Assessing the sustainable supply chains of tomato paste by fuzzy double frontier network DEA model

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.

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

$$ \begin{aligned} & ({\text{v}}_{{1{\text{k }}}}^{*} ,{\text{ v}}_{{2{\text{k }}}}^{*} ,{ } \ldots ,{\text{ v}}_{{\text{ik }}}^{*} ;{{ \upgamma }}_{{1{\text{k}}}}^{*} , {\upgamma }_{{2{\text{k}}}}^{*} ,{ } \ldots ,{{ \upgamma }}_{{{\text{ek}}}}^{*} ;{\text{u}}_{{1{\text{k}}}}^{*} ,{\text{ u}}_{{2{\text{k}}}}^{*} ,{ } \ldots ,{\text{ u}}_{{{\text{ek}}}}^{*} ;{{ \upmu }}_{{1{\text{k}}}}^{*} ,{{ \upmu }}_{{2{\text{k}}}}^{*} ,{ } \ldots ,{{ \upmu }}_{{{\text{bk}}}}^{*} ;{{ \upeta }}_{{1{\text{k}}}}^{*} ,{{ \upeta }}_{{2{\text{k}}}}^{*} ,{ } \ldots ,{{ \upeta }}_{{{\text{sk}}}}^{*} ; \\ & \quad \quad \overline{x}_{11}^{1*} ,{ }\overline{x}_{21}^{1*} ,{ } \ldots ,{ }\overline{x}_{i1}^{1*} ,{ }\overline{x}_{1j, }^{1*} \overline{x}_{2j}^{1*} ,{ } \ldots ,{ }\overline{x}_{ij}^{1*} ;{ }\overline{x}_{11}^{2p*} ,{ }\overline{x}_{21}^{2p*} ,{ } \ldots ,{ }\overline{x}_{e1}^{2p*} ,{ }\overline{x}_{1j, }^{2p*} \overline{x}_{2j}^{2p*} ,{ } \ldots ,{ }\overline{x}_{ej}^{2p*} ; \\ & \quad \quad \overline{y}_{11}^{1p*} ,{ }\overline{y}_{21}^{1p*} ,{ } \ldots ,{ }\overline{y}_{r1}^{1p*} ,{ }\overline{y}_{1j}^{1p*} ,{ }\overline{y}_{2j}^{1p*} ,{ } \ldots ,{ }\overline{{\overline{\overline{y}} }}_{rj}^{1p*} ;\overline{{\overline{\overline{y}} }}_{11}^{2p*} ,{ }\overline{{\overline{\overline{y}} }}_{21}^{2p*} ,{ } \ldots ,{ }\overline{{\overline{\overline{y}} }}_{b1}^{2p*} ,{ }\overline{{\overline{\overline{y}} }}_{1j}^{2p*} ,{ }\overline{{\overline{\overline{y}} }}_{2j}^{2p*} ,{ } \ldots ,{ }\overline{{\overline{\overline{y}} }}_{bj}^{2p*} ; \\ & \quad \quad \overline{z}_{11 }^{p*} ,{ }\overline{z}_{21 }^{p*} ,{ } \ldots ,{ }\overline{z}_{s1 }^{p*} ,{ }\overline{z}_{1j}^{p*} ,{ }\overline{z}_{2j}^{p*} ,{ } \ldots ,{ }\overline{z}_{sj}^{p*} ) \\ \end{aligned} $$

