Iteration final - PROBLEM_DESCRIPTION
Sequence: 5
Timestamp: 2025-07-25 22:28:35

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: assign[Event_ID, Participant_ID] ∈ {0, 1} (binary decision variable indicating whether a participant is assigned to an event)
- Operational parameters align with expected linear objective: maximize ∑(satisfaction_score[Event_ID, Participant_ID] × assign[Event_ID, Participant_ID])
- Business configuration includes: maximum number of participants allowed in an event (used for constraint bound for event capacity)
- 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": "local_govt_in_alabama",
  "iteration": 1,
  "business_context": "Optimizing the allocation of participants to events to maximize overall satisfaction while respecting event capacity and participant availability.",
  "optimization_problem_description": "Maximize the total satisfaction score of participants assigned to events, ensuring that no event exceeds its capacity and each participant is assigned to at most one event.",
  "optimization_formulation": {
    "objective": "maximize \u2211(satisfaction_score[Event_ID, Participant_ID] \u00d7 assign[Event_ID, Participant_ID])",
    "decision_variables": "assign[Event_ID, Participant_ID] \u2208 {0, 1} (binary decision variable indicating whether a participant is assigned to an event)",
    "constraints": "\u2211(assign[Event_ID, Participant_ID]) \u2264 Capacity[Event_ID] for each Event_ID, \u2211(assign[Event_ID, Participant_ID]) \u2264 1 for each Participant_ID"
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "satisfaction_score[Event_ID, Participant_ID]": {
        "currently_mapped_to": "Satisfaction_Scores.Score",
        "mapping_adequacy": "good",
        "description": "satisfaction score of participant for event"
      }
    },
    "constraint_bounds": {
      "Capacity[Event_ID]": {
        "currently_mapped_to": "Event_Capacities.Capacity",
        "mapping_adequacy": "good",
        "description": "maximum number of participants allowed in an event"
      }
    },
    "decision_variables": {
      "assign[Event_ID, Participant_ID]": {
        "currently_mapped_to": "Participants_in_Events",
        "mapping_adequacy": "good",
        "description": "binary decision variable indicating whether a participant is assigned to an event",
        "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 tables for satisfaction scores and event capacities, updated Participants_in_Events table, and created business_configuration_logic.json for scalar parameters and formulas.

CREATE TABLE Satisfaction_Scores (
  Event_ID INTEGER,
  Participant_ID INTEGER,
  Score FLOAT
);

CREATE TABLE Event_Capacities (
  Event_ID INTEGER,
  Capacity INTEGER
);

CREATE TABLE Participants_in_Events (
  Event_ID INTEGER,
  Participant_ID INTEGER
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic event sizes, participant preferences, and typical satisfaction scores in local government settings.

-- Realistic data for Satisfaction_Scores
INSERT INTO Satisfaction_Scores (Event_ID, Participant_ID, Score) VALUES (1, 101, 0.85);
INSERT INTO Satisfaction_Scores (Event_ID, Participant_ID, Score) VALUES (1, 102, 0.75);
INSERT INTO Satisfaction_Scores (Event_ID, Participant_ID, Score) VALUES (1, 103, 0.65);
INSERT INTO Satisfaction_Scores (Event_ID, Participant_ID, Score) VALUES (2, 101, 0.7);
INSERT INTO Satisfaction_Scores (Event_ID, Participant_ID, Score) VALUES (2, 102, 0.9);
INSERT INTO Satisfaction_Scores (Event_ID, Participant_ID, Score) VALUES (2, 103, 0.8);
INSERT INTO Satisfaction_Scores (Event_ID, Participant_ID, Score) VALUES (3, 101, 0.6);
INSERT INTO Satisfaction_Scores (Event_ID, Participant_ID, Score) VALUES (3, 102, 0.85);
INSERT INTO Satisfaction_Scores (Event_ID, Participant_ID, Score) VALUES (3, 103, 0.95);

