Computing fuzzy process efficiency in parallel systems

Abstract

This paper deals with parallel process systems in which the input and output data are fuzzy. The $\upalpha$-level based approach is used to compute the fuzzy system efficiency and a simple procedure is proposed to estimate the fuzzy efficiency of the different processes. The main contribution of the paper is estimating the latter taking into account the variability of the process efficiencies compatible with a given value of the system efficiency. This variability comes from the existence of alternative optimal weights in the system efficiency multiplier network DEA models. The computation of the fuzzy system efficiency involves one Linear and one Non-linear Program for each $\upalpha$-cut while the computation of each process efficiency requires solving just a couple of related Linear Programs for each $\upalpha$-cut. The proposed approach is illustrated with a parallel systems dataset extracted from the literature.

Appendices

Appendix A

In order to linearize model (6) just consider the new variables

$$\begin{aligned} \hat{\mathrm{x}}_{\mathrm{ij}}^\mathrm{p}&= \hbox {u}_\mathrm{i} \cdot \hbox {x}_{\mathrm{ij}}^\mathrm{p} \quad \forall \mathrm{j} \forall \mathrm{p} \forall \mathrm{i}\\ \hat{\mathrm{y}}_{\mathrm{kj}}^\mathrm{p}&= \hbox {v}_\mathrm{k} \cdot \hbox {y}_{\mathrm{kj}}^\mathrm{p} \quad \forall \mathrm{j} \forall \mathrm{p} \forall \mathrm{k} \end{aligned}$$

Expressing the model with these variables results in the following Linear Program (LP)

$$\begin{aligned}&\left( {\hbox {E}_0 } \right) _\upalpha ^\mathrm{U} =\hbox {max} \sum _\mathrm{p}\sum _\mathrm{k}{\hat{\mathrm{y}}_{\mathrm{k0}}^\mathrm{p} } \nonumber \\&\hbox {s.t.}\nonumber \\&\sum _\mathrm{p} {\sum _\mathrm{i} {\hat{\hbox {x}}_\mathrm{i0}^\mathrm{p}}} =1\nonumber \\&\sum _\mathrm{k}\hbox {v}_\mathrm{k} \cdot \hat{{\mathrm{y}}}_{\mathrm{kj}}^\mathrm{p} -\sum _\mathrm{i}\hbox {u}_\mathrm{i}\cdot \hat{{\mathrm{x}}}_{\mathrm{ij}}^\mathrm{p} \le 0\quad \forall \mathrm{j} \forall \mathrm{p}\\&\hbox {u}_\mathrm{i} \cdot \left( {\hbox {X}_{\mathrm{ij}}^\mathrm{p} } \right) _\upalpha ^\mathrm{L} \le \hat{\hbox {x}}_{\mathrm{ij}}^\mathrm{p} \le \hbox {u}_\mathrm{i} \cdot \left( {\hbox {X}_{\mathrm{ij}}^\mathrm{p} } \right) _\upalpha ^\mathrm{U} \quad \forall \mathrm{j}\forall \mathrm{p} \forall \mathrm{i}\nonumber \\&\hbox {v}_\mathrm{k} \cdot \left( {\hbox {Y}_{\mathrm{kj}}^\mathrm{p} } \right) _\upalpha ^\mathrm{L} \le \hat{{\hbox {y}}}_{\mathrm{kj}}^\mathrm{p} \le \hbox {v}_\mathrm{k} \cdot \left( {\hbox {Y}_{\mathrm{kj}}^\mathrm{p} } \right) _\upalpha ^\mathrm{U} \quad \forall \mathrm{j}\forall \mathrm{p} \forall \mathrm{k}\nonumber \\&\hbox {u}_\mathrm{i}, \hbox {v}_\mathrm{k} \ge 0\quad \forall \mathrm{i} \forall \mathrm{k}\nonumber \end{aligned}$$

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Appendix B

In order to linearize model (16) consider the same new variables as in Appendix A. The resulting LP is

$$\begin{aligned}&\left( {\hbox {E}_0^{\hat{{\mathrm{p}}}} } \right) _\upalpha ^\mathrm{U} =\hbox {max} \sum _\mathrm{k}{\hat{\hbox {y}}_{\mathrm{k0}}^{\hat{\mathrm{p}}} }\nonumber \\&\hbox {s.t.}\nonumber \\&\sum _\mathrm{i}\hat{{\hbox {x}}}_{\mathrm{i0}}^{\hat{\mathrm{p}}} =1\nonumber \\&\sum _\mathrm{k}\hbox {v}_\mathrm{k} \cdot \hat{\hbox {y}}_{\mathrm{kj}}^\mathrm{p} -\sum _\mathrm{i}\hbox {u}_\mathrm{i} \cdot \hat{\hbox {x}}_{\mathrm{ij}}^\mathrm{p} \le 0\quad \forall \mathrm{j} \forall \mathrm{p}\nonumber \\&\sum _\mathrm{p}\sum _\mathrm{k}\hat{\hbox {y}}_{\mathrm{k0}}^\mathrm{p} - \left( {\hbox {E}_0} \right) _\upalpha ^\mathrm{U} \cdot \left[ \sum _\mathrm{p}\sum _\mathrm{i}{\hat{\hbox {x}}_{\mathrm{i0}}^\mathrm{p} } \right] =0\\&\hbox {u}_\mathrm{i} \cdot \left( {\hbox {X}_{\mathrm{ij}}^\mathrm{p} } \right) _\upalpha ^\mathrm{L} \le \hat{{\hbox {x}}}_{\mathrm{ij}}^\mathrm{p} \le \hbox {u}_\mathrm{i} \cdot \left( {\hbox {X}_{\mathrm{ij}}^\mathrm{p}} \right) _\upalpha ^\mathrm{U} \quad \forall \mathrm{j} \forall \mathrm{p} \forall \mathrm{i}\nonumber \\&\hbox {v}_\mathrm{k} \cdot \left( {\hbox {Y}_{\mathrm{kj}}^\mathrm{p} } \right) _\upalpha ^\mathrm{L} \le \hat{{\hbox {y}}}_{\mathrm{kj}}^\mathrm{p} \le \hbox {v}_\mathrm{k} \cdot \left( {\hbox {Y}_{\mathrm{kj}}^\mathrm{p}} \right) _\upalpha ^\mathrm{U} \quad \forall \mathrm{j} \forall \mathrm{p} \forall \mathrm{k}\nonumber \\&\hbox {u}_\mathrm{i}, \hbox {v}_\mathrm{k} \ge 0\quad \forall \mathrm{i} \forall \mathrm{k}\nonumber \end{aligned}$$

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