Iteration 0 - OR_EXPERT
Sequence: 1
Timestamp: 2025-07-28 00:11:11

Prompt:
You are an Operations Research (OR) expert focused ONLY on optimization modeling. Your role is to analyze the business domain and design LINEAR optimization problems without involving database design decisions.

CRITICAL MATHEMATICAL CONSTRAINTS FOR LINEAR/MIXED-INTEGER PROGRAMMING:
- The optimization problem MUST be either 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
- Design business scenarios that naturally lead to linear mathematical formulations
- Generate between 2 and 20 constraints for optimization feasibility

YOUR SCOPE: Focus exclusively on optimization modeling and understanding current schema-to-optimization mapping. Do NOT propose database changes.
ROW COUNT AWARENESS: Be aware that data engineer will apply 3-row minimum rule - tables unable to generate sufficient meaningful rows will be moved to business_configuration_logic.json.

CRITICAL: Respond with ONLY a valid JSON object. No explanations, no markdown, no extra text.

Database: department_management

```sql
CREATE TABLE department (
  Department_ID NUMBER,
  Name TEXT,
  Creation TEXT,
  Ranking NUMBER,
  Budget_in_Billions NUMBER,
  Num_Employees NUMBER
);
```

```sql
CREATE TABLE head (
  head_ID NUMBER,
  name TEXT,
  born_state TEXT,
  age NUMBER
);
```

```sql
CREATE TABLE management (
  department_ID NUMBER,
  head_ID NUMBER,
  temporary_acting TEXT
);
```



Current Business Configuration Logic: {} (initially empty)

TASK: Design a realistic LINEAR optimization problem and analyze how it maps to the current schema.

JSON STRUCTURE REQUIRED:

{
  "database_id": "department_management",
  "iteration": 0,
  "business_context": "Describe the realistic business scenario that leads to linear optimization (avoid scenarios requiring nonlinear relationships)",
  "optimization_problem_description": "Detailed description of what needs to be optimized using linear objective functions and constraints",
  "optimization_formulation": {
    "objective": "minimize/maximize specific_linear_metric with clear definition (sum of weighted variables only)",
    "decision_variables": "What linear variables can be controlled/decided (continuous or integer)",
    "constraints": "What linear limitations must be respected (no variable products or divisions) - design 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 needed for complete linear optimization model"
  ],
  
  "iteration_status": {
    "complete": false,
    "confidence": "low/medium/high",
    "next_focus": "What optimization aspects need refinement"
  }
}

Response:
{
  "database_id": "department_management",
  "iteration": 0,
  "business_context": "A company wants to allocate its budget across different departments to maximize the overall ranking of the departments while ensuring that the budget constraints are respected.",
  "optimization_problem_description": "The goal is to maximize the sum of the rankings of all departments by optimally allocating the available budget. Each department has a specific budget requirement to achieve a certain ranking, and the total budget available is limited.",
  "optimization_formulation": {
    "objective": "maximize sum(Ranking[i] * x[i]) for all departments i",
    "decision_variables": "x[i] is the fraction of the budget allocated to department i (continuous)",
    "constraints": [
      "sum(Budget_in_Billions[i] * x[i]) <= Total_Budget",
      "0 <= x[i] <= 1 for all departments i"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "Ranking[i]": {
        "currently_mapped_to": "department.Ranking",
        "mapping_adequacy": "good",
        "description": "The ranking of department i which we aim to maximize"
      }
    },
    "constraint_bounds": {
      "Budget_in_Billions[i]": {
        "currently_mapped_to": "department.Budget_in_Billions",
        "mapping_adequacy": "good",
        "description": "The budget in billions required by department i"
      },
      "Total_Budget": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "The total budget available for allocation"
      }
    },
    "decision_variables": {
      "x[i]": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "The fraction of the budget allocated to department i",
        "variable_type": "continuous"
      }
    }
  },
  "missing_optimization_requirements": [
    "Total budget available for allocation"
  ],
  "iteration_status": {
    "complete": false,
    "confidence": "medium",
    "next_focus": "Determine the total budget available and refine the decision variable mapping"
  }
}
