Iteration 2 - OR_EXPERT_REFINEMENT
Sequence: 5
Timestamp: 2025-07-28 00:04:59

Prompt:
You are an Operations Research (OR) expert in iteration 2 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 1):
{
  "iteration": 2,
  "converged": false,
  "business_context": "A soccer league is organizing tryouts for players from various colleges. The goal is to select players for different positions while minimizing the total number of yellow cards among selected players, ensuring that each position is filled and that players are selected from a diverse set of colleges.",
  "optimization_problem": "Optimize the selection of players for different positions in a soccer team to minimize the total number of yellow cards among selected players. Ensure that each position is filled and that players are selected from a diverse set of colleges.",
  "objective": "minimize total_yellow_cards = \u2211(yCard[pID] * x[pID])",
  "table_count": 1,
  "key_changes": [
    "Schema adjustments were made to incorporate missing yellow card data and improve optimization mapping. Configuration logic was updated to include scalar parameters and formulas."
  ],
  "math_consistency": "high",
  "next_iteration_focus": "Incorporate missing yellow card data for players to complete the linear formulation",
  "mapping_adequacy_summary": "partially_adequate"
}

CURRENT SCHEMA:
```sql
-- Iteration 2 Database Schema
-- Objective: Schema adjustments were made to incorporate missing yellow card data and improve optimization mapping. Configuration logic was updated to include scalar parameters and formulas.

CREATE TABLE Tryout (
  pID INTEGER,
  pPos STRING,
  cName STRING,
  yCard INTEGER
);

CREATE TABLE PositionRequirements (
  position STRING,
  required_players INTEGER
);

CREATE TABLE CollegeLimits (
  college STRING,
  max_players INTEGER
);

CREATE TABLE PlayerYellowCards (
  pID INTEGER,
  yCard INTEGER
);


```

CURRENT DATA DICTIONARY:
{
  "tables": {
    "Tryout": {
      "business_purpose": "Stores tryout data for players",
      "optimization_role": "decision_variables",
      "columns": {
        "pID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each player",
          "optimization_purpose": "Used as decision variable index",
          "sample_values": "1, 2, 3"
        },
        "pPos": {
          "data_type": "STRING",
          "business_meaning": "Position the player is trying out for",
          "optimization_purpose": "Used in position constraints",
          "sample_values": "Goalkeeper, Defender, Midfielder, Forward"
        },
        "cName": {
          "data_type": "STRING",
          "business_meaning": "College the player is from",
          "optimization_purpose": "Used in college constraints",
          "sample_values": "College A, College B, College C"
        },
        "yCard": {
          "data_type": "INTEGER",
          "business_meaning": "Number of yellow cards for the player",
          "optimization_purpose": "Used in the objective function to minimize yellow cards",
          "sample_values": "0, 1, 2"
        }
      }
    },
    "PositionRequirements": {
      "business_purpose": "Defines player requirements for each position",
      "optimization_role": "constraint_bounds",
      "columns": {
        "position": {
          "data_type": "STRING",
          "business_meaning": "Position name",
          "optimization_purpose": "Defines position constraints",
          "sample_values": "Goalkeeper, Defender, Midfielder, Forward"
        },
        "required_players": {
          "data_type": "INTEGER",
          "business_meaning": "Number of players required for the position",
          "optimization_purpose": "Ensures each position is filled",
          "sample_values": "1, 4, 4, 2"
        }
      }
    },
    "CollegeLimits": {
      "business_purpose": "Defines limits on player selection from each college",
      "optimization_role": "constraint_bounds",
      "columns": {
        "college": {
          "data_type": "STRING",
          "business_meaning": "College name",
          "optimization_purpose": "Defines college constraints",
          "sample_values": "College A, College B, College C"
        },
        "max_players": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum number of players allowed from the college",
          "optimization_purpose": "Ensures diversity in player selection",
          "sample_values": "3"
        }
      }
    },
    "PlayerYellowCards": {
      "business_purpose": "Stores the number of yellow cards for each player",
      "optimization_role": "objective_coefficients",
      "columns": {
        "pID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each player",
          "optimization_purpose": "Links yellow card data to players",
          "sample_values": "1, 2, 3"
        },
        "yCard": {
          "data_type": "INTEGER",
          "business_meaning": "Number of yellow cards for the player",
          "optimization_purpose": "Used in the objective function to minimize yellow cards",
          "sample_values": "0, 1, 2"
        }
      }
    }
  }
}


CURRENT BUSINESS CONFIGURATION LOGIC:
{
  "max_players_per_college": {
    "sample_value": 3,
    "data_type": "INTEGER",
    "business_meaning": "Maximum number of players that can be selected from a single college",
    "optimization_role": "Used to enforce diversity in player selection",
    "configuration_type": "scalar_parameter"
  },
  "players_required_per_position": {
    "formula_expression": "Goalkeeper: 1, Defender: 4, Midfielder: 4, Forward: 2",
    "data_type": "STRING",
    "business_meaning": "Number of players required for each position",
    "optimization_role": "Ensures each position is filled with the required number of players",
    "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": "soccer_2",
  "iteration": 2,
  "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": "soccer_2",
  "iteration": 2,
  "business_context": "A soccer league is organizing tryouts for players from various colleges. The goal is to select players for different positions while minimizing the total number of yellow cards among selected players, ensuring that each position is filled and that players are selected from a diverse set of colleges.",
  "optimization_problem_description": "Optimize the selection of players for different positions in a soccer team to minimize the total number of yellow cards among selected players. Ensure that each position is filled and that players are selected from a diverse set of colleges.",
  "optimization_formulation": {
    "objective": "minimize total_yellow_cards = \u2211(yCard[pID] * x[pID])",
    "decision_variables": "x[pID] where x[pID] is a binary variable indicating if player pID is selected (1) or not (0)",
    "constraints": [
      "\u2211(x[pID] | pPos = position) = required_players for each position",
      "\u2211(x[pID] | cName = college) \u2264 max_players for each college"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "yCard[pID]": {
        "currently_mapped_to": "Tryout.yCard",
        "mapping_adequacy": "good",
        "description": "Number of yellow cards for player pID"
      }
    },
    "constraint_bounds": {
      "required_players[position]": {
        "currently_mapped_to": "PositionRequirements.required_players",
        "mapping_adequacy": "good",
        "description": "Number of players required for each position"
      },
      "max_players[college]": {
        "currently_mapped_to": "CollegeLimits.max_players",
        "mapping_adequacy": "good",
        "description": "Maximum number of players allowed from each college"
      }
    },
    "decision_variables": {
      "x[pID]": {
        "currently_mapped_to": "Tryout.pID",
        "mapping_adequacy": "good",
        "description": "Binary variable indicating if player pID is selected",
        "variable_type": "binary"
      }
    }
  },
  "missing_optimization_requirements": [],
  "iteration_status": {
    "complete": true,
    "confidence": "high",
    "next_focus": "Ready for convergence"
  }
}
