Iteration 1 - OR_EXPERT_REFINEMENT
Sequence: 3
Timestamp: 2025-07-25 22:25:45

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": "Optimize the allocation of singers to concerts in different stadiums to maximize total audience capacity while respecting singer availability and stadium constraints.",
  "optimization_problem": "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.",
  "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.",
  "table_count": 1,
  "key_changes": [
    "Schema changes include creating a singer_concert_assignment table for decision variables, adding singer_limit to business_configuration_logic, and ensuring stadium capacity is correctly mapped. Configuration logic updates include scalar parameters for singer limits and formulas for audience capacity calculations."
  ],
  "math_consistency": "high",
  "next_iteration_focus": "Refine constraints and decision variables to better align with the schema and business requirements.",
  "mapping_adequacy_summary": "partially_adequate"
}

CURRENT SCHEMA:
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating a singer_concert_assignment table for decision variables, adding singer_limit to business_configuration_logic, and ensuring stadium capacity is correctly mapped. Configuration logic updates include scalar parameters for singer limits and formulas for audience capacity calculations.

CREATE TABLE stadium (
  Capacity INTEGER
);

CREATE TABLE singer_concert_assignment (
  assignment BOOLEAN
);


```

CURRENT DATA DICTIONARY:
{
  "tables": {
    "stadium": {
      "business_purpose": "Stadium details including capacity.",
      "optimization_role": "objective_coefficients",
      "columns": {
        "Capacity": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum number of singers a stadium can accommodate.",
          "optimization_purpose": "Coefficient in the objective function.",
          "sample_values": "10000, 20000, 30000"
        }
      }
    },
    "singer_concert_assignment": {
      "business_purpose": "Assignments of singers to concerts in stadiums.",
      "optimization_role": "decision_variables",
      "columns": {
        "assignment": {
          "data_type": "BOOLEAN",
          "business_meaning": "Binary variable indicating if singer is assigned to concert in stadium.",
          "optimization_purpose": "Decision variable in the optimization model.",
          "sample_values": "0, 1"
        }
      }
    }
  }
}


CURRENT BUSINESS CONFIGURATION LOGIC:
{
  "singer_limit": {
    "sample_value": 1,
    "data_type": "INTEGER",
    "business_meaning": "Maximum number of concerts a singer can participate in.",
    "optimization_role": "Constraint bound for singer assignments.",
    "configuration_type": "scalar_parameter"
  },
  "total_audience_capacity_formula": {
    "formula_expression": "sum(Capacity_s * x_sc)",
    "data_type": "STRING",
    "business_meaning": "Total audience capacity across all concerts.",
    "optimization_role": "Objective function calculation.",
    "configuration_type": "business_logic_formula"
  }
}


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

JSON STRUCTURE REQUIRED:

{
  "database_id": "concert_singer",
  "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": "concert_singer",
  "iteration": 1,
  "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": "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": "1. \u2211(x_sc) \u2264 singer_limit for each singer c. 2. \u2211(x_sc) \u2264 Capacity_s for each stadium s."
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "Capacity_s": {
        "currently_mapped_to": "stadium.Capacity",
        "mapping_adequacy": "good",
        "description": "Capacity of stadium s used as coefficient in the objective function."
      }
    },
    "constraint_bounds": {
      "singer_limit": {
        "currently_mapped_to": "business_configuration_logic.singer_limit",
        "mapping_adequacy": "good",
        "description": "Maximum number of concerts a singer can participate in."
      },
      "Capacity_s": {
        "currently_mapped_to": "stadium.Capacity",
        "mapping_adequacy": "good",
        "description": "Maximum number of singers a stadium can accommodate."
      }
    },
    "decision_variables": {
      "x_sc": {
        "currently_mapped_to": "singer_concert_assignment.assignment",
        "mapping_adequacy": "good",
        "description": "Binary variable indicating if singer c is assigned to concert in stadium s.",
        "variable_type": "binary"
      }
    }
  },
  "missing_optimization_requirements": [],
  "iteration_status": {
    "complete": true,
    "confidence": "high",
    "next_focus": "Ready for convergence"
  }
}
