## 3 System Methodology
### Efficiency Assessment of Transportation Arcs Using DEA  
DEA emerges as a powerful non-parametric methodology for the comprehensive evaluation of DMUs in the intricate realm of multimodal transportation networks. Originating  
Figure 1: Multimodal Transportation Network in Iraq. Nodes A–I are interconnected using sea, raod, railways and air transportation sources. from the seminal work by [PERSON] et al. (2018), DEA provides an effective means to gauge the relative efficiency of peer units without requiring a predetermined functional form for a production function. The production frontier, constructed as a convex and piecewise function, is formulated by the linear combination of efficient units. Consider a scenario with p DMUs to be evaluated, indexed by \(j\in\{1,\ldots,p\}\). Each DMU is presumed to consume \(k\) input levels to produce s different outputs. The input and output vectors for DMUj are denoted as \(x_{j}=\left(x_{1j},\ldots,x_{kj}\right)\) and \(y_{j}=\left(y_{1j},\ldots,y_{sj}\right)\), respectively. All components of vectors \(x_{j}\) and \(y_{j}\) are non-negative for all DMUs, with each DMU having at least one strictly positive input and output. The relative efficiency score of DMUo is expressed as:  
\[\theta_{o}=max_{u,v}\left(\frac{\sum_{r=1}^{s}u_{r}y_{r0}}{\sum_{i=1}^{k}v_{i} x_{io}}\right) \tag{2}\]  
where \(u_{r}\) and \(v_{i}\) are the non-negative weights associated with output \(r\) and input \(i\), respectively. It is essential to ensure that these weights, when applied to other DMUs, do not result in an efficiency value exceeding one ([PERSON] and [PERSON], 2018). This requirement can be formally stated through the following constraints:  
\[\left(\frac{\sum_{r=1}^{s}u_{r}y_{rj}}{\sum_{i=1}^{k}v_{i}x_{ij}}\right)\leq 1,\ j=1,2,\ldots p \tag{3}\]  
Therefore, the relative efficiency score of DMUo is obtained by the following linear fractional model,  
\[\theta_{o}^{*}=Max\left(\frac{\sum_{r=1}^{s}u_{r}y_{r0}}{\sum_{i=1}^{k}v_{i} x_{io}}\right) \tag{4}\]  
subject to the constraints,  
\[\left(\frac{\sum_{r=1}^{s}u_{r}y_{rj}}{\sum_{i=1}^{k}v_{i}x_{ij}}\right)\leq 1,\ j=1,2,\ldots p \tag{5}\]  
\[v_{i}\geq\epsilon\ i=1,2,\ldots k \tag{6}\]  
\[u_{r}\geq\epsilon\ r=1,2,\ldots s \tag{7}\]  
In this context, the parameter \(\epsilon>0\) serves as a non-Archimedean element ensuring the existence of strongly efficient solutions ([PERSON] and [PERSON], 2018). It is important to highlight that \(0<\theta_{o}^{*}\leq 1\), and a DMU becomes efficient when \(\theta_{o}^{*}=1\). Each DMU is assessed based on its optimal weight. The outcomes of DEA models involve identifying the hyperplanes that delineate an envelope surface or Pareto frontier. It is noteworthy that an efficient DMU will be positioned on the Pareto frontier. The initial fractional model can be converted into the subsequent linear program using the Charnes-Cooper transformation (2018). \[\theta_{o}^{*}=Max{\sum_{r=1}^{s}u_{r}y_{r0}} \tag{8}\]  
subject to the constraints,  
\[\sum_{i=1}^{k}v_{i}x_{io}=1 \tag{9}\]  
\[\sum_{r=1}^{s}u_{r}y_{rj}-\sum_{i=1}^{k}v_{i}x_{ij}\leq 0,\ \ j=1,2,\ldots p \tag{10}\]  
\[\theta_{o}=max_{u,v}\left(\frac{\sum_{r=1}^{s}u_{r}y_{r0}}{\sum_{i=1}^{k}v_{i} x_{io}}\right) \tag{2}\]0  
\[\theta_{o}=max_{u,v}\left(\frac{\sum_{r=1}^{s}u_{r}y_{r0}}{\sum_{i=1}^{k}v_{i} x_{io}}\right) \tag{2}\]1  
Equation (8) represents the primary CCR model, introduced by [PERSON], [PERSON], and [PERSON]. Let \(u_{r}^{*}(r=1,2,\ldots s)\) and \(v_{i}^{*}(i=1,2,\ldots k)\) represent the optimal output and input weights in the aforementioned model. The term \(\theta_{o}^{*}=\sum_{r=1}^{s}u_{r}^{*}y_{r0}\) quantifies the best relative efficiency score of DMU\({}_{\circ}\), referred to as the CCR-efficiency of DMU\({}_{\circ}\). A DMU\({}_{\circ}\) is considered CCR-efficient if \(\theta_{o}^{*}=1\); otherwise, it is classified as CCR-inefficient.