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Notation

Parameters​

The following notation will be used throughout the mathematical descriptions of each of the ILPs.

S\mathcal{S}: set of sites

SPOP\mathcal{S}_{POP}: set of POPs

SDN\mathcal{S}_{DN}: set of DNs

SCN\mathcal{S}_{CN}: set of CNs

SDEM\mathcal{S}_{DEM}: set of demand sites

G\mathcal{G}: set of geographic locations for all the sites (some sites might have the same location)

Ki\mathcal{K}_{i}: set of candidate sectors on site i∈Si \in \mathcal{S}

L\mathcal{L}: set of links between sites

Λi,k\mathcal{\Lambda}_{i, k}: set of links connected to sector k∈Kik \in \mathcal{K_i} on site i∈Si \in \mathcal{S}

cic_i: cost of site ii (e.g., installation and other costs indepdendent of hardware)

c~i,k\tilde{c}_{i,k}: cost of sector kk on site ii (for nodes with multiple sectors, the node cost will only be counted once)

did_i: amount of demand at site i∈SDEMi \in \mathcal{S}_{DEM}

ti,jt_{i,j}: throughput capacity of link (i,j)∈L(i, j) \in \mathcal{L}

Decision Variables​

The following notation will be used for for the various decision variables in the ILPs.

si∈{0,1}s_i \in \{0,1\}: binary selection decision for site i∈S{i} \in \mathcal{S}

σi,k∈{0,1}\sigma_{i,k} \in \{0,1\}: binary selection decision for sector k∈Ki{k} \in \mathcal{K_i}

ℓi,j∈{0,1}\ell_{i,j} \in \{0,1\}: binary selection decision for link (i,j)∈L(i, j) \in \mathcal{L}

pi∈{0,1}p_i \in \{0,1\}: binary polarity (e.g., odd) decision for site i∈SPOP∪SDN{i} \in \mathcal{S}_{POP}\cup\mathcal{S}_{DN}

fi,j∈[0,ti,j]f_{i,j} \in [0, t_{i,j}] : flow through link (i,j)∈L(i, j) \in \mathcal{L}

τi,j∈[0,1]\tau_{i, j} \in [0, 1]: time division multiplexing for link (i,j)∈L(i, j) \in \mathcal{L}

ϕi∈[0,di]\phi_i \in [0, d_i]: amount of unsatisfied demand (or shortage) for demand site i∈SDEM{i} \in \mathcal{S}_{DEM}