Explore the Six 2026 MCM/ICM Modeling Challenges and How to Tackle Them

Problem A: Smartphone Battery Discharge Modeling

Develop a continuous‑time mathematical model for the state‑of‑charge (SOC) of a lithium‑ion smartphone battery and use it to predict the time until depletion under varying usage patterns.

Task Requirements

Continuous‑time model : Formulate differential equations for SOC. A basic formulation is

dSOC/dt = -\frac{P_{total}(t)}{C_{bat}}\quad\text{with}\quad P_{total}=P_{screen}+P_{cpu}+P_{net}+P_{gps}+P_{temp}

where C_{bat} is the usable capacity (Ah) and each power term is a function of measurable variables (brightness, CPU load, data rate, GPS usage, ambient temperature). Extend the model stepwise by adding terms for screen brightness, processor utilization, network activity, GPS, and temperature‑dependent efficiency.

Depletion‑time prediction : Solve the SOC ODE analytically for simple constant‑power scenarios or numerically (e.g., Runge‑Kutta) for time‑varying profiles to obtain the depletion time t_{dep} such that SOC(t_{dep})=0.

Sensitivity analysis : Vary parameters (e.g., battery internal resistance, temperature coefficient) and usage patterns to quantify their impact on t_{dep}. Use Monte‑Carlo sampling or one‑factor‑at‑a‑time analysis.

Model validation : Estimate parameters from empirical power‑draw datasets (e.g., Android BatteryStats) using least‑squares or Bayesian inference and compare predicted depletion times with measured values.

Solution Approach

Start from the physics of lithium‑ion cells (open‑circuit voltage, internal resistance). Treat total power draw as the sum of independent components, each modeled as a linear or nonlinear function of its driver variable. Calibrate the model with real‑world measurements, validate by computing root‑mean‑square error (RMSE) between predicted and observed SOC trajectories, and iterate to improve fidelity.

Problem B: Space‑Elevator Logistics for Lunar Colonization

Model the cost and schedule required to transport 100 million tonnes of material from Earth to the Moon using three alternative logistics chains:

Three independent space‑elevator ports delivering payloads to low‑Earth orbit (LEO) and then launching to lunar orbit.

Conventional chemical‑rocket launches from terrestrial launch sites.

A hybrid chain that combines elevator‑to‑LEO with rockets for the lunar transfer.

Task Requirements

Cost‑time model : Define

Cost = \sum_{i}\frac{m_i}{\eta_i}\,c_i,
\quad Time = \max_i\left(\frac{m_i}{\tau_i}\right)

where m_i is mass handled by mode i, \eta_i its efficiency, c_i unit cost, and \tau_i throughput (tonnes/day). Include launch‑vehicle specific impulse, elevator power consumption, and infrastructure amortization.

Failure impact : Introduce failure probabilities p_i for each subsystem (tether oscillation, rocket abort). Model expected additional cost and delay as ΔCost = Cost\times p_i/(1-p_i) and similarly for time.

Lunar water demand : Estimate water consumption per capita for a lunar settlement (e.g., 3 L/day) and compute the extra mass and cost to deliver the required water reserves.

Environmental impact : Quantify CO₂ equivalent emissions for each mode using emission factors (kg CO₂ per tonne‑km) and incorporate them into a multi‑objective optimization.

Solution Approach

Construct deterministic cost‑time equations for each logistics chain, then perform comparative analysis using Pareto fronts. Conduct sensitivity analysis on key parameters (elevator throughput, rocket specific impulse, failure rates) to assess robustness.

Problem C: Modeling Voting Dynamics in a Celebrity Dance Show

Estimate weekly fan votes for each contestant, compare two voting systems (ranking vs. percentage), and evaluate how contestant attributes affect outcomes.

Task Requirements

Fan‑vote estimation : Build a probabilistic model such as

V_{i,t}=\frac{e^{\beta_0+\beta_1\,age_i+\beta_2\,industry_i+\dots}}{\sum_j e^{\beta_0+\beta_1\,age_j+\dots}}

where V_{i,t} is the vote share of contestant i in week t. Calibrate β coefficients using historical vote totals.

Voting‑method comparison : Simulate elimination outcomes under (a) ranking‑based allocation (top‑k receive equal votes) and (b) percentage‑based allocation (votes proportional to judges' scores). Compute metrics such as elimination accuracy and fairness index.

