A Appendix
$$\begin{aligned}&\text{ NLMIP } \quad \text{ min } \sum _{j \in \mathcal {J}} \sum _{k \in \mathcal {K}} f_{j,k} X_{j,k} + \sum _{j' \in \mathcal {J}} \sum _{k' \in \mathcal {K}} f_{j',k'}^{R} X_{j',k'}^{R} \nonumber \\&\qquad - \sum _{j' \in \mathcal {J}} \sum _{k \in \mathcal {K}} \sum _{k' \in \mathcal {K}} \left( f_{j',k} + f_{j',k'}^{R} \right) \eta _{j',k,k'} Z_{j',k,k'} \nonumber \\&\qquad + \sum _{i \in \mathcal {I}} \sum _{j \in \mathcal {J}} \sum _{p \in \mathcal {P}} \left( t_{j,p} + t_{i,j,p} \right) \mu _{i,p} Y_{i,j,p} \nonumber \\&\qquad + \sum _{i \in \mathcal {I}} \sum _{j \in \mathcal {J}} \sum _{p \in \mathcal {P}} t_{i,j,p} \mu _{i,p} Y_{i,j,p}^{R} \nonumber \\&\qquad + \sum _{j \in \mathcal {J}} \sum _{p \in \mathcal {P}} o_{j,p} \frac{\sum _{i \in \mathcal {I}} \mu _{i,p} Y_{i,j,p}}{Q_{j,p}} \nonumber \\&\qquad + \sum _{j \in \mathcal {J}} \sum _{p \in \mathcal {P}} h_{j,p} \frac{Q_{j,p}}{2} \nonumber \\&\qquad + \sum _{j \in \mathcal {J}} \sum _{p \in \mathcal {P}} h_{j,p} z_{\alpha } \sqrt{ l_{j,p} \left( \sum _{i \in \mathcal {I}} \sigma _{i,p}^2 A_{i,j,p} + \sum _{i \in \mathcal {I}} \sum _{j' \in \mathcal {J}} \rho _{i,p}^2 \sigma _{i,p}^2 \left( Y_{i,j',j,p}^{R} \right) ^ 2 \right) } \nonumber \\&\qquad + \sum _{i \in \mathcal {I}} \sum _{j' \in \mathcal {J}} \sum _{j \in \mathcal {J}} \sum _{p \in \mathcal {P}} \left( t_{i,j',p}^{R} + r_{j',p} + h_{j',p}^{R} \tau _{j',p} \right. \nonumber \\ &\qquad \left. + t_{j',j,p}^{T} \right) \rho _{i,p} \mu _{i,p} Y_{i,j',j,p}^{R}\, + \sum _{i \in \mathcal {I}} \sum _{j' \in \mathcal {J}} \sum _{p \in \mathcal {P}} \left( t_{i,j',p}^{R} + h_{j',p}^{R} \, + s_{j',p} \right) \rho _{i,p} \mu _{i,p} \xi _{i,j',p} \nonumber \\ &\qquad \text{ s.t. } \quad Y_{i,j,p} + Y_{i,j,p}^{R} = A_{i,j,p} \qquad \forall i \in \mathcal {I}, \forall j \in \mathcal {J}, \forall p \in \mathcal {P} \end{aligned}$$
(1)
$$\begin{aligned}&\sum _{j \in \mathcal {J}} A_{i,j,p} = 1 \qquad \forall i \in \mathcal {I}, \forall p \in \mathcal {P} \end{aligned}$$
(2)
$$\begin{aligned}&A_{i,j,p} \le \sum _{k \in \mathcal {K}} X_{j,k} \qquad \forall i \in \mathcal {I}, \forall j \in \mathcal {J}, \forall p \in \mathcal {P} \end{aligned}$$
(3)
$$\begin{aligned}&\sum _{k \in \mathcal {K}} X_{j,k} \le 1 \qquad \forall j \in \mathcal {J} \end{aligned}$$
(4)
