Iteration final - PROBLEM_DESCRIPTION
Sequence: 7
Timestamp: 2025-07-25 22:29:13

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: transaction_amount[card_id] (continuous)
- Operational parameters align with expected linear objective: minimize sum(fee_per_card_type[card_type_code] * transaction_amount[card_id])
- Business configuration includes: fee associated with each card type (used for coefficient in the objective function), maximum transaction amount allowed per customer (used for bound in the constraint)
- Business logic formulas to express in natural language: total transaction fee calculation (calculation method for objective function)
- 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": "customers_card_transactions",
  "iteration": 2,
  "business_context": "A bank aims to minimize the total transaction fees incurred by its customers by optimizing the allocation of transactions across different card types, each with different fee structures, while ensuring that the total transaction amount per customer does not exceed their account limits.",
  "optimization_problem_description": "Minimize the total transaction fees by allocating transactions across card types with different fees, subject to the constraint that the total transaction amount per customer does not exceed their account limits.",
  "optimization_formulation": {
    "objective": "minimize sum(fee_per_card_type[card_type_code] * transaction_amount[card_id])",
    "decision_variables": "transaction_amount[card_id] (continuous)",
    "constraints": "sum(transaction_amount[card_id]) <= account_limit[customer_id]"
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "fee_per_card_type[card_type_code]": {
        "currently_mapped_to": "Card_Fees.fee",
        "mapping_adequacy": "good",
        "description": "fee associated with each card type"
      }
    },
    "constraint_bounds": {
      "account_limit[customer_id]": {
        "currently_mapped_to": "Account_Limits.account_limit",
        "mapping_adequacy": "good",
        "description": "maximum transaction amount allowed per customer"
      }
    },
    "decision_variables": {
      "transaction_amount[card_id]": {
        "currently_mapped_to": "Transaction_Allocations.transaction_amount",
        "mapping_adequacy": "good",
        "description": "amount of transaction allocated to each card type per customer",
        "variable_type": "continuous"
      }
    }
  },
  "missing_optimization_requirements": [],
  "iteration_status": {
    "complete": true,
    "confidence": "high",
    "next_focus": "Ready for convergence"
  }
}

FINAL DATABASE SCHEMA:
```sql
-- Iteration 2 Database Schema
-- Objective: Added Transaction_Allocations table to map missing decision variables, updated business configuration logic to include scalar parameters and formulas, and ensured schema follows normalization principles.

CREATE TABLE Card_Fees (
  card_type_code STRING,
  fee FLOAT
);

CREATE TABLE Account_Limits (
  customer_id INTEGER,
  account_limit FLOAT
);

CREATE TABLE Transaction_Allocations (
  customer_id INTEGER,
  card_id STRING,
  transaction_amount FLOAT
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical banking transaction fees, account limits, and transaction amounts, ensuring they align with real-world scenarios and enable a meaningful optimization problem.

-- Realistic data for Card_Fees
INSERT INTO Card_Fees (card_type_code, fee) VALUES ('VISA', 0.02);
INSERT INTO Card_Fees (card_type_code, fee) VALUES ('MASTERCARD', 0.03);
INSERT INTO Card_Fees (card_type_code, fee) VALUES ('AMEX', 0.035);

-- Realistic data for Account_Limits
INSERT INTO Account_Limits (customer_id, account_limit) VALUES (1, 1000);
INSERT INTO Account_Limits (customer_id, account_limit) VALUES (2, 1500);
INSERT INTO Account_Limits (customer_id, account_limit) VALUES (3, 2000);

-- Realistic data for Transaction_Allocations
INSERT INTO Transaction_Allocations (customer_id, card_id, transaction_amount) VALUES (1, 'CARD1', 100);
INSERT INTO Transaction_Allocations (customer_id, card_id, transaction_amount) VALUES (2, 'CARD2', 200);
INSERT INTO Transaction_Allocations (customer_id, card_id, transaction_amount) VALUES (3, 'CARD3', 300);


