Iteration final - PROBLEM_DESCRIPTION
Sequence: 9
Timestamp: 2025-07-25 22:35:24

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 3), 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): 1 if bridge i is built, 0 otherwise; y[j] (binary): 1 if mill j is built, 0 otherwise
- Operational parameters align with expected linear objective: minimize ∑(length_meters[i] * x[i] + length_feet[j] * y[j]) where x[i] and y[j] are binary decision variables indicating whether bridge i or mill j is built
- Business configuration includes: Minimum number of projects each architect must be assigned to (used for Constraint bound for minimum projects per architect), Maximum number of projects each architect can be assigned to (used for Constraint bound for maximum projects per architect)
- 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": "architecture",
  "iteration": 3,
  "business_context": "A construction company aims to minimize the total length of bridges and mills built by architects while ensuring each architect is assigned to at least one project and no architect is overburdened with more than three projects.",
  "optimization_problem_description": "Minimize the total length of bridges and mills built by architects, subject to constraints on the number of projects each architect can handle and ensuring each architect is assigned to at least one project.",
  "optimization_formulation": {
    "objective": "minimize \u2211(length_meters[i] * x[i] + length_feet[j] * y[j]) where x[i] and y[j] are binary decision variables indicating whether bridge i or mill j is built",
    "decision_variables": "x[i] (binary): 1 if bridge i is built, 0 otherwise; y[j] (binary): 1 if mill j is built, 0 otherwise",
    "constraints": "\u2211(x[i] + y[j]) >= min_projects[k] for each architect k; \u2211(x[i] + y[j]) <= max_projects[k] for each architect k"
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "length_meters[i]": {
        "currently_mapped_to": "bridge_lengths.length_meters",
        "mapping_adequacy": "good",
        "description": "Length of bridge i in meters"
      },
      "length_feet[j]": {
        "currently_mapped_to": "mill_lengths.length_feet",
        "mapping_adequacy": "good",
        "description": "Length of mill j in feet"
      }
    },
    "constraint_bounds": {
      "min_projects[k]": {
        "currently_mapped_to": "business_configuration_logic.min_projects_per_architect",
        "mapping_adequacy": "good",
        "description": "Minimum number of projects architect k must be assigned to"
      },
      "max_projects[k]": {
        "currently_mapped_to": "business_configuration_logic.max_projects_per_architect",
        "mapping_adequacy": "good",
        "description": "Maximum number of projects architect k can be assigned to"
      }
    },
    "decision_variables": {
      "x[i]": {
        "currently_mapped_to": "architect_project_assignments.bridge_assignment",
        "mapping_adequacy": "good",
        "description": "Binary decision variable indicating whether bridge i is built",
        "variable_type": "binary"
      },
      "y[j]": {
        "currently_mapped_to": "architect_project_assignments.mill_assignment",
        "mapping_adequacy": "good",
        "description": "Binary decision variable indicating whether mill j is built",
        "variable_type": "binary"
      }
    }
  },
  "missing_optimization_requirements": [],
  "iteration_status": {
    "complete": true,
    "confidence": "high",
    "next_focus": "Ready for convergence"
  }
}

FINAL DATABASE SCHEMA:
```sql
-- Iteration 3 Database Schema
-- Objective: Schema changes include adding tables for decision variables and constraint bounds, updating the data dictionary, and refining business configuration logic to include scalar parameters for minimum and maximum projects per architect.

CREATE TABLE bridge_lengths (
  bridge_id INTEGER,
  length_meters INTEGER
);

CREATE TABLE mill_lengths (
  mill_id INTEGER,
  length_feet INTEGER
);

CREATE TABLE architect_project_assignments (
  architect_id INTEGER,
  bridge_assignment BOOLEAN,
  mill_assignment BOOLEAN
);

CREATE TABLE architect_constraints (
  architect_id INTEGER,
  min_projects INTEGER,
  max_projects INTEGER
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 3 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical construction project lengths and architect workload capacities, ensuring realistic and meaningful data for optimization.

-- Realistic data for bridge_lengths
INSERT INTO bridge_lengths (bridge_id, length_meters) VALUES (1, 120);
INSERT INTO bridge_lengths (bridge_id, length_meters) VALUES (2, 250);
INSERT INTO bridge_lengths (bridge_id, length_meters) VALUES (3, 400);

-- Realistic data for mill_lengths
INSERT INTO mill_lengths (mill_id, length_feet) VALUES (1, 350);
INSERT INTO mill_lengths (mill_id, length_feet) VALUES (2, 450);
INSERT INTO mill_lengths (mill_id, length_feet) VALUES (3, 550);

-- Realistic data for architect_project_assignments
INSERT INTO architect_project_assignments (architect_id, bridge_assignment, mill_assignment) VALUES (1, True, False);
INSERT INTO architect_project_assignments (architect_id, bridge_assignment, mill_assignment) VALUES (2, False, True);
INSERT INTO architect_project_assignments (architect_id, bridge_assignment, mill_assignment) VALUES (3, True, True);

-- Realistic data for architect_constraints
INSERT INTO architect_constraints (architect_id, min_projects, max_projects) VALUES (1, 1, 3);
INSERT INTO architect_constraints (architect_id, min_projects, max_projects) VALUES (2, 1, 3);
INSERT INTO architect_constraints (architect_id, min_projects, max_projects) VALUES (3, 1, 3);


