Iteration final - PROBLEM_DESCRIPTION
Sequence: 7
Timestamp: 2025-07-25 22:29:20

Prompt:
You are a business analyst creating structured optimization problem documentation.

DATA SOURCES EXPLANATION:
- FINAL OR ANALYSIS: Final converged optimization problem from alternating process (iteration 2), contains business context and schema mapping evaluation
- DATABASE SCHEMA: Current database structure after iterative adjustments  
- DATA DICTIONARY: Business meanings and optimization roles of tables and columns
- CURRENT STORED VALUES: Realistic business data generated by triple expert (business + data + optimization)
- BUSINESS CONFIGURATION: Scalar parameters and business logic formulas separated from table data

CRITICAL REQUIREMENTS: 
- Ensure problem description naturally leads to LINEAR or MIXED-INTEGER optimization formulation
- Make business context consistent with the intended decision variables and objectives
- Align constraint descriptions with expected mathematical constraints
- Ensure data descriptions map clearly to expected coefficient sources
- Maintain business authenticity while fixing mathematical consistency issues
- Avoid business scenarios that would naturally require nonlinear relationships (variable products, divisions, etc.)

AUTO-EXTRACTED CONTEXT REQUIREMENTS:
- Business decisions match expected decision variables: Quantity_Produced[Furniture_ID] (integer)
- Operational parameters align with expected linear objective: maximize ∑((Market_Rate - Price_in_Dollar) × Quantity_Produced)
- Business configuration includes: Total production capacity of all factories (used for constraint bound), Total budget available for production (used for constraint bound)
- Use natural language to precisely describe linear mathematical relationships
- NO mathematical formulas, equations, or symbolic notation
- Present data as current operational information
- Focus on precise operational decision-making that leads to linear formulations
- Resource limitations match expected linear constraints
- Avoid scenarios requiring variable products, divisions, or other nonlinear relationships
- Include specific operational parameters that map to expected coefficient sources
- Reference business configuration parameters where appropriate

FINAL OR ANALYSIS:
{
  "database_id": "manufacturer",
  "iteration": 2,
  "business_context": "A furniture manufacturer aims to maximize its profit by determining the optimal number of each furniture type to produce, considering production costs, market rates, and factory capacity constraints. The problem is formulated as a linear optimization model.",
  "optimization_problem_description": "The manufacturer needs to decide how many units of each furniture type to produce to maximize profit, given the production costs, market rates, and limited factory capacity. The objective is to maximize the total profit, which is the difference between the revenue from selling the furniture and the production costs. The problem is formulated as a linear optimization model with linear constraints.",
  "optimization_formulation": {
    "objective": "maximize \u2211((Market_Rate - Price_in_Dollar) \u00d7 Quantity_Produced)",
    "decision_variables": "Quantity_Produced[Furniture_ID] (integer)",
    "constraints": [
      "\u2211(Quantity_Produced[Furniture_ID]) \u2264 Total_Factory_Capacity",
      "\u2211(Price_in_Dollar \u00d7 Quantity_Produced[Furniture_ID]) \u2264 Budget_Limit"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "Market_Rate[Furniture_ID]": {
        "currently_mapped_to": "furniture_market_rates.Market_Rate",
        "mapping_adequacy": "good",
        "description": "Market rate for each furniture type"
      },
      "Price_in_Dollar[Furniture_ID]": {
        "currently_mapped_to": "furniture_production_costs.Price_in_Dollar",
        "mapping_adequacy": "good",
        "description": "Production cost for each furniture type"
      }
    },
    "constraint_bounds": {
      "Total_Factory_Capacity": {
        "currently_mapped_to": "business_configuration_logic.Total_Factory_Capacity",
        "mapping_adequacy": "good",
        "description": "Total production capacity of all factories"
      },
      "Budget_Limit": {
        "currently_mapped_to": "business_configuration_logic.Budget_Limit",
        "mapping_adequacy": "good",
        "description": "Total budget available for production"
      }
    },
    "decision_variables": {
      "Quantity_Produced[Furniture_ID]": {
        "currently_mapped_to": "production_plan.Quantity_Produced",
        "mapping_adequacy": "good",
        "description": "Number of units to produce for each furniture type",
        "variable_type": "integer"
      }
    }
  },
  "missing_optimization_requirements": [],
  "iteration_status": {
    "complete": true,
    "confidence": "high",
    "next_focus": "Ready for convergence"
  }
}

FINAL 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
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical furniture manufacturing scenarios, considering production costs, market rates, and factory capacities. The data ensures a balanced optimization problem with feasible constraints.

