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\title{Resilience, Efficiency, and Ecology: Optimizing Hybrid Interplanetary Transport for the 2050 Moon Colony}
% \author{\small \href{http://www.latexstudio.net/}
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\date{\today}

\begin{document}


\begin{abstract}

To address the logistical requirements for establishing a 100,000-person Moon Colony by 2050, this study develops a comprehensive evaluation system spanning interplanetary transport decision-making, stochastic risk management, and multi-objective ecological balancing.

We first construct the \textbf{Universal Energy-Equivalent and Temporal Coordination Model (UETCM)}. By establishing \textbf{Universal Energy-Equivalent} measured in Joules as the primary cost proxy, the model bypasses the volatility of monetary valuations and enables a standardized comparison between chemical rockets and space elevator technologies. Grounded in the Tsiolkovsky equation and gravitational mechanics, the model delineates a linear mapping between energy expenditure and payload mass. To resolve the energy-time trade-off, we formulate a system burden function based on the \textbf{time opportunity cost}, which is calibrated at 504 PJ per year through weight equilibrium and geometric centroid methods. Simulation results demonstrate that hybrid transport can complete the 100-million-metric-ton mission in 100.7 years, significantly outperforming standalone elevator and rocket-only scenarios. Under the calibrated cost parameters, a balanced 139-year strategy is identified as the global deterministic optimum.

To manage real-world uncertainties, we extend the UETCM into a \textbf{Stochastic Risk and Robustness Framework} utilizing Conditional Value-at-Risk and 10,000 Monte Carlo iterations. This analysis identifies significant long-tail risks stemming from mechanical oscillations and system failures. To achieve a 95 percent completion confidence level, we recommend extending the operational timeline to a range of 155 to 160 years, adjusting the space elevator share to between 65 and 70 percent, and maintaining a 10 percent capacity reserve as a strategic buffer.

For the subsequent operational phase, we establish the \textbf{Life-Support Logistics and Stochastic Water Balance Model (LSL-SWBM)}. This model integrates multi-tier domestic demand with stochastic medical emergency needs approximated via normal distributions. Our findings indicate that annual water demand at the Luxury Standard is more than 35 times that of the Survival Standard. For the Comfort Standard, we recommend a hybrid scheme utilizing both the elevator and low-latitude rocket sites, which completes the initial filling in 4.57 days with a balanced energy footprint. 

Furthermore, environmental footprints are integrated into a \textbf{tri-objective optimization framework}. While rocket-only delivery incurs massive carbon dioxide and stratospheric water vapor emissions, the space elevator remains nearly carbon-neutral. Consequently, the 186-year standalone elevator scenario is proposed as the ultimate sustainable strategy, emitting only 846 million metric tons of carbon dioxide, which represents less than 6.5 percent of the rocket baseline.

Finally, we provide the Moon Colony Management Agency with actionable strategic recommendations. By proposing a phased hybrid transport plan with rebalancing triggers based on key performance indicators, this study offers a resilient pathway to lunar colonization that ensures logistical reliability while upholding ecological preservation.


\end{abstract}

\begin{keywords}
Universal Energy-Equivalent; Stochastic Risk Analysis ; Multi-objective Optimization
\end{keywords}
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\newpage

\section{Introduction}

\subsection{Background}
As human space exploration shifts from short-term landings to permanent settlement, Earth’s finite resources and fragile ecosystems motivate the development of scalable off-world habitats. Yet traditional chemical-propulsion rockets face persistent bottlenecks—limited payload ratios, high marginal launch costs, and non-negligible environmental impacts—making it difficult to sustain the mass logistics required for constructing and operating a 100,000-person Moon Colony. In this context, the Space Elevator System has been proposed as a transformative infrastructure \cite{penoyre2019spacelinepracticalspaceelevator} that can provide routine, energy-efficient access to space when paired with lunar transfer, but it also introduces new reliability and operational uncertainties. Therefore, the central challenge is to quantify and compare the energy (as a cost proxy), timeline, robustness under failures, and environmental footprint of rockets, elevators, and hybrid portfolios to support an actionable construction and sustainment strategy.


\begin{figure}[H] 
\centering
\includegraphics[width=12cm]{background.png}
\caption{background} 
\label{background}
\end{figure}



\subsection{Restatement of the Problem}

We develop a quantitative model to estimate the cost proxy (energy) and timeline of transporting materials to build a 100,000-person Moon Colony starting in 2050, and to support a defensible recommendation. Specifically, we:
\begin{itemize}
    \item Compare three delivery scenarios: elevators-only, rockets-only (selected sites), and a hybrid portfolio;
    \item Test robustness under non-ideal operations (e.g., tether dynamics, failures, downtime, and launch interruptions);
    \item Quantify one-year water resupply needs after habitation and map them to the delivery model;
    \item Evaluate and mitigate environmental impacts across scenarios;
    \item Deliver a one-page recommendation letter to the MCM Agency.
\end{itemize}

% \subsection{Literature Review}




\subsection{Our Work}

\begin{figure}[H]
    \centering
    \includegraphics[width=12cm]{Our Work.png}
    \caption{Our Work}
   
\end{figure}


% \begin{figure}[H] 
% \centering
% \includegraphics[width=12cm]{flow.png}
% \caption{the flow chart of our work} 
% \label{Site Distribution Map}
% \end{figure}
\section{Assumptions and Justifications}

To ensure model tractability while maintaining physical realism, we establish the following assumptions:

\begin{itemize}
    \item \textbf{Assumption 1: }Rocket payload capacity is assumed to be standardized at 100--150 metric tons per mission. \\
    \textbf{Justification: }Constant capacity avoids the volatility of payload metrics, facilitating a streamlined and efficient analysis of the total energy-cost required for construction.
    
    \item \textbf{Assumption 2: }Rocket propulsion systems are assumed to utilize liquid oxygen and methane (LOX/CH$_4$) as propellant. \\
    \textbf{Justification: }Current reusable heavy-lift standards, such as the SpaceX Starship, utilize LOX/CH$_4$ for its high energy density and minimal carbon buildup, representing future interplanetary transport norms.

    \item \textbf{Assumption 3: }Atmospheric drag and energy losses due to vehicle structural mass are neglected during rocket transit. \\
    \textbf{Justification: }These losses vary significantly per mission. Their exclusion reduces model complexity without compromising the macro-level accuracy of the energy expenditure framework.

    \item \textbf{Assumption 4: }Capital expenditures, including R\&D, manufacturing, and labor costs, are excluded from the assessment.\\
    \textbf{Justification: }This study focuses on the operational application phase of the logistics chain. Furthermore, such costs are often proprietary and difficult to estimate for future technologies.

    \item \textbf{Assumption 5: }Industrial water demand is assumed to be excluded from the water replenishment model.\\
    \textbf{Justification: }Lunar industrial scale data remains unavailable in current literature. Such demand is assumed to be integrated into the primary construction material transport modeled in Task 1.
\end{itemize}
\section{Notations}
\begin{center}
\begin{tabular}{clc}
\toprule[1pt]  % 顶部粗线
\textbf{Symbols} & \textbf{Description} & \textbf{Unit} \\
\midrule  % 中间线
$m_{fuel}$     & Total mass of the chemical fuel required for a mission & metric tons \\
$m_{p}$        & Mass of the effective payload being transported        & metric tons \\
$R$            & Mass ratio of initial mass to final mass               & - \\
$E$            & Total energy expenditure (Universal Energy-Equivalent) & Joules \\
$h$            & Length of the space elevator tether                    & km \\
$t$            & Total duration required for the construction timeline  & years  \\
$p$            & Probability of rocket launch failure                   & - \\
$W$            & Total water replenishment demand for the colony        & metric tons \\
\bottomrule[1pt]% 底部粗线
\end{tabular}
\end{center}
\section{Model I: Universal Energy-Equivalent and Temporal Coordination Model}
\subsection{Model Overview}


To evaluate Moon Colony logistics, we develop the Universal Energy-Equivalent and Temporal Coordination Model (UETCM). The model adopts Universal Energy-Equivalent (UEE), measured in Joules, as the primary metric to bypass volatile monetary valuations, facilitating a standardized thermodynamic comparison between the momentum transfer of chemical rockets and the gravitational potential gains of the Space Elevator System. 

