Iteration 0 - OR_EXPERT
Sequence: 1
Timestamp: 2025-07-27 23:21:31

Prompt:
You are an Operations Research (OR) expert focused ONLY on optimization modeling. Your role is to analyze the business domain and design LINEAR optimization problems without involving database design decisions.

CRITICAL MATHEMATICAL CONSTRAINTS FOR LINEAR/MIXED-INTEGER PROGRAMMING:
- The optimization problem MUST be either Linear Programming (LP) or Mixed-Integer Programming (MIP)
- Objective function MUST be linear: minimize/maximize ∑(coefficient × variable)
- All constraints MUST be linear: ∑(coefficient × variable) ≤/≥/= constant
- Decision variables can be continuous (LP) or mixed continuous/integer (MIP)
- NO variable products, divisions, or other nonlinear relationships
- Design business scenarios that naturally lead to linear mathematical formulations
- Generate between 2 and 20 constraints for optimization feasibility

YOUR SCOPE: Focus exclusively on optimization modeling and understanding current schema-to-optimization mapping. Do NOT propose database changes.
ROW COUNT AWARENESS: Be aware that data engineer will apply 3-row minimum rule - tables unable to generate sufficient meaningful rows will be moved to business_configuration_logic.json.

CRITICAL: Respond with ONLY a valid JSON object. No explanations, no markdown, no extra text.

Database: course_teach

```sql
CREATE TABLE course (
  Course_ID NUMBER,
  Staring_Date TEXT,
  Course TEXT
);
```

```sql
CREATE TABLE teacher (
  Teacher_ID NUMBER,
  Name TEXT,
  Age TEXT,
  Hometown TEXT
);
```

```sql
CREATE TABLE course_arrange (
  Course_ID NUMBER,
  Teacher_ID NUMBER,
  Grade NUMBER
);
```



Current Business Configuration Logic: {} (initially empty)

TASK: Design a realistic LINEAR optimization problem and analyze how it maps to the current schema.

JSON STRUCTURE REQUIRED:

{
  "database_id": "course_teach",
  "iteration": 0,
  "business_context": "Describe the realistic business scenario that leads to linear optimization (avoid scenarios requiring nonlinear relationships)",
  "optimization_problem_description": "Detailed description of what needs to be optimized using linear objective functions and constraints",
  "optimization_formulation": {
    "objective": "minimize/maximize specific_linear_metric with clear definition (sum of weighted variables only)",
    "decision_variables": "What linear variables can be controlled/decided (continuous or integer)",
    "constraints": "What linear limitations must be respected (no variable products or divisions) - design 2 to 20 constraints"
  },
  
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "coefficient_name[indices]": {
        "currently_mapped_to": "table.column OR business_configuration_logic.key OR missing",
        "mapping_adequacy": "good/missing/redundant/partial/inaccurate",
        "description": "what this coefficient represents in the optimization model"
      }
    },
    "constraint_bounds": {
      "constraint_name[indices]": {
        "currently_mapped_to": "table.column OR business_configuration_logic.key OR missing", 
        "mapping_adequacy": "good/missing/redundant/partial/inaccurate",
        "description": "what this constraint bound represents"
      }
    },
    "decision_variables": {
      "variable_name[indices]": {
        "currently_mapped_to": "table.column OR business_configuration_logic.key OR missing",
        "mapping_adequacy": "good/missing/redundant/partial/inaccurate", 
        "description": "what this decision variable represents",
        "variable_type": "continuous/integer/binary"
      }
    }
  },
  
  "missing_optimization_requirements": [
    "List what data/parameters are needed for complete linear optimization model"
  ],
  
  "iteration_status": {
    "complete": false,
    "confidence": "low/medium/high",
    "next_focus": "What optimization aspects need refinement"
  }
}

Response:
{
  "database_id": "course_teach",
  "iteration": 0,
  "business_context": "A university wants to optimize the assignment of teachers to courses to maximize the overall grade performance of students. Each teacher has a different impact on the grade performance based on their expertise and experience.",
  "optimization_problem_description": "The goal is to maximize the total expected grades of all courses by optimally assigning teachers to courses. Each teacher-course assignment has an associated expected grade improvement, and each teacher can only be assigned to one course at a time.",
  "optimization_formulation": {
    "objective": "maximize sum(Grade_Improvement[Course_ID, Teacher_ID] * x[Course_ID, Teacher_ID])",
    "decision_variables": "x[Course_ID, Teacher_ID] where x is binary, indicating whether a teacher is assigned to a course",
    "constraints": [
      "sum(x[Course_ID, Teacher_ID] for Teacher_ID) = 1 for each Course_ID",
      "sum(x[Course_ID, Teacher_ID] for Course_ID) <= 1 for each Teacher_ID"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "Grade_Improvement[Course_ID, Teacher_ID]": {
        "currently_mapped_to": "course_arrange.Grade",
        "mapping_adequacy": "partial",
        "description": "expected grade improvement when a specific teacher is assigned to a specific course"
      }
    },
    "constraint_bounds": {
      "sum(x[Course_ID, Teacher_ID] for Teacher_ID) = 1": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "ensures each course is assigned exactly one teacher"
      },
      "sum(x[Course_ID, Teacher_ID] for Course_ID) <= 1": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "ensures each teacher is assigned to at most one course"
      }
    },
    "decision_variables": {
      "x[Course_ID, Teacher_ID]": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "binary variable indicating if a teacher is assigned to a course",
        "variable_type": "binary"
      }
    }
  },
  "missing_optimization_requirements": [
    "Grade improvement data for each teacher-course pair",
    "Constraints ensuring each course is assigned exactly one teacher",
    "Constraints ensuring each teacher is assigned to at most one course"
  ],
  "iteration_status": {
    "complete": false,
    "confidence": "medium",
    "next_focus": "Refine the mapping of decision variables and constraints to the existing schema"
  }
}
