Results · IVRP

Instance Library

Every instance from 2 to 7 targets, solved with the exact MILP. Instance n is instance n−1 plus the next body outward, so the series isolates the effect of problem size. Every one was solved to proved optimality, so these are true optima rather than heuristic estimates.

These results are reported in v2 of the preprint, DOI 10.5281/zenodo.22710075, published 11 September 2026.

Targets	Bodies	Total Δv	Budget	Craft	Horizon	Variables	Solve	Status
2	Mercury, Venus	46.35 km/s	50 km/s	1	3000 d	1,374	0.0 s	optimal
3	Mercury, Venus, Mars	58.94 km/s	50 km/s	2	3000 d	3,360	0.2 s	optimal
4	Mercury, Venus, Mars, Jupiter	173.87 km/s	125 km/s	3	6000 d	8,452	4.5 s	optimal
5	Mercury, Venus, Mars, Jupiter, Saturn	161.28 km/s	110 km/s	2	9000 d	12,780	5.6 s	optimal
6	Mercury, Venus, Mars, Jupiter, Saturn, Uranus	274.96 km/s	200 km/s	2	14000 d	17,250	7.3 s	optimal
7	Mercury, Venus, Mars, Jupiter, Saturn, Uranus, Neptune	288.22 km/s	170 km/s	3	20000 d	24,311	9.6 s	optimal

Total Δv is not comparable across rows as a cost series: the mission horizon and the allowed transfer durations grow with the outermost body, so a larger instance can be cheaper than a smaller one. The rows are comparable for solver scaling, which is what they are here to show.

Mission Timelines

One timeline per instance, drawn from its own solve. Each bar is a transfer leg, so the gaps between bars are stay time: spacecraft waiting at a target for a cheap departure alignment, which is where most of the mission clock goes.

Mission timeline for the 2-target instance — 2 targets 46.3 km/s · 1 spacecraft · proved optimal

Mission timeline for the 3-target instance — 3 targets 58.9 km/s · 2 spacecraft · proved optimal

Mission timeline for the 4-target instance — 4 targets 173.9 km/s · 3 spacecraft · proved optimal

Mission timeline for the 5-target instance — 5 targets 161.3 km/s · 2 spacecraft · proved optimal

Mission timeline for the 6-target instance — 6 targets 275.0 km/s · 2 spacecraft · proved optimal

Mission timeline for the 7-target instance — 7 targets 288.2 km/s · 3 spacecraft · proved optimal

288.2

Total Δv (km/s)

7/7

Targets Visited

3

Vehicles Used

100%

Completion

Provenance. Solved with exact MILP (HiGHS via scipy.optimize.milp) on a 40-epoch time-expanded grid. Optimality was proved, so these figures are the true optimum of the model, not a heuristic estimate. Per-spacecraft budget 170 km/s.

Caveat. The model prices every leg as a direct Lambert transfer with no gravity assists, and requires each spacecraft to return to Earth. Real grand tours buy most of their Delta-v from planetary flybys, so these totals sit well above a flown mission profile.

Route Summary — 7-Target Instance

Vehicle 2

167.698 km/s

From	To	Depart (d)	TOF (d)	Δv (km/s)
Earth	Mercury	0.0	512.8	119.123
Mercury	Neptune	512.8	4615.4	28.783
Neptune	Earth	5128.2	6153.8	19.792

Vehicle 4

60.335 km/s

From	To	Depart (d)	TOF (d)	Δv (km/s)
Earth	Venus	0.0	512.8	23.939
Venus	Uranus	1025.6	5641.0	19.941
Uranus	Earth	6666.7	4615.4	16.456

Vehicle 5

60.187 km/s

From	To	Depart (d)	TOF (d)	Δv (km/s)
Earth	Mars	512.8	512.8	12.193
Mars	Jupiter	1538.5	1025.6	12.021
Jupiter	Saturn	2564.1	3076.9	19.991
Saturn	Earth	10769.2	2564.1	15.983

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