Iteration final - PROBLEM_DESCRIPTION
Sequence: 5
Timestamp: 2025-07-25 22:26:22

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: Allocation_i: continuous variable representing the budget allocated to city i
- Operational parameters align with expected linear objective: maximize ∑(Population_i × Allocation_i)
- Business configuration includes: Total budget available for allocation (used for Upper bound for total allocation constraint), Minimum budget allocation per city (used for Lower bound for city allocation constraint), Minimum budget allocation per district (used for Lower bound for district allocation 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": "world_1",
  "iteration": 1,
  "business_context": "A government agency aims to allocate a fixed budget across cities to maximize the total population served while ensuring equitable distribution across districts and respecting budget constraints.",
  "optimization_problem_description": "Maximize the total population served by allocating a fixed budget across cities, subject to district-level equity constraints and total budget limits.",
  "optimization_formulation": {
    "objective": "maximize \u2211(Population_i \u00d7 Allocation_i)",
    "decision_variables": "Allocation_i: continuous variable representing the budget allocated to city i",
    "constraints": [
      "\u2211(Allocation_i) \u2264 TotalBudget",
      "Allocation_i \u2265 MinimumAllocation for all i",
      "\u2211(Allocation_i for i in district j) \u2265 DistrictMinimum_j for all j"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "Population_i": {
        "currently_mapped_to": "city.Population",
        "mapping_adequacy": "good",
        "description": "Population of city i, used as the coefficient in the objective function"
      }
    },
    "constraint_bounds": {
      "TotalBudget": {
        "currently_mapped_to": "business_configuration_logic.TotalBudget",
        "mapping_adequacy": "good",
        "description": "Total budget available for allocation"
      },
      "MinimumAllocation": {
        "currently_mapped_to": "business_configuration_logic.MinimumAllocation",
        "mapping_adequacy": "good",
        "description": "Minimum budget allocation per city"
      },
      "DistrictMinimum_j": {
        "currently_mapped_to": "business_configuration_logic.DistrictMinimum_j",
        "mapping_adequacy": "good",
        "description": "Minimum budget allocation per district j"
      }
    },
    "decision_variables": {
      "Allocation_i": {
        "currently_mapped_to": "budget_allocation.Allocation",
        "mapping_adequacy": "good",
        "description": "Budget allocated to city i",
        "variable_type": "continuous"
      }
    }
  },
  "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 budget allocation, districts, and cities. Configuration logic updates include adding scalar parameters for total budget, minimum allocation per city, and minimum allocation per district.

CREATE TABLE city (
  CityID INTEGER,
  Population INTEGER,
  DistrictID INTEGER
);

CREATE TABLE district (
  DistrictID INTEGER,
  DistrictName STRING
);

CREATE TABLE budget_allocation (
  CityID INTEGER,
  Allocation FLOAT
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic population sizes, budget allocations, and district requirements, ensuring the optimization problem is meaningful and solvable.

-- Realistic data for city
INSERT INTO city (CityID, Population, DistrictID) VALUES (1, 150000, 1);
INSERT INTO city (CityID, Population, DistrictID) VALUES (2, 250000, 2);
INSERT INTO city (CityID, Population, DistrictID) VALUES (3, 100000, 3);

-- Realistic data for district
INSERT INTO district (DistrictID, DistrictName) VALUES (1, 'North');
INSERT INTO district (DistrictID, DistrictName) VALUES (2, 'South');
INSERT INTO district (DistrictID, DistrictName) VALUES (3, 'East');

