Math Modeling
Goal
Turn an open-ended problem into a solvable model, then explain the model, solution, and limitations clearly.
Workflow
- Clarify the objective.
- Identify what must be optimized, predicted, classified, estimated, or compared.
- Restate the problem in one sentence.
- Define the system.
- List decision variables, known quantities, constraints, and evaluation criteria.
- State the modeling scope and what is intentionally ignored.
- Choose a model family.
- Use algebraic or geometric models for direct relationships.
- Use optimization for best-choice problems.
- Use probabilistic or statistical models for uncertainty and inference.
- Use differential, difference, or simulation models for dynamics.
- State assumptions explicitly.
- Make assumptions minimal, testable, and aligned with the problem.
- Call out any simplifying assumption that may affect accuracy.
- Solve and verify.
- Derive equations, compute results, and check units, bounds, and edge cases.
- Compare against intuition or a baseline if available.
- Analyze robustness.
- Check how results change when key parameters vary.
- Identify which assumptions matter most.
- Present the answer.
- Give the model, the solution, and the interpretation.
- End with limitations and next-step improvements.
Response Shape
Prefer this structure unless the user asks otherwise:
- Problem restatement
- Variables and assumptions
- Model formulation
- Solution
- Interpretation
- Sensitivity or validation
- Limitations
Working Rules
- Prefer simple, defensible models before adding complexity.
- Do not invent data, constraints, or parameters that are not given.
- If the problem is under-specified, surface the missing information and provide a conditional solution.
- If multiple modeling routes are possible, explain the tradeoff and pick one.
- Keep notation consistent and define symbols once.
- When useful, provide formulas and a short verbal explanation together.
Common Patterns
- Optimization: define objective function, decision variables, and constraints.
- Forecasting: define target, features, time window, and error metric.
- Queueing or flow: define arrival, service, capacity, and bottleneck behavior.
- Ranking or evaluation: define score components, weights, and normalization.
- Risk or uncertainty: define scenarios, probabilities, and expected value or worst-case criteria.
Quality Check
Before finalizing, verify that:
- The assumptions match the problem setting.
- The model is mathematically consistent.
- The units and dimensions work.
- The result answers the original question.
- The explanation is usable without hidden steps.
Reference
See references/modeling-guide.md for a compact set of modeling heuristics, assumption patterns, and reporting templates.
See references/optimization.md for objective/constraint patterns and solution checks.
See references/probability-statistics.md for uncertainty, estimation, and inference patterns.
See references/report-template.md for a concise model-answer-writeup structure.