Iteration 0 - OR_EXPERT
Sequence: 1
Timestamp: 2025-07-27 23:56:55

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: game_1

```sql
CREATE TABLE Student (
  StuID NUMBER,
  LName TEXT,
  Fname TEXT,
  Age NUMBER,
  Sex TEXT,
  Major NUMBER,
  Advisor NUMBER,
  city_code TEXT
);
```

```sql
CREATE TABLE Video_Games (
  GameID NUMBER,
  GName TEXT,
  GType TEXT
);
```

```sql
CREATE TABLE Plays_Games (
  StuID NUMBER,
  GameID NUMBER,
  Hours_Played NUMBER
);
```

```sql
CREATE TABLE SportsInfo (
  StuID NUMBER,
  SportName TEXT,
  HoursPerWeek NUMBER,
  GamesPlayed NUMBER,
  OnScholarship 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": "game_1",
  "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": "game_1",
  "iteration": 0,
  "business_context": "A university wants to optimize the allocation of students' time between playing video games and participating in sports to maximize their overall engagement while ensuring academic responsibilities are not compromised.",
  "optimization_problem_description": "The goal is to maximize the total engagement score of students by allocating their available hours between playing video games and participating in sports, subject to constraints on maximum hours available, minimum academic hours required, and scholarship requirements.",
  "optimization_formulation": {
    "objective": "maximize total_engagement = sum(engagement_score[StuID, GameID] * x[StuID, GameID] + engagement_score[StuID, SportName] * y[StuID, SportName])",
    "decision_variables": "x[StuID, GameID]: hours allocated to playing a specific game, y[StuID, SportName]: hours allocated to participating in a specific sport",
    "constraints": [
      "sum(x[StuID, GameID] for all GameID) + sum(y[StuID, SportName] for all SportName) <= total_hours_available[StuID]",
      "sum(y[StuID, SportName] for all SportName) >= min_sport_hours[StuID] if OnScholarship == 'Yes'",
      "sum(x[StuID, GameID] for all GameID) <= max_game_hours[StuID]",
      "x[StuID, GameID] >= 0, y[StuID, SportName] >= 0"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "engagement_score[StuID, GameID]": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "represents the engagement score for a student playing a specific game"
      },
      "engagement_score[StuID, SportName]": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "represents the engagement score for a student participating in a specific sport"
      }
    },
    "constraint_bounds": {
      "total_hours_available[StuID]": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "total hours a student can allocate to games and sports"
      },
      "min_sport_hours[StuID]": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "minimum hours a student on scholarship must spend on sports"
      },
      "max_game_hours[StuID]": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "maximum hours a student can spend on games"
      }
    },
    "decision_variables": {
      "x[StuID, GameID]": {
        "currently_mapped_to": "Plays_Games.Hours_Played",
        "mapping_adequacy": "partial",
        "description": "hours allocated to playing a specific game by a student",
        "variable_type": "continuous"
      },
      "y[StuID, SportName]": {
        "currently_mapped_to": "SportsInfo.HoursPerWeek",
        "mapping_adequacy": "partial",
        "description": "hours allocated to participating in a specific sport by a student",
        "variable_type": "continuous"
      }
    }
  },
  "missing_optimization_requirements": [
    "Engagement scores for games and sports",
    "Total available hours for each student",
    "Minimum sport hours for scholarship students",
    "Maximum game hours for each student"
  ],
  "iteration_status": {
    "complete": false,
    "confidence": "medium",
    "next_focus": "Refine the mapping of engagement scores and constraints related to available hours and scholarship requirements"
  }
}
