Iteration 0 - OR_EXPERT
Sequence: 1
Timestamp: 2025-07-25 22:25:45

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": "Optimize the allocation of singers to concerts in different stadiums to maximize total audience capacity while respecting singer availability and stadium constraints.",
  "optimization_problem_description": "The goal is to maximize the total audience capacity by assigning singers to concerts in stadiums, ensuring that each singer is not overbooked and each concert does not exceed the stadium's capacity.",
  "optimization_formulation": {
    "objective": "maximize \u2211(Capacity_s \u00d7 x_sc) where Capacity_s is the capacity of stadium s and x_sc is a binary variable indicating if singer c is assigned to concert in stadium s.",
    "decision_variables": "x_sc: binary variable indicating if singer c is assigned to concert in stadium s.",
    "constraints": [
      "\u2211(x_sc) \u2264 1 for each singer c (each singer can be assigned to at most one concert).",
      "\u2211(x_sc) \u2264 Capacity_s for each stadium s (total singers assigned to a concert in stadium s cannot exceed its capacity).",
      "x_sc \u2208 {0, 1} for all s, c (binary decision variables)."
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "Capacity_s[s]": {
        "currently_mapped_to": "stadium.Capacity",
        "mapping_adequacy": "good",
        "description": "Capacity of stadium s."
      }
    },
    "constraint_bounds": {
      "singer_limit[c]": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "Maximum number of concerts a singer can participate in."
      },
      "stadium_capacity[s]": {
        "currently_mapped_to": "stadium.Capacity",
        "mapping_adequacy": "good",
        "description": "Maximum number of singers a stadium can accommodate."
      }
    },
    "decision_variables": {
      "x_sc[s, c]": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "Binary variable indicating if singer c is assigned to concert in stadium s.",
        "variable_type": "binary"
      }
    }
  },
  "missing_optimization_requirements": [
    "Maximum number of concerts a singer can participate in.",
    "Binary decision variables for singer-concert assignments."
  ],
  "iteration_status": {
    "complete": false,
    "confidence": "medium",
    "next_focus": "Refine constraints and decision variables to better align with the schema and business requirements."
  }
}
