Iteration final - PROBLEM_DESCRIPTION
Sequence: 5
Timestamp: 2025-07-25 22:30:48

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: Invite_Farm_i: binary variable indicating whether farm i is invited (1) or not (0)
- Operational parameters align with expected linear objective: maximize ∑(Quality_Score_Farm_i × Invite_Farm_i)
- Business configuration includes: Maximum number of farms the city can host. (used for Constraint bound for the total number of invited farms.), Maximum number of horses the city can accommodate. (used for Constraint bound for the total number of horses from invited farms.), Maximum number of cattle the city can accommodate. (used for Constraint bound for the total number of cattle from invited farms.), Maximum number of pigs the city can accommodate. (used for Constraint bound for the total number of pigs from invited farms.), Maximum number of sheep and goats the city can accommodate. (used for Constraint bound for the total number of sheep and goats from invited farms.)
- Business logic formulas to express in natural language: Weighted sum of animals on each farm to represent its quality. (calculation method for Objective coefficient for the quality score of each farm.)
- 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": "farm",
  "iteration": 1,
  "business_context": "A farm competition organizer aims to maximize the overall quality of the competition by selecting the best farms to participate, considering the number of animals each farm has and the city's capacity to host the event. The quality of each farm is determined by a weighted sum of the animals it possesses.",
  "optimization_problem_description": "The organizer needs to decide which farms to invite to the competition to maximize the total quality score of the invited farms, while respecting the hosting city's capacity constraints on the number of farms and animals.",
  "optimization_formulation": {
    "objective": "maximize \u2211(Quality_Score_Farm_i \u00d7 Invite_Farm_i)",
    "decision_variables": "Invite_Farm_i: binary variable indicating whether farm i is invited (1) or not (0)",
    "constraints": [
      "\u2211(Invite_Farm_i) \u2264 Max_Farms",
      "\u2211(horses_i \u00d7 Invite_Farm_i) \u2264 Max_Horses",
      "\u2211(cattle_i \u00d7 Invite_Farm_i) \u2264 Max_Cattle",
      "\u2211(pigs_i \u00d7 Invite_Farm_i) \u2264 Max_Pigs",
      "\u2211(sheep_and_goats_i \u00d7 Invite_Farm_i) \u2264 Max_Sheep_Goats"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "Quality_Score_Farm_i": {
        "currently_mapped_to": "business_configuration_logic.Quality_Score_Farm_i",
        "mapping_adequacy": "good",
        "description": "Weighted sum of animals on each farm to represent its quality."
      }
    },
    "constraint_bounds": {
      "Max_Farms": {
        "currently_mapped_to": "business_configuration_logic.Max_Farms",
        "mapping_adequacy": "good",
        "description": "Maximum number of farms the city can host."
      },
      "Max_Horses": {
        "currently_mapped_to": "business_configuration_logic.Max_Horses",
        "mapping_adequacy": "good",
        "description": "Maximum number of horses the city can accommodate."
      },
      "Max_Cattle": {
        "currently_mapped_to": "business_configuration_logic.Max_Cattle",
        "mapping_adequacy": "good",
        "description": "Maximum number of cattle the city can accommodate."
      },
      "Max_Pigs": {
        "currently_mapped_to": "business_configuration_logic.Max_Pigs",
        "mapping_adequacy": "good",
        "description": "Maximum number of pigs the city can accommodate."
      },
      "Max_Sheep_Goats": {
        "currently_mapped_to": "business_configuration_logic.Max_Sheep_Goats",
        "mapping_adequacy": "good",
        "description": "Maximum number of sheep and goats the city can accommodate."
      }
    },
    "decision_variables": {
      "Invite_Farm_i": {
        "currently_mapped_to": "farms.invite_status",
        "mapping_adequacy": "good",
        "description": "Decision variable for inviting the farm.",
        "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 farms, animals, and city capacities, and updating business configuration logic to handle scalar parameters and formulas for optimization.

CREATE TABLE farms (
  farm_id INTEGER,
  invite_status BOOLEAN
);

CREATE TABLE animals (
  farm_id INTEGER,
  horses INTEGER,
  cattle INTEGER,
  pigs INTEGER,
  sheep_and_goats INTEGER
);

CREATE TABLE city_capacities (
  max_farms INTEGER,
  max_horses INTEGER,
  max_cattle INTEGER,
  max_pigs INTEGER,
  max_sheep_goats INTEGER
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical farm sizes and city hosting capacities, ensuring a balance between farm quality and city constraints.

