Appendix 1: The proof of the proposed FNSBM-DEA model
The fuzzy efficiency of the kth DMU with network structure can be calculated from the following FNSBM-DEA model:
$$ {\rho}_k=\min \frac{{\mathrm{W}}^{\mathrm{P}}\left[1-\frac{1}{\mathrm{m}+{\mathrm{s}}^{\prime }}\left(\sum \limits_{\mathrm{i}=1}^{\mathrm{m}}\frac{{\mathrm{s}}_{\mathrm{i}}^{\mathrm{p}-}}{{\tilde{\mathrm{x}}}_{\mathrm{i}\mathrm{k}}^{\mathrm{p}}}+\sum \limits_{{\mathrm{r}}^{\prime }=1}^{{\mathrm{s}}^{\prime }}\frac{{\mathrm{f}}_{{\mathrm{r}}^{\prime}}^{\mathrm{p}}}{{\tilde{\mathrm{y}}}_{{\mathrm{r}}^{\prime}\mathrm{k}}^{\mathrm{up}-\mathrm{c}}}\right)\right]+{\mathrm{W}}^{\mathrm{C}}\left[1-\frac{1}{\mathrm{s}+{\mathrm{m}}^{\prime }+{\mathrm{E}}^{\prime }}\left(\sum \limits_{\mathrm{r}=1}^{\mathrm{s}}\frac{{\mathrm{s}}_{\mathrm{r}}^{\mathrm{p}-}}{{\tilde{\mathrm{y}}}_{\mathrm{r}\mathrm{k}}^{\mathrm{DP}-\mathrm{C}}}+\sum \limits_{{\mathrm{i}}^{\prime }=1}^{{\mathrm{m}}^{\prime }}\frac{{\mathrm{s}}_{{\mathrm{i}}^{\prime}}^{\mathrm{C}-}}{{\tilde{\mathrm{x}}}_{{\mathrm{i}}^{\prime}\mathrm{k}}^{\mathrm{C}}}+\sum \limits_{{\mathrm{e}}^{\prime }=1}^{{\mathrm{E}}^{\prime }}\frac{{\mathrm{f}}_{{\mathrm{e}}^{\prime}}^{\mathrm{C}}}{{\tilde{\mathrm{z}}}_{{\mathrm{e}}^{\prime}\mathrm{k}}^{\mathrm{uc}}}\right)\right]}{{\mathrm{W}}^{\mathrm{P}}\left[1+\frac{1}{\mathrm{s}}\left(\sum \limits_{\mathrm{r}=1}^{\mathrm{s}}\frac{{\mathrm{s}}_{\mathrm{r}}^{\mathrm{p}+}}{{\tilde{\mathrm{y}}}_{\mathrm{r}\mathrm{k}}^{\mathrm{DP}-\mathrm{C}}}\right)\right]+{\mathrm{W}}^{\mathrm{C}}\left[1+\frac{1}{\mathrm{E}}\left(\sum \limits_{\mathrm{e}=1}^{\mathrm{E}}\frac{{\mathrm{s}}_{{\mathrm{e}}^{\prime}}^{\mathrm{C}}}{{\tilde{\mathrm{z}}}_{\mathrm{e}\mathrm{k}}^{\mathrm{p}\mathrm{c}}}\right)\right]} $$
(84)
S.t:
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{P}}{\tilde{\mathrm{x}}}_{\mathrm{i}\mathrm{j}}^{\mathrm{P}}+{\mathrm{S}}_{\mathrm{i}}^{\mathrm{P}-}={\tilde{\mathrm{x}}}_{\mathrm{i}\mathrm{k}}^{\mathrm{P}}, $$
(85)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{C}}{\tilde{x}}_{i^{\prime }j}^c+{\mathrm{S}}_{{\mathrm{i}}^{\prime}}^{\mathrm{C}-}={\tilde{x}}_{i^{\prime }k}^C, $$
(86)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{P}}{\tilde{\mathrm{y}}}_{\mathrm{r}\mathrm{j}}^{\mathrm{DP}-\mathrm{C}}-{\mathrm{S}}_{\mathrm{r}}^{\mathrm{P}+}={\tilde{\mathrm{y}}}_{\mathrm{r}\mathrm{k}}^{\mathrm{DP}-\mathrm{C}}, $$
(87)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{P}}{\tilde{y}}_{r^{\prime }j}^{UP-C}+{\mathrm{f}}_{{\mathrm{r}}^{\prime}}^{\mathrm{P}}={\tilde{y}}_{r^{\prime }k}^{UP-C}, $$
(88)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{P}}{\tilde{\mathrm{y}}}_{\mathrm{r}\mathrm{j}}^{\mathrm{DP}-\mathrm{C}}+{\mathrm{S}}_{\mathrm{r}}^{\mathrm{P}-}={\tilde{\mathrm{y}}}_{\mathrm{r}\mathrm{k}}^{\mathrm{DP}-\mathrm{C}} $$
(89)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{C}}{\tilde{\mathrm{z}}}_{\mathrm{e}\mathrm{j}}^{\mathrm{DC}}+{\mathrm{S}}_{\mathrm{e}}^{\mathrm{C}+}={\tilde{\mathrm{z}}}_{\mathrm{rk}}^{\mathrm{DC}}, $$
(90)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{C}}{\tilde{z}}_{e^{\prime }j}^{UC}+{\mathrm{f}}_{{\mathrm{e}}^{\prime}}^{\mathrm{C}}={\tilde{z}}_{e^{\prime }k}^{UC} $$
(91)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{P}}{\tilde{\mathrm{y}}}_{\mathrm{rj}}^{\mathrm{DP}-\mathrm{C}}\ge \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{C}}{\tilde{\mathrm{y}}}_{\mathrm{rj}}^{\mathrm{DP}-\mathrm{C}} $$
(92)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{P}}=1 $$
(93)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{C}}=1 $$
(94)
$$ {\mathrm{W}}^{\mathrm{P}}+{\mathrm{W}}^{\mathrm{c}}=1 $$
(95)
$$ {\uplambda}_{\mathrm{j}}^{\mathrm{P}},{\uplambda}_{\mathrm{j}}^{\mathrm{C}},{\mathrm{S}}_{\mathrm{i}}^{\mathrm{P}-},,{\mathrm{S}}_{{\mathrm{i}}^{\prime}}^{\mathrm{C}-},{\mathrm{f}}_{{\mathrm{r}}^{\prime}}^{\mathrm{P}},{\mathrm{f}}_{{\mathrm{e}}^{\prime}}^{\mathrm{C}},{\mathrm{S}}_{\mathrm{e}}^{\mathrm{C}+}\ge 0,{\mathrm{S}}_{\mathrm{r}}^{\mathrm{P}+},{\mathrm{S}}_{\mathrm{r}}^{\mathrm{P}-}:\mathrm{free}\ \mathrm{in} \operatorname {sign},\mathrm{j}=1,\dots, \mathrm{n} $$
where “˜” indicates fuzziness. Note that all the fuzzy inputs and outputs are TFNs. Thus, we have:
$$ {\rho}_k=\min \left[\frac{{\mathrm{W}}^{\mathrm{P}}\left[1-\frac{1}{\mathrm{m}+{\mathrm{s}}^{\prime }}\left(\sum \limits_{\mathrm{i}=1}^{\mathrm{m}}\frac{{\mathrm{s}}_{\mathrm{i}}^{\mathrm{p}-}}{{\mathrm{x}}_{\mathrm{i}\mathrm{k}}^{\left(\mathrm{p}\right)\mathrm{L}}+{\mathrm{x}}_{\mathrm{i}\mathrm{k}}^{\left(\mathrm{p}\right)\mathrm{M}}+{\mathrm{x}}_{\mathrm{i}\mathrm{k}}^{\left(\mathrm{p}\right)\mathrm{R}}}+\sum \limits_{{\mathrm{r}}^{\prime }=1}^{{\mathrm{s}}^{\prime }}\frac{{\mathrm{f}}_{{\mathrm{r}}^{\prime}}^{\mathrm{p}}}{{\mathrm{y}}_{{\mathrm{r}}^{\prime}\mathrm{k}}^{\left(\mathrm{up}-\mathrm{c}\right)\mathrm{L}}+{\mathrm{y}}_{{\mathrm{r}}^{\prime}\mathrm{k}}^{\left(\mathrm{up}-\mathrm{c}\right)\mathrm{M}}+{\mathrm{y}}_{{\mathrm{r}}^{\prime}\mathrm{k}}^{\left(\mathrm{up}-\mathrm{c}\right)\mathrm{R}}}\right)\right]}{{\mathrm{W}}^{\mathrm{P}}\left[1+\frac{1}{\mathrm{s}}\left(\sum \limits_{\mathrm{r}=1}^{\mathrm{s}}\frac{{\mathrm{s}}_{\mathrm{r}}^{\mathrm{p}+}}{{\mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{L}}+{\mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+{\mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{R}}}\right)\right]+{\mathrm{W}}^{\mathrm{C}}\left[1+\frac{1}{\mathrm{E}}\left(\sum \limits_{\mathrm{e}=1}^{\mathrm{E}}\frac{{\mathrm{s}}_{{\mathrm{e}}^{\prime}}^{\mathrm{C}}}{{\mathrm{z}}_{\mathrm{e}\mathrm{k}}^{\left(\mathrm{p}\mathrm{c}\right)\mathrm{L}}+{\mathrm{z}}_{\mathrm{e}\mathrm{k}}^{\left(\mathrm{p}\mathrm{c}\right)\mathrm{M}}+{\mathrm{z}}_{\mathrm{e}\mathrm{k}}^{\left(\mathrm{p}\mathrm{c}\right)\mathrm{R}}}\right)\right]}+\frac{+{\mathrm{W}}^{\mathrm{C}}\left[1-\frac{1}{\mathrm{s}+{\mathrm{m}}^{\prime }+{\mathrm{E}}^{\prime }}\left(\sum \limits_{\mathrm{r}=1}^{\mathrm{s}}\frac{{\mathrm{s}}_{\mathrm{r}}^{\mathrm{p}-}}{{\mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{L}}+{\mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+{\mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{R}}}+\sum \limits_{{\mathrm{i}}^{\prime }=1}^{{\mathrm{m}}^{\prime }}\frac{{\mathrm{s}}_{{\mathrm{i}}^{\prime}}^{\mathrm{C}-}}{{\mathrm{x}}_{{\mathrm{i}}^{\prime}\mathrm{k}}^{\left(\mathrm{C}\right)\mathrm{L}}+{\mathrm{x}}_{{\mathrm{i}}^{\prime}\mathrm{k}}^{\left(\mathrm{C}\right)\mathrm{M}}+{\mathrm{x}}_{{\mathrm{i}}^{\prime}\mathrm{k}}^{\left(\mathrm{C}\right)\mathrm{R}}}+\sum \limits_{{\mathrm{e}}^{\prime }=1}^{{\mathrm{E}}^{\prime }}\frac{{\mathrm{f}}_{{\mathrm{e}}^{\prime}}^{\mathrm{C}}}{{\mathrm{z}}_{{\mathrm{e}}^{\prime}\mathrm{k}}^{\left(\mathrm{uc}\right)\mathrm{l}}+{\mathrm{z}}_{{\mathrm{e}}^{\prime}\mathrm{k}}^{\left(\mathrm{uc}\right)\mathrm{M}}+{\mathrm{z}}_{{\mathrm{e}}^{\prime}\mathrm{k}}^{\left(\mathrm{uc}\right)\mathrm{R}}}\right)\right]}{{\mathrm{W}}^{\mathrm{P}}\left[1+\frac{1}{\mathrm{s}}\left(\sum \limits_{\mathrm{r}=1}^{\mathrm{s}}\frac{{\mathrm{s}}_{\mathrm{r}}^{\mathrm{p}+}}{{\mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{L}}+{\mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+{\mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{R}}}\right)\right]+{\mathrm{W}}^{\mathrm{C}}\left[1+\frac{1}{\mathrm{E}}\left(\sum \limits_{\mathrm{e}=1}^{\mathrm{E}}\frac{{\mathrm{s}}_{{\mathrm{e}}^{\prime}}^{\mathrm{C}}}{{\mathrm{z}}_{\mathrm{e}\mathrm{k}}^{\left(\mathrm{p}\mathrm{c}\right)\mathrm{L}}+{\mathrm{z}}_{\mathrm{e}\mathrm{k}}^{\left(\mathrm{p}\mathrm{c}\right)\mathrm{M}}+{\mathrm{z}}_{\mathrm{e}\mathrm{k}}^{\left(\mathrm{p}\mathrm{c}\right)\mathrm{R}}}\right)\right]}\right] $$
(96)
S.t:
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{P}}\left({\mathrm{x}}_{\mathrm{i}\mathrm{j}}^{\mathrm{P}\mathrm{L}}+{\mathrm{x}}_{\mathrm{i}\mathrm{j}}^{\mathrm{P}\mathrm{M}}+{\mathrm{x}}_{\mathrm{i}\mathrm{j}}^{\mathrm{P}\mathrm{R}}\right)+{\mathrm{S}}_{\mathrm{i}}^{\mathrm{P}-}=\left({\mathrm{x}}_{\mathrm{i}\mathrm{k}}^{\mathrm{P}\mathrm{L}}+{\mathrm{x}}_{\mathrm{i}\mathrm{k}}^{\mathrm{P}\mathrm{M}}+{\mathrm{x}}_{\mathrm{i}\mathrm{k}}^{\mathrm{P}\mathrm{R}}\right), $$
(97)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{C}}\left({x}_{i^{\prime }j}^{cL}+{x}_{i^{\prime }j}^{cM}+{x}_{i^{\prime }j}^{cR}\right)+{\mathrm{S}}_{{\mathrm{i}}^{\prime}}^{\mathrm{C}-}=\left({x}_{i^{\prime }k}^{cL}+{x}_{i^{\prime }k}^{cM}+{x}_{i^{\prime }k}^{cR}\right), $$
(98)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{P}}\left({\mathrm{y}}_{\mathrm{r}\mathrm{j}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{L}}+{\mathrm{y}}_{\mathrm{r}\mathrm{j}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+{\mathrm{y}}_{\mathrm{r}\mathrm{j}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{R}}\right)-{\mathrm{S}}_{\mathrm{r}}^{\mathrm{P}+}=\left({\mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{L}}+{\mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+{\mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{R}}\right), $$
(99)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{P}}\left({y}_{r^{\prime }j}^{\left( UP-C\right)L}+{y}_{r^{\prime }j}^{\left( UP-C\right)M}+{y}_{r^{\prime }j}^{\left( UP-C\right)R}\right)+{\mathrm{f}}_{{\mathrm{r}}^{\prime}}^{\mathrm{P}}=\left({y}_{r^{\prime }k}^{\left( UP-C\right)L}+{y}_{r^{\prime }k}^{\left( UP-C\right)M}+{y}_{r^{\prime }k}^{\left( UP-C\right)R}\right), $$
(100)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{P}}\left({\mathrm{y}}_{\mathrm{r}\mathrm{j}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{L}}{\mathrm{y}}_{\mathrm{r}\mathrm{j}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+{\mathrm{y}}_{\mathrm{r}\mathrm{j}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{R}}\right)+{\mathrm{S}}_{\mathrm{r}}^{\mathrm{P}-}=\left({\mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{L}}+{\mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+{\mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{R}}\right), $$
(101)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{C}}\left({\mathrm{z}}_{\mathrm{e}\mathrm{j}}^{\left(\mathrm{DC}\right)\mathrm{L}}+{\mathrm{z}}_{\mathrm{e}\mathrm{j}}^{\left(\mathrm{DC}\right)\mathrm{M}}+{\mathrm{z}}_{\mathrm{e}\mathrm{j}}^{\left(\mathrm{DC}\right)\mathrm{R}}\right)+{\mathrm{S}}_{\mathrm{e}}^{\mathrm{C}+}=\left({\mathrm{z}}_{\mathrm{rk}}^{\left(\mathrm{DC}\right)\mathrm{L}}+{\mathrm{z}}_{\mathrm{rk}}^{\left(\mathrm{DC}\right)\mathrm{M}}+{\mathrm{z}}_{\mathrm{rk}}^{\left(\mathrm{DC}\right)\mathrm{R}}\right), $$
(102)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{C}}\left({z}_{e^{\prime }j}^{UCL}+{z}_{e^{\prime }j}^{UCM}+{z}_{e^{\prime }j}^{UCR}\right)+{\mathrm{f}}_{{\mathrm{e}}^{\prime}}^{\mathrm{C}}=\left({z}_{e^{\prime }k}^{UCL}+{z}_{e^{\prime }k}^{UCM}+{z}_{e^{\prime }k}^{UCR}\right) $$
