Iteration final - PROBLEM_DESCRIPTION
Sequence: 5
Timestamp: 2025-07-25 22:30:59

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 1), 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[i] ∈ {0, 1} for each user i, indicating whether the user is selected to tweet.
- Operational parameters align with expected linear objective: maximize ∑(followers[i] * x[i]) where x[i] is a binary decision variable indicating whether user i is selected to tweet.
- Business configuration includes: Maximum total number of tweets allowed. (used for Constraint bound for total number of tweets.), Maximum number of tweets allowed per user. (used for Constraint bound for tweets per user.)
- 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": "twitter_1",
  "iteration": 1,
  "business_context": "Maximize the reach of tweets by selecting a subset of users to tweet, considering their follower count and ensuring that no user is overloaded with tweets, while adhering to a total tweet limit.",
  "optimization_problem_description": "Maximize the total reach of tweets by selecting a subset of users to tweet, where the reach is defined as the sum of followers of the selected users. The selection is constrained by the maximum number of tweets each user can post and the total number of tweets allowed.",
  "optimization_formulation": {
    "objective": "maximize \u2211(followers[i] * x[i]) where x[i] is a binary decision variable indicating whether user i is selected to tweet.",
    "decision_variables": "x[i] \u2208 {0, 1} for each user i, indicating whether the user is selected to tweet.",
    "constraints": "\u2211(x[i]) \u2264 total_tweets_allowed, x[i] \u2264 max_tweets_per_user[i] for each user i"
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "followers[i]": {
        "currently_mapped_to": "user_profiles.followers",
        "mapping_adequacy": "good",
        "description": "Number of followers of user i, used as the coefficient in the objective function."
      }
    },
    "constraint_bounds": {
      "total_tweets_allowed": {
        "currently_mapped_to": "business_configuration_logic.total_tweets_allowed",
        "mapping_adequacy": "good",
        "description": "Maximum total number of tweets allowed."
      },
      "max_tweets_per_user[i]": {
        "currently_mapped_to": "user_profiles.max_tweets_per_user",
        "mapping_adequacy": "good",
        "description": "Maximum number of tweets allowed per user i."
      }
    },
    "decision_variables": {
      "x[i]": {
        "currently_mapped_to": "tweet_selection.is_selected",
        "mapping_adequacy": "good",
        "description": "Binary decision variable indicating whether user i is selected to tweet.",
        "variable_type": "binary"
      }
    }
  },
  "missing_optimization_requirements": [],
  "iteration_status": {
    "complete": true,
    "confidence": "high",
    "next_focus": "Ready for convergence"
  }
}

FINAL DATABASE SCHEMA:
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating new tables for decision variables and constraint bounds, and updating business configuration logic to include scalar parameters and formulas for optimization constraints.

CREATE TABLE user_profiles (
  user_id INTEGER,
  followers INTEGER,
  max_tweets_per_user INTEGER
);

CREATE TABLE tweet_selection (
  user_id INTEGER,
  is_selected BOOLEAN
);

CREATE TABLE user_tweet_limits (
  user_id INTEGER,
  max_tweets INTEGER
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic user profiles, follower counts, and tweet limits, ensuring a balance between user engagement and tweet frequency constraints.

-- Realistic data for user_profiles
INSERT INTO user_profiles (user_id, followers, max_tweets_per_user) VALUES (1, 1500, 3);
INSERT INTO user_profiles (user_id, followers, max_tweets_per_user) VALUES (2, 2500, 5);
INSERT INTO user_profiles (user_id, followers, max_tweets_per_user) VALUES (3, 1000, 2);

-- Realistic data for tweet_selection
INSERT INTO tweet_selection (user_id, is_selected) VALUES (1, False);
INSERT INTO tweet_selection (user_id, is_selected) VALUES (2, True);
INSERT INTO tweet_selection (user_id, is_selected) VALUES (3, False);

