Iteration final - PROBLEM_DESCRIPTION
Sequence: 7
Timestamp: 2025-07-27 22:26:58

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 2), 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: DriverAssignments.Assigned[i][j] where i is DriverID and j is SchoolID, binary
- Operational parameters align with expected linear objective: minimize total_experience_mismatch = sum(ExperienceMismatch.YearsMismatch[i][j] * DriverAssignments.Assigned[i][j] for all i, j)
- Business configuration includes: Number of drivers required by each school (used for Used in constraint to ensure each school gets required drivers), Indicates if a driver is full-time (used for Used in constraint to ensure only full-time drivers are assigned)
- 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": "school_bus",
  "iteration": 2,
  "business_context": "Optimize the allocation of full-time drivers to schools to minimize the total years of experience mismatch while ensuring all schools have the required number of drivers.",
  "optimization_problem_description": "Assign full-time drivers to schools such that the total mismatch in years of experience is minimized. Each school requires a specific number of drivers, and each driver can be assigned to only one school.",
  "optimization_formulation": {
    "objective": "minimize total_experience_mismatch = sum(ExperienceMismatch.YearsMismatch[i][j] * DriverAssignments.Assigned[i][j] for all i, j)",
    "decision_variables": "DriverAssignments.Assigned[i][j] where i is DriverID and j is SchoolID, binary",
    "constraints": [
      "sum(DriverAssignments.Assigned[i][j] for all j) <= 1 for all i where Drivers.IsFullTime[i] = true",
      "sum(DriverAssignments.Assigned[i][j] for all i) = SchoolRequirements.RequiredDrivers[j] for all j"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "YearsMismatch[i][j]": {
        "currently_mapped_to": "ExperienceMismatch.YearsMismatch",
        "mapping_adequacy": "good",
        "description": "Years of experience mismatch between driver i and school j"
      }
    },
    "constraint_bounds": {
      "RequiredDrivers[j]": {
        "currently_mapped_to": "SchoolRequirements.RequiredDrivers",
        "mapping_adequacy": "good",
        "description": "Number of drivers required by school j"
      }
    },
    "decision_variables": {
      "Assigned[i][j]": {
        "currently_mapped_to": "DriverAssignments.Assigned",
        "mapping_adequacy": "good",
        "description": "Binary variable indicating if driver i is assigned to school j",
        "variable_type": "binary"
      }
    }
  },
  "missing_optimization_requirements": [],
  "iteration_status": {
    "complete": true,
    "confidence": "high",
    "next_focus": "Ready for convergence"
  }
}

FINAL DATABASE SCHEMA:
```sql
-- Iteration 2 Database Schema
-- Objective: Schema changes include creating a new table for years of experience mismatch and updating existing tables to fill mapping gaps. Business configuration logic is updated to handle scalar parameters and formulas.

CREATE TABLE DriverAssignments (
  DriverID INTEGER,
  SchoolID INTEGER,
  Assigned BOOLEAN
);

CREATE TABLE SchoolRequirements (
  SchoolID INTEGER,
  RequiredDrivers INTEGER
);

CREATE TABLE Drivers (
  DriverID INTEGER,
  IsFullTime BOOLEAN
);

CREATE TABLE ExperienceMismatch (
  DriverID INTEGER,
  SchoolID INTEGER,
  YearsMismatch INTEGER
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical school bus operations, ensuring a mix of driver experience levels and school requirements to create a realistic and challenging optimization problem.

-- Realistic data for DriverAssignments
INSERT INTO DriverAssignments (DriverID, SchoolID, Assigned) VALUES (1, 101, False);
INSERT INTO DriverAssignments (DriverID, SchoolID, Assigned) VALUES (2, 102, True);
INSERT INTO DriverAssignments (DriverID, SchoolID, Assigned) VALUES (3, 103, True);

-- Realistic data for SchoolRequirements
INSERT INTO SchoolRequirements (SchoolID, RequiredDrivers) VALUES (101, 3);
INSERT INTO SchoolRequirements (SchoolID, RequiredDrivers) VALUES (102, 4);
INSERT INTO SchoolRequirements (SchoolID, RequiredDrivers) VALUES (103, 5);

-- Realistic data for Drivers
INSERT INTO Drivers (DriverID, IsFullTime) VALUES (1, True);
INSERT INTO Drivers (DriverID, IsFullTime) VALUES (2, True);
INSERT INTO Drivers (DriverID, IsFullTime) VALUES (3, True);

-- Realistic data for ExperienceMismatch
INSERT INTO ExperienceMismatch (DriverID, SchoolID, YearsMismatch) VALUES (1, 101, 2);
INSERT INTO ExperienceMismatch (DriverID, SchoolID, YearsMismatch) VALUES (2, 102, 1);
INSERT INTO ExperienceMismatch (DriverID, SchoolID, YearsMismatch) VALUES (3, 103, 0);


