Iteration final - PROBLEM_DESCRIPTION
Sequence: 5
Timestamp: 2025-07-25 22:43:29

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_ij: binary variable indicating whether person i is friends with person j
- Operational parameters align with expected linear objective: maximize ∑(x_ij) where x_ij is a binary decision variable indicating whether person i is friends with person j
- Business configuration includes: maximum number of friendships allowed per person (used for constraint bound), minimum number of friendships required per age group (used for constraint bound)
- Business logic formulas to express in natural language: ensures each person has a balanced number of friends across different age groups (calculation method for constraint)
- 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": "network_2",
  "iteration": 1,
  "business_context": "A social network aims to maximize the number of friendships formed while ensuring each person has a balanced number of friends across different age groups and does not exceed a maximum number of friendships.",
  "optimization_problem_description": "Maximize the total number of friendships formed, subject to constraints ensuring each person has a balanced number of friends across different age groups and does not exceed a maximum number of friendships.",
  "optimization_formulation": {
    "objective": "maximize \u2211(x_ij) where x_ij is a binary decision variable indicating whether person i is friends with person j",
    "decision_variables": "x_ij: binary variable indicating whether person i is friends with person j",
    "constraints": [
      "\u2211(x_ij for j in age_group_k) >= min_friendships_per_age_group for each person i and age group k",
      "\u2211(x_ij for all j) <= max_friendships_per_person for each person i",
      "x_ij = x_ji for all i, j to ensure mutual friendships"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "1": {
        "currently_mapped_to": "friendships.is_friends",
        "mapping_adequacy": "good",
        "description": "coefficient for the objective function, representing the presence of a friendship"
      }
    },
    "constraint_bounds": {
      "max_friendships_per_person": {
        "currently_mapped_to": "business_configuration_logic.max_friendships_per_person",
        "mapping_adequacy": "good",
        "description": "maximum number of friendships allowed per person"
      },
      "min_friendships_per_age_group": {
        "currently_mapped_to": "business_configuration_logic.min_friendships_per_age_group",
        "mapping_adequacy": "good",
        "description": "minimum number of friendships required per age group"
      }
    },
    "decision_variables": {
      "x_ij": {
        "currently_mapped_to": "friendships.is_friends",
        "mapping_adequacy": "good",
        "description": "binary decision variable indicating whether person i is friends with person j",
        "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 tables for decision variables, age groups, and friendships. Configuration logic updates include scalar parameters for max friendships per person and min friendships per age group, and a formula for friendship balance.

CREATE TABLE friendships (
  person_i INTEGER,
  person_j INTEGER,
  is_friends BOOLEAN
);

CREATE TABLE age_groups (
  age_group_id INTEGER,
  age_range STRING
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic social network scenarios, ensuring diversity in age groups and balanced friendships while respecting constraints.

-- Realistic data for friendships
INSERT INTO friendships (person_i, person_j, is_friends) VALUES (1, 2, True);
INSERT INTO friendships (person_i, person_j, is_friends) VALUES (1, 3, False);
INSERT INTO friendships (person_i, person_j, is_friends) VALUES (2, 3, True);
INSERT INTO friendships (person_i, person_j, is_friends) VALUES (3, 4, True);
INSERT INTO friendships (person_i, person_j, is_friends) VALUES (4, 5, False);

