## 2 Methodology
### DEA Model  
The concept of efficiency is related to the economy of resources, efficiency is often defined as the relationship between the results obtained (outputs) and the resources used (inputs). Numerically, an efficiency score can be obtained as a relation between output and input, called technical efficiency. DEA uses linear programming models to determine weights and calculate the efficiency score for each DMUs [29; 30]. The only requirements are that the DMUs convert inputs into outputs, both similar and quantifiable. Simultaneously, the multiple inputs and outputs are analyzed in their natural physical units instead of having to convert them to a common denominator [31]. It is assumed that if a die-nozzle DMU, named \(die-nozzle_{1}\) can produce or generate \(Y_{1}\) output units with \(X_{1}\) input units, then other die-nozzle must also be able to do the same if they are operated efficiently. Similarly, if \(die-nozzle_{2}\) can produce \(Y_{2}\) output units with \(X_{2}\) input units, then the other die-nozzle must also be able to do the same. \(Die-nozzle_{1}\) and \(die-nozzle_{2}\) can be combined to generate a die-nozzle composed of inputs and outputs of them. This virtual die-nozzle is used as a standard of performance for the die-nozzle. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**Production Line A**} \\  & **January** & **February** & **March** & **April** & **May** & **June** \\ \hline
**Production quantity in km** & 44,889 & 40,454 & 46,736 & 45,650 & 43,647 & 47,956 \\
**Number of die-nozzles/month** & 151 & 145 & 152 & 143 & 159 & 138 \\ \hline  & \multicolumn{6}{c}{**Production Line B**} \\  & **January** & **February** & **March** & **April** & **May** & **June** \\ \hline
**Production quantity in km** & 44,597 & 39,794 & 45,476 & 47,487 & 43,794 & 46,433 \\
**Number of die-nozzles/month** & 124 & 139 & 134 & 121 & 135 & 145 \\ \hline  & \multicolumn{6}{c}{**Production Line C**} \\  & **January** & **February** & **March** & **April** & **May** & **June** \\ \hline
**Production quantity in km** & 42,753 & 38,738 & 45,488 & 43,876 & 41,376 & 44,289 \\
**Number of die-nozzles/month** & 108 & 74 & 111 & 112 & 113 & 88 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Number of km produced by cable vs. the number of die-nozzles/month, used for each production line. Equation (3) describes the mathematical model of DEA. It allowed to validate the results obtained in the analysis of the performance of the five die-nozzle types. See Section 3.7 Computational results of the DEA model. Formulation of DEA Model for Die-Nozzle Pairs  
\[ \begin{array}{ll}&\min\theta^{*},\\ &\text{Subject to}\\ &\sum\limits_{j=1,j\
eq 0}^{n}\lambda_{j}x_{ij}\leq\theta^{*}x_{io}\text{ for }i=1,2,\ldots,m;\\ &\sum\limits_{j=1,j\
eq 0}^{n}\lambda_{j}y_{rj}\leq y_{r0}\text{ for }r=1,2,\ldots,s;\\ &\lambda_{j}\geq 0\text{ for }j\
eq 0;\\ &\sum\limits_{j\
eq 0}^{n}\lambda_{j}\geq 1.\end{array}  \tag{3}\]  
where: \(DMU_{0}\) represents one of the n DMUs under evaluation, and \(x_{io}\) and \(y_{ro}\) are the \(i_{th}\) input and \(r_{th}\) output for \(DMU_{0}\), respectively. To rate the performance of a particular die-nozzle pair, DEA forms a weighted average of all the observations. The average represents a composite die-nozzle pair. The weights are to be determined so that the composite die-nozzle pair is located on the frontier. It represents _best practice_. The actual die-nozzle pair at hand is then compared with its _best practice_. Denoting the weights to be attached to a particular die-nozzle pair by \(\lambda_{j}\). The inputs and outputs for this model are indicated in Table 4. In Table 4, the inputs and outputs of the DMUs are determined, the meaning of these is as follow:  
* Productivity: meters produced with a specific die-nozzle pair/total meters by period (%). * Machine: numbers of machine-hours used with a specific die-nozzle pair/total of machine hours/period (%). * Time: time in hours using a specific die-nozzle pair/total of hours by period (%).