# Complete Optimization Problem and Solution: epinions_1

## 1. Problem Context and Goals

### Context  
In the context of a review platform, the goal is to optimize the selection of user reviews to maximize the overall trust score between users. Each review is associated with a binary decision variable indicating whether it is selected. The selection process aims to prioritize reviews that contribute most significantly to the trust relationships between users. The operational parameters are designed to align with a linear objective, focusing on maximizing the sum of the product of trust scores and review ratings for the selected reviews. The business configuration includes a constraint on the maximum number of reviews that can be selected, ensuring that the optimization process remains manageable and focused. This constraint is represented by a scalar parameter that limits the total number of reviews chosen. The problem is structured to avoid nonlinear relationships, ensuring that all decision-making processes lead to linear formulations. The specific operational parameters, such as trust scores and review ratings, serve as coefficients in the optimization model, directly influencing the selection of reviews.

### Goals  
The primary goal of this optimization problem is to maximize the overall trust score on the review platform. This is achieved by selecting reviews that enhance trust relationships between users. The metric to optimize is the sum of the products of trust scores and review ratings for the selected reviews. Success is measured by the extent to which the selected reviews contribute to the trust score, with the optimization process guided by the coefficients derived from trust scores and ratings. The objective is clearly defined in linear terms, focusing on maximizing the cumulative impact of selected reviews on user trust.

## 2. Constraints    

The optimization process is subject to a key constraint: the total number of selected reviews must not exceed a predefined maximum. This constraint ensures that the selection process remains within practical limits, reflecting typical resource allocation constraints in business operations. The constraint is expressed in linear terms, where the sum of the binary decision variables representing selected reviews is limited by the maximum number of reviews allowed. This approach ensures that the optimization model remains linear and manageable.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating a new table for decision variables and updating the business configuration logic to handle scalar parameters and formulas. This addresses the OR expert's mapping gaps and missing requirements.

CREATE TABLE trust (
  source_u_id INTEGER,
  target_u_id INTEGER,
  trust FLOAT
);

CREATE TABLE review (
  a_id INTEGER,
  rating INTEGER
);

CREATE TABLE decision_variables (
  a_id INTEGER,
  selected BOOLEAN
);
```

### Data Dictionary  
The data dictionary provides a comprehensive mapping of tables and columns to their business purposes and roles in the optimization process:

- **Trust Table**: This table stores the trust scores between users, which serve as coefficients in the optimization objective. Each entry includes the source and target user IDs and the trust score, reflecting the strength of the trust relationship.

- **Review Table**: This table contains reviews and their associated ratings. The ratings are used as coefficients in the optimization objective, influencing the selection of reviews based on their potential impact on trust.

- **Decision Variables Table**: This table records the decision variables for review selection. Each entry indicates whether a review is selected, represented by a binary variable. This table plays a crucial role in the optimization process, determining which reviews are prioritized.

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical user interactions on review platforms, ensuring a mix of high and low trust scores and ratings to reflect realistic user behavior.

-- Realistic data for trust
INSERT INTO trust (source_u_id, target_u_id, trust) VALUES (1, 4, 0.7);
INSERT INTO trust (source_u_id, target_u_id, trust) VALUES (2, 5, 0.9);
INSERT INTO trust (source_u_id, target_u_id, trust) VALUES (3, 6, 0.4);

-- Realistic data for review
INSERT INTO review (a_id, rating) VALUES (101, 5);
INSERT INTO review (a_id, rating) VALUES (102, 2);
INSERT INTO review (a_id, rating) VALUES (103, 4);
INSERT INTO review (a_id, rating) VALUES (104, 3);
INSERT INTO review (a_id, rating) VALUES (105, 1);

-- Realistic data for decision_variables
INSERT INTO decision_variables (a_id, selected) VALUES (101, True);
INSERT INTO decision_variables (a_id, selected) VALUES (102, False);
INSERT INTO decision_variables (a_id, selected) VALUES (103, True);
```