$$\begin{aligned}&\sum _{k \in \mathcal {K}} X_{j,k} \le A_{j,j,p} \qquad \forall j \in \mathcal {J}, \forall p \in \mathcal {P} \end{aligned}$$
(5)
$$\begin{aligned}&\sum _{p \in \mathcal {P}} \Bigg ( Q_{j,p} + z_{\alpha } \sqrt{ l_{j,p} \left( \sum _{i \in \mathcal {I}} \sigma _{i,p}^2 A_{i,j,p} + \sum _{i \in \mathcal {I}} \sum _{j' \in \mathcal {J}} \rho _{i,p}^2 \sigma _{i,p}^2 \left( Y_{i,j',j,p}^{R} \right) ^ 2 \right) } \nonumber \\&\qquad + l_{j,p} \sum _{i \in \mathcal {I}} \mu _{i,p} A_{i,j,p} \Bigg ) \le \sum _{k \in \mathcal {K}} c_{j,k} X_{j,k} \qquad \forall j \in \mathcal {J} \end{aligned}$$
(6)
$$\begin{aligned}&\sum _{i \in \mathcal {I}} \sum _{j' \in \mathcal {J}} \rho _{i,p} \mu _{i,p} Y_{i,j',j,p}^{R}\nonumber \\&\quad = \sum _{i \in \mathcal {I}} \mu _{i,p} Y_{i,j,p}^{R} \qquad \forall j \in \mathcal {J}, \forall p \in \mathcal {P}&\end{aligned}$$
(7)
$$\begin{aligned}&\sum _{j \in \mathcal {J}} Y_{i,j',j,p}^{R} + \xi _{i,j',p} \nonumber \\&\quad = B_{i,j',p} \qquad \forall i \in \mathcal {I}, \forall j' \in \mathcal {J}, \forall p \in \mathcal {P} \end{aligned}$$
(8)
$$\begin{aligned}&\sum _{j' \in \mathcal {J}} B_{i,j',p} = 1 \qquad \forall i \in \mathcal {I}, \forall p \in \mathcal {P} \end{aligned}$$
(9)
$$\begin{aligned}&B_{i,j',p} \le \sum _{k' \in \mathcal {K}} X_{j',k'}^{R} \qquad \forall i \in \mathcal {I}, \forall j' \in \mathcal {J}, \forall p \in \mathcal {P} \end{aligned}$$
(10)
$$\begin{aligned}&\sum _{k' \in \mathcal {K}} X_{j',k'}^{R} \le 1 \qquad \forall j' \in \mathcal {J} \end{aligned}$$
(11)
$$\begin{aligned}&\sum _{i \in \mathcal {I}} \sum _{p \in \mathcal {P}} \left( \rho _{i,p} \mu _{i,p} \xi _{i,j',p} + \sum _{j \in \mathcal {J}} \tau _{j',p} \rho _{i,p} \mu _{i,p} Y_{i,j',j,p}^{R} \right) \nonumber \\&\quad \le \sum _{k' \in \mathcal {K}} c_{j',k'}^{R} X_{j',k'}^{R} \quad \forall j' \in \mathcal {J} \end{aligned}$$
(12)
$$\begin{aligned}&X_{j',k} \cdot X_{j',k'}^{R} = Z_{j',k,k'} \qquad \forall j' \in \mathcal {J}, \forall k \in \mathcal {K}, \forall k' \in \mathcal {K} \end{aligned}$$
(13)
$$\begin{aligned}&X_{j,k} \in \left\{ 0, 1 \right\} \qquad \forall j \in \mathcal {J}, \forall k \in \mathcal {K} \end{aligned}$$
(14)
$$\begin{aligned}&A_{i,j,p} \in \left\{ 0, 1 \right\} \qquad \forall i \in \mathcal {I}, \forall j \in \mathcal {J}, \forall p \in \mathcal {P} \end{aligned}$$
(15)
$$\begin{aligned}&Y_{i,j,p} \in \left[ 0, 1 \right] \qquad \forall i \in \mathcal {I}, \forall j \in \mathcal {J}, \forall p \in \mathcal {P} \end{aligned}$$