to be the optimal solution of Model-6 given α = α₂. Therefore,

$$ \left. {\begin{array}{*{20}l} {\left\{ {\mathop \sum \limits_{{{\text{i}} = 1}}^{{\text{I}}} {\overline{\text{x}}}_{{{\text{ik}}}}^{1} + \mathop \sum \limits_{{{\text{e}} = 1}}^{{\text{E}}} {\overline{\text{x}}}_{{1{\text{k}}}}^{21} + \mathop \sum \limits_{{{\text{p}} = 2}}^{{\text{P}}} \left\{ {\mathop \sum \limits_{{{\text{r}} = 1}}^{{\text{R}}} {\overline{\text{y}}}_{{{\text{rk}}}}^{{1{\text{p}}}} + \mathop \sum \limits_{{{\text{e}} = 1}}^{{\text{E}}} {\overline{\text{x}}}_{{{\text{ek}}}}^{{2{\text{p}}}} } \right\}} \right\} = 1} \hfill \\ {\left\{ {\mathop \sum \limits_{{{\text{i}} = 1}}^{{\text{I}}} {\overline{\text{x}}}_{{{\text{ik}}}}^{1} + \mathop \sum \limits_{{{\text{e}} = 1}}^{{\text{E}}} {\overline{\text{x}}}_{{1{\text{k}}}}^{21} + \mathop \sum \limits_{{{\text{p}} = 2}}^{{\text{P}}} \left\{ {\mathop \sum \limits_{{{\text{r}} = 1}}^{{\text{R}}} {\overline{\text{y}}}_{{{\text{rk}}}}^{{1{\text{p}}}} + \mathop \sum \limits_{{{\text{e}} = 1}}^{{\text{E}}} {\overline{\text{x}}}_{{{\text{ek}}}}^{{2{\text{p}}}} } \right\}} \right\} = 1} \hfill \\ {\mathop \sum \limits_{{{\text{p}} = 2}}^{{\text{P}}} \left\{ {\mathop \sum \limits_{{{\text{r}} = 1}}^{{\text{R}}} {\overline{\text{y}}}_{{{\text{rj}}}}^{{1{\text{p}}}} + \mathop \sum \limits_{{{\text{s}} = 1}}^{{\text{S}}} {\overline{\text{z}}}_{{{\text{sj}}}}^{{\text{p}}} + \mathop \sum \limits_{{{\text{b}} = 1}}^{{\text{B}}} \overline{{\overline{\overline{y}} }}_{{{\text{bj}}}}^{{2{\text{p}}}} } \right\} - \mathop \sum \limits_{{{\text{p}} = 2}}^{{\text{P}}} \left\{ {\mathop \sum \limits_{{{\text{r}} = 1}}^{{{\text{R}}_{{1 - {\text{p}}}} }} {\overline{\text{y}}}_{{{\text{r}}_{{1 - {\text{p}}}} {\text{j}}}}^{{1{\text{p}} - 1}} + \mathop \sum \limits_{{{\text{e}} = 1}}^{{\text{E}}} {\overline{\text{x}}}_{{{\text{ej}}}}^{{2{\text{p}}}} } \right\} \le 0,} \hfill \\ \end{array} } \right\} $$

(Expression 10)

$$ \left. {\begin{array}{*{20}l} {{\text{v}}_{{{\text{ik}}}} \left( {{\alpha x}_{{{\text{ij}}}}^{{1{\text{M}}}} + \left( {1 - {\upalpha }} \right){\text{x}}_{{{\text{ij}}}}^{{1{\text{L}}}} } \right) \le \overline{x}_{ij}^{1} \le {\text{v}}_{{{\text{ik}}}} \left( {{\alpha x}_{{{\text{ij}}}}^{{1{\text{M}}}} + \left( {1 - {\upalpha }} \right){\text{x}}_{{{\text{ij}}}}^{{1{\text{U}}}} } \right)} \hfill \\ {{\upgamma }_{{{\text{ek}}}} \left( {{\alpha x}_{{{\text{ej}}}}^{{2{\text{pM}}}} + \left( {1 - {\upalpha }} \right){\text{x}}_{{{\text{ej}}}}^{{2{\text{pL}}}} } \right) \le \overline{x}_{ej}^{2p} \le {\upgamma }_{{{\text{ek}}}} \left( {{\alpha x}_{{{\text{ej}}}}^{{2{\text{pM}}}} + \left( {1 - {\upalpha }} \right){\text{x}}_{{{\text{ej}}}}^{{2{\text{pU}}}} } \right)} \hfill \\ {{\text{u}}_{{{\text{rk}}}} \left( {{\alpha y}_{{{\text{rj}}}}^{{1{\text{pM}}}} + \left( {1 - {\upalpha }} \right){\text{y}}_{{{\text{rj}}}}^{{1{\text{pL}}}} } \right) \le \overline{y}_{rj}^{1p} \le {\text{u}}_{{{\text{rk}}}} \left( {{\alpha y}_{{{\text{rj}}}}^{{1{\text{pM}}}} + \left( {1 - {\upalpha }} \right){\text{y}}_{{{\text{rj}}}}^{{1{\text{pU}}}} } \right)} \hfill \\ {{\upmu }_{{{\text{bk}}}} \left( {{\upalpha }\overline{\overline{y}}_{{{\text{bj}}}}^{{2{\text{pM}}}} + \left( {1 - {\upalpha }} \right)\overline{\overline{y}}_{{{\text{bj}}}}^{{2{\text{pL}}}} } \right) \le \overline{{\overline{\overline{y}} }}_{bj}^{2p} \le {\upmu }_{{{\text{bk}}}} \left( {{\upalpha }\overline{\overline{y}}_{{{\text{bj}}}}^{{2{\text{pM}}}} + \left( {1 - {\upalpha }} \right)\overline{\overline{y}}_{{{\text{bj}}}}^{{2{\text{pU}}}} } \right)} \hfill \\ {{\upeta }_{{{\text{sk}}}} \left( {{\alpha z}_{{{\text{sj}}}}^{{{\text{pM}}}} + \left( {1 - {\upalpha }} \right){\text{z}}_{{{\text{sj}}}}^{{{\text{pL}}}} } \right) \le \overline{z}_{sj }^{p} \le {\upeta }_{{{\text{sk}}}} \left( {{\alpha z}_{{{\text{sj}}}}^{{{\text{pM}}}} + \left( {1 - {\upalpha }} \right){\text{z}}_{{{\text{sj}}}}^{{{\text{pU}}}} } \right)} \hfill \\ \end{array} } \right\} $$

