Iteration 3 - OR_EXPERT_REFINEMENT
Sequence: 7
Timestamp: 2025-07-25 22:36:07

Prompt:
You are an Operations Research (OR) expert in iteration 3 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.


CONVERGENCE PRESSURE (Iteration 3/5):
- Focus on finalizing the linear optimization model
- Set complete=true if formulation is mathematically sound and linear
- Minimize further schema change requests


CURRENT STATE (iteration 2):
{
  "iteration": 3,
  "converged": false,
  "business_context": "A retail chain aims to optimize employee hiring and allocation across shops to minimize total hiring costs while ensuring each shop has sufficient staff and employees are fairly evaluated based on their bonuses. The optimization problem is linear, focusing on minimizing costs and meeting staffing requirements.",
  "optimization_problem": "Minimize the total hiring costs by deciding the number of full-time and part-time employees to hire for each shop, ensuring that each shop meets its staffing requirements and that employees receive bonuses based on their evaluations. The problem is formulated as a linear optimization problem with linear constraints.",
  "objective": "minimize \u2211(cost_full_time[shop_id] \u00d7 x_full_time[shop_id] + cost_part_time[shop_id] \u00d7 x_part_time[shop_id])",
  "table_count": 1,
  "key_changes": [
    "Schema changes include creating a new table for objective coefficients (cost_full_time and cost_part_time) and updating business configuration logic to ensure all optimization requirements are mapped. Existing tables and configuration logic are reviewed for completeness and alignment with OR expert's analysis."
  ],
  "math_consistency": "high",
  "next_iteration_focus": "Define and map the missing objective coefficients (cost_full_time and cost_part_time) to the schema or business configuration logic.",
  "mapping_adequacy_summary": "partially_adequate"
}

CURRENT SCHEMA:
```sql
-- Iteration 3 Database Schema
-- Objective: Schema changes include creating a new table for objective coefficients (cost_full_time and cost_part_time) and updating business configuration logic to ensure all optimization requirements are mapped. Existing tables and configuration logic are reviewed for completeness and alignment with OR expert's analysis.

CREATE TABLE shop_staff_requirements (
  shop_id INTEGER,
  min_staff INTEGER,
  max_full_time INTEGER,
  max_part_time INTEGER
);

CREATE TABLE employee_bonus_decisions (
  employee_id INTEGER,
  y BOOLEAN
);

CREATE TABLE employee_hiring_decisions (
  shop_id INTEGER,
  x_full_time INTEGER,
  x_part_time INTEGER
);

CREATE TABLE employee_costs (
  shop_id INTEGER,
  cost_full_time INTEGER,
  cost_part_time INTEGER
);


```

CURRENT DATA DICTIONARY:
{
  "tables": {
    "shop_staff_requirements": {
      "business_purpose": "Staffing requirements for each shop",
      "optimization_role": "constraint_bounds",
      "columns": {
        "shop_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each shop",
          "optimization_purpose": "Index for shop-specific constraints",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "min_staff": {
          "data_type": "INTEGER",
          "business_meaning": "Minimum staff required for the shop",
          "optimization_purpose": "Lower bound for staffing constraint",
          "sample_values": [
            5,
            6,
            7
          ]
        },
        "max_full_time": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum full-time employees allowed for the shop",
          "optimization_purpose": "Upper bound for full-time hiring constraint",
          "sample_values": [
            10,
            12,
            15
          ]
        },
        "max_part_time": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum part-time employees allowed for the shop",
          "optimization_purpose": "Upper bound for part-time hiring constraint",
          "sample_values": [
            8,
            10,
            12
          ]
        }
      }
    },
    "employee_bonus_decisions": {
      "business_purpose": "Decisions to award bonuses to employees",
      "optimization_role": "decision_variables",
      "columns": {
        "employee_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each employee",
          "optimization_purpose": "Index for bonus decision variable",
          "sample_values": [
            101,
            102,
            103
          ]
        },
        "y": {
          "data_type": "BOOLEAN",
          "business_meaning": "Binary decision to award bonus",
          "optimization_purpose": "Decision variable for bonus allocation",
          "sample_values": [
            true,
            false,
            true
          ]
        }
      }
    },
    "employee_hiring_decisions": {
      "business_purpose": "Decisions on the number of full-time and part-time employees to hire for each shop",
      "optimization_role": "decision_variables",
      "columns": {
        "shop_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each shop",
          "optimization_purpose": "Index for hiring decision variables",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "x_full_time": {
          "data_type": "INTEGER",
          "business_meaning": "Number of full-time employees to hire for the shop",
          "optimization_purpose": "Decision variable for full-time hiring",
          "sample_values": [
            2,
            3,
            4
          ]
        },
        "x_part_time": {
          "data_type": "INTEGER",
          "business_meaning": "Number of part-time employees to hire for the shop",
          "optimization_purpose": "Decision variable for part-time hiring",
          "sample_values": [
            3,
            4,
            5
          ]
        }
      }
    },
    "employee_costs": {
      "business_purpose": "Cost per full-time and part-time employee for each shop",
      "optimization_role": "objective_coefficients",
      "columns": {
        "shop_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each shop",
          "optimization_purpose": "Index for shop-specific costs",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "cost_full_time": {
          "data_type": "INTEGER",
          "business_meaning": "Cost per full-time employee for the shop",
          "optimization_purpose": "Objective coefficient for full-time hiring cost",
          "sample_values": [
            1500,
            1600,
            1700
          ]
        },
        "cost_part_time": {
          "data_type": "INTEGER",
          "business_meaning": "Cost per part-time employee for the shop",
          "optimization_purpose": "Objective coefficient for part-time hiring cost",
          "sample_values": [
            800,
            850,
            900
          ]
        }
      }
    }
  }
}