The modeling process begins with establishing a linear mapping between payload mass and energy consumption via the Tsiolkovsky equation and gravitational gradients, integrated with physio-geographical constraints to identify the feasibility gap within the delivery window. To integrate the competing dimensions of timeline and cost, we define a Time Opportunity Cost parameter $\lambda$—derived through a consensus of four independent analytical methodologies—to scalarize the initial multi-dimensional problem into a unified objective function. This allows for the identification of a globally optimal operating point that minimizes the total system burden ($J$). 

Finally, UETCM is extended into a stochastic model incorporating Risk-Adjusted Optimization and Conditional Value-at-Risk (CVaR). Through 10,000 Monte Carlo simulations, we quantify not only the performance impact of perturbations such as tether swaying and system failures but also derive modified strategies with robust safety margins, providing a comprehensive framework that spans from physical mechanisms to resilient strategic decision-making.

\begin{figure}[H] 
\centering
\includegraphics[width=11cm]{Flow Chart of Model I.png}
\caption{Flow Chart of Model I} 
\end{figure}
\subsection{Energy-Equivalent as a Cost Proxy}

Monetary costs are ill-conditioned for an Earth--Moon logistics program spanning many decades due to inflation, technology shocks, and geopolitical uncertainty \cite{chatgpt}. We therefore use \emph{energy consumption} as a stable, physics-grounded proxy for marginal cost, enabling a consistent comparison between rockets and the Space Elevator System.

\subsubsection{Cost Convergence Analysis}
The operational cost per mission, $Cost(x)$, is modeled as the sum of hardware amortization, energy expenditure, and maintenance expenses:
\begin{equation}
    Cost(x) = \frac{C_{hardware}}{x} + C_{energy} + C_{operations}
\end{equation}
where $x$ is the number of reuse cycles. As reuse increases in a mature post-2050 transport regime, $\frac{C_{hardware}}{x}$ diminishes and operations become more automated, leaving $C_{energy}$ as the dominant term. Thus, differences in energy efficiency largely determine the marginal cost of sustained high-throughput transport.

\subsubsection{Empirical Industry Validation}

Evidence from mature terrestrial transport supports this modeling choice: in commercial aviation, fuel remains a persistent and material share (15--25\%) of operating expenses, indicating that energy is a primary driver once hardware is amortized and operations are standardized.

\begin{figure}[H]
    \centering
    \includegraphics[width=10cm]{fuel_share_trend.png}
    \caption{Trend of fuel cost as a percentage of total operating expenses}
    \label{fig:fuel_share_trend}
\end{figure}

In summary, we adopt Universal Energy-Equivalent as the primary metric to avoid long-horizon monetary ambiguity and to compare transport architectures on a common thermodynamic baseline (i.e., overcoming gravitational wells and imparting the required kinetic energy).

\subsection{Ideal Energy Cost Modeling}
\subsubsection{Rocket Momentum Dynamics}

For traditional rocket transport, energy demand is dominated by propellant required to deliver a payload to the Earth--Moon transfer trajectory. Neglecting air resistance, the Tsiolkovsky rocket equation gives the mass ratio $R$:
\[ R = \frac{m_0}{m_f} = e^{\Delta V / v_e} \]
where $v_e$ is the effective exhaust velocity and $\Delta V$ (e.g., for Trans-Lunar Injection) is treated as a mission constant. Accounting for structural mass via a structural coefficient $\alpha$ (structure-to-fuel ratio), the fuel mass scales linearly with payload:
\[ m_{fuel} = \underbrace{\frac{R-1}{1-\alpha(R-1)}}_{k} \cdot m_p \]
which defines the proportional coefficient $k$. Using the idealized chemical energy release proxy, the rocket energy is:
\[ E_{rocket} = \frac{1}{2} m_{fuel} \cdot v_e^2 = \frac{1}{2} k \cdot v_e^2 \cdot m_p \]
Therefore, under fixed mission and vehicle parameters, rocket energy scales linearly with payload mass. For completeness, we also express the total energy requirement as the sum of gravitational and transition terms:
\[ \Delta E_{total} = m_p \cdot \left[ \left( \frac{G M_E}{R_E} - \frac{G M_E}{d_{EM}} \right) - \frac{G M_M}{2 r_M} \right] \]
which is likewise proportional to $m_p$.

\subsubsection{Space Elevator Mechanics}

Compared with direct rockets, the Space Elevator System adds an Earth-surface-to-apex segment along the tether before the lunar-transfer rocket leg (already covered above). We model only the elevator segment energy change.


\begin{figure}[H] 
\centering
\includegraphics[width=10cm]{stage.png}
\caption{ Schematic of the Space Elevator Infrastructure and Earth-Moon Transport Phases} 
\end{figure}

From the Earth Port to the Apex Anchor, the energy change includes gravitational potential gain and rotation-related kinetic terms:
\[ E_{elevator} = m_p \left[ \left( \frac{GM_E}{R_E} - \frac{GM_E}{r_{0}} \right) + \frac{1}{2}\omega^2 r_{0}^2 \right] \]
where $r_0$ is the distance from the Apex Anchor to Earth's center and $\omega$ is Earth's angular velocity; thus $E_{elevator}$ also scales linearly with $m_p$.


\subsection{Ideal Timeline and Logistic Efficiency Modeling}

After resolving the physical energy efficiency, the model determines the construction timeline. The complexity arises from the fact that 100 million metric tons of material cannot be delivered instantaneously; the progress is constrained by the geographical distribution of launch sites, frequency limits, and the inherent capacities of the space elevator and rocket systems.

\subsubsection{Transport Progress Formulas}

To quantify the construction timeline, the model defines $M_{total} = 10^8$ metric tons as the total demand. Based on the three scenarios provided, the completion time $t$ depends on the annual cumulative capacity $C$:

\begin{itemize}
    \item Space elevator only (Scenario a): $t_{a} = \frac{M_{total}}{C_{e}}$
    \item Rockets only (Scenario b): $t_{b} = \frac{M_{total}}{C_{r}}$
    \item Hybrid transport (Scenario c): $t_{c} = \frac{M_{total}}{\lambda C_{e} + \mu C_{r}}$
\end{itemize}

where $C_e$ is the annual throughput of the space elevator system and $C_r$ is the integrated annual capacity of the rocket system. Shortening the timeline requires increasing annual capacity. By adjusting the hybrid weights $\lambda$ and $\mu$, we seek a dynamic balance between construction speed and energy cost.