(Expression 11)

$$ \left. {v_{ij} ,\gamma_{ej} , u_{rj} , \mu_{bj} ,{\upeta }_{sj} \ge {\upvarepsilon }} \right\} $$

(Expression 12)

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,

$$ \left. {\begin{array}{*{20}l} {{\text{v}}_{{{\text{ik}}}} \left( {{\alpha x}_{{{\text{ij}}}}^{{1{\text{M}}}} + \left( {1 - {\upalpha }} \right){\text{x}}_{{{\text{ij}}}}^{{1{\text{L}}}} } \right) \le \overline{x}_{ij}^{1} \le {\text{v}}_{{{\text{ik}}}} \left( {{\alpha x}_{{{\text{ij}}}}^{{1{\text{M}}}} + \left( {1 - {\upalpha }} \right){\text{x}}_{{{\text{ij}}}}^{{1{\text{U}}}} } \right)} \hfill \\ {{\upgamma }_{{{\text{ek}}}} \left( {{\alpha x}_{{{\text{ej}}}}^{{2{\text{pM}}}} + \left( {1 - {\upalpha }} \right){\text{x}}_{{{\text{ej}}}}^{{2{\text{pL}}}} } \right) \le \overline{x}_{ej}^{2p} \le {\upgamma }_{{{\text{ek}}}} \left( {{\alpha x}_{{{\text{ej}}}}^{{2{\text{pM}}}} + \left( {1 - {\upalpha }} \right){\text{x}}_{{{\text{ej}}}}^{{2{\text{pU}}}} } \right)} \hfill \\ {{\text{u}}_{{{\text{rk}}}} \left( {{\alpha y}_{{{\text{rj}}}}^{{1{\text{pM}}}} + \left( {1 - {\upalpha }} \right){\text{y}}_{{{\text{rj}}}}^{{1{\text{pL}}}} } \right) \le \overline{y}_{rj}^{1p} \le {\text{u}}_{{{\text{rk}}}} \left( {{\alpha y}_{{{\text{rj}}}}^{{1{\text{pM}}}} + \left( {1 - {\upalpha }} \right){\text{y}}_{{{\text{rj}}}}^{{1{\text{pU}}}} } \right)} \hfill \\ {{\upmu }_{{{\text{bk}}}} \left( {{\upalpha }\overline{\overline{y}}_{{{\text{bj}}}}^{{2{\text{pM}}}} + \left( {1 - {\upalpha }} \right)\overline{\overline{y}}_{{{\text{bj}}}}^{{2{\text{pL}}}} } \right) \le \overline{{\overline{\overline{y}} }}_{bj}^{2p} \le {\upmu }_{{{\text{bk}}}} \left( {{\upalpha }\overline{\overline{y}}_{{{\text{bj}}}}^{{2{\text{pM}}}} + \left( {1 - {\upalpha }} \right)\overline{\overline{y}}_{{{\text{bj}}}}^{{2{\text{pU}}}} } \right)} \hfill \\ {{\upeta }_{{{\text{sk}}}} \left( {{\alpha z}_{{{\text{sj}}}}^{{{\text{pM}}}} + \left( {1 - {\upalpha }} \right){\text{z}}_{{{\text{sj}}}}^{{{\text{pL}}}} } \right) \le \overline{z}_{sj }^{p} \le {\upeta }_{{{\text{sk}}}} \left( {{\alpha z}_{{{\text{sj}}}}^{{{\text{pM}}}} + \left( {1 - {\upalpha }} \right){\text{z}}_{{{\text{sj}}}}^{{{\text{pU}}}} } \right)} \hfill \\ \end{array} } \right\} $$