```

DATA DICTIONARY:
{
  "tables": {
    "bridge_lengths": {
      "business_purpose": "Stores the length of each bridge in meters for optimization",
      "optimization_role": "objective_coefficients",
      "columns": {
        "bridge_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each bridge",
          "optimization_purpose": "Links bridges to their lengths",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "length_meters": {
          "data_type": "INTEGER",
          "business_meaning": "Length of the bridge in meters",
          "optimization_purpose": "Objective coefficient for bridge length",
          "sample_values": [
            100,
            150,
            200
          ]
        }
      }
    },
    "mill_lengths": {
      "business_purpose": "Stores the length of each mill in feet for optimization",
      "optimization_role": "objective_coefficients",
      "columns": {
        "mill_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each mill",
          "optimization_purpose": "Links mills to their lengths",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "length_feet": {
          "data_type": "INTEGER",
          "business_meaning": "Length of the mill in feet",
          "optimization_purpose": "Objective coefficient for mill length",
          "sample_values": [
            328,
            400,
            500
          ]
        }
      }
    },
    "architect_project_assignments": {
      "business_purpose": "Stores the assignment of architects to projects, indicating whether a bridge or mill is built",
      "optimization_role": "decision_variables",
      "columns": {
        "architect_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each architect",
          "optimization_purpose": "Links architects to their project assignments",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "bridge_assignment": {
          "data_type": "BOOLEAN",
          "business_meaning": "Indicates whether the architect is assigned to build a bridge",
          "optimization_purpose": "Decision variable for bridge assignment",
          "sample_values": [
            true,
            false,
            true
          ]
        },
        "mill_assignment": {
          "data_type": "BOOLEAN",
          "business_meaning": "Indicates whether the architect is assigned to build a mill",
          "optimization_purpose": "Decision variable for mill assignment",
          "sample_values": [
            false,
            true,
            false
          ]
        }
      }
    },
    "architect_constraints": {
      "business_purpose": "Stores the minimum and maximum number of projects each architect can handle",
      "optimization_role": "constraint_bounds",
      "columns": {
        "architect_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each architect",
          "optimization_purpose": "Links architects to their project constraints",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "min_projects": {
          "data_type": "INTEGER",
          "business_meaning": "Minimum number of projects the architect must be assigned to",
          "optimization_purpose": "Constraint bound for minimum projects per architect",
          "sample_values": [
            1,
            1,
            1
          ]
        },
        "max_projects": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum number of projects the architect can be assigned to",
          "optimization_purpose": "Constraint bound for maximum projects per architect",
          "sample_values": [
            3,
            3,
            3
          ]
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "min_projects_per_architect": {
    "data_type": "INTEGER",
    "business_meaning": "Minimum number of projects each architect must be assigned to",
    "optimization_role": "Constraint bound for minimum projects per architect",
    "configuration_type": "scalar_parameter",
    "value": 1,
    "business_justification": "Ensures each architect is assigned to at least one project, maintaining workload balance."
  },
  "max_projects_per_architect": {
    "data_type": "INTEGER",
    "business_meaning": "Maximum number of projects each architect can be assigned to",
    "optimization_role": "Constraint bound for maximum projects per architect",
    "configuration_type": "scalar_parameter",
    "value": 3,
    "business_justification": "Prevents any architect from being overburdened with more than three projects, ensuring manageable workloads."
  }
}

