Iteration 0 - OR_EXPERT
Sequence: 1
Timestamp: 2025-07-25 22:27:23

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 aims to optimize the assignment of teachers to courses to maximize the overall teaching quality, measured by the sum of grades assigned to each course-teacher pair, while respecting constraints on teacher availability and course requirements.",
  "optimization_problem_description": "The objective is to maximize the total teaching quality by assigning teachers to courses based on their grades. Constraints include ensuring each course is assigned exactly one teacher and each teacher is assigned to no more than a specified number of courses.",
  "optimization_formulation": {
    "objective": "maximize \u2211(Grade_ij \u00d7 X_ij) where Grade_ij is the grade for teacher i teaching course j, and X_ij is a binary decision variable indicating whether teacher i is assigned to course j.",
    "decision_variables": "X_ij: binary variable indicating if teacher i is assigned to course j.",
    "constraints": [
      "\u2211(X_ij) = 1 for each course j (each course must have exactly one teacher).",
      "\u2211(X_ij) \u2264 Max_Courses_i for each teacher i (each teacher can teach at most Max_Courses_i courses)."
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "Grade_ij": {
        "currently_mapped_to": "course_arrange.Grade",
        "mapping_adequacy": "good",
        "description": "Grade represents the teaching quality of teacher i for course j."
      }
    },
    "constraint_bounds": {
      "Max_Courses_i": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "Maximum number of courses a teacher can handle."
      }
    },
    "decision_variables": {
      "X_ij": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "Binary decision variable indicating if teacher i is assigned to course j.",
        "variable_type": "binary"
      }
    }
  },
  "missing_optimization_requirements": [
    "Maximum number of courses each teacher can handle (Max_Courses_i).",
    "Binary decision variables (X_ij) to represent teacher-course assignments."
  ],
  "iteration_status": {
    "complete": false,
    "confidence": "medium",
    "next_focus": "Define and map the maximum number of courses each teacher can handle (Max_Courses_i) and ensure the binary decision variables (X_ij) are properly represented."
  }
}
