Iteration final - PROBLEM_DESCRIPTION
Sequence: 7
Timestamp: 2025-07-27 23:56:55

Prompt:
You are a business analyst creating structured optimization problem documentation.

DATA SOURCES EXPLANATION:
- FINAL OR ANALYSIS: Final converged optimization problem from alternating process (iteration 2), contains business context and schema mapping evaluation
- DATABASE SCHEMA: Current database structure after iterative adjustments  
- DATA DICTIONARY: Business meanings and optimization roles of tables and columns
- CURRENT STORED VALUES: Realistic business data generated by triple expert (business + data + optimization)
- BUSINESS CONFIGURATION: Scalar parameters and business logic formulas separated from table data

CRITICAL REQUIREMENTS: 
- Ensure problem description naturally leads to LINEAR or MIXED-INTEGER optimization formulation
- Make business context consistent with the intended decision variables and objectives
- Align constraint descriptions with expected mathematical constraints
- Ensure data descriptions map clearly to expected coefficient sources
- Maintain business authenticity while fixing mathematical consistency issues
- Avoid business scenarios that would naturally require nonlinear relationships (variable products, divisions, etc.)

AUTO-EXTRACTED CONTEXT REQUIREMENTS:
- Business decisions match expected decision variables: x[StuID, GameID] and y[StuID, SportName] are continuous variables representing hours allocated to games and sports respectively
- Operational parameters align with expected linear objective: maximize total_engagement = sum(GameEngagementScore[StuID, GameID] * x[StuID, GameID] + SportEngagementScore[StuID, SportName] * y[StuID, SportName])
- Business configuration includes: Total hours a student can allocate to games and sports. (used for Used as a constraint in optimization model.), Minimum hours a student on scholarship must spend on sports. (used for Used as a constraint in optimization model.), Maximum hours a student can spend on games. (used for Used as a constraint in optimization model.)
- Use natural language to precisely describe linear mathematical relationships
- NO mathematical formulas, equations, or symbolic notation
- Present data as current operational information
- Focus on precise operational decision-making that leads to linear formulations
- Resource limitations match expected linear constraints
- Avoid scenarios requiring variable products, divisions, or other nonlinear relationships
- Include specific operational parameters that map to expected coefficient sources
- Reference business configuration parameters where appropriate

FINAL OR ANALYSIS:
{
  "database_id": "game_1",
  "iteration": 2,
  "business_context": "A university is optimizing 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(GameEngagementScore[StuID, GameID] * x[StuID, GameID] + SportEngagementScore[StuID, SportName] * y[StuID, SportName])",
    "decision_variables": "x[StuID, GameID] and y[StuID, SportName] are continuous variables representing hours allocated to games and sports respectively",
    "constraints": [
      "sum(x[StuID, GameID] + y[StuID, SportName]) <= total_hours_available for each StuID",
      "sum(y[StuID, SportName]) >= min_sport_hours for each StuID on scholarship",
      "sum(x[StuID, GameID]) <= max_game_hours for each StuID"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "GameEngagementScore[StuID, GameID]": {
        "currently_mapped_to": "EngagementScores.GameEngagementScore",
        "mapping_adequacy": "good",
        "description": "Engagement score for games used as coefficient in objective function"
      },
      "SportEngagementScore[StuID, SportName]": {
        "currently_mapped_to": "EngagementScores.SportEngagementScore",
        "mapping_adequacy": "good",
        "description": "Engagement score for sports used as coefficient in objective function"
      }
    },
    "constraint_bounds": {
      "total_hours_available": {
        "currently_mapped_to": "business_configuration_logic.total_hours_available",
        "mapping_adequacy": "good",
        "description": "Total hours a student can allocate to games and sports"
      },
      "min_sport_hours": {
        "currently_mapped_to": "business_configuration_logic.min_sport_hours",
        "mapping_adequacy": "good",
        "description": "Minimum hours a student on scholarship must spend on sports"
      },
      "max_game_hours": {
        "currently_mapped_to": "business_configuration_logic.max_game_hours",
        "mapping_adequacy": "good",
        "description": "Maximum hours a student can spend on games"
      }
    },
    "decision_variables": {
      "x[StuID, GameID]": {
        "currently_mapped_to": "GameAllocation.HoursAllocated",
        "mapping_adequacy": "good",
        "description": "Hours allocated by student to game",
        "variable_type": "continuous"
      },
      "y[StuID, SportName]": {
        "currently_mapped_to": "SportAllocation.HoursAllocated",
        "mapping_adequacy": "good",
        "description": "Hours allocated by student to sport",
        "variable_type": "continuous"
      }
    }
  },
  "missing_optimization_requirements": [],
  "iteration_status": {
    "complete": true,
    "confidence": "high",
    "next_focus": "Ready for convergence"
  }
}

