Iteration final - PROBLEM_DESCRIPTION
Sequence: 5
Timestamp: 2025-07-25 22:29:33

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: balance_savings[custid] (continuous), balance_checking[custid] (continuous)
- Operational parameters align with expected linear objective: maximize ∑(interest_rate_savings × balance_savings[custid] + interest_rate_checking × balance_checking[custid])
- Business configuration includes: Interest rate for savings accounts (used for Objective coefficient for savings account balances), Interest rate for checking accounts (used for Objective coefficient for checking account balances), Minimum balance required for savings accounts (used for Lower bound constraint for savings account balances), Minimum balance required for checking accounts (used for Lower bound constraint for checking account balances), Total funds available in the bank (used for Upper bound constraint for total account balances), Maximum balance allowed for savings accounts (used for Upper bound constraint for savings account balances), Maximum balance allowed for checking accounts (used for Upper bound constraint for checking account balances)
- 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": "small_bank_1",
  "iteration": 1,
  "business_context": "A bank aims to optimize the allocation of funds between savings and checking accounts to maximize interest income while ensuring liquidity and regulatory compliance. The optimization must maintain a linear structure to ensure computational efficiency and scalability.",
  "optimization_problem_description": "Maximize the total interest income from savings and checking accounts, subject to constraints on minimum balances, total funds, and regulatory requirements. The problem is formulated as a linear programming (LP) model.",
  "optimization_formulation": {
    "objective": "maximize \u2211(interest_rate_savings \u00d7 balance_savings[custid] + interest_rate_checking \u00d7 balance_checking[custid])",
    "decision_variables": "balance_savings[custid] (continuous), balance_checking[custid] (continuous)",
    "constraints": [
      "balance_savings[custid] + balance_checking[custid] \u2264 total_funds[custid] (total funds constraint)",
      "balance_savings[custid] \u2265 min_balance_savings (minimum savings balance constraint)",
      "balance_checking[custid] \u2265 min_balance_checking (minimum checking balance constraint)",
      "balance_savings[custid] \u2264 max_balance_savings (maximum savings balance constraint)",
      "balance_checking[custid] \u2264 max_balance_checking (maximum checking balance constraint)",
      "\u2211(balance_savings[custid] + balance_checking[custid]) \u2264 total_bank_funds (total bank funds constraint)"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "interest_rate_savings": {
        "currently_mapped_to": "business_configuration_logic.interest_rate_savings",
        "mapping_adequacy": "good",
        "description": "Interest rate for savings accounts"
      },
      "interest_rate_checking": {
        "currently_mapped_to": "business_configuration_logic.interest_rate_checking",
        "mapping_adequacy": "good",
        "description": "Interest rate for checking accounts"
      }
    },
    "constraint_bounds": {
      "total_funds[custid]": {
        "currently_mapped_to": "CUSTOMER_FUNDS.total_funds",
        "mapping_adequacy": "good",
        "description": "Total funds available per customer"
      },
      "min_balance_savings": {
        "currently_mapped_to": "business_configuration_logic.min_balance_savings",
        "mapping_adequacy": "good",
        "description": "Minimum balance required for savings accounts"
      },
      "min_balance_checking": {
        "currently_mapped_to": "business_configuration_logic.min_balance_checking",
        "mapping_adequacy": "good",
        "description": "Minimum balance required for checking accounts"
      },
      "max_balance_savings": {
        "currently_mapped_to": "business_configuration_logic.max_balance_savings",
        "mapping_adequacy": "good",
        "description": "Maximum balance allowed for savings accounts"
      },
      "max_balance_checking": {
        "currently_mapped_to": "business_configuration_logic.max_balance_checking",
        "mapping_adequacy": "good",
        "description": "Maximum balance allowed for checking accounts"
      },
      "total_bank_funds": {
        "currently_mapped_to": "business_configuration_logic.total_bank_funds",
        "mapping_adequacy": "good",
        "description": "Total funds available in the bank"
      }
    },
    "decision_variables": {
      "balance_savings[custid]": {
        "currently_mapped_to": "SAVINGS.balance",
        "mapping_adequacy": "good",
        "description": "Balance in savings account",
        "variable_type": "continuous"
      },
      "balance_checking[custid]": {
        "currently_mapped_to": "CHECKING.balance",
        "mapping_adequacy": "good",
        "description": "Balance in checking account",
        "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 missing optimization requirements and updating business configuration logic for scalar parameters and formulas.

