Iteration 0 - OR_EXPERT
Sequence: 1
Timestamp: 2025-07-27 23:05:38

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: performance_attendance

```sql
CREATE TABLE member (
  Member_ID TEXT,
  Name TEXT,
  Nationality TEXT,
  Role TEXT
);
```

```sql
CREATE TABLE performance (
  Performance_ID NUMBER,
  Date TEXT,
  Host TEXT,
  Location TEXT,
  Attendance NUMBER
);
```

```sql
CREATE TABLE member_attendance (
  Member_ID NUMBER,
  Performance_ID NUMBER,
  Num_of_Pieces 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": "performance_attendance",
  "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": "performance_attendance",
  "iteration": 0,
  "business_context": "A theater company wants to optimize the allocation of its members to performances to maximize the total number of pieces performed, while considering constraints such as member availability and performance attendance limits.",
  "optimization_problem_description": "The goal is to maximize the total number of pieces performed by members across all performances, subject to constraints on member availability and performance attendance limits.",
  "optimization_formulation": {
    "objective": "maximize total_pieces = \u2211(Num_of_Pieces[Member_ID, Performance_ID])",
    "decision_variables": "Num_of_Pieces[Member_ID, Performance_ID] - integer variables representing the number of pieces a member performs in a specific performance",
    "constraints": [
      "\u2211(Num_of_Pieces[Member_ID, Performance_ID]) <= Attendance[Performance_ID] for each Performance_ID",
      "\u2211(Num_of_Pieces[Member_ID, Performance_ID]) <= availability[Member_ID] for each Member_ID",
      "Num_of_Pieces[Member_ID, Performance_ID] >= 0 for all Member_ID, Performance_ID"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "Num_of_Pieces[Member_ID, Performance_ID]": {
        "currently_mapped_to": "member_attendance.Num_of_Pieces",
        "mapping_adequacy": "good",
        "description": "Number of pieces a member performs in a specific performance"
      }
    },
    "constraint_bounds": {
      "Attendance[Performance_ID]": {
        "currently_mapped_to": "performance.Attendance",
        "mapping_adequacy": "good",
        "description": "Maximum number of pieces that can be performed in a performance"
      },
      "availability[Member_ID]": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "Maximum number of pieces a member is available to perform"
      }
    },
    "decision_variables": {
      "Num_of_Pieces[Member_ID, Performance_ID]": {
        "currently_mapped_to": "member_attendance.Num_of_Pieces",
        "mapping_adequacy": "good",
        "description": "Number of pieces a member performs in a specific performance",
        "variable_type": "integer"
      }
    }
  },
  "missing_optimization_requirements": [
    "availability data for each member to determine how many pieces they can perform"
  ],
  "iteration_status": {
    "complete": false,
    "confidence": "medium",
    "next_focus": "Incorporate member availability data into the optimization model"
  }
}
