=== Problem Context ===
# Complete Optimization Problem and Solution: shop_membership

## 1. Problem Context and Goals

### Context  
A retail chain is focused on optimizing the allocation of membership benefits across its branches to maximize customer satisfaction. Customer satisfaction is measured by the total pounds spent by members at each branch. The business must make decisions on how to allocate resources to each branch to achieve this goal while adhering to two key operational limitations:  
1. **Total Membership Benefits Budget**: The retail chain has a fixed total budget allocated for membership benefits across all branches. This budget cannot be exceeded when distributing benefits.  
2. **Branch Capacity Limits**: Each branch has a maximum number of members it can handle, which is determined by its operational capacity. The total pounds spent by members at a branch must not exceed this capacity.  

The business configuration includes:  
- A total budget of 500,000 units allocated for membership benefits across all branches.  
- A formula to calculate the total capacity across all branches, which is the sum of the maximum number of members each branch can handle.  

The decision variables in this problem are the total pounds spent by members at each branch, which are continuous values. The goal is to maximize the sum of these values across all branches while respecting the budget and capacity constraints.  

### Goals  
The primary goal of this optimization problem is to maximize the total pounds spent by members across all branches. This metric directly reflects customer satisfaction and is aligned with the retail chain’s objective of enhancing member engagement. Success is measured by achieving the highest possible sum of total pounds spent while staying within the allocated budget and respecting each branch’s capacity limits.  

## 2. Constraints  

The optimization problem is subject to the following constraints:  
1. **Total Membership Benefits Budget**: The sum of the membership amounts allocated to all branches must not exceed the total budget of 500,000 units. This ensures that the retail chain operates within its financial limits.  
2. **Branch Capacity Limits**: The sum of the total pounds spent by members at all branches must not exceed the total capacity across all branches. This ensures that no branch is overloaded beyond its operational capabilities.  

These constraints are designed to ensure that the optimization problem remains linear, avoiding any scenarios that would require nonlinear relationships such as variable products or divisions.  

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating a new table for branch capacity, modifying the purchase table to better map constraints, and adding configuration logic for budget and capacity limits.

CREATE TABLE purchase (
  Total_pounds FLOAT,
  branch_id INTEGER
);

CREATE TABLE branch (
  membership_amount FLOAT
);

CREATE TABLE branch_capacity (
  max_members INTEGER
);
```

### Data Dictionary  
- **purchase Table**:  
  - **Total_pounds**: Represents the total pounds spent by members at a specific branch. This is the key decision variable and objective coefficient in the optimization problem.  
  - **branch_id**: Identifies the branch where the purchase was made. This is used to map constraints to specific branches.  

- **branch Table**:  
  - **membership_amount**: Represents the maximum membership amount allocated to a branch. This is used as a constraint bound to ensure the total budget is not exceeded.  

- **branch_capacity Table**:  
  - **max_members**: Represents the maximum number of members a branch can handle. This is used as a constraint bound to ensure branch capacities are respected.  


=== Schema ===
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating a new table for branch capacity, modifying the purchase table to better map constraints, and adding configuration logic for budget and capacity limits.

CREATE TABLE purchase (
  Total_pounds FLOAT,
  branch_id INTEGER
);

CREATE TABLE branch (
  membership_amount FLOAT
);

CREATE TABLE branch_capacity (
  max_members INTEGER
);



=== LLM Response ===
```sql
-- Query Description: Retrieve the total pounds spent by members at each branch. This is the key decision variable in the optimization problem.
-- Purpose: To understand the current distribution of pounds spent across branches, which will be optimized.
SELECT branch_id, SUM(Total_pounds) AS total_pounds_spent
FROM purchase
GROUP BY branch_id;

-- Query Description: Retrieve the membership amount allocated to each branch. This is needed to ensure the total budget constraint is respected.
-- Purpose: To gather data on the current allocation of membership benefits, which will be used to enforce the budget constraint.
SELECT branch_id, membership_amount
FROM branch;

-- Query Description: Retrieve the maximum number of members each branch can handle. This is needed to ensure the branch capacity constraint is respected.
-- Purpose: To gather data on the operational capacity of each branch, which will be used to enforce the capacity constraint.
SELECT branch_id, max_members
FROM branch_capacity;

-- Query Description: Calculate the total budget allocated for membership benefits across all branches.
-- Purpose: To ensure that the sum of membership amounts across all branches does not exceed the total budget of 500,000 units.
SELECT SUM(membership_amount) AS total_budget
FROM branch;

-- Query Description: Calculate the total capacity across all branches, which is the sum of the maximum number of members each branch can handle.
-- Purpose: To ensure that the sum of total pounds spent by members across all branches does not exceed the total capacity.
SELECT SUM(max_members) AS total_capacity
FROM branch_capacity;

-- Query Description: Retrieve the total pounds spent by members at each branch along with the branch's membership amount and capacity.
-- Purpose: To provide a comprehensive view of the current state of each branch, including pounds spent, membership benefits, and capacity.
-- This data will be used to make informed decisions on how to allocate resources optimally.
SELECT p.branch_id, 
       SUM(p.Total_pounds) AS total_pounds_spent, 
       b.membership_amount, 
       bc.max_members
FROM purchase p
JOIN branch b ON p.branch_id = b.branch_id
JOIN branch_capacity bc ON p.branch_id = bc.branch_id
GROUP BY p.branch_id, b.membership_amount, bc.max_members;

-- Query Description: Retrieve the total pounds spent by members across all branches.
-- Purpose: To understand the overall customer satisfaction metric, which is the sum of total pounds spent by members across all branches.
SELECT SUM(Total_pounds) AS total_pounds_spent_all_branches
FROM purchase;

-- Query Description: Retrieve the total membership amount allocated across all branches.
-- Purpose: To ensure that the total membership benefits budget constraint is respected.
SELECT SUM(membership_amount) AS total_membership_amount
FROM branch;

-- Query Description: Retrieve the total capacity across all branches.
-- Purpose: To ensure that the total capacity constraint is respected.
SELECT SUM(max_members) AS total_capacity
FROM branch_capacity;
```

### Explanation:
1. **Decision Variables**: The total pounds spent by members at each branch (`Total_pounds`) are the key decision variables. The first query retrieves this data grouped by `branch_id`.
2. **Objective Function Coefficients**: The total pounds spent by members across all branches is the objective to maximize. The seventh query retrieves this aggregated data.
3. **Constraint Parameters**: 
   - The total membership benefits budget is constrained by the sum of `membership_amount` across all branches. The fourth and eighth queries retrieve this data.
   - The branch capacity limits are constrained by the sum of `max_members` across all branches. The fifth and ninth queries retrieve this data.
4. **Comprehensive View**: The sixth query provides a detailed view of each branch, including pounds spent, membership benefits, and capacity, which is crucial for making informed allocation decisions.

These queries collectively provide the necessary data to formulate and solve the optimization problem while respecting the given constraints.
