Abstract
This study develops a double frontier fuzzy network data envelopment analysis (FNDEA) model for assessing the sustainable supply chains. The proposed FNDEA model evaluates the optimistic and pessimistic sustainability of supply chains. The α-cut approach is used to solve the proposed models. The main contribution of this paper is to develop a novel double frontier FNDEA model in the presence of undesirable outputs. To demonstrate the applicability of the proposed approach, the sustainable supply chains of tomato paste are assessed.
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Authors would like to appreciate the Guest Editor Professor Arijit Bhattacharya and four anonymous Reviewers for their constructive comments.
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Appendices
Appendix A
Theorem 1
\( \theta_{\alpha 2}^{*k} \le \theta_{\alpha 1}^{*k} \) for any α1, \(\alpha_{2} \in \left( {0,1} \right]\) and \(\alpha_{1} \le \alpha_{2}\).\(\theta_{\alpha 2}^{*k}\) and \( \theta_{\alpha 1}^{*k}\) are the optimum objective functions of Model (6) given the α1 and α2, respectively.
Proof
Let
to be the optimal solution of Model-6 given α = α2. Therefore,
Since \(\left( {{\tilde{\text{x}}}_{{{\text{ij}}}}^{1} } \right)_{{{\upalpha }_{1} }} = \left[ {\left( {{\upalpha }_{1} {\text{x}}_{{{\text{ij}}}}^{{1{\text{M}}}} + \left( {1 - {\upalpha }_{1} } \right){\text{x}}_{{{\text{ij}}}}^{{1{\text{L}}}} } \right),\left( {{\upalpha }_{1} {\text{x}}_{{{\text{ij}}}}^{{1{\text{M}}}} + \left( {1 - {\upalpha }_{1} } \right){\text{x}}_{{{\text{ij}}}}^{{1{\text{U}}}} } \right)} \right]\forall_{{\text{i}}}\) and \(\left( {{\tilde{\text{x}}}_{{{\text{ij}}}}^{1} } \right)_{{{\upalpha }_{2} }} = \left[ {\left( {{\upalpha }_{2} {\text{x}}_{{{\text{ij}}}}^{{1{\text{M}}}} + \left( {1 - {\upalpha }_{2} } \right){\text{x}}_{{{\text{ij}}}}^{{1{\text{L}}}} } \right),\left( {{\upalpha }_{2} {\text{x}}_{{{\text{ij}}}}^{{1{\text{M}}}} + \left( {1 - {\upalpha }_{2} } \right){\text{x}}_{{{\text{ij}}}}^{{1{\text{U}}}} } \right)} \right]{ }\forall_{{\text{i}}}\), so for \(\alpha_{1} \le {\upalpha }_{2}\), we get \(\left( {{\upalpha }_{1} {\text{x}}_{{{\text{ij}}}}^{{1{\text{M}}}} + \left( {1 - {\upalpha }_{1} } \right){\text{x}}_{{{\text{ij}}}}^{{1{\text{U}}}} } \right) \le \left( {{\upalpha }_{2} {\text{x}}_{{{\text{ij}}}}^{{1{\text{M}}}} + \left( {1 - {\upalpha }_{2} } \right){\text{x}}_{{{\text{ij}}}}^{{1{\text{L}}}} } \right)\forall_{{\text{i}}}\) and \(\left( {{\upalpha }_{1} {\text{x}}_{{{\text{ij}}}}^{{1{\text{M}}}} + \left( {1 - {\upalpha }_{1} } \right){\text{x}}_{{{\text{ij}}}}^{{1{\text{U}}}} } \right) \le \left( {{\upalpha }_{2} {\text{x}}_{{{\text{ij}}}}^{{1{\text{M}}}} + \left( {1 - {\upalpha }_{2} } \right){\text{x}}_{{{\text{ij}}}}^{{1{\text{U}}}} } \right)\forall_{{\text{i}}}\).
