First, the scope of the problem and the theoretical foundations are presented. The considered ISC network is a layered network in which nodes represent interaction points between the two layers. The two interacting networks are respectively PN that delivers cargo from Earth to Mars and SN that is responsible for propellant supply along the journey. They share the same nodes but comprise different arcs according to their distinct objectives. The nodes are defined as surface nodes (celestial bodies where supplies and demands are generated), orbital nodes (relatively stable orbits accessible from surface nodes), Lagrange points (potential positions for on-orbit infrastructure) and auxiliary nodes (LTO and DTO that serve as gateways for inter-cluster travel). The propellant consumption is given by a propellant mass fraction that is directly derived from the rocket equation.

Next, the author formulates a multistage stochastic programming (MSP) model. The proposed MSP model for ISC planning considers the demands of a Mars base as a source of uncertainty. The uncertainty is represented by random variables, which follow specified probability distributions. And, a set of scenarios that is essentially a discrete approximation of the original probability distribution allows MSP to reflect the uncertainty at multiple points in time. Two classes of decision variables are also differentiated in the developed MSP model: scenario-independent strategic variables, which represent decisions that must be made before the uncertainty occurs and cannot be changed during the planning horizon, and scenario-dependent tactical variables, which are made after the uncertain parameters have materialized and can be adjusted over time. Based on the above model description, a multistage stochastic MILP model is developed, in which the objective function shown in equation (3) minimizes the total launch mass (TLM).

(3)

Spacecraft, payload and propellant flows in the PN, propellant flows in the SN and other constraints are also shown in detail.

Next, numerical studies are discussed. In the optimization, the planning horizon considered includes 6 years (on Earth) divided into 24 periods of 3 months each. The demands are assumed to be a seasonal pattern, affected by the rather harsh seasonal conditions on Mars, and do not occur before period 9. Based on the approach used by Hahn and Kuhn [20]Seasonal demands for each period and scenario are calculated as shown in equation (22), where the scenario factor ft,s is used to generate the number of scenarios from the base case, the base level of demand is estimated at dibase=50 t for i=March, and the remaining terms generate the seasonal demand model from the base level of demand assuming a harmonic oscillation with an amplitude of amp=0.1.

To account for uncertainty in the models, a discrete set of scenarios is generated, each comprising discrete realizations of each uncertain parameter, which is captured in the scenario factor ft,s in equation (22) and obtained by the moment-matching scenario generation heuristic proposed by Hoyland et al. [23]. The entire MILP stochastic model, which includes 564,726 constraints, 19,225 integer variables, and 76,268 continuous variables, is implemented in IBM ILOG CPLEX Optimization Studio v20.1.0. The base case results for total launched mass show that the space infrastructure facility is responsible for over 37% of the total launch mass, with the In Situ Resource Utilization (ISRU) facilities and the on-orbit depots sharing roughly equal shares. Approximately 170 times more propellant than dry mass is required to support the Mars base over the planning horizon. The tasks are somewhat divided as some vehicles are responsible for the majority of the cargo while others retrieve propellant from the depot. Regarding infrastructure, ISRU facilities are installed on all planetary satellites, i.e., the Moon, Phobos and Deimos, while depots are established at L2, LDO and LMO. And the capacity of the depots to store propellant allows reducing the size of these expensive ISRU facilities. To explore the effects of technological progress on the SC network, sensitivity analyses regarding the spacecraft payload capacity (cap^{Q}), its propulsive capacity (cap^{P}), and the specific impulse (I_{p}) of the propulsion system used are carried out. The increase in the ceiling^{Q} has the weakest effects but nevertheless leads to a constant decrease in the TLM. Increasing the ceiling^{P} gives similar results but the trend expresses a much more pronounced decline. Increasing values of I_{p} have by far the greatest impact on TLM. The most significant effects on infrastructure decisions result, once again, from_{p}because the sizes of ISRU facilities and depots follow an exponential decline synchronous with I_{p} Increasing. As these results indicate, improving propulsion systems could be essential to making ISCs a reality. Furthermore, the results also show that increased availability of ISRU locations significantly improves ISC efficiency.

Finally, the author provides concluding remarks. In this paper, a multistage stochastic MILP model is developed that integrates production-distribution and infrastructure allocation decisions in space in the presence of demand uncertainty. A layered network structure is considered in this model, composed of a PN and a SN, which are responsible for cargo and propellant supply, respectively. To validate the model, case studies were constructed based on real Earth-Mars transfer window data and flight trajectories obtained via NASA’s Trajectory Browser. The results show that: 1) a division of labor approach is often applied in which some spacecraft carry the bulk of the payload while others retrieve propellant from on-orbit depots to supply the entire fleet; 2) spacecraft launched together from Earth would use different transfer windows to stagger their arrival times, effectively allowing for smaller ISRU facilities and repositories in the Mars cluster; 3) advances in propulsion technology would have a significant impact on the optimal array design while advances in other spacecraft properties would have lesser effects; and 4) the availability of ISRU sites strongly influences the optimal solution, indicating that the technological capability to manufacture propellant on other celestial bodies is essential for an efficient ISC. Overall, the proposed model provides valuable insights and expands our understanding of SC planning from a space logistics perspective.

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