Iteration 1 - OR_EXPERT_REFINEMENT
Sequence: 3
Timestamp: 2025-07-25 22:48:34

Prompt:
You are an Operations Research (OR) expert in iteration 1 of an alternating optimization process. The algorithm alternates between OR expert analysis and data engineering implementation until convergence.

CRITICAL MATHEMATICAL CONSTRAINTS FOR LINEAR/MIXED-INTEGER PROGRAMMING:
- The optimization problem MUST remain 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
- If previous iteration introduced nonlinear elements, redesign as linear formulation
- Maintain between 2 and 20 constraints for optimization feasibility

YOUR SCOPE: Focus exclusively on optimization modeling and mapping analysis. Do NOT propose database changes.
ROW COUNT AWARENESS: Understand that data engineer applies 3-row minimum rule - insufficient table data gets moved to business_configuration_logic.json.


DATA AVAILABILITY CHECK: 
Before listing missing requirements, verify:
- Check current schema for required data columns
- Check business configuration logic for required parameters  
- Only list as "missing" if data is truly unavailable
- If all mappings are "good", missing_requirements should be []

CONSISTENCY RULES:
- IF all mapping_adequacy == "good" THEN missing_optimization_requirements = []
- IF missing_optimization_requirements = [] THEN complete CAN be true
- IF complete == true THEN confidence should be "high"

SELF-CHECK: Before responding, verify:
1. Does current schema contain the data I claim is missing?
2. Are my mapping assessments consistent with missing requirements?
3. Is my complete status consistent with missing requirements?

MAPPING COMPLETENESS CHECK: Ensure logical consistency between:
- All objective coefficients mapped with adequacy evaluation
- All constraint bounds mapped with adequacy evaluation  
- All decision variables mapped with adequacy evaluation
- Missing requirements list matches inadequate mappings only


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



CURRENT STATE (iteration 0):
{
  "iteration": 1,
  "converged": false,
  "business_context": "A sports league aims to optimize the allocation of players to teams to maximize the overall performance of the league. The performance is measured by the total wins of all teams, considering the players' historical performance and their positions.",
  "optimization_problem": "The objective is to maximize the total wins across all teams by assigning players to teams based on their historical performance and positions. Constraints include ensuring each player is assigned to only one team, each team has a minimum and maximum number of players, and the total number of players assigned does not exceed the league's capacity.",
  "objective": "maximize \u2211(w_ij * x_ij) where w_ij is the historical win rate of player i in position j and x_ij is a binary decision variable indicating if player i is assigned to position j in a team.",
  "table_count": 3,
  "key_changes": [
    "Schema changes include creating tables for player positions, team constraints, and league capacity. Configuration logic updates include scalar parameters for team size constraints and league capacity, and a formula for calculating historical win rates by position."
  ],
  "math_consistency": "high",
  "next_iteration_focus": "Refine the mapping of historical win rates by position and gather data on team size constraints and league capacity.",
  "mapping_adequacy_summary": "needs_improvement"
}

CURRENT SCHEMA:
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating tables for player positions, team constraints, and league capacity. Configuration logic updates include scalar parameters for team size constraints and league capacity, and a formula for calculating historical win rates by position.

CREATE TABLE player_position (
  player_id INTEGER,
  position_id INTEGER,
  historical_win_rate FLOAT,
  player_assigned BOOLEAN
);

CREATE TABLE team_constraints (
  team_id INTEGER,
  min_players INTEGER,
  max_players INTEGER
);