Also, we have \(\left( {{\upalpha }_{1} {\text{x}}_{{{\text{ej}}}}^{{2{\text{pM}}}} + \left( {1 - {\upalpha }_{1} } \right){\text{x}}_{{{\text{ej}}}}^{{2{\text{pL}}}} } \right) \le \left( {{\upalpha }_{2} {\text{x}}_{{{\text{ej}}}}^{{2{\text{pM}}}} + \left( {1 - {\upalpha }_{2} } \right){\text{x}}_{{{\text{ej}}}}^{{2{\text{pL}}}} } \right)\forall_{{\text{e}}} \) and \(\left( {{\upalpha }_{1} {\text{x}}_{{{\text{ej}}}}^{{2{\text{pM}}}} + \left( {1 - {\upalpha }_{1} } \right){\text{x}}_{{{\text{ej}}}}^{{2{\text{pU}}}} } \right) \le \left( {{\upalpha }_{2} {\text{x}}_{{{\text{ej}}}}^{{2{\text{pM}}}} + \left( {1 - {\upalpha }_{2} } \right){\text{x}}_{{{\text{ej}}}}^{{2{\text{pU}}}} } \right){ }\forall_{{\text{e}}}\). Similarly, \(\left( {{\upalpha }_{1} {\text{y}}_{{{\text{rj}}}}^{{1{\text{pM}}}} + \left( {1 - {\upalpha }_{1} } \right){\text{y}}_{{{\text{rj}}}}^{{1{\text{pL}}}} } \right) \le \left( {{\upalpha }_{2} {\text{y}}_{{{\text{rj}}}}^{{1{\text{pM}}}} + \left( {1 - {\upalpha }_{2} } \right){\text{y}}_{{{\text{rj}}}}^{{1{\text{pL}}}} } \right)\forall_{{\text{r}}}\) and \(\left( {{\upalpha }_{1} {\text{y}}_{{{\text{rj}}}}^{{1{\text{pM}}}} + \left( {1 - {\upalpha }_{1} } \right){\text{y}}_{{{\text{rj}}}}^{{1{\text{pU}}}} } \right) \le \left( {{\upalpha }_{2} {\text{y}}_{{{\text{rj}}}}^{{1{\text{pM}}}} + \left( {1 - {\upalpha }_{2} } \right){\text{y}}_{{{\text{rj}}}}^{{1{\text{pU}}}} } \right)\forall_{{\text{r}}} . \) In a similar way, \(\left( {{\upalpha }_{1} \overline{\overline{y}}_{{{\text{bj}}}}^{{2{\text{pM}}}} + \left( {1 - {\upalpha }_{1} } \right)\overline{\overline{y}}_{{{\text{bj}}}}^{{2{\text{pL}}}} } \right) \le \left( {{\upalpha }_{2} \overline{\overline{y}}_{{{\text{bj}}}}^{{2{\text{pM}}}} + \left( {1 - {\upalpha }_{2} } \right)\overline{\overline{y}}_{{{\text{bj}}}}^{{2{\text{pL}}}} } \right)\forall_{{\text{b }}}\) and \(\left( {{\upalpha }_{1} \overline{\overline{y}}_{{{\text{bj}}}}^{{2{\text{pM}}}} + \left( {1 - {\upalpha }_{1} } \right)\overline{\overline{y}}_{{{\text{bj}}}}^{{2{\text{pU}}}} } \right) \le \left( {{\upalpha }_{2} \overline{\overline{y}}_{{{\text{bj}}}}^{{2{\text{pM}}}} + \left( {1 - {\upalpha }_{2} } \right)\overline{\overline{y}}_{{{\text{bj}}}}^{{2{\text{pU}}}} } \right)\forall_{{\text{b}}}\). Finally, \(\left( {{\upalpha }_{1} {\text{z}}_{{{\text{sj}}}}^{{{\text{pM}}}} + \left( {1 - {\upalpha }_{1} } \right){\text{z}}_{{{\text{sj}}}}^{{{\text{pL}}}} } \right) \le \left( {{\upalpha }_{2} {\text{z}}_{{{\text{sj}}}}^{{{\text{pM}}}} + \left( {1 - {\upalpha }_{2} } \right){\text{z}}_{{{\text{sj}}}}^{{{\text{pL}}}} } \right)\forall_{{\text{s}}} \) and \( \left( {{\upalpha }_{1} {\text{z}}_{{{\text{sj}}}}^{{{\text{pM}}}} + \left( {1 - {\upalpha }_{1} } \right){\text{z}}_{{{\text{sj}}}}^{{{\text{pU}}}} } \right) \le \left( {{\upalpha }_{2} {\text{z}}_{{{\text{sj}}}}^{{{\text{pM}}}} + \left( {1 - {\upalpha }_{2} } \right){\text{z}}_{{{\text{sj}}}}^{{{\text{pU}}}} } \right){ }\forall_{{\text{s}}} .\)
Consequently,
Given expressions (10), (12), and (13), we find out that
is a feasible solution of Model (6) in α = α1.
Therefore, \(\theta_{\alpha 2}^{k*} \le \theta_{\alpha 1}^{k*}\), where \(\theta_{\alpha 2}^{k*}\) and \(\theta_{\alpha 1}^{k*}\) are the optimum objective function of Model (6) in α1 and α2, respectively. Hence,\( \theta_{\alpha 2}^{k*} \le \theta_{\alpha 1}^{k*}\) for any α1, \(\alpha_{2} \in \left( {0,1} \right]\), and \(\alpha_{1} \le \alpha_{2}\).
Appendix B
To find the unique optimal solution, Shoja et al. (2008) modified CCR model. They proved that if the primal/dual of a DEA model is non-degenerate, then the dual/primal has unique optimal solution. To guarantee the unique optimality of the proposed models, we modify models (6) and (7) based on the approach of Shoja et al. (2008), which is as follows:
Model-9
Subject to:
Model-10
Subject to:
Note that by imposing ε, for each 0 < ε < ε*, models (9) and (10) have no multiple optimal solutions. For proof, see Shoja et al. (2008).
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Tavassoli, M., Fathi, A. & Saen, R.F. Assessing the sustainable supply chains of tomato paste by fuzzy double frontier network DEA model. Ann Oper Res (2021). https://doi.org/10.1007/s10479-021-04139-4
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DOI: https://doi.org/10.1007/s10479-021-04139-4