Attribute impact : Perform regression or ANOVA to quantify the effect of age, professional background, and prior dance experience on both judges' scores and fan votes.

Solution Approach

Use maximum‑likelihood estimation to fit the vote‑share model, then run Monte‑Carlo simulations for each voting rule. Compare simulated elimination sequences with actual show data to assess model fidelity and propose a hybrid voting scheme that balances expert and public input.

Problem D: Dynamic Decision Modeling for Sports Team Management

Design a multi‑period optimization model that helps team owners and general managers adjust roster, pricing, and investment decisions in response to performance metrics and economic conditions.

Task Requirements

Dynamic decision framework : Formulate a stochastic dynamic program

max_{x_t}\; \sum_{t=1}^{T}\beta^{t-1}\,\Pi_t(x_t,\xi_t)
\text{s.t.}\; x_{t+1}=f(x_t, a_t, \xi_t)

where \Pi_t is profit in season t, x_t the state vector (roster, ticket price, arena capacity), a_t decisions, and \xi_t random shocks (player injuries, market trends).

Player recruitment strategy : Include binary variables for signing each candidate, with constraints on salary cap and roster size. Evaluate marginal profit contribution via shadow prices.

League rule changes : Model scenario trees for possible expansions or rule modifications (e.g., salary‑cap adjustments) and compute optimal policy under each scenario.

Commercial decision analysis : Incorporate ticket‑pricing elasticity, arena‑investment depreciation, and sponsorship revenue into the profit function.

Solution Approach

Implement the model in a mixed‑integer linear programming (MILP) solver, use scenario analysis for uncertainty, and perform sensitivity analysis on key parameters (attendance elasticity, salary‑cap limits). The optimal policy provides season‑by‑season recommendations for roster composition, pricing, and capital projects.

Problem E: Passive Solar Shading for Campus Buildings

Design passive solar shading systems for two university buildings to improve heating and cooling performance while accounting for projected climate change.

Task Requirements

Sungrove redesign : Create a shading configuration for the North Academic Hall that minimizes annual cooling load while preserving winter solar gain. Model solar heat gain Q_{sol}=I\,A_{win}\,SHGC\,\cos\theta where I is irradiance, A_{win} window area, SHGC solar‑heat‑gain coefficient, and \theta incidence angle. Optimize over shading depth and overhang angle.

Borealis extension : Adapt the design to a building with higher latitude and greater winter heating demand. Emphasize low‑angle winter overhangs and high‑angle summer louvers.

Solar heat‑gain prediction : Use a simplified thermal model

C_{b}\frac{dT_{in}}{dt}=Q_{sol}+Q_{int}-U\,A_{env}(T_{in}-T_{out})

where C_b is building thermal mass, U overall heat‑transfer coefficient, and Q_{int} internal gains. Run the model for typical meteorological year (TMY) data to evaluate seasonal performance.

Solution Approach

Develop a parametric geometry model of overhangs and louvers, couple it with the thermal balance equation, and perform a parametric sweep (e.g., using Python or MATLAB) to identify configurations that minimize total energy use under future climate scenarios (e.g., IPCC RCP4.5).

Problem F: Future of Generative AI (Gen‑AI)

Analyze how generative AI will transform three occupational domains—one STEM, one technical‑trade, and one artistic profession—and propose curriculum adjustments for higher‑education institutions.

Task Requirements

Occupation selection : Choose representative jobs (e.g., data‑science engineer, CNC machinist, visual‑effects artist) and collect labor‑market indicators (employment growth, skill demand).

Data‑driven impact model : Build a regression or scenario model such as

Impact = \alpha_0 + \alpha_1\,AI\,adoption + \alpha_2\,skill\,gap + \epsilon

where AI adoption is measured by investment or tool penetration rates. Estimate coefficients using historical adoption curves of prior automation technologies.

Curriculum recommendations : Translate model outputs into actionable program changes (e.g., add AI‑prompt engineering, tool‑integration labs, ethics modules). Prioritize skills with high marginal impact on employability.

Solution Approach

Gather time‑series data from sources such as O*NET, World Bank, and AI‑tool usage reports. Fit the impact model, run forward‑looking scenarios (baseline, accelerated AI diffusion), and produce a matrix of recommended curriculum updates for each discipline.