Iteration final - PROBLEM_DESCRIPTION
Sequence: 5
Timestamp: 2025-07-25 22:30:28

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: binary decision variable indicating whether city i is selected as a host
- Operational parameters align with expected linear objective: maximize ∑(w1 * Population_i + w2 * GDP_i + w3 * Temperature_i) * x_i
- Business configuration includes: Weight for regional population in the objective function (used for Objective coefficient), Weight for GDP in the objective function (used for Objective coefficient), Weight for average temperature in the objective function (used for Objective coefficient), Total number of matches to be hosted (used for Constraint bound), Total budget constraint for hosting (used for Constraint bound), Maximum number of matches per city (used for Constraint bound)
- Business logic formulas to express in natural language: Calculation of the benefit for selecting a city as a host (calculation method for Objective function component)
- 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": "city_record",
  "iteration": 1,
  "business_context": "A sports organization aims to optimize the selection of host cities for matches in a given year, considering factors like regional population, GDP, and average temperature during the match month. The optimization must respect constraints such as the maximum number of matches per city and the total budget allocated for hosting.",
  "optimization_problem_description": "The goal is to maximize the overall benefit of selecting host cities for matches, where the benefit is a weighted sum of regional population, GDP, and average temperature during the match month. The selection must respect constraints such as the maximum number of matches per city and the total budget allocated for hosting.",
  "optimization_formulation": {
    "objective": "maximize \u2211(w1 * Population_i + w2 * GDP_i + w3 * Temperature_i) * x_i",
    "decision_variables": "x_i: binary decision variable indicating whether city i is selected as a host",
    "constraints": [
      "\u2211(x_i) \u2264 Total_Matches",
      "\u2211(hosting_cost_i * x_i) \u2264 Total_Budget",
      "x_i \u2264 Max_Matches_Per_City for all i"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "w1": {
        "currently_mapped_to": "objective_coefficients.weight_population",
        "mapping_adequacy": "good",
        "description": "Weight for regional population in the objective function"
      },
      "w2": {
        "currently_mapped_to": "objective_coefficients.weight_gdp",
        "mapping_adequacy": "good",
        "description": "Weight for GDP in the objective function"
      },
      "w3": {
        "currently_mapped_to": "objective_coefficients.weight_temperature",
        "mapping_adequacy": "good",
        "description": "Weight for average temperature in the objective function"
      }
    },
    "constraint_bounds": {
      "Total_Matches": {
        "currently_mapped_to": "business_configuration_logic.Total_Matches",
        "mapping_adequacy": "good",
        "description": "Total number of matches to be hosted"
      },
      "Total_Budget": {
        "currently_mapped_to": "match_constraints.total_budget",
        "mapping_adequacy": "good",
        "description": "Total budget constraint for hosting"
      },
      "Max_Matches_Per_City": {
        "currently_mapped_to": "match_constraints.max_matches_per_city",
        "mapping_adequacy": "good",
        "description": "Maximum number of matches per city"
      }
    },
    "decision_variables": {
      "x_i": {
        "currently_mapped_to": "city_data.city_id",
        "mapping_adequacy": "good",
        "description": "Binary decision variable indicating whether city i is selected as a host",
        "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 city data, match constraints, and objective coefficients. Configuration logic updates include scalar parameters for weights and constraints, and formulas for benefit calculation.

CREATE TABLE city_data (
  city_id INTEGER,
  population INTEGER,
  gdp FLOAT,
  avg_temperature FLOAT,
  hosting_cost FLOAT
);

CREATE TABLE match_constraints (
  constraint_id INTEGER,
  max_matches_per_city INTEGER,
  total_budget FLOAT
);

CREATE TABLE objective_coefficients (
  coefficient_id INTEGER,
  weight_population FLOAT,
  weight_gdp FLOAT,
  weight_temperature FLOAT
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic city demographics, economic data, and climate conditions, ensuring they align with the optimization problem's requirements and constraints.

-- Realistic data for city_data
INSERT INTO city_data (city_id, population, gdp, avg_temperature, hosting_cost) VALUES (1, 800000, 75000.0, 22.0, 120000.0);
INSERT INTO city_data (city_id, population, gdp, avg_temperature, hosting_cost) VALUES (2, 1200000, 110000.0, 26.0, 180000.0);
INSERT INTO city_data (city_id, population, gdp, avg_temperature, hosting_cost) VALUES (3, 600000, 60000.0, 20.0, 100000.0);

-- Realistic data for match_constraints
INSERT INTO match_constraints (constraint_id, max_matches_per_city, total_budget) VALUES (1, 2, 1000000.0);
INSERT INTO match_constraints (constraint_id, max_matches_per_city, total_budget) VALUES (2, 2, 1000000.0);
INSERT INTO match_constraints (constraint_id, max_matches_per_city, total_budget) VALUES (3, 2, 1000000.0);

