Iteration 1 - OR_EXPERT_REFINEMENT
Sequence: 3
Timestamp: 2025-07-27 23:05:38

Prompt:
You are an Operations Research (OR) expert in iteration 1 of an alternating optimization process. The algorithm alternates between OR expert analysis and data engineering implementation until convergence.

CRITICAL MATHEMATICAL CONSTRAINTS FOR LINEAR/MIXED-INTEGER PROGRAMMING:
- The optimization problem MUST remain 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
- If previous iteration introduced nonlinear elements, redesign as linear formulation
- Maintain between 2 and 20 constraints for optimization feasibility

YOUR SCOPE: Focus exclusively on optimization modeling and mapping analysis. Do NOT propose database changes.
ROW COUNT AWARENESS: Understand that data engineer applies 3-row minimum rule - insufficient table data gets moved to business_configuration_logic.json.


DATA AVAILABILITY CHECK: 
Before listing missing requirements, verify:
- Check current schema for required data columns
- Check business configuration logic for required parameters  
- Only list as "missing" if data is truly unavailable
- If all mappings are "good", missing_requirements should be []

CONSISTENCY RULES:
- IF all mapping_adequacy == "good" THEN missing_optimization_requirements = []
- IF missing_optimization_requirements = [] THEN complete CAN be true
- IF complete == true THEN confidence should be "high"

SELF-CHECK: Before responding, verify:
1. Does current schema contain the data I claim is missing?
2. Are my mapping assessments consistent with missing requirements?
3. Is my complete status consistent with missing requirements?

MAPPING COMPLETENESS CHECK: Ensure logical consistency between:
- All objective coefficients mapped with adequacy evaluation
- All constraint bounds mapped with adequacy evaluation  
- All decision variables mapped with adequacy evaluation
- Missing requirements list matches inadequate mappings only


CRITICAL: Respond with ONLY a valid JSON object. No explanations, no markdown, no extra text.



CURRENT STATE (iteration 0):
{
  "iteration": 1,
  "converged": false,
  "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": "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.",
  "objective": "maximize total_pieces = \u2211(Num_of_Pieces[Member_ID, Performance_ID])",
  "table_count": 1,
  "key_changes": [
    "Incorporated member availability data into the schema and moved scalar parameters to configuration logic. Adjusted schema to align with optimization requirements."
  ],
  "math_consistency": "high",
  "next_iteration_focus": "Incorporate member availability data into the optimization model",
  "mapping_adequacy_summary": "partially_adequate"
}

CURRENT SCHEMA:
```sql
-- Iteration 1 Database Schema
-- Objective: Incorporated member availability data into the schema and moved scalar parameters to configuration logic. Adjusted schema to align with optimization requirements.

CREATE TABLE member_attendance (
  Member_ID INTEGER,
  Performance_ID INTEGER,
  Num_of_Pieces INTEGER
);

CREATE TABLE performance (
  Performance_ID INTEGER,
  Attendance INTEGER
);

CREATE TABLE member_availability (
  Member_ID INTEGER,
  Availability INTEGER
);


```

CURRENT DATA DICTIONARY:
{
  "tables": {
    "member_attendance": {
      "business_purpose": "Tracks the number of pieces each member performs in each performance",
      "optimization_role": "decision_variables",
      "columns": {
        "Member_ID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each member",
          "optimization_purpose": "Index for decision variables",
          "sample_values": "1, 2, 3"
        },
        "Performance_ID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each performance",
          "optimization_purpose": "Index for decision variables",
          "sample_values": "101, 102, 103"
        },
        "Num_of_Pieces": {
          "data_type": "INTEGER",
          "business_meaning": "Number of pieces performed by a member in a performance",
          "optimization_purpose": "Decision variable value",
          "sample_values": "0, 1, 2"
        }
      }
    },
    "performance": {
      "business_purpose": "Stores information about each performance",
      "optimization_role": "constraint_bounds",
      "columns": {
        "Performance_ID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each performance",
          "optimization_purpose": "Index for constraints",
          "sample_values": "101, 102, 103"
        },
        "Attendance": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum number of pieces that can be performed in a performance",
          "optimization_purpose": "Constraint bound",
          "sample_values": "50, 60, 70"
        }
      }
    },
    "member_availability": {
      "business_purpose": "Stores the availability of each member",
      "optimization_role": "constraint_bounds",
      "columns": {
        "Member_ID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each member",
          "optimization_purpose": "Index for constraints",
          "sample_values": "1, 2, 3"
        },
        "Availability": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum number of pieces a member is available to perform",
          "optimization_purpose": "Constraint bound",
          "sample_values": "5, 10, 15"
        }
      }
    }
  }
}


CURRENT BUSINESS CONFIGURATION LOGIC:
{
  "attendance_limit": {
    "sample_value": "100",
    "data_type": "INTEGER",
    "business_meaning": "Maximum number of attendees allowed per performance",
    "optimization_role": "Used as a constraint bound in optimization model",
    "configuration_type": "scalar_parameter"
  },
  "availability_threshold": {
    "sample_value": "10",
    "data_type": "INTEGER",
    "business_meaning": "Maximum number of pieces a member can perform",
    "optimization_role": "Used as a constraint bound in optimization model",
    "configuration_type": "scalar_parameter"
  }
}


TASK: Refine the optimization problem formulation by analyzing current data schema mapping and identifying requirements while maintaining LINEAR structure.

JSON STRUCTURE REQUIRED:

{
  "database_id": "performance_attendance",
  "iteration": 1,
  "business_context": "Updated realistic business scenario description that supports linear optimization",
  "optimization_problem_description": "Refined description of LINEAR optimization problem", 
  "optimization_formulation": {
    "objective": "refined linear minimize/maximize with mathematical precision (sum of weighted variables only)",
    "decision_variables": "clearly defined controllable linear variables (continuous or integer)",
    "constraints": "mathematically precise LINEAR constraint definitions (no variable products or divisions) - maintain 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 still needed for linear formulation (should be [] if all mappings are good)"
  ],
  
  "iteration_status": {
    "complete": true/false based on linear formulation completeness,
    "confidence": "updated confidence level",
    "next_focus": "What needs refinement for linear optimization OR 'Ready for convergence'"
  }
}

Response:
{
  "database_id": "performance_attendance",
  "iteration": 1,
  "business_context": "A theater company is optimizing the allocation of its members to performances to maximize the total number of pieces performed, 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] are integer variables representing the number of pieces performed by a member in a performance",
    "constraints": [
      "\u2211(Num_of_Pieces[Member_ID, Performance_ID]) \u2264 Attendance[Performance_ID] for each Performance_ID",
      "\u2211(Num_of_Pieces[Member_ID, Performance_ID]) \u2264 Availability[Member_ID] for each Member_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 performed by a member in a 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": "member_availability.Availability",
        "mapping_adequacy": "good",
        "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 performed by a member in a performance",
        "variable_type": "integer"
      }
    }
  },
  "missing_optimization_requirements": [],
  "iteration_status": {
    "complete": true,
    "confidence": "high",
    "next_focus": "Ready for convergence"
  }
}