-- Realistic data for Event_Capacities
INSERT INTO Event_Capacities (Event_ID, Capacity) VALUES (1, 50);
INSERT INTO Event_Capacities (Event_ID, Capacity) VALUES (2, 100);
INSERT INTO Event_Capacities (Event_ID, Capacity) VALUES (3, 75);

-- Realistic data for Participants_in_Events
INSERT INTO Participants_in_Events (Event_ID, Participant_ID) VALUES (1, 101);
INSERT INTO Participants_in_Events (Event_ID, Participant_ID) VALUES (1, 102);
INSERT INTO Participants_in_Events (Event_ID, Participant_ID) VALUES (1, 103);
INSERT INTO Participants_in_Events (Event_ID, Participant_ID) VALUES (2, 101);
INSERT INTO Participants_in_Events (Event_ID, Participant_ID) VALUES (2, 102);
INSERT INTO Participants_in_Events (Event_ID, Participant_ID) VALUES (2, 103);
INSERT INTO Participants_in_Events (Event_ID, Participant_ID) VALUES (3, 101);
INSERT INTO Participants_in_Events (Event_ID, Participant_ID) VALUES (3, 102);
INSERT INTO Participants_in_Events (Event_ID, Participant_ID) VALUES (3, 103);


```

DATA DICTIONARY:
{
  "tables": {
    "Satisfaction_Scores": {
      "business_purpose": "satisfaction score of each participant for each event",
      "optimization_role": "objective_coefficients",
      "columns": {
        "Event_ID": {
          "data_type": "INTEGER",
          "business_meaning": "unique identifier for the event",
          "optimization_purpose": "links to event in optimization model",
          "sample_values": "1, 2, 3"
        },
        "Participant_ID": {
          "data_type": "INTEGER",
          "business_meaning": "unique identifier for the participant",
          "optimization_purpose": "links to participant in optimization model",
          "sample_values": "101, 102, 103"
        },
        "Score": {
          "data_type": "FLOAT",
          "business_meaning": "satisfaction score of the participant for the event",
          "optimization_purpose": "coefficient in the objective function",
          "sample_values": "0.8, 0.9, 1.0"
        }
      }
    },
    "Event_Capacities": {
      "business_purpose": "maximum number of participants allowed in each event",
      "optimization_role": "constraint_bounds",
      "columns": {
        "Event_ID": {
          "data_type": "INTEGER",
          "business_meaning": "unique identifier for the event",
          "optimization_purpose": "links to event in optimization model",
          "sample_values": "1, 2, 3"
        },
        "Capacity": {
          "data_type": "INTEGER",
          "business_meaning": "maximum number of participants allowed in the event",
          "optimization_purpose": "bound in the capacity constraint",
          "sample_values": "50, 100, 150"
        }
      }
    },
    "Participants_in_Events": {
      "business_purpose": "assignment of participants to events",
      "optimization_role": "decision_variables",
      "columns": {
        "Event_ID": {
          "data_type": "INTEGER",
          "business_meaning": "unique identifier for the event",
          "optimization_purpose": "links to event in optimization model",
          "sample_values": "1, 2, 3"
        },
        "Participant_ID": {
          "data_type": "INTEGER",
          "business_meaning": "unique identifier for the participant",
          "optimization_purpose": "links to participant in optimization model",
          "sample_values": "101, 102, 103"
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "event_capacity": {
    "data_type": "INTEGER",
    "business_meaning": "maximum number of participants allowed in an event",
    "optimization_role": "constraint bound for event capacity",
    "configuration_type": "scalar_parameter",
    "value": 50,