\subsubsection{Spatial and Geographical Optimization}

The primary obstacles to construction progress are geographical constraints and frequency limits, which define the upper bound of $C_r$.
\begin{itemize}
    \item \textbf{Frequency constraints: } 
\end{itemize}

 To estimate the global launch capacity ceiling, we apply a Richards growth model to historical launch data. As shown in Figure \ref{fig:richards_fit}, the model predicts a theoretical saturation limit of $K = 4298$ annual launches. However, accounting for practical constraints such as maintenance cycles and meteorological windows, we adopt a conservative operational cadence of one launch per day per site (365 annually). This calibrated cap serves as the primary constraint for determining the minimum construction duration.

\begin{figure}[H] 
\centering
\includegraphics[width=8cm]{richards_curve_2010_K4298.png} 
\caption{Richards growth model fitting and projected capacity limit $K$} 
\label{fig:richards_fit}
\end{figure}

\begin{itemize}
   \item \textbf{Latitude-based efficiency correction: }
\end{itemize}

Earth's rotation provides rockets with a tangential initial velocity $v_{0i}$. The energy gain varies by latitude as follows:
\[ v_{0i} = v_{0E} \cdot \cos \theta_{i} \]
As illustrated in Figure \ref{latitudeeffects}, higher latitudes result in lower initial velocities, necessitating more chemical fuel. As latitude increases, the non-linear loss of tangential velocity causes the fuel-to-payload ratio to increase from 31.8 at equatorial sites to 34.0 at high-latitude sites.
\begin{figure}[H] 
\centering
\includegraphics[width=8cm]{latitude_effects.png}
\caption{Rotation Velocity and Loss vs Latitude} 
\label{latitudeeffects}
\end{figure}


Consequently, the model implements a low-latitude priority principle: low-latitude sites such as Kourou are prioritized to minimize global energy consumption when the number of launches has not reached the limit. This ensures that for a given timeline, the efficiency of material delivery is maximized.
\subsection{Unified Cost Optimization via Time Opportunity Scaling}
To select an actionable transport strategy, we scalarize the two objectives---energy consumption ($E$) and construction duration ($T$)---using a time opportunity cost parameter $\lambda$.

\subsubsection{Reformulation via Opportunity Cost}

We define the Total System Burden as:
\begin{equation}
    J(T)=E(T)+\lambda T, \qquad \frac{dE}{dT}\bigg|_{T=T^*}=-\lambda
\label{eq:scalarization_logic}
\end{equation}
Here $\lambda$ (PJ/year) is an energy-equivalent penalty per additional year, capturing the value of schedule acceleration (e.g., earlier utilization and reduced long-horizon overhead). The optimal duration $T^*$ is reached when the marginal energy saving rate equals $\lambda$.

\subsubsection{Calibration of Cost Parameter $\lambda$}


\begin{figure}[H]
\centering
\includegraphics[width=0.7\textwidth]{total_cost_curves.png}
\caption{Sensitivity analysis of optimal timeline $T^*$ versus time opportunity cost $\lambda$}
\label{fig:lambda_sensitivity}
\end{figure}

Rather than choosing $\lambda$ subjectively, we cross-check it using four independent heuristics (normalized trade-off scaling, marginal-slope matching, phase-boundary location, and feasible-region geometry). These estimates converge near $\lambda\approx 504$ PJ/year, so we use $\lambda=504$ PJ/year as the baseline and treat the spread as sensitivity.



\subsection{Results of Task 1}

Through numerical simulation of the 100 million metric ton material transport mission, this study elucidates the intrinsic correlation between construction progress and resource consumption under diverse technical constraints. The findings demonstrate that the optimal transport strategy is not a simple superposition of individual methods but a dynamic equilibrium based on the trade-off between physical energy efficiency and temporal costs.

\subsubsection{Comparison of Baseline Scenarios}

Based on the model outputs, Table \ref{tab:comparison} summarizes the core indicators for the three baseline scenarios. 
\begin{table}[H]
    \small
\centering
\caption{Multi-indicator Comparison of Earth-Moon Material Transport Scenarios}
\label{tab:comparison}
\begin{tabular}{lcccc}
\toprule
Scenario Type & Min Duration (a) & Total Energy (PJ) & Unit Energy (GJ/t) \\ \midrule
Elevator-Only (a) & 186.2 & 15,720 & 157.2 \\
Rocket-Only (b) & 219.2 & 50,609 & 506.1 \\
Hybrid (c) & 100.7 & 31,537 & 315.4 \\ \bottomrule
\end{tabular}
\end{table}
Integrating Table \ref{tab:comparison} and Figure \ref{Three Transport Scenarios Feasibility Comparison}, a clear duration-energy trade-off emerges. Rocket-only is limited to 219.2 years with peak energy demand, while elevator-only provides the lowest energy footprint  but requires 186.2 years. The hybrid scenario overcomes these bottlenecks, compressing the timeline to 100.7 years. It is the sole solution for the 100--186 year window and converges to elevator efficiency as duration increases. Ultimately, hybrid transport is indispensable for speed, while the space elevator represents the optimal long-term balance of time and energy.
\begin{figure}[H] 
\centering
\includegraphics[width=10cm]{three_scenarios_comparison.png}
\caption{Three Transport Scenarios Feasibility Comparison} 
\label{Three Transport Scenarios Feasibility Comparison}
\end{figure}
\subsubsection{Optimal Strategy Selection and Allocation}
To determine the optimal operating point, we analyze the behavior of the total cost function $J(T) = E(T) + \lambda T$. Based on the consensus value of $\lambda = 504$ PJ/year, three strategic configurations are evaluated within the feasible optimization space (100.7--186 years), as detailed in Table \ref{tab:strategies}.

\begin{table}[H]
    \small
\centering
\caption{Core Metrics of Typical Optimization Strategies for Scenario C}
\label{tab:strategies}
\begin{tabular}{lcccc}
\toprule
Strategy Type & Duration (a) & Total Energy (PJ) & Elevator Share & Energy Saving  \\ \midrule
Strategy A (Cost-Prioritized) & 186.0 & 15,826 & 99.9\% & 85.5\% \\
Strategy B (Time-Prioritized) & 101.0 & 58,391 & 54.2\% & 46.4\% \\
\rowcolor{recommendColor}
Strategy C (Balanced) & 139.0 & \textbf{38,729} & 74.6\% & \textbf{64.4\%} \\ \bottomrule
\end{tabular}
\end{table}

These strategies represent key operational configurations: Strategy A minimizes physical energy via full-load elevator operation; Strategy B achieves the shortest timeline through maximum energy input; and Strategy C represents the global minimum of the total cost function $J$, balancing construction efficiency with energy expenditure.

\begin{enumerate}
    \item \textbf{Marginal Energy Saving}: As shown in Figure \ref{fig:energy_time_tradeoff}, the marginal saving curve exhibits a step-wise diminishing trend. Between years 101 and 139, each additional year allocated to the timeline reduces energy demand by approximately 210 PJ.
    \item \textbf{Optimal Operating Point}: The 139-year duration is identified as the optimal point where the marginal reduction in energy expenditure equals the marginal time opportunity cost $\lambda$. At this point, the total system burden $J$ is minimized.
\end{enumerate}


\subsubsection{Sensitivity and Parameter Stability}

To assess the robustness of our model, we perform sensitivity analysis 
on five key parameters. Figure \ref{fig:sensitivity} summarizes the results.

\begin{figure}[H] 
\centering
\includegraphics[width=8cm]{tornado_chart.png}
\caption{Sensitivity Analysis} 
\label{fig:sensitivity}
\end{figure}

\begin{itemize}
    \item \textbf{Rocket Payload Capacity (100--150 tons):}
    As specified in the problem statement, we analyze the full payload range. 
Results show that higher capacity reduces minimum timeline from 111 to 93 
years, but has limited impact on the optimal hybrid solution at $\lambda=504$.
 \item \textbf{Elevator Capacity (±20\%):}
 Elevator capacity emerges as the most sensitive parameter. A 20\% reduction 
extends the elevator-only timeline from 186 to 233 years, fundamentally 
altering the hybrid strategy's feasible region.
\item \textbf{Launch Frequency:}
Doubling launch frequency from 1 to 2 per day reduces minimum timeline 
from 101 to 69 years, enabling more aggressive time-optimized strategies.