(103)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{P}}{\mathrm{y}}_{\mathrm{rj}}^{\mathrm{DP}-\mathrm{C}}\ge \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{C}}{\mathrm{y}}_{\mathrm{rj}}^{\mathrm{DP}-\mathrm{C}} $$
(104)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{P}}=1 $$
(105)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{C}}=1 $$
(106)
$$ {\mathrm{W}}^{\mathrm{P}}+{\mathrm{W}}^{\mathrm{c}}=1 $$
(107)
$$ {\uplambda}_{\mathrm{j}}^{\mathrm{P}},{\uplambda}_{\mathrm{j}}^{\mathrm{C}},{\mathrm{S}}_{\mathrm{i}}^{\mathrm{P}-},,{\mathrm{S}}_{{\mathrm{i}}^{\prime}}^{\mathrm{C}-},{\mathrm{f}}_{{\mathrm{r}}^{\prime}}^{\mathrm{P}},{\mathrm{f}}_{{\mathrm{e}}^{\prime}}^{\mathrm{C}},{\mathrm{S}}_{\mathrm{e}}^{\mathrm{C}+}\ge 0,{\mathrm{S}}_{\mathrm{r}}^{\mathrm{P}+},{\mathrm{S}}_{\mathrm{r}}^{\mathrm{P}-}:\mathrm{free}\ \mathrm{in} \operatorname {sign},\mathrm{j}=1,\dots, \mathrm{n} $$
To transform the proposed fuzzy model into a crisp model, we apply the α-cut approach proposed by Saati et al. [41]. As discussed by Puri and Yadav [40], the α-cut method is used more than other methods as this approach can measure the performance of DMUs given α ∈ (0, 1]. Using the α-cut method, the proposed model is changed to an interval programming. By applying α-cut method, model (96–107) can be expressed as follows:
$$ {\rho}_k=\min \left[\frac{{\mathrm{W}}^{\mathrm{P}}\left[1-\frac{1}{\mathrm{m}+{\mathrm{s}}^{\prime }}\left(\sum \limits_{\mathrm{i}=1}^{\mathrm{m}}\frac{{\mathrm{s}}_{\mathrm{i}}^{\mathrm{p}-}}{\left[\left({\upalpha \mathrm{x}}_{\mathrm{i}\mathrm{k}}^{\left(\mathrm{p}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{x}}_{\mathrm{i}\mathrm{k}}^{\left(\mathrm{p}\right)\mathrm{L}}\right),\left({\upalpha \mathrm{x}}_{\mathrm{i}\mathrm{k}}^{\left(\mathrm{p}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{x}}_{\mathrm{i}\mathrm{k}}^{\left(\mathrm{p}\right)\mathrm{R}}\right)\right]}+\sum \limits_{{\mathrm{r}}^{\prime }=1}^{{\mathrm{s}}^{\prime }}\frac{{\mathrm{f}}_{{\mathrm{r}}^{\prime}}^{\mathrm{p}}}{\left[\left(\upalpha {\mathrm{y}}_{{\mathrm{r}}^{\prime}\mathrm{k}}^{\left(\mathrm{up}-\mathrm{c}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{y}}_{{\mathrm{r}}^{\prime}\mathrm{k}}^{\left(\mathrm{up}-\mathrm{c}\right)\mathrm{L}}\right),\left(\upalpha {\mathrm{y}}_{{\mathrm{r}}^{\prime}\mathrm{k}}^{\left(\mathrm{up}-\mathrm{c}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{y}}_{{\mathrm{r}}^{\prime}\mathrm{k}}^{\left(\mathrm{up}-\mathrm{c}\right)\mathrm{R}}\right)\right]}\right)\right]}{{\mathrm{W}}^{\mathrm{P}}\left[1+\frac{1}{\mathrm{s}}\left(\sum \limits_{\mathrm{r}=1}^{\mathrm{s}}\frac{{\mathrm{s}}_{\mathrm{r}}^{\mathrm{p}+}}{\left[\left(\upalpha {\mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{L}}\right),\left(\upalpha {\mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{R}}\right)\right]}\right)\right]+{\mathrm{W}}^{\mathrm{C}}\left[1+\frac{1}{\mathrm{E}}\left(\sum \limits_{\mathrm{e}=1}^{\mathrm{E}}\frac{{\mathrm{s}}_{{\mathrm{e}}^{\prime}}^{\mathrm{C}}}{\left({\upalpha \mathrm{z}}_{\mathrm{e}\mathrm{k}}^{\left(\mathrm{p}\mathrm{c}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{z}}_{\mathrm{e}\mathrm{k}}^{\left(\mathrm{p}\mathrm{c}\right)\mathrm{L}}\right),\left({\upalpha \mathrm{z}}_{\mathrm{e}\mathrm{k}}^{\left(\mathrm{p}\mathrm{c}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{z}}_{\mathrm{e}\mathrm{k}}^{\left(\mathrm{p}\mathrm{c}\right)\mathrm{L}}\right)}\right)\right]}+\frac{+{\mathrm{W}}^{\mathrm{C}}\left[1-\frac{1}{\mathrm{s}+{\mathrm{m}}^{\prime }+{\mathrm{E}}^{\prime }}\left(\sum \limits_{\mathrm{r}=1}^{\mathrm{s}}\frac{{\mathrm{s}}_{\mathrm{r}}^{\mathrm{p}-}}{\left[\left(\upalpha {\mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+\left(1-\upalpha \right)\upalpha {\mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{L}}\right),\left(\upalpha {\mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+\left(1-\upalpha \right)\upalpha {\mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{R}}\right)\right]}+\sum \limits_{{\mathrm{i}}^{\prime }=1}^{{\mathrm{m}}^{\prime }}\frac{{\mathrm{s}}_{{\mathrm{i}}^{\prime}}^{\mathrm{C}-}}{\left[\upalpha {\mathrm{x}}_{{\mathrm{i}}^{\prime}\mathrm{k}}^{\left(\mathrm{C}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{x}}_{{\mathrm{i}}^{\prime}\mathrm{k}}^{\left(\mathrm{C}\right)\mathrm{L}}\Big),\left({\upalpha \mathrm{x}}_{\mathrm{i}\mathrm{k}}^{\left(\mathrm{p}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{x}}_{\mathrm{i}\mathrm{k}}^{\left(\mathrm{p}\right)\mathrm{R}}\right)\right]}+\sum \limits_{{\mathrm{e}}^{\prime }=1}^{{\mathrm{E}}^{\prime }}\frac{{\mathrm{f}}_{{\mathrm{e}}^{\prime}}^{\mathrm{C}}}{\left[\left(\upalpha {\mathrm{z}}_{{\mathrm{e}}^{\prime}\mathrm{k}}^{\left(\mathrm{uc}\right)\mathrm{M}}+\left(1-\upalpha \right)\upalpha {\mathrm{z}}_{{\mathrm{e}}^{\prime}\mathrm{k}}^{\left(\mathrm{uc}\right)\mathrm{L}}\right),\left(\upalpha {\mathrm{z}}_{{\mathrm{e}}^{\prime}\mathrm{k}}^{\left(\mathrm{uc}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{z}}_{{\mathrm{e}}^{\prime}\mathrm{k}}^{\left(\mathrm{uc}\right)\mathrm{R}}\right)\right]}\right)\right]}{{\mathrm{W}}^{\mathrm{P}}\left[1+\frac{1}{\mathrm{s}}\left(\sum \limits_{\mathrm{r}=1}^{\mathrm{s}}\frac{{\mathrm{s}}_{\mathrm{r}}^{\mathrm{p}+}}{\left[\left(\upalpha {\mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{L}}\right),\upalpha {\mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{R}}\Big),\right]}\right)\right]+{\mathrm{W}}^{\mathrm{C}}\left[1+\frac{1}{\mathrm{E}}\left(\sum \limits_{\mathrm{e}=1}^{\mathrm{E}}\frac{{\mathrm{s}}_{{\mathrm{e}}^{\prime}}^{\mathrm{C}}}{\left[\left({\upalpha \mathrm{z}}_{\mathrm{e}\mathrm{k}}^{\left(\mathrm{p}\mathrm{c}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{z}}_{\mathrm{e}\mathrm{k}}^{\left(\mathrm{p}\mathrm{c}\right)\mathrm{L}}\right),\left(\left({\upalpha \mathrm{z}}_{\mathrm{e}\mathrm{k}}^{\left(\mathrm{p}\mathrm{c}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{z}}_{\mathrm{e}\mathrm{k}}^{\left(\mathrm{p}\mathrm{c}\right)\mathrm{R}}\right)\right)\right]}\right)\right]}\right] $$
