Iteration final - PROBLEM_DESCRIPTION
Sequence: 5
Timestamp: 2025-07-27 23:29:46

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: Invested_i: amount to be invested in school i, continuous
- Operational parameters align with expected linear objective: maximize total_invested = ∑(Invested_i) for all schools i
- Business configuration includes: Minimum percentage of the budgeted amount that must be invested in each school (used for Used as a constraint in the optimization model)
- 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_finance",
  "iteration": 1,
  "business_context": "A school district aims to optimize its budget allocation across different schools to maximize the total invested amount while ensuring that each school receives a minimum percentage of its budgeted amount.",
  "optimization_problem_description": "The objective is to maximize the total invested amount across all schools, ensuring that each school receives at least a specified minimum percentage of its budgeted amount. The decision variables are the amounts to be invested in each school, with constraints ensuring investments do not exceed budgeted amounts and meet minimum investment requirements.",
  "optimization_formulation": {
    "objective": "maximize total_invested = \u2211(Invested_i) for all schools i",
    "decision_variables": "Invested_i: amount to be invested in school i, continuous",
    "constraints": [
      "Invested_i \u2264 Budgeted_i for all schools i",
      "Invested_i \u2265 min_investment_percent_i \u00d7 Budgeted_i for all schools i"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "Invested[i]": {
        "currently_mapped_to": "budget.Invested",
        "mapping_adequacy": "good",
        "description": "Amount invested in school i"
      }
    },
    "constraint_bounds": {
      "max_investment[i]": {
        "currently_mapped_to": "budget.Budgeted",
        "mapping_adequacy": "good",
        "description": "Maximum budgeted amount for school i"
      },
      "min_investment[i]": {
        "currently_mapped_to": "budget.min_investment_percent",
        "mapping_adequacy": "good",
        "description": "Minimum percentage of budget that must be invested in school i"
      }
    },
    "decision_variables": {
      "Invested[i]": {
        "currently_mapped_to": "budget.Invested",
        "mapping_adequacy": "good",
        "description": "Amount to be invested in school i",
        "variable_type": "continuous"
      }
    }
  },
  "missing_optimization_requirements": [],
  "iteration_status": {
    "complete": true,
    "confidence": "high",
    "next_focus": "Ready for convergence"
  }
}

FINAL DATABASE SCHEMA:
```sql
-- Iteration 1 Database Schema
-- Objective: Schema adjustments include adding missing configuration logic for minimum investment percentage and ensuring all optimization requirements are mapped correctly. Business configuration logic is updated to include scalar parameters and formulas.

CREATE TABLE budget (
  Invested FLOAT,
  Budgeted FLOAT,
  min_investment_percent FLOAT
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical budget allocations for schools, ensuring that the minimum investment percentages are realistic and allow for a meaningful optimization problem.

-- Realistic data for budget
INSERT INTO budget (Invested, Budgeted, min_investment_percent) VALUES (4500.0, 5000.0, 0.1);
INSERT INTO budget (Invested, Budgeted, min_investment_percent) VALUES (5400.0, 6000.0, 0.15);
INSERT INTO budget (Invested, Budgeted, min_investment_percent) VALUES (6300.0, 7000.0, 0.2);


```

DATA DICTIONARY:
{
  "tables": {
    "budget": {
      "business_purpose": "Stores budget and investment data for each school",
      "optimization_role": "decision_variables/constraint_bounds",
      "columns": {
        "Invested": {
          "data_type": "FLOAT",
          "business_meaning": "Amount invested in school i",
          "optimization_purpose": "Decision variable representing investment amount",
          "sample_values": "1000.0, 2000.0, 3000.0"
        },
        "Budgeted": {
          "data_type": "FLOAT",
          "business_meaning": "Total budgeted amount for school i",
          "optimization_purpose": "Constraint bound for maximum investment",
          "sample_values": "5000.0, 6000.0, 7000.0"
        },
        "min_investment_percent": {
          "data_type": "FLOAT",
          "business_meaning": "Minimum percentage of budget that must be invested",
          "optimization_purpose": "Constraint bound for minimum investment",
          "sample_values": "0.1, 0.15, 0.2"
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "min_investment_percent": {
    "data_type": "FLOAT",
    "business_meaning": "Minimum percentage of the budgeted amount that must be invested in each school",
    "optimization_role": "Used as a constraint in the optimization model",
    "configuration_type": "scalar_parameter",
    "value": 0.15,
    "business_justification": "A 15% minimum investment ensures that each school receives a meaningful portion of its budget, allowing for essential expenditures."
  }
}

