Iteration final - PROBLEM_DESCRIPTION
Sequence: 7
Timestamp: 2025-07-25 22:40:05

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: x_ij: number of pieces member i performs in performance j (integer)
- Operational parameters align with expected linear objective: maximize ∑(Attendance_j × ∑(x_ij)) where x_ij is the number of pieces member i performs in performance j
- Business configuration includes: Attendance for performance j (used for Objective coefficient for attendance in optimization model)
- 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": "performance_attendance",
  "iteration": 2,
  "business_context": "A music ensemble aims to optimize the number of pieces performed by each member across various performances to maximize overall attendance, while ensuring fair distribution of pieces among members and respecting performance capacity constraints.",
  "optimization_problem_description": "Maximize the total attendance across all performances by optimizing the number of pieces each member performs, subject to constraints on the maximum number of pieces a member can perform, the total number of pieces per performance, and ensuring each member is assigned at least one piece.",
  "optimization_formulation": {
    "objective": "maximize \u2211(Attendance_j \u00d7 \u2211(x_ij)) where x_ij is the number of pieces member i performs in performance j",
    "decision_variables": "x_ij: number of pieces member i performs in performance j (integer)",
    "constraints": [
      "\u2211(x_ij) \u2264 max_pieces_i for each member i (maximum pieces per member)",
      "\u2211(x_ij) \u2264 max_pieces_j for each performance j (maximum pieces per performance)",
      "\u2211(x_ij) \u2265 1 for each member i (minimum pieces per member)"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "Attendance_j[j]": {
        "currently_mapped_to": "attendance_data.attendance",
        "mapping_adequacy": "good",
        "description": "Attendance for performance j, used as coefficient in the objective function"
      }
    },
    "constraint_bounds": {
      "max_pieces_i[i]": {
        "currently_mapped_to": "member_constraints.max_pieces",
        "mapping_adequacy": "good",
        "description": "Maximum number of pieces member i can perform"
      },
      "max_pieces_j[j]": {
        "currently_mapped_to": "performance_constraints.max_pieces",
        "mapping_adequacy": "good",
        "description": "Maximum number of pieces allowed in performance j"
      }
    },
    "decision_variables": {
      "x_ij[i,j]": {
        "currently_mapped_to": "member_attendance.num_of_pieces",
        "mapping_adequacy": "good",
        "description": "Number of pieces member i performs in performance j",
        "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: Added attendance_data table to capture attendance per performance, updated business configuration logic with attendance-related scalar parameters, and ensured all mappings align with OR expert requirements.

CREATE TABLE member_constraints (
  member_id INTEGER,
  max_pieces INTEGER
);

CREATE TABLE performance_constraints (
  performance_id INTEGER,
  max_pieces INTEGER
);

CREATE TABLE member_attendance (
  member_id INTEGER,
  performance_id INTEGER,
  num_of_pieces INTEGER
);

CREATE TABLE attendance_data (
  performance_id INTEGER,
  attendance INTEGER
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic scenarios for a music ensemble, considering typical attendance numbers, member capabilities, and performance constraints. The data ensures a balanced distribution of pieces among members and performances, while respecting the optimization constraints.

-- Realistic data for member_constraints
INSERT INTO member_constraints (member_id, max_pieces) VALUES (1, 5);
INSERT INTO member_constraints (member_id, max_pieces) VALUES (2, 4);
INSERT INTO member_constraints (member_id, max_pieces) VALUES (3, 3);

-- Realistic data for performance_constraints
INSERT INTO performance_constraints (performance_id, max_pieces) VALUES (1, 10);
INSERT INTO performance_constraints (performance_id, max_pieces) VALUES (2, 8);
INSERT INTO performance_constraints (performance_id, max_pieces) VALUES (3, 6);

-- Realistic data for member_attendance
INSERT INTO member_attendance (member_id, performance_id, num_of_pieces) VALUES (1, 1, 3);
INSERT INTO member_attendance (member_id, performance_id, num_of_pieces) VALUES (2, 1, 2);
INSERT INTO member_attendance (member_id, performance_id, num_of_pieces) VALUES (3, 1, 1);