(108)
S.t:
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{P}}\left[\left({\upalpha \mathrm{x}}_{\mathrm{i}\mathrm{j}}^{\mathrm{P}\mathrm{M}}+\left(1-\upalpha \right){\mathrm{x}}_{\mathrm{i}\mathrm{j}}^{\mathrm{P}\mathrm{L}}\right),\left({\upalpha \mathrm{x}}_{\mathrm{i}\mathrm{j}}^{\mathrm{P}\mathrm{M}}+\left(1-\upalpha \right){\mathrm{x}}_{\mathrm{i}\mathrm{j}}^{\mathrm{P}\mathrm{r}}\right)\right]+{\mathrm{S}}_{\mathrm{i}}^{\mathrm{P}-}=\left[\left({\upalpha \mathrm{x}}_{\mathrm{i}\mathrm{k}}^{\mathrm{P}\mathrm{M}}+\left(1-\upalpha \right){\mathrm{x}}_{\mathrm{i}\mathrm{k}}^{\mathrm{P}\mathrm{L}}\right),\left({\upalpha \mathrm{x}}_{\mathrm{i}\mathrm{k}}^{\mathrm{P}\mathrm{M}}+\left(1-\upalpha \right){\mathrm{x}}_{\mathrm{i}\mathrm{k}}^{\mathrm{P}\mathrm{r}}\right)\right] $$
(109)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{C}}\left[\left({\alpha x}_{i^{\prime }j}^{cM}+\left(1-\alpha \right){x}_{i^{\prime }j}^{cL}\right),\Big({\alpha x}_{i^{\prime }j}^{cM}+\left(1-\alpha \right){x}_{i^{\prime }j}^{cR},\Big)\right]+{\mathrm{S}}_{{\mathrm{i}}^{\prime}}^{\mathrm{C}-}=\left[\left({\alpha x}_{i^{\prime }k}^{cM}+\left(1-\alpha \right){x}_{i^{\prime }k}^{cL}\right),\left({\alpha x}_{i^{\prime }k}^{cM}+\left(1-\alpha \right){x}_{i^{\prime }k}^{cR}\right)\right] $$
(110)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{P}}\left[\left(\upalpha {\mathrm{y}}_{\mathrm{r}\mathrm{j}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{y}}_{\mathrm{r}\mathrm{j}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{L}}\right),\left(\upalpha {\mathrm{y}}_{\mathrm{r}\mathrm{j}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{y}}_{\mathrm{r}\mathrm{j}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{R}}\right)\right]-{\mathrm{S}}_{\mathrm{r}}^{\mathrm{P}+}=\left[\left(\upalpha {\mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{L}}\right),\left(\upalpha {\mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{R}}\right)\right] $$
(111)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{P}}\left[\left({\alpha y}_{r^{\prime }j}^{\left( UP-C\right)M}+\left(1-\alpha \right){y}_{r^{\prime }j}^{\left( UP-C\right)L}\right),\left({\alpha y}_{r^{\prime }j}^{\left( UP-C\right)M}+\left(1-\alpha \right){y}_{r^{\prime }j}^{\left( UP-C\right)R}\right)\right]+{\mathrm{f}}_{{\mathrm{r}}^{\prime}}^{\mathrm{P}}=\left[\left({\alpha y}_{r^{\prime }k}^{\left( UP-C\right)M}+\left(1-\alpha \right){y}_{r^{\prime }k}^{\left( UP-C\right)L}\right),\left({\alpha y}_{r^{\prime }k}^{\left( UP-C\right)M}+\left(1-\alpha \right){y}_{r^{\prime }k}^{\left( UP-C\right)R}\right)\right] $$
(112)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{P}}\left[\left({\upalpha \mathrm{y}}_{\mathrm{r}\mathrm{j}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{y}}_{\mathrm{r}\mathrm{j}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{L}}\right),\left({\upalpha \mathrm{y}}_{\mathrm{r}\mathrm{j}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{y}}_{\mathrm{r}\mathrm{j}}^{\left(\mathrm{DP}-\mathrm{C}\right)}\right)\right]+{\mathrm{S}}_{\mathrm{r}}^{\mathrm{P}-}=\left[\left({\upalpha \mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{L}}\right),\left({\upalpha \mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{y}}_{\mathrm{r}\mathrm{k}}^{\left(\mathrm{DP}-\mathrm{C}\right)}\right)\right] $$
(113)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{C}}\left[\left(\upalpha {\mathrm{z}}_{\mathrm{e}\mathrm{j}}^{\left(\mathrm{DC}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{z}}_{\mathrm{e}\mathrm{j}}^{\left(\mathrm{DC}\right)\mathrm{L}}\right),\left(\upalpha {\mathrm{z}}_{\mathrm{e}\mathrm{j}}^{\left(\mathrm{DC}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{z}}_{\mathrm{e}\mathrm{j}}^{\left(\mathrm{DC}\right)\mathrm{R}}\right)\right]+{\mathrm{S}}_{\mathrm{e}}^{\mathrm{C}+}=\left[\left(\upalpha {\mathrm{z}}_{\mathrm{e}\mathrm{k}}^{\left(\mathrm{DC}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{z}}_{\mathrm{e}\mathrm{k}}^{\left(\mathrm{DC}\right)\mathrm{L}}\right),\left(\upalpha {\mathrm{z}}_{\mathrm{e}\mathrm{k}}^{\left(\mathrm{DC}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{z}}_{\mathrm{e}\mathrm{k}}^{\left(\mathrm{DC}\right)\mathrm{R}}\right)\right] $$
(114)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{C}}\left[\left(\upalpha {z}_{e^{\prime }j}^{UCM}+\left(1-\upalpha \right)\upalpha {z}_{e^{\prime }j}^{UCL},\upalpha {z}_{e^{\prime }j}^{UCM}+\left(1-\upalpha \right)\upalpha {z}_{e^{\prime }j}^{UCR}\right)\right]+{\mathrm{f}}_{{\mathrm{e}}^{\prime}}^{\mathrm{C}}=\left[\left(\upalpha {z}_{e^{\prime }k}^{UCM}+\left(1-\upalpha \right)\upalpha {z}_{e^{\prime }k}^{UCL},\upalpha {z}_{e^{\prime }k}^{UCM}+\left(1-\upalpha \right)\upalpha {z}_{e^{\prime }k}^{UCR}\right)\right] $$
