Iteration final - PROBLEM_DESCRIPTION
Sequence: 5
Timestamp: 2025-07-25 22:27:36

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_i: binary decision variable indicating whether to place a new restaurant in city i.
- Operational parameters align with expected linear objective: maximize ∑(RATING_i * X_i) where X_i is a binary decision variable indicating whether to place a new restaurant in city i.
- Business configuration includes: Maximum number of new restaurants allowed per city. (used for Constraint bound for city-level restaurant placement.), Minimum number of new restaurants required per region. (used for Constraint bound for regional restaurant distribution.), Total budget constraint for the number of new restaurants. (used for Global constraint on the total number of new restaurants.)
- 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": "restaurants",
  "iteration": 1,
  "business_context": "A restaurant chain aims to optimize the placement of new restaurants across different cities to maximize overall customer satisfaction, represented by the sum of ratings, while ensuring a balanced distribution across regions and limiting the number of new restaurants per city.",
  "optimization_problem_description": "Maximize the total customer satisfaction (sum of ratings) from new restaurant placements, subject to constraints on the number of new restaurants per city, ensuring a minimum number of restaurants in each region, and adhering to a total budget for new restaurants.",
  "optimization_formulation": {
    "objective": "maximize \u2211(RATING_i * X_i) where X_i is a binary decision variable indicating whether to place a new restaurant in city i.",
    "decision_variables": "X_i: binary decision variable indicating whether to place a new restaurant in city i.",
    "constraints": [
      "\u2211(X_i) \u2264 TOTAL_BUDGET: Total number of new restaurants cannot exceed the budget.",
      "\u2211(X_i for city i in region j) \u2265 MIN_RESTAURANTS_PER_REGION: Minimum number of new restaurants required per region.",
      "X_i \u2264 MAX_RESTAURANTS_PER_CITY: Maximum number of new restaurants allowed per city."
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "RATING_i": {
        "currently_mapped_to": "RESTAURANT.RATING",
        "mapping_adequacy": "good",
        "description": "Customer satisfaction rating of the restaurant, used as the coefficient in the objective function."
      }
    },
    "constraint_bounds": {
      "TOTAL_BUDGET": {
        "currently_mapped_to": "business_configuration_logic.TOTAL_BUDGET",
        "mapping_adequacy": "good",
        "description": "Total budget constraint for the number of new restaurants."
      },
      "MIN_RESTAURANTS_PER_REGION": {
        "currently_mapped_to": "business_configuration_logic.MIN_RESTAURANTS_PER_REGION",
        "mapping_adequacy": "good",
        "description": "Minimum number of new restaurants required per region."
      },
      "MAX_RESTAURANTS_PER_CITY": {
        "currently_mapped_to": "business_configuration_logic.MAX_RESTAURANTS_PER_CITY",
        "mapping_adequacy": "good",
        "description": "Maximum number of new restaurants allowed per city."
      }
    },
    "decision_variables": {
      "X_i": {
        "currently_mapped_to": "DECISION_VARIABLES.X_i",
        "mapping_adequacy": "good",
        "description": "Binary decision variable indicating whether to place a new restaurant in city i.",
        "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 creating new tables for decision variables and constraint bounds, moving scalar parameters to business configuration logic, and updating the data dictionary to reflect optimization mappings.

CREATE TABLE RESTAURANT (
  RATING FLOAT,
  REGION STRING
);

CREATE TABLE DECISION_VARIABLES (
  X_i BOOLEAN
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic business scenarios, ensuring that the optimization problem remains meaningful and solvable. The data respects the constraints and relationships defined in the schema and business configuration logic.

-- Realistic data for RESTAURANT
INSERT INTO RESTAURANT (RATING, REGION) VALUES (4.5, 'North');
INSERT INTO RESTAURANT (RATING, REGION) VALUES (3.8, 'South');
INSERT INTO RESTAURANT (RATING, REGION) VALUES (4.2, 'East');
INSERT INTO RESTAURANT (RATING, REGION) VALUES (3.9, 'West');
INSERT INTO RESTAURANT (RATING, REGION) VALUES (4.1, 'North');