-- Realistic data for objective_coefficients
INSERT INTO objective_coefficients (coefficient_id, weight_population, weight_gdp, weight_temperature) VALUES (1, 0.4, 0.3, 0.3);
INSERT INTO objective_coefficients (coefficient_id, weight_population, weight_gdp, weight_temperature) VALUES (2, 0.4, 0.3, 0.3);
INSERT INTO objective_coefficients (coefficient_id, weight_population, weight_gdp, weight_temperature) VALUES (3, 0.4, 0.3, 0.3);


```

DATA DICTIONARY:
{
  "tables": {
    "city_data": {
      "business_purpose": "Contains data about potential host cities, including population, GDP, and average temperature during match month.",
      "optimization_role": "business_data",
      "columns": {
        "city_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each city",
          "optimization_purpose": "Decision variable identifier",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "population": {
          "data_type": "INTEGER",
          "business_meaning": "Regional population of the city",
          "optimization_purpose": "Objective function component",
          "sample_values": [
            500000,
            1000000,
            1500000
          ]
        },
        "gdp": {
          "data_type": "FLOAT",
          "business_meaning": "GDP of the city",
          "optimization_purpose": "Objective function component",
          "sample_values": [
            50000.0,
            100000.0,
            150000.0
          ]
        },
        "avg_temperature": {
          "data_type": "FLOAT",
          "business_meaning": "Average temperature during the match month",
          "optimization_purpose": "Objective function component",
          "sample_values": [
            20.0,
            25.0,
            30.0
          ]
        },
        "hosting_cost": {
          "data_type": "FLOAT",
          "business_meaning": "Cost of hosting a match in the city",
          "optimization_purpose": "Constraint component",
          "sample_values": [
            100000.0,
            150000.0,
            200000.0
          ]
        }
      }
    },
    "match_constraints": {
      "business_purpose": "Stores constraints related to hosting matches, such as maximum number of matches per city and total budget.",
      "optimization_role": "constraint_bounds",
      "columns": {
        "constraint_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each constraint",
          "optimization_purpose": "Constraint identifier",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "max_matches_per_city": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum number of matches per city",
          "optimization_purpose": "Constraint bound",
          "sample_values": [
            2,
            2,
            2
          ]
        },
        "total_budget": {
          "data_type": "FLOAT",
          "business_meaning": "Total budget constraint for hosting",
          "optimization_purpose": "Constraint bound",
          "sample_values": [
            1000000.0,
            1000000.0,
            1000000.0
          ]
        }
      }
    },
    "objective_coefficients": {
      "business_purpose": "Stores the weights for the objective function components (population, GDP, temperature).",
      "optimization_role": "objective_coefficients",
      "columns": {
        "coefficient_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each coefficient",
          "optimization_purpose": "Coefficient identifier",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "weight_population": {
          "data_type": "FLOAT",
          "business_meaning": "Weight for regional population in the objective function",
          "optimization_purpose": "Objective coefficient",
          "sample_values": [
            0.4,
            0.4,
            0.4
          ]
        },
        "weight_gdp": {
          "data_type": "FLOAT",
          "business_meaning": "Weight for GDP in the objective function",
          "optimization_purpose": "Objective coefficient",
          "sample_values": [
            0.3,
            0.3,
            0.3
          ]
        },
        "weight_temperature": {
          "data_type": "FLOAT",
          "business_meaning": "Weight for average temperature in the objective function",
          "optimization_purpose": "Objective coefficient",
          "sample_values": [
            0.3,
            0.3,
            0.3
          ]
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "w1": {
    "data_type": "FLOAT",
    "business_meaning": "Weight for regional population in the objective function",
    "optimization_role": "Objective coefficient",
    "configuration_type": "scalar_parameter",
    "value": 0.4,
    "business_justification": "Population is a significant factor in determining the benefit of hosting matches."
  },
  "w2": {
    "data_type": "FLOAT",
    "business_meaning": "Weight for GDP in the objective function",
    "optimization_role": "Objective coefficient",
    "configuration_type": "scalar_parameter",
    "value": 0.3,