-- Realistic data for user_tweet_limits
INSERT INTO user_tweet_limits (user_id, max_tweets) VALUES (1, 3);
INSERT INTO user_tweet_limits (user_id, max_tweets) VALUES (2, 5);
INSERT INTO user_tweet_limits (user_id, max_tweets) VALUES (3, 2);


```

DATA DICTIONARY:
{
  "tables": {
    "user_profiles": {
      "business_purpose": "Stores user profile information including follower count.",
      "optimization_role": "objective_coefficients",
      "columns": {
        "user_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each user.",
          "optimization_purpose": "Index for decision variables and constraints.",
          "sample_values": "1, 2, 3"
        },
        "followers": {
          "data_type": "INTEGER",
          "business_meaning": "Number of followers of the user.",
          "optimization_purpose": "Coefficient in the objective function.",
          "sample_values": "1000, 2000, 3000"
        },
        "max_tweets_per_user": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum number of tweets allowed per user.",
          "optimization_purpose": "Constraint bound for tweets per user.",
          "sample_values": "5, 5, 5"
        }
      }
    },
    "tweet_selection": {
      "business_purpose": "Binary decision variable indicating whether user i is selected to tweet.",
      "optimization_role": "decision_variables",
      "columns": {
        "user_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each user.",
          "optimization_purpose": "Index for decision variables.",
          "sample_values": "1, 2, 3"
        },
        "is_selected": {
          "data_type": "BOOLEAN",
          "business_meaning": "Whether the user is selected to tweet.",
          "optimization_purpose": "Binary decision variable in the optimization model.",
          "sample_values": "true, false, true"
        }
      }
    },
    "user_tweet_limits": {
      "business_purpose": "Maximum number of tweets allowed per user.",
      "optimization_role": "constraint_bounds",
      "columns": {
        "user_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each user.",
          "optimization_purpose": "Index for constraint bounds.",
          "sample_values": "1, 2, 3"
        },
        "max_tweets": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum number of tweets allowed per user.",
          "optimization_purpose": "Constraint bound for tweets per user.",
          "sample_values": "5, 5, 5"
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "total_tweets_allowed": {
    "data_type": "INTEGER",
    "business_meaning": "Maximum total number of tweets allowed.",
    "optimization_role": "Constraint bound for total number of tweets.",
    "configuration_type": "scalar_parameter",
    "value": 10,
    "business_justification": "A realistic total tweet limit that allows for meaningful selection of users without overloading the system."
  },
  "max_tweets_per_user": {
    "data_type": "INTEGER",
    "business_meaning": "Maximum number of tweets allowed per user.",
    "optimization_role": "Constraint bound for tweets per user.",
    "configuration_type": "scalar_parameter",
    "value": 5,
    "business_justification": "A reasonable maximum tweet limit per user to prevent over-tweeting and maintain user engagement."
  }
}

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: twitter_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[i] ∈ {0, 1} for each user i, indicating whether the user is selected to tweet.
- Operational parameters align with expected linear objective: maximize ∑(followers[i] * x[i]) where x[i] is a binary decision variable indicating whether user i is selected to tweet.
- Business configuration includes: Maximum total number of tweets allowed. (used for Constraint bound for total number of tweets.), Maximum number of tweets allowed per user. (used for Constraint bound for tweets per user.)
- 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 ∑(followers[i] * x[i]) where x[i] is a binary decision variable indicating whether user i is selected to tweet.
- 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: ∑(x[i]) ≤ total_tweets_allowed, x[i] ≤ max_tweets_per_user[i] for each user i (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 1 Database Schema
-- Objective: Schema changes include creating new tables for decision variables and constraint bounds, and updating business configuration logic to include scalar parameters and formulas for optimization constraints.

CREATE TABLE user_profiles (
  user_id INTEGER,
  followers INTEGER,
  max_tweets_per_user INTEGER
);

CREATE TABLE tweet_selection (
  user_id INTEGER,
  is_selected BOOLEAN
);

CREATE TABLE user_tweet_limits (
  user_id INTEGER,
  max_tweets INTEGER
);


```

### 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 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic user profiles, follower counts, and tweet limits, ensuring a balance between user engagement and tweet frequency constraints.

-- Realistic data for user_profiles
INSERT INTO user_profiles (user_id, followers, max_tweets_per_user) VALUES (1, 1500, 3);
INSERT INTO user_profiles (user_id, followers, max_tweets_per_user) VALUES (2, 2500, 5);
INSERT INTO user_profiles (user_id, followers, max_tweets_per_user) VALUES (3, 1000, 2);

-- Realistic data for tweet_selection
INSERT INTO tweet_selection (user_id, is_selected) VALUES (1, False);
INSERT INTO tweet_selection (user_id, is_selected) VALUES (2, True);
INSERT INTO tweet_selection (user_id, is_selected) VALUES (3, False);