CREATE TABLE CUSTOMER_FUNDS (
  custid INTEGER,
  total_funds FLOAT
);

CREATE TABLE SAVINGS (
  custid INTEGER,
  balance FLOAT
);

CREATE TABLE CHECKING (
  custid INTEGER,
  balance FLOAT
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical banking practices, ensuring realistic interest rates, balance thresholds, and total funds that align with a small bank's operational scale.

-- Realistic data for CUSTOMER_FUNDS
INSERT INTO CUSTOMER_FUNDS (custid, total_funds) VALUES (1, 1000.0);
INSERT INTO CUSTOMER_FUNDS (custid, total_funds) VALUES (2, 1500.0);
INSERT INTO CUSTOMER_FUNDS (custid, total_funds) VALUES (3, 2000.0);

-- Realistic data for SAVINGS
INSERT INTO SAVINGS (custid, balance) VALUES (1, 500.0);
INSERT INTO SAVINGS (custid, balance) VALUES (2, 750.0);
INSERT INTO SAVINGS (custid, balance) VALUES (3, 1000.0);

-- Realistic data for CHECKING
INSERT INTO CHECKING (custid, balance) VALUES (1, 300.0);
INSERT INTO CHECKING (custid, balance) VALUES (2, 450.0);
INSERT INTO CHECKING (custid, balance) VALUES (3, 600.0);


```

DATA DICTIONARY:
{
  "tables": {
    "CUSTOMER_FUNDS": {
      "business_purpose": "Total funds available per customer for allocation between savings and checking accounts",
      "optimization_role": "constraint_bounds",
      "columns": {
        "custid": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each customer",
          "optimization_purpose": "Index for decision variables and constraints",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "total_funds": {
          "data_type": "FLOAT",
          "business_meaning": "Total funds available for the customer",
          "optimization_purpose": "Upper bound constraint for total account balances",
          "sample_values": [
            1000.0,
            1500.0,
            2000.0
          ]
        }
      }
    },
    "SAVINGS": {
      "business_purpose": "Savings account balances for each customer",
      "optimization_role": "decision_variables",
      "columns": {
        "custid": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each customer",
          "optimization_purpose": "Index for decision variables",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "balance": {
          "data_type": "FLOAT",
          "business_meaning": "Balance in savings account",
          "optimization_purpose": "Decision variable for savings account allocation",
          "sample_values": [
            500.0,
            750.0,
            1000.0
          ]
        }
      }
    },
    "CHECKING": {
      "business_purpose": "Checking account balances for each customer",
      "optimization_role": "decision_variables",
      "columns": {
        "custid": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each customer",
          "optimization_purpose": "Index for decision variables",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "balance": {
          "data_type": "FLOAT",
          "business_meaning": "Balance in checking account",
          "optimization_purpose": "Decision variable for checking account allocation",
          "sample_values": [
            300.0,
            450.0,
            600.0
          ]
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "interest_rate_savings": {
    "data_type": "FLOAT",
    "business_meaning": "Interest rate for savings accounts",
    "optimization_role": "Objective coefficient for savings account balances",
    "configuration_type": "scalar_parameter",
    "value": 0.03,
    "business_justification": "Typical interest rate for savings accounts in a small bank."
  },
  "interest_rate_checking": {
    "data_type": "FLOAT",
    "business_meaning": "Interest rate for checking accounts",
    "optimization_role": "Objective coefficient for checking account balances",
    "configuration_type": "scalar_parameter",
    "value": 0.01,
    "business_justification": "Typical interest rate for checking accounts in a small bank."
  },
  "min_balance_savings": {
    "data_type": "FLOAT",
    "business_meaning": "Minimum balance required for savings accounts",
    "optimization_role": "Lower bound constraint for savings account balances",
    "configuration_type": "scalar_parameter",
    "value": 100.0,
    "business_justification": "Minimum balance required to avoid fees and maintain account activity."
  },
  "min_balance_checking": {
    "data_type": "FLOAT",
    "business_meaning": "Minimum balance required for checking accounts",
    "optimization_role": "Lower bound constraint for checking account balances",
    "configuration_type": "scalar_parameter",
    "value": 50.0,
    "business_justification": "Minimum balance required to avoid fees and maintain account activity."
  },
  "total_bank_funds": {
    "data_type": "FLOAT",
    "business_meaning": "Total funds available in the bank",
    "optimization_role": "Upper bound constraint for total account balances",
