Iteration 1 - OR_EXPERT_REFINEMENT
Sequence: 3
Timestamp: 2025-07-25 22:26:20

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 music label wants to maximize the total sales of songs by selecting the optimal set of singers to promote, considering their net worth and the potential sales of their songs.",
  "optimization_problem": "The objective is to maximize the total sales of songs by selecting a subset of singers, subject to constraints on the total net worth of the selected singers and the number of singers that can be promoted.",
  "objective": "maximize \u2211(Sales_i * x_i) where x_i is a binary decision variable indicating whether singer i is selected",
  "table_count": 1,
  "key_changes": [
    "Schema changes include creating a new table for decision variables and updating the song table to include sales data. Configuration logic updates include adding scalar parameters for Total_Budget and Max_Singers."
  ],
  "math_consistency": "high",
  "next_iteration_focus": "Define the missing parameters (Total_Budget and Max_Singers) and ensure the decision variables are properly mapped.",
  "mapping_adequacy_summary": "needs_improvement"
}

CURRENT SCHEMA:
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating a new table for decision variables and updating the song table to include sales data. Configuration logic updates include adding scalar parameters for Total_Budget and Max_Singers.

CREATE TABLE singer_selection (
  singer_id INTEGER,
  is_selected BOOLEAN
);

CREATE TABLE song (
  song_id INTEGER,
  singer_id INTEGER,
  Sales INTEGER
);


```

CURRENT DATA DICTIONARY:
{
  "tables": {
    "singer_selection": {
      "business_purpose": "Represents the selection of singers for promotion.",
      "optimization_role": "decision_variables",
      "columns": {
        "singer_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each singer.",
          "optimization_purpose": "Links to the singer being considered for selection.",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "is_selected": {
          "data_type": "BOOLEAN",
          "business_meaning": "Indicates whether the singer is selected for promotion.",
          "optimization_purpose": "Binary decision variable x_i.",
          "sample_values": [
            true,
            false,
            true
          ]
        }
      }
    },
    "song": {
      "business_purpose": "Represents the songs and their sales data.",
      "optimization_role": "objective_coefficients",
      "columns": {
        "song_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each song.",
          "optimization_purpose": "Links to the song being considered.",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "singer_id": {
          "data_type": "INTEGER",
          "business_meaning": "Identifier for the singer of the song.",
          "optimization_purpose": "Links the song to the singer.",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "Sales": {
          "data_type": "INTEGER",
          "business_meaning": "Sales of the song.",
          "optimization_purpose": "Objective coefficient Sales_i.",
          "sample_values": [
            1000,
            2000,
            3000
          ]
        }
      }
    }
  }
}


CURRENT BUSINESS CONFIGURATION LOGIC:
{
  "Total_Budget": {
    "sample_value": 100,
    "data_type": "INTEGER",
    "business_meaning": "The total budget available for promoting singers.",
    "optimization_role": "Constraint bound for total net worth of selected singers.",
    "configuration_type": "scalar_parameter"
  },
  "Max_Singers": {
    "sample_value": 5,
    "data_type": "INTEGER",
    "business_meaning": "The maximum number of singers that can be promoted.",
    "optimization_role": "Constraint bound for the number of selected singers.",
    "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": "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": "singer",
  "iteration": 1,
  "business_context": "A music label aims to maximize the total sales of songs by selecting the optimal set of singers to promote, considering their net worth and the potential sales of their songs, while adhering to budget and promotional capacity constraints.",
  "optimization_problem_description": "Maximize the total sales of songs by selecting a subset of singers, subject to constraints on the total net worth of the selected singers and the maximum number of singers that can be promoted.",
  "optimization_formulation": {
    "objective": "maximize \u2211(Sales_i * x_i) where x_i is a binary decision variable indicating whether singer i is selected",
    "decision_variables": "x_i \u2208 {0, 1} for each singer i, indicating selection status",
    "constraints": "\u2211(Net_Worth_i * x_i) \u2264 Total_Budget, \u2211(x_i) \u2264 Max_Singers"
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "Sales_i": {
        "currently_mapped_to": "song.Sales",
        "mapping_adequacy": "good",
        "description": "Sales of the song associated with singer i"
      }
    },
    "constraint_bounds": {
      "Total_Budget": {
        "currently_mapped_to": "business_configuration_logic.Total_Budget",
        "mapping_adequacy": "good",
        "description": "Total budget available for promoting singers"
      },
      "Max_Singers": {
        "currently_mapped_to": "business_configuration_logic.Max_Singers",
        "mapping_adequacy": "good",
        "description": "Maximum number of singers that can be promoted"
      }
    },
    "decision_variables": {
      "x_i": {
        "currently_mapped_to": "singer_selection.is_selected",
        "mapping_adequacy": "good",
        "description": "Binary decision variable indicating whether singer i is selected",
        "variable_type": "binary"
      }
    }
  },
  "missing_optimization_requirements": [],
  "iteration_status": {
    "complete": true,
    "confidence": "high",
    "next_focus": "Ready for convergence"
  }
}
