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

## 1. Problem Context and Goals

### Context  
A furniture manufacturer is tasked with determining the optimal number of units to produce for each furniture type to maximize profit. The decision involves balancing production costs, market rates, and resource limitations. The manufacturer operates under two key constraints: the total production capacity across all factories is limited to 1,000 units, and the total budget available for production is capped at $50,000.  

The profit for each furniture type is calculated as the difference between its market rate and its production cost. The manufacturer aims to allocate production quantities in a way that maximizes total profit while staying within the factory capacity and budget limits. This problem is formulated as a linear optimization model, ensuring that all relationships and constraints are linear and directly tied to operational parameters.  

### Goals  
The primary goal is to maximize the total profit from furniture production. This is achieved by optimizing the number of units produced for each furniture type, considering the difference between the market rate and the production cost for each item. Success is measured by the total profit generated, which is directly influenced by the production quantities and the operational parameters of market rates, production costs, and resource constraints.  

## 2. Constraints  

The manufacturer must adhere to the following constraints:  
1. **Total Production Capacity**: The sum of units produced across all furniture types must not exceed the total factory capacity of 1,000 units. This ensures that production does not surpass the manufacturer's operational capabilities.  
2. **Budget Limit**: The total cost of production, calculated as the sum of the production cost multiplied by the number of units produced for each furniture type, must not exceed the available budget of $50,000. This ensures that production remains financially feasible.  

These constraints are linear and directly tied to the manufacturer's operational limitations, ensuring a realistic and feasible optimization problem.  

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 2 Database Schema
-- Objective: Schema changes include creating tables for objective coefficients (Market_Rate and Price_in_Dollar) and updating business configuration logic to include scalar parameters and formulas for missing optimization requirements.

CREATE TABLE production_plan (
  Furniture_ID INTEGER,
  Quantity_Produced INTEGER
);

CREATE TABLE furniture_market_rates (
  Furniture_ID INTEGER,
  Market_Rate FLOAT
);

CREATE TABLE furniture_production_costs (
  Furniture_ID INTEGER,
  Price_in_Dollar FLOAT
);
```

### Data Dictionary  
- **production_plan**:  
  - **Furniture_ID**: Unique identifier for each furniture type.  
  - **Quantity_Produced**: Number of units to produce for each furniture type. This is the decision variable in the optimization model.  

- **furniture_market_rates**:  
  - **Furniture_ID**: Unique identifier for each furniture type.  
  - **Market_Rate**: Market rate for each furniture type. This is used as an objective coefficient in the optimization model.  

- **furniture_production_costs**:  
  - **Furniture_ID**: Unique identifier for each furniture type.  
  - **Price_in_Dollar**: Production cost for each furniture type. This is used as an objective coefficient in the optimization model.  


=== Schema ===
-- Iteration 2 Database Schema
-- Objective: Schema changes include creating tables for objective coefficients (Market_Rate and Price_in_Dollar) and updating business configuration logic to include scalar parameters and formulas for missing optimization requirements.

CREATE TABLE production_plan (
  Furniture_ID INTEGER,
  Quantity_Produced INTEGER
);

CREATE TABLE furniture_market_rates (
  Furniture_ID INTEGER,
  Market_Rate FLOAT
);

CREATE TABLE furniture_production_costs (
  Furniture_ID INTEGER,
  Price_in_Dollar FLOAT
);



=== LLM Response ===
```sql
-- Query Description: Retrieve the market rate and production cost for each furniture type.
-- This is essential for calculating the profit per unit for each furniture type, which is the objective function coefficient.
SELECT 
    fmr.Furniture_ID, 
    fmr.Market_Rate, 
    fpc.Price_in_Dollar
FROM 
    furniture_market_rates fmr
JOIN 
    furniture_production_costs fpc 
ON 
    fmr.Furniture_ID = fpc.Furniture_ID;

-- Query Description: Retrieve the current production plan (quantity produced for each furniture type).
-- This is the decision variable in the optimization model, representing the number of units to produce for each furniture type.
SELECT 
    Furniture_ID, 
    Quantity_Produced
FROM 
    production_plan;

-- Query Description: Calculate the total production cost for the current production plan.
-- This is needed to ensure the budget constraint is not exceeded.
SELECT 
    SUM(pp.Quantity_Produced * fpc.Price_in_Dollar) AS Total_Production_Cost
FROM 
    production_plan pp
JOIN 
    furniture_production_costs fpc 
ON 
    pp.Furniture_ID = fpc.Furniture_ID;

-- Query Description: Calculate the total number of units produced across all furniture types.
-- This is needed to ensure the production capacity constraint is not exceeded.
SELECT 
    SUM(Quantity_Produced) AS Total_Units_Produced
FROM 
    production_plan;

-- Query Description: Retrieve the profit per unit for each furniture type.
-- This is crucial for the objective function, as it represents the profit generated per unit for each furniture type.
SELECT 
    fmr.Furniture_ID, 
    (fmr.Market_Rate - fpc.Price_in_Dollar) AS Profit_Per_Unit
FROM 
    furniture_market_rates fmr
JOIN 
    furniture_production_costs fpc 
ON 
    fmr.Furniture_ID = fpc.Furniture_ID;

-- Query Description: Calculate the total profit for the current production plan.
-- This is the primary goal of the optimization problem, representing the total profit generated from the current production quantities.
SELECT 
    SUM(pp.Quantity_Produced * (fmr.Market_Rate - fpc.Price_in_Dollar)) AS Total_Profit
FROM 
    production_plan pp
JOIN 
    furniture_market_rates fmr 
ON 
    pp.Furniture_ID = fmr.Furniture_ID
JOIN 
    furniture_production_costs fpc 
ON 
    pp.Furniture_ID = fpc.Furniture_ID;

-- Query Description: Retrieve the list of all furniture types with their respective market rates and production costs.
-- This is useful for understanding the range of market rates and production costs across different furniture types.
SELECT 
    fmr.Furniture_ID, 
    fmr.Market_Rate, 
    fpc.Price_in_Dollar
FROM 
    furniture_market_rates fmr
JOIN 
    furniture_production_costs fpc 
ON 
    fmr.Furniture_ID = fpc.Furniture_ID;

-- Query Description: Retrieve the maximum and minimum market rates and production costs across all furniture types.
-- This helps in understanding the variability in market rates and production costs, which can influence the optimization strategy.
SELECT 
    MAX(fmr.Market_Rate) AS Max_Market_Rate, 
    MIN(fmr.Market_Rate) AS Min_Market_Rate, 
    MAX(fpc.Price_in_Dollar) AS Max_Production_Cost, 
    MIN(fpc.Price_in_Dollar) AS Min_Production_Cost
FROM 
    furniture_market_rates fmr
JOIN 
    furniture_production_costs fpc 
ON 
    fmr.Furniture_ID = fpc.Furniture_ID;
```

These queries are designed to retrieve the necessary data for formulating and solving the optimization problem. They provide insights into the decision variables, objective function coefficients, and constraint parameters, ensuring that the optimization model is both accurate and feasible.
