Iteration final - PROBLEM_DESCRIPTION
Sequence: 5
Timestamp: 2025-07-25 22:33:14

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: AcceptReservation: binary decision variable indicating whether to accept (1) or reject (0) a reservation
- Operational parameters align with expected linear objective: maximize ∑(Rate × AcceptReservation)
- Business configuration includes: Maximum number of guests allowed in a room (used for Constraint bound for room capacity)
- Business logic formulas to express in natural language: Detects overlapping reservation dates (calculation method for Constraint to prevent double-booking)
- 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": "inn_1",
  "iteration": 1,
  "business_context": "Optimize room allocation to maximize revenue while respecting room capacities and reservation constraints, ensuring linearity in the optimization model.",
  "optimization_problem_description": "Maximize total revenue from room reservations by deciding which reservations to accept, ensuring that the number of guests does not exceed room capacities and that each room is not double-booked, using a linear formulation.",
  "optimization_formulation": {
    "objective": "maximize \u2211(Rate \u00d7 AcceptReservation)",
    "decision_variables": "AcceptReservation: binary decision variable indicating whether to accept (1) or reject (0) a reservation",
    "constraints": "1. \u2211(AcceptReservation) \u2264 maxOccupancy (room capacity constraint), 2. AcceptReservation[i] + AcceptReservation[j] \u2264 1 for all overlapping reservations i and j (date overlap constraint)"
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "Rate": {
        "currently_mapped_to": "Reservations.Rate",
        "mapping_adequacy": "good",
        "description": "Revenue generated from each reservation, used as the coefficient in the objective function"
      }
    },
    "constraint_bounds": {
      "maxOccupancy": {
        "currently_mapped_to": "business_configuration_logic.maxOccupancy",
        "mapping_adequacy": "good",
        "description": "Maximum number of guests allowed in a room, used as the bound for the room capacity constraint"
      }
    },
    "decision_variables": {
      "AcceptReservation": {
        "currently_mapped_to": "ReservationDecisions.AcceptReservation",
        "mapping_adequacy": "good",
        "description": "Binary decision variable indicating whether to accept or reject a reservation",
        "variable_type": "binary"
      }
    }
  },
  "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 adding a new table for decision variables and updating configuration logic to handle date overlap constraints. Business configuration logic now includes scalar parameters and formulas for optimization.

CREATE TABLE ReservationDecisions (
  AcceptReservation BOOLEAN
);

CREATE TABLE Reservations (
  Rate FLOAT,
  AcceptReservation BOOLEAN
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical hotel room capacities, reservation rates, and realistic booking scenarios to ensure the optimization problem is meaningful and solvable.

-- Realistic data for ReservationDecisions
INSERT INTO ReservationDecisions (AcceptReservation) VALUES (1);
INSERT INTO ReservationDecisions (AcceptReservation) VALUES (0);
INSERT INTO ReservationDecisions (AcceptReservation) VALUES (1);

-- Realistic data for Reservations
INSERT INTO Reservations (Rate, AcceptReservation) VALUES (120.0, 1);
INSERT INTO Reservations (Rate, AcceptReservation) VALUES (100.0, 0);
INSERT INTO Reservations (Rate, AcceptReservation) VALUES (150.0, 1);
INSERT INTO Reservations (Rate, AcceptReservation) VALUES (110.0, 1);
INSERT INTO Reservations (Rate, AcceptReservation) VALUES (130.0, 0);


```

DATA DICTIONARY:
{
  "tables": {
    "ReservationDecisions": {
      "business_purpose": "Stores the decision to accept or reject each reservation",
      "optimization_role": "decision_variables",
      "columns": {
        "AcceptReservation": {
          "data_type": "BOOLEAN",
          "business_meaning": "Decision to accept or reject a reservation",
          "optimization_purpose": "Binary decision variable in optimization model",
          "sample_values": "0, 1"
        }
      }
    },
    "Reservations": {
      "business_purpose": "Stores details of each reservation",
      "optimization_role": "business_data",
      "columns": {
        "Rate": {
          "data_type": "FLOAT",
          "business_meaning": "Revenue generated from each reservation",
          "optimization_purpose": "Coefficient in the objective function",
          "sample_values": "100.0, 150.0"
        },
        "AcceptReservation": {
          "data_type": "BOOLEAN",
          "business_meaning": "Decision to accept or reject a reservation",
          "optimization_purpose": "Binary decision variable in optimization model",
          "sample_values": "0, 1"
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "maxOccupancy": {
    "data_type": "INTEGER",
    "business_meaning": "Maximum number of guests allowed in a room",
    "optimization_role": "Constraint bound for room capacity",
    "configuration_type": "scalar_parameter",
    "value": 4,
    "business_justification": "This value represents a typical maximum occupancy for a standard hotel room, ensuring realistic capacity constraints."
  },
  "dateOverlapFormula": {
    "data_type": "STRING",
    "business_meaning": "Detects overlapping reservation dates",
    "optimization_role": "Constraint to prevent double-booking",
    "configuration_type": "business_logic_formula",
    "formula_expression": "CheckIn[Code1] < CheckOut[Code2] && CheckIn[Code2] < CheckOut[Code1]"
  }
}

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: inn_1

## 1. Problem Context and Goals

### Context  
[Regenerate business context that naturally aligns with LINEAR optimization formulation. Ensure:]
- Business decisions match expected decision variables: AcceptReservation: binary decision variable indicating whether to accept (1) or reject (0) a reservation
- Operational parameters align with expected linear objective: maximize ∑(Rate × AcceptReservation)
- Business configuration includes: Maximum number of guests allowed in a room (used for Constraint bound for room capacity)
- Business logic formulas to express in natural language: Detects overlapping reservation dates (calculation method for Constraint to prevent double-booking)
- 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 ∑(Rate × AcceptReservation)
- 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: 1. ∑(AcceptReservation) ≤ maxOccupancy (room capacity constraint), 2. AcceptReservation[i] + AcceptReservation[j] ≤ 1 for all overlapping reservations i and j (date overlap constraint) (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 adding a new table for decision variables and updating configuration logic to handle date overlap constraints. Business configuration logic now includes scalar parameters and formulas for optimization.

CREATE TABLE ReservationDecisions (
  AcceptReservation BOOLEAN
);

CREATE TABLE Reservations (
  Rate FLOAT,
  AcceptReservation BOOLEAN
);


```

### 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 hotel room capacities, reservation rates, and realistic booking scenarios to ensure the optimization problem is meaningful and solvable.