FINAL DATABASE SCHEMA:
```sql
-- Iteration 2 Database Schema
-- Objective: Schema changes include creating new tables for decision variables, modifying existing tables to improve mapping adequacy, and moving scalar parameters to configuration logic.

CREATE TABLE EngagementScores (
  StuID INTEGER,
  GameID INTEGER,
  SportName STRING,
  GameEngagementScore FLOAT,
  SportEngagementScore FLOAT
);

CREATE TABLE GameAllocation (
  StuID INTEGER,
  GameID INTEGER,
  HoursAllocated FLOAT
);

CREATE TABLE SportAllocation (
  StuID INTEGER,
  SportName STRING,
  HoursAllocated FLOAT
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical student schedules, engagement levels, and scholarship requirements to ensure a balanced allocation between games and sports.

-- Realistic data for EngagementScores
INSERT INTO EngagementScores (StuID, GameID, SportName, GameEngagementScore, SportEngagementScore) VALUES (1, 101, 'Basketball', 1.8, 1.5);
INSERT INTO EngagementScores (StuID, GameID, SportName, GameEngagementScore, SportEngagementScore) VALUES (2, 102, 'Soccer', 2.2, 1.8);
INSERT INTO EngagementScores (StuID, GameID, SportName, GameEngagementScore, SportEngagementScore) VALUES (3, 103, 'Tennis', 1.5, 2.0);

-- Realistic data for GameAllocation
INSERT INTO GameAllocation (StuID, GameID, HoursAllocated) VALUES (1, 101, 10.0);
INSERT INTO GameAllocation (StuID, GameID, HoursAllocated) VALUES (2, 102, 12.0);
INSERT INTO GameAllocation (StuID, GameID, HoursAllocated) VALUES (3, 103, 8.0);

-- Realistic data for SportAllocation
INSERT INTO SportAllocation (StuID, SportName, HoursAllocated) VALUES (1, 'Basketball', 8.0);
INSERT INTO SportAllocation (StuID, SportName, HoursAllocated) VALUES (2, 'Soccer', 6.0);
INSERT INTO SportAllocation (StuID, SportName, HoursAllocated) VALUES (3, 'Tennis', 10.0);