```

DATA DICTIONARY:
{
  "tables": {
    "DriverAssignments": {
      "business_purpose": "Tracks driver assignments to schools",
      "optimization_role": "decision_variables",
      "columns": {
        "DriverID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each driver",
          "optimization_purpose": "Identifies driver in assignment decision variable",
          "sample_values": "1, 2, 3"
        },
        "SchoolID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each school",
          "optimization_purpose": "Identifies school in assignment decision variable",
          "sample_values": "101, 102, 103"
        },
        "Assigned": {
          "data_type": "BOOLEAN",
          "business_meaning": "Indicates if a driver is assigned to a school",
          "optimization_purpose": "Binary decision variable for driver assignment",
          "sample_values": "true, false"
        }
      }
    },
    "SchoolRequirements": {
      "business_purpose": "Stores driver requirements for each school",
      "optimization_role": "constraint_bounds",
      "columns": {
        "SchoolID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each school",
          "optimization_purpose": "Identifies school in driver requirement constraint",
          "sample_values": "101, 102, 103"
        },
        "RequiredDrivers": {
          "data_type": "INTEGER",
          "business_meaning": "Number of drivers required by the school",
          "optimization_purpose": "Constraint bound for driver assignment",
          "sample_values": "3, 4, 5"
        }
      }
    },
    "Drivers": {
      "business_purpose": "Stores information about drivers",
      "optimization_role": "business_data",
      "columns": {
        "DriverID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each driver",
          "optimization_purpose": "Identifies driver in optimization model",
          "sample_values": "1, 2, 3"
        },
        "IsFullTime": {
          "data_type": "BOOLEAN",
          "business_meaning": "Indicates if a driver is full-time",
          "optimization_purpose": "Used in constraint to ensure only full-time drivers are assigned",
          "sample_values": "true, false"
        }
      }
    },
    "ExperienceMismatch": {
      "business_purpose": "Stores years of experience mismatch between drivers and schools",
      "optimization_role": "objective_coefficients",
      "columns": {
        "DriverID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each driver",
          "optimization_purpose": "Identifies driver in experience mismatch calculation",
          "sample_values": "1, 2, 3"
        },
        "SchoolID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each school",
          "optimization_purpose": "Identifies school in experience mismatch calculation",
          "sample_values": "101, 102, 103"
        },
        "YearsMismatch": {
          "data_type": "INTEGER",
          "business_meaning": "Years of experience mismatch between driver and school",
          "optimization_purpose": "Coefficient in objective function to minimize experience mismatch",
          "sample_values": "0, 1, 2"
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "required_drivers": {
    "data_type": "INTEGER",
    "business_meaning": "Number of drivers required by each school",
    "optimization_role": "Used in constraint to ensure each school gets required drivers",
    "configuration_type": "scalar_parameter",
    "value": 4,
    "business_justification": "An average of 4 drivers per school reflects typical operational needs and ensures constraints are challenging yet solvable."
  },
  "is_full_time": {
    "data_type": "BOOLEAN",
    "business_meaning": "Indicates if a driver is full-time",
    "optimization_role": "Used in constraint to ensure only full-time drivers are assigned",
    "configuration_type": "scalar_parameter",
    "value": true,
    "business_justification": "All drivers being full-time ensures maximum flexibility in meeting school requirements."
  }
}