(16)
$$\begin{aligned}&Y_{i,j,p}^{R} \in \left[ 0, 1 \right] \qquad \forall i \in \mathcal {I}, \forall j \in \mathcal {J}, \forall p \in \mathcal {P} \end{aligned}$$
(17)
$$\begin{aligned}&Q_{j,p} \ge 0 \qquad \forall j \in \mathcal {J}, \forall p \in \mathcal {P} \end{aligned}$$
(18)
$$\begin{aligned}&X_{j',k'}^{R} \in \left\{ 0, 1 \right\} \qquad \forall j' \in \mathcal {J}, \forall k' \in \mathcal {K} \end{aligned}$$
(19)
$$\begin{aligned}&B_{i,j',p} \in \left\{ 0, 1 \right\} \qquad \forall i \in \mathcal {I}, \forall j' \in \mathcal {J}, \forall p \in \mathcal {P} \end{aligned}$$
(20)
$$\begin{aligned}&Y_{i,j',j,p}^{R} \in \left[ 0, 1 \right] \qquad \forall i \in \mathcal {I}, \forall j \in \mathcal {J}, \forall j' \in \mathcal {J}, \forall p \in \mathcal {P} \end{aligned}$$
(21)
$$\begin{aligned}&\xi _{i,j',p} \in \left[ 0, 1 \right] \qquad \forall i \in \mathcal {I}, \forall j' \in \mathcal {J}, \forall p \in \mathcal {P} \end{aligned}$$
(22)
$$\begin{aligned}&Z_{j',k,k'} \in \left\{ 0, 1 \right\} \qquad \forall j' \in \mathcal {J}, \forall k \in \mathcal {K}, \forall k' \in \mathcal {K} \end{aligned}$$
(23)
B Appendix
$$\begin{aligned}&\text{ SOCP } \quad \text{ min } \sum _{j \in \mathcal {J}} \sum _{k \in \mathcal {K}} f_{j,k} X_{j,k} + \sum _{j' \in \mathcal {J}} \sum _{k' \in \mathcal {K}} f_{j',k'}^{R} X_{j',k'}^{R} \nonumber \\&\qquad - \sum _{j' \in \mathcal {J}} \sum _{k \in \mathcal {K}} \sum _{k' \in \mathcal {K}} \left( f_{j',k} + f_{j',k'}^{R} \right) \eta _{j',k,k'} Z_{j',k,k'} \nonumber \\&\qquad + \sum _{i \in \mathcal {I}} \sum _{j \in \mathcal {J}} \sum _{p \in \mathcal {P}} \left( t_{j,p} + t_{i,j,p} \right) \mu _{i,p} Y_{i,j,p}\nonumber \\&\qquad + \sum _{i \in \mathcal {I}} \sum _{j \in \mathcal {J}} \sum _{p \in \mathcal {P}} t_{i,j,p} \mu _{i,p} Y_{i,j,p}^{R} \nonumber \\&\qquad + \sum _{j \in \mathcal {J}} \sum _{p \in \mathcal {P}} h_{j,p} \left( z_{\alpha } S_{j,p} + \frac{\zeta _{j,p}}{2} \right) \nonumber \\&\qquad + \sum _{i \in \mathcal {I}} \sum _{j' \in \mathcal {J}} \sum _{j \in \mathcal {J}} \sum _{p \in \mathcal {P}} \left( t_{i,j',p}^{R} + r_{j',p} + h_{j',p}^{R} \tau _{j',p} \right. \nonumber \\&\qquad \left. +\, t_{j',j,p}^{T} \right) \rho _{i,p} \mu _{i,p} Y_{i,j',j,p}^{R} \nonumber \\&\qquad + \sum _{i \in \mathcal {I}} \sum _{j' \in \mathcal {J}} \sum _{p \in \mathcal {P}} \left( t_{i,j',p}^{R} + h_{j',p}^{R} + s_{j',p} \right) \rho _{i,p} \mu _{i,p} \xi _{i,j',p} \nonumber \\&\qquad \text{ s.t. } \quad Y_{i,j,p} + Y_{i,j,p}^{R} = A_{i,j,p} \qquad \forall i \in \mathcal {I}, \forall j \in \mathcal {J}, \forall p \in \mathcal {P} \end{aligned}$$
