Iteration final - PROBLEM_DESCRIPTION
Sequence: 5
Timestamp: 2025-07-25 22:31:04

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 1), 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: Num_of_Ticket[Museum_ID] (integer)
- Operational parameters align with expected linear objective: maximize ∑(Total_spent[Museum_ID] × Num_of_Ticket[Museum_ID])
- Business configuration includes: Maximum number of tickets that can be sold at each museum (used for Upper bound constraint for ticket sales), Maximum number of visitors that can be handled by the staff at each museum (used for Upper bound constraint for ticket sales based on staff availability)
- 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": "museum_visit",
  "iteration": 1,
  "business_context": "A museum chain aims to maximize its total revenue from ticket sales across all museums while ensuring that the number of staff members is sufficient to handle the visitors and that the number of tickets sold does not exceed a certain limit per museum.",
  "optimization_problem_description": "Determine the optimal number of tickets to sell at each museum to maximize total revenue, considering constraints on staff availability and ticket limits.",
  "optimization_formulation": {
    "objective": "maximize \u2211(Total_spent[Museum_ID] \u00d7 Num_of_Ticket[Museum_ID])",
    "decision_variables": "Num_of_Ticket[Museum_ID] (integer)",
    "constraints": "Num_of_Ticket[Museum_ID] \u2264 Total_Ticket_Limit[Museum_ID], Num_of_Ticket[Museum_ID] \u2264 Staff_Capacity[Museum_ID]"
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "Total_spent[Museum_ID]": {
        "currently_mapped_to": "visit.Total_spent",
        "mapping_adequacy": "good",
        "description": "Revenue generated from ticket sales at each museum"
      }
    },
    "constraint_bounds": {
      "Total_Ticket_Limit[Museum_ID]": {
        "currently_mapped_to": "museum_constraints.Total_Ticket_Limit",
        "mapping_adequacy": "good",
        "description": "Maximum number of tickets that can be sold at each museum"
      },
      "Staff_Capacity[Museum_ID]": {
        "currently_mapped_to": "museum_constraints.Staff_Capacity",
        "mapping_adequacy": "good",
        "description": "Maximum number of visitors that can be handled by the staff at each museum"
      }
    },
    "decision_variables": {
      "Num_of_Ticket[Museum_ID]": {
        "currently_mapped_to": "visit.Num_of_Ticket",
        "mapping_adequacy": "good",
        "description": "Number of tickets sold at each museum",
        "variable_type": "integer"
      }
    }
  },
  "missing_optimization_requirements": [],
  "iteration_status": {
    "complete": true,
    "confidence": "high",
    "next_focus": "Ready for convergence"
  }
}

FINAL DATABASE SCHEMA:
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating tables for missing constraints (Total_Ticket_Limit and Staff_Capacity) and updating business configuration logic for scalar parameters and formulas.

CREATE TABLE museum_constraints (
  Museum_ID INTEGER,
  Total_Ticket_Limit INTEGER,
  Staff_Capacity INTEGER
);

CREATE TABLE visit (
  Museum_ID INTEGER,
  Num_of_Ticket INTEGER,
  Total_spent FLOAT
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical museum capacities, staff-to-visitor ratios, and ticket pricing strategies to ensure realistic and meaningful optimization.

-- Realistic data for museum_constraints
INSERT INTO museum_constraints (Museum_ID, Total_Ticket_Limit, Staff_Capacity) VALUES (1, 1200, 600);
INSERT INTO museum_constraints (Museum_ID, Total_Ticket_Limit, Staff_Capacity) VALUES (2, 1500, 750);
INSERT INTO museum_constraints (Museum_ID, Total_Ticket_Limit, Staff_Capacity) VALUES (3, 1000, 500);