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: school_bus

## 1. Problem Context and Goals

### Context  
[Regenerate business context that naturally aligns with LINEAR optimization formulation. Ensure:]
- Business decisions match expected decision variables: DriverAssignments.Assigned[i][j] where i is DriverID and j is SchoolID, binary
- Operational parameters align with expected linear objective: minimize total_experience_mismatch = sum(ExperienceMismatch.YearsMismatch[i][j] * DriverAssignments.Assigned[i][j] for all i, j)
- Business configuration includes: Number of drivers required by each school (used for Used in constraint to ensure each school gets required drivers), Indicates if a driver is full-time (used for Used in constraint to ensure only full-time drivers are assigned)
- 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 total_experience_mismatch = sum(ExperienceMismatch.YearsMismatch[i][j] * DriverAssignments.Assigned[i][j] for all i, j)
- Success measurement aligned with expected coefficient sources
- Use natural language to precisely describe linear optimization goal
- NO mathematical formulas, equations, or symbolic notation

## 2. Constraints    

[Regenerate constraints that directly match expected LINEAR mathematical constraints:]
- Expected constraint: ['sum(DriverAssignments.Assigned[i][j] for all j) <= 1 for all i where Drivers.IsFullTime[i] = true', 'sum(DriverAssignments.Assigned[i][j] for all i) = SchoolRequirements.RequiredDrivers[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 2 Database Schema
-- Objective: Schema changes include creating a new table for years of experience mismatch and updating existing tables to fill mapping gaps. Business configuration logic is updated to handle scalar parameters and formulas.

CREATE TABLE DriverAssignments (
  DriverID INTEGER,
  SchoolID INTEGER,
  Assigned BOOLEAN
);

CREATE TABLE SchoolRequirements (
  SchoolID INTEGER,
  RequiredDrivers INTEGER
);

CREATE TABLE Drivers (
  DriverID INTEGER,
  IsFullTime BOOLEAN
);

CREATE TABLE ExperienceMismatch (
  DriverID INTEGER,
  SchoolID INTEGER,
  YearsMismatch 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 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical school bus operations, ensuring a mix of driver experience levels and school requirements to create a realistic and challenging optimization problem.

-- Realistic data for DriverAssignments
INSERT INTO DriverAssignments (DriverID, SchoolID, Assigned) VALUES (1, 101, False);
INSERT INTO DriverAssignments (DriverID, SchoolID, Assigned) VALUES (2, 102, True);
INSERT INTO DriverAssignments (DriverID, SchoolID, Assigned) VALUES (3, 103, True);

-- Realistic data for SchoolRequirements
INSERT INTO SchoolRequirements (SchoolID, RequiredDrivers) VALUES (101, 3);
INSERT INTO SchoolRequirements (SchoolID, RequiredDrivers) VALUES (102, 4);
INSERT INTO SchoolRequirements (SchoolID, RequiredDrivers) VALUES (103, 5);

-- Realistic data for Drivers
INSERT INTO Drivers (DriverID, IsFullTime) VALUES (1, True);
INSERT INTO Drivers (DriverID, IsFullTime) VALUES (2, True);
INSERT INTO Drivers (DriverID, IsFullTime) VALUES (3, True);

