mm-modeling

Stage 2: Modeling & Decomposition

Purpose

Design a comprehensive mathematical modeling solution, decompose it into manageable subtasks, and establish task dependencies via DAG analysis.

Input

Read mm-workspace/01_analysis.json for the problem analysis results.

Process

Step 1: High-Level Modeling (Actor-Critic, 2 rounds)

Load references/actor_critic.md for the Actor-Critic mechanism guide.

Actor: Generate Modeling Solution

Design a complete modeling solution covering:

• Assumption system: Clear, justified assumptions
• Variable definitions: All variables, parameters, and constants
• Constraints: Mathematical constraints from the problem
• Key equations: Governing equations and relationships
• Solution strategy: Analytical, numerical, or simulation approach
• Validation plan: How to verify the model's correctness
• Innovation: Novel formulations or extensions

Write as structured modeling solution: use numbered lists for assumptions, tables for variable definitions, and numbered LaTeX equations. Use coherent paragraphs for reasoning.

Critic: Evaluate the Modeling Solution (Independent Subagent)

Use the Agent tool to dispatch an independent subagent for the Critic role. The subagent must NOT inherit the Actor's reasoning context — it only receives the Actor's final output.

Dispatch an Agent with the following prompt structure:

你是一名严格的数学建模评审专家（Critic 角色）。请对以下建模方案进行批评。

## 审查标准
- 假设合理性：假设是否有充分的理由支撑？
- 技术匹配度：所选方法是否匹配问题类型？
- 数据兼容性：可用数据能否支持该模型？
- 可计算性：该模型在实际中是否可解？
- 完整性：是否覆盖了所有问题要求？

## 问题背景
{Insert brief problem description, 1-2 paragraphs}

## 被审查的建模方案
{Insert Actor's complete modeling solution}

## 输出要求
1. 逐一指出具体问题（附位置引用）
2. 每个问题给出改进方向（指出方向，不提供完整方案）
3. 最后给出总体评价：是否存在重大问题需要追加一轮
4. 直接输出批评内容，不要有多余的寒暄

Receive the Critic feedback and proceed to Improvement.

Improvement: Refine the Modeling Solution

Produce an improved version addressing all critiques. Complete standalone document.

Round 2: Repeat Critic → Improvement once more.

Step 2: Problem Decomposition

Split the modeling solution into 3-6 subtasks (default: 5, including sensitivity analysis).

Decomposition principles:

• Each subtask addresses a distinct aspect of the overall problem
• Subtasks should map to the competition's question structure when possible
• Each subtask has clear objectives, methods, and expected outputs
• No redundancy between subtasks
• All aspects of the modeling solution are covered
• The last subtask should be a sensitivity analysis / robustness testing task that depends on all model-building tasks, testing parameter perturbations, assumption variations, and result stability

For each subtask, provide:

• ID: Sequential number (1, 2, 3, ...)
• Description: Detailed task description (one paragraph, comprehensive)
• Method: Expected modeling/approach method
• Expected output: What the task should produce

Refinement: After initial decomposition, refine each subtask description to be more specific and actionable. Ensure each can be understood independently.

Step 3: DAG Construction

Analyze dependencies between subtasks:

1. Identify dependencies for each task:

   • Data dependencies (output of task A feeds into task B)
   • Method dependencies (task B builds on task A's framework)
   • Computational dependencies (task B needs task A's numerical results)

2. Build adjacency list:

   {
     "1": [],
     "2": ["1"],
     "3": ["1"],
     "4": ["2", "3"]
   }
   

3. Compute topological order: Determine the execution sequence. For the example above: [1, 2, 3, 4] (tasks 2 and 3 can be parallel)

4. Generate dependency analysis: For each task, write a brief paragraph explaining its dependencies.

Step 4: Save Output

Write to mm-workspace/02_modeling.json:

{
  "modeling_solution": "final improved modeling solution text",
  "tasks": [
    {
      "id": 1,
      "description": "detailed task description",
      "method": "expected method",
      "expected_output": "what this task produces",
      "dependencies": []
    }
  ],
  "dag_order": [1, 2, 3, 4],
  "dependency_analysis": ["task 1 dependency analysis...", "task 2 dependency analysis..."],
  "stage": "modeling_complete"
}

Then commit: cd mm-workspace && git add -A && git commit -m "feat(s2): modeling and decomposition complete"

Step 5: Present to User

Display:

• Summary of the modeling approach
• Task decomposition overview (list each task with its objective)
• Dependency graph and execution order
• Which tasks can be parallelized

Then ask: "建模方案和任务分解完成。是否需要修改？确认后开始逐任务求解。"