    "business_justification": "The default event capacity is set to 50, which is a typical size for local government events, ensuring manageable group sizes."
  }
}

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: local_govt_in_alabama

## 1. Problem Context and Goals

### Context  
[Regenerate business context that naturally aligns with LINEAR optimization formulation. Ensure:]
- Business decisions match expected decision variables: assign[Event_ID, Participant_ID] ∈ {0, 1} (binary decision variable indicating whether a participant is assigned to an event)
- Operational parameters align with expected linear objective: maximize ∑(satisfaction_score[Event_ID, Participant_ID] × assign[Event_ID, Participant_ID])
- Business configuration includes: maximum number of participants allowed in an event (used for constraint bound for event capacity)
- 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 ∑(satisfaction_score[Event_ID, Participant_ID] × assign[Event_ID, Participant_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: ∑(assign[Event_ID, Participant_ID]) ≤ Capacity[Event_ID] for each Event_ID, ∑(assign[Event_ID, Participant_ID]) ≤ 1 for each Participant_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: Added tables for satisfaction scores and event capacities, updated Participants_in_Events table, and created business_configuration_logic.json for scalar parameters and formulas.

CREATE TABLE Satisfaction_Scores (
  Event_ID INTEGER,
  Participant_ID INTEGER,
  Score FLOAT
);

CREATE TABLE Event_Capacities (
  Event_ID INTEGER,
  Capacity INTEGER
);

CREATE TABLE Participants_in_Events (
  Event_ID INTEGER,
  Participant_ID 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 realistic event sizes, participant preferences, and typical satisfaction scores in local government settings.

-- Realistic data for Satisfaction_Scores
INSERT INTO Satisfaction_Scores (Event_ID, Participant_ID, Score) VALUES (1, 101, 0.85);
INSERT INTO Satisfaction_Scores (Event_ID, Participant_ID, Score) VALUES (1, 102, 0.75);
INSERT INTO Satisfaction_Scores (Event_ID, Participant_ID, Score) VALUES (1, 103, 0.65);
INSERT INTO Satisfaction_Scores (Event_ID, Participant_ID, Score) VALUES (2, 101, 0.7);
INSERT INTO Satisfaction_Scores (Event_ID, Participant_ID, Score) VALUES (2, 102, 0.9);
INSERT INTO Satisfaction_Scores (Event_ID, Participant_ID, Score) VALUES (2, 103, 0.8);
INSERT INTO Satisfaction_Scores (Event_ID, Participant_ID, Score) VALUES (3, 101, 0.6);
INSERT INTO Satisfaction_Scores (Event_ID, Participant_ID, Score) VALUES (3, 102, 0.85);
INSERT INTO Satisfaction_Scores (Event_ID, Participant_ID, Score) VALUES (3, 103, 0.95);

-- Realistic data for Event_Capacities
INSERT INTO Event_Capacities (Event_ID, Capacity) VALUES (1, 50);
INSERT INTO Event_Capacities (Event_ID, Capacity) VALUES (2, 100);
INSERT INTO Event_Capacities (Event_ID, Capacity) VALUES (3, 75);

-- Realistic data for Participants_in_Events
INSERT INTO Participants_in_Events (Event_ID, Participant_ID) VALUES (1, 101);
INSERT INTO Participants_in_Events (Event_ID, Participant_ID) VALUES (1, 102);
INSERT INTO Participants_in_Events (Event_ID, Participant_ID) VALUES (1, 103);
INSERT INTO Participants_in_Events (Event_ID, Participant_ID) VALUES (2, 101);
INSERT INTO Participants_in_Events (Event_ID, Participant_ID) VALUES (2, 102);
INSERT INTO Participants_in_Events (Event_ID, Participant_ID) VALUES (2, 103);
INSERT INTO Participants_in_Events (Event_ID, Participant_ID) VALUES (3, 101);
INSERT INTO Participants_in_Events (Event_ID, Participant_ID) VALUES (3, 102);
INSERT INTO Participants_in_Events (Event_ID, Participant_ID) VALUES (3, 103);