-- Realistic data for production_plan
INSERT INTO production_plan (Furniture_ID, Quantity_Produced) VALUES (1, 0);
INSERT INTO production_plan (Furniture_ID, Quantity_Produced) VALUES (2, 0);
INSERT INTO production_plan (Furniture_ID, Quantity_Produced) VALUES (3, 0);

-- Realistic data for furniture_market_rates
INSERT INTO furniture_market_rates (Furniture_ID, Market_Rate) VALUES (1, 120.0);
INSERT INTO furniture_market_rates (Furniture_ID, Market_Rate) VALUES (2, 180.0);
INSERT INTO furniture_market_rates (Furniture_ID, Market_Rate) VALUES (3, 250.0);

-- Realistic data for furniture_production_costs
INSERT INTO furniture_production_costs (Furniture_ID, Price_in_Dollar) VALUES (1, 60.0);
INSERT INTO furniture_production_costs (Furniture_ID, Price_in_Dollar) VALUES (2, 90.0);
INSERT INTO furniture_production_costs (Furniture_ID, Price_in_Dollar) VALUES (3, 130.0);


```

DATA DICTIONARY:
{
  "tables": {
    "production_plan": {
      "business_purpose": "Number of units to produce for each furniture type",
      "optimization_role": "decision_variables",
      "columns": {
        "Furniture_ID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each furniture type",
          "optimization_purpose": "Index for decision variables",
          "sample_values": "1, 2, 3"
        },
        "Quantity_Produced": {
          "data_type": "INTEGER",
          "business_meaning": "Number of units to produce for each furniture type",
          "optimization_purpose": "Decision variable in optimization model",
          "sample_values": "10, 20, 30"
        }
      }
    },
    "furniture_market_rates": {
      "business_purpose": "Market rate for each furniture type",
      "optimization_role": "objective_coefficients",
      "columns": {
        "Furniture_ID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each furniture type",
          "optimization_purpose": "Index for objective coefficients",
          "sample_values": "1, 2, 3"
        },
        "Market_Rate": {
          "data_type": "FLOAT",
          "business_meaning": "Market rate for each furniture type",
          "optimization_purpose": "Objective coefficient in optimization model",
          "sample_values": "100.0, 150.0, 200.0"
        }
      }
    },
    "furniture_production_costs": {
      "business_purpose": "Production cost for each furniture type",
      "optimization_role": "objective_coefficients",
      "columns": {
        "Furniture_ID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each furniture type",
          "optimization_purpose": "Index for objective coefficients",
          "sample_values": "1, 2, 3"
        },
        "Price_in_Dollar": {
          "data_type": "FLOAT",
          "business_meaning": "Production cost for each furniture type",
          "optimization_purpose": "Objective coefficient in optimization model",
          "sample_values": "50.0, 75.0, 100.0"
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "Total_Factory_Capacity": {
    "data_type": "INTEGER",
    "business_meaning": "Total production capacity of all factories",
    "optimization_role": "constraint bound",
    "configuration_type": "scalar_parameter",
    "value": 1000,
    "business_justification": "Total factory capacity is set to 1000 units, reflecting the manufacturer's production capability."
  },
  "Budget_Limit": {
    "data_type": "FLOAT",
    "business_meaning": "Total budget available for production",
    "optimization_role": "constraint bound",
    "configuration_type": "scalar_parameter",
    "value": 50000.0,
    "business_justification": "Budget limit is set to $50,000, reflecting the manufacturer's available budget for production."
  }
}

Business Configuration Design: 
Our system separates business logic design from value determination:
- Configuration Logic (business_configuration_logic.json): Templates designed by data engineers with sample_value for scalars and actual formulas for business logic
- Configuration Values (business_configuration.json): Realistic values determined by domain experts for scalar parameters only
- Design Rationale: Ensures business logic consistency while allowing flexible parameter tuning


TASK: Create structured markdown documentation for SECTIONS 1-3 ONLY (Problem Description).

EXACT MARKDOWN STRUCTURE TO FOLLOW:

# Complete Optimization Problem and Solution: manufacturer

## 1. Problem Context and Goals

### Context  
[Regenerate business context that naturally aligns with LINEAR optimization formulation. Ensure:]
- Business decisions match expected decision variables: Quantity_Produced[Furniture_ID] (integer)
- Operational parameters align with expected linear objective: maximize ∑((Market_Rate - Price_in_Dollar) × Quantity_Produced)
- Business configuration includes: Total production capacity of all factories (used for constraint bound), Total budget available for production (used for constraint bound)
- Use natural language to precisely describe linear mathematical relationships
- NO mathematical formulas, equations, or symbolic notation
- Present data as current operational information
- Focus on precise operational decision-making that leads to linear formulations
- Resource limitations match expected linear constraints
- Avoid scenarios requiring variable products, divisions, or other nonlinear relationships
- Include specific operational parameters that map to expected coefficient sources
- Reference business configuration parameters where appropriate
- CRITICAL: Include ALL business configuration information (scalar parameters AND business logic formulas) in natural business language

### Goals  
[Regenerate goals that clearly lead to LINEAR mathematical objective:]
- Optimization goal: maximize
- Metric to optimize: maximize ∑((Market_Rate - Price_in_Dollar) × Quantity_Produced)
- Success measurement aligned with expected coefficient sources
- Use natural language to precisely describe linear optimization goal
- NO mathematical formulas, equations, or symbolic notation

## 2. Constraints    

[Regenerate constraints that directly match expected LINEAR mathematical constraints:]
- Expected constraint: ['∑(Quantity_Produced[Furniture_ID]) ≤ Total_Factory_Capacity', '∑(Price_in_Dollar × Quantity_Produced[Furniture_ID]) ≤ Budget_Limit'] (Form: Standard constraint form based on business requirements)

[Each constraint should be described in business terms that naturally lead to LINEAR mathematical forms (no variable products or divisions)]

## 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  
[Create comprehensive business-oriented data dictionary mapping tables and columns to their business purposes and optimization roles - NOT technical database terms. Base this on the data_dictionary provided but rewrite in business language that clearly connects to the expected linear mathematical formulation]

### Current Stored Values  
```sql
-- Iteration 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical furniture manufacturing scenarios, considering production costs, market rates, and factory capacities. The data ensures a balanced optimization problem with feasible constraints.

