# Complete Optimization Problem and Solution: epinions_1

## 1. Problem Context and Goals

### Context  
The platform operates in a business environment where users provide reviews, and other users rely on these reviews to make informed decisions. To enhance the reliability of these reviews, the platform aims to assign trust weights to reviews based on the trust levels between users. These trust weights reflect the confidence a user has in the reviewer's opinion. The platform's goal is to maximize the overall trust-weighted ratings of reviews, ensuring that the weights are non-negative and do not exceed the trust levels between users. This approach ensures that reviews from highly trusted users have a more significant impact on the overall rating.

The business configuration includes a maximum allowable trust weight for a review, which is set to 1.0. This parameter ensures that the trust weights remain realistic and feasible, aligning with the highest possible trust level between users. The decision variables in this optimization problem are the continuous trust weights assigned to each review, and the objective is to maximize the sum of the trust-weighted ratings of all reviews.

### Goals  
The primary goal of this optimization problem is to maximize the overall trust-weighted ratings of reviews on the platform. This is achieved by assigning optimal trust weights to each review based on the trust levels between users. The success of this optimization is measured by the total sum of the trust-weighted ratings, where each review's rating is multiplied by its corresponding trust weight. The platform aims to ensure that the trust weights are non-negative and do not exceed the trust levels between users, thereby maintaining the integrity and reliability of the reviews.

## 2. Constraints    

The optimization problem is subject to two main constraints:

1. **Non-Negative Trust Weights**: The trust weight assigned to each review must be non-negative. This ensures that the impact of a review on the overall rating is always positive or neutral, reflecting the platform's commitment to maintaining the integrity of the review system.

2. **Trust Weight Upper Bound**: The trust weight assigned to each review must not exceed the trust level between the reviewer and the user. This constraint ensures that the trust weights are realistic and feasible, aligning with the actual trust levels between users. The maximum allowable trust weight for a review is set to 1.0, as defined in the business configuration.

These constraints ensure that the trust weights are within acceptable limits and that the optimization problem remains linear, avoiding any nonlinear relationships such as variable products or divisions.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include adding a trust_weight table for decision variables, updating the review table to include optimization-relevant data, and moving scalar parameters to business_configuration_logic.json.

CREATE TABLE trust_weight (
  weight FLOAT
);

CREATE TABLE review (
  rating FLOAT
);

CREATE TABLE trust (
  trust FLOAT
);
```

### Data Dictionary  
- **trust_weight Table**:  
  - **Business Purpose**: Stores the trust weights assigned to reviews.  
  - **Optimization Role**: Decision variables in the optimization model.  
  - **Columns**:  
    - **weight**: The weight assigned to a review based on trust. This is a continuous variable that directly influences the optimization objective.

- **review Table**:  
  - **Business Purpose**: Stores review data, including ratings provided by users.  
  - **Optimization Role**: Provides the coefficients for the objective function.  
  - **Columns**:  
    - **rating**: The rating of the review by the author. This value is used as a coefficient in the objective function to calculate the trust-weighted ratings.

- **trust Table**:  
  - **Business Purpose**: Stores the trust levels between users.  
  - **Optimization Role**: Provides the bounds for the constraints.  
  - **Columns**:  
    - **trust**: The trust level between the source and target user. This value is used to set the upper bound for the trust weights in the constraints.


### Retrieved Values

**Query 1: These weights are the variables that will be optimized to maximize the overall trust-weighted ratings.**

```sql
SELECT weight FROM trust_weight;
```

**Results (CSV format):**
```csv
weight
0.5
0.7
0.9
```

**Query 2: These ratings are multiplied by the trust weights to calculate the trust-weighted ratings.**

```sql
SELECT rating FROM review;
```

**Results (CSV format):**
```csv
rating
4.5
3.8
5.0
3.0
4.0
```

**Query 3: These trust levels ensure that the trust weights do not exceed the actual trust between users.**

```sql
SELECT trust FROM trust;
```

**Results (CSV format):**
```csv
trust
0.6
0.8
1.0
0.7
```

**Query 4: For the purpose of this query, we assume it is stored in a table named `business_configuration`.**

```sql
SELECT max_trust_weight FROM business_configuration WHERE parameter_name = 'max_trust_weight';
```

**Error:** Execution failed on sql 'SELECT max_trust_weight FROM business_configuration WHERE parameter_name = 'max_trust_weight';': no such table: business_configuration

**Query 5: This aggregated data helps in understanding the scale of the optimization problem.**

```sql
SELECT SUM(rating) AS total_rating FROM review;
```

**Results (CSV format):**
```csv
total_rating
20.3
```

**Query 6: This summary statistic provides insight into the typical trust levels, which can inform the optimization process.**

```sql
SELECT AVG(trust) AS average_trust FROM trust;
```

**Results (CSV format):**
```csv
average_trust
0.775
```

**Query 7: This information helps in setting realistic bounds for the trust weights.**

```sql
SELECT MIN(trust) AS min_trust, MAX(trust) AS max_trust FROM trust;
```

**Results (CSV format):**
```csv
min_trust,max_trust
0.6,1.0
```

**Query 8: This information helps in assessing the complexity of the optimization problem.**

```sql
SELECT COUNT(*) AS total_reviews FROM review;
```

**Results (CSV format):**
```csv
total_reviews
5
```

**Query 9: This information helps in assessing the complexity of the constraints.**

```sql
SELECT COUNT(*) AS total_trust_relationships FROM trust;
```

**Results (CSV format):**
```csv
total_trust_relationships
4
```

**Query 10: This information helps in understanding the potential impact of different reviews on the optimization objective.**

```sql
SELECT rating, COUNT(*) AS frequency FROM review GROUP BY rating ORDER BY rating;
```

**Results (CSV format):**
```csv
rating,frequency
3.0,1
3.8,1
4.0,1
4.5,1
5.0,1
```

**Query 11: This information helps in understanding the potential variability in the constraints.**

```sql
SELECT trust, COUNT(*) AS frequency FROM trust GROUP BY trust ORDER BY trust;
```

**Results (CSV format):**
```csv
trust,frequency
0.6,1
0.7,1
0.8,1
1.0,1
```

**Query 12: For the purpose of this query, we assume a join can be made based on user IDs or review IDs.**

```sql
SELECT r.rating, t.trust FROM review r JOIN trust t ON r.user_id = t.source_user_id ORDER BY r.rating, t.trust;
```

**Error:** Execution failed on sql 'SELECT r.rating, t.trust FROM review r JOIN trust t ON r.user_id = t.source_user_id ORDER BY r.rating, t.trust;': no such column: r.user_id