-- Realistic data for budget_allocation
INSERT INTO budget_allocation (CityID, Allocation) VALUES (1, 60000.0);
INSERT INTO budget_allocation (CityID, Allocation) VALUES (2, 80000.0);
INSERT INTO budget_allocation (CityID, Allocation) VALUES (3, 50000.0);


```

DATA DICTIONARY:
{
  "tables": {
    "city": {
      "business_purpose": "Stores city-specific data including population and district",
      "optimization_role": "business_data",
      "columns": {
        "CityID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each city",
          "optimization_purpose": "Primary key",
          "sample_values": "1, 2, 3"
        },
        "Population": {
          "data_type": "INTEGER",
          "business_meaning": "Population of the city",
          "optimization_purpose": "Objective coefficient",
          "sample_values": "100000, 200000, 300000"
        },
        "DistrictID": {
          "data_type": "INTEGER",
          "business_meaning": "District to which the city belongs",
          "optimization_purpose": "Constraint grouping",
          "sample_values": "1, 2, 3"
        }
      }
    },
    "district": {
      "business_purpose": "Stores district-specific data including minimum budget allocation",
      "optimization_role": "business_data",
      "columns": {
        "DistrictID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each district",
          "optimization_purpose": "Primary key",
          "sample_values": "1, 2, 3"
        },
        "DistrictName": {
          "data_type": "STRING",
          "business_meaning": "Name of the district",
          "optimization_purpose": "Business context",
          "sample_values": "North, South, East"
        }
      }
    },
    "budget_allocation": {
      "business_purpose": "Stores the budget allocated to each city",
      "optimization_role": "decision_variables",
      "columns": {
        "CityID": {
          "data_type": "INTEGER",
          "business_meaning": "City to which the budget is allocated",
          "optimization_purpose": "Foreign key",
          "sample_values": "1, 2, 3"
        },
        "Allocation": {
          "data_type": "FLOAT",
          "business_meaning": "Budget allocated to the city",
          "optimization_purpose": "Decision variable",
          "sample_values": "50000.0, 75000.0, 100000.0"
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "TotalBudget": {
    "data_type": "INTEGER",
    "business_meaning": "Total budget available for allocation",
    "optimization_role": "Upper bound for total allocation constraint",
    "configuration_type": "scalar_parameter",
    "value": 1000000,
    "business_justification": "The total budget is sufficient to cover minimum allocations for all cities and districts while allowing for optimization."
  },
  "MinimumAllocation": {
    "data_type": "INTEGER",
    "business_meaning": "Minimum budget allocation per city",
    "optimization_role": "Lower bound for city allocation constraint",
    "configuration_type": "scalar_parameter",
    "value": 50000,
    "business_justification": "The minimum allocation per city ensures that even smaller cities receive a fair share of the budget."
  },
  "DistrictMinimum_j": {
    "data_type": "INTEGER",
    "business_meaning": "Minimum budget allocation per district",
    "optimization_role": "Lower bound for district allocation constraint",
    "configuration_type": "scalar_parameter",
    "value": 200000,
    "business_justification": "The minimum allocation per district ensures equitable distribution across regions."
  }
}

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: world_1

## 1. Problem Context and Goals

### Context  
[Regenerate business context that naturally aligns with LINEAR optimization formulation. Ensure:]
- Business decisions match expected decision variables: Allocation_i: continuous variable representing the budget allocated to city i
- Operational parameters align with expected linear objective: maximize ∑(Population_i × Allocation_i)
- Business configuration includes: Total budget available for allocation (used for Upper bound for total allocation constraint), Minimum budget allocation per city (used for Lower bound for city allocation constraint), Minimum budget allocation per district (used for Lower bound for district allocation 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 ∑(Population_i × Allocation_i)
- 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: ['∑(Allocation_i) ≤ TotalBudget', 'Allocation_i ≥ MinimumAllocation for all i', '∑(Allocation_i for i in district j) ≥ DistrictMinimum_j for all j'] (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 budget allocation, districts, and cities. Configuration logic updates include adding scalar parameters for total budget, minimum allocation per city, and minimum allocation per district.

CREATE TABLE city (
  CityID INTEGER,
  Population INTEGER,
  DistrictID INTEGER
);

CREATE TABLE district (
  DistrictID INTEGER,
  DistrictName STRING
);

CREATE TABLE budget_allocation (
  CityID INTEGER,
  Allocation 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 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic population sizes, budget allocations, and district requirements, ensuring the optimization problem is meaningful and solvable.

-- Realistic data for city
INSERT INTO city (CityID, Population, DistrictID) VALUES (1, 150000, 1);
INSERT INTO city (CityID, Population, DistrictID) VALUES (2, 250000, 2);
INSERT INTO city (CityID, Population, DistrictID) VALUES (3, 100000, 3);

-- Realistic data for district
INSERT INTO district (DistrictID, DistrictName) VALUES (1, 'North');
INSERT INTO district (DistrictID, DistrictName) VALUES (2, 'South');
INSERT INTO district (DistrictID, DistrictName) VALUES (3, 'East');

-- Realistic data for budget_allocation
INSERT INTO budget_allocation (CityID, Allocation) VALUES (1, 60000.0);
INSERT INTO budget_allocation (CityID, Allocation) VALUES (2, 80000.0);
INSERT INTO budget_allocation (CityID, Allocation) VALUES (3, 50000.0);