-- Realistic data for farms
INSERT INTO farms (farm_id, invite_status) VALUES (1, False);
INSERT INTO farms (farm_id, invite_status) VALUES (2, False);
INSERT INTO farms (farm_id, invite_status) VALUES (3, False);

-- Realistic data for animals
INSERT INTO animals (farm_id, horses, cattle, pigs, sheep_and_goats) VALUES (1, 5, 20, 50, 30);
INSERT INTO animals (farm_id, horses, cattle, pigs, sheep_and_goats) VALUES (2, 10, 30, 60, 40);
INSERT INTO animals (farm_id, horses, cattle, pigs, sheep_and_goats) VALUES (3, 15, 40, 70, 50);

-- Realistic data for city_capacities
INSERT INTO city_capacities (max_farms, max_horses, max_cattle, max_pigs, max_sheep_goats) VALUES (10, 50, 100, 200, 150);


```

DATA DICTIONARY:
{
  "tables": {
    "farms": {
      "business_purpose": "Represents farms eligible for the competition.",
      "optimization_role": "business_data",
      "columns": {
        "farm_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each farm.",
          "optimization_purpose": "Identifier for decision variables.",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "invite_status": {
          "data_type": "BOOLEAN",
          "business_meaning": "Whether the farm is invited to the competition.",
          "optimization_purpose": "Decision variable for inviting the farm.",
          "sample_values": [
            true,
            false,
            true
          ]
        }
      }
    },
    "animals": {
      "business_purpose": "Represents the number of each type of animal on each farm.",
      "optimization_role": "business_data",
      "columns": {
        "farm_id": {
          "data_type": "INTEGER",
          "business_meaning": "Foreign key linking to the farms table.",
          "optimization_purpose": "Identifier for decision variables.",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "horses": {
          "data_type": "INTEGER",
          "business_meaning": "Number of horses on the farm.",
          "optimization_purpose": "Used in quality score calculation and constraints.",
          "sample_values": [
            5,
            10,
            15
          ]
        },
        "cattle": {
          "data_type": "INTEGER",
          "business_meaning": "Number of cattle on the farm.",
          "optimization_purpose": "Used in quality score calculation and constraints.",
          "sample_values": [
            20,
            30,
            40
          ]
        },
        "pigs": {
          "data_type": "INTEGER",
          "business_meaning": "Number of pigs on the farm.",
          "optimization_purpose": "Used in quality score calculation and constraints.",
          "sample_values": [
            50,
            60,
            70
          ]
        },
        "sheep_and_goats": {
          "data_type": "INTEGER",
          "business_meaning": "Number of sheep and goats on the farm.",
          "optimization_purpose": "Used in quality score calculation and constraints.",
          "sample_values": [
            30,
            40,
            50
          ]
        }
      }
    },
    "city_capacities": {
      "business_purpose": "Represents the city's capacity constraints for hosting the competition.",
      "optimization_role": "constraint_bounds",
      "columns": {
        "max_farms": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum number of farms the city can host.",
          "optimization_purpose": "Constraint bound for the total number of invited farms.",
          "sample_values": [
            10
          ]
        },
        "max_horses": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum number of horses the city can accommodate.",
          "optimization_purpose": "Constraint bound for the total number of horses from invited farms.",
          "sample_values": [
            50
          ]
        },
        "max_cattle": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum number of cattle the city can accommodate.",
          "optimization_purpose": "Constraint bound for the total number of cattle from invited farms.",
          "sample_values": [
            100
          ]
        },
        "max_pigs": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum number of pigs the city can accommodate.",
          "optimization_purpose": "Constraint bound for the total number of pigs from invited farms.",
          "sample_values": [
            200
          ]
        },
        "max_sheep_goats": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum number of sheep and goats the city can accommodate.",
          "optimization_purpose": "Constraint bound for the total number of sheep and goats from invited farms.",
          "sample_values": [
            150
          ]
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "Max_Farms": {
    "data_type": "INTEGER",
    "business_meaning": "Maximum number of farms the city can host.",
    "optimization_role": "Constraint bound for the total number of invited farms.",
    "configuration_type": "scalar_parameter",
    "value": 10,
    "business_justification": "The city can reasonably host up to 10 farms."
  },
  "Max_Horses": {
    "data_type": "INTEGER",
    "business_meaning": "Maximum number of horses the city can accommodate.",
    "optimization_role": "Constraint bound for the total number of horses from invited farms.",
