Iteration final - PROBLEM_DESCRIPTION
Sequence: 5
Timestamp: 2025-07-25 22:39:32

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_{ij} (binary): 1 if scientist i is assigned to project j, 0 otherwise
- Operational parameters align with expected linear objective: minimize ∑(Hours × x_{ij}) where x_{ij} is a binary variable indicating if scientist i is assigned to project j
- Business configuration includes: Hours required for a project (used for Used in the objective function to minimize total project hours)
- Business logic formulas to express in natural language: Average hours per scientist (calculation method for Used to evaluate performance metrics)
- 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": "scientist_1",
  "iteration": 1,
  "business_context": "A research institute needs to assign scientists to projects in a way that minimizes the total project hours while ensuring that each scientist is assigned to at least one project and no project exceeds its maximum allowed hours.",
  "optimization_problem_description": "Minimize the total project hours by optimally assigning scientists to projects. Constraints include ensuring each scientist is assigned to at least one project, no project exceeds its maximum allowed hours, and each project has at least one scientist assigned.",
  "optimization_formulation": {
    "objective": "minimize \u2211(Hours \u00d7 x_{ij}) where x_{ij} is a binary variable indicating if scientist i is assigned to project j",
    "decision_variables": "x_{ij} (binary): 1 if scientist i is assigned to project j, 0 otherwise",
    "constraints": [
      "\u2211 x_{ij} \u2265 1 for all i (each scientist is assigned to at least one project)",
      "\u2211(Hours \u00d7 x_{ij}) \u2264 MaxHours for all j (no project exceeds its maximum allowed hours)",
      "\u2211 x_{ij} \u2265 1 for all j (each project has at least one scientist assigned)"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "Hours[i][j]": {
        "currently_mapped_to": "Projects.Hours",
        "mapping_adequacy": "good",
        "description": "Hours required for project j"
      }
    },
    "constraint_bounds": {
      "MaxHours[j]": {
        "currently_mapped_to": "ProjectMaxHours.MaxHours",
        "mapping_adequacy": "good",
        "description": "Maximum allowed hours for project j"
      }
    },
    "decision_variables": {
      "x_{ij}": {
        "currently_mapped_to": "AssignedTo.is_assigned",
        "mapping_adequacy": "good",
        "description": "Binary decision variable indicating if scientist i is assigned to project j",
        "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 adding a table for maximum allowed project hours and refining the assignment table to fully map binary decision variables. Configuration logic updated to include scalar parameters for project hours and formulas for performance metrics.

CREATE TABLE Projects (
  Hours INTEGER
);

CREATE TABLE ProjectMaxHours (
  MaxHours INTEGER
);

CREATE TABLE AssignedTo (
  is_assigned BOOLEAN
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical project hours and maximum allowed hours in research settings, ensuring a balance between project complexity and resource constraints.

-- Realistic data for Projects
INSERT INTO Projects (Hours) VALUES (120);
INSERT INTO Projects (Hours) VALUES (150);
INSERT INTO Projects (Hours) VALUES (100);

-- Realistic data for ProjectMaxHours
INSERT INTO ProjectMaxHours (MaxHours) VALUES (200);
INSERT INTO ProjectMaxHours (MaxHours) VALUES (250);
INSERT INTO ProjectMaxHours (MaxHours) VALUES (180);