-- Realistic data for visit
INSERT INTO visit (Museum_ID, Num_of_Ticket, Total_spent) VALUES (1, 500, 2500.0);
INSERT INTO visit (Museum_ID, Num_of_Ticket, Total_spent) VALUES (2, 700, 3500.0);
INSERT INTO visit (Museum_ID, Num_of_Ticket, Total_spent) VALUES (3, 300, 1500.0);


```

DATA DICTIONARY:
{
  "tables": {
    "museum_constraints": {
      "business_purpose": "Maximum ticket limits and staff capacity for each museum",
      "optimization_role": "constraint_bounds",
      "columns": {
        "Museum_ID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each museum",
          "optimization_purpose": "Index for constraints",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "Total_Ticket_Limit": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum number of tickets that can be sold at each museum",
          "optimization_purpose": "Upper bound constraint for ticket sales",
          "sample_values": [
            1000,
            1500,
            2000
          ]
        },
        "Staff_Capacity": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum number of visitors that can be handled by the staff at each museum",
          "optimization_purpose": "Upper bound constraint for ticket sales based on staff availability",
          "sample_values": [
            500,
            750,
            1000
          ]
        }
      }
    },
    "visit": {
      "business_purpose": "Revenue generated from ticket sales at each museum",
      "optimization_role": "objective_coefficients",
      "columns": {
        "Museum_ID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each museum",
          "optimization_purpose": "Index for objective coefficients",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "Num_of_Ticket": {
          "data_type": "INTEGER",
          "business_meaning": "Number of tickets sold at each museum",
          "optimization_purpose": "Decision variable for ticket sales",
          "sample_values": [
            200,
            300,
            400
          ]
        },
        "Total_spent": {
          "data_type": "FLOAT",
          "business_meaning": "Revenue generated from ticket sales at each museum",
          "optimization_purpose": "Objective coefficient for revenue maximization",
          "sample_values": [
            1000.0,
            1500.0,
            2000.0
          ]
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "Total_Ticket_Limit": {
    "data_type": "INTEGER",
    "business_meaning": "Maximum number of tickets that can be sold at each museum",
    "optimization_role": "Upper bound constraint for ticket sales",
    "configuration_type": "scalar_parameter",
    "value": 1500,
    "business_justification": "This value represents a reasonable upper limit for ticket sales across all museums, ensuring that no museum exceeds its capacity."
  },
  "Staff_Capacity": {
    "data_type": "INTEGER",
    "business_meaning": "Maximum number of visitors that can be handled by the staff at each museum",
    "optimization_role": "Upper bound constraint for ticket sales based on staff availability",
    "configuration_type": "scalar_parameter",
    "value": 750,
    "business_justification": "This value ensures that staff can handle the maximum number of visitors, maintaining a safe and manageable environment."
  }
}

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: museum_visit

## 1. Problem Context and Goals

### Context  
[Regenerate business context that naturally aligns with LINEAR optimization formulation. Ensure:]
- Business decisions match expected decision variables: Num_of_Ticket[Museum_ID] (integer)
- Operational parameters align with expected linear objective: maximize ∑(Total_spent[Museum_ID] × Num_of_Ticket[Museum_ID])
- Business configuration includes: Maximum number of tickets that can be sold at each museum (used for Upper bound constraint for ticket sales), Maximum number of visitors that can be handled by the staff at each museum (used for Upper bound constraint for ticket sales based on staff availability)
- 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 ∑(Total_spent[Museum_ID] × Num_of_Ticket[Museum_ID])
- 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: Num_of_Ticket[Museum_ID] ≤ Total_Ticket_Limit[Museum_ID], Num_of_Ticket[Museum_ID] ≤ Staff_Capacity[Museum_ID] (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 1 Database Schema
-- Objective: Schema changes include creating tables for missing constraints (Total_Ticket_Limit and Staff_Capacity) and updating business configuration logic for scalar parameters and formulas.

CREATE TABLE museum_constraints (
  Museum_ID INTEGER,
  Total_Ticket_Limit INTEGER,
  Staff_Capacity INTEGER
);

CREATE TABLE visit (
  Museum_ID INTEGER,
  Num_of_Ticket INTEGER,
  Total_spent 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 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical museum capacities, staff-to-visitor ratios, and ticket pricing strategies to ensure realistic and meaningful optimization.

-- Realistic data for museum_constraints
INSERT INTO museum_constraints (Museum_ID, Total_Ticket_Limit, Staff_Capacity) VALUES (1, 1200, 600);
INSERT INTO museum_constraints (Museum_ID, Total_Ticket_Limit, Staff_Capacity) VALUES (2, 1500, 750);
INSERT INTO museum_constraints (Museum_ID, Total_Ticket_Limit, Staff_Capacity) VALUES (3, 1000, 500);

-- Realistic data for visit
INSERT INTO visit (Museum_ID, Num_of_Ticket, Total_spent) VALUES (1, 500, 2500.0);
INSERT INTO visit (Museum_ID, Num_of_Ticket, Total_spent) VALUES (2, 700, 3500.0);
INSERT INTO visit (Museum_ID, Num_of_Ticket, Total_spent) VALUES (3, 300, 1500.0);