-- Realistic data for ExperienceMismatch
INSERT INTO ExperienceMismatch (DriverID, SchoolID, YearsMismatch) VALUES (1, 101, 2);
INSERT INTO ExperienceMismatch (DriverID, SchoolID, YearsMismatch) VALUES (2, 102, 1);
INSERT INTO ExperienceMismatch (DriverID, SchoolID, YearsMismatch) VALUES (3, 103, 0);


```

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

Response:
# Complete Optimization Problem and Solution: school_bus

## 1. Problem Context and Goals

### Context  
The school bus operation aims to efficiently allocate full-time drivers to various schools. The primary decision involves determining which drivers are assigned to which schools. Each driver can be assigned to only one school, and the assignment is represented as a binary decision variable. The operational goal is to minimize the mismatch in years of experience between drivers and the schools they are assigned to. This mismatch is quantified and used as a coefficient in the optimization process. 

Each school has a specific requirement for the number of drivers, which must be met. Additionally, only full-time drivers are considered for assignments to ensure that the schools' needs are met consistently. The business configuration includes scalar parameters such as the number of drivers required by each school and whether a driver is full-time, which are crucial for forming the constraints in the optimization model. These parameters are set to reflect typical operational needs and ensure the problem remains challenging yet solvable.

### Goals  
The primary goal of this optimization problem is to minimize the total mismatch in years of experience between the drivers and the schools they are assigned to. This is achieved by carefully selecting driver assignments that reduce the overall experience mismatch. The success of this optimization is measured by how well the assignments minimize the total experience mismatch, aligning with the coefficients derived from the experience mismatch data. The objective is clearly defined in linear terms, focusing on reducing the sum of mismatches across all driver-school assignments.

## 2. Constraints    

The optimization problem is subject to several constraints that ensure the solution is feasible and meets the operational requirements:

- Each driver can be assigned to at most one school, and this constraint applies only to full-time drivers. This ensures that drivers are not over-allocated and that only eligible drivers are considered for assignments.
  
- Each school must receive the exact number of drivers it requires. This constraint ensures that the operational needs of each school are met, maintaining the balance between supply and demand.

These constraints are expressed in linear terms, focusing on the allocation of drivers to schools without involving any nonlinear relationships.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 2 Database Schema
-- Objective: Schema changes include creating a new table for years of experience mismatch and updating existing tables to fill mapping gaps. Business configuration logic is updated to handle scalar parameters and formulas.

CREATE TABLE DriverAssignments (
  DriverID INTEGER,
  SchoolID INTEGER,
  Assigned BOOLEAN
);

CREATE TABLE SchoolRequirements (
  SchoolID INTEGER,
  RequiredDrivers INTEGER
);

CREATE TABLE Drivers (
  DriverID INTEGER,
  IsFullTime BOOLEAN
);

CREATE TABLE ExperienceMismatch (
  DriverID INTEGER,
  SchoolID INTEGER,
  YearsMismatch INTEGER
);
```

### Data Dictionary  
The data dictionary provides a comprehensive mapping of tables and columns to their business purposes and roles in the optimization process:

- **DriverAssignments**: This table tracks which drivers are assigned to which schools. It plays a crucial role in the decision-making process, with each assignment represented as a binary variable indicating whether a driver is assigned to a school.

- **SchoolRequirements**: This table stores the number of drivers required by each school. It is essential for defining the constraints that ensure each school receives the necessary number of drivers.

- **Drivers**: This table contains information about the drivers, specifically whether they are full-time. This information is used to filter eligible drivers for assignments, ensuring that only full-time drivers are considered.

- **ExperienceMismatch**: This table records the years of experience mismatch between drivers and schools. It provides the coefficients used in the objective function to minimize the total experience mismatch.

### Current Stored Values  
```sql
-- Iteration 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical school bus operations, ensuring a mix of driver experience levels and school requirements to create a realistic and challenging optimization problem.

-- Realistic data for DriverAssignments
INSERT INTO DriverAssignments (DriverID, SchoolID, Assigned) VALUES (1, 101, False);
INSERT INTO DriverAssignments (DriverID, SchoolID, Assigned) VALUES (2, 102, True);
INSERT INTO DriverAssignments (DriverID, SchoolID, Assigned) VALUES (3, 103, True);

-- Realistic data for SchoolRequirements
INSERT INTO SchoolRequirements (SchoolID, RequiredDrivers) VALUES (101, 3);
INSERT INTO SchoolRequirements (SchoolID, RequiredDrivers) VALUES (102, 4);
INSERT INTO SchoolRequirements (SchoolID, RequiredDrivers) VALUES (103, 5);

-- Realistic data for Drivers
INSERT INTO Drivers (DriverID, IsFullTime) VALUES (1, True);
INSERT INTO Drivers (DriverID, IsFullTime) VALUES (2, True);
INSERT INTO Drivers (DriverID, IsFullTime) VALUES (3, True);

-- Realistic data for ExperienceMismatch
INSERT INTO ExperienceMismatch (DriverID, SchoolID, YearsMismatch) VALUES (1, 101, 2);
INSERT INTO ExperienceMismatch (DriverID, SchoolID, YearsMismatch) VALUES (2, 102, 1);
INSERT INTO ExperienceMismatch (DriverID, SchoolID, YearsMismatch) VALUES (3, 103, 0);
```