(115)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{P}}\left[\left({\upalpha \mathrm{y}}_{\mathrm{rj}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{y}}_{\mathrm{rj}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{L}}\right),\left({\upalpha \mathrm{y}}_{\mathrm{rj}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{y}}_{\mathrm{rj}}^{\left(\mathrm{DP}-\mathrm{C}\right)}\right)\right]\ge \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{C}}\left[\left({\upalpha \mathrm{y}}_{\mathrm{rj}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{y}}_{\mathrm{rj}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{L}}\right),\left({\upalpha \mathrm{y}}_{\mathrm{rj}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{y}}_{\mathrm{rj}}^{\left(\mathrm{DP}-\mathrm{C}\right)}\right)\right] $$
(116)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{P}}=1 $$
(117)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{C}}=1 $$
(118)
$$ {\mathrm{W}}^{\mathrm{P}}+{\mathrm{W}}^{\mathrm{c}}=1 $$
(119)
$$ {\uplambda}_{\mathrm{j}}^{\mathrm{P}},{\uplambda}_{\mathrm{j}}^{\mathrm{C}},{\mathrm{S}}_{\mathrm{i}}^{\mathrm{P}-},{\mathrm{S}}_{\mathrm{r}}^{\mathrm{P}+},{\mathrm{S}}_{\mathrm{r}}^{\mathrm{P}-},{\mathrm{S}}_{{\mathrm{i}}^{\prime}}^{\mathrm{C}-},{\mathrm{f}}_{{\mathrm{r}}^{\prime}}^{\mathrm{P}},{\mathrm{f}}_{{\mathrm{e}}^{\prime}}^{\mathrm{C}},{\mathrm{S}}_{\mathrm{e}}^{\mathrm{C}+}\kern0.75em \ge 0,\kern0.75em \mathrm{j}=1,\dots, \mathrm{n} $$
Obviously, all the coefficients in the fuzzy model (108–119) are intervals. Thus, we have:
$$ \left\{\begin{array}{c}{\hat{x}}_{ij}^p\in \left\{\upalpha {\mathrm{x}}_{\mathrm{ij}}^{\mathrm{PM}}+\left(1-\upalpha \right){\mathrm{x}}_{\mathrm{ij}}^{\mathrm{PL}},\upalpha {\mathrm{x}}_{\mathrm{ij}}^{\mathrm{PM}}+\left(1-\upalpha \right){\mathrm{x}}_{\mathrm{ij}}^{\mathrm{PR}}\right\}\forall \mathrm{i},\mathrm{j},\\ {}{\hat{x}}_{i^{\prime }j}^C\in \left\{\upalpha {\mathrm{x}}_{i^{\prime}\mathrm{j}}^{\mathrm{CM}}+\left(1-\upalpha \right){\mathrm{x}}_{i^{\prime}\mathrm{j}}^{\mathrm{CL}},\upalpha {\mathrm{x}}_{i^{\prime}\mathrm{j}}^{\mathrm{CM}}+\left(1-\upalpha \right)\upalpha {\mathrm{x}}_{i^{\prime}\mathrm{j}}^{\mathrm{CR}}\right\}\forall \mathrm{i},\mathrm{j},\\ {}{\hat{\mathrm{y}}}_{\mathrm{rj}}^{\mathrm{DP}-\mathrm{C}}\in \left\{\upalpha {\mathrm{y}}_{\mathrm{rj}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{y}}_{\mathrm{rj}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{L}},\upalpha {\mathrm{y}}_{\mathrm{rj}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{y}}_{\mathrm{rj}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{R}}\right\}\kern1em \forall \mathrm{i},\mathrm{j},\\ {}{\hat{y}}_{r^{\prime }j}^{UP-C}\in \left\{{\alpha y}_{r^{\prime }j}^{\left( UP-C\right)M}+\left(1-\alpha \right){y}_{r^{\prime }j}^{\left( UP-C\right)L},{\alpha y}_{r^{\prime }j}^{\left( UP-C\right)M}+\left(1-\alpha \right){y}_{r^{\prime }j}^{\left( UP-C\right)R}\right\}\forall \mathrm{i},\mathrm{j},\\ {}{\hat{\mathrm{z}}}_{\mathrm{ej}}^{\mathrm{DC}}\kern0.5em \in \left\{\upalpha {\mathrm{z}}_{\mathrm{ej}}^{\left(\mathrm{DC}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{z}}_{\mathrm{ej}}^{\left(\mathrm{DC}\right)\mathrm{L}},\upalpha {\mathrm{z}}_{\mathrm{ej}}^{\left(\mathrm{DC}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{z}}_{\mathrm{ej}}^{\left(\mathrm{DC}\right)\mathrm{R}}\right\}\forall \mathrm{i},\mathrm{j},\\ {}{\hat{z}}_{e^{\prime }j}^{UC}\in \left\{\upalpha {z}_{e^{\prime }j}^{UC M}+\left(1-\upalpha \right)\upalpha {z}_{e^{\prime }j}^{UC L},\upalpha {z}_{e^{\prime }j}^{UC M}+\left(1-\upalpha \right)\upalpha {z}_{e^{\prime }j}^{UC R}\right\}\forall \mathrm{i},\mathrm{j},\end{array}\right. $$
By substituting the variables \( {\hat{x}}_{ij}^p,{\hat{x}}_{i^{\prime }j}^C,{\hat{\mathrm{y}}}_{\mathrm{rj}}^{\mathrm{DP}-\mathrm{C}},{\hat{y}}_{r^{\prime }j}^{UP-C},{\hat{\mathrm{z}}}_{\mathrm{ej}}^{\mathrm{DC}},\mathrm{and}\ {\hat{z}}_{e^{\prime }j}^{UC} \) in model (108–119), we have:
$$ {\rho}_k=\min \frac{{\mathrm{W}}^{\mathrm{P}}\left[1-\frac{1}{\mathrm{m}+{\mathrm{s}}^{\prime }}\left(\sum \limits_{\mathrm{i}=1}^{\mathrm{m}}\frac{{\mathrm{s}}_{\mathrm{i}}^{\mathrm{p}-}}{{\hat{\mathrm{x}}}_{\mathrm{i}\mathrm{k}}^{\mathrm{p}}}+\sum \limits_{{\mathrm{r}}^{\prime }=1}^{{\mathrm{s}}^{\prime }}\frac{{\mathrm{f}}_{{\mathrm{r}}^{\prime}}^{\mathrm{p}}}{{\hat{\mathrm{y}}}_{{\mathrm{r}}^{\prime}\mathrm{k}}^{\mathrm{up}-\mathrm{c}}}\right)\right]+{\mathrm{W}}^{\mathrm{C}}\left[1-\frac{1}{\mathrm{s}+{\mathrm{m}}^{\prime }+{\mathrm{E}}^{\prime }}\left(\sum \limits_{\mathrm{r}=1}^{\mathrm{s}}\frac{{\mathrm{s}}_{\mathrm{r}}^{\mathrm{p}-}}{{\hat{\mathrm{y}}}_{\mathrm{r}\mathrm{k}}^{\mathrm{DP}-\mathrm{C}}}+\sum \limits_{{\mathrm{i}}^{\prime }=1}^{{\mathrm{m}}^{\prime }}\frac{{\mathrm{s}}_{{\mathrm{i}}^{\prime}}^{\mathrm{C}-}}{{\hat{\mathrm{x}}}_{{\mathrm{i}}^{\prime}\mathrm{k}}^{\mathrm{C}}}+\sum \limits_{{\mathrm{e}}^{\prime }=1}^{{\mathrm{E}}^{\prime }}\frac{{\mathrm{f}}_{{\mathrm{e}}^{\prime}}^{\mathrm{C}}}{{\hat{\mathrm{z}}}_{{\mathrm{e}}^{\prime}\mathrm{k}}^{\mathrm{uc}}}\right)\right]}{{\mathrm{W}}^{\mathrm{P}}\left[1+\frac{1}{\mathrm{s}}\left(\sum \limits_{\mathrm{r}=1}^{\mathrm{s}}\frac{{\mathrm{s}}_{\mathrm{r}}^{\mathrm{p}+}}{{\hat{\mathrm{y}}}_{\mathrm{r}\mathrm{k}}^{\mathrm{DP}-\mathrm{C}}}\right)\right]+{\mathrm{W}}^{\mathrm{C}}\left[1+\frac{1}{\mathrm{E}}\left(\sum \limits_{\mathrm{e}=1}^{\mathrm{E}}\frac{{\mathrm{s}}_{{\mathrm{e}}^{\prime}}^{\mathrm{C}}}{{\hat{\mathrm{z}}}_{\mathrm{e}\mathrm{k}}^{\mathrm{p}\mathrm{c}}}\right)\right]} $$