-- Realistic data for attendance_data
INSERT INTO attendance_data (performance_id, attendance) VALUES (1, 200);
INSERT INTO attendance_data (performance_id, attendance) VALUES (2, 150);
INSERT INTO attendance_data (performance_id, attendance) VALUES (3, 100);


```

DATA DICTIONARY:
{
  "tables": {
    "member_constraints": {
      "business_purpose": "Maximum number of pieces each member can perform",
      "optimization_role": "constraint_bounds",
      "columns": {
        "member_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each member",
          "optimization_purpose": "Links to member_attendance table",
          "sample_values": "1, 2, 3"
        },
        "max_pieces": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum number of pieces the member can perform",
          "optimization_purpose": "Constraint bound for maximum pieces per member",
          "sample_values": "5, 6, 7"
        }
      }
    },
    "performance_constraints": {
      "business_purpose": "Maximum number of pieces allowed per performance",
      "optimization_role": "constraint_bounds",
      "columns": {
        "performance_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each performance",
          "optimization_purpose": "Links to member_attendance table",
          "sample_values": "1, 2, 3"
        },
        "max_pieces": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum number of pieces allowed in the performance",
          "optimization_purpose": "Constraint bound for maximum pieces per performance",
          "sample_values": "10, 12, 15"
        }
      }
    },
    "member_attendance": {
      "business_purpose": "Number of pieces each member performs in each performance",
      "optimization_role": "decision_variables",
      "columns": {
        "member_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each member",
          "optimization_purpose": "Links to member_constraints table",
          "sample_values": "1, 2, 3"
        },
        "performance_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each performance",
          "optimization_purpose": "Links to performance_constraints table",
          "sample_values": "1, 2, 3"
        },
        "num_of_pieces": {
          "data_type": "INTEGER",
          "business_meaning": "Number of pieces the member performs in the performance",
          "optimization_purpose": "Decision variable in optimization model",
          "sample_values": "1, 2, 3"
        }
      }
    },
    "attendance_data": {
      "business_purpose": "Attendance for each performance, used in the optimization objective function",
      "optimization_role": "objective_coefficients",
      "columns": {
        "performance_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each performance",
          "optimization_purpose": "Links to performance_constraints table",
          "sample_values": "1, 2, 3"
        },
        "attendance": {
          "data_type": "INTEGER",
          "business_meaning": "Attendance for the performance",
          "optimization_purpose": "Objective coefficient in optimization model",
          "sample_values": "100, 150, 200"
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "Attendance_j": {
    "data_type": "INTEGER",
    "business_meaning": "Attendance for performance j",
    "optimization_role": "Objective coefficient for attendance in optimization model",
    "configuration_type": "scalar_parameter",
    "value": 150,
    "business_justification": "The average attendance across performances is used as a scalar parameter to simplify the objective function."
  }
}

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: performance_attendance

## 1. Problem Context and Goals

### Context  
[Regenerate business context that naturally aligns with LINEAR optimization formulation. Ensure:]
- Business decisions match expected decision variables: x_ij: number of pieces member i performs in performance j (integer)
- Operational parameters align with expected linear objective: maximize ∑(Attendance_j × ∑(x_ij)) where x_ij is the number of pieces member i performs in performance j
- Business configuration includes: Attendance for performance j (used for Objective coefficient for attendance in optimization model)
- 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 ∑(Attendance_j × ∑(x_ij)) where x_ij is the number of pieces member i performs in performance j
- 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: ['∑(x_ij) ≤ max_pieces_i for each member i (maximum pieces per member)', '∑(x_ij) ≤ max_pieces_j for each performance j (maximum pieces per performance)', '∑(x_ij) ≥ 1 for each member i (minimum pieces per member)'] (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: Added attendance_data table to capture attendance per performance, updated business configuration logic with attendance-related scalar parameters, and ensured all mappings align with OR expert requirements.

CREATE TABLE member_constraints (
  member_id INTEGER,
  max_pieces INTEGER
);

CREATE TABLE performance_constraints (
  performance_id INTEGER,
  max_pieces INTEGER
);

CREATE TABLE member_attendance (
  member_id INTEGER,
  performance_id INTEGER,
  num_of_pieces INTEGER
);

CREATE TABLE attendance_data (
  performance_id INTEGER,
  attendance INTEGER
);


```

### 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 realistic scenarios for a music ensemble, considering typical attendance numbers, member capabilities, and performance constraints. The data ensures a balanced distribution of pieces among members and performances, while respecting the optimization constraints.

-- Realistic data for member_constraints
INSERT INTO member_constraints (member_id, max_pieces) VALUES (1, 5);
INSERT INTO member_constraints (member_id, max_pieces) VALUES (2, 4);
INSERT INTO member_constraints (member_id, max_pieces) VALUES (3, 3);

-- Realistic data for performance_constraints
INSERT INTO performance_constraints (performance_id, max_pieces) VALUES (1, 10);
INSERT INTO performance_constraints (performance_id, max_pieces) VALUES (2, 8);
INSERT INTO performance_constraints (performance_id, max_pieces) VALUES (3, 6);

-- Realistic data for member_attendance
INSERT INTO member_attendance (member_id, performance_id, num_of_pieces) VALUES (1, 1, 3);
INSERT INTO member_attendance (member_id, performance_id, num_of_pieces) VALUES (2, 1, 2);
INSERT INTO member_attendance (member_id, performance_id, num_of_pieces) VALUES (3, 1, 1);