-- Realistic data for age_groups
INSERT INTO age_groups (age_group_id, age_range) VALUES (1, 18-25);
INSERT INTO age_groups (age_group_id, age_range) VALUES (2, 26-35);
INSERT INTO age_groups (age_group_id, age_range) VALUES (3, 36-45);


```

DATA DICTIONARY:
{
  "tables": {
    "friendships": {
      "business_purpose": "binary variable indicating whether person i is friends with person j",
      "optimization_role": "decision_variables",
      "columns": {
        "person_i": {
          "data_type": "INTEGER",
          "business_meaning": "ID of person i",
          "optimization_purpose": "index for decision variable",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "person_j": {
          "data_type": "INTEGER",
          "business_meaning": "ID of person j",
          "optimization_purpose": "index for decision variable",
          "sample_values": [
            2,
            3,
            4
          ]
        },
        "is_friends": {
          "data_type": "BOOLEAN",
          "business_meaning": "whether person i is friends with person j",
          "optimization_purpose": "binary decision variable",
          "sample_values": [
            true,
            false,
            true
          ]
        }
      }
    },
    "age_groups": {
      "business_purpose": "definitions of age groups for friendship balance constraints",
      "optimization_role": "business_data",
      "columns": {
        "age_group_id": {
          "data_type": "INTEGER",
          "business_meaning": "ID of age group",
          "optimization_purpose": "index for age group constraints",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "age_range": {
          "data_type": "STRING",
          "business_meaning": "age range for the group",
          "optimization_purpose": "defines age group for constraints",
          "sample_values": [
            "18-25",
            "26-35",
            "36-45"
          ]
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "max_friendships_per_person": {
    "data_type": "INTEGER",
    "business_meaning": "maximum number of friendships allowed per person",
    "optimization_role": "constraint bound",
    "configuration_type": "scalar_parameter",
    "value": 15,
    "business_justification": "A realistic maximum number of friendships per person, allowing for a diverse network without overwhelming any single user."
  },
  "min_friendships_per_age_group": {
    "data_type": "INTEGER",
    "business_meaning": "minimum number of friendships required per age group",
    "optimization_role": "constraint bound",
    "configuration_type": "scalar_parameter",
    "value": 3,
    "business_justification": "Ensures each person has a minimum number of friends across different age groups, promoting a balanced and inclusive network."
  },
  "friendship_balance_formula": {
    "data_type": "STRING",
    "business_meaning": "ensures each person has a balanced number of friends across different age groups",
    "optimization_role": "constraint",
    "configuration_type": "business_logic_formula",
    "formula_expression": "sum(x_ij for j in age_group_k) >= min_friendships_per_age_group"
  }
}

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: network_2

## 1. Problem Context and Goals

### Context  
[Regenerate business context that naturally aligns with LINEAR optimization formulation. Ensure:]
- Business decisions match expected decision variables: x_ij: binary variable indicating whether person i is friends with person j
- Operational parameters align with expected linear objective: maximize ∑(x_ij) where x_ij is a binary decision variable indicating whether person i is friends with person j
- Business configuration includes: maximum number of friendships allowed per person (used for constraint bound), minimum number of friendships required per age group (used for constraint bound)
- Business logic formulas to express in natural language: ensures each person has a balanced number of friends across different age groups (calculation method for constraint)
- 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 ∑(x_ij) where x_ij is a binary decision variable indicating whether person i is friends with person j
- 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_ij for j in age_group_k) >= min_friendships_per_age_group for each person i and age group k', '∑(x_ij for all j) <= max_friendships_per_person for each person i', 'x_ij = x_ji for all i, j to ensure mutual friendships'] (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 tables for decision variables, age groups, and friendships. Configuration logic updates include scalar parameters for max friendships per person and min friendships per age group, and a formula for friendship balance.

CREATE TABLE friendships (
  person_i INTEGER,
  person_j INTEGER,
  is_friends BOOLEAN
);

CREATE TABLE age_groups (
  age_group_id INTEGER,
  age_range STRING
);


```

### 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 social network scenarios, ensuring diversity in age groups and balanced friendships while respecting constraints.

-- Realistic data for friendships
INSERT INTO friendships (person_i, person_j, is_friends) VALUES (1, 2, True);
INSERT INTO friendships (person_i, person_j, is_friends) VALUES (1, 3, False);
INSERT INTO friendships (person_i, person_j, is_friends) VALUES (2, 3, True);
INSERT INTO friendships (person_i, person_j, is_friends) VALUES (3, 4, True);
INSERT INTO friendships (person_i, person_j, is_friends) VALUES (4, 5, False);