(120)
S.t:
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{P}}{\hat{\mathrm{x}}}_{\mathrm{i}\mathrm{j}}^{\mathrm{P}}+{\mathrm{S}}_{\mathrm{i}}^{\mathrm{P}-}={\hat{\mathrm{x}}}_{\mathrm{i}\mathrm{k}}^{\mathrm{P}}, $$
(121)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{C}}{\hat{x}}_{i^{\prime }j}^c+{\mathrm{S}}_{{\mathrm{i}}^{\prime}}^{\mathrm{C}-}={\hat{x}}_{i^{\prime }k}^C $$
(122)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{P}}{\hat{\mathrm{y}}}_{\mathrm{r}\mathrm{j}}^{\mathrm{DP}-\mathrm{C}}-{\mathrm{S}}_{\mathrm{r}}^{\mathrm{P}+}={\hat{\mathrm{y}}}_{\mathrm{r}\mathrm{k}}^{\mathrm{DP}-\mathrm{C}}, $$
(123)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{P}}{\hat{y}}_{r^{\prime }j}^{UP-C}+{\mathrm{f}}_{{\mathrm{r}}^{\prime}}^{\mathrm{P}}={\hat{y}}_{r^{\prime }k}^{UP-C}, $$
(124)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{P}}{\hat{\mathrm{y}}}_{\mathrm{r}\mathrm{j}}^{\mathrm{DP}-\mathrm{C}}+{\mathrm{S}}_{\mathrm{r}}^{\mathrm{P}-}={\hat{\mathrm{y}}}_{\mathrm{r}\mathrm{k}}^{\mathrm{DP}-\mathrm{C}} $$
(125)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{C}}{\hat{\mathrm{z}}}_{\mathrm{e}\mathrm{j}}^{\mathrm{DC}}+{\mathrm{S}}_{\mathrm{e}}^{\mathrm{C}+}={\hat{\mathrm{z}}}_{\mathrm{rk}}^{\mathrm{DC}}, $$
(126)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{C}}{\hat{z}}_{e^{\prime }j}^{UC}+{\mathrm{f}}_{{\mathrm{e}}^{\prime}}^{\mathrm{C}}={\hat{z}}_{e^{\prime }k}^{UC} $$
(127)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{P}}{\hat{\mathrm{y}}}_{\mathrm{rj}}^{\mathrm{DP}-\mathrm{C}}\ge \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{C}}{\hat{\mathrm{y}}}_{\mathrm{rj}}^{\mathrm{DP}-\mathrm{C}} $$
(128)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{P}}=1 $$
(129)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{C}}=1 $$
(130)
$$ {\mathrm{W}}^{\mathrm{P}}+{\mathrm{W}}^{\mathrm{c}}=1 $$
(131)
$$ \kern0.5em \upalpha {\mathrm{x}}_{\mathrm{ij}}^{\mathrm{PM}}+\left(1-\upalpha \right){\mathrm{x}}_{\mathrm{ij}}^{\mathrm{PL}}\le {\hat{x}}_{ij}^p\le \upalpha {\mathrm{x}}_{\mathrm{ij}}^{\mathrm{PM}}+\left(1-\upalpha \right){\mathrm{x}}_{\mathrm{ij}}^{\mathrm{PR}} $$
(132)
$$ \upalpha {\mathrm{x}}_{\mathrm{ik}}^{\mathrm{PM}}+\left(1-\upalpha \right){\mathrm{x}}_{\mathrm{ik}}^{\mathrm{PL}}\le {\hat{x}}_{ik}^p\le \upalpha {\mathrm{x}}_{\mathrm{ik}}^{\mathrm{PM}}+\left(1-\upalpha \right){\mathrm{x}}_{\mathrm{ik}}^{\mathrm{PR}} $$
(133)
$$ \upalpha {\mathrm{x}}_{i^{\prime}\mathrm{j}}^{\mathrm{CM}}+\left(1-\upalpha \right){\mathrm{x}}_{i^{\prime}\mathrm{j}}^{\mathrm{CL}}\le {\hat{x}}_{i^{\prime }j}^C\le \upalpha {\mathrm{x}}_{i^{\prime}\mathrm{j}}^{\mathrm{CM}}+\left(1-\upalpha \right)\upalpha {\mathrm{x}}_{i^{\prime}\mathrm{j}}^{\mathrm{CR}} $$
(134)
$$ \upalpha {\mathrm{x}}_{i^{\prime}\mathrm{k}}^{\mathrm{CM}}+\left(1-\upalpha \right){\mathrm{x}}_{i^{\prime}\mathrm{k}}^{\mathrm{CL}}\le {\hat{x}}_{i^{\prime }k}^C\le \upalpha {\mathrm{x}}_{i^{\prime}\mathrm{k}}^{\mathrm{CM}}+\left(1-\upalpha \right)\upalpha {\mathrm{x}}_{i^{\prime}\mathrm{k}}^{\mathrm{CR}} $$
(135)
$$ \upalpha {\mathrm{y}}_{\mathrm{rj}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{y}}_{\mathrm{rj}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{L}}\le {\hat{\mathrm{y}}}_{\mathrm{rj}}^{\mathrm{DP}-\mathrm{C}}\le \upalpha {\mathrm{y}}_{\mathrm{rj}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{y}}_{\mathrm{rj}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{R}} $$
(136)
$$ \upalpha {\mathrm{y}}_{\mathrm{rk}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{y}}_{\mathrm{rk}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{L}}\le {\hat{\mathrm{y}}}_{\mathrm{rk}}^{\mathrm{DP}-\mathrm{C}}\le \upalpha {\mathrm{y}}_{\mathrm{rk}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{y}}_{\mathrm{rk}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{R}} $$
(137)
$$ {\alpha y}_{r^{\prime }j}^{\left( UP-C\right)M}+\left(1-\alpha \right){y}_{r^{\prime }j}^{\left( UP-C\right)L}\le {\hat{y}}_{r^{\prime }j}^{UP-C}\le {\alpha y}_{r^{\prime }j}^{\left( UP-C\right)M}+\left(1-\alpha \right){y}_{r^{\prime }j}^{\left( UP-C\right)R} $$
(138)
$$ {\alpha y}_{r^{\prime }k}^{\left( UP-C\right)M}+\left(1-\alpha \right){y}_{r^{\prime }k}^{\left( UP-C\right)L}\le {\hat{y}}_{r^{\prime }k}^{UP-C}\le {\alpha y}_{r^{\prime }k}^{\left( UP-C\right)M}+\left(1-\alpha \right){y}_{r^{\prime }k}^{\left( UP-C\right)R} $$
(139)
$$ \upalpha {\mathrm{z}}_{\mathrm{ej}}^{\left(\mathrm{DC}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{z}}_{\mathrm{ej}}^{\left(\mathrm{DC}\right)\mathrm{L}}\le {\hat{\mathrm{z}}}_{\mathrm{ej}}^{\mathrm{DC}}\le \upalpha {\mathrm{z}}_{\mathrm{ej}}^{\left(\mathrm{DC}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{z}}_{\mathrm{ej}}^{\left(\mathrm{DC}\right)\mathrm{R}} $$
(140)
$$ \upalpha {\mathrm{z}}_{\mathrm{ek}}^{\left(\mathrm{DC}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{z}}_{\mathrm{ek}}^{\left(\mathrm{DC}\right)\mathrm{L}}\le {\hat{\mathrm{z}}}_{\mathrm{ek}}^{\mathrm{DC}}\le \upalpha {\mathrm{z}}_{\mathrm{ek}}^{\left(\mathrm{DC}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{z}}_{\mathrm{ek}}^{\left(\mathrm{DC}\right)\mathrm{R}} $$