    "configuration_type": "scalar_parameter",
    "value": 1000000.0,
    "business_justification": "Total funds available in the bank, reflecting a small bank's scale."
  },
  "max_balance_savings": {
    "data_type": "FLOAT",
    "business_meaning": "Maximum balance allowed for savings accounts",
    "optimization_role": "Upper bound constraint for savings account balances",
    "configuration_type": "scalar_parameter",
    "value": 50000.0,
    "business_justification": "Maximum balance allowed to manage risk and liquidity."
  },
  "max_balance_checking": {
    "data_type": "FLOAT",
    "business_meaning": "Maximum balance allowed for checking accounts",
    "optimization_role": "Upper bound constraint for checking account balances",
    "configuration_type": "scalar_parameter",
    "value": 30000.0,
    "business_justification": "Maximum balance allowed to manage risk and liquidity."
  }
}

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: small_bank_1

## 1. Problem Context and Goals

### Context  
[Regenerate business context that naturally aligns with LINEAR optimization formulation. Ensure:]
- Business decisions match expected decision variables: balance_savings[custid] (continuous), balance_checking[custid] (continuous)
- Operational parameters align with expected linear objective: maximize ∑(interest_rate_savings × balance_savings[custid] + interest_rate_checking × balance_checking[custid])
- Business configuration includes: Interest rate for savings accounts (used for Objective coefficient for savings account balances), Interest rate for checking accounts (used for Objective coefficient for checking account balances), Minimum balance required for savings accounts (used for Lower bound constraint for savings account balances), Minimum balance required for checking accounts (used for Lower bound constraint for checking account balances), Total funds available in the bank (used for Upper bound constraint for total account balances), Maximum balance allowed for savings accounts (used for Upper bound constraint for savings account balances), Maximum balance allowed for checking accounts (used for Upper bound constraint for checking account balances)
- 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 ∑(interest_rate_savings × balance_savings[custid] + interest_rate_checking × balance_checking[custid])
- 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: ['balance_savings[custid] + balance_checking[custid] ≤ total_funds[custid] (total funds constraint)', 'balance_savings[custid] ≥ min_balance_savings (minimum savings balance constraint)', 'balance_checking[custid] ≥ min_balance_checking (minimum checking balance constraint)', 'balance_savings[custid] ≤ max_balance_savings (maximum savings balance constraint)', 'balance_checking[custid] ≤ max_balance_checking (maximum checking balance constraint)', '∑(balance_savings[custid] + balance_checking[custid]) ≤ total_bank_funds (total bank funds constraint)'] (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 missing optimization requirements and updating business configuration logic for scalar parameters and formulas.

CREATE TABLE CUSTOMER_FUNDS (
  custid INTEGER,
  total_funds FLOAT
);

CREATE TABLE SAVINGS (
  custid INTEGER,
  balance FLOAT
);

CREATE TABLE CHECKING (
  custid INTEGER,
  balance 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 banking practices, ensuring realistic interest rates, balance thresholds, and total funds that align with a small bank's operational scale.

-- Realistic data for CUSTOMER_FUNDS
INSERT INTO CUSTOMER_FUNDS (custid, total_funds) VALUES (1, 1000.0);
INSERT INTO CUSTOMER_FUNDS (custid, total_funds) VALUES (2, 1500.0);
INSERT INTO CUSTOMER_FUNDS (custid, total_funds) VALUES (3, 2000.0);

-- Realistic data for SAVINGS
INSERT INTO SAVINGS (custid, balance) VALUES (1, 500.0);
INSERT INTO SAVINGS (custid, balance) VALUES (2, 750.0);
INSERT INTO SAVINGS (custid, balance) VALUES (3, 1000.0);

-- Realistic data for CHECKING
INSERT INTO CHECKING (custid, balance) VALUES (1, 300.0);
INSERT INTO CHECKING (custid, balance) VALUES (2, 450.0);
INSERT INTO CHECKING (custid, balance) VALUES (3, 600.0);


```

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

Response:
# Complete Optimization Problem and Solution: small_bank_1

## 1. Problem Context and Goals

### Context  
A small bank is focused on optimizing the allocation of funds between savings and checking accounts for its customers to maximize interest income while ensuring liquidity and regulatory compliance. The bank must decide how much to allocate to each customer's savings and checking accounts based on their total available funds. The interest rates for savings and checking accounts are fixed at 3% and 1%, respectively, and these rates directly influence the bank's income.  