```

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

Response:
# Complete Optimization Problem and Solution: museum_visit

## 1. Problem Context and Goals

### Context  
A museum chain is focused on maximizing its total revenue from ticket sales across all its locations. The key decision involves determining the optimal number of tickets to sell at each museum. This decision must consider two critical operational constraints:  
1. **Ticket Sales Limit**: Each museum has a maximum number of tickets it can sell, ensuring that ticket sales do not exceed the museum's capacity.  
2. **Staff Availability**: Each museum has a maximum number of visitors that its staff can effectively manage, ensuring a safe and enjoyable experience for all guests.  

The business configuration includes:  
- **Total_Ticket_Limit**: The maximum number of tickets that can be sold at each museum, which serves as an upper bound for ticket sales.  
- **Staff_Capacity**: The maximum number of visitors that can be handled by the staff at each museum, which also serves as an upper bound for ticket sales based on staff availability.  

The goal is to make decisions that align with these constraints while maximizing revenue. The revenue generated at each museum is directly proportional to the number of tickets sold, ensuring a linear relationship between the decision variables and the objective.  

### Goals  
The primary optimization goal is to maximize the total revenue generated from ticket sales across all museums. This is achieved by determining the optimal number of tickets to sell at each museum, considering the constraints on ticket sales and staff availability. Success is measured by the total revenue, which is calculated as the sum of the revenue generated at each museum. The revenue at each museum is determined by multiplying the number of tickets sold by the revenue per ticket, ensuring a straightforward and linear relationship.  

## 2. Constraints  

The optimization problem is subject to the following constraints:  
1. **Ticket Sales Limit**: The number of tickets sold at each museum cannot exceed the maximum number of tickets that the museum is allowed to sell. This ensures that ticket sales remain within the museum's capacity.  
2. **Staff Availability**: The number of tickets sold at each museum cannot exceed the maximum number of visitors that the museum's staff can handle. This ensures that the staff can effectively manage the number of visitors, maintaining a safe and enjoyable environment.  

These constraints are designed to ensure that the optimization problem remains linear, with no variable products or divisions involved.  

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating tables for missing constraints (Total_Ticket_Limit and Staff_Capacity) and updating business configuration logic for scalar parameters and formulas.

CREATE TABLE museum_constraints (
  Museum_ID INTEGER,
  Total_Ticket_Limit INTEGER,
  Staff_Capacity INTEGER
);

CREATE TABLE visit (
  Museum_ID INTEGER,
  Num_of_Ticket INTEGER,
  Total_spent FLOAT
);
```

### Data Dictionary  
- **museum_constraints**: This table contains the maximum ticket limits and staff capacity for each museum.  
  - **Museum_ID**: A unique identifier for each museum.  
  - **Total_Ticket_Limit**: The maximum number of tickets that can be sold at each museum.  
  - **Staff_Capacity**: The maximum number of visitors that can be handled by the staff at each museum.  

- **visit**: This table contains the revenue generated from ticket sales at each museum.  
  - **Museum_ID**: A unique identifier for each museum.  
  - **Num_of_Ticket**: The number of tickets sold at each museum.  
  - **Total_spent**: The revenue generated from ticket sales at each museum.  

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical museum capacities, staff-to-visitor ratios, and ticket pricing strategies to ensure realistic and meaningful optimization.

-- Realistic data for museum_constraints
INSERT INTO museum_constraints (Museum_ID, Total_Ticket_Limit, Staff_Capacity) VALUES (1, 1200, 600);
INSERT INTO museum_constraints (Museum_ID, Total_Ticket_Limit, Staff_Capacity) VALUES (2, 1500, 750);
INSERT INTO museum_constraints (Museum_ID, Total_Ticket_Limit, Staff_Capacity) VALUES (3, 1000, 500);

-- Realistic data for visit
INSERT INTO visit (Museum_ID, Num_of_Ticket, Total_spent) VALUES (1, 500, 2500.0);
INSERT INTO visit (Museum_ID, Num_of_Ticket, Total_spent) VALUES (2, 700, 3500.0);
INSERT INTO visit (Museum_ID, Num_of_Ticket, Total_spent) VALUES (3, 300, 1500.0);
```