    "business_justification": "GDP contributes to the economic benefit of hosting matches."
  },
  "w3": {
    "data_type": "FLOAT",
    "business_meaning": "Weight for average temperature in the objective function",
    "optimization_role": "Objective coefficient",
    "configuration_type": "scalar_parameter",
    "value": 0.3,
    "business_justification": "Temperature affects the comfort and appeal of hosting matches."
  },
  "Total_Matches": {
    "data_type": "INTEGER",
    "business_meaning": "Total number of matches to be hosted",
    "optimization_role": "Constraint bound",
    "configuration_type": "scalar_parameter",
    "value": 10,
    "business_justification": "A reasonable number of matches to be hosted within the given constraints."
  },
  "Total_Budget": {
    "data_type": "FLOAT",
    "business_meaning": "Total budget constraint for hosting",
    "optimization_role": "Constraint bound",
    "configuration_type": "scalar_parameter",
    "value": 1000000.0,
    "business_justification": "A realistic budget that allows for hosting matches across multiple cities."
  },
  "Max_Matches_Per_City": {
    "data_type": "INTEGER",
    "business_meaning": "Maximum number of matches per city",
    "optimization_role": "Constraint bound",
    "configuration_type": "scalar_parameter",
    "value": 2,
    "business_justification": "Ensures no single city dominates the hosting schedule."
  },
  "Benefit_Calculation": {
    "data_type": "STRING",
    "business_meaning": "Calculation of the benefit for selecting a city as a host",
    "optimization_role": "Objective function component",
    "configuration_type": "business_logic_formula",
    "formula_expression": "w1 * Population_i + w2 * GDP_i + w3 * Temperature_i"
  }
}

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: city_record

## 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: binary decision variable indicating whether city i is selected as a host
- Operational parameters align with expected linear objective: maximize ∑(w1 * Population_i + w2 * GDP_i + w3 * Temperature_i) * x_i
- Business configuration includes: Weight for regional population in the objective function (used for Objective coefficient), Weight for GDP in the objective function (used for Objective coefficient), Weight for average temperature in the objective function (used for Objective coefficient), Total number of matches to be hosted (used for Constraint bound), Total budget constraint for hosting (used for Constraint bound), Maximum number of matches per city (used for Constraint bound)
- Business logic formulas to express in natural language: Calculation of the benefit for selecting a city as a host (calculation method for Objective function component)
- 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 ∑(w1 * Population_i + w2 * GDP_i + w3 * Temperature_i) * x_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: ['∑(x_i) ≤ Total_Matches', '∑(hosting_cost_i * x_i) ≤ Total_Budget', 'x_i ≤ Max_Matches_Per_City for all 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 tables for city data, match constraints, and objective coefficients. Configuration logic updates include scalar parameters for weights and constraints, and formulas for benefit calculation.

CREATE TABLE city_data (
  city_id INTEGER,
  population INTEGER,
  gdp FLOAT,
  avg_temperature FLOAT,
  hosting_cost FLOAT
);

CREATE TABLE match_constraints (
  constraint_id INTEGER,
  max_matches_per_city INTEGER,
  total_budget FLOAT
);

CREATE TABLE objective_coefficients (
  coefficient_id INTEGER,
  weight_population FLOAT,
  weight_gdp FLOAT,
  weight_temperature 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 city demographics, economic data, and climate conditions, ensuring they align with the optimization problem's requirements and constraints.

-- Realistic data for city_data
INSERT INTO city_data (city_id, population, gdp, avg_temperature, hosting_cost) VALUES (1, 800000, 75000.0, 22.0, 120000.0);
INSERT INTO city_data (city_id, population, gdp, avg_temperature, hosting_cost) VALUES (2, 1200000, 110000.0, 26.0, 180000.0);
INSERT INTO city_data (city_id, population, gdp, avg_temperature, hosting_cost) VALUES (3, 600000, 60000.0, 20.0, 100000.0);

-- Realistic data for match_constraints
INSERT INTO match_constraints (constraint_id, max_matches_per_city, total_budget) VALUES (1, 2, 1000000.0);
INSERT INTO match_constraints (constraint_id, max_matches_per_city, total_budget) VALUES (2, 2, 1000000.0);
INSERT INTO match_constraints (constraint_id, max_matches_per_city, total_budget) VALUES (3, 2, 1000000.0);

-- Realistic data for objective_coefficients
INSERT INTO objective_coefficients (coefficient_id, weight_population, weight_gdp, weight_temperature) VALUES (1, 0.4, 0.3, 0.3);
INSERT INTO objective_coefficients (coefficient_id, weight_population, weight_gdp, weight_temperature) VALUES (2, 0.4, 0.3, 0.3);
INSERT INTO objective_coefficients (coefficient_id, weight_population, weight_gdp, weight_temperature) VALUES (3, 0.4, 0.3, 0.3);