\item \textbf{Engine Technology:}
Engine Isp primarily affects energy consumption rather than timeline. 
LOX/Hydrogen (Isp=450s) reduces fuel ratio by 64\% compared to LOX/Methane, 
but practical considerations favor the latter.
\end{itemize}


\begin{table}[H]
    \small
\centering
\caption{Sensitivity Analysis Summary}
\begin{tabular}{lccc}
\toprule
Parameter & Range & $\Delta T^*$ & $\Delta E^*$ \\
\midrule
Payload Capacity & 100--150 t & $-7$ y ($-5.0\%$) & $+3246$ PJ ($+8.4\%$) \\
Elevator Capacity & $\pm 20\%$ & $+42$ y ($+30.2\%$) & $-47902$ PJ ($-123.5\%$) \\
Launch Frequency & 0.5--2/day & $-43$ y ($-30.9\%$) & $+20867$ PJ ($+53.8\%$) \\
Structural Coef. $\alpha$ & 0.06--0.14 & $+85$ y ($+61.4\%$) & $-28149$ PJ ($-72.6\%$) \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Stochastic Risk and Robustness Analysis}
Task 2 asks a fundamentally different question from Task 1: not merely how much performance degrades, but how our recommended strategy should change in response to system uncertainties. We achieve this through a three-stage methodology: quantifying perturbation impacts, identifying risk-adjusted decision boundaries, and finally deriving modified optimal strategies with robust safety margins.
\subsubsection{Stochastic Perturbation Modeling}

Given the unprecedented nature of the Space Elevator System, we acknowledge significant epistemic uncertainty in failure parameters. We adopt a conservative approach by establishing parameter ranges rather than point estimates:
\begin{table}[H]
\small
\centering
\caption{Perturbation Parameter Ranges and Sources}
\begin{tabular}{lccc}
\toprule
\textbf{Parameter} & \textbf{Baseline} & \textbf{Range} & \textbf{Basis} \\
\midrule
Tether swaying ($\Delta\theta$) & $0^\circ$ & $[-0.5^\circ, 0.5^\circ]$ & Coriolis-induced oscillations during release \\
Rocket failure rate & 0.75\% & [0.3\%, 1.5\%] & Falcon 9 historical \cite{skyrocket2026falcon9} + technology maturation \\
Elevator breaks/year & 2 & [1, 4] & IAA Space Elevator Assessment \cite{swan2013space}\\
Downtime per break & 14 days & [7, 30] days & Scaled from offshore platform repairs \\
Weather cancellation & 10\% & [5\%, 20\%] & Site-specific meteorological data \\
\bottomrule
\end{tabular}
\end{table}


\subsubsection{Risk-Adjusted Optimization}

Under uncertainty, the deterministic objective function must be extended to incorporate risk preferences. We reformulate the optimization as:
\begin{equation}
    \min_{T} \quad J_{robust}(T) = \mathbb{E}[E(T)] + \lambda \cdot \mathbb{E}[T] + \gamma \cdot \text{CVaR}_{\alpha}(T)
    \label{eq:robust_cost}
\end{equation}
where $\text{CVaR}_{\alpha}(T)$ denotes the Conditional Value-at-Risk at confidence level $\alpha$ (typically 95\%), representing the expected completion time in the worst $1-\alpha$ scenarios \cite{chatgpt}. The risk aversion coefficient $\gamma$ (PJ/year) quantifies the MCM Agency's willingness to pay for reduced schedule uncertainty.


\subsubsection{Monte Carlo Lifecycle Assessment}

Due to the highly non-linear interactions between these stochastic variables, we utilized a Monte Carlo algorithm to execute 10,000 simulations of the construction cycle. 


\begin{itemize}
    \item \textbf{Simulation Logic}: For each simulated day, the algorithm samples the system state (Normal/Failure/Weather). In the event of interference, payload distribution is adjusted and incomplete tasks are rescheduled.
    \item \textbf{Distribution Analysis}: As shown in Figure \ref{Completion Time Distribution Under Perturbations}, the results exhibit a significant right-skewed distribution. This implies that real-world timelines are prone to a long tail of cumulative delays—the probability of massive delays is low but the impact is profound, serving as a critical basis for risk assessment.
\end{itemize}

\begin{figure}[H] 
\centering
\includegraphics[width=12cm]{completion_time_distribution.png}
\caption{Completion Time Distribution Under Perturbations} 
\label{Completion Time Distribution Under Perturbations}
\end{figure}

\subsection{Results of Task 2}

Idealized models define the theoretical limits, but stochastic variables such as tether swaying, equipment failure, and weather interference cause performance to deviate. We evaluate system performance under disturbance based on 10,000 Monte Carlo iterations.

\subsubsection{Assessment of Performance Degradation}

Table \ref{tab:stochastic_comparison} compares the deterministic solutions with stochastic means, quantifying the costs of system perturbations.
\begin{table}[H]
  \small
  \centering
  \caption{Comparison of Core Metrics Under Perfect and Non-Perfect Conditions}
  \label{tab:stochastic_comparison}
  % 缩放至单栏宽度，保留1em内边距避免贴边（与示例格式一致）
  \resizebox{\dimexpr\linewidth-1em\relax}{!}{
    \begin{tabular}{lcccc}
      \toprule[1pt] % 加粗顶线（与示例一致）
      \rowcolor{white} % 表头底色为白（与示例一致）
      Strategy Type & Dimension & Perfect Condition & Non-Perfect (Mean $\pm$ Std) & Change Rate \\
      \midrule[0.6pt] % 中等粗细中间线（与示例一致）
      % Strategy A：设置基线浅灰底色
      \rowcolor{baselineColor}
      Strategy A & Time (a) & 186.2 & 202.3 $\pm$ 0.6 & +8.6\% \\
      \rowcolor{baselineColor}
      & Energy (PJ) & 15,720 & 15,738 $\pm$ 0 & +0.1\% \\
      % Strategy B：设置基线浅灰底色
      \rowcolor{baselineColor}
      Strategy B & Time (a) & 100.7 & 120.7 $\pm$ 0.4 & +19.9\% \\
      \rowcolor{baselineColor}
      & Energy (PJ) & 31,537 & 30,217 $\pm$ 30 & -4.2\% \\
      % Strategy C：设置推荐方案浅蓝底色（突出核心方案，与示例一致）
      \rowcolor{recommendColor}
      Strategy C & Time (a) & 139.0 & 155.3 $\pm$ 0.5 & +11.7\% \\
      \rowcolor{recommendColor}
      & Energy (PJ) & \textbf{38,729} & \textbf{38,285 $\pm$ 45} & -1.2\% \\
      \bottomrule[1pt] % 加粗底线（与示例一致）
    \end{tabular}
  }
\end{table}
\subsubsection{Resilient Strategic Adjustments}

The Monte Carlo results reveal that perturbations do not merely shift performance metrics---they fundamentally alter the risk-return profile of each strategy:

\textbf{(1) Optimal Timeline Adjustment}

Under perfect conditions, the balanced strategy yields $T^* = 139$ years. However, incorporating the 95th percentile delay, the effective planning horizon becomes:
\begin{equation}
    T^*_{adjusted} = T^* + \Delta T_{buffer} = 139 + 16.3 \times 1.15 \approx 158 \text{ years}
\end{equation}
where the 15\% margin accounts for tail-risk scenarios. Recommendation: MCM Agency should plan for a 155-160 year timeline to achieve 95\% confidence in completion.