(24)
$$\begin{aligned}&\sum _{j \in \mathcal {J}} A_{i,j,p} = 1 \qquad \forall i \in \mathcal {I}, \forall p \in \mathcal {P} \end{aligned}$$
(25)
$$\begin{aligned}&A_{i,j,p} \le \sum _{k \in \mathcal {K}} X_{j,k} \qquad \forall i \in \mathcal {I}, \forall j \in \mathcal {J}, \forall p \in \mathcal {P} \end{aligned}$$
(26)
$$\begin{aligned}&\sum _{k \in \mathcal {K}} X_{j,k} \le 1 \qquad \forall j \in \mathcal {J} \end{aligned}$$
(27)
$$\begin{aligned}&\sum _{k \in \mathcal {K}} X_{j,k} \le A_{j,j,p} \qquad \forall j \in \mathcal {J}, \forall p \in \mathcal {P} \end{aligned}$$
(28)
$$\begin{aligned}&l_{j,p} \left( \sum _{i \in \mathcal {I}} \sigma _{i,p}^2 A_{i,j,p}^2 + \sum _{i \in \mathcal {I}} \sum _{j' \in \mathcal {J}} \rho _{i,p}^2 \sigma _{i,p}^2 \left( Y_{i,j',j,p}^{R} \right) ^ 2 \right) \nonumber \\&\quad \le S_{j,p}^2 \qquad \forall j \in \mathcal {J}, \forall p \in \mathcal {P} \end{aligned}$$
(29)
$$\begin{aligned}&\frac{2 o_{j,p}}{h_{j,p}} \sum _{i \in \mathcal {I}} \mu _{i,p} \varPsi _{i,j,p} ^ 2 + \phi _{j,p} ^ 2 \nonumber \\&\qquad - \frac{\zeta _{j,p} ^ 2}{4} \le 0 \qquad \forall j \in \mathcal {J}, \forall p \in \mathcal {P} \end{aligned}$$
(30)
$$\begin{aligned}&Q_{j,p} - \frac{\zeta _{j,p}}{2} = \phi _{j,p} \qquad \forall j \in \mathcal {J}, \forall p \in \mathcal {P} \end{aligned}$$
(31)
$$\begin{aligned}&\sum _{p \in \mathcal {P}} \left( Q_{j,p} + z_{\alpha } S_{j,p} + l_{j,p} \sum _{i \in \mathcal {I}} \mu _{i,p} A_{i,j,p} \right) \nonumber \\&\quad \le \sum _{k \in \mathcal {K}} c_{j,k} X_{j,k} \qquad \forall j \in \mathcal {J} \end{aligned}$$
(32)
$$\begin{aligned}&\sum _{i \in \mathcal {I}} \sum _{j' \in \mathcal {J}} \rho _{i,p} \mu _{i,p} Y_{i,j',j,p}^{R}\nonumber \\&\quad = \sum _{i \in \mathcal {I}} \mu _{i,p} Y_{i,j,p}^{R} \qquad \forall j \in \mathcal {J}, \forall p \in \mathcal {P} \end{aligned}$$
(33)
$$\begin{aligned}&\sum _{j \in \mathcal {J}} Y_{i,j',j,p}^{R} + \xi _{i,j',p} \nonumber \\&\quad =B_{i,j',p} \qquad \forall i \in \mathcal {I}, \forall j' \in \mathcal {J}, \forall p \in \mathcal {P} \end{aligned}$$
(34)
