Iteration final - PROBLEM_DESCRIPTION
Sequence: 5
Timestamp: 2025-07-25 22:28:10

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: {'student_participation[stuid, actid]': {'variable_type': 'binary', 'description': '1 if student stuid participates in activity actid, 0 otherwise'}, 'faculty_participation[facid, actid]': {'variable_type': 'binary', 'description': '1 if faculty facid participates in activity actid, 0 otherwise'}}
- Operational parameters align with expected linear objective: maximize ∑(student_participation[stuid, actid] + faculty_participation[facid, actid])
- Business configuration includes: Maximum number of activities a faculty member can participate in (used for Used in faculty participation constraint), Maximum number of activities a student can participate in (used for Used in student participation constraint)
- Business logic formulas to express in natural language: Total participation in an activity (calculation method for Used in activity capacity constraint)
- 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": "activity_1",
  "iteration": 1,
  "business_context": "A university aims to optimize the allocation of students and faculty to extracurricular activities to maximize overall participation while respecting constraints on faculty availability, student preferences, and activity capacity limits.",
  "optimization_problem_description": "Maximize the total participation in activities by students and faculty, ensuring that each student is assigned to at most one activity, each faculty member is assigned to at most two activities, and the total number of participants in each activity does not exceed a predefined limit.",
  "optimization_formulation": {
    "objective": "maximize \u2211(student_participation[stuid, actid] + faculty_participation[facid, actid])",
    "decision_variables": {
      "student_participation[stuid, actid]": {
        "variable_type": "binary",
        "description": "1 if student stuid participates in activity actid, 0 otherwise"
      },
      "faculty_participation[facid, actid]": {
        "variable_type": "binary",
        "description": "1 if faculty facid participates in activity actid, 0 otherwise"
      }
    },
    "constraints": [
      "\u2211(student_participation[stuid, actid]) \u2264 1 for all stuid",
      "\u2211(faculty_participation[facid, actid]) \u2264 2 for all facid",
      "\u2211(student_participation[stuid, actid] + faculty_participation[facid, actid]) \u2264 max_participants[actid] for all actid"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "student_participation[stuid, actid]": {
        "currently_mapped_to": "Participates_in.stuid",
        "mapping_adequacy": "good",
        "description": "Student participation in activities"
      },
      "faculty_participation[facid, actid]": {
        "currently_mapped_to": "Faculty_Participates_in.FacID",
        "mapping_adequacy": "good",
        "description": "Faculty participation in activities"
      }
    },
    "constraint_bounds": {
      "max_participants[actid]": {
        "currently_mapped_to": "Activity_Capacity.max_participants",
        "mapping_adequacy": "good",
        "description": "Maximum number of participants allowed in each activity"
      },
      "faculty_availability_limit": {
        "currently_mapped_to": "business_configuration_logic.faculty_availability_limit",
        "mapping_adequacy": "good",
        "description": "Maximum number of activities a faculty member can participate in"
      },
      "student_preference_threshold": {
        "currently_mapped_to": "business_configuration_logic.student_preference_threshold",
        "mapping_adequacy": "good",
        "description": "Maximum number of activities a student can participate in"
      }
    },
    "decision_variables": {
      "student_participation[stuid, actid]": {
        "currently_mapped_to": "Participates_in.stuid",
        "mapping_adequacy": "good",
        "description": "Student participation in activities",
        "variable_type": "binary"
      },
      "faculty_participation[facid, actid]": {
        "currently_mapped_to": "Faculty_Participates_in.FacID",
        "mapping_adequacy": "good",
        "description": "Faculty participation in activities",
        "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: Added Activity_Capacity table to address missing max_participants constraint. Updated business_configuration_logic.json with scalar parameters and formulas. Ensured schema follows normalization principles and industry standards.

CREATE TABLE Participates_in (
  stuid INTEGER,
  actid INTEGER
);

CREATE TABLE Faculty_Participates_in (
  FacID INTEGER,
  actid INTEGER
);

CREATE TABLE Activity_Capacity (
  actid INTEGER,
  max_participants INTEGER
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical university extracurricular activity participation rates, faculty workload considerations, and activity capacity limits. The data ensures that the optimization problem is meaningful and solvable by respecting constraints and business logic.

-- Realistic data for Participates_in
INSERT INTO Participates_in (stuid, actid) VALUES (101, 1);
INSERT INTO Participates_in (stuid, actid) VALUES (102, 2);
INSERT INTO Participates_in (stuid, actid) VALUES (103, 3);

-- Realistic data for Faculty_Participates_in
INSERT INTO Faculty_Participates_in (FacID, actid) VALUES (201, 1);
INSERT INTO Faculty_Participates_in (FacID, actid) VALUES (202, 2);
INSERT INTO Faculty_Participates_in (FacID, actid) VALUES (203, 3);