-- Realistic data for attendance_data
INSERT INTO attendance_data (performance_id, attendance) VALUES (1, 200);
INSERT INTO attendance_data (performance_id, attendance) VALUES (2, 150);
INSERT INTO attendance_data (performance_id, attendance) VALUES (3, 100);


```

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

Response:
# Complete Optimization Problem and Solution: performance_attendance

## 1. Problem Context and Goals

### Context  
A music ensemble is focused on optimizing the number of pieces each member performs across various performances to maximize overall attendance. The ensemble must make decisions about how many pieces each member should perform in each performance, ensuring that these decisions align with the goal of attracting the largest possible audience. The attendance for each performance is a key factor in this optimization, as it directly influences the overall success of the ensemble's performances.  

The ensemble operates under specific constraints: each member has a maximum number of pieces they can perform, and each performance has a maximum number of pieces that can be included. Additionally, every member must perform at least one piece to ensure fair participation. The attendance for each performance is a known value and is used to weight the importance of each performance in the optimization process.  

The business configuration includes a scalar parameter for attendance, which represents the average attendance across performances. This parameter simplifies the objective function by providing a consistent measure of attendance impact.  

### Goals  
The primary goal of this optimization is to maximize the total attendance across all performances by strategically assigning the number of pieces each member performs. Success is measured by the overall attendance achieved, which is directly influenced by the number of pieces performed by each member in each performance. The optimization ensures that the ensemble's performances are as attractive as possible to the audience while respecting the operational constraints of the members and performances.  

## 2. Constraints  

The optimization must adhere to the following constraints:  
1. **Maximum Pieces per Member**: Each member cannot perform more than a specified maximum number of pieces across all performances. This ensures that no member is overburdened.  
2. **Maximum Pieces per Performance**: Each performance cannot include more than a specified maximum number of pieces. This ensures that performances remain manageable and focused.  
3. **Minimum Pieces per Member**: Each member must perform at least one piece across all performances. This ensures fair participation and engagement from all members.  

These constraints are designed to maintain operational feasibility while achieving the optimization goal.  

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 2 Database Schema
-- Objective: Added attendance_data table to capture attendance per performance, updated business configuration logic with attendance-related scalar parameters, and ensured all mappings align with OR expert requirements.

CREATE TABLE member_constraints (
  member_id INTEGER,
  max_pieces INTEGER
);

CREATE TABLE performance_constraints (
  performance_id INTEGER,
  max_pieces INTEGER
);

CREATE TABLE member_attendance (
  member_id INTEGER,
  performance_id INTEGER,
  num_of_pieces INTEGER
);

CREATE TABLE attendance_data (
  performance_id INTEGER,
  attendance INTEGER
);
```

### Data Dictionary  
- **member_constraints**:  
  - **member_id**: Unique identifier for each member.  
  - **max_pieces**: Maximum number of pieces the member can perform across all performances.  

- **performance_constraints**:  
  - **performance_id**: Unique identifier for each performance.  
  - **max_pieces**: Maximum number of pieces allowed in the performance.  

- **member_attendance**:  
  - **member_id**: Unique identifier for each member.  
  - **performance_id**: Unique identifier for each performance.  
  - **num_of_pieces**: Number of pieces the member performs in the performance.  

- **attendance_data**:  
  - **performance_id**: Unique identifier for each performance.  
  - **attendance**: Attendance for the performance, used to weight the importance of the performance in the optimization.  

### Current Stored Values  
```sql
-- Iteration 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic scenarios for a music ensemble, considering typical attendance numbers, member capabilities, and performance constraints. The data ensures a balanced distribution of pieces among members and performances, while respecting the optimization constraints.

-- Realistic data for member_constraints
INSERT INTO member_constraints (member_id, max_pieces) VALUES (1, 5);
INSERT INTO member_constraints (member_id, max_pieces) VALUES (2, 4);
INSERT INTO member_constraints (member_id, max_pieces) VALUES (3, 3);

-- Realistic data for performance_constraints
INSERT INTO performance_constraints (performance_id, max_pieces) VALUES (1, 10);
INSERT INTO performance_constraints (performance_id, max_pieces) VALUES (2, 8);
INSERT INTO performance_constraints (performance_id, max_pieces) VALUES (3, 6);

-- Realistic data for member_attendance
INSERT INTO member_attendance (member_id, performance_id, num_of_pieces) VALUES (1, 1, 3);
INSERT INTO member_attendance (member_id, performance_id, num_of_pieces) VALUES (2, 1, 2);
INSERT INTO member_attendance (member_id, performance_id, num_of_pieces) VALUES (3, 1, 1);

-- Realistic data for attendance_data
INSERT INTO attendance_data (performance_id, attendance) VALUES (1, 200);
INSERT INTO attendance_data (performance_id, attendance) VALUES (2, 150);
INSERT INTO attendance_data (performance_id, attendance) VALUES (3, 100);
```