$$\begin{aligned}&\sum _{j' \in \mathcal {J}} B_{i,j',p} = 1 \qquad \forall i \in \mathcal {I}, \forall p \in \mathcal {P} \end{aligned}$$
(35)
$$\begin{aligned}&B_{i,j',p} \le \sum _{k' \in \mathcal {K}} X_{j',k'}^{R} \qquad \forall i \in \mathcal {I}, \forall j' \in \mathcal {J}, \forall p \in \mathcal {P} \end{aligned}$$
(36)
$$\begin{aligned}&\sum _{k' \in \mathcal {K}} X_{j',k'}^{R} \le 1 \qquad \forall j' \in \mathcal {J} \end{aligned}$$
(37)
$$\begin{aligned}&\sum _{i \in \mathcal {I}} \sum _{p \in \mathcal {P}} \left( \rho _{i,p} \mu _{i,p} \xi _{i,j',p} + \sum _{j \in \mathcal {J}} \tau _{j',p} \rho _{i,p} \mu _{i,p} Y_{i,j',j,p}^{R} \right) \nonumber \\&\quad \le \sum _{k' \in \mathcal {K}} c_{j',k'}^{R} X_{j',k'}^{R} \qquad \forall j' \in \mathcal {J} \end{aligned}$$
(38)
$$\begin{aligned}&X_{j',k} + X_{j',k'}^{R} - 2 Z_{j',k,k'} \ge 0 \qquad \forall j' \in \mathcal {J}, \forall k \in \mathcal {K}, \forall k' \in \mathcal {K} \end{aligned}$$
(39)
$$\begin{aligned}&X_{j',k} + X_{j',k'}^{R} - Z_{j',k,k'} \le 1 \qquad \forall j' \in \mathcal {J}, \forall k \in \mathcal {K}, \forall k' \in \mathcal {K} \end{aligned}$$
(40)
$$\begin{aligned}&\varPsi _{i,j,p} = \sum _{s \in \mathcal {S}} \left( a_s U_{i,j,p,s} \,+\, b_s V_{i,j,p,s} \right) \qquad \forall i \in \mathcal {I}, \forall j \in \mathcal {J}, \forall p \in \mathcal {P} \end{aligned}$$
(41)
$$\begin{aligned}&low_s U_{i,j,p,s} \nonumber \\&\quad \le Y_{i,j,p} \qquad \forall i \in \mathcal {I}, \forall j \in \mathcal {J}, \forall p \in \mathcal {P}, \forall s \in \mathcal {S} \end{aligned}$$
(42)
$$\begin{aligned}&Y_{i,j,p} \le up_s U_{i,j,p,s} \nonumber \\&\qquad + \text{ M } \left( 1 - U_{i,j,p,s} \right) \qquad \forall i \in \mathcal {I},\nonumber \\&\qquad \forall j \in \mathcal {J}, \forall p \in \mathcal {P}, \forall s \in \mathcal {S} \end{aligned}$$
(43)
$$\begin{aligned}&V_{i,j,p,s} \le U_{i,j,p,s} \qquad \forall i \in \mathcal {I}, \forall j \in \mathcal {J}, \forall p \in \mathcal {P}, \forall s \in \mathcal {S} \end{aligned}$$
(44)
$$\begin{aligned}&V_{i,j,p,s} \le Y_{i,j,p} \qquad \forall i \in \mathcal {I}, \forall j \in \mathcal {J}, \forall p \in \mathcal {P}, \forall s \in \mathcal {S} \end{aligned}$$
(45)
$$\begin{aligned}&V_{i,j,p,s} \ge Y_{i,j,p}\nonumber \\&\qquad - \left( 1 - U_{i,j,p,s} \right) \qquad \forall i \in \mathcal {I}, \forall j \in \mathcal {J}, \forall p \in \mathcal {P}, \forall s \in \mathcal {S} \end{aligned}$$