-- Realistic data for Activity_Capacity
INSERT INTO Activity_Capacity (actid, max_participants) VALUES (1, 10);
INSERT INTO Activity_Capacity (actid, max_participants) VALUES (2, 15);
INSERT INTO Activity_Capacity (actid, max_participants) VALUES (3, 20);


```

DATA DICTIONARY:
{
  "tables": {
    "Participates_in": {
      "business_purpose": "Tracks student participation in activities",
      "optimization_role": "decision_variables",
      "columns": {
        "stuid": {
          "data_type": "INTEGER",
          "business_meaning": "Student ID",
          "optimization_purpose": "Identifier for student participation decision variable",
          "sample_values": [
            101,
            102,
            103
          ]
        },
        "actid": {
          "data_type": "INTEGER",
          "business_meaning": "Activity ID",
          "optimization_purpose": "Identifier for activity in student participation decision variable",
          "sample_values": [
            1,
            2,
            3
          ]
        }
      }
    },
    "Faculty_Participates_in": {
      "business_purpose": "Tracks faculty participation in activities",
      "optimization_role": "decision_variables",
      "columns": {
        "FacID": {
          "data_type": "INTEGER",
          "business_meaning": "Faculty ID",
          "optimization_purpose": "Identifier for faculty participation decision variable",
          "sample_values": [
            201,
            202,
            203
          ]
        },
        "actid": {
          "data_type": "INTEGER",
          "business_meaning": "Activity ID",
          "optimization_purpose": "Identifier for activity in faculty participation decision variable",
          "sample_values": [
            1,
            2,
            3
          ]
        }
      }
    },
    "Activity_Capacity": {
      "business_purpose": "Maximum number of participants allowed in each activity",
      "optimization_role": "constraint_bounds",
      "columns": {
        "actid": {
          "data_type": "INTEGER",
          "business_meaning": "Activity ID",
          "optimization_purpose": "Identifier for activity in capacity constraint",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "max_participants": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum number of participants allowed",
          "optimization_purpose": "Upper bound for activity participation constraint",
          "sample_values": [
            10,
            15,
            20
          ]
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "faculty_availability_limit": {
    "data_type": "INTEGER",
    "business_meaning": "Maximum number of activities a faculty member can participate in",
    "optimization_role": "Used in faculty participation constraint",
    "configuration_type": "scalar_parameter",
    "value": 2,
    "business_justification": "Faculty members are limited to participating in 2 activities to balance their workload and ensure availability."
  },
  "student_preference_threshold": {
    "data_type": "INTEGER",
    "business_meaning": "Maximum number of activities a student can participate in",
    "optimization_role": "Used in student participation constraint",
    "configuration_type": "scalar_parameter",
    "value": 1,
    "business_justification": "Students are allowed to participate in only 1 activity to ensure focused engagement and avoid overcommitment."
  },
  "total_participation_calculation": {
    "data_type": "STRING",
    "business_meaning": "Total participation in an activity",
    "optimization_role": "Used in activity capacity constraint",
    "configuration_type": "business_logic_formula",
    "formula_expression": "student_participation[stuid, actid] + faculty_participation[facid, actid]"
  }
}

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: activity_1

## 1. Problem Context and Goals

### Context  
[Regenerate business context that naturally aligns with LINEAR optimization formulation. Ensure:]
- Business decisions match expected decision variables: {'student_participation[stuid, actid]': {'variable_type': 'binary', 'description': '1 if student stuid participates in activity actid, 0 otherwise'}, 'faculty_participation[facid, actid]': {'variable_type': 'binary', 'description': '1 if faculty facid participates in activity actid, 0 otherwise'}}
- Operational parameters align with expected linear objective: maximize ∑(student_participation[stuid, actid] + faculty_participation[facid, actid])
- Business configuration includes: Maximum number of activities a faculty member can participate in (used for Used in faculty participation constraint), Maximum number of activities a student can participate in (used for Used in student participation constraint)
- Business logic formulas to express in natural language: Total participation in an activity (calculation method for Used in activity capacity constraint)
- 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 ∑(student_participation[stuid, actid] + faculty_participation[facid, actid])
- 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: ['∑(student_participation[stuid, actid]) ≤ 1 for all stuid', '∑(faculty_participation[facid, actid]) ≤ 2 for all facid', '∑(student_participation[stuid, actid] + faculty_participation[facid, actid]) ≤ max_participants[actid] for all actid'] (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: Added Activity_Capacity table to address missing max_participants constraint. Updated business_configuration_logic.json with scalar parameters and formulas. Ensured schema follows normalization principles and industry standards.

CREATE TABLE Participates_in (
  stuid INTEGER,
  actid INTEGER
);

CREATE TABLE Faculty_Participates_in (
  FacID INTEGER,
  actid INTEGER
);

CREATE TABLE Activity_Capacity (
  actid INTEGER,
  max_participants 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 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical university extracurricular activity participation rates, faculty workload considerations, and activity capacity limits. The data ensures that the optimization problem is meaningful and solvable by respecting constraints and business logic.

-- Realistic data for Participates_in
INSERT INTO Participates_in (stuid, actid) VALUES (101, 1);
INSERT INTO Participates_in (stuid, actid) VALUES (102, 2);
INSERT INTO Participates_in (stuid, actid) VALUES (103, 3);

-- Realistic data for Faculty_Participates_in
INSERT INTO Faculty_Participates_in (FacID, actid) VALUES (201, 1);
INSERT INTO Faculty_Participates_in (FacID, actid) VALUES (202, 2);
INSERT INTO Faculty_Participates_in (FacID, actid) VALUES (203, 3);

-- Realistic data for Activity_Capacity
INSERT INTO Activity_Capacity (actid, max_participants) VALUES (1, 10);
INSERT INTO Activity_Capacity (actid, max_participants) VALUES (2, 15);
INSERT INTO Activity_Capacity (actid, max_participants) VALUES (3, 20);


