\({\Gamma _s}=FC+V{C_s},\)
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(2)
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\(\begin{gathered} FC=\sum\limits_{i} {f{i_i}x{i_i}} +\sum\limits_{m} {f{m_m}x{m_m}} +\sum\limits_{d} {f{c_d}x{d_d}} \hfill \\ \,\,\,\,\,\,\,\,\,\,+\sum\limits_{u} {f{u_u}x{u_u}} +\sum\limits_{k} {f{k_k}x{k_k}} +\sum\limits_{e} {f{e_e}x{e_e}} , \hfill \\ \end{gathered}\)
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(3)
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\(\begin{gathered} V{C_s}=\sum\limits_{p} {\sum\limits_{t} ( } \sum\limits_{i} {\sum\limits_{m} {vs{m_{impts}}qi{m_{impts}}} } +\sum\limits_{m} {\sum\limits_{d} {vm{c_{mdpts}}qm{d_{mdpts}}} } \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\sum\limits_{d} {\sum\limits_{r} {vd{r_{drpts}}qd{r_{drpts}}} } +\sum\limits_{r} {\sum\limits_{u} {vr{u_{rupts}}qr{u_{rupts}}} } \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,+\sum\limits_{c} {\sum\limits_{k} {vc{k_{ckpts}}qc{k_{ckpts}}} } \,+\sum\limits_{k} {\sum\limits_{e} {vk{e_{kepts}}qk{e_{kepts}}} } \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,+\sum\limits_{k} {\sum\limits_{m} {vk{m_{kmpts}}qk{m_{kmpts}}+\sum\limits_{k} {\sum\limits_{v} {vk{v_{kvpts}}qk{v_{kvpts}}} } } } ), \hfill \\ \end{gathered}\)
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\(\forall s\)
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(4)
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subject to:
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Forward flow quantity constraints:
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\(\sum\limits_{d} {qd{r_{drpts}}} +ww{w_{drpts}} \geqslant {d_{rpts}},\)
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\(\forall r,p,t,s\)
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(5)
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\(\sum\limits_{m} {qm{d_{mdpts}}} =\sum\limits_{r} {qd{r_{drpts}}} ,\)
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\(\forall d,p,t,s\)
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(6)
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\(\sum\limits_{i} {qi{m_{impts}}} +\sum\limits_{k} {qk{m_{kmpts}}} \geqslant \sum\limits_{d} {qm{d_{mdpts}}} ,\)
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\(\forall m,p,t,s\)
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(7)
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Backward flow quantity constraints:
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\(\sum\limits_{u} {qr{u_{rupts}}} \geqslant \rho {d_{rpts}},\)
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\(\forall r,p,t,s\)
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(8)
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\(\sum\limits_{k} {qu{k_{ukpts}}} =\sum\limits_{r} {qr{u_{rupts}}} ,\)
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\(\forall u,p,t,s\)
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(9)
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\(\sum\limits_{m} {qk{m_{kmpts}}} \geqslant {\rho ^{\prime}_{pt}}\sum\limits_{u} {qu{k_{ukpts}}} ,\)
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\(\forall k,p,t,s\)
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(10)
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\(\sum\limits_{e} {qk{e_{kepts}}} \geqslant {\rho ^{\prime\prime}_{pt}}\sum\limits_{u} {qu{k_{ukpts}}} ,\)
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\(\forall k,p,t,s\)
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(11)
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\(\sum\limits_{v} {qk{v_{kvpts}}} \geqslant {\rho ^{\prime\prime\prime}_{pt}}\sum\limits_{u} {qu{k_{ukpts}}} ,\)
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\(\forall k,p,t,s\)
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(12)
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\(\sum\limits_{i} {qi{m_{impts}}} \geqslant \rho \rho \sum\limits_{d} {qm{d_{mdpts}}} ,\)
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\(\forall m,p,t,s\)
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(13)
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Resiliency strategy with multi-resource strategy and flexible capacity:
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\(\sum\limits_{m} {qi{m_{impts}}} \leqslant pr{i_i}Cap{s_{ipts}}x{i_i},\)
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\(\forall i,p,t,s\)
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(14)
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\(\sum\limits_{d} {qm{d_{mdpts}}} \leqslant pr{m_m}Cap{m_{mpts}}x{m_m},\)
