Iteration 0 - OR_EXPERT
Sequence: 1
Timestamp: 2025-07-25 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: 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 music ensemble aims to optimize the number of pieces performed by each member across various performances to maximize overall attendance, while ensuring fair distribution of pieces among members and respecting performance capacity constraints.",
  "optimization_problem_description": "The objective is to maximize the total attendance across all performances by optimizing the number of pieces each member performs, subject to constraints on the maximum number of pieces a member can perform, the total number of pieces per performance, and ensuring each member is assigned at least one piece.",
  "optimization_formulation": {
    "objective": "maximize \u2211(Attendance_j \u00d7 \u2211(x_ij)) where x_ij is the number of pieces member i performs in performance j",
    "decision_variables": "x_ij: number of pieces member i performs in performance j (integer)",
    "constraints": [
      "\u2211(x_ij) \u2264 Max_Pieces_i for each member i (maximum pieces per member)",
      "\u2211(x_ij) \u2264 Capacity_j for each performance j (performance capacity)",
      "\u2211(x_ij) \u2265 1 for each member i (minimum pieces per member)",
      "x_ij \u2265 0 for all i, j (non-negativity)"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "Attendance_j": {
        "currently_mapped_to": "performance.Attendance",
        "mapping_adequacy": "good",
        "description": "Attendance at performance j"
      }
    },
    "constraint_bounds": {
      "Max_Pieces_i": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "Maximum number of pieces member i can perform"
      },
      "Capacity_j": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "Maximum number of pieces allowed in performance j"
      }
    },
    "decision_variables": {
      "x_ij": {
        "currently_mapped_to": "member_attendance.Num_of_Pieces",
        "mapping_adequacy": "partial",
        "description": "Number of pieces member i performs in performance j",
        "variable_type": "integer"
      }
    }
  },
  "missing_optimization_requirements": [
    "Max_Pieces_i: Maximum number of pieces each member can perform",
    "Capacity_j: Maximum number of pieces allowed per performance"
  ],
  "iteration_status": {
    "complete": false,
    "confidence": "medium",
    "next_focus": "Refine constraints and ensure all necessary parameters are available for a complete optimization model"
  }
}
