Iteration 1 - OR_EXPERT_REFINEMENT
Sequence: 3
Timestamp: 2025-07-25 22:32:00

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 theme gallery aims to maximize its annual revenue from exhibitions by optimizing the allocation of artists to exhibitions based on their popularity and ticket prices, while respecting constraints on artist availability and exhibition capacity.",
  "optimization_problem": "The gallery needs to decide how many exhibitions each artist should participate in to maximize total revenue, considering ticket prices and expected attendance, while ensuring no artist is overbooked and each exhibition has a minimum number of artists.",
  "objective": "maximize \u2211(Ticket_Price \u00d7 Attendance \u00d7 x_ij) where x_ij is the number of exhibitions artist i participates in in year j",
  "table_count": 3,
  "key_changes": [
    "Schema changes include creating tables for missing optimization requirements (Max_Exhibitions_Per_Artist, Min_Artists_Per_Exhibition, and decision variables x_ij). Business configuration logic updated with scalar parameters and formulas for optimization constraints and calculations."
  ],
  "math_consistency": "high",
  "next_iteration_focus": "Define and map missing constraint bounds and decision variables",
  "mapping_adequacy_summary": "needs_improvement"
}

CURRENT SCHEMA:
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating tables for missing optimization requirements (Max_Exhibitions_Per_Artist, Min_Artists_Per_Exhibition, and decision variables x_ij). Business configuration logic updated with scalar parameters and formulas for optimization constraints and calculations.

CREATE TABLE artist_exhibition_limits (
  Artist_ID INTEGER,
  Max_Exhibitions INTEGER
);

CREATE TABLE exhibition_artist_requirements (
  Exhibition_ID INTEGER,
  Min_Artists INTEGER
);

CREATE TABLE artist_exhibition_assignments (
  Artist_ID INTEGER,
  Exhibition_ID INTEGER,
  Num_Exhibitions INTEGER
);


```

CURRENT DATA DICTIONARY:
{
  "tables": {
    "artist_exhibition_limits": {
      "business_purpose": "Maximum number of exhibitions each artist can participate in",
      "optimization_role": "constraint_bounds",
      "columns": {
        "Artist_ID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each artist",
          "optimization_purpose": "Index for artist constraints",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "Max_Exhibitions": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum number of exhibitions the artist can participate in",
          "optimization_purpose": "Constraint bound for artist participation",
          "sample_values": [
            5,
            6,
            7
          ]
        }
      }
    },
    "exhibition_artist_requirements": {
      "business_purpose": "Minimum number of artists required for each exhibition",
      "optimization_role": "constraint_bounds",
      "columns": {
        "Exhibition_ID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each exhibition",
          "optimization_purpose": "Index for exhibition constraints",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "Min_Artists": {
          "data_type": "INTEGER",
          "business_meaning": "Minimum number of artists required for the exhibition",
          "optimization_purpose": "Constraint bound for exhibition requirements",
          "sample_values": [
            3,
            4,
            5
          ]
        }
      }
    },
    "artist_exhibition_assignments": {
      "business_purpose": "Number of exhibitions each artist participates in per year",
      "optimization_role": "decision_variables",
      "columns": {
        "Artist_ID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each artist",
          "optimization_purpose": "Index for artist participation",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "Exhibition_ID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each exhibition",
          "optimization_purpose": "Index for exhibition participation",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "Num_Exhibitions": {
          "data_type": "INTEGER",
          "business_meaning": "Number of exhibitions the artist participates in",
          "optimization_purpose": "Decision variable for optimization",
          "sample_values": [
            1,
            2,
            3
          ]
        }
      }
    }
  }
}


CURRENT BUSINESS CONFIGURATION LOGIC:
{
  "Max_Exhibitions_Per_Artist": {
    "sample_value": 5,
    "data_type": "INTEGER",
    "business_meaning": "Maximum number of exhibitions an artist can participate in",
    "optimization_role": "Constraint bound for artist participation",
    "configuration_type": "scalar_parameter"
  },
  "Min_Artists_Per_Exhibition": {
    "sample_value": 3,
    "data_type": "INTEGER",
    "business_meaning": "Minimum number of artists required for an exhibition",
    "optimization_role": "Constraint bound for exhibition requirements",
    "configuration_type": "scalar_parameter"
  },
  "Revenue_Calculation": {
    "formula_expression": "Ticket_Price * Attendance * x_ij",
    "data_type": "STRING",
    "business_meaning": "Total revenue calculation for the gallery",
    "optimization_role": "Objective function for optimization",
    "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": "theme_gallery",
  "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": "theme_gallery",
  "iteration": 1,
  "business_context": "A theme gallery aims to maximize its annual revenue from exhibitions by optimizing the allocation of artists to exhibitions based on their popularity and ticket prices, while respecting constraints on artist availability and exhibition capacity.",
  "optimization_problem_description": "The gallery needs to decide how many exhibitions each artist should participate in to maximize total revenue, considering ticket prices and expected attendance, while ensuring no artist is overbooked and each exhibition has a minimum number of artists.",
  "optimization_formulation": {
    "objective": "maximize \u2211(Ticket_Price_j * Attendance_ij * x_ij) where x_ij is the number of exhibitions artist i participates in in year j",
    "decision_variables": "x_ij: number of exhibitions artist i participates in in year j (integer)",
    "constraints": "\u2211(x_ij) \u2264 Max_Exhibitions_Per_Artist_i for all i, \u2211(x_ij) \u2265 Min_Artists_Per_Exhibition_j for all j, x_ij \u2265 0 for all i, j"
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "Ticket_Price_j": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "Ticket price for exhibition j"
      },
      "Attendance_ij": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "Expected attendance for artist i in exhibition j"
      }
    },
    "constraint_bounds": {
      "Max_Exhibitions_Per_Artist_i": {
        "currently_mapped_to": "artist_exhibition_limits.Max_Exhibitions",
        "mapping_adequacy": "good",
        "description": "Maximum number of exhibitions artist i can participate in"
      },
      "Min_Artists_Per_Exhibition_j": {
        "currently_mapped_to": "exhibition_artist_requirements.Min_Artists",
        "mapping_adequacy": "good",
        "description": "Minimum number of artists required for exhibition j"
      }
    },
    "decision_variables": {
      "x_ij": {
        "currently_mapped_to": "artist_exhibition_assignments.Num_Exhibitions",
        "mapping_adequacy": "good",
        "description": "Number of exhibitions artist i participates in in year j",
        "variable_type": "integer"
      }
    }
  },
  "missing_optimization_requirements": [
    "Ticket_Price_j",
    "Attendance_ij"
  ],
  "iteration_status": {
    "complete": false,
    "confidence": "medium",
    "next_focus": "Define and map missing objective coefficients (Ticket_Price_j and Attendance_ij)"
  }
}