\textbf{(2) Hybrid Ratio Adjustment}

Given that elevator breaks dominate schedule risk (correlation = 0.836), the optimal elevator share should be reduced to enhance redundancy:
\begin{equation}
    \text{Elevator share}_{adjusted} = 74.6\% - \Delta_{redundancy} \approx 65\%-70\%
\end{equation}
This increases rocket utilization as a ``backup channel,'' accepting a 5-8\% energy penalty in exchange for improved schedule resilience.

\textbf{(3) Strategic Reserve Capacity}

We recommend maintaining a 10\% reserve margin in annual transport capacity:
\begin{equation}
    C_{reserve} = 0.1 \times (\lambda C_e + \mu C_r)
\end{equation}
This reserve can absorb localized disruptions without triggering cascading delays.

\subsubsection{Evolution of Key Indicators}

Stochastic disturbances transform deterministic values into probability distributions, but the strategic ranking remains robust.

\begin{figure}[H] 
\centering
\includegraphics[width=10cm]{energy_distribution.png}
\caption{Energy Distribution} 
\label{Energy Distribution}
\end{figure}

\begin{itemize}
    \item \textbf{Energy Stability}: As illustrated in Figure \ref{Energy Distribution}, energy consumption follows a narrow-peak normal distribution with a standard deviation of less than 0.2\%. This confirms that tether swaying does not cause energy collapse. The slight mean shift in Strategies B and C is due to payload loss from rocket failures offsetting energy increments from swaying.
   
    \item \textbf{Temporal Right-Skewed Characteristics}: Median durations delay by 8.6\% to 19.9\%. However, the duration intervals for each strategy remain strictly separated, ensuring that the fundamental decision logic—Strategy A for cost and Strategy B for speed—holds true even under perturbation.
\end{itemize}



\section{Model II: Life-Support Logistics and Stochastic Water Balance Model}

\subsection{Model Overview}

As the Moon Colony transitions from the construction phase to the operational phase, the logistical focus shifts from structural materials to life-support supplies. We develop the Life-Support Logistics and Stochastic Water Balance Model (LSL-SWBM) to quantify the water security boundaries of the settlement during its first year of operation. This model accounts for multi-level demand functions based on psychological comfort and incorporates a normal approximation to address stochastic medical emergency needs. By mapping these water requirements onto the transport framework established in Model I, we evaluate the additional pressure exerted by different comfort factors on the Earth-Moon logistics chain.



\begin{figure}[H] 
\centering
\includegraphics[width=11cm]{Flow Chart of Model II.png}
\caption{Flow Chart of Model II} 
\label{Flow Chart of Model II}
\end{figure}



\subsection{Water Demand Architecture}

In an isolated lunar ecosystem, water consumption is primarily sustained by a recycling system \cite{LIU2021113}. However, logistical replenishment must compensate for system losses and sudden medical surges. This model assumes that lunar water use is restricted to domestic and medical emergency purposes, excluding industrial use, to define the core demand framework.

\subsubsection{Domestic Water Evolution}

Within the colony, domestic water consists of survival and hygiene components. The daily demand is formulated as:
\begin{equation}
W_{r} = N \times (w_{s} + 0.4\alpha)
\end{equation}
where $N$ is the population, $w_{s}$ represents the survival baseline, and $\alpha$ is the comfort factor. The following table summarizes the water standards across different demand tiers \cite{francisco2025nasa}\cite{whitmore2012nasa}:

\begin{table}[H]
    \small
\centering
\caption{Water Demand Standards Across Different Comfort Tiers}
\begin{tabular}{lccl}
\toprule
\textbf{Demand Tier} & \textbf{$\alpha$ Value} & \textbf{Daily Use per Capita} & \textbf{Description} \\ \midrule
Survival Standard & 1 & 2.9 L & Minimum survival threshold \\
Comfort Standard & 50 & 22.5 L & Moderate domestic comfort \\
Luxury Standard & 250 & 102.5 L & Equivalent to Earth-like usage \\ \bottomrule
\end{tabular}
\end{table}

\subsubsection{Medical Emergency Demand Modeling}

Given the large population , the daily number of patients $X$ (assuming a daily incidence rate $p=2\%$) follows a binomial distribution, which is accurately approximated by a normal distribution $X \sim N(Np, Np(1-p))$. To ensure medical safety under extreme conditions, the model adopts the peak demand at a 99\% confidence level as the daily reserve target:
\begin{equation}
W_{medical} = (E(X) + Z_{0.99} \cdot \sigma) \times 5 \text{ kg}
\end{equation}
This indicator remains stable across different domestic comfort scenarios, ensuring the robustness of the medical support system.

\subsection{Replenishment and Buffer Strategies}

Unlike construction materials, water is highly recyclable. Given the maturity of water recycling technology, the cumulative rate of daily demand after the initial transport is relatively slow. Since a daily delivery schedule would be prohibitively expensive, we adopt a monthly supply mode based on a dual-stage replenishment logic.

\subsubsection{Initial Month Filling}

During the first month, recycled water from the previous month is unavailable. Thus, the initial supply must satisfy two criteria: providing domestic water for 30 days and maximizing medical reserves for potential surges. The initial transport volume is defined as:
\begin{equation}
W_{initial} = (W_{r} + W_{mi}) \cdot T
\end{equation}
where $T=30$ days and $W_{mi}$ is the daily emergency medical supply at a 99\% confidence level. This strategy establishes the system’s circulating base while creating a 30-day emergency buffer.

\subsubsection{Monthly Routine Compensation}

In subsequent months, domestic demand is met through a combination of Earth-based replenishment and water recycling. Simultaneously, medical reserves need only cover the mean incidence rate due to the confidence buffer established initially. With a recycling efficiency $\eta$ (set at 0.9), the routine monthly supply model is:
\begin{equation}
W_{routine} = (W_{r}(1-\eta) + W_{m}) \cdot T
\end{equation}
This model precisely offsets recycling losses and daily medical consumption to maintain a dynamic water balance.
\subsection{Results of Task 3}

To evaluate the operational sustainability of the Moon Colony, we perform a quantitative decomposition of the water replenishment mission across temporal and energetic dimensions. By assuming ideal conditions and excluding the perturbations analyzed in Task 2, this approach isolates the specific influence of varying living standards to facilitate a definitive baseline comparison.
\subsubsection{Demand Scaling Across Comfort Tiers}

Minute variations in the comfort factor $\alpha$ trigger significant shifts in logistical scale. The water demand metrics for the three simulated tiers are summarized in Table \ref{tab:water_demand_stats}.

\begin{table}[H]
    \small
    \centering
    \caption{Water Demand Metrics under Different Comfort Scenarios}
    \label{tab:water_demand_stats}
    \begin{tabular}{lccccc}
        \toprule
        \textbf{Demand Tier} & \textbf{$\alpha$} & \makecell{\textbf{Water Inventory} \\ \textbf{(metric tons)}} & \makecell{\textbf{Daily Rep.} \\ \textbf{(metric tons)}} & \makecell{\textbf{Initial Trans.} \\ \textbf{(metric tons)}} & \makecell{\textbf{Annual Total} \\ \textbf{(metric tons)}} \\
        \midrule
        Survival Standard & 1   & 290    & 29.1    & 1,163  & 10,622  \\
        Comfort Standard  & 50  & 2,250  & 225.1   & 9,003  & 82,162  \\
        Luxury Standard   & 250 & 10,250 & 1,025.1 & 41,003 & 374,162 \\
        \bottomrule
    \end{tabular}
\end{table}

Table \ref{tab:water_demand_stats} illustrates a multiplier effect: as the standard shifts from Survival to Luxury, the annual requirement surges by over 35-fold. Supported by high-efficiency recycling, the daily replenishment required from Earth remains a small fraction of the total demand, emphasizing the value of closed-loop systems.