-- Realistic data for production_plan
INSERT INTO production_plan (Furniture_ID, Quantity_Produced) VALUES (1, 0);
INSERT INTO production_plan (Furniture_ID, Quantity_Produced) VALUES (2, 0);
INSERT INTO production_plan (Furniture_ID, Quantity_Produced) VALUES (3, 0);

-- Realistic data for furniture_market_rates
INSERT INTO furniture_market_rates (Furniture_ID, Market_Rate) VALUES (1, 120.0);
INSERT INTO furniture_market_rates (Furniture_ID, Market_Rate) VALUES (2, 180.0);
INSERT INTO furniture_market_rates (Furniture_ID, Market_Rate) VALUES (3, 250.0);

-- Realistic data for furniture_production_costs
INSERT INTO furniture_production_costs (Furniture_ID, Price_in_Dollar) VALUES (1, 60.0);
INSERT INTO furniture_production_costs (Furniture_ID, Price_in_Dollar) VALUES (2, 90.0);
INSERT INTO furniture_production_costs (Furniture_ID, Price_in_Dollar) VALUES (3, 130.0);


```

STOP HERE - Do not include section 4 (Mathematical Optimization Formulation).

Response:
# 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.  

### Current Stored Values  
```sql
-- Iteration 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical furniture manufacturing scenarios, considering production costs, market rates, and factory capacities. The data ensures a balanced optimization problem with feasible constraints.

-- Realistic data for production_plan
INSERT INTO production_plan (Furniture_ID, Quantity_Produced) VALUES (1, 0);
INSERT INTO production_plan (Furniture_ID, Quantity_Produced) VALUES (2, 0);
INSERT INTO production_plan (Furniture_ID, Quantity_Produced) VALUES (3, 0);

-- Realistic data for furniture_market_rates
INSERT INTO furniture_market_rates (Furniture_ID, Market_Rate) VALUES (1, 120.0);
INSERT INTO furniture_market_rates (Furniture_ID, Market_Rate) VALUES (2, 180.0);
INSERT INTO furniture_market_rates (Furniture_ID, Market_Rate) VALUES (3, 250.0);

-- Realistic data for furniture_production_costs
INSERT INTO furniture_production_costs (Furniture_ID, Price_in_Dollar) VALUES (1, 60.0);
INSERT INTO furniture_production_costs (Furniture_ID, Price_in_Dollar) VALUES (2, 90.0);
INSERT INTO furniture_production_costs (Furniture_ID, Price_in_Dollar) VALUES (3, 130.0);
```