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: architecture

## 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): 1 if bridge i is built, 0 otherwise; y[j] (binary): 1 if mill j is built, 0 otherwise
- Operational parameters align with expected linear objective: minimize ∑(length_meters[i] * x[i] + length_feet[j] * y[j]) where x[i] and y[j] are binary decision variables indicating whether bridge i or mill j is built
- Business configuration includes: Minimum number of projects each architect must be assigned to (used for Constraint bound for minimum projects per architect), Maximum number of projects each architect can be assigned to (used for Constraint bound for maximum projects per architect)
- 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: minimize
- Metric to optimize: minimize ∑(length_meters[i] * x[i] + length_feet[j] * y[j]) where x[i] and y[j] are binary decision variables indicating whether bridge i or mill j is built
- 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] + y[j]) >= min_projects[k] for each architect k; ∑(x[i] + y[j]) <= max_projects[k] for each architect k (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 3 Database Schema
-- Objective: Schema changes include adding tables for decision variables and constraint bounds, updating the data dictionary, and refining business configuration logic to include scalar parameters for minimum and maximum projects per architect.

CREATE TABLE bridge_lengths (
  bridge_id INTEGER,
  length_meters INTEGER
);

CREATE TABLE mill_lengths (
  mill_id INTEGER,
  length_feet INTEGER
);

CREATE TABLE architect_project_assignments (
  architect_id INTEGER,
  bridge_assignment BOOLEAN,
  mill_assignment BOOLEAN
);

CREATE TABLE architect_constraints (
  architect_id INTEGER,
  min_projects INTEGER,
  max_projects 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 3 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical construction project lengths and architect workload capacities, ensuring realistic and meaningful data for optimization.

-- Realistic data for bridge_lengths
INSERT INTO bridge_lengths (bridge_id, length_meters) VALUES (1, 120);
INSERT INTO bridge_lengths (bridge_id, length_meters) VALUES (2, 250);
INSERT INTO bridge_lengths (bridge_id, length_meters) VALUES (3, 400);

-- Realistic data for mill_lengths
INSERT INTO mill_lengths (mill_id, length_feet) VALUES (1, 350);
INSERT INTO mill_lengths (mill_id, length_feet) VALUES (2, 450);
INSERT INTO mill_lengths (mill_id, length_feet) VALUES (3, 550);

-- Realistic data for architect_project_assignments
INSERT INTO architect_project_assignments (architect_id, bridge_assignment, mill_assignment) VALUES (1, True, False);
INSERT INTO architect_project_assignments (architect_id, bridge_assignment, mill_assignment) VALUES (2, False, True);
INSERT INTO architect_project_assignments (architect_id, bridge_assignment, mill_assignment) VALUES (3, True, True);

