Iteration 1 - OR_EXPERT_REFINEMENT
Sequence: 3
Timestamp: 2025-07-27 22:02:29

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 cinema chain wants to maximize its revenue by optimally scheduling films across its cinemas, considering the capacity of each cinema and the number of show times available per day.",
  "optimization_problem": "The goal is to maximize the total revenue from film screenings across all cinemas by deciding how many times each film should be shown in each cinema, subject to constraints on cinema capacity and available show times.",
  "objective": "maximize total_revenue = sum(schedule[Cinema_ID, Film_ID] * schedule[Price])",
  "table_count": 1,
  "key_changes": [
    "Schema changes include adding a new table for decision variables, modifying existing tables to fill mapping gaps, and updating configuration logic for scalar parameters and formulas."
  ],
  "math_consistency": "high",
  "next_iteration_focus": "Refine the estimation of demand and incorporate additional constraints based on historical data",
  "mapping_adequacy_summary": "partially_adequate"
}

CURRENT SCHEMA:
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include adding a new table for decision variables, modifying existing tables to fill mapping gaps, and updating configuration logic for scalar parameters and formulas.

CREATE TABLE cinema (
  Cinema_ID INTEGER,
  Capacity INTEGER,
  Max_Screenings_Per_Day INTEGER
);

CREATE TABLE film_schedule (
  Cinema_ID_Film_ID INTEGER,
  Show_Times INTEGER
);


```

CURRENT DATA DICTIONARY:
{
  "tables": {
    "cinema": {
      "business_purpose": "Stores information about each cinema",
      "optimization_role": "constraint_bounds",
      "columns": {
        "Cinema_ID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each cinema",
          "optimization_purpose": "Used to link cinema data with schedules",
          "sample_values": "1, 2, 3"
        },
        "Capacity": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum number of people that can be accommodated",
          "optimization_purpose": "Constraint for scheduling",
          "sample_values": "100, 150, 200"
        },
        "Max_Screenings_Per_Day": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum screenings possible per day",
          "optimization_purpose": "Constraint for scheduling",
          "sample_values": "5, 6, 7"
        }
      }
    },
    "film_schedule": {
      "business_purpose": "Stores the number of times each film is shown in each cinema",
      "optimization_role": "decision_variables",
      "columns": {
        "Cinema_ID_Film_ID": {
          "data_type": "INTEGER",
          "business_meaning": "Composite key for cinema and film",
          "optimization_purpose": "Decision variable for scheduling",
          "sample_values": "101, 102, 103"
        },
        "Show_Times": {
          "data_type": "INTEGER",
          "business_meaning": "Number of times a film is shown",
          "optimization_purpose": "Decision variable for scheduling",
          "sample_values": "2, 3, 4"
        }
      }
    }
  }
}


CURRENT BUSINESS CONFIGURATION LOGIC:
{
  "max_screenings_per_day": {
    "sample_value": "5",
    "data_type": "INTEGER",
    "business_meaning": "Maximum number of screenings possible per day for each cinema",
    "optimization_role": "Constraint parameter for scheduling",
    "configuration_type": "scalar_parameter"
  },
  "demand_estimation": {
    "formula_expression": "historical_demand * adjustment_factor",
    "data_type": "STRING",
    "business_meaning": "Estimates demand for each film based on historical data",
    "optimization_role": "Used to refine revenue estimation",
    "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": "cinema",
  "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": "cinema",
  "iteration": 1,
  "business_context": "A cinema chain aims to maximize its revenue by optimally scheduling films across its cinemas, considering the capacity of each cinema and the number of show times available per day.",
  "optimization_problem_description": "The goal is to maximize the total revenue from film screenings across all cinemas by deciding how many times each film should be shown in each cinema, subject to constraints on cinema capacity and available show times.",
  "optimization_formulation": {
    "objective": "maximize total_revenue = sum(schedule[Cinema_ID_Film_ID] * price[Cinema_ID_Film_ID])",
    "decision_variables": "schedule[Cinema_ID_Film_ID] represents the number of times a film is shown in a cinema, integer",
    "constraints": [
      "sum(schedule[Cinema_ID_Film_ID]) <= cinema[Max_Screenings_Per_Day] for each Cinema_ID",
      "schedule[Cinema_ID_Film_ID] * average_attendance <= cinema[Capacity] for each Cinema_ID_Film_ID"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "price[Cinema_ID_Film_ID]": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "The price per screening for each film in each cinema"
      }
    },
    "constraint_bounds": {
      "Max_Screenings_Per_Day[Cinema_ID]": {
        "currently_mapped_to": "cinema.Max_Screenings_Per_Day",
        "mapping_adequacy": "good",
        "description": "Maximum number of screenings allowed per day for each cinema"
      },
      "Capacity[Cinema_ID]": {
        "currently_mapped_to": "cinema.Capacity",
        "mapping_adequacy": "good",
        "description": "Maximum capacity of each cinema"
      }
    },
    "decision_variables": {
      "schedule[Cinema_ID_Film_ID]": {
        "currently_mapped_to": "film_schedule.Show_Times",
        "mapping_adequacy": "good",
        "description": "Number of times a film is shown in a cinema",
        "variable_type": "integer"
      }
    }
  },
  "missing_optimization_requirements": [
    "price[Cinema_ID_Film_ID]"
  ],
  "iteration_status": {
    "complete": false,
    "confidence": "medium",
    "next_focus": "Incorporate pricing data for each film in each cinema to complete the linear formulation"
  }
}
