Iteration 0 - OR_EXPERT
Sequence: 1
Timestamp: 2025-07-25 22:50:52

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: school_player

```sql
CREATE TABLE school (
  School_ID NUMBER,
  School TEXT,
  Location TEXT,
  Enrollment NUMBER,
  Founded NUMBER,
  Denomination TEXT,
  Boys_or_Girls TEXT,
  Day_or_Boarding TEXT,
  Year_Entered_Competition NUMBER,
  School_Colors TEXT
);
```

```sql
CREATE TABLE school_details (
  School_ID NUMBER,
  Nickname TEXT,
  Colors TEXT,
  League TEXT,
  Class TEXT,
  Division TEXT
);
```

```sql
CREATE TABLE school_performance (
  School_Id NUMBER,
  School_Year TEXT,
  Class_A TEXT,
  Class_AA TEXT
);
```

```sql
CREATE TABLE player (
  Player_ID NUMBER,
  Player TEXT,
  Team TEXT,
  Age NUMBER,
  Position TEXT,
  School_ID 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": "school_player",
  "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": "school_player",
  "iteration": 0,
  "business_context": "Optimize the allocation of players to schools based on school performance and player positions to maximize overall team performance.",
  "optimization_problem_description": "The goal is to maximize the overall performance of teams by assigning players to schools in a way that aligns with the schools' historical performance and the players' positions. The objective is to maximize the sum of weighted performance scores, where weights are based on school performance and player positions.",
  "optimization_formulation": {
    "objective": "maximize \u2211(performance_weight[School_ID, Position] \u00d7 assign[Player_ID, School_ID])",
    "decision_variables": "assign[Player_ID, School_ID] (binary: 1 if player is assigned to school, 0 otherwise)",
    "constraints": [
      "Each player must be assigned to exactly one school: \u2211(assign[Player_ID, School_ID]) = 1 for all Player_ID",
      "Each school cannot exceed its enrollment capacity: \u2211(assign[Player_ID, School_ID]) \u2264 Enrollment[School_ID] for all School_ID",
      "The number of players assigned to a school must be at least the minimum required for a team: \u2211(assign[Player_ID, School_ID]) \u2265 min_team_size for all School_ID",
      "The number of players in each position must be within a specified range: min_positions[Position] \u2264 \u2211(assign[Player_ID, School_ID]) \u2264 max_positions[Position] for all Position"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "performance_weight[School_ID, Position]": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "Weight representing the performance of a school in a specific position"
      }
    },
    "constraint_bounds": {
      "Enrollment[School_ID]": {
        "currently_mapped_to": "school.Enrollment",
        "mapping_adequacy": "good",
        "description": "Maximum number of players a school can enroll"
      },
      "min_team_size": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "Minimum number of players required to form a team"
      },
      "min_positions[Position]": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "Minimum number of players required in a specific position"
      },
      "max_positions[Position]": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "Maximum number of players allowed in a specific position"
      }
    },
    "decision_variables": {
      "assign[Player_ID, School_ID]": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "Binary variable indicating if a player is assigned to a school",
        "variable_type": "binary"
      }
    }
  },
  "missing_optimization_requirements": [
    "Performance weights for schools based on positions",
    "Minimum team size requirement",
    "Minimum and maximum number of players per position",
    "Historical performance data for schools"
  ],
  "iteration_status": {
    "complete": false,
    "confidence": "medium",
    "next_focus": "Refine the constraints and objective function to better align with available data and business requirements"
  }
}
