18 | 11.8 | 12.3 | 11.4 | 0.5 | 0.4 | |

19 | 11.5 | 12.1 | 11.7 | 0.6 | 0.2 | |

20 | 11 | 11.8 | 11.9 | 0.8 | 0.9 | |

21 | 10.2 | 11.4 | 11.9 | 1.2 | 1.7 | |

22 | 10.3 | 10.9 | 11.5 | 0.6 | 1.2 | |

23 | 10.9 | 10.5 | 11.2 | 0.4 | 0.2 | |

24 | 11.2 | 10.5 | 11.0 | 0.7 | 0.2 | |

25 | 12.5 | 10.8 | 10.9 | 1.7 | 1.7 | |

26 | 13.4 | 11.5 | 11.0 | 1.9 | 2.4 | |

27 | 14.7 | 12.4 | 11.4 | 2.3 | 3.3 | |

28 | 14.1 | 13.5 | 12.2 | 0.6 | 1.9 | |

29 | 14 | 14.1 | 12.8 | 0.1 | 1.2 | |

30 | 13.5 | 14.3 | 13.3 | 0.8 | 0.2 | |

31 | 13.5 | 13.9 | 13.7 | 0.4 | 0.2 | |

32 | 13.1 | 13.7 | 13.9 | 0.6 | 0.8 | |

33 | 12.5 | 13.4 | 13.8 | 0.9 | 1.3 | |

34 |
13.0 |
13.5 |
MAD = |
0.9 |
1.3 |

Since MAD of MA(3) is less than that of MA(6), we should be preferring MA(3) over MA(6). However, Moving average may not be a good choice for predicting the demand because there is a clear periodicity of demand as found in the time series plot of drawn earlier. We should go for a method which can capture both trend and seasonality together.

PERIOD | DEMAND | Forecast (Exp Smoothing) |
Absolute Error | α = | 0.9 | Forecast | MAD | |

1 | 9.4 | 11.35 | 1.95 | 12.56 | 0.65 | |||

2 | 10.3 | 9.59 | 0.71 | 0.1 | 12.43 | 1.20 | ||

3 | 11.5 | 10.23 | 1.27 | 0.2 | 13.00 | 1.04 | ||

4 | 11.1 | 11.37 | 0.27 | 0.3 | 13.10 | 0.95 | ||

5 | 11 | 11.13 | 0.13 | 0.4 | 13.04 | 0.88 | ||

6 | 10.5 | 11.01 | 0.51 | 0.5 | 12.93 | 0.82 | ||

7 | 10.2 | 10.55 | 0.35 | 0.6 | 12.82 | 0.76 | ||

8 | 8.9 | 10.24 | 1.34 | 0.7 | 12.72 | 0.72 | ||

9 | 8.3 | 9.03 | 0.73 | 0.8 | 12.64 | 0.69 | ||

10 | 8.12 | 8.37 | 0.25 | 0.9 | 12.56 | 0.65 | ||

11 | 8.8 | 8.15 | 0.65 | |||||

32 | 13.1 | 13.51 | 0.41 | |||||

33 | 12.5 | 13.14 | 0.64 | |||||

34 | 12.56 | 0.65 | <—MAD |

The MAD is least when α = 0.9

- Use Exponential smoothing forecasts with alpha of 0.1, 0.2, …, 0.9 to predict October 2022 demand. Identify the value of alpha that results in the lowest MAD.

- Find the monthly seasonal indices for the demand values using Simple Average (SA) method. Find the de-seasonalized demand values by dividing monthly demand by respective seasonal indices.

- Use regression to perform trend analysis on the de-seasonalized demand values. Is trend analysis suitable for this data? Find MAD and explain the Excel Regression output (trend equation, r, r-squared, goodness of model).

- Find the seasonally adjusted trend forecasts for October through December 2022.

- Perform simple linear regression analysis with ADV as the independent variable to predict demand. Write the complete equation, find MAD and explain the Excel Regression output.
__Make sure to use the de-seasonalized demand data for this model and all future models.__

- Repeat part (10) with DIFF as the independent variable.

- Construct multiple linear regression model with Period, AIP, DIFF, and ADV as independent variables. Formulate the equation, find MAD, and explain the output. Rank variables based on their degree of contribution to the model. Observe significant F, R-squared, and p-values and explain.

- Perform multiple linear regression analysis with Period, DIFF, and ADV as independent variables. Formulate the equation and find MAD. Which variable is the most significant predictor of demand? Rank the independent variables based on their degree of contribution to the model. Observe significant F, R-squared, and p-values and explain.

- Use the model obtained in parts 13 and make forecasts for the following months. Make sure to seasonalize final forecasts.

__Period Price AIP ADV ____ __

Oct. 2022 $7.30 $7.65 $10.9

Nov. 2022 $7.45 $7.70 $11.5

Dec. 2022 $7.80 $7.95 $11.7

- Provide a case conclusion based on above analysis.

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**Grading Criteria **

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- Completeness/Correctness 70%
- Quality of Interpretations/Analysis 20%
- General Quality of Report/graphs/writing 10%