Iteration final - PROBLEM_DESCRIPTION
Sequence: 5
Timestamp: 2025-07-25 22:26:34

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: investment_school_i: continuous variable representing the amount invested in school i
- Operational parameters align with expected linear objective: maximize ∑(weight_school_i × investment_school_i)
- Business configuration includes: total budget available for investment across all schools (used for upper bound for total investment constraint), minimum investment required for each school (used for lower bound for investment per school constraint)
- 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 allocate its budget across multiple schools to maximize the total educational impact, measured by the sum of weighted investments, while ensuring that the total investment does not exceed the total budget and that each school receives a minimum investment.",
  "optimization_problem_description": "Maximize the total educational impact by allocating investments across schools, subject to a total budget constraint and minimum investment requirements for each school.",
  "optimization_formulation": {
    "objective": "maximize \u2211(weight_school_i \u00d7 investment_school_i)",
    "decision_variables": "investment_school_i: continuous variable representing the amount invested in school i",
    "constraints": "\u2211(investment_school_i) \u2264 total_budget; investment_school_i \u2265 minimum_investment_school_i for all i"
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "weight_school_i[school_id]": {
        "currently_mapped_to": "school_weights.weight",
        "mapping_adequacy": "good",
        "description": "weight representing the educational impact per dollar invested in school i"
      }
    },
    "constraint_bounds": {
      "total_budget": {
        "currently_mapped_to": "budget.total_budget",
        "mapping_adequacy": "good",
        "description": "upper bound for total investment across all schools"
      },
      "minimum_investment_school_i[school_id]": {
        "currently_mapped_to": "school_minimum_investments.minimum_investment",
        "mapping_adequacy": "good",
        "description": "lower bound for investment in school i"
      }
    },
    "decision_variables": {
      "investment_school_i[school_id]": {
        "currently_mapped_to": "budget.Invested",
        "mapping_adequacy": "good",
        "description": "amount 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 changes include creating tables for school weights and minimum investments, updating the budget table, and adding configuration logic for scalar parameters and formulas.

CREATE TABLE school_weights (
  school_id INTEGER,
  weight FLOAT
);

CREATE TABLE school_minimum_investments (
  school_id INTEGER,
  minimum_investment FLOAT
);

CREATE TABLE budget (
  school_id INTEGER,
  Invested FLOAT,
  total_budget FLOAT
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic school district budgets, typical minimum investment requirements, and weights reflecting varying educational impact per dollar invested across schools.

-- Realistic data for school_weights
INSERT INTO school_weights (school_id, weight) VALUES (1, 0.6);
INSERT INTO school_weights (school_id, weight) VALUES (2, 0.8);
INSERT INTO school_weights (school_id, weight) VALUES (3, 0.5);

-- Realistic data for school_minimum_investments
INSERT INTO school_minimum_investments (school_id, minimum_investment) VALUES (1, 60000);
INSERT INTO school_minimum_investments (school_id, minimum_investment) VALUES (2, 70000);
INSERT INTO school_minimum_investments (school_id, minimum_investment) VALUES (3, 50000);

