Iteration 0 - OR_EXPERT
Sequence: 1
Timestamp: 2025-07-27 22:37:51

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: concert_singer

```sql
CREATE TABLE stadium (
  Stadium_ID NUMBER,
  Location TEXT,
  Name TEXT,
  Capacity NUMBER,
  Highest NUMBER,
  Lowest NUMBER,
  Average NUMBER
);
```

```sql
CREATE TABLE singer (
  Singer_ID NUMBER,
  Name TEXT,
  Country TEXT,
  Song_Name TEXT,
  Song_release_year TEXT,
  Age NUMBER,
  Is_male OTHERS
);
```

```sql
CREATE TABLE concert (
  concert_ID NUMBER,
  concert_Name TEXT,
  Theme TEXT,
  Stadium_ID TEXT,
  Year TEXT
);
```

```sql
CREATE TABLE singer_in_concert (
  concert_ID NUMBER,
  Singer_ID 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": "concert_singer",
  "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": "concert_singer",
  "iteration": 0,
  "business_context": "A concert organizer wants to maximize the total attendance across multiple concerts while considering the capacity limitations of each stadium and ensuring that each concert has at least one singer.",
  "optimization_problem_description": "The goal is to maximize the total number of attendees at concerts by selecting which concerts to hold at which stadiums, subject to stadium capacity constraints and ensuring each concert has at least one singer.",
  "optimization_formulation": {
    "objective": "maximize total_attendance = \u2211(attendance_coefficient[concert_ID] \u00d7 x[concert_ID])",
    "decision_variables": "x[concert_ID] is a binary variable indicating whether concert_ID is held (1) or not (0)",
    "constraints": [
      "\u2211(x[concert_ID] \u00d7 capacity[Stadium_ID]) \u2264 capacity[Stadium_ID] for each Stadium_ID",
      "\u2211(x[concert_ID]) \u2265 1 for each concert_ID"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "attendance_coefficient[concert_ID]": {
        "currently_mapped_to": "stadium.Average",
        "mapping_adequacy": "partial",
        "description": "average attendance expected if the concert is held at a stadium"
      }
    },
    "constraint_bounds": {
      "capacity[Stadium_ID]": {
        "currently_mapped_to": "stadium.Capacity",
        "mapping_adequacy": "good",
        "description": "maximum number of attendees a stadium can hold"
      }
    },
    "decision_variables": {
      "x[concert_ID]": {
        "currently_mapped_to": "concert.concert_ID",
        "mapping_adequacy": "good",
        "description": "binary decision variable indicating if a concert is held",
        "variable_type": "binary"
      }
    }
  },
  "missing_optimization_requirements": [
    "Specific attendance coefficients for each concert and stadium combination",
    "Mapping of concerts to specific stadiums"
  ],
  "iteration_status": {
    "complete": false,
    "confidence": "medium",
    "next_focus": "Refine attendance coefficients and ensure mapping of concerts to specific stadiums"
  }
}