CURRENT BUSINESS CONFIGURATION LOGIC:
{
  "cost_full_time": {
    "sample_value": 1500,
    "data_type": "INTEGER",
    "business_meaning": "Cost per full-time employee for each shop",
    "optimization_role": "Objective coefficient for full-time hiring cost",
    "configuration_type": "scalar_parameter"
  },
  "cost_part_time": {
    "sample_value": 800,
    "data_type": "INTEGER",
    "business_meaning": "Cost per part-time employee for each shop",
    "optimization_role": "Objective coefficient for part-time hiring cost",
    "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": "employee_hire_evaluation",
  "iteration": 3,
  "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": "employee_hire_evaluation",
  "iteration": 3,
  "business_context": "A retail chain aims to optimize employee hiring and allocation across shops to minimize total hiring costs while ensuring each shop has sufficient staff and employees are fairly evaluated based on their bonuses. The optimization problem is linear, focusing on minimizing costs and meeting staffing requirements.",
  "optimization_problem_description": "Minimize the total hiring costs by deciding the number of full-time and part-time employees to hire for each shop, ensuring that each shop meets its staffing requirements and that employees receive bonuses based on their evaluations. The problem is formulated as a linear optimization problem with linear constraints.",
  "optimization_formulation": {
    "objective": "minimize \u2211(cost_full_time[shop_id] \u00d7 x_full_time[shop_id] + cost_part_time[shop_id] \u00d7 x_part_time[shop_id])",
    "decision_variables": "x_full_time[shop_id]: number of full-time employees to hire for shop_id (integer), x_part_time[shop_id]: number of part-time employees to hire for shop_id (integer), y[employee_id]: binary decision to award bonus to employee_id (binary)",
    "constraints": "x_full_time[shop_id] + x_part_time[shop_id] \u2265 min_staff[shop_id] for all shop_id, x_full_time[shop_id] \u2264 max_full_time[shop_id] for all shop_id, x_part_time[shop_id] \u2264 max_part_time[shop_id] for all shop_id"
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "cost_full_time[shop_id]": {
        "currently_mapped_to": "employee_costs.cost_full_time",
        "mapping_adequacy": "good",
        "description": "Cost per full-time employee for each shop"
      },
      "cost_part_time[shop_id]": {
        "currently_mapped_to": "employee_costs.cost_part_time",
        "mapping_adequacy": "good",
        "description": "Cost per part-time employee for each shop"
      }
    },
    "constraint_bounds": {
      "min_staff[shop_id]": {
        "currently_mapped_to": "shop_staff_requirements.min_staff",
        "mapping_adequacy": "good",
        "description": "Minimum staff required for each shop"
      },
      "max_full_time[shop_id]": {
        "currently_mapped_to": "shop_staff_requirements.max_full_time",
        "mapping_adequacy": "good",
        "description": "Maximum full-time employees allowed for each shop"
      },
      "max_part_time[shop_id]": {
        "currently_mapped_to": "shop_staff_requirements.max_part_time",
        "mapping_adequacy": "good",
        "description": "Maximum part-time employees allowed for each shop"
      }
    },
    "decision_variables": {
      "x_full_time[shop_id]": {
        "currently_mapped_to": "employee_hiring_decisions.x_full_time",
        "mapping_adequacy": "good",
        "description": "Number of full-time employees to hire for each shop",
        "variable_type": "integer"
      },
      "x_part_time[shop_id]": {
        "currently_mapped_to": "employee_hiring_decisions.x_part_time",
        "mapping_adequacy": "good",
        "description": "Number of part-time employees to hire for each shop",
        "variable_type": "integer"
      },
      "y[employee_id]": {
        "currently_mapped_to": "employee_bonus_decisions.y",
        "mapping_adequacy": "good",
        "description": "Binary decision to award bonus to each employee",
        "variable_type": "binary"
      }
    }
  },
  "missing_optimization_requirements": [],
  "iteration_status": {
    "complete": true,
    "confidence": "high",
    "next_focus": "Ready for convergence"
  }
}