    "configuration_type": "scalar_parameter",
    "value": 50,
    "business_justification": "The city can accommodate up to 50 horses."
  },
  "Max_Cattle": {
    "data_type": "INTEGER",
    "business_meaning": "Maximum number of cattle the city can accommodate.",
    "optimization_role": "Constraint bound for the total number of cattle from invited farms.",
    "configuration_type": "scalar_parameter",
    "value": 100,
    "business_justification": "The city can accommodate up to 100 cattle."
  },
  "Max_Pigs": {
    "data_type": "INTEGER",
    "business_meaning": "Maximum number of pigs the city can accommodate.",
    "optimization_role": "Constraint bound for the total number of pigs from invited farms.",
    "configuration_type": "scalar_parameter",
    "value": 200,
    "business_justification": "The city can accommodate up to 200 pigs."
  },
  "Max_Sheep_Goats": {
    "data_type": "INTEGER",
    "business_meaning": "Maximum number of sheep and goats the city can accommodate.",
    "optimization_role": "Constraint bound for the total number of sheep and goats from invited farms.",
    "configuration_type": "scalar_parameter",
    "value": 150,
    "business_justification": "The city can accommodate up to 150 sheep and goats."
  },
  "Quality_Score_Farm_i": {
    "data_type": "STRING",
    "business_meaning": "Weighted sum of animals on each farm to represent its quality.",
    "optimization_role": "Objective coefficient for the quality score of each farm.",
    "configuration_type": "business_logic_formula",
    "formula_expression": "(horses * 0.3) + (cattle * 0.25) + (pigs * 0.2) + (sheep_and_goats * 0.25)"
  }
}

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: farm

## 1. Problem Context and Goals

### Context  
[Regenerate business context that naturally aligns with LINEAR optimization formulation. Ensure:]
- Business decisions match expected decision variables: Invite_Farm_i: binary variable indicating whether farm i is invited (1) or not (0)
- Operational parameters align with expected linear objective: maximize ∑(Quality_Score_Farm_i × Invite_Farm_i)
- Business configuration includes: Maximum number of farms the city can host. (used for Constraint bound for the total number of invited farms.), Maximum number of horses the city can accommodate. (used for Constraint bound for the total number of horses from invited farms.), Maximum number of cattle the city can accommodate. (used for Constraint bound for the total number of cattle from invited farms.), Maximum number of pigs the city can accommodate. (used for Constraint bound for the total number of pigs from invited farms.), Maximum number of sheep and goats the city can accommodate. (used for Constraint bound for the total number of sheep and goats from invited farms.)
- Business logic formulas to express in natural language: Weighted sum of animals on each farm to represent its quality. (calculation method for Objective coefficient for the quality score of each farm.)
- 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 ∑(Quality_Score_Farm_i × Invite_Farm_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: ['∑(Invite_Farm_i) ≤ Max_Farms', '∑(horses_i × Invite_Farm_i) ≤ Max_Horses', '∑(cattle_i × Invite_Farm_i) ≤ Max_Cattle', '∑(pigs_i × Invite_Farm_i) ≤ Max_Pigs', '∑(sheep_and_goats_i × Invite_Farm_i) ≤ Max_Sheep_Goats'] (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 farms, animals, and city capacities, and updating business configuration logic to handle scalar parameters and formulas for optimization.

CREATE TABLE farms (
  farm_id INTEGER,
  invite_status BOOLEAN
);

CREATE TABLE animals (
  farm_id INTEGER,
  horses INTEGER,
  cattle INTEGER,
  pigs INTEGER,
  sheep_and_goats INTEGER
);

CREATE TABLE city_capacities (
  max_farms INTEGER,
  max_horses INTEGER,
  max_cattle INTEGER,
  max_pigs INTEGER,
  max_sheep_goats 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 typical farm sizes and city hosting capacities, ensuring a balance between farm quality and city constraints.

-- Realistic data for farms
INSERT INTO farms (farm_id, invite_status) VALUES (1, False);
INSERT INTO farms (farm_id, invite_status) VALUES (2, False);
INSERT INTO farms (farm_id, invite_status) VALUES (3, False);

-- Realistic data for animals
INSERT INTO animals (farm_id, horses, cattle, pigs, sheep_and_goats) VALUES (1, 5, 20, 50, 30);
INSERT INTO animals (farm_id, horses, cattle, pigs, sheep_and_goats) VALUES (2, 10, 30, 60, 40);
INSERT INTO animals (farm_id, horses, cattle, pigs, sheep_and_goats) VALUES (3, 15, 40, 70, 50);

-- Realistic data for city_capacities
INSERT INTO city_capacities (max_farms, max_horses, max_cattle, max_pigs, max_sheep_goats) VALUES (10, 50, 100, 200, 150);


```

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

Response:
# Complete Optimization Problem and Solution: farm

## 1. Problem Context and Goals

### Context  
A farm competition organizer is tasked with selecting the best farms to participate in an event hosted by a city. The goal is to maximize the overall quality of the competition by inviting farms that contribute the most to the event's success. The quality of each farm is determined by a weighted sum of the animals it possesses, including horses, cattle, pigs, sheep, and goats. The weights assigned to each animal type reflect their relative importance in the competition.  