-- Realistic data for AssignedTo
INSERT INTO AssignedTo (is_assigned) VALUES (True);
INSERT INTO AssignedTo (is_assigned) VALUES (False);
INSERT INTO AssignedTo (is_assigned) VALUES (True);
INSERT INTO AssignedTo (is_assigned) VALUES (True);
INSERT INTO AssignedTo (is_assigned) VALUES (False);


```

DATA DICTIONARY:
{
  "tables": {
    "Projects": {
      "business_purpose": "Details of each project",
      "optimization_role": "objective_coefficients",
      "columns": {
        "Hours": {
          "data_type": "INTEGER",
          "business_meaning": "Hours required for the project",
          "optimization_purpose": "Used in the objective function to minimize total project hours",
          "sample_values": "100, 150, 200"
        }
      }
    },
    "ProjectMaxHours": {
      "business_purpose": "Maximum allowed hours for each project",
      "optimization_role": "constraint_bounds",
      "columns": {
        "MaxHours": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum allowed hours for the project",
          "optimization_purpose": "Used in the constraint to ensure no project exceeds its maximum allowed hours",
          "sample_values": "200, 250, 300"
        }
      }
    },
    "AssignedTo": {
      "business_purpose": "Assignment of scientists to projects",
      "optimization_role": "decision_variables",
      "columns": {
        "is_assigned": {
          "data_type": "BOOLEAN",
          "business_meaning": "Indicates if a scientist is assigned to a project",
          "optimization_purpose": "Used as a binary decision variable in the optimization model",
          "sample_values": "true, false"
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "project_hours": {
    "data_type": "INTEGER",
    "business_meaning": "Hours required for a project",
    "optimization_role": "Used in the objective function to minimize total project hours",
    "configuration_type": "scalar_parameter",
    "value": 120,
    "business_justification": "Represents the average hours required for a typical project, aligning with the generated data"
  },
  "performance_metric_formula": {
    "data_type": "STRING",
    "business_meaning": "Average hours per scientist",
    "optimization_role": "Used to evaluate performance metrics",
    "configuration_type": "business_logic_formula",
    "formula_expression": "total_hours / number_of_scientists"
  }
}

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: scientist_1

## 1. Problem Context and Goals

### Context  
[Regenerate business context that naturally aligns with LINEAR optimization formulation. Ensure:]
- Business decisions match expected decision variables: x_{ij} (binary): 1 if scientist i is assigned to project j, 0 otherwise
- Operational parameters align with expected linear objective: minimize ∑(Hours × x_{ij}) where x_{ij} is a binary variable indicating if scientist i is assigned to project j
- Business configuration includes: Hours required for a project (used for Used in the objective function to minimize total project hours)
- Business logic formulas to express in natural language: Average hours per scientist (calculation method for Used to evaluate performance metrics)
- 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 ∑(Hours × x_{ij}) where x_{ij} is a binary variable indicating if scientist i is assigned to project 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: ['∑ x_{ij} ≥ 1 for all i (each scientist is assigned to at least one project)', '∑(Hours × x_{ij}) ≤ MaxHours for all j (no project exceeds its maximum allowed hours)', '∑ x_{ij} ≥ 1 for all j (each project has at least one scientist assigned)'] (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 adding a table for maximum allowed project hours and refining the assignment table to fully map binary decision variables. Configuration logic updated to include scalar parameters for project hours and formulas for performance metrics.

CREATE TABLE Projects (
  Hours INTEGER
);

CREATE TABLE ProjectMaxHours (
  MaxHours INTEGER
);

CREATE TABLE AssignedTo (
  is_assigned BOOLEAN
);


```

### 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 project hours and maximum allowed hours in research settings, ensuring a balance between project complexity and resource constraints.

-- Realistic data for Projects
INSERT INTO Projects (Hours) VALUES (120);
INSERT INTO Projects (Hours) VALUES (150);
INSERT INTO Projects (Hours) VALUES (100);

-- Realistic data for ProjectMaxHours
INSERT INTO ProjectMaxHours (MaxHours) VALUES (200);
INSERT INTO ProjectMaxHours (MaxHours) VALUES (250);
INSERT INTO ProjectMaxHours (MaxHours) VALUES (180);