```

DATA DICTIONARY:
{
  "tables": {
    "EngagementScores": {
      "business_purpose": "Stores engagement scores for students.",
      "optimization_role": "objective_coefficients",
      "columns": {
        "StuID": {
          "data_type": "INTEGER",
          "business_meaning": "Student identifier.",
          "optimization_purpose": "Links engagement scores to students.",
          "sample_values": "1, 2, 3"
        },
        "GameID": {
          "data_type": "INTEGER",
          "business_meaning": "Game identifier.",
          "optimization_purpose": "Links engagement scores to games.",
          "sample_values": "101, 102, 103"
        },
        "SportName": {
          "data_type": "STRING",
          "business_meaning": "Name of the sport.",
          "optimization_purpose": "Links engagement scores to sports.",
          "sample_values": "Basketball, Soccer, Tennis"
        },
        "GameEngagementScore": {
          "data_type": "FLOAT",
          "business_meaning": "Engagement score for games.",
          "optimization_purpose": "Coefficient in objective function.",
          "sample_values": "1.5, 2.0, 2.5"
        },
        "SportEngagementScore": {
          "data_type": "FLOAT",
          "business_meaning": "Engagement score for sports.",
          "optimization_purpose": "Coefficient in objective function.",
          "sample_values": "1.0, 1.8, 2.2"
        }
      }
    },
    "GameAllocation": {
      "business_purpose": "Stores hours allocated by students to games.",
      "optimization_role": "decision_variables",
      "columns": {
        "StuID": {
          "data_type": "INTEGER",
          "business_meaning": "Student identifier.",
          "optimization_purpose": "Links allocation to students.",
          "sample_values": "1, 2, 3"
        },
        "GameID": {
          "data_type": "INTEGER",
          "business_meaning": "Game identifier.",
          "optimization_purpose": "Links allocation to games.",
          "sample_values": "101, 102, 103"
        },
        "HoursAllocated": {
          "data_type": "FLOAT",
          "business_meaning": "Hours allocated by student to game.",
          "optimization_purpose": "Decision variable in optimization model.",
          "sample_values": "2.0, 3.5, 4.0"
        }
      }
    },
    "SportAllocation": {
      "business_purpose": "Stores hours allocated by students to sports.",
      "optimization_role": "decision_variables",
      "columns": {
        "StuID": {
          "data_type": "INTEGER",
          "business_meaning": "Student identifier.",
          "optimization_purpose": "Links allocation to students.",
          "sample_values": "1, 2, 3"
        },
        "SportName": {
          "data_type": "STRING",
          "business_meaning": "Name of the sport.",
          "optimization_purpose": "Links allocation to sports.",
          "sample_values": "Basketball, Soccer, Tennis"
        },
        "HoursAllocated": {
          "data_type": "FLOAT",
          "business_meaning": "Hours allocated by student to sport.",
          "optimization_purpose": "Decision variable in optimization model.",
          "sample_values": "1.5, 2.0, 2.5"
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "total_hours_available": {
    "data_type": "INTEGER",
    "business_meaning": "Total hours a student can allocate to games and sports.",
    "optimization_role": "Used as a constraint in optimization model.",
    "configuration_type": "scalar_parameter",
    "value": 40,
    "business_justification": "Reflects a typical weekly schedule for students balancing academics and extracurriculars."
  },
  "min_sport_hours": {
    "data_type": "INTEGER",
    "business_meaning": "Minimum hours a student on scholarship must spend on sports.",
    "optimization_role": "Used as a constraint in optimization model.",
    "configuration_type": "scalar_parameter",
    "value": 5,
    "business_justification": "Ensures students on scholarships meet minimum participation requirements in sports."
  },
  "max_game_hours": {
    "data_type": "INTEGER",
    "business_meaning": "Maximum hours a student can spend on games.",
    "optimization_role": "Used as a constraint in optimization model.",
    "configuration_type": "scalar_parameter",
    "value": 15,
    "business_justification": "Prevents excessive gaming, ensuring students maintain academic focus."
  }
}

Business Configuration Design: 
Our system separates business logic design from value determination:
- Configuration Logic (business_configuration_logic.json): Templates designed by data engineers with sample_value for scalars and actual formulas for business logic
- Configuration Values (business_configuration.json): Realistic values determined by domain experts for scalar parameters only
- Design Rationale: Ensures business logic consistency while allowing flexible parameter tuning


TASK: Create structured markdown documentation for SECTIONS 1-3 ONLY (Problem Description).

EXACT MARKDOWN STRUCTURE TO FOLLOW:

# Complete Optimization Problem and Solution: game_1

## 1. Problem Context and Goals

### Context  
[Regenerate business context that naturally aligns with LINEAR optimization formulation. Ensure:]
- Business decisions match expected decision variables: x[StuID, GameID] and y[StuID, SportName] are continuous variables representing hours allocated to games and sports respectively
- Operational parameters align with expected linear objective: maximize total_engagement = sum(GameEngagementScore[StuID, GameID] * x[StuID, GameID] + SportEngagementScore[StuID, SportName] * y[StuID, SportName])
- Business configuration includes: Total hours a student can allocate to games and sports. (used for Used as a constraint in optimization model.), Minimum hours a student on scholarship must spend on sports. (used for Used as a constraint in optimization model.), Maximum hours a student can spend on games. (used for Used as a constraint in optimization model.)
- Use natural language to precisely describe linear mathematical relationships
- NO mathematical formulas, equations, or symbolic notation
- Present data as current operational information
- Focus on precise operational decision-making that leads to linear formulations
- Resource limitations match expected linear constraints
- Avoid scenarios requiring variable products, divisions, or other nonlinear relationships
- Include specific operational parameters that map to expected coefficient sources
- Reference business configuration parameters where appropriate
- CRITICAL: Include ALL business configuration information (scalar parameters AND business logic formulas) in natural business language

### Goals  
[Regenerate goals that clearly lead to LINEAR mathematical objective:]
- Optimization goal: maximize
- Metric to optimize: maximize total_engagement = sum(GameEngagementScore[StuID, GameID] * x[StuID, GameID] + SportEngagementScore[StuID, SportName] * y[StuID, SportName])
- Success measurement aligned with expected coefficient sources
- Use natural language to precisely describe linear optimization goal
- NO mathematical formulas, equations, or symbolic notation