-- Realistic data for architect_constraints
INSERT INTO architect_constraints (architect_id, min_projects, max_projects) VALUES (1, 1, 3);
INSERT INTO architect_constraints (architect_id, min_projects, max_projects) VALUES (2, 1, 3);
INSERT INTO architect_constraints (architect_id, min_projects, max_projects) VALUES (3, 1, 3);


```

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

Response:
# Complete Optimization Problem and Solution: architecture

## 1. Problem Context and Goals

### Context  
A construction company is tasked with optimizing the allocation of projects to architects to minimize the total length of bridges and mills built. The company must decide which bridges and mills to construct, represented by binary decision variables indicating whether a specific bridge or mill is built. Each bridge has a defined length in meters, and each mill has a defined length in feet, which directly contribute to the total length to be minimized.  

The company operates under specific business rules to ensure fair and manageable workloads for architects. Each architect must be assigned to at least one project, as defined by the minimum number of projects per architect, and no architect can be assigned more than three projects, as defined by the maximum number of projects per architect. These rules ensure a balanced distribution of work while preventing overburdening any single architect.  

The operational parameters, such as the lengths of bridges and mills, are sourced from the current database, which includes realistic data generated by experts in business, data, and optimization. The business configuration parameters, including the minimum and maximum number of projects per architect, are defined as scalar values and are critical for enforcing the constraints in the optimization problem.  

### Goals  
The primary goal of this optimization problem is to minimize the total length of bridges and mills built by architects. This is achieved by strategically selecting which bridges and mills to construct, ensuring that the sum of their lengths is as small as possible. Success is measured by the reduction in total length, which directly impacts resource efficiency and cost savings for the company.  

The optimization process ensures that all architects are assigned to at least one project and that no architect is assigned more than three projects. These constraints are critical for maintaining workload balance and operational feasibility.  

## 2. Constraints  

The optimization problem is subject to the following constraints:  
1. **Minimum Projects per Architect**: Each architect must be assigned to at least one project. This ensures that all architects are actively contributing to the construction efforts.  
2. **Maximum Projects per Architect**: No architect can be assigned more than three projects. This prevents overburdening any single architect and ensures manageable workloads.  

These constraints are enforced using the minimum and maximum number of projects per architect, which are defined as scalar parameters in the business configuration.  

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 3 Database Schema
-- Objective: Schema changes include adding tables for decision variables and constraint bounds, updating the data dictionary, and refining business configuration logic to include scalar parameters for minimum and maximum projects per architect.

CREATE TABLE bridge_lengths (
  bridge_id INTEGER,
  length_meters INTEGER
);

CREATE TABLE mill_lengths (
  mill_id INTEGER,
  length_feet INTEGER
);

CREATE TABLE architect_project_assignments (
  architect_id INTEGER,
  bridge_assignment BOOLEAN,
  mill_assignment BOOLEAN
);

CREATE TABLE architect_constraints (
  architect_id INTEGER,
  min_projects INTEGER,
  max_projects INTEGER
);
```

### Data Dictionary  
- **bridge_lengths**: Stores the length of each bridge in meters, which is used to calculate the total length of bridges built.  
  - `bridge_id`: Unique identifier for each bridge.  
  - `length_meters`: Length of the bridge in meters, used as a coefficient in the optimization objective.  

- **mill_lengths**: Stores the length of each mill in feet, which is used to calculate the total length of mills built.  
  - `mill_id`: Unique identifier for each mill.  
  - `length_feet`: Length of the mill in feet, used as a coefficient in the optimization objective.  

- **architect_project_assignments**: Tracks the assignment of architects to projects, indicating whether a bridge or mill is built.  
  - `architect_id`: Unique identifier for each architect.  
  - `bridge_assignment`: Indicates whether the architect is assigned to build a bridge.  
  - `mill_assignment`: Indicates whether the architect is assigned to build a mill.  

- **architect_constraints**: Defines the minimum and maximum number of projects each architect can handle.  
  - `architect_id`: Unique identifier for each architect.  
  - `min_projects`: Minimum number of projects the architect must be assigned to.  
  - `max_projects`: Maximum number of projects the architect can be assigned to.  

### Current Stored Values  
```sql
-- Iteration 3 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical construction project lengths and architect workload capacities, ensuring realistic and meaningful data for optimization.

-- Realistic data for bridge_lengths
INSERT INTO bridge_lengths (bridge_id, length_meters) VALUES (1, 120);
INSERT INTO bridge_lengths (bridge_id, length_meters) VALUES (2, 250);
INSERT INTO bridge_lengths (bridge_id, length_meters) VALUES (3, 400);

-- Realistic data for mill_lengths
INSERT INTO mill_lengths (mill_id, length_feet) VALUES (1, 350);
INSERT INTO mill_lengths (mill_id, length_feet) VALUES (2, 450);
INSERT INTO mill_lengths (mill_id, length_feet) VALUES (3, 550);

-- Realistic data for architect_project_assignments
INSERT INTO architect_project_assignments (architect_id, bridge_assignment, mill_assignment) VALUES (1, True, False);
INSERT INTO architect_project_assignments (architect_id, bridge_assignment, mill_assignment) VALUES (2, False, True);
INSERT INTO architect_project_assignments (architect_id, bridge_assignment, mill_assignment) VALUES (3, True, True);

-- Realistic data for architect_constraints
INSERT INTO architect_constraints (architect_id, min_projects, max_projects) VALUES (1, 1, 3);
INSERT INTO architect_constraints (architect_id, min_projects, max_projects) VALUES (2, 1, 3);
INSERT INTO architect_constraints (architect_id, min_projects, max_projects) VALUES (3, 1, 3);
```