\subsubsection{Integrated Trade-off Analysis of Transport Schemes}

To satisfy these requirements, we mapped the demand indicators onto the transport framework of Model I. Four distinct schemes were evaluated based on their baseline capacities, as defined in Table \ref{tab:transport_schemes_def}.

\begin{table}[H]
    \small
    \centering
    \caption{Definition and Baseline Capacity of Transport Schemes}
    \label{tab:transport_schemes_def}
    \begin{tabular}{clcc}
        \toprule
        \textbf{Scheme} & \textbf{Description} & \textbf{Daily Capacity (t/d)} & \textbf{Specific Energy (GJ/t)} \\
        \midrule
        1 & Space Elevator Only (3 units)  & 1,471 & 157.2 \\
        2 & Rocket-Only (10 sites)         & 1,250 & 506.1 \\
        3 & Hybrid (Elevator + 10 sites)   & 2,721 & 317.5 \\
        4 & Hybrid (Elevator + Low-lat.)   & 1,971 & 243.5 \\
        \bottomrule
    \end{tabular}
\end{table}

By consolidating the simulation outputs for all scenarios, Table \ref{tab:consolidated_performance} presents a comprehensive comparison of scheme performance across $\alpha$ tiers.

\begin{table}[H]
    \small
    \centering
    \caption{Consolidated Performance Metrics of Transport Schemes across Comfort Tiers}
    \label{tab:consolidated_performance}
    \begin{tabular}{lccccc}
        \toprule
        \textbf{Scenario} & \textbf{Scheme} & \textbf{Initial Days} & \textbf{Initial En. (TJ)} & \textbf{Monthly Days} & \textbf{Annual En. (PJ)} \\
        \midrule
        \multirow{4}{*}{Survival ($\alpha=1$)} 
        & Scheme 1 & 0.79 & 182.8 & 0.59 & 1.67 \\
        & Scheme 2 & 0.93 & 588.6 & 0.69 & 5.38 \\
        & Scheme 3 & 0.43 & 369.2 & 0.32 & 3.37 \\
        & Scheme 4 & 0.59 & 283.1 & 0.44 & 2.59 \\
        \midrule
        \rowcolor{recommendColor} \multirow{4}{*}{Comfort ($\alpha=50$)} 
        & Scheme 1 & 6.12 & 1,415.3 & 4.59 & 12.92 \\
        & Scheme 2 & 7.20 & 4,556.3 & 5.40 & 41.58 \\
        & Scheme 3 & 3.31 & 2,858.1 & 2.48 & 26.08 \\
        & Scheme 4 & 4.57 & 2,191.9 & 3.43 & 20.00 \\
        \midrule
        \multirow{4}{*}{Luxury ($\alpha=250$)} 
        & Scheme 1 & 27.87 & 6,445.7 & 20.90 & 58.82 \\
        & Scheme 2 & 32.80 & 20,751.2 & 24.60 & 189.36 \\
        & Scheme 3 & 15.07 & 13,016.9 & 11.30 & 118.78 \\
        & Scheme 4 & 20.80 & 9,982.5 & 15.60 & 91.09 \\
        \bottomrule
    \end{tabular}
\end{table}
While water transport is highly feasible for survival, the Comfort Scenario ($\alpha=50$) serves as the planning representative where efficiency-timeline trade-offs intensify. In this tier, Scheme 1 defines the efficiency baseline, Scheme 3 maximizes throughput to minimize delivery time, and Scheme 4 offers the optimal operational balance. Under the Luxury Scenario ($\alpha=250$), energy demand for rocket transport peaks at 189.36 PJ. Crucially, even this extreme demand represents only 0.37 percent of the total construction energy, confirming that post-2050 logistical constraints stem from capacity allocation rather than aggregate energy availability.

\subsection{Sensitivity Analysis for Water Supply Model}

To assess the robustness of Model II and identify the dominant levers in water logistics, we conduct a sensitivity analysis over five variables: recycling efficiency ($\eta$), comfort factor ($\alpha$), population ($N$), medical demand parameters, and buffer duration.

\subsubsection{Parameter Sensitivity Ranking via Tornado Analysis}

We systematically varied each parameter within its feasible range while holding others at baseline values ($\alpha=50$, $\eta=90\%$, $N=100,000$, sickness rate $p=2\%$, buffer $=30$ days). The resulting impacts on annual water supply are visualized through a tornado diagram (Figure \ref{fig:tornado}).

\begin{figure}[h]
    \centering
    \includegraphics[width=12cm]{tornado_analysis.png}
    \caption{Tornado Diagram}
    \label{fig:tornado}
\end{figure}
\begin{enumerate}
    \item \textbf{Recycling efficiency ($\eta$)} dominates: modest degradations cause large increases in annual resupply because losses scale with $(1-\eta)$.
    \item \textbf{Secondary drivers}: comfort ($\alpha$) is the next lever (tightening provides headroom), population ($N$) is near-linear, while medical/buffer settings have limited effect on annual totals but affect initial fill and resilience.
\end{enumerate}

\subsubsection{Quantitative Sensitivity Coefficients}

Table \ref{tab:sensitivity_coefficients} presents the normalized sensitivity coefficients, defined as the percentage change in output per percentage change in input parameter:

\begin{table}[H]
    \small
    \centering
    \caption{Sensitivity Coefficients for Key Model Parameters}
    \label{tab:sensitivity_coefficients}
    \begin{tabular}{lccccc}
        \toprule
        \textbf{Parameter} & \textbf{Baseline} & \textbf{Range} & \textbf{$\Delta$ Annual Supply} & \textbf{Sensitivity} & \textbf{Rank} \\
        \midrule
        Recycling Rate ($\eta$) & 90\% & 80\%--95\% & $-$52\% to +96\% & $-$9.6 & 1 \\
        Comfort Factor ($\alpha$) & 50 & 25--100 & $-$71\% to +17\% & +0.88 & 2 \\
        Population ($N$) & 100k & 80k--120k & $\pm$20\% & +1.0 & 3 \\
        Sickness Rate ($p$) & 2\% & 1\%--4\% & $-$0.3\% to +0.6\% & +0.30 & 4 \\
        Buffer Days & 30 & 15--45 & $\pm$0\%$^*$ & 0 & 5 \\
        \bottomrule
    \end{tabular}
    \begin{tablenotes}
        \small
        \item $^*$ Buffer days affect only initial transport, not annual replenishment.
    \end{tablenotes}
\end{table}

The sensitivity coefficient for $\eta$ reaches $-9.6$: a 1\% decrease in recycling efficiency increases annual resupply by about 9.6\%, making recycling integrity the primary control variable.