## 2. Constraints    

[Regenerate constraints that directly match expected LINEAR mathematical constraints:]
- Expected constraint: ['sum(x[StuID, GameID] + y[StuID, SportName]) <= total_hours_available for each StuID', 'sum(y[StuID, SportName]) >= min_sport_hours for each StuID on scholarship', 'sum(x[StuID, GameID]) <= max_game_hours for each StuID'] (Form: Standard constraint form based on business requirements)

[Each constraint should be described in business terms that naturally lead to LINEAR mathematical forms (no variable products or divisions)]

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 2 Database Schema
-- Objective: Schema changes include creating new tables for decision variables, modifying existing tables to improve mapping adequacy, and moving scalar parameters to configuration logic.

CREATE TABLE EngagementScores (
  StuID INTEGER,
  GameID INTEGER,
  SportName STRING,
  GameEngagementScore FLOAT,
  SportEngagementScore FLOAT
);

CREATE TABLE GameAllocation (
  StuID INTEGER,
  GameID INTEGER,
  HoursAllocated FLOAT
);

CREATE TABLE SportAllocation (
  StuID INTEGER,
  SportName STRING,
  HoursAllocated FLOAT
);


```

### Data Dictionary  
[Create comprehensive business-oriented data dictionary mapping tables and columns to their business purposes and optimization roles - NOT technical database terms. Base this on the data_dictionary provided but rewrite in business language that clearly connects to the expected linear mathematical formulation]

### Current Stored Values  
```sql
-- Iteration 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical student schedules, engagement levels, and scholarship requirements to ensure a balanced allocation between games and sports.

-- Realistic data for EngagementScores
INSERT INTO EngagementScores (StuID, GameID, SportName, GameEngagementScore, SportEngagementScore) VALUES (1, 101, 'Basketball', 1.8, 1.5);
INSERT INTO EngagementScores (StuID, GameID, SportName, GameEngagementScore, SportEngagementScore) VALUES (2, 102, 'Soccer', 2.2, 1.8);
INSERT INTO EngagementScores (StuID, GameID, SportName, GameEngagementScore, SportEngagementScore) VALUES (3, 103, 'Tennis', 1.5, 2.0);

-- Realistic data for GameAllocation
INSERT INTO GameAllocation (StuID, GameID, HoursAllocated) VALUES (1, 101, 10.0);
INSERT INTO GameAllocation (StuID, GameID, HoursAllocated) VALUES (2, 102, 12.0);
INSERT INTO GameAllocation (StuID, GameID, HoursAllocated) VALUES (3, 103, 8.0);

-- Realistic data for SportAllocation
INSERT INTO SportAllocation (StuID, SportName, HoursAllocated) VALUES (1, 'Basketball', 8.0);
INSERT INTO SportAllocation (StuID, SportName, HoursAllocated) VALUES (2, 'Soccer', 6.0);
INSERT INTO SportAllocation (StuID, SportName, HoursAllocated) VALUES (3, 'Tennis', 10.0);


```

STOP HERE - Do not include section 4 (Mathematical Optimization Formulation).

Response:
# Complete Optimization Problem and Solution: game_1

## 1. Problem Context and Goals

### Context  
The university is focused on optimizing how students allocate their time between playing video games and participating in sports. The aim is to enhance student engagement while ensuring that academic responsibilities are not neglected. The decision variables in this context are the hours students allocate to games and sports, represented as continuous variables. Specifically, x[StuID, GameID] denotes the hours a student allocates to a particular game, and y[StuID, SportName] represents the hours allocated to a specific sport. The operational parameters are designed to align with the objective of maximizing total engagement. This involves summing the product of game engagement scores and hours allocated to games, along with the product of sport engagement scores and hours allocated to sports. The business configuration includes constraints such as the total hours a student can allocate to games and sports, the minimum hours a student on scholarship must spend on sports, and the maximum hours a student can spend on games. These constraints ensure that the allocation of time is balanced and adheres to the university's academic and extracurricular policies.

### Goals  
The primary goal of this optimization problem is to maximize student engagement. This is achieved by optimizing the allocation of hours to games and sports in a way that maximizes the total engagement score. The engagement score is calculated by summing the products of game engagement scores with the hours allocated to games and sport engagement scores with the hours allocated to sports. Success in this optimization is measured by the total engagement score, which reflects the effectiveness of the time allocation in enhancing student engagement. The objective is clearly defined in linear terms, focusing on maximizing this engagement score without involving any nonlinear relationships.

## 2. Constraints    

The constraints in this optimization problem are designed to ensure that the allocation of time adheres to the university's policies and supports student success. The first constraint ensures that the total hours a student allocates to games and sports do not exceed the available hours, reflecting a balanced schedule. The second constraint mandates that students on scholarships spend a minimum number of hours on sports, ensuring they meet participation requirements. The third constraint limits the hours a student can spend on games, preventing excessive gaming and promoting academic focus. These constraints are expressed in linear terms, directly aligning with the expected mathematical formulation.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 2 Database Schema
-- Objective: Schema changes include creating new tables for decision variables, modifying existing tables to improve mapping adequacy, and moving scalar parameters to configuration logic.

CREATE TABLE EngagementScores (
  StuID INTEGER,
  GameID INTEGER,
  SportName STRING,
  GameEngagementScore FLOAT,
  SportEngagementScore FLOAT
);

CREATE TABLE GameAllocation (
  StuID INTEGER,
  GameID INTEGER,
  HoursAllocated FLOAT
);

CREATE TABLE SportAllocation (
  StuID INTEGER,
  SportName STRING,
  HoursAllocated FLOAT
);
```

