# Measuring Efficiency and Effectiveness of Highway Management in Sustainability
## 3 Data, Methods, and Results
### Measuring Efficiency: Data Entvelopment Analysis Approach  
[PERSON], [PERSON], and [PERSON] et al. (2019) (pp. 14-28) distinguished concepts of productivity and efficiency. Following their explanation, we suppose two governments (\(A\) and \(B\)) who use \(x\) input and produce \(y\) output. The average productivities (\(AP\)) are defined as \(AP(A)=\frac{y_{A}}{\dot{x}_{A}}\) and \(AP(B)=\frac{y_{B}}{\dot{x}_{B}}\) for governments \(A\) and \(B\), respectively. Higher AP means a higher level of productivity. Thus, the productivity index of government \(A\) is measured as \(\Pi_{A,B}=\frac{AP_{A}}{AP_{B}}=\frac{y_{A}/x_{A}}{y_{B}/x_{B}}\). If \(\Pi\) exceeds one, it means that government \(A\) is relatively more productive than the reference government. If we suppose that technology is known, then the maximum level of output is expressed as \(y^{*}=f(x)\). Technical efficiency (TE) is measured as the ratio of actual output to \"the maximum producible quantity,\" \(\text{TE}^{A}_{O}=\frac{V_{A}}{\dot{y}_{A}}\leq 1\) and \(\text{TE}^{B}_{O}=\frac{y_{B}}{\dot{y}_{B}}\leq 1\). In addition, the equation can be expressed as \(\text{TE}_{O}=\frac{y}{\dot{y}^{*}}=\frac{y/x}{y^{*}/x}=\frac{AP}{AP^{*}}\). This technical efficiency is output-oriented.  
Data Envelopment Analysis (DEA) is useful for comparing the efficiency of each decision-making unit (DMU) [18; 20; 33]. Thus, DEA calculates the degree of efficiency of DMUs that produce the same kind of product, regarding the relative level of efficiency rather than an idealized standard of efficiency [18] (p. 471). As seen in Figure 3, which assumes one output with two inputs, five DMUs have efficient bundles of input among seven DMUs. Since the isoquant line envelops the inefficient DMUs A and E, it is called \"data envelopment analysis\" [20].  
Figure 3: Graph for data envelopment analysis (DEA) adapted from [20] (p. 83).  
The basic function of DEA, according to [PERSON] and [PERSON] [34] (p. 64), is expressed as Max \(\theta=\frac{\sum_{i=1}^{\infty}u_{i}y_{in}}{\sum_{i=1}^{\infty}v_{i}x_{in}}\). The efficiency rating of the service unit is evaluated by DEA: \(y\) is amount of output, \(x\) is amount of input, \(i\) means number of inputs, \(r\) represents number of outputs, \(j\) indicates service unit, \(u\) is the coefficient or weight assigned by DEA to output \(r\); and \(v\) is the coefficient or weight assigned by DEA to input \(i\).  
Data envelopment analysis was conducted to identify variations in the efficiency scores of 48 states. In the panel data of 11 years, we entered the length of lanes and the daily vehicle-miles traveled (DVMT) as output factors, and investment and maintenance expenditure of state highways were entered as input factors. Table 2 shows the information of data used in DEA.  
Assuming constant return to scale and output-oriented DEA, the analysis yielded efficiency scores and their rankings. DEA efficiency scores are relative values among which the highest score is one. Table 3 presents a part of the DEA results, which has each state's relative technical efficiency scores (TES) and its rankings. For example, in 2000, Georgia, Idaho, Ohio, and South Dakota are ranked first, but New Jersey is ranked the lowest. This efficiency score is more useful when comparing cases, whereas previous literature used the ratio of vehicle-miles to expenditures as an efficiency measure.