\subsubsection{Worst-Case Stress Testing}

Beyond univariate analysis, we evaluate system performance under compound adverse conditions through Monte Carlo-informed scenario construction. Table \ref{tab:worst_case} summarizes nine representative scenarios spanning from optimal to catastrophic conditions:

\begin{table}[H]
    \small
    \centering
    \caption{Worst-Case Scenario Analysis: Water Supply System Stress Test}
    \label{tab:worst_case}
    \resizebox{\textwidth}{!}{
    \begin{tabular}{lcccccccc}
        \toprule
        \textbf{Scenario} & \textbf{$\alpha$} & \textbf{$\eta$} & \textbf{$N$ (k)} & \textbf{$p$} & \textbf{Buffer} & \textbf{Annual (kt)} & \textbf{Capacity (\%)} & \textbf{Feasible} \\
        \midrule
        Survival Mode & 1 & 90\% & 100 & 2\% & 30 & 14.2 & 2.7\% & \checkmark \\
        \rowcolor{recommendColor}
        Comfort Mode (Baseline) & 50 & 90\% & 100 & 2\% & 30 & 85.8 & 16.0\% & \checkmark \\
        Luxury Mode & 250 & 90\% & 100 & 2\% & 30 & 377.8 & 70.3\% & \checkmark \\
        Recycle Degradation & 50 & 80\% & 100 & 2\% & 30 & 167.9 & 31.3\% & \checkmark \\
        Epidemic Outbreak & 50 & 90\% & 100 & 5\% & 45 & 100.4 & 18.7\% & \checkmark \\
        Population Surge & 50 & 90\% & 120 & 2\% & 30 & 102.9 & 19.2\% & \checkmark \\
        Moderate Stress & 100 & 85\% & 110 & 3\% & 40 & 264.4 & 49.2\% & \checkmark \\
        \rowcolor{baselineColor}
        \textbf{Worst Case} & \textbf{250} & \textbf{80\%} & \textbf{120} & \textbf{5\%} & \textbf{45} & \textbf{919.8} & \textbf{171.3\%} & \ding{55} \\
        \bottomrule
    \end{tabular}
    }
\end{table}

The stress test shows that feasibility is controlled mainly by the joint condition \textbf{$\eta<82\%$ with $\alpha>200$}, under which annual water demand can exceed elevator capacity (worst case: \textbf{171.3\%}), whereas moderate stress remains well within capacity (49.2\%).

\subsection{Conclusion and Strategic Insights}

\begin{enumerate}
    \item \textbf{Trade-off between Comfort and Capacity}: Quantitative analysis confirms that quality of life acts as a logistical amplifier. At the Luxury tier, the annual water replenishment occupies 69.68\% of the theoretical annual capacity of the space elevator system. Maintaining high living standards significantly constricts the transport window for other critical infrastructure and scientific materials.
    
    \item \textbf{Tiered Strategy Recommendations}:
    \begin{itemize}
        \item \textbf{Initial Operations}: We recommend Survival tier combined with Scheme 1. This minimizes energy consumption while ensuring the survival threshold for 100,000 residents, reserving hybrid capacity for urgent infrastructure tasks.
        \item \textbf{Mature Operations}: We recommend Comfort tier combined with Scheme 4. This provides a balance between speed and cost, completing the initial filling in 4.57 days with a reasonable energy footprint.
    \end{itemize}
    
    \item \textbf{System Resilience}: Thanks to the 90\% efficient recycling system and the 30-day emergency buffer, the water logistics chain exhibits high resilience. Even during a full month of transport disruption, the survival of the colony remains uncompromised, allowing for logistical maintenance and error recovery.
\end{enumerate}


\section{Model Extension}

Following the assessment of construction timelines and life-support logistics, this chapter extends the model to incorporate the environmental footprint as a critical optimization boundary. By quantifying the ecological externalities of various transport scenarios, we refine previous strategic selections to promote symbiosis between Earth's conservation and lunar sustainability.

\subsection{Quantification of Baseline Environmental Impacts}

We extended the environmental sub-module based on the energy conversion logic of Model I. For liquid oxygen/methane ($\text{LOX/CH}_4$) propulsion systems, fuel consumption is mapped to emission mass via the combustion equation:
\begin{equation}
    \text{CH}_{4} + 2\text{O}_{2} \longrightarrow \text{CO}_{2} + 2\text{H}_{2}\text{O}
\end{equation}
A life-cycle assessment (LCA), including fuel production and power generation, was conducted for four baseline scenarios, as summarized in Table \ref{tab:environmental_metrics}.

\begin{table}[H]
\small
\centering
\caption{Environmental Metrics Across Four Baseline Scenarios}
\label{tab:environmental_metrics}
\begin{tabular}{lccccc}
\toprule
\textbf{Scenario} & \textbf{Timeline} & \textbf{Total Fuel} & \textbf{Total $\text{CO}_2$} & \textbf{Stratospheric $\text{H}_2\text{O}$} & \textbf{Intensity} \\
 & \textbf{(a)} & \textbf{(Mt)} & \textbf{(Mt)} & \textbf{(Mt)} & \textbf{(Mt/yr)} \\
\midrule
Rocket-Only & 219 & 9,596 & 12,968 & 1,919 & 59.2 \\
Elevator-Only & 186 & 69 & 846 & 0 & 4.5 \\
Hybrid (Time-Prioritized) & 101 & 4,445 & 6,414 & 882 & 63.7 \\
\rowcolor{recommendColor} Hybrid (Balanced) & 139 & 2,438 & 3,856 & 477 & 27.7 \\
\bottomrule
\end{tabular}
\end{table}

\begin{figure}[H]
    \centering
    \includegraphics[width=8cm]{environmental_comparison.png}
    \caption{Multi-dimensional environmental impact analysis}
    \label{fig:environmental_analysis}
\end{figure}

The analysis in Figure \ref{fig:environmental_analysis} clearly distinguishes the ecological costs:
\begin{itemize}
    \item The \textbf{Rocket-Only scenario} incurs the highest $\text{CO}_2$ emissions and stratospheric $\text{H}_2\text{O}$ injection, posing the greatest risk to global climate and the ozone layer.
    \item The \textbf{Elevator-Only scenario} completely circumvents stratospheric pollution, with total $\text{CO}_2$ emissions being only $6.5\%$ of the rocket scenario.
    \item The \textbf{Balanced Hybrid scenario} achieves a $40\%$ reduction in $\text{CO}_2$ and a $46\%$ reduction in stratospheric $\text{H}_2\text{O}$ compared to the Time-Prioritized plan, successfully equilibrating construction efficiency with environmental load.
\end{itemize}

\subsection{Earth Port Assessment}

While the space elevator facilitates zero-emission transit, its Earth Port—a permanent maritime structure in equatorial waters—may impact local marine ecosystems. Drawing on research from the International Space Elevator Consortium (ISEC) \cite{isec2021greenroad}, we analogized the Earth Port to offshore oil rigs, the most mature existing maritime facilities.

