=== Problem Context ===
# 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.  


=== Schema ===
-- 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
);



=== LLM Response ===
```sql
-- Query Description: Retrieve the list of regions and the number of restaurants in each region.
-- This is important for the regional distribution constraint, ensuring at least one new restaurant is placed in each region.
SELECT REGION, COUNT(*) AS NUM_RESTAURANTS
FROM RESTAURANT
GROUP BY REGION;

-- Query Description: Retrieve the average rating of restaurants in each region.
-- This helps in understanding the potential customer satisfaction impact of placing a new restaurant in each region.
SELECT REGION, AVG(RATING) AS AVG_RATING
FROM RESTAURANT
GROUP BY REGION;

-- Query Description: Retrieve the maximum rating of restaurants in each city.
-- This is useful for identifying the cities with the highest potential customer satisfaction impact.
SELECT REGION, MAX(RATING) AS MAX_RATING
FROM RESTAURANT
GROUP BY REGION;

-- Query Description: Retrieve the list of restaurants with their ratings and regions.
-- This data is crucial for the objective function, as it provides the coefficients (ratings) to maximize total customer satisfaction.
SELECT RATING, REGION
FROM RESTAURANT;

-- Query Description: Retrieve the count of decision variables (X_i) that are set to TRUE (indicating a new restaurant placement).
-- This is important for the total budget constraint, ensuring the total number of new restaurants does not exceed the budget.
SELECT COUNT(*) AS TOTAL_NEW_RESTAURANTS
FROM DECISION_VARIABLES
WHERE X_i = TRUE;

-- Query Description: Retrieve the count of decision variables (X_i) set to TRUE per region.
-- This helps in enforcing the regional distribution constraint, ensuring at least one new restaurant is placed in each region.
SELECT REGION, COUNT(*) AS NEW_RESTAURANTS_PER_REGION
FROM DECISION_VARIABLES
JOIN RESTAURANT ON DECISION_VARIABLES.X_i = TRUE
GROUP BY REGION;

-- Query Description: Retrieve the count of decision variables (X_i) set to TRUE per city.
-- This is important for the city-level placement constraint, ensuring no more than 2 new restaurants are placed in any single city.
SELECT CITY, COUNT(*) AS NEW_RESTAURANTS_PER_CITY
FROM DECISION_VARIABLES
JOIN RESTAURANT ON DECISION_VARIABLES.X_i = TRUE
GROUP BY CITY;

-- Query Description: Retrieve the list of cities with their respective regions and restaurant ratings.
-- This data is useful for making informed decisions on where to place new restaurants based on regional and city-level data.
SELECT CITY, REGION, RATING
FROM RESTAURANT;

-- Query Description: Retrieve the total number of restaurants per city.
-- This helps in understanding the current saturation level of restaurants in each city, which is important for the city-level placement constraint.
SELECT CITY, COUNT(*) AS TOTAL_RESTAURANTS
FROM RESTAURANT
GROUP BY CITY;

-- Query Description: Retrieve the list of regions with less than the required minimum number of new restaurants.
-- This is crucial for enforcing the regional distribution constraint, ensuring at least one new restaurant is placed in each region.
SELECT REGION
FROM (
    SELECT REGION, COUNT(*) AS NEW_RESTAURANTS_PER_REGION
    FROM DECISION_VARIABLES
    JOIN RESTAURANT ON DECISION_VARIABLES.X_i = TRUE
    GROUP BY REGION
) AS REGION_COUNTS
WHERE NEW_RESTAURANTS_PER_REGION < 1;
```

These queries are designed to retrieve the necessary data for the optimization problem, including information for the objective function, decision variables, and constraints. They help in making informed decisions on where to place new restaurants to maximize customer satisfaction while adhering to the operational and financial constraints.