### Data Dictionary  
The data dictionary provides a comprehensive overview of the tables and columns used in this optimization problem, mapping them to their business purposes and roles in the optimization process. 

- **EngagementScores**: This table stores the engagement scores for each student, game, and sport. The engagement scores serve as coefficients in the objective function, linking the scores to the students, games, and sports.
  - **StuID**: Represents the student identifier, linking engagement scores to individual students.
  - **GameID**: Represents the game identifier, linking engagement scores to specific games.
  - **SportName**: Represents the name of the sport, linking engagement scores to specific sports.
  - **GameEngagementScore**: Represents the engagement score for games, used as a coefficient in the objective function.
  - **SportEngagementScore**: Represents the engagement score for sports, used as a coefficient in the objective function.

- **GameAllocation**: This table records the hours students allocate to games, serving as decision variables in the optimization model.
  - **StuID**: Represents the student identifier, linking allocation to individual students.
  - **GameID**: Represents the game identifier, linking allocation to specific games.
  - **HoursAllocated**: Represents the hours allocated by a student to a game, serving as a decision variable.

- **SportAllocation**: This table records the hours students allocate to sports, serving as decision variables in the optimization model.
  - **StuID**: Represents the student identifier, linking allocation to individual students.
  - **SportName**: Represents the name of the sport, linking allocation to specific sports.
  - **HoursAllocated**: Represents the hours allocated by a student to a sport, serving as a decision variable.

### Current Stored Values  
```sql
-- Iteration 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical student schedules, engagement levels, and scholarship requirements to ensure a balanced allocation between games and sports.

-- Realistic data for EngagementScores
INSERT INTO EngagementScores (StuID, GameID, SportName, GameEngagementScore, SportEngagementScore) VALUES (1, 101, 'Basketball', 1.8, 1.5);
INSERT INTO EngagementScores (StuID, GameID, SportName, GameEngagementScore, SportEngagementScore) VALUES (2, 102, 'Soccer', 2.2, 1.8);
INSERT INTO EngagementScores (StuID, GameID, SportName, GameEngagementScore, SportEngagementScore) VALUES (3, 103, 'Tennis', 1.5, 2.0);

-- Realistic data for GameAllocation
INSERT INTO GameAllocation (StuID, GameID, HoursAllocated) VALUES (1, 101, 10.0);
INSERT INTO GameAllocation (StuID, GameID, HoursAllocated) VALUES (2, 102, 12.0);
INSERT INTO GameAllocation (StuID, GameID, HoursAllocated) VALUES (3, 103, 8.0);

-- Realistic data for SportAllocation
INSERT INTO SportAllocation (StuID, SportName, HoursAllocated) VALUES (1, 'Basketball', 8.0);
INSERT INTO SportAllocation (StuID, SportName, HoursAllocated) VALUES (2, 'Soccer', 6.0);
INSERT INTO SportAllocation (StuID, SportName, HoursAllocated) VALUES (3, 'Tennis', 10.0);
```