```

DATA DICTIONARY:
{
  "tables": {
    "Card_Fees": {
      "business_purpose": "fee associated with each card type",
      "optimization_role": "objective_coefficients",
      "columns": {
        "card_type_code": {
          "data_type": "STRING",
          "business_meaning": "code representing the card type",
          "optimization_purpose": "identifier for the card type fee",
          "sample_values": "VISA, MASTERCARD"
        },
        "fee": {
          "data_type": "FLOAT",
          "business_meaning": "fee associated with the card type",
          "optimization_purpose": "coefficient in the objective function",
          "sample_values": "0.02, 0.03"
        }
      }
    },
    "Account_Limits": {
      "business_purpose": "maximum transaction amount allowed per customer",
      "optimization_role": "constraint_bounds",
      "columns": {
        "customer_id": {
          "data_type": "INTEGER",
          "business_meaning": "unique identifier for the customer",
          "optimization_purpose": "identifier for the account limit",
          "sample_values": "1, 2, 3"
        },
        "account_limit": {
          "data_type": "FLOAT",
          "business_meaning": "maximum transaction amount allowed",
          "optimization_purpose": "bound in the constraint",
          "sample_values": "1000, 1500, 2000"
        }
      }
    },
    "Transaction_Allocations": {
      "business_purpose": "amount of transaction allocated to each card type per customer",
      "optimization_role": "decision_variables",
      "columns": {
        "customer_id": {
          "data_type": "INTEGER",
          "business_meaning": "unique identifier for the customer",
          "optimization_purpose": "identifier for the transaction allocation",
          "sample_values": "1, 2, 3"
        },
        "card_id": {
          "data_type": "STRING",
          "business_meaning": "unique identifier for the card",
          "optimization_purpose": "identifier for the transaction allocation",
          "sample_values": "CARD1, CARD2"
        },
        "transaction_amount": {
          "data_type": "FLOAT",
          "business_meaning": "amount of transaction allocated to the card",
          "optimization_purpose": "decision variable in the optimization model",
          "sample_values": "100, 200, 300"
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "fee_per_card_type": {
    "data_type": "FLOAT",
    "business_meaning": "fee associated with each card type",
    "optimization_role": "coefficient in the objective function",
    "configuration_type": "scalar_parameter",
    "value": 0.025,
    "business_justification": "Average transaction fee across card types, ensuring a balanced optimization problem."
  },
  "account_limit": {
    "data_type": "FLOAT",
    "business_meaning": "maximum transaction amount allowed per customer",
    "optimization_role": "bound in the constraint",
    "configuration_type": "scalar_parameter",
    "value": 1500,
    "business_justification": "Median account limit, ensuring constraints are neither too tight nor too loose."
  },
  "total_transaction_fee_formula": {
    "data_type": "STRING",
    "business_meaning": "total transaction fee calculation",
    "optimization_role": "objective function",
    "configuration_type": "business_logic_formula",
    "formula_expression": "sum(fee_per_card_type[card_type_code] * transaction_amount[card_id])"
  }
}

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: customers_card_transactions

## 1. Problem Context and Goals

### Context  
[Regenerate business context that naturally aligns with LINEAR optimization formulation. Ensure:]
- Business decisions match expected decision variables: transaction_amount[card_id] (continuous)
- Operational parameters align with expected linear objective: minimize sum(fee_per_card_type[card_type_code] * transaction_amount[card_id])
- Business configuration includes: fee associated with each card type (used for coefficient in the objective function), maximum transaction amount allowed per customer (used for bound in the constraint)
- Business logic formulas to express in natural language: total transaction fee calculation (calculation method for objective function)
- 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 sum(fee_per_card_type[card_type_code] * transaction_amount[card_id])
- 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(transaction_amount[card_id]) <= account_limit[customer_id] (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: Added Transaction_Allocations table to map missing decision variables, updated business configuration logic to include scalar parameters and formulas, and ensured schema follows normalization principles.

CREATE TABLE Card_Fees (
  card_type_code STRING,
  fee FLOAT
);

CREATE TABLE Account_Limits (
  customer_id INTEGER,
  account_limit FLOAT
);

CREATE TABLE Transaction_Allocations (
  customer_id INTEGER,
  card_id STRING,
  transaction_amount 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 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical banking transaction fees, account limits, and transaction amounts, ensuring they align with real-world scenarios and enable a meaningful optimization problem.

-- Realistic data for Card_Fees
INSERT INTO Card_Fees (card_type_code, fee) VALUES ('VISA', 0.02);
INSERT INTO Card_Fees (card_type_code, fee) VALUES ('MASTERCARD', 0.03);
INSERT INTO Card_Fees (card_type_code, fee) VALUES ('AMEX', 0.035);

-- Realistic data for Account_Limits
INSERT INTO Account_Limits (customer_id, account_limit) VALUES (1, 1000);
INSERT INTO Account_Limits (customer_id, account_limit) VALUES (2, 1500);
INSERT INTO Account_Limits (customer_id, account_limit) VALUES (3, 2000);