-- Realistic data for DECISION_VARIABLES
INSERT INTO DECISION_VARIABLES (X_i) VALUES (True);
INSERT INTO DECISION_VARIABLES (X_i) VALUES (False);
INSERT INTO DECISION_VARIABLES (X_i) VALUES (True);


```

DATA DICTIONARY:
{
  "tables": {
    "RESTAURANT": {
      "business_purpose": "Stores information about existing and potential new restaurants.",
      "optimization_role": "business_data",
      "columns": {
        "RATING": {
          "data_type": "FLOAT",
          "business_meaning": "Customer satisfaction rating of the restaurant.",
          "optimization_purpose": "Objective coefficient for maximizing total satisfaction.",
          "sample_values": "4.5, 3.8, 4.2"
        },
        "REGION": {
          "data_type": "STRING",
          "business_meaning": "Geographical region where the restaurant is located.",
          "optimization_purpose": "Supports regional constraints.",
          "sample_values": "North, South, East, West"
        }
      }
    },
    "DECISION_VARIABLES": {
      "business_purpose": "Binary decision variables for new restaurant placements.",
      "optimization_role": "decision_variables",
      "columns": {
        "X_i": {
          "data_type": "BOOLEAN",
          "business_meaning": "Indicates whether to place a new restaurant in city i.",
          "optimization_purpose": "Decision variable in optimization model.",
          "sample_values": "true, false"
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "MAX_RESTAURANTS_PER_CITY": {
    "data_type": "INTEGER",
    "business_meaning": "Maximum number of new restaurants allowed per city.",
    "optimization_role": "Constraint bound for city-level restaurant placement.",
    "configuration_type": "scalar_parameter",
    "value": 2,
    "business_justification": "Limiting new restaurants to 2 per city ensures a balanced distribution and prevents over-saturation in any single city."
  },
  "MIN_RESTAURANTS_PER_REGION": {
    "data_type": "INTEGER",
    "business_meaning": "Minimum number of new restaurants required per region.",
    "optimization_role": "Constraint bound for regional restaurant distribution.",
    "configuration_type": "scalar_parameter",
    "value": 1,
    "business_justification": "Ensuring at least 1 new restaurant per region maintains a fair distribution across all regions."
  },
  "TOTAL_BUDGET": {
    "data_type": "INTEGER",
    "business_meaning": "Total budget constraint for the number of new restaurants.",
    "optimization_role": "Global constraint on the total number of new restaurants.",
    "configuration_type": "scalar_parameter",
    "value": 5,
    "business_justification": "A total budget of 5 new restaurants allows for strategic placement while adhering to financial constraints."
  }
}

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: restaurants

## 1. Problem Context and Goals

### Context  
[Regenerate business context that naturally aligns with LINEAR optimization formulation. Ensure:]
- Business decisions match expected decision variables: X_i: binary decision variable indicating whether to place a new restaurant in city i.
- Operational parameters align with expected linear objective: maximize ∑(RATING_i * X_i) where X_i is a binary decision variable indicating whether to place a new restaurant in city i.
- Business configuration includes: Maximum number of new restaurants allowed per city. (used for Constraint bound for city-level restaurant placement.), Minimum number of new restaurants required per region. (used for Constraint bound for regional restaurant distribution.), Total budget constraint for the number of new restaurants. (used for Global constraint on the total number of new restaurants.)
- 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 ∑(RATING_i * X_i) where X_i is a binary decision variable indicating whether to place a new restaurant in city 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: ['∑(X_i) ≤ TOTAL_BUDGET: Total number of new restaurants cannot exceed the budget.', '∑(X_i for city i in region j) ≥ MIN_RESTAURANTS_PER_REGION: Minimum number of new restaurants required per region.', 'X_i ≤ MAX_RESTAURANTS_PER_CITY: Maximum number of new restaurants allowed per city.'] (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 new tables for decision variables and constraint bounds, moving scalar parameters to business configuration logic, and updating the data dictionary to reflect optimization mappings.

CREATE TABLE RESTAURANT (
  RATING FLOAT,
  REGION STRING
);

CREATE TABLE DECISION_VARIABLES (
  X_i 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 realistic business scenarios, ensuring that the optimization problem remains meaningful and solvable. The data respects the constraints and relationships defined in the schema and business configuration logic.

-- Realistic data for RESTAURANT
INSERT INTO RESTAURANT (RATING, REGION) VALUES (4.5, 'North');
INSERT INTO RESTAURANT (RATING, REGION) VALUES (3.8, 'South');
INSERT INTO RESTAURANT (RATING, REGION) VALUES (4.2, 'East');
INSERT INTO RESTAURANT (RATING, REGION) VALUES (3.9, 'West');
INSERT INTO RESTAURANT (RATING, REGION) VALUES (4.1, 'North');

-- Realistic data for DECISION_VARIABLES
INSERT INTO DECISION_VARIABLES (X_i) VALUES (True);
INSERT INTO DECISION_VARIABLES (X_i) VALUES (False);
INSERT INTO DECISION_VARIABLES (X_i) VALUES (True);