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\(\forall m,p,t,s\)
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(15)
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\(\sum\limits_{r} {qd{r_{drpts}}} \leqslant pr{d_d}Cap{d_{dpts}}x{d_d},\)
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\(\forall d,p,t,s\)
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(16)
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\(\sum\limits_{r} {qr{u_{rupts}}} \leqslant pr{u_u}Cap{u_{upts}}x{u_u},\)
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\(\forall u,p,t,s\)
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(17)
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\(\sum\limits_{u} {qu{k_{ukpts}}} \leqslant pr{k_k}Cap{k_{kpts}}x{k_k},\)
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\(\forall k,p,t,s\)
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(18)
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\(\sum\limits_{k} {qk{e_{kepts}}} \leqslant pr{e_e}Cap{e_{epts}}x{e_e},\)
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\(\forall e,p,t,s\)
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(19)
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\(\sum\limits_{i} {x{i_i}} \geqslant {\psi _1},\)
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(20)
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\(\sum\limits_{d} {x{d_d}} \geqslant {\psi _2},\)
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(21)
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Sustainability constraint (allowed energy consumption):
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\({\Gamma ^{\prime}_{ts}} \leqslant {\text{Ema}}{{\text{x}}_{ts}},\)
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\(\forall t,s\)
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(22)
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\({\Gamma ^{\prime}_{ts}}=FE+V{E_{ts}},\)
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\(\forall t,s\)
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(23)
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\(\begin{gathered} FE=\sum\limits_{i} {e{i_i}x{i_i}} +\sum\limits_{m} {e{m_m}x{m_m}} +\sum\limits_{d} {e{c_d}x{d_d}} \hfill \\ \,\,\,\,\,\,\,\,+\sum\limits_{u} {e{u_u}x{u_u}} +\sum\limits_{k} {e{k_k}x{k_k}} +\sum\limits_{e} {e{e_e}x{e_e}} , \hfill \\ \end{gathered}\)
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(24)
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\(\begin{gathered} V{E_{ts}}=\sum\limits_{p} ( \sum\limits_{i} {\sum\limits_{m} {es{m_{impts}}qi{m_{impts}}} } +\sum\limits_{m} {\sum\limits_{d} {em{c_{mdpts}}qm{d_{mdpts}}} } \hfill \\ \,\,+\sum\limits_{d} {\sum\limits_{r} {ed{r_{drpts}}qd{r_{drpts}}} } +\sum\limits_{r} {\sum\limits_{u} {er{u_{rupts}}qr{u_{rupts}}} } \hfill \\ \,\,+\sum\limits_{c} {\sum\limits_{k} {ec{k_{ckpts}}qc{k_{ckpts}}} } \,+\sum\limits_{k} {\sum\limits_{e} {ek{e_{kepts}}qk{e_{kepts}}} } \hfill \\ \,\,+\sum\limits_{k} {\sum\limits_{m} {ek{m_{kmpts}}qk{m_{kmpts}}+\sum\limits_{k} {\sum\limits_{v} {ek{v_{kvpts}}qk{v_{kvpts}}} } } } ), \hfill \\ \end{gathered}\)
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\(\forall t,s\)
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(25)
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Agile constraint (satisfaction demand):
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\(\frac{{\sum\limits_{d} {\sum\limits_{r} {\sum\limits_{p} {\sum\limits_{t} {\sum\limits_{s} {qd{r_{drpts}}} } } } } }}{{\sum\limits_{r} {\sum\limits_{p} {\sum\limits_{t} {\sum\limits_{s} {{d_{rpts}}} } } } }} \geqslant \gamma ,\)
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(26)
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Decision variables:
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\(x{i_i},x{m_m},x{d_d},x{r_r}\,,x{u_u},\,x{k_k},x{e_e} \in \{ 0,1\} ,\)
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\(\begin{gathered} \forall i,m,d, \hfill \\ r,u,k,e \hfill \\ \end{gathered}\)
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(27)
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\(\begin{gathered} qi{m_{impts}},qm{d_{mdpts}},qd{r_{drpts}},qr{u_{rupts}}, \hfill \\ qc{k_{ckpts}},qk{e_{kepts}},qk{m_{kmpts}},qk{v_{kvpts}} \geqslant 0, \hfill \\ \end{gathered}\)
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\(\begin{gathered} \forall i,m,d, \hfill \\ r,u,k,e \hfill \\ \end{gathered}\)
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(28)
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