-- Realistic data for user_tweet_limits
INSERT INTO user_tweet_limits (user_id, max_tweets) VALUES (1, 3);
INSERT INTO user_tweet_limits (user_id, max_tweets) VALUES (2, 5);
INSERT INTO user_tweet_limits (user_id, max_tweets) VALUES (3, 2);


```

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

Response:
# Complete Optimization Problem and Solution: twitter_1

## 1. Problem Context and Goals

### Context  
The business aims to maximize the reach of tweets by strategically selecting a subset of users to post tweets. The reach is determined by the total number of followers of the selected users. Each user has a specific follower count, and the decision to select a user for tweeting is binary—either they are selected or not. The selection process must adhere to two key operational constraints:  
1. **Total Tweet Limit**: The total number of tweets across all selected users cannot exceed a predefined maximum limit. This ensures that the system is not overwhelmed with tweets.  
2. **User Tweet Limit**: Each user has a maximum number of tweets they are allowed to post. This prevents any single user from being overburdened with tweeting responsibilities.  

The business configuration includes two critical scalar parameters:  
- **Maximum Total Tweets Allowed**: The system-wide limit on the total number of tweets that can be posted.  
- **Maximum Tweets Per User**: The individual limit on the number of tweets each user can post.  

These parameters are designed to balance user engagement with operational feasibility, ensuring a realistic and manageable tweet distribution strategy.

### Goals  
The primary goal of this optimization problem is to maximize the total reach of the tweets. This is achieved by selecting users whose combined follower count is as large as possible, while respecting the operational constraints. Success is measured by the total number of followers reached through the selected tweets. The optimization process ensures that the selection of users is both strategic and compliant with the predefined limits on tweet volume.

## 2. Constraints  

The optimization problem is subject to the following constraints:  
1. **Total Tweet Constraint**: The sum of tweets posted by all selected users must not exceed the maximum total number of tweets allowed. This ensures that the system operates within its capacity.  
2. **User Tweet Constraint**: Each selected user cannot post more tweets than their individual maximum limit. This prevents overloading any single user with tweeting responsibilities.  

These constraints are designed to maintain operational efficiency and user engagement, ensuring that the tweet distribution strategy is both effective and sustainable.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating new tables for decision variables and constraint bounds, and updating business configuration logic to include scalar parameters and formulas for optimization constraints.

CREATE TABLE user_profiles (
  user_id INTEGER,
  followers INTEGER,
  max_tweets_per_user INTEGER
);

CREATE TABLE tweet_selection (
  user_id INTEGER,
  is_selected BOOLEAN
);

CREATE TABLE user_tweet_limits (
  user_id INTEGER,
  max_tweets INTEGER
);
```

### Data Dictionary  
- **user_profiles**:  
  - **user_id**: Unique identifier for each user.  
  - **followers**: Number of followers of the user, used to determine the reach of their tweets.  
  - **max_tweets_per_user**: Maximum number of tweets allowed per user, used to constrain the number of tweets each user can post.  

- **tweet_selection**:  
  - **user_id**: Unique identifier for each user.  
  - **is_selected**: Binary indicator of whether the user is selected to tweet.  

- **user_tweet_limits**:  
  - **user_id**: Unique identifier for each user.  
  - **max_tweets**: Maximum number of tweets allowed per user, used to constrain the number of tweets each user can post.  

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic user profiles, follower counts, and tweet limits, ensuring a balance between user engagement and tweet frequency constraints.

-- Realistic data for user_profiles
INSERT INTO user_profiles (user_id, followers, max_tweets_per_user) VALUES (1, 1500, 3);
INSERT INTO user_profiles (user_id, followers, max_tweets_per_user) VALUES (2, 2500, 5);
INSERT INTO user_profiles (user_id, followers, max_tweets_per_user) VALUES (3, 1000, 2);

-- Realistic data for tweet_selection
INSERT INTO tweet_selection (user_id, is_selected) VALUES (1, False);
INSERT INTO tweet_selection (user_id, is_selected) VALUES (2, True);
INSERT INTO tweet_selection (user_id, is_selected) VALUES (3, False);

-- Realistic data for user_tweet_limits
INSERT INTO user_tweet_limits (user_id, max_tweets) VALUES (1, 3);
INSERT INTO user_tweet_limits (user_id, max_tweets) VALUES (2, 5);
INSERT INTO user_tweet_limits (user_id, max_tweets) VALUES (3, 2);
```