(Expression 13)

Given expressions (10), (12), and (13), we find out that

$$ \begin{aligned} & (v_{1k }^{*} , v_{2k }^{*} , \ldots , v_{ik }^{*} ; \gamma_{1k}^{*} , \gamma_{2k}^{*} , \ldots , \gamma_{ek}^{*} ;u_{1k}^{*} , u_{2k}^{*} , \ldots , u_{ek}^{*} ; \mu_{1k}^{*} , \mu_{2k}^{*} , \ldots , \mu_{bk}^{*} ; \eta_{1k}^{*} , \eta_{2k}^{*} , \ldots , \eta_{sk}^{*} ; \\ & \quad \overline{x}_{11}^{1*} , \overline{x}_{21}^{1*} , \ldots , \overline{x}_{i1}^{1*} , \overline{x}_{1j, }^{1*} \overline{x}_{2j}^{1*} , \ldots , \overline{x}_{ij}^{1*} ; \overline{x}_{11}^{2p*} , \overline{x}_{21}^{2p*} , \ldots , \overline{x}_{e1}^{2p*} , \overline{x}_{1j, }^{2p*} \overline{x}_{2j}^{2p*} , \ldots , \overline{x}_{ej}^{2p*} ; \\ & \quad \overline{y}_{11}^{1p*} , \overline{y}_{21}^{1p*} , \ldots , \overline{y}_{r1}^{1p*} , \overline{y}_{1j}^{1p*} , \overline{y}_{2j}^{1p*} , \ldots , \overline{y}_{rj}^{1p*} ;\overline{{\overline{\overline{y}} }}_{11}^{2p*} , \overline{{\overline{\overline{y}} }}_{21}^{2p*} , \ldots , \overline{{\overline{\overline{y}} }}_{b1}^{2p*} , \overline{{\overline{\overline{y}} }}_{1j}^{2p*} , \overline{{\overline{\overline{y}} }}_{2j}^{2p*} , \ldots , \overline{{\overline{\overline{y}} }}_{bj}^{2p*} ; \\ & \quad \overline{z}_{11 }^{p*} ,{ }\overline{z}_{21 }^{p*} ,{ } \ldots ,{ }\overline{z}_{s1 }^{p*} ,{ }\overline{z}_{1j}^{p*} ,{ }\overline{z}_{2j}^{p*} ,{ } \ldots ,{ }\overline{z}_{sj}^{p*} ) \\ \end{aligned} $$

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

$$ {\text{Max}}\quad \tilde{\theta }_{Optimistic} = \left\{ {\mathop \sum \limits_{{{\text{p}} = 1}}^{{\text{P}}} \left\{ {\mathop \sum \limits_{{{\text{r}} = 1}}^{{\text{R}}} \overline{y}_{rj}^{1p} + \mathop \sum \limits_{{{\text{s}} = 1}}^{{\text{S}}} \overline{z}_{sj}^{p} + \mathop \sum \limits_{{{\text{b}} = 1}}^{{\text{B}}} \overline{{\overline{\overline{y}} }}_{{{\text{bk}}}}^{{2{\text{p}}}} } \right\}} \right\} $$

Subject to:

$$ \begin{aligned} & \left\{ {\mathop \sum \limits_{{{\text{i}} = 1}}^{{\text{I}}} {\overline{\text{x}}}_{{{\text{ik}}}}^{1} + \mathop \sum \limits_{{{\text{e}} = 1}}^{{\text{E}}} {\overline{\text{x}}}_{{{\text{ek}}}}^{21} + \mathop \sum \limits_{{{\text{p}} = 2}}^{{\text{P}}} \left\{ {\mathop \sum \limits_{{{\text{r}} = 1}}^{{\text{R}}} {\overline{\text{y}}}_{{{\text{rk}}}}^{{1{\text{p}} - 1}} + \mathop \sum \limits_{{{\text{e}} = 1}}^{{\text{E}}} {\overline{\text{x}}}_{{{\text{ek}}}}^{{2{\text{p}}}} } \right\}} \right\} = 1 + \in \\ & \quad \quad \left\{ {\mathop \sum \limits_{{{\text{r}} = 1}}^{{\text{R}}} {\overline{\text{y}}}_{{{\text{rj}}}}^{11} + \mathop \sum \limits_{{{\text{b}} = 1}}^{{\text{B}}} \overline{{\overline{\overline{y}} }}_{{{\text{bj}}}}^{21} } \right\} - \left\{ {\mathop \sum \limits_{{{\text{i}} = 1}}^{{\text{I}}} {\overline{\text{x}}}_{{{\text{ij}}}}^{1} + \mathop \sum \limits_{{{\text{e}} = 1}}^{{\text{E}}} {\overline{\text{x}}}_{{{\text{ej}}}}^{21} } \right\} \le 0 + \in,\quad j = 1, \ldots ,n \\ & \quad \quad \mathop \sum \limits_{{{\text{p}} = 2}}^{{\text{P}}} \left\{ {\mathop \sum \limits_{{{\text{r}} = 1}}^{{\text{R}}} {\overline{\text{y}}}_{{{\text{rj}}}}^{{1{\text{p}}}} + \mathop \sum \limits_{{{\text{s}} = 1}}^{{\text{S}}} {\overline{\text{z}}}_{{{\text{sj}}}}^{{\text{p}}} + \mathop \sum \limits_{{{\text{b}} = 1}}^{{\text{B}}} \overline{{\overline{\overline{y}} }}_{{{\text{bj}}}}^{{2{\text{p}}}} } \right\} - \mathop \sum \limits_{{{\text{p}} = 2}}^{{\text{P}}} \left\{ {\mathop \sum \limits_{{{\text{r}} = 1}}^{{{\text{R}}_{{1 - {\text{p}}}} }} {\overline{\text{y}}}_{{{\text{r}}_{{1 - {\text{p}}}} {\text{j}}}}^{{1{\text{p}} - 1}} + \mathop \sum \limits_{{{\text{e}} = 1}}^{{\text{E}}} {\overline{\text{x}}}_{{{\text{ej}}}}^{{2{\text{p}}}} } \right\} \le 0,\quad j = 1, \ldots ,n \\ & \quad \quad {\text{v}}_{{{\text{ik}}}} \left( {{\alpha x}_{{{\text{ij}}}}^{{1{\text{M}}}} + \left( {1 - {\upalpha }} \right){\text{x}}_{{{\text{ij}}}}^{{1{\text{L}}}} } \right) \le \overline{x}_{ij}^{1} \le {\text{v}}_{{{\text{ik}}}} \left( {{\alpha x}_{{{\text{ij}}}}^{{1{\text{M}}}} + \left( {1 - {\upalpha }} \right){\text{x}}_{{{\text{ij}}}}^{{1{\text{U}}}} } \right) \\ & \quad \quad {\upgamma }_{{{\text{ek}}}} \left( {{\alpha x}_{{{\text{ej}}}}^{{2{\text{pM}}}} + \left( {1 - {\upalpha }} \right){\text{x}}_{{{\text{ej}}}}^{{2{\text{pL}}}} } \right) \le \overline{x}_{ej}^{2p} \le {\upgamma }_{{{\text{ek}}}} \left( {{\alpha x}_{{{\text{ej}}}}^{{2{\text{pM}}}} + \left( {1 - {\upalpha }} \right){\text{x}}_{{{\text{ej}}}}^{{2{\text{pU}}}} } \right) \\ & \quad \quad {\text{u}}_{{{\text{rk}}}} \left( {{\alpha y}_{{{\text{rj}}}}^{{1{\text{pM}}}} + \left( {1 - {\upalpha }} \right){\text{y}}_{{{\text{rj}}}}^{{1{\text{pL}}}} } \right) \le \overline{y}_{rj}^{1p} \le {\text{u}}_{{{\text{rk}}}} \left( {{\alpha y}_{{{\text{rj}}}}^{{1{\text{pM}}}} + \left( {1 - {\upalpha }} \right){\text{y}}_{{{\text{rj}}}}^{{1{\text{pU}}}} } \right) \\ & \quad \quad {\upmu }_{{{\text{bk}}}} \left( {{\upalpha }\overline{\overline{{\text{y}}}}_{{{\text{bj}}}}^{{2{\text{pM}}}} + \left( {1 - {\upalpha }} \right)\overline{\overline{{\text{y}}}}_{{{\text{bj}}}}^{{2{\text{pL}}}} } \right) \le \overline{{\overline{\overline{y}} }}_{bj}^{2p} \le {\upmu }_{{{\text{bk}}}} \left( {{\upalpha }\overline{\overline{{\text{y}}}}_{{{\text{bj}}}}^{{2{\text{pM}}}} + \left( {1 - {\upalpha }} \right)\overline{\overline{{\text{y}}}}_{{{\text{bj}}}}^{{2{\text{pU}}}} } \right) \\ & \quad \quad {\upeta }_{{{\text{sk}}}} \left( {{\alpha z}_{{{\text{sj}}}}^{{{\text{pM}}}} + \left( {1 - {\upalpha }} \right){\text{z}}_{{{\text{sj}}}}^{{{\text{pL}}}} } \right) \le \overline{z}_{sj }^{p} \le {\upeta }_{{{\text{sk}}}} \left( {{\alpha z}_{{{\text{sj}}}}^{{{\text{pM}}}} + \left( {1 - {\upalpha }} \right){\text{z}}_{{{\text{sj}}}}^{{{\text{pU}}}} } \right) \\ & \quad \quad v_{ij} ,\gamma_{ej} , u_{rj} , \mu_{bj} ,{\upeta }_{sj} \ge {\upvarepsilon } \\ \end{aligned} $$