CREATE TABLE league_capacity (
  league_id INTEGER,
  capacity INTEGER
);


```

CURRENT DATA DICTIONARY:
{
  "tables": {
    "player_position": {
      "business_purpose": "Mapping of players to positions and their historical win rates",
      "optimization_role": "decision_variables",
      "columns": {
        "player_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for a player",
          "optimization_purpose": "Identifier for player i",
          "sample_values": "1, 2, 3"
        },
        "position_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for a position",
          "optimization_purpose": "Identifier for position j",
          "sample_values": "1, 2, 3"
        },
        "historical_win_rate": {
          "data_type": "FLOAT",
          "business_meaning": "Historical win rate of player i in position j",
          "optimization_purpose": "Coefficient in the objective function",
          "sample_values": "0.75, 0.80, 0.85"
        },
        "player_assigned": {
          "data_type": "BOOLEAN",
          "business_meaning": "Indicates if player i is assigned to position j in a team",
          "optimization_purpose": "Binary decision variable x_ij",
          "sample_values": "true, false"
        }
      }
    },
    "team_constraints": {
      "business_purpose": "Minimum and maximum number of players allowed per team",
      "optimization_role": "constraint_bounds",
      "columns": {
        "team_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for a team",
          "optimization_purpose": "Identifier for team j",
          "sample_values": "1, 2, 3"
        },
        "min_players": {
          "data_type": "INTEGER",
          "business_meaning": "Minimum number of players required for team j",
          "optimization_purpose": "Lower bound constraint for team size",
          "sample_values": "5, 6, 7"
        },
        "max_players": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum number of players allowed for team j",
          "optimization_purpose": "Upper bound constraint for team size",
          "sample_values": "10, 11, 12"
        }
      }
    },
    "league_capacity": {
      "business_purpose": "Total number of players the league can accommodate",
      "optimization_role": "constraint_bounds",
      "columns": {
        "league_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for the league",
          "optimization_purpose": "Identifier for the league",
          "sample_values": "1"
        },
        "capacity": {
          "data_type": "INTEGER",
          "business_meaning": "Total number of players the league can accommodate",
          "optimization_purpose": "Upper bound constraint for total players",
          "sample_values": "100"
        }
      }
    }
  }
}


CURRENT BUSINESS CONFIGURATION LOGIC:
{
  "min_players": {
    "sample_value": 5,
    "data_type": "INTEGER",
    "business_meaning": "Minimum number of players required for a team",
    "optimization_role": "Lower bound constraint for team size",
    "configuration_type": "scalar_parameter"
  },
  "max_players": {
    "sample_value": 10,
    "data_type": "INTEGER",
    "business_meaning": "Maximum number of players allowed for a team",
    "optimization_role": "Upper bound constraint for team size",
    "configuration_type": "scalar_parameter"
  },
  "league_capacity": {
    "sample_value": 100,
    "data_type": "INTEGER",
    "business_meaning": "Total number of players the league can accommodate",
    "optimization_role": "Upper bound constraint for total players",
    "configuration_type": "scalar_parameter"
  },
  "historical_win_rate_formula": {
    "formula_expression": "w_ij = (wins_ij / total_games_ij)",
    "data_type": "STRING",
    "business_meaning": "Historical win rate of player i in position j",
    "optimization_role": "Coefficient in the objective function",
    "configuration_type": "business_logic_formula"
  }
}


TASK: Refine the optimization problem formulation by analyzing current data schema mapping and identifying requirements while maintaining LINEAR structure.

JSON STRUCTURE REQUIRED:

{
  "database_id": "match_season",
  "iteration": 1,
  "business_context": "Updated realistic business scenario description that supports linear optimization",
  "optimization_problem_description": "Refined description of LINEAR optimization problem", 
  "optimization_formulation": {
    "objective": "refined linear minimize/maximize with mathematical precision (sum of weighted variables only)",
    "decision_variables": "clearly defined controllable linear variables (continuous or integer)",
    "constraints": "mathematically precise LINEAR constraint definitions (no variable products or divisions) - maintain 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 still needed for linear formulation (should be [] if all mappings are good)"
  ],
  
  "iteration_status": {
    "complete": true/false based on linear formulation completeness,
    "confidence": "updated confidence level",
    "next_focus": "What needs refinement for linear optimization OR 'Ready for convergence'"
  }
}

Response:
{
  "database_id": "match_season",
  "iteration": 1,
  "business_context": "A sports league aims to optimize the allocation of players to teams to maximize the overall performance of the league. The performance is measured by the total wins of all teams, considering the players' historical performance and their positions.",
  "optimization_problem_description": "Maximize the total wins across all teams by assigning players to teams based on their historical performance and positions. Constraints include ensuring each player is assigned to only one team, each team has a minimum and maximum number of players, and the total number of players assigned does not exceed the league's capacity.",
  "optimization_formulation": {
    "objective": "maximize \u2211(w_ij * x_ij) where w_ij is the historical win rate of player i in position j and x_ij is a binary decision variable indicating if player i is assigned to position j in a team.",
    "decision_variables": "x_ij: binary variable indicating if player i is assigned to position j in a team.",
    "constraints": "1. \u2211(x_ij) \u2264 1 for all i (each player is assigned to at most one position), 2. min_players_j \u2264 \u2211(x_ij) \u2264 max_players_j for all j (team size constraints), 3. \u2211(x_ij) \u2264 league_capacity (total players assigned does not exceed league capacity)."
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "w_ij[i,j]": {
        "currently_mapped_to": "player_position.historical_win_rate",
        "mapping_adequacy": "good",
        "description": "Historical win rate of player i in position j."
      }
    },
    "constraint_bounds": {
      "min_players_j[j]": {
        "currently_mapped_to": "team_constraints.min_players",
        "mapping_adequacy": "good",
        "description": "Minimum number of players required for team j."
      },
      "max_players_j[j]": {
        "currently_mapped_to": "team_constraints.max_players",
        "mapping_adequacy": "good",
        "description": "Maximum number of players allowed for team j."
      },
      "league_capacity": {
        "currently_mapped_to": "league_capacity.capacity",
        "mapping_adequacy": "good",
        "description": "Total number of players the league can accommodate."
      }
    },
    "decision_variables": {
      "x_ij[i,j]": {
        "currently_mapped_to": "player_position.player_assigned",
        "mapping_adequacy": "good",
        "description": "Binary decision variable indicating if player i is assigned to position j in a team.",
        "variable_type": "binary"
      }
    }
  },
  "missing_optimization_requirements": [],
  "iteration_status": {
    "complete": true,
    "confidence": "high",
    "next_focus": "Ready for convergence"
  }
}
