Iteration 0 - OR_EXPERT
Sequence: 1
Timestamp: 2025-07-25 22:48:08

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 across all clubs, while respecting constraints on the number of players per club and the total budget for player salaries.",
  "optimization_problem_description": "The league aims to maximize the total points scored by all clubs by optimally assigning players to clubs. The constraints include a maximum number of players per club, a minimum number of players per club, and a total budget limit for player salaries.",
  "optimization_formulation": {
    "objective": "maximize \u2211(Points_ij * X_ij) where Points_ij is the points scored by player i in club j, and X_ij is a binary decision variable indicating whether player i is assigned to club j.",
    "decision_variables": "X_ij: binary variable indicating if player i is assigned to club j.",
    "constraints": [
      "\u2211(X_ij) \u2264 Max_Players_j for each club j (maximum players per club)",
      "\u2211(X_ij) \u2265 Min_Players_j for each club j (minimum players per club)",
      "\u2211(Salary_i * X_ij) \u2264 Budget_j for each club j (budget constraint)",
      "\u2211(X_ij) = 1 for each player i (each player must be assigned to exactly one club)"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "Points_ij": {
        "currently_mapped_to": "player.Points",
        "mapping_adequacy": "good",
        "description": "Points scored by player i"
      }
    },
    "constraint_bounds": {
      "Max_Players_j": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "Maximum number of players allowed per club"
      },
      "Min_Players_j": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "Minimum number of players required per club"
      },
      "Budget_j": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "Total budget available for player salaries per club"
      }
    },
    "decision_variables": {
      "X_ij": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "Binary decision variable indicating if player i is assigned to club j",
        "variable_type": "binary"
      }
    }
  },
  "missing_optimization_requirements": [
    "Maximum number of players per club (Max_Players_j)",
    "Minimum number of players per club (Min_Players_j)",
    "Total budget for player salaries per club (Budget_j)",
    "Player salaries (Salary_i)"
  ],
  "iteration_status": {
    "complete": false,
    "confidence": "medium",
    "next_focus": "Define and map the missing constraints and parameters, such as Max_Players_j, Min_Players_j, Budget_j, and Salary_i."
  }
}