```

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

Response:
# Complete Optimization Problem and Solution: restaurants

## 1. Problem Context and Goals

### Context  
A restaurant chain is strategically planning the placement of new restaurants across various cities to maximize overall customer satisfaction. Customer satisfaction is measured by the restaurant's rating, which serves as a key indicator of success. The decision to place a new restaurant in a specific city is represented by a binary choice: either to place the restaurant or not. The goal is to ensure that the total customer satisfaction across all new placements is maximized while adhering to several operational constraints.  

The business has established specific configuration parameters to guide this decision-making process:  
- **Maximum number of new restaurants allowed per city**: To prevent over-saturation, no more than 2 new restaurants can be placed in any single city.  
- **Minimum number of new restaurants required per region**: To maintain a balanced distribution, at least 1 new restaurant must be placed in each geographical region.  
- **Total budget constraint for the number of new restaurants**: The total number of new restaurants placed across all cities cannot exceed 5, ensuring financial feasibility.  

These parameters ensure that the optimization problem remains linear and avoids scenarios requiring complex nonlinear relationships, such as variable products or divisions.  

### Goals  
The primary goal of this optimization problem is to maximize the total customer satisfaction derived from the placement of new restaurants. This is achieved by selecting the optimal set of cities for new restaurant placements, where each placement contributes to the overall satisfaction based on the restaurant's rating. Success is measured by the sum of the ratings of all selected restaurants, ensuring that the highest possible customer satisfaction is achieved within the defined constraints.  

## 2. Constraints  

The optimization problem is subject to the following constraints, which ensure that the solution aligns with the business's operational and financial limitations:  
1. **Total budget constraint**: The total number of new restaurants placed across all cities must not exceed the budget of 5. This ensures that the financial resources are not overextended.  
2. **Regional distribution constraint**: Each geographical region must have at least 1 new restaurant placed. This ensures a fair and balanced distribution of new restaurants across all regions.  
3. **City-level placement constraint**: No more than 2 new restaurants can be placed in any single city. This prevents over-saturation and ensures a diverse placement strategy.  

These constraints are designed to naturally lead to linear mathematical forms, avoiding any complexity that would require nonlinear relationships.  

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating new tables for decision variables and constraint bounds, moving scalar parameters to business configuration logic, and updating the data dictionary to reflect optimization mappings.

CREATE TABLE RESTAURANT (
  RATING FLOAT,
  REGION STRING
);

CREATE TABLE DECISION_VARIABLES (
  X_i BOOLEAN
);
```

### Data Dictionary  
- **RESTAURANT Table**:  
  - **RATING**: Represents the customer satisfaction rating of the restaurant. This value is used as the coefficient in the objective function to maximize total satisfaction.  
  - **REGION**: Indicates the geographical region where the restaurant is located. This supports the regional distribution constraint.  

- **DECISION_VARIABLES Table**:  
  - **X_i**: A binary decision variable that indicates whether to place a new restaurant in city i. This is the key variable in the optimization model.  

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic business scenarios, ensuring that the optimization problem remains meaningful and solvable. The data respects the constraints and relationships defined in the schema and business configuration logic.

-- Realistic data for RESTAURANT
INSERT INTO RESTAURANT (RATING, REGION) VALUES (4.5, 'North');
INSERT INTO RESTAURANT (RATING, REGION) VALUES (3.8, 'South');
INSERT INTO RESTAURANT (RATING, REGION) VALUES (4.2, 'East');
INSERT INTO RESTAURANT (RATING, REGION) VALUES (3.9, 'West');
INSERT INTO RESTAURANT (RATING, REGION) VALUES (4.1, 'North');

-- Realistic data for DECISION_VARIABLES
INSERT INTO DECISION_VARIABLES (X_i) VALUES (True);
INSERT INTO DECISION_VARIABLES (X_i) VALUES (False);
INSERT INTO DECISION_VARIABLES (X_i) VALUES (True);
```