-- Realistic data for Transaction_Allocations
INSERT INTO Transaction_Allocations (customer_id, card_id, transaction_amount) VALUES (1, 'CARD1', 100);
INSERT INTO Transaction_Allocations (customer_id, card_id, transaction_amount) VALUES (2, 'CARD2', 200);
INSERT INTO Transaction_Allocations (customer_id, card_id, transaction_amount) VALUES (3, 'CARD3', 300);


```

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

Response:
# Complete Optimization Problem and Solution: customers_card_transactions

## 1. Problem Context and Goals

### Context  
A bank is focused on reducing the total transaction fees incurred by its customers by strategically allocating transactions across different card types. Each card type has a distinct fee structure, and the bank aims to optimize these allocations to minimize costs. The bank must ensure that the total transaction amount per customer does not exceed their predefined account limits. This operational decision-making process involves determining the appropriate transaction amounts for each card type while adhering to customer-specific constraints.  

The business configuration includes key parameters such as the fee associated with each card type, which directly influences the total transaction fee calculation. Additionally, the maximum transaction amount allowed per customer serves as a critical constraint to ensure compliance with account limits. The total transaction fee is calculated by summing the product of the fee for each card type and the corresponding transaction amount allocated to that card. This approach ensures a linear relationship between the decision variables and the objective, avoiding any nonlinear complexities such as variable products or divisions.  

### Goals  
The primary goal of this optimization problem is to minimize the total transaction fees incurred by the bank’s customers. This is achieved by optimizing the allocation of transaction amounts across different card types, each with its own fee structure. Success is measured by the reduction in the total fee, which is directly influenced by the fees associated with each card type and the transaction amounts allocated to them. The optimization process ensures that the total transaction amount per customer remains within their account limits, maintaining operational feasibility while achieving cost efficiency.  

## 2. Constraints  

The optimization problem is subject to one key constraint: the total transaction amount allocated to all card types for a given customer must not exceed their account limit. This ensures that the bank’s operational decisions remain within the bounds of each customer’s financial capacity. The constraint is expressed as the sum of transaction amounts across all card types for a customer being less than or equal to their account limit. This linear constraint aligns with the business requirement of maintaining customer-specific transaction limits while optimizing fee costs.  

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 2 Database Schema
-- Objective: Added Transaction_Allocations table to map missing decision variables, updated business configuration logic to include scalar parameters and formulas, and ensured schema follows normalization principles.

CREATE TABLE Card_Fees (
  card_type_code STRING,
  fee FLOAT
);

CREATE TABLE Account_Limits (
  customer_id INTEGER,
  account_limit FLOAT
);

CREATE TABLE Transaction_Allocations (
  customer_id INTEGER,
  card_id STRING,
  transaction_amount FLOAT
);
```

### Data Dictionary  
- **Card_Fees**:  
  - **Business Purpose**: Stores the fee associated with each card type.  
  - **Optimization Role**: Provides the coefficients for the objective function.  
  - **Columns**:  
    - `card_type_code`: Represents the type of card (e.g., VISA, MASTERCARD).  
    - `fee`: The fee associated with the card type, used to calculate the total transaction fee.  

- **Account_Limits**:  
  - **Business Purpose**: Defines the maximum transaction amount allowed per customer.  
  - **Optimization Role**: Provides the bounds for the constraints.  
  - **Columns**:  
    - `customer_id`: Unique identifier for the customer.  
    - `account_limit`: The maximum transaction amount allowed for the customer.  

- **Transaction_Allocations**:  
  - **Business Purpose**: Tracks the amount of transaction allocated to each card type per customer.  
  - **Optimization Role**: Represents the decision variables in the optimization model.  
  - **Columns**:  
    - `customer_id`: Unique identifier for the customer.  
    - `card_id`: Unique identifier for the card.  
    - `transaction_amount`: The amount of transaction allocated to the card.  

### Current Stored Values  
```sql
-- Iteration 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical banking transaction fees, account limits, and transaction amounts, ensuring they align with real-world scenarios and enable a meaningful optimization problem.

-- Realistic data for Card_Fees
INSERT INTO Card_Fees (card_type_code, fee) VALUES ('VISA', 0.02);
INSERT INTO Card_Fees (card_type_code, fee) VALUES ('MASTERCARD', 0.03);
INSERT INTO Card_Fees (card_type_code, fee) VALUES ('AMEX', 0.035);

-- Realistic data for Account_Limits
INSERT INTO Account_Limits (customer_id, account_limit) VALUES (1, 1000);
INSERT INTO Account_Limits (customer_id, account_limit) VALUES (2, 1500);
INSERT INTO Account_Limits (customer_id, account_limit) VALUES (3, 2000);

-- Realistic data for Transaction_Allocations
INSERT INTO Transaction_Allocations (customer_id, card_id, transaction_amount) VALUES (1, 'CARD1', 100);
INSERT INTO Transaction_Allocations (customer_id, card_id, transaction_amount) VALUES (2, 'CARD2', 200);
INSERT INTO Transaction_Allocations (customer_id, card_id, transaction_amount) VALUES (3, 'CARD3', 300);
```
