## 4 Experimental Results
### Deriving Operational Efficiencies and Performance Assessment  
To conduct performance assessment, the correlation coefficients between input variables (COGS- cost of goods sold, full-time employees, OE- operating expenses) and the output variable (sales revenue) are shown in Table 8. Positive coefficients imply that the output is proportional to the input, thus justifying the validity of the input and output variables. In terms of BCC measures, the operational efficiencies for Walmart, Costco, and Kroger are shown in Figure 6. Clearly, Kroger has significantly lagged behind Walmart and Costco since 2009. To address hidden causalities, all input and output variables are displayed in Figures 7-10. Not surprisingly, Walmart exhibits the largest scales of input and output variables. Kroger shows almost equivalent sales (Figure 7) and COGS (Figure 8) to Costco, though it has more full-time employees (Figure 9) and higher operating expenses (Figure 10) than Costco. These observations clearly account for Kroger's operational efficiency ranking being the lowest, because it consumes more input resources without generating higher sales revenues. Although Walmart expresses worry about Amazon's move to retail markets, Kroger can potentially be more impacted by Amazon because it is a chain supermarket with community-based stores. To elicit more insights, partial operational efficiencies with respect to a single input variable are derived and shown in Table 9: Costco performs poorly in \"COGS\" and Kroger performs the worst in both \"full-time employees\" and \"operating expenses\".  
To help Kroger improve operational efficiencies, Table 10 shows the required adjustments of input resources in percentages. In 2005, no adjustments were required for any retailer. As shown in Figure 6, Kroger performed the worst from 2008 to 2019, and hence, it needed to concurrently reduce COGS, full-time employees, and operating expenses during these years. In 2010, 2012, and 2021, Costco and Walmart performed efficiently, and thus, no adjustments were required. Further, the reduction of COGS and operating expenses was more critical to Costco and Kroger, while Walmart seemed to focus on decreasing full-time employees. Although Walmart had the greatest scales of input resources and out revenues, it performed efficiently in many years: 2005, 2010, 2011, 2012, 2013, 2016, 2017, and 2021. In 2018, Walmart spent a lot of money to merge an e-commerce platform because it wanted to defend its territory and compete with Amazon. This event, coupled with the US-China trade war, can explain Walmart's inefficiencies in 2018 and 2019. On average, Costco did not perform as well as Walmart but it was still more efficient than Kroger. In practice, to enhance operational efficiencies, cost reduction is easier than increasing sales revenues.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Sales** & **COGS** & **Full-Time** & **Operating** \\  & **Revenues** & **Employees** & **Expenses** \\ \hline Sales revenues & 1 & & & \\ COGS & 0.999 & 1 & & \\ Employees & 0.978 & 0.97 & 1 & \\ Operating expenses & 0.988 & 0.98 & 0.984 & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Correlation coefficients between input and output variables.  
Figure 8: Cost of goods sold (COGS) in millions $USD.  
Figure 6: Operational efficiencies.  
Figure 7: Sales revenues in millions $USD.  
Figure 10: Operating expenses in millions $USD.  
Figure 9: Full-time employees.