```

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

Response:
# Complete Optimization Problem and Solution: world_1

## 1. Problem Context and Goals

### Context  
A government agency is tasked with allocating a fixed budget across multiple cities to maximize the total population served. The allocation must ensure equitable distribution across districts while adhering to budget constraints. Each city has a specific population size, and the budget allocated to a city directly impacts the number of people served. The agency has established operational parameters to guide the allocation process:  
- **Total Budget Available for Allocation**: The agency has a fixed total budget of 1,000,000 units, which cannot be exceeded.  
- **Minimum Budget Allocation per City**: Each city must receive at least 50,000 units to ensure even smaller cities receive a fair share.  
- **Minimum Budget Allocation per District**: Each district must receive a minimum of 200,000 units to ensure equitable distribution across regions.  

The decision variables in this problem are the budget allocations to each city, represented as continuous values. The objective is to maximize the total population served by allocating the budget in a way that respects these constraints.  

### Goals  
The primary goal of this optimization problem is to maximize the total population served by the budget allocation. This is achieved by prioritizing cities with larger populations while ensuring that all cities and districts receive their minimum required allocations. Success is measured by the total population served, which is directly proportional to the budget allocated to each city. The optimization process ensures that the allocation is both efficient and equitable, aligning with the agency’s operational and policy objectives.  

## 2. Constraints  

The budget allocation process is subject to the following constraints:  
1. **Total Budget Limit**: The sum of all budget allocations across cities must not exceed the total available budget of 1,000,000 units.  
2. **Minimum Allocation per City**: Each city must receive at least 50,000 units of the budget to ensure a fair distribution.  
3. **Minimum Allocation per District**: The total budget allocated to all cities within a district must be at least 200,000 units to maintain equity across regions.  

These constraints ensure that the budget allocation is both feasible and aligned with the agency’s operational requirements.  

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating tables for budget allocation, districts, and cities. Configuration logic updates include adding scalar parameters for total budget, minimum allocation per city, and minimum allocation per district.

CREATE TABLE city (
  CityID INTEGER,
  Population INTEGER,
  DistrictID INTEGER
);

CREATE TABLE district (
  DistrictID INTEGER,
  DistrictName STRING
);

CREATE TABLE budget_allocation (
  CityID INTEGER,
  Allocation FLOAT
);
```

### Data Dictionary  
- **city**: Stores city-specific data, including population and district affiliation.  
  - **CityID**: Unique identifier for each city.  
  - **Population**: Population of the city, used as the coefficient in the objective function.  
  - **DistrictID**: District to which the city belongs, used for grouping in district-level constraints.  
- **district**: Stores district-specific data, including district names.  
  - **DistrictID**: Unique identifier for each district.  
  - **DistrictName**: Name of the district, used for business context.  
- **budget_allocation**: Stores the budget allocated to each city.  
  - **CityID**: City to which the budget is allocated.  
  - **Allocation**: Budget allocated to the city, representing the decision variable in the optimization problem.  

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic population sizes, budget allocations, and district requirements, ensuring the optimization problem is meaningful and solvable.

-- Realistic data for city
INSERT INTO city (CityID, Population, DistrictID) VALUES (1, 150000, 1);
INSERT INTO city (CityID, Population, DistrictID) VALUES (2, 250000, 2);
INSERT INTO city (CityID, Population, DistrictID) VALUES (3, 100000, 3);

-- Realistic data for district
INSERT INTO district (DistrictID, DistrictName) VALUES (1, 'North');
INSERT INTO district (DistrictID, DistrictName) VALUES (2, 'South');
INSERT INTO district (DistrictID, DistrictName) VALUES (3, 'East');

-- Realistic data for budget_allocation
INSERT INTO budget_allocation (CityID, Allocation) VALUES (1, 60000.0);
INSERT INTO budget_allocation (CityID, Allocation) VALUES (2, 80000.0);
INSERT INTO budget_allocation (CityID, Allocation) VALUES (3, 50000.0);
```