(141)
$$ {z}_{e^{\prime }j}^{UC M}+\left(1-\upalpha \right)\upalpha {z}_{e^{\prime }j}^{UC L}\le {\hat{z}}_{e^{\prime }j}^{UC}\le \upalpha {z}_{e^{\prime }j}^{UC M}+\left(1-\upalpha \right)\upalpha {z}_{e^{\prime }j}^{UC R} $$
(142)
$$ {z}_{e^{\prime }k}^{UC M}+\left(1-\upalpha \right)\upalpha {z}_{e^{\prime }k}^{UC L}\le {\hat{z}}_{e^{\prime }k}^{UC}\le \upalpha {z}_{e^{\prime }k}^{UC M}+\left(1-\upalpha \right)\upalpha {z}_{e^{\prime }k}^{UC R} $$
(143)
$$ {\uplambda}_{\mathrm{j}}^{\mathrm{P}},{\uplambda}_{\mathrm{j}}^{\mathrm{C}},{\mathrm{S}}_{\mathrm{i}}^{\mathrm{P}-},,{\mathrm{S}}_{{\mathrm{i}}^{\prime}}^{\mathrm{C}-},{\mathrm{f}}_{{\mathrm{r}}^{\prime}}^{\mathrm{P}},{\mathrm{f}}_{{\mathrm{e}}^{\prime}}^{\mathrm{C}},{\mathrm{S}}_{\mathrm{e}}^{\mathrm{C}+}\ge 0,{\mathrm{S}}_{\mathrm{r}}^{\mathrm{P}+},{\mathrm{S}}_{\mathrm{r}}^{\mathrm{P}-}:\mathrm{free}\ \mathrm{in} \operatorname {sign},\mathrm{j}=1,\dots, \mathrm{n}\kern0.75em \ge 0,\kern0.75em \mathrm{j}=1,\dots, \mathrm{n} $$
Note that model (120–143) is a nonlinear programming problem. To convert nonlinear programming to linear programming, we use the following variable substitutions:
$$ \left\{\begin{array}{c}{\overline{x}}_{ij}^p={\lambda}_j^p{\hat{x}}_{ij}^p\forall \left(i,j\right),\\ {}{\overline{x}}_{i^{\prime }j}^C={\lambda}_j^C{\hat{x}}_{i^{\prime }j}^C\forall \left(i,j\right),\\ {}{\overline{y}}_{rj}^{DP-C}={\lambda}_j^p{\hat{y}}_{rj}^{DP-C}\forall \left(i,j\right),\\ {}{\overline{y}}_{r^{\prime }j}^{UP-C}={\lambda}_j^p{\hat{y}}_{r^{\prime }j}^{UP-C}\forall \left(i,j\right),\\ {}{\overline{z}}_{ej}^{DC}={\lambda}_j^C{\hat{z}}_{ej}^{DC}\forall \left(i,j\right)\\ {}{\overline{z}}_{e^{\prime }j}^{UC}={\hat{\lambda_j^Cz}}_{e^{\prime }j}^{UC}\forall \left(i,j\right)\end{array}\right. $$
By substituting the above variables, model (120–143) is expressed as the following linear programming problem:
$$ {\rho}_k=\min \frac{{\mathrm{W}}^{\mathrm{P}}\left[1-\frac{1}{\mathrm{m}+{\mathrm{s}}^{\prime }}\left(\sum \limits_{\mathrm{i}=1}^{\mathrm{m}}\frac{{\mathrm{s}}_{\mathrm{i}}^{\mathrm{p}-}}{{\hat{\mathrm{x}}}_{\mathrm{i}\mathrm{k}}^{\mathrm{p}}}+\sum \limits_{{\mathrm{r}}^{\prime }=1}^{{\mathrm{s}}^{\prime }}\frac{{\mathrm{f}}_{{\mathrm{r}}^{\prime}}^{\mathrm{p}}}{{\hat{\mathrm{y}}}_{{\mathrm{r}}^{\prime}\mathrm{k}}^{\mathrm{up}-\mathrm{c}}}\right)\right]+{\mathrm{W}}^{\mathrm{C}}\left[1-\frac{1}{\mathrm{s}+{\mathrm{m}}^{\prime }+{\mathrm{E}}^{\prime }}\left(\sum \limits_{\mathrm{r}=1}^{\mathrm{s}}\frac{{\mathrm{s}}_{\mathrm{r}}^{\mathrm{p}-}}{{\hat{\mathrm{y}}}_{\mathrm{r}\mathrm{k}}^{\mathrm{DP}-\mathrm{C}}}+\sum \limits_{{\mathrm{i}}^{\prime }=1}^{{\mathrm{m}}^{\prime }}\frac{{\mathrm{s}}_{{\mathrm{i}}^{\prime}}^{\mathrm{C}-}}{{\hat{\mathrm{x}}}_{{\mathrm{i}}^{\prime}\mathrm{k}}^{\mathrm{C}}}+\sum \limits_{{\mathrm{e}}^{\prime }=1}^{{\mathrm{E}}^{\prime }}\frac{{\mathrm{f}}_{{\mathrm{e}}^{\prime}}^{\mathrm{C}}}{{\hat{\mathrm{z}}}_{{\mathrm{e}}^{\prime}\mathrm{k}}^{\mathrm{uc}}}\right)\right]}{{\mathrm{W}}^{\mathrm{P}}\left[1+\frac{1}{\mathrm{s}}\left(\sum \limits_{\mathrm{r}=1}^{\mathrm{s}}\frac{{\mathrm{s}}_{\mathrm{r}}^{\mathrm{p}+}}{{\hat{\mathrm{y}}}_{\mathrm{r}\mathrm{k}}^{\mathrm{DP}-\mathrm{C}}}\right)\right]+{\mathrm{W}}^{\mathrm{C}}\left[1+\frac{1}{\mathrm{E}}\left(\sum \limits_{\mathrm{e}=1}^{\mathrm{E}}\frac{{\mathrm{s}}_{{\mathrm{e}}^{\prime}}^{\mathrm{C}}}{{\hat{\mathrm{z}}}_{\mathrm{e}\mathrm{k}}^{\mathrm{p}\mathrm{c}}}\right)\right]} $$
(144)
S.t:
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\overline{x}}_{ij}^p+{\mathrm{S}}_{\mathrm{i}}^{\mathrm{P}-}={\hat{\mathrm{x}}}_{\mathrm{i}\mathrm{k}}^{\mathrm{P}}, $$
(145)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\overline{x}}_{i^{\prime }j}^C+{\mathrm{S}}_{{\mathrm{i}}^{\prime}}^{\mathrm{C}-}={\hat{x}}_{i^{\prime }k}^C, $$
(146)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\overline{\mathrm{y}}}_{\mathrm{r}\mathrm{j}}^{\mathrm{DP}-\mathrm{C}}-{\mathrm{S}}_{\mathrm{r}}^{\mathrm{P}+}={\hat{\mathrm{y}}}_{\mathrm{r}\mathrm{k}}^{\mathrm{DP}-\mathrm{C}}, $$
(147)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\overline{\mathrm{y}}}_{r^{\prime }j}^{UP-C}+{\mathrm{f}}_{{\mathrm{r}}^{\prime}}^{\mathrm{P}}={\hat{y}}_{r^{\prime }k}^{UP-C}, $$
(148)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\overline{\mathrm{y}}}_{\mathrm{r}\mathrm{j}}^{\mathrm{DP}-\mathrm{C}}+{\mathrm{S}}_{\mathrm{r}}^{\mathrm{P}-}={\hat{\mathrm{y}}}_{\mathrm{r}\mathrm{k}}^{\mathrm{DP}-\mathrm{C}} $$
(149)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\overline{\mathrm{z}}}_{\mathrm{e}\mathrm{j}}^{\mathrm{DC}}+{\mathrm{S}}_{\mathrm{e}}^{\mathrm{C}+}={\hat{\mathrm{z}}}_{\mathrm{rk}}^{\mathrm{DC}}, $$
(150)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\overline{z}}_{e^{\prime }j}^{UC}+{\mathrm{f}}_{{\mathrm{e}}^{\prime}}^{\mathrm{C}}={\hat{z}}_{e^{\prime }k}^{UC} $$
(151)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\overline{\mathrm{y}}}_{\mathrm{rj}}^{\mathrm{DP}-\mathrm{C}}\ge \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\overline{\mathrm{y}}}_{\mathrm{rj}}^{{\left(\mathrm{DP}-\mathrm{C}\right)}^{\prime }} $$
(152)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{P}}=1 $$
(153)
$$ \sum \limits_{\mathrm{j}=1}^{\mathrm{n}}{\uplambda}_{\mathrm{j}}^{\mathrm{C}}=1 $$
(154)
$$ {\mathrm{W}}^{\mathrm{P}}+{\mathrm{W}}^{\mathrm{c}}=1 $$
(155)
