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A shipping company is attempting to determine how its shipping costs for a month depend on the number of units shipped during a month. The number of units shipped and total shipping cost for the last 15 months are given in the file P12_63.xlsx.

a. Determine a relationship between units shipped and monthly shipping cost.

b. Plot the errors for the predictions in order of time sequence. Is there any unusual pattern?

c. It turns out that there was a trucking strike during months 11 through 15, and you believe that this might have influenced shipping costs. How can the answer to part a be modified to account for the effect of the strike? After accounting for this effect, does the unusual pattern in part b disappear?

Paper For Above instruction

Analyzing Shipping Costs and the Impact of a Strike

Understanding the relationship between shipping costs and the number of units shipped is vital for logistical and financial planning within transportation companies. The data provided for the last 15 months, encapsulating units shipped and total shipping costs, serves as the basis for establishing this relationship. The analysis proceeds through several stages: identifying the underlying model, evaluating prediction errors, and accounting for extraordinary events such as strikes that influence operational costs.

Establishing the Relationship between Units Shipped and Shipping Costs

To determine how shipping costs depend on the number of units shipped, a statistical modeling approach such as linear regression is typically employed. Assuming the data in the Excel file is accessible, the first step is to extract the number of units shipped (independent variable) and total shipping costs (dependent variable). Plotting these data points can provide visual insights into the nature of their relationship—whether linear or nonlinear.

Preliminary analysis suggests a proportional (linear) relationship is likely, given that shipping costs often increase with volume due to fixed and variable costs. Applying least squares regression yields an equation of the form:

ShippingCost = a + b * UnitsShipped

where 'a' represents fixed costs and 'b' indicates the incremental cost per unit shipped. The coefficients are estimated from the data using statistical software or Excel's regression tools.

It's essential to validate the model's fit by examining the R-squared value, residual plots, and significance levels of the predictors. These metrics confirm whether the linear model appropriately captures the dependency or if a more complex model is necessary.

Analyzing Prediction Errors Over Time

Once the model is established, predictions for each month’s shipping cost can be generated based on the units shipped that month. The residuals (errors) are calculated as the difference between actual and predicted costs:

Residual = ActualCost - PredictedCost

Plotting these residuals sequentially over the 15 months reveals patterns that might indicate model inadequacies or external influences. In particular, consistent positive or negative residuals during specific periods could suggest anomalies or structural breaks in the data.

In this dataset, months 11 through 15 are suspected to have been affected by an external factor—the trucking strike—which likely inflated costs. The residual pattern during these months may appear unusually large or display a distinct trend, highlighting the impact of this strike on the transportation costs.

Incorporating the Effect of the Strike into the Model

To account for the disruption caused by the strike, the model can be extended by introducing a categorical variable that indicates whether a month was affected by the strike. Define a dummy variable:

StrikeMonth = 1 if month 11-15, 0 otherwise

This variable is incorporated into the regression model, resulting in:

ShippingCost = a + b  UnitsShipped + c  StrikeMonth

where 'c' captures the additional cost attributable to the strike. Estimating this model allows us to isolate the effect of the strike from the baseline cost structure.

After including this variable, re-estimating the model generally leads to a reduction in residuals during the strike months, thereby improving forecast accuracy and explanatory power. The adjusted model should diminish the unusual residual pattern observed previously, signifying that accounting for the strike accounts for the anomalous cost increases.

Conclusion

In sum, analyzing the relationship between units shipped and shipping costs typically involves linear regression modeling, which effectively captures the cost behavior under normal circumstances. Residual analysis over time illuminates periods where external factors—such as a trucking strike—impact costs. Incorporating such factors into the model by introducing dummy variables enhances accuracy and explicability. The approach outlined demonstrates how external disruptions can be quantitatively integrated into predictive models, leading to more reliable decision-making in logistics and cost management.

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