# Evaluating 3D Printers Using Data Envelopment Analysis
## 2 Theoretical Framework  
DEA is one of the most popular and best documented techniques of mathematical programming, and several variations of the technique have been developed [38; 39]. In the context of the present study, three DEA models/variations were examined: the CCR (Charnes-Cooper-Rhodes) model [13], the BCC (Banker-Charnes-Cooper) model [40] and the context-dependent DEA model [41]. The CCR or 'constant returns to scale' model is the oldest and most commonly employed variation of DEA [42]. In the CCR formulation of DEA, the technical efficiency \(\theta^{*}\) of each DMU (DMU\({}_{0}\)) is evaluated by solving the following minimization problem [43]. \[\left\{ \begin{array}{c}\theta^{*}=min\theta\\ \text{Subject to}\\ \sum\limits_{j=1}^{n}x_{ij}\lambda_{j}\leq\theta x_{i0},\;i=1,\;2,\;\ldots,\;m \\ \sum\limits_{j=1}^{n}y_{j}\lambda_{j}\geq y_{r0},\;r=1,\;2,\;\ldots,\;s\\ \lambda_{j}\geq 0,\;\;j=1,\;2,\;\ldots,\;n\end{array} \right. \tag{1}\]  
where \(n\) is the number of DMUs, \(m\) is the number of inputs, \(s\) the number of outputs and \(\lambda_{j}\) are the corresponding input/output coefficients (nonnegative scalars) assigned to each DMU. The above model is, also, commonly referred as the Farrell model [44]. The above formulation of the problem implies that the value of \(\theta^{*}\) cannot exceed 1. Accordingly, all DMUs for which \(\theta^{*}=1\) are considered efficient, while all other DMUs are considered inefficient. An efficiency score of 1 does not guarantee 'full efficiency' in the sense that some inputs could be reduced or some outputs could be increased without affecting the need for other inputs or the production of other outputs [45]. To assess this possibility slack values for DMUs are usually evaluated. The slack values \(s-\) and \(s+\) of DMU\({}_{0}\) represent the amount that resource consumption or output production should be improved for the specific unit to reach the efficient frontier, and are evaluated by solving the following maximization problem. \[\left\{ \begin{array}{c}max\sum\limits_{i=1}^{m}s_{i}^{-}+\sum\limits_{r=1}^{ s}s_{r}^{+}\\ \text{Subject to}\\ \sum\limits_{j=1}^{n}x_{ij}\lambda_{i}+s_{i}^{-}=\theta^{*}x_{i0},\ i=1,\ 2,\ \ldots,\ m\\ \sum\limits_{j=1}^{n}y_{ij}\lambda_{j}-s_{r}^{+}=y_{r0},\ r=1,\ 2,\ \ldots,\ s\\ \lambda_{j},\ s_{i}^{-},\ s_{r}^{+}\geq 0\ \forall\ i,j,r\end{array} \right. \tag{2}\]  
Following slack evaluation, the identification of fully efficient DMUs is possible, for which both efficiency \(\theta^{*}=1\) and all slack values are equal to zero (\(s_{i}^{-*}=s_{r}^{+*}=0\)). The BCC or 'variable returns to scale' model is formulated as the CCR model described above, but incorporates the following additional constraint:  
\[\sum\limits_{j=1}^{n}\lambda_{j}=1 \tag{3}\]  
Efficient units, in this model also, are those that have the maximum efficiency score and zero slack values. Classic DEA models help in evaluating the efficiency of DMUs and provide an indication of the distance to the best practice frontier, but do not provide specific guidance regarding how inefficient units may become efficient, i.e., whether outputs should be increased, inputs be reduced or both, and to what extent. Furthermore, as has been stressed in several studies [46; 47], the performance ranking is often influenced by the context, i.e., the set of DMUs a specific unit is rated against. Especially in the case of product selection it has been argued that the choice context, i.e., what other options are available, is especially important for determining the \"attractiveness\" of an option [48]. For addressing the above considerations, the method of context-dependent DEA has been proposed [41]. In context-dependent DEA DMUs are classified in several subgroups/-sets which are considered efficient in specific contexts. This is achieved through successive optimization runs (levels), where efficient DMUs are assigned to a context-specific group and are subsequently removed from further DEA evaluation. The process is repeated until all DMUs are assigned to a specific group.