The bank must adhere to several operational rules:  
- Each customer's total allocation to savings and checking accounts cannot exceed their total available funds.  
- Savings accounts must maintain a minimum balance of $100 to avoid fees and ensure account activity.  
- Checking accounts must maintain a minimum balance of $50 for the same reasons.  
- Savings accounts cannot exceed a maximum balance of $50,000 to manage risk and liquidity.  
- Checking accounts cannot exceed a maximum balance of $30,000 for the same purpose.  
- The bank’s total funds across all accounts cannot exceed $1,000,000, reflecting its operational scale.  

These rules ensure that the bank’s decisions are both profitable and compliant with regulatory requirements. The problem is structured to maintain a linear relationship between decision variables and constraints, ensuring computational efficiency and scalability.

### Goals  
The primary goal of this optimization is to maximize the bank’s total interest income from savings and checking accounts. This is achieved by strategically allocating each customer’s funds between their savings and checking accounts, weighted by the respective interest rates. Success is measured by the total interest income generated, which is directly influenced by the interest rates and the allocated balances. The optimization ensures that all operational and regulatory constraints are met while achieving the highest possible income.

## 2. Constraints  

The optimization must adhere to the following constraints:  
1. **Total Funds Constraint**: For each customer, the sum of their savings and checking account balances cannot exceed their total available funds.  
2. **Minimum Savings Balance Constraint**: Each customer’s savings account balance must be at least $100.  
3. **Minimum Checking Balance Constraint**: Each customer’s checking account balance must be at least $50.  
4. **Maximum Savings Balance Constraint**: Each customer’s savings account balance cannot exceed $50,000.  
5. **Maximum Checking Balance Constraint**: Each customer’s checking account balance cannot exceed $30,000.  
6. **Total Bank Funds Constraint**: The sum of all savings and checking account balances across all customers cannot exceed the bank’s total available funds of $1,000,000.  

These constraints ensure that the bank’s allocations are both feasible and compliant with operational and regulatory requirements.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating tables for missing optimization requirements and updating business configuration logic for scalar parameters and formulas.

CREATE TABLE CUSTOMER_FUNDS (
  custid INTEGER,
  total_funds FLOAT
);

CREATE TABLE SAVINGS (
  custid INTEGER,
  balance FLOAT
);

CREATE TABLE CHECKING (
  custid INTEGER,
  balance FLOAT
);
```

### Data Dictionary  
- **CUSTOMER_FUNDS**:  
  - **Purpose**: Tracks the total funds available for each customer to allocate between savings and checking accounts.  
  - **Columns**:  
    - `custid`: Unique identifier for each customer.  
    - `total_funds`: Total funds available for the customer, used to constrain the sum of their account balances.  

- **SAVINGS**:  
  - **Purpose**: Tracks the balance in each customer’s savings account.  
  - **Columns**:  
    - `custid`: Unique identifier for each customer.  
    - `balance`: The balance in the savings account, which is a decision variable in the optimization.  

- **CHECKING**:  
  - **Purpose**: Tracks the balance in each customer’s checking account.  
  - **Columns**:  
    - `custid`: Unique identifier for each customer.  
    - `balance`: The balance in the checking account, which is a decision variable in the optimization.  

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical banking practices, ensuring realistic interest rates, balance thresholds, and total funds that align with a small bank's operational scale.

-- Realistic data for CUSTOMER_FUNDS
INSERT INTO CUSTOMER_FUNDS (custid, total_funds) VALUES (1, 1000.0);
INSERT INTO CUSTOMER_FUNDS (custid, total_funds) VALUES (2, 1500.0);
INSERT INTO CUSTOMER_FUNDS (custid, total_funds) VALUES (3, 2000.0);

-- Realistic data for SAVINGS
INSERT INTO SAVINGS (custid, balance) VALUES (1, 500.0);
INSERT INTO SAVINGS (custid, balance) VALUES (2, 750.0);
INSERT INTO SAVINGS (custid, balance) VALUES (3, 1000.0);

-- Realistic data for CHECKING
INSERT INTO CHECKING (custid, balance) VALUES (1, 300.0);
INSERT INTO CHECKING (custid, balance) VALUES (2, 450.0);
INSERT INTO CHECKING (custid, balance) VALUES (3, 600.0);
```
