Math Modeling

Goal

Turn an open-ended problem into a solvable model, then explain the model, solution, and limitations clearly.

Workflow

1. Clarify the objective.

   • Identify what must be optimized, predicted, classified, estimated, or compared.
   • Restate the problem in one sentence.

2. Define the system.

   • List decision variables, known quantities, constraints, and evaluation criteria.
   • State the modeling scope and what is intentionally ignored.

3. 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.

4. State assumptions explicitly.

   • Make assumptions minimal, testable, and aligned with the problem.
   • Call out any simplifying assumption that may affect accuracy.

5. Solve and verify.

   • Derive equations, compute results, and check units, bounds, and edge cases.
   • Compare against intuition or a baseline if available.

6. Analyze robustness.

   • Check how results change when key parameters vary.
   • Identify which assumptions matter most.

7. 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.