-- Realistic data for age_groups
INSERT INTO age_groups (age_group_id, age_range) VALUES (1, 18-25);
INSERT INTO age_groups (age_group_id, age_range) VALUES (2, 26-35);
INSERT INTO age_groups (age_group_id, age_range) VALUES (3, 36-45);


```

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

Response:
# Complete Optimization Problem and Solution: network_2

## 1. Problem Context and Goals

### Context  
The social network aims to maximize the number of friendships formed among its users while ensuring a balanced and inclusive network. Each user can form friendships with others, but the network enforces specific rules to maintain balance and prevent overloading any single user. 

Key operational parameters include:  
- **Maximum number of friendships per person**: Each user is allowed to have up to 15 friendships. This ensures that no single user is overwhelmed with too many connections.  
- **Minimum number of friendships per age group**: Each user must have at least 3 friendships with users from different age groups. This promotes diversity and inclusivity across the network.  

The network also ensures that friendships are mutual, meaning if person i is friends with person j, then person j must also be friends with person i. This mutual relationship is a fundamental requirement for all friendships in the network.  

The goal is to maximize the total number of friendships while adhering to these constraints, ensuring a balanced and sustainable social network.

### Goals  
The primary optimization goal is to maximize the total number of friendships formed across the network. Success is measured by the total count of friendships, ensuring that the network grows while maintaining balance and inclusivity.  

The network aims to achieve this by:  
- Encouraging as many friendships as possible within the allowed limits.  
- Ensuring that each user has a minimum number of friendships across different age groups.  
- Maintaining mutual friendships to foster genuine connections.  

This goal aligns with the operational parameters and ensures that the network remains diverse, inclusive, and scalable.

## 2. Constraints  

The optimization problem is subject to the following constraints:  
1. **Minimum friendships per age group**: Each user must have at least 3 friendships with users from each defined age group. This ensures that users have diverse connections across different age ranges.  
2. **Maximum friendships per person**: No user can have more than 15 friendships. This prevents any single user from being overwhelmed with too many connections.  
3. **Mutual friendships**: If person i is friends with person j, then person j must also be friends with person i. This ensures that all friendships are mutual and genuine.  

These constraints ensure that the network remains balanced, inclusive, and sustainable while maximizing the total number of friendships.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating tables for decision variables, age groups, and friendships. Configuration logic updates include scalar parameters for max friendships per person and min friendships per age group, and a formula for friendship balance.

CREATE TABLE friendships (
  person_i INTEGER,
  person_j INTEGER,
  is_friends BOOLEAN
);

CREATE TABLE age_groups (
  age_group_id INTEGER,
  age_range STRING
);
```

### Data Dictionary  
- **Friendships Table**:  
  - **Purpose**: Tracks whether a friendship exists between two users.  
  - **Columns**:  
    - `person_i`: The ID of the first user in the friendship.  
    - `person_j`: The ID of the second user in the friendship.  
    - `is_friends`: Indicates whether the two users are friends (True or False).  

- **Age Groups Table**:  
  - **Purpose**: Defines the age groups used to ensure balanced friendships across different age ranges.  
  - **Columns**:  
    - `age_group_id`: The ID of the age group.  
    - `age_range`: The age range for the group (e.g., 18-25, 26-35).  

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic social network scenarios, ensuring diversity in age groups and balanced friendships while respecting constraints.

-- Realistic data for friendships
INSERT INTO friendships (person_i, person_j, is_friends) VALUES (1, 2, True);
INSERT INTO friendships (person_i, person_j, is_friends) VALUES (1, 3, False);
INSERT INTO friendships (person_i, person_j, is_friends) VALUES (2, 3, True);
INSERT INTO friendships (person_i, person_j, is_friends) VALUES (3, 4, True);
INSERT INTO friendships (person_i, person_j, is_friends) VALUES (4, 5, False);

-- Realistic data for age_groups
INSERT INTO age_groups (age_group_id, age_range) VALUES (1, 18-25);
INSERT INTO age_groups (age_group_id, age_range) VALUES (2, 26-35);
INSERT INTO age_groups (age_group_id, age_range) VALUES (3, 36-45);
```
