Egalitarian Peer-to-Peer Satellite Refueling Strategy

bi = supply/demand corresponding to a node in Vn C M = cost of an egalitarian peer-to-peer solution M CLB = optimal value of objective function of the optimization problem cij = cost of an edge in Gn cij = cost of an edge in G‘ c i; j; k = cost of a triplet i; j; k representing an egalitarian peer-to-peer maneuver d = distance between two triplets tp and tq E‘ = set of edges in G‘ Ef = set of edges in Gn representing forward trips of active satellites En = set of edges in Gn Er = set of edges in Gn representing return trips of active satellites Es = set of source arcs in Gn Et = set of sink arcs in Gn fi = maximum fuel capacity of satellite si f i = minimum fuel requirement by satellite si to remain operational fi;t = fuel content of satellite si at time t G = constellation digraph G‘ = bipartite graph used for calculating lower bound on cost of optimal e-p2p solution Gn = constellation network g0 = acceleration due to gravity at surface of the earth Isp = specific thrust of satellite s J = index set for satellites/orbital slots J a = index set for orbital slots of active satellites J p = index set for orbital slots of passive satellites J r = index set of orbital slots available for active satellites to return J d;t = index set for orbital slots of fuel-deficient satellites at time t J s;t = index set for orbital slots of fuel-sufficient satellites at time t M = egalitarian peer-to-peer solution composed of a set of triplets M = optimal egalitarian peer-to-peer solution MH = egalitarian peer-to-peer solution obtained after local search onMIP MIP = Egalitarian peer-to-peer solution yield by the optimization problem MP2P = optimal egalitarian peer-to-peer solution ms = mass of permanent structure of satellite s N2 tp; tq = two-exchange neighborhood of a triplet pair comprising tp and tq N M = neighborhood of an egalitarian peer-to-peer solution M p ij = fuel expenditure required for an orbital transfer by satellite s from slot i to slot j Q i; j; k = edge in E‘ corresponding to a triplet i; j; k si = satellite with index i 2 J T = total time allotted for refueling T = set of feasible triplets in the constellation graph tp = triplet ip; jp; kp with index p Vn = set of vertices in Gn xij = binary variable corresponding to an arc i; j 2 En or an edge hi; ji in E‘ Vij = velocity change required for a transfer from slot i to slot j = suboptimality measure