Model-10

$$ {\text{Min }}\quad \varphi_{pesimmistic} = \left\{ {\mathop \sum \limits_{{{\text{p}} = 1}}^{{\text{P}}} \left\{ {\mathop \sum \limits_{{{\text{r}} = 1}}^{{\text{R}}} \overline{y}_{rj}^{1p} + \mathop \sum \limits_{{{\text{s}} = 1}}^{{\text{S}}} \overline{z}_{sj}^{p} + \mathop \sum \limits_{{{\text{b}} = 1}}^{{\text{B}}} \overline{{\overline{\overline{y}} }}_{{{\text{bk}}}}^{{2{\text{p}}}} } \right\}} \right\} $$

Subject to:

$$ \begin{aligned} & \left\{ {\mathop \sum \limits_{{{\text{i}} = 1}}^{{\text{I}}} {\overline{\text{x}}}_{{{\text{ik}}}}^{1} + \mathop \sum \limits_{{{\text{e}} = 1}}^{{\text{E}}} {\overline{\text{x}}}_{{{\text{ek}}}}^{21} + \mathop \sum \limits_{{{\text{p}} = 2}}^{{\text{P}}} \left\{ {\mathop \sum \limits_{{{\text{r}} = 1}}^{{\text{R}}} {\overline{\text{y}}}_{{{\text{rk}}}}^{{1{\text{p}} - 1}} + \mathop \sum \limits_{{{\text{e}} = 1}}^{{\text{E}}} {\overline{\text{x}}}_{{{\text{ek}}}}^{{2{\text{p}}}} } \right\}} \right\} = 1 + \in \\ & \quad \quad \left\{ {\mathop \sum \limits_{{{\text{r}} = 1}}^{{\text{R}}} {\overline{\text{y}}}_{{{\text{rj}}}}^{11} + \mathop \sum \limits_{{{\text{b}} = 1}}^{{\text{B}}} \overline{{\overline{\overline{y}} }}_{{{\text{bj}}}}^{21} } \right\} - \left\{ {\mathop \sum \limits_{{{\text{i}} = 1}}^{{\text{I}}} {\overline{\text{x}}}_{{{\text{ij}}}}^{1} + \mathop \sum \limits_{{{\text{e}} = 1}}^{{\text{E}}} {\overline{\text{x}}}_{{{\text{ej}}}}^{21} } \right\} \ge 0 + \in,\quad j = 1, \ldots ,n \\ & \quad \quad \mathop \sum \limits_{{{\text{p}} = 2}}^{{\text{P}}} \left\{ {\mathop \sum \limits_{{{\text{r}} = 1}}^{{\text{R}}} {\overline{\text{y}}}_{{{\text{rj}}}}^{{1{\text{p}}}} + \mathop \sum \limits_{{{\text{s}} = 1}}^{{\text{S}}} {\overline{\text{z}}}_{{{\text{sj}}}}^{{\text{p}}} + \mathop \sum \limits_{{{\text{b}} = 1}}^{{\text{B}}} \overline{{\overline{\overline{y}} }}_{{{\text{bj}}}}^{{2{\text{p}}}} } \right\} - \mathop \sum \limits_{{{\text{p}} = 2}}^{{\text{P}}} \left\{ {\mathop \sum \limits_{{{\text{r}} = 1}}^{{{\text{R}}_{{1 - {\text{p}}}} }} {\overline{\text{y}}}_{{{\text{r}}_{{1 - {\text{p}}}} {\text{j}}}}^{{1{\text{p}} - 1}} + \mathop \sum \limits_{{{\text{e}} = 1}}^{{\text{E}}} {\overline{\text{x}}}_{{{\text{ej}}}}^{{2{\text{p}}}} } \right\} \ge 