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_finance

## 1. Problem Context and Goals

### Context  
[Regenerate business context that naturally aligns with LINEAR optimization formulation. Ensure:]
- Business decisions match expected decision variables: Invested_i: amount to be invested in school i, continuous
- Operational parameters align with expected linear objective: maximize total_invested = ∑(Invested_i) for all schools i
- Business configuration includes: Minimum percentage of the budgeted amount that must be invested in each school (used for Used as a constraint in the optimization model)
- 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 total_invested = ∑(Invested_i) for all schools 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: ['Invested_i ≤ Budgeted_i for all schools i', 'Invested_i ≥ min_investment_percent_i × Budgeted_i for all schools 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 adjustments include adding missing configuration logic for minimum investment percentage and ensuring all optimization requirements are mapped correctly. Business configuration logic is updated to include scalar parameters and formulas.

CREATE TABLE budget (
  Invested FLOAT,
  Budgeted FLOAT,
  min_investment_percent 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 typical budget allocations for schools, ensuring that the minimum investment percentages are realistic and allow for a meaningful optimization problem.

-- Realistic data for budget
INSERT INTO budget (Invested, Budgeted, min_investment_percent) VALUES (4500.0, 5000.0, 0.1);
INSERT INTO budget (Invested, Budgeted, min_investment_percent) VALUES (5400.0, 6000.0, 0.15);
INSERT INTO budget (Invested, Budgeted, min_investment_percent) VALUES (6300.0, 7000.0, 0.2);


```

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

Response:
# Complete Optimization Problem and Solution: school_finance

## 1. Problem Context and Goals

### Context  
The school district is focused on optimizing its budget allocation across various schools to ensure the most effective use of resources. The primary decision involves determining the amount to be invested in each school, denoted as Invested_i, which is a continuous variable. The operational goal is to maximize the total amount invested across all schools. This is achieved by ensuring that each school receives at least a minimum percentage of its budgeted amount, as specified in the business configuration. The business configuration includes a scalar parameter that dictates the minimum percentage of the budget that must be invested in each school, serving as a constraint in the optimization model. The decision-making process is linear, focusing on straightforward allocation without involving complex interactions such as variable products or divisions. The operational parameters are clearly defined, aligning with the expected coefficient sources, and the business configuration parameters are referenced appropriately to maintain consistency and clarity.

### Goals  
The optimization goal is to maximize the total amount invested across all schools. The metric for optimization is the total invested amount, which is the sum of investments in each school. Success is measured by the alignment of investments with the expected coefficient sources, ensuring that the total invested amount is maximized. The goal is articulated in natural language, focusing on the linear optimization objective without resorting to mathematical formulas or symbolic notation.

## 2. Constraints    

The constraints are designed to ensure that the investment decisions adhere to the budgetary limits and minimum investment requirements. Each school must receive an investment amount that does not exceed its budgeted amount. Additionally, each school must receive at least a specified minimum percentage of its budgeted amount. These constraints are expressed in business terms that naturally lead to linear mathematical forms, avoiding any complex interactions such as variable products or divisions. The constraints are as follows:
- The investment in each school should not exceed the budgeted amount for that school.
- The investment in each school should be at least the minimum percentage of the budgeted amount, as specified in the business configuration.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema adjustments include adding missing configuration logic for minimum investment percentage and ensuring all optimization requirements are mapped correctly. Business configuration logic is updated to include scalar parameters and formulas.

CREATE TABLE budget (
  Invested FLOAT,
  Budgeted FLOAT,
  min_investment_percent FLOAT
);
```

### Data Dictionary  
The data dictionary provides a comprehensive mapping of tables and columns to their business purposes and optimization roles. The "budget" table stores essential data for each school, including the amount invested, the total budgeted amount, and the minimum investment percentage. Each column serves a specific role in the optimization process:
- **Invested**: Represents the amount invested in each school. It is a decision variable in the optimization model, indicating the investment amount.
- **Budgeted**: Indicates the total budgeted amount for each school. It serves as a constraint bound, ensuring that investments do not exceed this amount.
- **min_investment_percent**: Specifies the minimum percentage of the budget that must be invested in each school. It acts as a constraint bound, ensuring that investments meet the minimum requirement.

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical budget allocations for schools, ensuring that the minimum investment percentages are realistic and allow for a meaningful optimization problem.

-- Realistic data for budget
INSERT INTO budget (Invested, Budgeted, min_investment_percent) VALUES (4500.0, 5000.0, 0.1);
INSERT INTO budget (Invested, Budgeted, min_investment_percent) VALUES (5400.0, 6000.0, 0.15);
INSERT INTO budget (Invested, Budgeted, min_investment_percent) VALUES (6300.0, 7000.0, 0.2);
```