```

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

Response:
# Complete Optimization Problem and Solution: activity_1

## 1. Problem Context and Goals

### Context  
A university is managing the allocation of students and faculty to extracurricular activities with the goal of maximizing overall participation. The decision-making process involves determining which students and faculty members should participate in which activities. Each student can participate in at most one activity, and each faculty member can participate in at most two activities. Additionally, each activity has a predefined maximum number of participants that cannot be exceeded.  

The business configuration includes the following operational parameters:  
- **Faculty Availability Limit**: Faculty members are limited to participating in a maximum of two activities to balance their workload and ensure availability.  
- **Student Preference Threshold**: Students are allowed to participate in only one activity to ensure focused engagement and avoid overcommitment.  
- **Total Participation Calculation**: The total participation in an activity is calculated as the sum of student and faculty participation in that activity.  

The optimization problem is designed to ensure that these constraints are respected while maximizing the total number of participants across all activities.  

### Goals  
The primary goal of this optimization problem is to maximize the total participation in extracurricular activities by both students and faculty. Success is measured by the total number of participants across all activities, which is the sum of student and faculty participation. This goal aligns with the operational parameters and ensures that the allocation respects the constraints on faculty availability, student preferences, and activity capacity limits.  

## 2. Constraints  

The optimization problem must adhere to the following constraints:  
1. **Student Participation Limit**: Each student can participate in at most one activity. This ensures that students are not overcommitted and can focus on their chosen activity.  
2. **Faculty Participation Limit**: Each faculty member can participate in at most two activities. This constraint balances faculty workload and ensures their availability across activities.  
3. **Activity Capacity Limit**: The total number of participants in each activity, including both students and faculty, must not exceed the predefined maximum capacity for that activity. This ensures that activities are not overcrowded and can operate effectively.  

These constraints are designed to ensure that the allocation of participants is feasible and aligns with the operational capabilities of the university.  

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Added Activity_Capacity table to address missing max_participants constraint. Updated business_configuration_logic.json with scalar parameters and formulas. Ensured schema follows normalization principles and industry standards.

CREATE TABLE Participates_in (
  stuid INTEGER,
  actid INTEGER
);

CREATE TABLE Faculty_Participates_in (
  FacID INTEGER,
  actid INTEGER
);

CREATE TABLE Activity_Capacity (
  actid INTEGER,
  max_participants INTEGER
);
```

### Data Dictionary  
- **Participates_in**: Tracks student participation in activities.  
  - **stuid**: Unique identifier for a student. Used to determine which students participate in which activities.  
  - **actid**: Unique identifier for an activity. Used to link students to specific activities.  
- **Faculty_Participates_in**: Tracks faculty participation in activities.  
  - **FacID**: Unique identifier for a faculty member. Used to determine which faculty members participate in which activities.  
  - **actid**: Unique identifier for an activity. Used to link faculty members to specific activities.  
- **Activity_Capacity**: Defines the maximum number of participants allowed in each activity.  
  - **actid**: Unique identifier for an activity. Used to link capacity limits to specific activities.  
  - **max_participants**: The maximum number of participants allowed in the activity. Used to enforce capacity constraints.  

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical university extracurricular activity participation rates, faculty workload considerations, and activity capacity limits. The data ensures that the optimization problem is meaningful and solvable by respecting constraints and business logic.

-- Realistic data for Participates_in
INSERT INTO Participates_in (stuid, actid) VALUES (101, 1);
INSERT INTO Participates_in (stuid, actid) VALUES (102, 2);
INSERT INTO Participates_in (stuid, actid) VALUES (103, 3);

-- Realistic data for Faculty_Participates_in
INSERT INTO Faculty_Participates_in (FacID, actid) VALUES (201, 1);
INSERT INTO Faculty_Participates_in (FacID, actid) VALUES (202, 2);
INSERT INTO Faculty_Participates_in (FacID, actid) VALUES (203, 3);

-- Realistic data for Activity_Capacity
INSERT INTO Activity_Capacity (actid, max_participants) VALUES (1, 10);
INSERT INTO Activity_Capacity (actid, max_participants) VALUES (2, 15);
INSERT INTO Activity_Capacity (actid, max_participants) VALUES (3, 20);
```