0 + \in,\quad { }j = 1, \ldots ,n \\ & \quad \quad {\text{v}}_{{{\text{ik}}}} \left( {{\alpha x}_{{{\text{ij}}}}^{{1{\text{M}}}} + \left( {1 - {\upalpha }} \right){\text{x}}_{{{\text{ij}}}}^{{1{\text{L}}}} } \right) \le \overline{x}_{ij}^{1} \le {\text{v}}_{{{\text{ik}}}} \left( {{\alpha x}_{{{\text{ij}}}}^{{1{\text{M}}}} + \left( {1 - {\upalpha }} \right){\text{x}}_{{{\text{ij}}}}^{{1{\text{U}}}} } \right) \\ & \quad \quad {\upgamma }_{{{\text{ek}}}} \left( {{\alpha x}_{{{\text{ej}}}}^{{2{\text{pM}}}} + \left( {1 - {\upalpha }} \right){\text{x}}_{{{\text{ej}}}}^{{2{\text{pL}}}} } \right) \le \overline{x}_{ej}^{2p} \le {\upgamma }_{{{\text{ek}}}} \left( {{\alpha x}_{{{\text{ej}}}}^{{2{\text{pM}}}} + \left( {1 - {\upalpha }} \right){\text{x}}_{{{\text{ej}}}}^{{2{\text{pU}}}} } \right) \\ & \quad \quad {\text{u}}_{{{\text{rk}}}} \left( {{\alpha y}_{{{\text{rj}}}}^{{1{\text{pM}}}} + \left( {1 - {\upalpha }} \right){\text{y}}_{{{\text{rj}}}}^{{1{\text{pL}}}} } \right) \le \overline{y}_{rj}^{1p} \le {\text{u}}_{{{\text{rk}}}} \left( {{\alpha y}_{{{\text{rj}}}}^{{1{\text{pM}}}} + \left( {1 - {\upalpha }} \right){\text{y}}_{{{\text{rj}}}}^{{1{\text{pU}}}} } \right) \\ & \quad \quad {\upmu }_{{{\text{bk}}}} \left( {{\upalpha }\overline{\overline{{\text{y}}}}_{{{\text{bj}}}}^{{2{\text{pM}}}} + \left( {1 - {\upalpha }} \right)\overline{\overline{{\text{y}}}}_{{{\text{bj}}}}^{{2{\text{pL}}}} } \right) \le \overline{{\overline{\overline{y}} }}_{bj}^{2p} \le {\upmu }_{{{\text{bk}}}} \left( {{\upalpha }\overline{\overline{{\text{y}}}}_{{{\text{bj}}}}^{{2{\text{pM}}}} + \left( {1 - {\upalpha }} \right)\overline{\overline{{\text{y}}}}_{{{\text{bj}}}}^{{2{\text{pU}}}} } \right) \\ & \quad \quad {\upeta }_{{{\text{sk}}}} \left( {{\alpha z}_{{{\text{sj}}}}^{{{\text{pM}}}} + \left( {1 - {\upalpha }} \right){\text{z}}_{{{\text{sj}}}}^{{{\text{pL}}}} } \right) \le \overline{z}_{sj }^{p} \le {\upeta }_{{{\text{sk}}}} \left( {{\alpha z}_{{{\text{sj}}}}^{{{\text{pM}}}} + \left( {1 - {\upalpha }} \right){\text{z}}_{{{\text{sj}}}}^{{{\text{pU}}}} } \right) \\ & \quad \quad v_{ij} ,\gamma_{ej} , u_{rj} , \mu_{bj} ,{\upeta }_{sj} \ge {\upvarepsilon } \\ \end{aligned} $$

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).