-- Realistic data for budget
INSERT INTO budget (school_id, Invested, total_budget) VALUES (1, 120000, 1000000);
INSERT INTO budget (school_id, Invested, total_budget) VALUES (2, 180000, 1000000);
INSERT INTO budget (school_id, Invested, total_budget) VALUES (3, 80000, 1000000);


```

DATA DICTIONARY:
{
  "tables": {
    "school_weights": {
      "business_purpose": "weights for educational impact per dollar invested in each school",
      "optimization_role": "objective_coefficients",
      "columns": {
        "school_id": {
          "data_type": "INTEGER",
          "business_meaning": "unique identifier for each school",
          "optimization_purpose": "index for school weights",
          "sample_values": "1, 2, 3"
        },
        "weight": {
          "data_type": "FLOAT",
          "business_meaning": "weight representing the educational impact per dollar invested in the school",
          "optimization_purpose": "coefficient in the objective function",
          "sample_values": "0.5, 0.7, 0.9"
        }
      }
    },
    "school_minimum_investments": {
      "business_purpose": "minimum investment required for each school",
      "optimization_role": "constraint_bounds",
      "columns": {
        "school_id": {
          "data_type": "INTEGER",
          "business_meaning": "unique identifier for each school",
          "optimization_purpose": "index for minimum investments",
          "sample_values": "1, 2, 3"
        },
        "minimum_investment": {
          "data_type": "FLOAT",
          "business_meaning": "minimum investment required for the school",
          "optimization_purpose": "lower bound for investment per school constraint",
          "sample_values": "50000, 60000, 70000"
        }
      }
    },
    "budget": {
      "business_purpose": "budget allocation for schools",
      "optimization_role": "decision_variables",
      "columns": {
        "school_id": {
          "data_type": "INTEGER",
          "business_meaning": "unique identifier for each school",
          "optimization_purpose": "index for investment decisions",
          "sample_values": "1, 2, 3"
        },
        "Invested": {
          "data_type": "FLOAT",
          "business_meaning": "amount invested in the school",
          "optimization_purpose": "decision variable in the optimization model",
          "sample_values": "100000, 150000, 200000"
        },
        "total_budget": {
          "data_type": "FLOAT",
          "business_meaning": "total budget available for investment across all schools",
          "optimization_purpose": "upper bound for total investment constraint",
          "sample_values": "1000000"
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "total_budget": {
    "data_type": "FLOAT",
    "business_meaning": "total budget available for investment across all schools",
    "optimization_role": "upper bound for total investment constraint",
    "configuration_type": "scalar_parameter",
    "value": 1000000,
    "business_justification": "This total budget is realistic for a medium-sized school district and allows for meaningful allocation decisions."
  },
  "minimum_investment_school_i": {
    "data_type": "FLOAT",
    "business_meaning": "minimum investment required for each school",
    "optimization_role": "lower bound for investment per school constraint",
    "configuration_type": "scalar_parameter",
    "value": 50000,
    "business_justification": "This minimum investment ensures that even the smallest schools receive adequate funding to operate."
  }
}

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: investment_school_i: continuous variable representing the amount invested in school i
- Operational parameters align with expected linear objective: maximize ∑(weight_school_i × investment_school_i)
- Business configuration includes: total budget available for investment across all schools (used for upper bound for total investment constraint), minimum investment required for each school (used for lower bound for investment per school constraint)
- 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 ∑(weight_school_i × investment_school_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: ∑(investment_school_i) ≤ total_budget; investment_school_i ≥ minimum_investment_school_i for all 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 changes include creating tables for school weights and minimum investments, updating the budget table, and adding configuration logic for scalar parameters and formulas.

CREATE TABLE school_weights (
  school_id INTEGER,
  weight FLOAT
);

CREATE TABLE school_minimum_investments (
  school_id INTEGER,
  minimum_investment FLOAT
);

CREATE TABLE budget (
  school_id INTEGER,
  Invested FLOAT,
  total_budget 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 realistic school district budgets, typical minimum investment requirements, and weights reflecting varying educational impact per dollar invested across schools.

-- Realistic data for school_weights
INSERT INTO school_weights (school_id, weight) VALUES (1, 0.6);
INSERT INTO school_weights (school_id, weight) VALUES (2, 0.8);
INSERT INTO school_weights (school_id, weight) VALUES (3, 0.5);

-- Realistic data for school_minimum_investments
INSERT INTO school_minimum_investments (school_id, minimum_investment) VALUES (1, 60000);
INSERT INTO school_minimum_investments (school_id, minimum_investment) VALUES (2, 70000);
INSERT INTO school_minimum_investments (school_id, minimum_investment) VALUES (3, 50000);

-- Realistic data for budget
INSERT INTO budget (school_id, Invested, total_budget) VALUES (1, 120000, 1000000);
INSERT INTO budget (school_id, Invested, total_budget) VALUES (2, 180000, 1000000);
INSERT INTO budget (school_id, Invested, total_budget) VALUES (3, 80000, 1000000);