\begin{table}[H]
\small
\centering
\caption{Comparative Impact: Earth Port vs. Offshore Oil Platforms}
\label{tab:earth_port_impact}
\begin{tabular}{@{}l@{\hspace{0.8em}}l@{\hspace{0.8em}}l@{\hspace{0.8em}}c@{}}
\toprule
\textbf{Impact Type} & \textbf{Oil Platform} & \textbf{Earth Port} & \textbf{Range} \\
\midrule
Physical Damage & Benthic disruption (anchoring) & Anchor zone-only & Localized \\
Chemical Discharge & Drilling fluids/water & No discharge & Negligible \\
Interference & Light/acoustic disturbance & Navigation light + low-freq noise & Localized \\
Habitat Alteration & Artificial reef effect & Hard substrate (sessile colonies) & Positive \\
Accident Risk & Hydrocarbon spills & No leakage risk & Negligible \\
\bottomrule
\end{tabular}
\end{table}

The analogy confirms that the Earth Port's environmental footprint is significantly lower than that of existing oil platforms. Compared to the widespread atmospheric damage caused by rockets, the localized impacts of the space elevator are nearly negligible, reinforcing its ecological superiority.
\subsection{Tri-objective Decision Framework}
We expand the bi-objective optimization to a tri-objective framework ($J_{total}$) by integrating environmental externalities as primary constraints:
\begin{equation}
    \min J_{total} = \alpha \cdot \text{Time} + \beta \cdot \text{Energy} + \gamma \cdot \text{Emission}_{\text{CO}_2} + \delta \cdot \text{Emission}_{\text{H}_2\text{O, strat}}
\end{equation}

\begin{figure}[H] 
\centering
\includegraphics[width=0.6\textwidth]{marginal_benefit.png}
\caption{Energy and $\text{CO}_2$ Reduction Potential Analysis.} 
\label{fig:reduction_analysis}
\end{figure}

Simulation analysis reveals a significant coupling between environmental benefits and energy efficiency. Notably, the system reaches its peak marginal environmental benefit at approximately the 186-year mark, representing an "Efficiency Cliff". Beyond this threshold, any further extension of the construction timeline yields diminishing returns in environmental preservation.

\subsection{Result of Task 4}


Conclusively, the 186-year standalone space elevator scenario is proposed as the optimal strategy for lunar colonization. This decision integrates construction duration, energy efficiency, environmental impact, and system reliability.

\begin{figure}[htbp]
    \centering
    \includegraphics[width=0.6\textwidth]{radar_chart.png}
    \caption{Multi-dimensional Environmental Impact (Radar Chart)}
    \label{fig:radar_analysis}
\end{figure}

\begin{itemize}
    \item The rocket-only scenario imposes the most severe environmental risks, with carbon emissions and stratospheric water vapor injection threatening the global climate and ozone stability. 
    \item Furthermore, although the hybrid scenario offers temporal advantages, it incurs substantial emissions and introduces heightened system complexity and failure risks. 
    \item Consequently, the 186-year elevator-only plan represents the most sustainable choice, as it completely eliminates rocket-based pollution by trading temporal efficiency for ecological preservation.

\end{itemize}

\section{Strengths and Weaknesses}

\subsection{Strengths}
\begin{itemize}
  \item \textbf{Comparable cost proxy across technologies.}
  Using Universal Energy-Equivalent (UEE, Joules) avoids long-horizon monetary uncertainty and enables a thermodynamically consistent comparison between chemical rockets and space-elevator lifting.
  \item \textbf{Interpretable multi-objective decision logic.}
  The time-opportunity parameter $\lambda$ converts the energy--time trade-off into a single objective with a clear optimality condition ($dE/dT=-\lambda$), making stakeholder preferences explicit and auditable.
  \item \textbf{Risk-aware recommendation.}
  CVaR-style risk adjustment and Monte Carlo simulations translate uncertainties (failures, downtime, weather) into a confidence-bounded schedule and actionable operational policies (strategy band, reserve margin, trigger actions).
  \item \textbf{Actionable sensitivity insights.}
  Sensitivity analysis identifies dominant levers (e.g., elevator capacity and recycling efficiency), directly informing engineering priorities and governance constraints.
\end{itemize}

\subsection{Weaknesses and Possible Improvement}
\begin{itemize}
  \item \textbf{Parameter uncertainty and long-horizon extrapolation.}
  Key inputs (elevator throughput, failure/downtime rates, future launch cadence) are necessarily assumed or extrapolated.

  \textit{Improvement:} expand scenario sets and report threshold/robust regions where the recommendation remains unchanged.
  \item \textbf{UEE is not a complete economic cost.}
  Energy captures a physical lower bound but does not fully represent CAPEX, labor, insurance, and supply-chain constraints.

  \textit{Improvement:} add a parallel monetary sensitivity layer (order-of-magnitude CAPEX/OPEX ranges) to validate ranking stability.
  
\end{itemize}


% \section{Conclusion}




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\section*{Report on Use of AI}

\begin{enumerate}
\item OpenAI ChatGPT (accessed 2026--02--01)
\begin{description}[leftmargin=!,labelwidth=2.6cm]
\item[Purpose:] We used a generative AI assistant only for (i) outlining and language polishing, (ii) checking for clarity/consistency of model descriptions, and (iii) LaTeX presentation suggestions (tables/figures). We did \textbf{not} use AI to generate final numerical results, figures, or citations; all computations and plots were produced by the team using our own scripts and datasets, and all claims were verified by the team before inclusion.

\item[Where AI was potentially useful in the paper:] background framing (Introduction), simplifying and explaining assumptions (Assumptions), verifying the expression/units of key equations (Model I \& II), drafting/condensing the robustness narrative (stochastic/CVaR), and improving readability of the recommendation/strengths-weaknesses sections.

\item[Query 1:] ``Rewrite the following paragraph in concise academic English, keep the meaning, do not add new claims: \emph{Monetary costs are ill-conditioned for an Earth--Moon logistics program spanning many decades due to inflation, technology shocks, and geopolitical uncertainty. We therefore use energy consumption as a stable proxy for marginal cost, enabling a consistent comparison between rockets and the Space Elevator System.}'' 
\item[Output 1:] ``Because monetary costs over multi-decade programs are highly uncertain (inflation, technology shocks, and geopolitics), we adopt energy consumption as a physics-grounded proxy for marginal cost, enabling consistent comparison between rockets and the Space Elevator System.''

\item[Query 2:] ``Give a short, publication-style definition of CVaR at confidence level \( \alpha \) and one sentence on how it is used in schedule-risk planning.'' 
\item[Output 2:] ``Conditional Value-at-Risk (CVaR\(_\alpha\)) is the expected value of a random variable conditional on outcomes in the worst \(1-\alpha\) tail of its distribution. In schedule-risk planning, CVaR\(_\alpha\) summarizes ``expected worst-case'' delays beyond the \( \alpha \)-quantile and can be penalized to obtain risk-averse plans.''

\item[Query 3:] ``Provide a minimal LaTeX example (xcolor + booktabs) to highlight one recommended row in light blue and keep the rest plain, suitable for an MCM paper table.'' 
\item[Output 3:] ``Use \verb|\usepackage[table]{xcolor}| and define a color, then apply \verb|\rowcolor{...}| to the recommended row. Example:
\begin{verbatim}
\definecolor{recommendColor}{RGB}{217,234,249}
\begin{tabular}{lccc}
\toprule
... \\
\midrule
Baseline & ... \\
\rowcolor{recommendColor} Recommended & ... \\
\bottomrule
\end{tabular}
\end{verbatim}''
\end{description}

\item Cursor IDE Autocomplete (Free plan, accessed 2026--02--01)
\begin{description}[leftmargin=!,labelwidth=2.6cm]
\item[Purpose:] Auto-completions for code used in preparing our models.
\end{description}
\end{enumerate}

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\end{document}
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