Iteration 0 - OR_EXPERT
Sequence: 1
Timestamp: 2025-07-28 00:20:44

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: party_host

```sql
CREATE TABLE party (
  Party_ID NUMBER,
  Party_Theme TEXT,
  Location TEXT,
  First_year TEXT,
  Last_year TEXT,
  Number_of_hosts NUMBER
);
```

```sql
CREATE TABLE host (
  Host_ID NUMBER,
  Name TEXT,
  Nationality TEXT,
  Age TEXT
);
```

```sql
CREATE TABLE party_host (
  Party_ID NUMBER,
  Host_ID NUMBER,
  Is_Main_in_Charge OTHERS
);
```



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": "party_host",
  "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": "party_host",
  "iteration": 0,
  "business_context": "A company organizes themed parties and wants to optimize the allocation of hosts to parties to minimize the total number of hosts required while ensuring each party has at least one main host in charge.",
  "optimization_problem_description": "Minimize the total number of hosts assigned to parties while ensuring each party has at least one main host in charge and respecting the maximum number of hosts that can be assigned to a party.",
  "optimization_formulation": {
    "objective": "minimize \u2211(1 \u00d7 x_ij) where x_ij is a binary variable indicating if host j is assigned to party i",
    "decision_variables": "x_ij: binary variable indicating if host j is assigned to party i",
    "constraints": [
      "\u2211(x_ij) \u2265 1 for each party i (ensures at least one host per party)",
      "\u2211(x_ij) \u2264 Number_of_hosts for each party i (respects maximum hosts per party)",
      "x_ij = 1 if host j is the main in charge for party i"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "1[ij]": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "represents the inclusion of host j in party i"
      }
    },
    "constraint_bounds": {
      "1[i]": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "ensures at least one host is assigned to each party"
      },
      "Number_of_hosts[i]": {
        "currently_mapped_to": "party.Number_of_hosts",
        "mapping_adequacy": "good",
        "description": "maximum number of hosts that can be assigned to party i"
      }
    },
    "decision_variables": {
      "x_ij": {
        "currently_mapped_to": "party_host.Is_Main_in_Charge",
        "mapping_adequacy": "partial",
        "description": "binary variable indicating if host j is assigned to party i",
        "variable_type": "binary"
      }
    }
  },
  "missing_optimization_requirements": [
    "Data on which host is assigned to which party",
    "Binary indicator for main host in charge"
  ],
  "iteration_status": {
    "complete": false,
    "confidence": "medium",
    "next_focus": "Refine mapping of decision variables and ensure all necessary data is available"
  }
}