$$ {\uplambda}_{\mathrm{j}}^{\mathrm{P}}\ \left(\upalpha {\mathrm{x}}_{\mathrm{ij}}^{\mathrm{P}\mathrm{M}}+\left(1-\upalpha \right){\mathrm{x}}_{\mathrm{ij}}^{\mathrm{P}\mathrm{L}}\right)\le {\overline{x}}_{ij}^p\le {\uplambda}_{\mathrm{j}}^{\mathrm{P}}\left(\upalpha {\mathrm{x}}_{\mathrm{ij}}^{\mathrm{P}\mathrm{M}}+\left(1-\upalpha \right){\mathrm{x}}_{\mathrm{ij}}^{\mathrm{P}\mathrm{R}}\right) $$
(156)
$$ \upalpha {\mathrm{x}}_{\mathrm{ik}}^{\mathrm{PM}}+\left(1-\upalpha \right){\mathrm{x}}_{\mathrm{ik}}^{\mathrm{PL}}\le {\hat{x}}_{ik}^p\le \upalpha {\mathrm{x}}_{\mathrm{ik}}^{\mathrm{PM}}+\left(1-\upalpha \right){\mathrm{x}}_{\mathrm{ik}}^{\mathrm{PR}} $$
(157)
$$ {\uplambda}_{\mathrm{j}}^{\mathrm{C}}\left(\upalpha {\mathrm{x}}_{i^{\prime}\mathrm{j}}^{\mathrm{C}\mathrm{M}}+\left(1-\upalpha \right){\mathrm{x}}_{i^{\prime}\mathrm{j}}^{\mathrm{C}\mathrm{L}}\right)\le {\overline{x}}_{i^{\prime }j}^C\le {\uplambda}_{\mathrm{j}}^{\mathrm{C}}\left(\upalpha {\mathrm{x}}_{i^{\prime}\mathrm{j}}^{\mathrm{C}\mathrm{M}}+\left(1-\upalpha \right)\upalpha {\mathrm{x}}_{i^{\prime}\mathrm{j}}^{\mathrm{C}\mathrm{R}}\right) $$
(158)
$$ \upalpha {\mathrm{x}}_{i^{\prime}\mathrm{k}}^{\mathrm{CM}}+\left(1-\upalpha \right){\mathrm{x}}_{i^{\prime}\mathrm{k}}^{\mathrm{CL}}\le {\hat{x}}_{i^{\prime }k}^C\le \upalpha {\mathrm{x}}_{i^{\prime}\mathrm{k}}^{\mathrm{CM}}+\left(1-\upalpha \right)\upalpha {\mathrm{x}}_{i^{\prime}\mathrm{k}}^{\mathrm{CR}} $$
(159)
$$ {\uplambda}_{\mathrm{j}}^{\mathrm{P}}\left(\upalpha {\mathrm{y}}_{\mathrm{rj}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{y}}_{\mathrm{rj}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{L}}\right)\le {\overline{\mathrm{y}}}_{\mathrm{rj}}^{\mathrm{DP}-\mathrm{C}}\le {\uplambda}_{\mathrm{j}}^{\mathrm{P}}\left(\upalpha {\mathrm{y}}_{\mathrm{rj}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{y}}_{\mathrm{rj}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{R}}\right) $$
(160)
$$ {\uplambda}_{\mathrm{j}}^{\mathrm{C}}\left(\upalpha {\mathrm{y}}_{\mathrm{rj}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{y}}_{\mathrm{rj}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{L}}\right)\le {\overline{\mathrm{y}}}_{\mathrm{rj}}^{{\left(\mathrm{DP}-\mathrm{C}\right)}^{\prime }}\le {\uplambda}_{\mathrm{j}}^{\mathrm{C}}\left(\upalpha {\mathrm{y}}_{\mathrm{rj}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{y}}_{\mathrm{rj}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{R}}\right) $$
(161)
$$ \upalpha {\mathrm{y}}_{\mathrm{rk}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{y}}_{\mathrm{rk}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{L}}\le {\hat{\mathrm{y}}}_{\mathrm{rk}}^{\mathrm{DP}-\mathrm{C}}\le \upalpha {\mathrm{y}}_{\mathrm{rk}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{y}}_{\mathrm{rk}}^{\left(\mathrm{DP}-\mathrm{C}\right)\mathrm{R}} $$
(162)
$$ {\uplambda}_{\mathrm{j}}^{\mathrm{P}}\left({\alpha y}_{r^{\prime }j}^{\left( UP-C\right)M}+\left(1-\alpha \right){y}_{r^{\prime }j}^{\left( UP-C\right)L}\right)\le {\overline{\mathrm{y}}}_{r^{\prime }j}^{UP-C}\le {\uplambda}_{\mathrm{j}}^{\mathrm{P}}\left({\alpha y}_{r^{\prime }j}^{\left( UP-C\right)M}+\left(1-\alpha \right){y}_{r^{\prime }j}^{\left( UP-C\right)R}\right) $$
(163)
$$ {\alpha y}_{r^{\prime }k}^{\left( UP-C\right)M}+\left(1-\alpha \right){y}_{r^{\prime }k}^{\left( UP-C\right)L}\le {\hat{y}}_{r^{\prime }k}^{UP-C}\le {\alpha y}_{r^{\prime }k}^{\left( UP-C\right)M}+\left(1-\alpha \right){y}_{r^{\prime }k}^{\left( UP-C\right)R} $$
(164)
$$ {\uplambda}_{\mathrm{j}}^{\mathrm{C}}\left(\upalpha {\mathrm{z}}_{\mathrm{ej}}^{\left(\mathrm{DC}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{z}}_{\mathrm{ej}}^{\left(\mathrm{DC}\right)\mathrm{L}}\right)\le {\overline{\mathrm{z}}}_{\mathrm{ej}}^{\mathrm{DC}}\le {\uplambda}_{\mathrm{j}}^{\mathrm{C}}\left(\upalpha {\mathrm{z}}_{\mathrm{ej}}^{\left(\mathrm{DC}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{z}}_{\mathrm{ej}}^{\left(\mathrm{DC}\right)\mathrm{R}}\right) $$
(165)
$$ \upalpha {\mathrm{z}}_{\mathrm{ek}}^{\left(\mathrm{DC}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{z}}_{\mathrm{ek}}^{\left(\mathrm{DC}\right)\mathrm{L}}\le {\hat{\mathrm{z}}}_{\mathrm{ek}}^{\mathrm{DC}}\le \upalpha {\mathrm{z}}_{\mathrm{ek}}^{\left(\mathrm{DC}\right)\mathrm{M}}+\left(1-\upalpha \right){\mathrm{z}}_{\mathrm{ek}}^{\left(\mathrm{DC}\right)\mathrm{R}} $$
(166)
$$ {\uplambda}_{\mathrm{j}}^{\mathrm{C}}\left({z}_{e^{\prime }j}^{UC M}+\left(1-\upalpha \right)\upalpha {z}_{e^{\prime }j}^{UC L}\right)\le {\overline{z}}_{e^{\prime }j}^{UC}\le {\uplambda}_{\mathrm{j}}^{\mathrm{C}}\left(\upalpha {z}_{e^{\prime }j}^{UC M}+\left(1-\upalpha \right)\upalpha {z}_{e^{\prime }j}^{UC R}\right) $$
(167)
$$ {z}_{e^{\prime }k}^{UC M}+\left(1-\upalpha \right)\upalpha {z}_{e^{\prime }k}^{UC L}\le {\hat{z}}_{e^{\prime }k}^{UC}\le \upalpha {z}_{e^{\prime }k}^{UC M}+\left(1-\upalpha \right)\upalpha {z}_{e^{\prime }k}^{UC R} $$
(168)
$$ {\uplambda}_{\mathrm{j}}^{\mathrm{P}},{\uplambda}_{\mathrm{j}}^{\mathrm{C}},{\mathrm{S}}_{\mathrm{i}}^{\mathrm{P}-},,{\mathrm{S}}_{{\mathrm{i}}^{\prime}}^{\mathrm{C}-},{\mathrm{f}}_{{\mathrm{r}}^{\prime}}^{\mathrm{P}},{\mathrm{f}}_{{\mathrm{e}}^{\prime}}^{\mathrm{C}},{\mathrm{S}}_{\mathrm{e}}^{\mathrm{C}+}\ge 0,{\mathrm{S}}_{\mathrm{r}}^{\mathrm{P}+},{\mathrm{S}}_{\mathrm{r}}^{\mathrm{P}-}:\mathrm{free}\ \mathrm{in} \operatorname {sign},\mathrm{j}=1,\dots, \mathrm{n}. $$
