Iteration 0 - OR_EXPERT
Sequence: 1
Timestamp: 2025-07-27 22:40:05

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: sports_competition

```sql
CREATE TABLE club (
  Club_ID NUMBER,
  name TEXT,
  Region TEXT,
  Start_year TEXT
);
```

```sql
CREATE TABLE club_rank (
  Rank NUMBER,
  Club_ID NUMBER,
  Gold NUMBER,
  Silver NUMBER,
  Bronze NUMBER,
  Total NUMBER
);
```

```sql
CREATE TABLE player (
  Player_ID NUMBER,
  name TEXT,
  Position TEXT,
  Club_ID NUMBER,
  Apps NUMBER,
  Tries NUMBER,
  Goals TEXT,
  Points NUMBER
);
```

```sql
CREATE TABLE competition (
  Competition_ID NUMBER,
  Year NUMBER,
  Competition_type TEXT,
  Country TEXT
);
```

```sql
CREATE TABLE competition_result (
  Competition_ID NUMBER,
  Club_ID_1 NUMBER,
  Club_ID_2 NUMBER,
  Score TEXT
);
```



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": "sports_competition",
  "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": "sports_competition",
  "iteration": 0,
  "business_context": "A sports league wants to optimize the allocation of players to clubs to maximize the total points scored by all clubs in the league, considering constraints on player applications and club capacities.",
  "optimization_problem_description": "Optimize the assignment of players to clubs to maximize the total points scored by all clubs, subject to constraints on the number of applications a player can make and the maximum number of players a club can have.",
  "optimization_formulation": {
    "objective": "maximize total_points = \u2211(points[i] * x[i,j]) for all players i and clubs j",
    "decision_variables": "x[i,j] = 1 if player i is assigned to club j, 0 otherwise (binary)",
    "constraints": [
      "\u2211(x[i,j]) <= 1 for all players i (each player can be assigned to at most one club)",
      "\u2211(x[i,j]) <= club_capacity[j] for all clubs j (each club has a maximum capacity)",
      "\u2211(x[i,j] * apps[i]) <= max_apps for all players i (each player has a maximum number of applications)"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "points[i]": {
        "currently_mapped_to": "player.Points",
        "mapping_adequacy": "good",
        "description": "Points scored by player i"
      }
    },
    "constraint_bounds": {
      "club_capacity[j]": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "Maximum number of players club j can have"
      },
      "max_apps": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "Maximum number of applications a player can make"
      }
    },
    "decision_variables": {
      "x[i,j]": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "Binary variable indicating if player i is assigned to club j",
        "variable_type": "binary"
      }
    }
  },
  "missing_optimization_requirements": [
    "Club capacity data for each club",
    "Maximum number of applications a player can make"
  ],
  "iteration_status": {
    "complete": false,
    "confidence": "medium",
    "next_focus": "Gather missing data on club capacities and player application limits"
  }
}