(46)
$$\begin{aligned}&\sum _{s \in \mathcal {S}} U_{i,j,p,s} = 1 \qquad \forall i \in \mathcal {I}, \forall j \in \mathcal {J}, \forall p \in \mathcal {P} \end{aligned}$$
(47)
$$\begin{aligned}&X_{j,k} \in \left\{ 0, 1 \right\} \qquad \forall j \in \mathcal {J}, \forall k \in \mathcal {K} \end{aligned}$$
(48)
$$\begin{aligned}&A_{i,j,p} \in \left\{ 0, 1 \right\} \qquad \forall i \in \mathcal {I}, \forall j \in \mathcal {J}, \forall p \in \mathcal {P} \end{aligned}$$
(49)
$$\begin{aligned}&Y_{i,j,p} \in \left[ 0, 1 \right] \qquad \forall i \in \mathcal {I}, \forall j \in \mathcal {J}, \forall p \in \mathcal {P} \end{aligned}$$
(50)
$$\begin{aligned}&Y_{i,j,p}^{R} \in \left[ 0, 1 \right] \qquad \forall i \in \mathcal {I}, \forall j \in \mathcal {J}, \forall p \in \mathcal {P} \end{aligned}$$
(51)
$$\begin{aligned}&\varPsi _{i,j,p} \in \left[ 0, 1 \right] \qquad \forall i \in \mathcal {I}, \forall j \in \mathcal {J}, \forall p \in \mathcal {P} \end{aligned}$$
(52)
$$\begin{aligned}&S_{j,p}, Q_{j,p}, \zeta _{j,p}, \phi _{j,p} \ge 0 \qquad \forall j \in \mathcal {J}, \forall p \in \mathcal {P} \end{aligned}$$
(53)
$$\begin{aligned}&X_{j',k'}^{R} \in \left\{ 0, 1 \right\} \qquad \forall j' \in \mathcal {J}, \forall k' \in \mathcal {K} \end{aligned}$$
(54)
$$\begin{aligned}&B_{i,j',p} \in \left\{ 0, 1 \right\} \qquad \forall i \in \mathcal {I}, \forall j' \in \mathcal {J}, \forall p \in \mathcal {P} \end{aligned}$$
(55)
$$\begin{aligned}&Y_{i,j',j,p}^{R} \in \left[ 0, 1 \right] \qquad \forall i \in \mathcal {I}, \forall j \in \mathcal {J}, \forall j' \in \mathcal {J}, \forall p \in \mathcal {P} \end{aligned}$$
(56)
$$\begin{aligned}&\xi _{i,j',p} \in \left[ 0, 1 \right] \qquad \forall i \in \mathcal {I}, \forall j' \in \mathcal {J}, \forall p \in \mathcal {P} \end{aligned}$$
(57)
$$\begin{aligned}&Z_{j',k,k'} \in \left\{ 0, 1 \right\} \qquad \forall j' \in \mathcal {J}, \forall k \in \mathcal {K}, \forall k' \in \mathcal {K} \end{aligned}$$
(58)
$$\begin{aligned}&U_{i,j,p,s} \in \left\{ 0,1 \right\} \qquad \forall i \in \mathcal {I}, \forall j \in \mathcal {J}, \forall p \in \mathcal {P}, \forall s \in \mathcal {S} \end{aligned}$$
(59)
$$\begin{aligned}&V_{i,j,p,s} \in \left[ 0,1 \right] \qquad \forall i \in \mathcal {I}, \forall j \in \mathcal {J}, \forall p \in \mathcal {P}, \forall s \in \mathcal {S} \end{aligned}$$
(60)
