Iteration 1 - OR_EXPERT_REFINEMENT
Sequence: 3
Timestamp: 2025-07-25 22:41:49

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 students to clubs to maximize overall student engagement while respecting club capacity and student preferences.",
  "optimization_problem": "The goal is to maximize the total engagement score of students in clubs, where engagement is a weighted sum of student participation and club importance. Constraints include club capacity limits and ensuring each student is assigned to at most one club.",
  "objective": "maximize \u2211(engagement_score[StuID, ClubID] * x[StuID, ClubID])",
  "table_count": 2,
  "key_changes": [
    "Schema changes include creating tables for engagement scores and club capacities, and updating configuration logic for scalar parameters and formulas."
  ],
  "math_consistency": "high",
  "next_iteration_focus": "Define engagement scores and club capacities for complete optimization model",
  "mapping_adequacy_summary": "needs_improvement"
}

CURRENT SCHEMA:
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating tables for engagement scores and club capacities, and updating configuration logic for scalar parameters and formulas.

CREATE TABLE Engagement_Scores (
  StuID INTEGER,
  ClubID INTEGER,
  score FLOAT
);

CREATE TABLE Club_Capacities (
  ClubID INTEGER,
  capacity INTEGER
);

CREATE TABLE Member_of_club (
  StuID INTEGER,
  ClubID INTEGER
);


```

CURRENT DATA DICTIONARY:
{
  "tables": {
    "Engagement_Scores": {
      "business_purpose": "Stores engagement scores for each student-club pair",
      "optimization_role": "objective_coefficients",
      "columns": {
        "StuID": {
          "data_type": "INTEGER",
          "business_meaning": "Student ID",
          "optimization_purpose": "Identifies the student in the engagement score",
          "sample_values": "1, 2, 3"
        },
        "ClubID": {
          "data_type": "INTEGER",
          "business_meaning": "Club ID",
          "optimization_purpose": "Identifies the club in the engagement score",
          "sample_values": "101, 102, 103"
        },
        "score": {
          "data_type": "FLOAT",
          "business_meaning": "Engagement score of the student in the club",
          "optimization_purpose": "Used as a coefficient in the objective function",
          "sample_values": "0.5, 0.7, 0.9"
        }
      }
    },
    "Club_Capacities": {
      "business_purpose": "Stores maximum number of students allowed in each club",
      "optimization_role": "constraint_bounds",
      "columns": {
        "ClubID": {
          "data_type": "INTEGER",
          "business_meaning": "Club ID",
          "optimization_purpose": "Identifies the club in the capacity constraint",
          "sample_values": "101, 102, 103"
        },
        "capacity": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum number of students allowed in the club",
          "optimization_purpose": "Used as a bound in the capacity constraint",
          "sample_values": "20, 25, 30"
        }
      }
    },
    "Member_of_club": {
      "business_purpose": "Stores which students are members of which clubs",
      "optimization_role": "decision_variables",
      "columns": {
        "StuID": {
          "data_type": "INTEGER",
          "business_meaning": "Student ID",
          "optimization_purpose": "Identifies the student in the decision variable",
          "sample_values": "1, 2, 3"
        },
        "ClubID": {
          "data_type": "INTEGER",
          "business_meaning": "Club ID",
          "optimization_purpose": "Identifies the club in the decision variable",
          "sample_values": "101, 102, 103"
        }
      }
    }
  }
}


CURRENT BUSINESS CONFIGURATION LOGIC:
{
  "default_club_capacity": {
    "sample_value": 20,
    "data_type": "INTEGER",
    "business_meaning": "Default maximum number of students allowed in a club",
    "optimization_role": "Used as a constraint bound in the optimization model",
    "configuration_type": "scalar_parameter"
  },
  "engagement_score_formula": {
    "formula_expression": "participation * club_importance",
    "data_type": "STRING",
    "business_meaning": "Formula to calculate engagement score",
    "optimization_role": "Used to compute objective coefficients",
    "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": "club_1",
  "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": "club_1",
  "iteration": 1,
  "business_context": "Optimize the allocation of students to clubs to maximize overall student engagement while respecting club capacity and ensuring each student is assigned to at most one club.",
  "optimization_problem_description": "Maximize the total engagement score of students in clubs, where engagement is a weighted sum of student participation and club importance. Constraints include club capacity limits and ensuring each student is assigned to at most one club.",
  "optimization_formulation": {
    "objective": "maximize \u2211(engagement_score[StuID, ClubID] * x[StuID, ClubID])",
    "decision_variables": "x[StuID, ClubID] \u2208 {0, 1} (binary variable indicating if student StuID is assigned to club ClubID)",
    "constraints": [
      "\u2211(x[StuID, ClubID] for all StuID) \u2264 capacity[ClubID] for all ClubID (club capacity constraint)",
      "\u2211(x[StuID, ClubID] for all ClubID) \u2264 1 for all StuID (student assignment constraint)"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "engagement_score[StuID, ClubID]": {
        "currently_mapped_to": "Engagement_Scores.score",
        "mapping_adequacy": "good",
        "description": "Engagement score of student StuID in club ClubID"
      }
    },
    "constraint_bounds": {
      "capacity[ClubID]": {
        "currently_mapped_to": "Club_Capacities.capacity",
        "mapping_adequacy": "good",
        "description": "Maximum number of students allowed in club ClubID"
      }
    },
    "decision_variables": {
      "x[StuID, ClubID]": {
        "currently_mapped_to": "Member_of_club",
        "mapping_adequacy": "good",
        "description": "Binary decision variable indicating if student StuID is assigned to club ClubID",
        "variable_type": "binary"
      }
    }
  },
  "missing_optimization_requirements": [],
  "iteration_status": {
    "complete": true,
    "confidence": "high",
    "next_focus": "Ready for convergence"
  }
}