-- Realistic data for ReservationDecisions
INSERT INTO ReservationDecisions (AcceptReservation) VALUES (1);
INSERT INTO ReservationDecisions (AcceptReservation) VALUES (0);
INSERT INTO ReservationDecisions (AcceptReservation) VALUES (1);

-- Realistic data for Reservations
INSERT INTO Reservations (Rate, AcceptReservation) VALUES (120.0, 1);
INSERT INTO Reservations (Rate, AcceptReservation) VALUES (100.0, 0);
INSERT INTO Reservations (Rate, AcceptReservation) VALUES (150.0, 1);
INSERT INTO Reservations (Rate, AcceptReservation) VALUES (110.0, 1);
INSERT INTO Reservations (Rate, AcceptReservation) VALUES (130.0, 0);


```

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

Response:
# Complete Optimization Problem and Solution: inn_1

## 1. Problem Context and Goals

### Context  
The business operates a hotel and seeks to optimize room allocation decisions to maximize revenue while adhering to operational constraints. The primary decision involves determining whether to accept or reject each reservation request. Each reservation has an associated rate, which represents the revenue generated if the reservation is accepted. The hotel has a fixed maximum occupancy per room, ensuring that the number of guests does not exceed the room's capacity. Additionally, the system must prevent double-booking by ensuring that no two reservations for the same room overlap in their check-in and check-out dates.  

The business configuration includes a maximum occupancy limit of 4 guests per room, which serves as a constraint on the total number of guests that can be accommodated. The system also employs a formula to detect overlapping reservation dates, ensuring that only one reservation is accepted for any given room during the same time period. These constraints are designed to maintain operational feasibility while maximizing revenue.  

The optimization problem is formulated linearly, focusing on straightforward relationships between decision variables, revenue, and constraints. This ensures that the problem remains computationally tractable and aligned with business needs.  

### Goals  
The primary goal of this optimization problem is to maximize the total revenue generated from accepted reservations. This is achieved by strategically deciding which reservations to accept, based on their associated rates, while ensuring that room capacities are not exceeded and that no double-booking occurs. Success is measured by the total revenue generated, which is directly tied to the rates of the accepted reservations. The optimization process aims to make decisions that align with these objectives, ensuring that the hotel operates efficiently and profitably.  

## 2. Constraints  

The optimization problem is subject to the following constraints:  

1. **Room Capacity Constraint**: The total number of guests across all accepted reservations for a room must not exceed the maximum occupancy limit of 4 guests. This ensures that the room is not overbooked and that guest comfort and safety are maintained.  

2. **Date Overlap Constraint**: For any two reservations that overlap in their check-in and check-out dates, only one of them can be accepted. This prevents double-booking and ensures that each room is allocated to only one reservation at any given time.  

These constraints are designed to reflect real-world operational limitations and are expressed in a way that naturally leads to a linear optimization formulation.  

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include adding a new table for decision variables and updating configuration logic to handle date overlap constraints. Business configuration logic now includes scalar parameters and formulas for optimization.

CREATE TABLE ReservationDecisions (
  AcceptReservation BOOLEAN
);

CREATE TABLE Reservations (
  Rate FLOAT,
  AcceptReservation BOOLEAN
);
```

### Data Dictionary  
- **ReservationDecisions Table**:  
  - **Purpose**: Stores the decision to accept or reject each reservation.  
  - **AcceptReservation Column**:  
    - **Business Meaning**: Indicates whether a reservation is accepted (1) or rejected (0).  
    - **Optimization Role**: Serves as the binary decision variable in the optimization model.  

- **Reservations Table**:  
  - **Purpose**: Stores details of each reservation, including the revenue it generates.  
  - **Rate Column**:  
    - **Business Meaning**: The revenue generated from each reservation.  
    - **Optimization Role**: Used as the coefficient in the objective function to maximize total revenue.  
  - **AcceptReservation Column**:  
    - **Business Meaning**: Indicates whether a reservation is accepted (1) or rejected (0).  
    - **Optimization Role**: Serves as the binary decision variable in the optimization model.  

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical hotel room capacities, reservation rates, and realistic booking scenarios to ensure the optimization problem is meaningful and solvable.

-- Realistic data for ReservationDecisions
INSERT INTO ReservationDecisions (AcceptReservation) VALUES (1);
INSERT INTO ReservationDecisions (AcceptReservation) VALUES (0);
INSERT INTO ReservationDecisions (AcceptReservation) VALUES (1);

-- Realistic data for Reservations
INSERT INTO Reservations (Rate, AcceptReservation) VALUES (120.0, 1);
INSERT INTO Reservations (Rate, AcceptReservation) VALUES (100.0, 0);
INSERT INTO Reservations (Rate, AcceptReservation) VALUES (150.0, 1);
INSERT INTO Reservations (Rate, AcceptReservation) VALUES (110.0, 1);
INSERT INTO Reservations (Rate, AcceptReservation) VALUES (130.0, 0);
```