-- Realistic data for AssignedTo
INSERT INTO AssignedTo (is_assigned) VALUES (True);
INSERT INTO AssignedTo (is_assigned) VALUES (False);
INSERT INTO AssignedTo (is_assigned) VALUES (True);
INSERT INTO AssignedTo (is_assigned) VALUES (True);
INSERT INTO AssignedTo (is_assigned) VALUES (False);


```

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

Response:
# Complete Optimization Problem and Solution: scientist_1

## 1. Problem Context and Goals

### Context  
A research institute is tasked with efficiently assigning scientists to projects to optimize resource utilization. The primary business decision involves determining whether a scientist is assigned to a specific project, represented as a binary choice. The operational parameters include the hours required for each project, which are used to calculate the total project hours. The goal is to minimize the total project hours while ensuring that each scientist is assigned to at least one project and that no project exceeds its maximum allowed hours.  

The business configuration includes scalar parameters such as the hours required for a project, which directly influence the objective function. Additionally, a business logic formula calculates the average hours per scientist, providing a performance metric to evaluate the efficiency of the assignment process. The problem is designed to avoid nonlinear relationships, ensuring that all decisions and constraints are linear in nature.  

### Goals  
The optimization goal is to minimize the total project hours by optimally assigning scientists to projects. Success is measured by achieving the lowest possible total project hours while adhering to the constraints. The metric to optimize is the sum of the hours required for each project multiplied by the binary decision variable indicating whether a scientist is assigned to that project. This goal aligns with the operational parameters and ensures efficient resource allocation.  

## 2. Constraints  

The problem is subject to the following constraints:  
1. **Scientist Assignment Constraint**: Each scientist must be assigned to at least one project. This ensures that all scientists are actively contributing to the research efforts.  
2. **Project Hours Constraint**: The total hours assigned to a project must not exceed its maximum allowed hours. This ensures that projects are completed within their resource limits.  
3. **Project Assignment Constraint**: Each project must have at least one scientist assigned to it. This ensures that all projects are staffed and progress is made.  

These constraints are designed to align with linear mathematical forms, avoiding any nonlinear relationships such as variable products or divisions.  

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include adding a table for maximum allowed project hours and refining the assignment table to fully map binary decision variables. Configuration logic updated to include scalar parameters for project hours and formulas for performance metrics.

CREATE TABLE Projects (
  Hours INTEGER
);

CREATE TABLE ProjectMaxHours (
  MaxHours INTEGER
);

CREATE TABLE AssignedTo (
  is_assigned BOOLEAN
);
```

### Data Dictionary  
- **Projects**:  
  - **Business Purpose**: Details of each project, including the hours required for completion.  
  - **Optimization Role**: Provides the coefficients for the objective function, which aims to minimize total project hours.  
  - **Columns**:  
    - **Hours**: Represents the hours required for the project. This value is used in the objective function to calculate total project hours.  

- **ProjectMaxHours**:  
  - **Business Purpose**: Specifies the maximum allowed hours for each project.  
  - **Optimization Role**: Provides the bounds for the constraint that ensures no project exceeds its maximum allowed hours.  
  - **Columns**:  
    - **MaxHours**: Represents the maximum allowed hours for the project. This value is used in the constraint to limit project hours.  

- **AssignedTo**:  
  - **Business Purpose**: Tracks the assignment of scientists to projects.  
  - **Optimization Role**: Represents the binary decision variables in the optimization model.  
  - **Columns**:  
    - **is_assigned**: Indicates whether a scientist is assigned to a project. This binary variable is used in the optimization model to determine assignments.  

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical project hours and maximum allowed hours in research settings, ensuring a balance between project complexity and resource constraints.

-- Realistic data for Projects
INSERT INTO Projects (Hours) VALUES (120);
INSERT INTO Projects (Hours) VALUES (150);
INSERT INTO Projects (Hours) VALUES (100);

-- Realistic data for ProjectMaxHours
INSERT INTO ProjectMaxHours (MaxHours) VALUES (200);
INSERT INTO ProjectMaxHours (MaxHours) VALUES (250);
INSERT INTO ProjectMaxHours (MaxHours) VALUES (180);

-- Realistic data for AssignedTo
INSERT INTO AssignedTo (is_assigned) VALUES (True);
INSERT INTO AssignedTo (is_assigned) VALUES (False);
INSERT INTO AssignedTo (is_assigned) VALUES (True);
INSERT INTO AssignedTo (is_assigned) VALUES (True);
INSERT INTO AssignedTo (is_assigned) VALUES (False);
```
