# Developing Productivity-Safety Effectiveness Index Using Data Envelopment Analysis (DEA)
## 2 Background
### Methodological Background
#### 2.2.2 Data Envelopment Analysis (DEA)  
DEA is a linear programming model for measuring the relative efficiency or performance of decision-making units (DMUs) [27]. The DMU represents each analysis target with multiple inputs and outputs. It reflects multiple outputs, allowing each DMU to select the optimal weights of multiple variables, which maximize the DMU's efficiency [27; 28]. With an efficiency range from 0% to 100%, the DMUs at 100% are completely efficient, but other DMUs with efficiency levels below 100% are relatively inefficient. As shown in Figure 1, the left-top elements can be better units because they have less input and high output. DEA solves the relative distance between better units and worse units using linear programming. This is the main concept of relative efficiency. This concept also extends to multiple inputs and outputs as one of the multiple criteria decision-making techniques. According to these advantages, DEA has been widely used in many studies to measure relative efficiency as well [28]. However, methodologically, DEA faces challenges in identifying cause-and-effect relationships between inputs and outputs, as it is a model \begin{table}
\begin{tabular}{c c c} \hline \hline
**Variables** & **Operational Definition** & **Normal Formula** \\ \hline \multirow{4}{*}{Labor Productivity (LP)} & The amount of output produced per unit & \multirow{4}{*}{(Total outputs/total man-hours of year)/(Total outputs/total man-hours of base year) \(\times\) multiplier(100)} \\  & It measures the efficiency of workers in & \\  & producing goods or services. & \\  & The maximum amount of goods or services that a business can produce given & \\  & its resources. It measures the capability of & \\  & a company or a facility to produce output & \\  & within a specified period. & \\  & & (Production outputs of year/Production capacity of year)/(production outputs of base year/Production capacity of year) \(\times\) weight of \\  & & base year/Production capacity of base year \(\times\) weight of \\  & & items)/(Total weight of all \\  & & production items) \(\times\) multiplier(100) \\  & & (Production outputs of year/Production capacity of year)/(production outputs of base year/Production capacity of year) \(\times\) weight of operation ratio of items)/(Total weight of all operation ratio \\ \hline \multirow{4}{*}{Injury Rate (IR)} & The frequency of workplace injuries and illnesses within a specific time frame. & The number of workers injured by incidents/The number of employees \(\times\) multiplier(1000) \\  & It measures the number of workplace & \\  & injuries or incidents relative to a standard & \\  & unit of exposure, typically expressed per a & \\  & certain number of hours worked or per a & \\  & specific number of employees. & \\  & The number of lost workdays associated & \\  & with each incident. & \\  & The total severity scores are divided by & \\  & the total number of hours worked or & \\  & another unit of exposure. & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Operational definition of efficiency indexes. free approach. Additionally, when the number of DMUs with large inputs and outputs increases, the DEA model becomes more complex, making it difficult to compute the relative efficiency. Model-based approaches, such as regression analysis and neural networks, which construct relationships between inputs and outputs, are among the alternatives to DEA. These approaches offer a different perspective compared to DEA by modeling the functional relationships between variables. These approaches provide valuable insights into the relationship between productivity and safety performance. However, they require well-structured data and, in the case of neural networks, significant computational power for training the models. Unlike DEA, these model-based approaches typically do not provide a direct measure of relative efficiency across decision-making units (DMUs) but, rather, focus on understanding and predicting the relationships between inputs and outputs.