```

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

Response:
# Complete Optimization Problem and Solution: city_record

## 1. Problem Context and Goals

### Context  
A sports organization is tasked with selecting the optimal set of host cities for a series of matches in a given year. The decision-making process is guided by a set of key factors: the regional population of each city, its GDP, and the average temperature during the match month. These factors are weighted to reflect their relative importance in determining the overall benefit of selecting a city as a host. The weights are as follows: 40% for population, 30% for GDP, and 30% for temperature. The organization must also adhere to several operational constraints: the total number of matches to be hosted is capped at 10, the total budget for hosting cannot exceed $1,000,000, and no single city can host more than 2 matches. The goal is to maximize the overall benefit of the selected host cities while respecting these constraints.

### Goals  
The primary goal of this optimization problem is to maximize the total benefit derived from selecting host cities for the matches. This benefit is calculated as a weighted sum of each city's regional population, GDP, and average temperature during the match month. The weights assigned to these factors ensure that the selection process aligns with the organization's priorities. Success is measured by the ability to select a set of cities that collectively provide the highest possible benefit without exceeding the total number of matches, the budget, or the maximum number of matches per city.

## 2. Constraints    

The optimization process must respect the following constraints:  
1. **Total Matches Constraint**: The total number of matches hosted across all selected cities must not exceed 10.  
2. **Budget Constraint**: The total cost of hosting matches in the selected cities must not exceed the allocated budget of $1,000,000.  
3. **Maximum Matches Per City Constraint**: No single city can host more than 2 matches.  

These constraints ensure that the selection process remains feasible and aligns with the organization's operational limitations.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating tables for city data, match constraints, and objective coefficients. Configuration logic updates include scalar parameters for weights and constraints, and formulas for benefit calculation.

CREATE TABLE city_data (
  city_id INTEGER,
  population INTEGER,
  gdp FLOAT,
  avg_temperature FLOAT,
  hosting_cost FLOAT
);

CREATE TABLE match_constraints (
  constraint_id INTEGER,
  max_matches_per_city INTEGER,
  total_budget FLOAT
);

CREATE TABLE objective_coefficients (
  coefficient_id INTEGER,
  weight_population FLOAT,
  weight_gdp FLOAT,
  weight_temperature FLOAT
);
```

### Data Dictionary  
- **city_data**: Contains information about potential host cities, including their unique identifier, regional population, GDP, average temperature during the match month, and the cost of hosting a match in the city.  
  - **city_id**: Unique identifier for each city.  
  - **population**: Regional population of the city, used as a component in the benefit calculation.  
  - **gdp**: GDP of the city, used as a component in the benefit calculation.  
  - **avg_temperature**: Average temperature during the match month, used as a component in the benefit calculation.  
  - **hosting_cost**: Cost of hosting a match in the city, used to enforce the budget constraint.  

- **match_constraints**: Stores operational constraints related to hosting matches, including the maximum number of matches per city and the total budget.  
  - **constraint_id**: Unique identifier for each constraint.  
  - **max_matches_per_city**: Maximum number of matches that can be hosted in a single city.  
  - **total_budget**: Total budget allocated for hosting matches across all cities.  

- **objective_coefficients**: Contains the weights assigned to each factor in the benefit calculation.  
  - **coefficient_id**: Unique identifier for each coefficient.  
  - **weight_population**: Weight assigned to the regional population in the benefit calculation (40%).  
  - **weight_gdp**: Weight assigned to the GDP in the benefit calculation (30%).  
  - **weight_temperature**: Weight assigned to the average temperature in the benefit calculation (30%).  

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic city demographics, economic data, and climate conditions, ensuring they align with the optimization problem's requirements and constraints.

-- Realistic data for city_data
INSERT INTO city_data (city_id, population, gdp, avg_temperature, hosting_cost) VALUES (1, 800000, 75000.0, 22.0, 120000.0);
INSERT INTO city_data (city_id, population, gdp, avg_temperature, hosting_cost) VALUES (2, 1200000, 110000.0, 26.0, 180000.0);
INSERT INTO city_data (city_id, population, gdp, avg_temperature, hosting_cost) VALUES (3, 600000, 60000.0, 20.0, 100000.0);

-- Realistic data for match_constraints
INSERT INTO match_constraints (constraint_id, max_matches_per_city, total_budget) VALUES (1, 2, 1000000.0);
INSERT INTO match_constraints (constraint_id, max_matches_per_city, total_budget) VALUES (2, 2, 1000000.0);
INSERT INTO match_constraints (constraint_id, max_matches_per_city, total_budget) VALUES (3, 2, 1000000.0);

-- Realistic data for objective_coefficients
INSERT INTO objective_coefficients (coefficient_id, weight_population, weight_gdp, weight_temperature) VALUES (1, 0.4, 0.3, 0.3);
INSERT INTO objective_coefficients (coefficient_id, weight_population, weight_gdp, weight_temperature) VALUES (2, 0.4, 0.3, 0.3);
INSERT INTO objective_coefficients (coefficient_id, weight_population, weight_gdp, weight_temperature) VALUES (3, 0.4, 0.3, 0.3);
```