```

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

Response:
# Complete Optimization Problem and Solution: school_finance

## 1. Problem Context and Goals

### Context  
A school district is tasked with allocating its budget across multiple schools to maximize the overall educational impact. The impact is measured by the sum of weighted investments, where each school has a specific weight representing the educational benefit per dollar invested. The district has a total budget of $1,000,000 available for investment across all schools, and each school must receive a minimum investment to ensure basic operational needs are met. The minimum investment required for each school varies, with values such as $60,000, $70,000, and $50,000 depending on the school. The goal is to determine the optimal amount to invest in each school, ensuring that the total investment does not exceed the available budget and that each school receives at least its minimum required investment.

### Goals  
The primary goal of this optimization problem is to maximize the total educational impact across all schools. This is achieved by allocating investments in a way that the sum of the weighted investments is as large as possible. The weights reflect the relative educational impact per dollar invested in each school, and the success of the allocation is measured by the total weighted sum of investments. The optimization process ensures that the district’s budget is used efficiently to achieve the highest possible educational benefit.

## 2. Constraints    

The optimization problem is subject to two key constraints:  
1. **Total Budget Constraint**: The sum of investments across all schools must not exceed the total available budget of $1,000,000. This ensures that the district does not overspend its allocated resources.  
2. **Minimum Investment Constraint**: Each school must receive at least its specified minimum investment. For example, School 1 must receive at least $60,000, School 2 at least $70,000, and School 3 at least $50,000. This ensures that all schools have the necessary funding to operate effectively.  

These constraints ensure that the investment decisions are both financially feasible and operationally sound.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating tables for school weights and minimum investments, updating the budget table, and adding configuration logic for scalar parameters and formulas.

CREATE TABLE school_weights (
  school_id INTEGER,
  weight FLOAT
);

CREATE TABLE school_minimum_investments (
  school_id INTEGER,
  minimum_investment FLOAT
);

CREATE TABLE budget (
  school_id INTEGER,
  Invested FLOAT,
  total_budget FLOAT
);
```

### Data Dictionary  
- **school_weights**:  
  - **Purpose**: Stores the weights representing the educational impact per dollar invested in each school.  
  - **Columns**:  
    - `school_id`: Unique identifier for each school.  
    - `weight`: The weight reflecting the educational impact per dollar invested in the school.  

- **school_minimum_investments**:  
  - **Purpose**: Stores the minimum investment required for each school to ensure basic operational needs.  
  - **Columns**:  
    - `school_id`: Unique identifier for each school.  
    - `minimum_investment`: The minimum amount of funding required for the school.  

- **budget**:  
  - **Purpose**: Tracks the current investments in each school and the total budget available for allocation.  
  - **Columns**:  
    - `school_id`: Unique identifier for each school.  
    - `Invested`: The amount currently invested in the school.  
    - `total_budget`: The total budget available for investment across all schools.  

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic school district budgets, typical minimum investment requirements, and weights reflecting varying educational impact per dollar invested across schools.

-- Realistic data for school_weights
INSERT INTO school_weights (school_id, weight) VALUES (1, 0.6);
INSERT INTO school_weights (school_id, weight) VALUES (2, 0.8);
INSERT INTO school_weights (school_id, weight) VALUES (3, 0.5);

-- Realistic data for school_minimum_investments
INSERT INTO school_minimum_investments (school_id, minimum_investment) VALUES (1, 60000);
INSERT INTO school_minimum_investments (school_id, minimum_investment) VALUES (2, 70000);
INSERT INTO school_minimum_investments (school_id, minimum_investment) VALUES (3, 50000);

-- Realistic data for budget
INSERT INTO budget (school_id, Invested, total_budget) VALUES (1, 120000, 1000000);
INSERT INTO budget (school_id, Invested, total_budget) VALUES (2, 180000, 1000000);
INSERT INTO budget (school_id, Invested, total_budget) VALUES (3, 80000, 1000000);
```