The organizer must decide which farms to invite, ensuring that the total number of invited farms does not exceed the city's capacity to host them. Additionally, the city has specific limits on the total number of each type of animal that can be accommodated during the event. These limits include the maximum number of horses, cattle, pigs, and sheep and goats.  

The decision to invite a farm is represented by a binary choice: a farm is either invited (1) or not invited (0). The quality score of each farm is calculated as a weighted sum of its animals, with weights assigned as follows: horses (30%), cattle (25%), pigs (20%), and sheep and goats (25%).  

The city's capacity constraints are defined by the following parameters:  
- The maximum number of farms that can be hosted is 10.  
- The maximum number of horses that can be accommodated is 50.  
- The maximum number of cattle that can be accommodated is 100.  
- The maximum number of pigs that can be accommodated is 200.  
- The maximum number of sheep and goats that can be accommodated is 150.  

These constraints ensure that the event remains manageable and aligns with the city's hosting capabilities.  

### Goals  
The primary goal of the optimization problem is to maximize the total quality score of the invited farms. This is achieved by selecting a combination of farms whose collective quality score is the highest possible, while respecting the city's capacity constraints.  

Success is measured by the total quality score of the invited farms, which is calculated as the sum of the quality scores of each invited farm. The quality score of each farm is derived from the weighted sum of its animals, ensuring that farms with a higher number of animals, particularly those with higher weights, contribute more to the overall score.  

## 2. Constraints  

The optimization problem must adhere to the following constraints:  
1. The total number of invited farms cannot exceed the city's maximum hosting capacity of 10 farms.  
2. The total number of horses from all invited farms cannot exceed the city's maximum capacity of 50 horses.  
3. The total number of cattle from all invited farms cannot exceed the city's maximum capacity of 100 cattle.  
4. The total number of pigs from all invited farms cannot exceed the city's maximum capacity of 200 pigs.  
5. The total number of sheep and goats from all invited farms cannot exceed the city's maximum capacity of 150 sheep and goats.  

These constraints ensure that the event remains within the city's logistical and operational limits, while still allowing for the selection of high-quality farms.  

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating tables for farms, animals, and city capacities, and updating business configuration logic to handle scalar parameters and formulas for optimization.

CREATE TABLE farms (
  farm_id INTEGER,
  invite_status BOOLEAN
);

CREATE TABLE animals (
  farm_id INTEGER,
  horses INTEGER,
  cattle INTEGER,
  pigs INTEGER,
  sheep_and_goats INTEGER
);

CREATE TABLE city_capacities (
  max_farms INTEGER,
  max_horses INTEGER,
  max_cattle INTEGER,
  max_pigs INTEGER,
  max_sheep_goats INTEGER
);
```

### Data Dictionary  
- **farms**: Represents farms eligible for the competition.  
  - **farm_id**: Unique identifier for each farm.  
  - **invite_status**: Indicates whether the farm is invited to the competition (True/False).  

- **animals**: Represents the number of each type of animal on each farm.  
  - **farm_id**: Links to the farms table to identify which farm the animals belong to.  
  - **horses**: Number of horses on the farm.  
  - **cattle**: Number of cattle on the farm.  
  - **pigs**: Number of pigs on the farm.  
  - **sheep_and_goats**: Number of sheep and goats on the farm.  

- **city_capacities**: Represents the city's capacity constraints for hosting the competition.  
  - **max_farms**: Maximum number of farms the city can host.  
  - **max_horses**: Maximum number of horses the city can accommodate.  
  - **max_cattle**: Maximum number of cattle the city can accommodate.  
  - **max_pigs**: Maximum number of pigs the city can accommodate.  
  - **max_sheep_goats**: Maximum number of sheep and goats the city can accommodate.  

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical farm sizes and city hosting capacities, ensuring a balance between farm quality and city constraints.

-- Realistic data for farms
INSERT INTO farms (farm_id, invite_status) VALUES (1, False);
INSERT INTO farms (farm_id, invite_status) VALUES (2, False);
INSERT INTO farms (farm_id, invite_status) VALUES (3, False);

-- Realistic data for animals
INSERT INTO animals (farm_id, horses, cattle, pigs, sheep_and_goats) VALUES (1, 5, 20, 50, 30);
INSERT INTO animals (farm_id, horses, cattle, pigs, sheep_and_goats) VALUES (2, 10, 30, 60, 40);
INSERT INTO animals (farm_id, horses, cattle, pigs, sheep_and_goats) VALUES (3, 15, 40, 70, 50);

-- Realistic data for city_capacities
INSERT INTO city_capacities (max_farms, max_horses, max_cattle, max_pigs, max_sheep_goats) VALUES (10, 50, 100, 200, 150);
```
