## Appendix A The Relationship between Standard QAP and QSAP-C  
Given three \(n\times n\) input matrices with real elements \(F=\left(f_{ij}\right)\), \(D=\left(d_{kl}\right)\), and \(B=\left(b_{ik}\right)\), where \(f_{ij}\) is the flow between the facility \(i\) and facility \(j\), \(d_{kl}\) is the distance between the location \(k\) and location \(l\), and \(b_{ik}\) is the cost of placing facility \(i\) at location \(k\). The Koopmans-Beckmann version of the QAP can be formulated as follows:  
\[\text{Min}{\sum_{i=1}^{N}\sum_{k=-1}^{N}\sum_{l=1}^{N}\sum_{j=1}^{N}f_{ij}d_{ kl}x_{ik}x_{lj}}+\sum_{i=1}^{N}\sum_{k=1}^{N}b_{ik}x_{ik} \tag{1}\]  
\[\sum_{i=1}^{N}x_{ik}=1\ \forall\ k=1,2,\ldots N \tag{2}\]  
\[\sum_{k=1}^{N}x_{ik}=1\ \forall\ i=1,2,\ldots N \tag{3}\]  
Ref. [59] generalize traditional QAP as the network-based QAP (NET-QAP). Their generic QAP network depicts a QAP model with \(N\) origins defined as \(N_{S}\), \(N\) candidate locations for origins defined as \(N_{L}\), \(M\) candidate locations for destinations defined as \(M_{S}\)and \(M\) destinations defined as \(M_{L}\). Using the same indices, let \(i\in N_{S}\), \(k\in N_{L}\), \(l\in M_{L}\), \(j\in M_{s}\), and the NET-QAP model can be presented as follows:  
\[\text{Min}\sum_{i=1}^{N_{S}}\sum_{k=1}^{N_{L}}\sum_{l=1}^{M_{S}}\sum_{j=1}^{M_{L }}f_{ij}d_{kl}x_{ik}x_{lj}+\sum_{i=1}^{N_{S}}\sum_{k=1}^{N_{L}}b_{ik}x_{ik}+ \sum_{l=1}^{M_{S}}\sum_{j=1}^{M_{L}}b_{lj}x_{lj} \tag{11}\]  
\[\sum_{i=1}^{N_{S}}x_{ik}=1\ \forall\ k=1,2,\ldots N \tag{12}\]  
\[\sum_{k=1}^{N_{L}}x_{ik}=1\ \forall\ i=1,2,\ldots N \tag{13}\]  
\[\sum_{l=1}^{M_{S}}x_{lj}=1\ \forall\ j=1,2,\ldots M \tag{14}\]  
\[\sum_{j=1}^{M_{L}}x_{lj}=1\ \forall\ l=1,2,\ldots M \tag{15}\]  
In this paper, we further generalize the NET-QAP model: (1) the number of nodes is different in four echelons (that is \(|N_{S}|\
eq|N_{L}|\), \(|M_{L}|\
eq|M_{s}|\)); (2) the assignment between \(N_{S}(M_{s})\) and \(N_{L}(M_{L})\) is general assignment; (3) capacities of \(N_{L}(M_{L})\) are considered; (4) the cost on links \((i,k)\) and \((l,j)\) is not fixed and it is also the product of distance and freight flow. It is immediate that we can write QSAP-C as the NET-QAP of the form.  
\[\text{Min}\sum_{i=1}^{N_{S}}\sum_{k=1}^{N_{L}}\sum_{l=1}^{M_{S}}\sum_{l=1}^{M _{L}}f_{ij}d_{kl}x_{ik}x_{lj}+\sum_{i=1}^{N_{S}}\sum_{k=1}^{N_{L}}\sum_{j=1}^ {M_{L}}f_{ij}d_{ik}x_{ik}+\sum_{l=1}^{M_{S}}\sum_{j=1}^{M_{L}}\sum_{i=1}^{N_{ S}}f_{ij}d_{lj}x_{lj} \tag{16}\]  
\[\sum_{k=1}^{N_{L}}x_{ik}=1\ \forall\ i=1,2,\ldots N_{S} \tag{17}\]  
\[\sum_{l=1}^{M_{L}}x_{lj}=1\ \forall\ j=1,2,\ldots M_{S} \tag{18}\]  
\[\sum_{i\in I}\sum_{j\in I}x_{ik}f_{ij}+\sum_{i\in I}\sum_{j\in I}x_{lj}f_{ij}- \sum_{i\in I}\sum_{j\in J}x_{ik}x_{lj}f_{kj}\leq\]  
_Appendix A.2 The Quality of Different Lower Bounds_  
\begin{table}
\begin{tabular}{l c c c c c c c c c} \hline
**Instance** & **GLB62** & **RRD95** & **HG98** & **KCCEB99** & **AB01** & **RRRP02** & **RS03** & **R04** & **BV04** & **HH01** \\ \hline Had16 & 9.7\% & - & 4.4\% & 4.5\% & 3.4\% & - & 0.6\% & 0 & 1.3\% & 0 \\ Had18 & 10.9\% & - & 5.1\% & 5.2\% & 4.0\% & - & 0.8\% & 0.04\% & 1.1\% & 0 \\ Had20 & 10.9\% & - & 5.1\% & 5.1\% & 3.5\% & - & 0.5\% & 0.03\% & 1.6\% & 0 \\ Kra30a & 23.1\% & 14.5\% & 14.7\% & 15.0\% & 22.9\% & - & 12.7\% & - & 2.5\% & 3.0\% \\ Kra30a & 24.5\% & 16.0\% & 16.3\% & 16.6\% & 24.5\% & - & 11.2\% & - & 4.1\% & 4.7\% \\ Nug12 & 14.7\% & 9.5\% & 9.5\% & 9.9\% & 13.8\% & 0 & 3.6\% & 1.9\% & 1.7\% & 0 \\ Nug15 & 16.3\% & 9.5\% & 9.7\% & 10.2\% & 13.0\% & - & 2.4\% & 1.0\% & 0.8\% & 0 \\ Nug18 & - & - & - & - & - & - & - & - & - & - \\ Nug20 & 20.0\% & 15.1\% & 15.2\% & 15.4\% & 10.9\% & - & 4.6\% & 3.0\% & 2.5\% & 2.4\% \\ Nug22 & - & - & - & - & - & - & - & - & - & 2.4\% \\ Nug30 & 25.9\% & 21.5\% & 21.7\% & 21.9\% & 12.4\% & - & 5.2\% & - & 3.1\% & 5.8\% \\ Rou15 & 15.7\% & 8.3\% & 8.5\% & 8.6\% & 14.2\% & - & 5.9\% & 1.5\% & 1.1\% & 0 \\ \hline \end{tabular}
\end{table}
Table 1: Comparison of lower bounds for standard QAP.