Iteration 1 - OR_EXPERT_REFINEMENT
Sequence: 3
Timestamp: 2025-07-25 22:28:37

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 maximize the total audience engagement across multiple musicals by optimally assigning actors to roles based on their age and performance duration, while ensuring that each actor is assigned to at most one role and each role is filled by exactly one actor.",
  "optimization_problem": "The objective is to maximize the total audience engagement, which is a weighted sum of the actors' ages and their performance durations. The constraints ensure that each actor is assigned to at most one role, each role is filled by exactly one actor, and the total duration of all performances does not exceed a given limit.",
  "objective": "maximize \u2211(w1 * age[i] + w2 * duration[i]) * x[i,j]",
  "table_count": 3,
  "key_changes": [
    "Schema changes include creating tables for roles, actors, and assignments. Configuration logic updates include adding weights for engagement metric and maximum duration."
  ],
  "math_consistency": "high",
  "next_iteration_focus": "Define the weights for the engagement metric and the maximum allowed total duration D",
  "mapping_adequacy_summary": "needs_improvement"
}

CURRENT SCHEMA:
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating tables for roles, actors, and assignments. Configuration logic updates include adding weights for engagement metric and maximum duration.

CREATE TABLE roles (
  role_id INTEGER,
  musical_id INTEGER,
  role_name STRING
);

CREATE TABLE actors (
  actor_id INTEGER,
  age INTEGER,
  performance_duration INTEGER
);

CREATE TABLE assignments (
  assignment_id INTEGER,
  actor_id INTEGER,
  role_id INTEGER,
  assignment_status BOOLEAN
);


```

CURRENT DATA DICTIONARY:
{
  "tables": {
    "roles": {
      "business_purpose": "List of roles and their corresponding musicals",
      "optimization_role": "business_data",
      "columns": {
        "role_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each role",
          "optimization_purpose": "identifier for roles in assignments",
          "sample_values": "1, 2, 3"
        },
        "musical_id": {
          "data_type": "INTEGER",
          "business_meaning": "Identifier for the musical the role belongs to",
          "optimization_purpose": "grouping roles by musical",
          "sample_values": "1, 2, 3"
        },
        "role_name": {
          "data_type": "STRING",
          "business_meaning": "Name of the role",
          "optimization_purpose": "descriptive information",
          "sample_values": "Lead, Supporting, Chorus"
        }
      }
    },
    "actors": {
      "business_purpose": "List of actors with their ages and performance durations",
      "optimization_role": "business_data",
      "columns": {
        "actor_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each actor",
          "optimization_purpose": "identifier for actors in assignments",
          "sample_values": "1, 2, 3"
        },
        "age": {
          "data_type": "INTEGER",
          "business_meaning": "Age of the actor",
          "optimization_purpose": "used in the engagement metric",
          "sample_values": "25, 30, 35"
        },
        "performance_duration": {
          "data_type": "INTEGER",
          "business_meaning": "Duration of the actor's performance",
          "optimization_purpose": "used in the engagement metric",
          "sample_values": "10, 15, 20"
        }
      }
    },
    "assignments": {
      "business_purpose": "Assignments of actors to roles",
      "optimization_role": "decision_variables",
      "columns": {
        "assignment_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each assignment",
          "optimization_purpose": "identifier for assignments",
          "sample_values": "1, 2, 3"
        },
        "actor_id": {
          "data_type": "INTEGER",
          "business_meaning": "Identifier for the actor assigned",
          "optimization_purpose": "links to actors table",
          "sample_values": "1, 2, 3"
        },
        "role_id": {
          "data_type": "INTEGER",
          "business_meaning": "Identifier for the role assigned",
          "optimization_purpose": "links to roles table",
          "sample_values": "1, 2, 3"
        },
        "assignment_status": {
          "data_type": "BOOLEAN",
          "business_meaning": "Indicates if the actor is assigned to the role",
          "optimization_purpose": "binary decision variable",
          "sample_values": "true, false"
        }
      }
    }
  }
}


CURRENT BUSINESS CONFIGURATION LOGIC:
{
  "w1": {
    "sample_value": 0.5,
    "data_type": "FLOAT",
    "business_meaning": "weight for actor's age in the engagement metric",
    "optimization_role": "coefficient in the objective function",
    "configuration_type": "scalar_parameter"
  },
  "w2": {
    "sample_value": 0.5,
    "data_type": "FLOAT",
    "business_meaning": "weight for actor's performance duration in the engagement metric",
    "optimization_role": "coefficient in the objective function",
    "configuration_type": "scalar_parameter"
  },
  "D": {
    "sample_value": 120,
    "data_type": "INTEGER",
    "business_meaning": "maximum allowed total duration of all performances",
    "optimization_role": "bound in the duration constraint",
    "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": "musical",
  "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": "musical",
  "iteration": 1,
  "business_context": "A theater company wants to maximize the total audience engagement across multiple musicals by optimally assigning actors to roles based on their age and performance duration, while ensuring that each actor is assigned to at most one role and each role is filled by exactly one actor, and the total duration of all performances does not exceed a given limit.",
  "optimization_problem_description": "The objective is to maximize the total audience engagement, which is a weighted sum of the actors' ages and their performance durations. The constraints ensure that each actor is assigned to at most one role, each role is filled by exactly one actor, and the total duration of all performances does not exceed a given limit.",
  "optimization_formulation": {
    "objective": "maximize \u2211(w1 * age[i] + w2 * performance_duration[i]) * x[i,j]",
    "decision_variables": "x[i,j] = binary decision variable indicating if actor i is assigned to role j",
    "constraints": [
      "\u2211x[i,j] <= 1 for all i (each actor is assigned to at most one role)",
      "\u2211x[i,j] = 1 for all j (each role is filled by exactly one actor)",
      "\u2211(performance_duration[i] * x[i,j]) <= D (total duration of all performances does not exceed D)"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "w1": {
        "currently_mapped_to": "business_configuration_logic.w1",
        "mapping_adequacy": "good",
        "description": "weight for actor's age in the engagement metric"
      },
      "w2": {
        "currently_mapped_to": "business_configuration_logic.w2",
        "mapping_adequacy": "good",
        "description": "weight for actor's performance duration in the engagement metric"
      }
    },
    "constraint_bounds": {
      "D": {
        "currently_mapped_to": "business_configuration_logic.D",
        "mapping_adequacy": "good",
        "description": "maximum allowed total duration of all performances"
      }
    },
    "decision_variables": {
      "x[i,j]": {
        "currently_mapped_to": "assignments.assignment_status",
        "mapping_adequacy": "good",
        "description": "binary decision variable indicating if actor i is assigned to role j",
        "variable_type": "binary"
      }
    }
  },
  "missing_optimization_requirements": [],
  "iteration_status": {
    "complete": true,
    "confidence": "high",
    "next_focus": "Ready for convergence"
  }
}
