Iteration 0 - OR_EXPERT
Sequence: 1
Timestamp: 2025-07-25 22:48:34

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: match_season

```sql
CREATE TABLE country (
  Country_id NUMBER,
  Country_name TEXT,
  Capital TEXT,
  Official_native_language TEXT
);
```

```sql
CREATE TABLE team (
  Team_id NUMBER,
  Name TEXT
);
```

```sql
CREATE TABLE match_season (
  Season NUMBER,
  Player TEXT,
  Position TEXT,
  Country NUMBER,
  Team NUMBER,
  Draft_Pick_Number NUMBER,
  Draft_Class TEXT,
  College TEXT
);
```

```sql
CREATE TABLE player (
  Player_ID NUMBER,
  Player TEXT,
  Years_Played TEXT,
  Total_WL TEXT,
  Singles_WL TEXT,
  Doubles_WL TEXT,
  Team 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": "match_season",
  "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": "match_season",
  "iteration": 0,
  "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": "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.",
  "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": [
      "\u2211(x_ij) \u2264 1 for all i (each player is assigned to at most one position in one team)",
      "\u2211(x_ij) \u2265 min_players for all j (each team has a minimum number of players)",
      "\u2211(x_ij) \u2264 max_players for all j (each team has a maximum number of players)",
      "\u2211(x_ij) \u2264 league_capacity (total number of players assigned does not exceed the league's capacity)"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "w_ij[i,j]": {
        "currently_mapped_to": "player.Total_WL",
        "mapping_adequacy": "partial",
        "description": "Historical win rate of player i in position j"
      }
    },
    "constraint_bounds": {
      "min_players[j]": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "Minimum number of players required for team j"
      },
      "max_players[j]": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "Maximum number of players allowed for team j"
      },
      "league_capacity": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "Total number of players the league can accommodate"
      }
    },
    "decision_variables": {
      "x_ij[i,j]": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "Binary decision variable indicating if player i is assigned to position j in a team",
        "variable_type": "binary"
      }
    }
  },
  "missing_optimization_requirements": [
    "Historical win rate of players by position",
    "Minimum and maximum number of players per team",
    "League capacity",
    "Mapping of players to positions"
  ],
  "iteration_status": {
    "complete": false,
    "confidence": "low",
    "next_focus": "Refine the mapping of historical win rates by position and gather data on team size constraints and league capacity."
  }
}