```

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

Response:
# Complete Optimization Problem and Solution: local_govt_in_alabama

## 1. Problem Context and Goals

### Context  
The local government in Alabama is tasked with optimizing the allocation of participants to events to ensure the highest possible satisfaction among attendees while adhering to operational constraints. The decision to assign a participant to an event is represented as a binary choice: either the participant is assigned (1) or not (0). The goal is to maximize the total satisfaction score across all assignments, where the satisfaction score reflects how much a participant is expected to enjoy a specific event. 

Each event has a predefined capacity, which limits the number of participants that can be assigned to it. This capacity is a critical operational parameter, ensuring that events are not overcrowded and remain manageable. Additionally, each participant can be assigned to at most one event, ensuring fairness and preventing overcommitment. 

The satisfaction scores are derived from historical data and participant preferences, providing a realistic measure of expected enjoyment. The default event capacity is set to 50 participants, a typical size for local government events, ensuring manageable group sizes. This configuration ensures that the optimization problem remains linear, avoiding complex scenarios such as variable products or divisions.

### Goals  
The primary goal of this optimization problem is to maximize the overall satisfaction of participants by assigning them to events in the most effective way. This is achieved by summing the satisfaction scores of all assigned participant-event pairs. Success is measured by the total satisfaction score, which directly reflects the quality of the assignments. The optimization process ensures that the assignments respect event capacities and participant availability, leading to a fair and efficient allocation.

## 2. Constraints  

The optimization problem is subject to two key constraints:  
1. **Event Capacity Constraint**: The total number of participants assigned to any single event must not exceed the event's predefined capacity. This ensures that events remain within manageable limits and do not become overcrowded.  
2. **Participant Assignment Constraint**: Each participant can be assigned to at most one event. This ensures fairness and prevents participants from being overcommitted to multiple events simultaneously.  

These constraints are designed to align with the operational realities of event planning, ensuring that the solution is both practical and effective.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Added tables for satisfaction scores and event capacities, updated Participants_in_Events table, and created business_configuration_logic.json for scalar parameters and formulas.

CREATE TABLE Satisfaction_Scores (
  Event_ID INTEGER,
  Participant_ID INTEGER,
  Score FLOAT
);

CREATE TABLE Event_Capacities (
  Event_ID INTEGER,
  Capacity INTEGER
);

CREATE TABLE Participants_in_Events (
  Event_ID INTEGER,
  Participant_ID INTEGER
);
```

### Data Dictionary  
- **Satisfaction_Scores**:  
  - **Business Purpose**: Captures the satisfaction score of each participant for each event, reflecting how much a participant is expected to enjoy a specific event.  
  - **Optimization Role**: Provides the coefficients for the objective function, representing the value of assigning a participant to an event.  
  - **Columns**:  
    - **Event_ID**: Unique identifier for the event, linking it to the optimization model.  
    - **Participant_ID**: Unique identifier for the participant, linking them to the optimization model.  
    - **Score**: The satisfaction score of the participant for the event, used as a coefficient in the objective function.  

- **Event_Capacities**:  
  - **Business Purpose**: Defines the maximum number of participants allowed in each event, ensuring events remain within manageable limits.  
  - **Optimization Role**: Provides the bounds for the event capacity constraints.  
  - **Columns**:  
    - **Event_ID**: Unique identifier for the event, linking it to the optimization model.  
    - **Capacity**: The maximum number of participants allowed in the event, used as a bound in the capacity constraint.  

- **Participants_in_Events**:  
  - **Business Purpose**: Represents the assignment of participants to events, capturing the decision to assign a participant to an event.  
  - **Optimization Role**: Represents the binary decision variables in the optimization model.  
  - **Columns**:  
    - **Event_ID**: Unique identifier for the event, linking it to the optimization model.  
    - **Participant_ID**: Unique identifier for the participant, linking them to the optimization model.  

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic event sizes, participant preferences, and typical satisfaction scores in local government settings.

-- Realistic data for Satisfaction_Scores
INSERT INTO Satisfaction_Scores (Event_ID, Participant_ID, Score) VALUES (1, 101, 0.85);
INSERT INTO Satisfaction_Scores (Event_ID, Participant_ID, Score) VALUES (1, 102, 0.75);
INSERT INTO Satisfaction_Scores (Event_ID, Participant_ID, Score) VALUES (1, 103, 0.65);
INSERT INTO Satisfaction_Scores (Event_ID, Participant_ID, Score) VALUES (2, 101, 0.7);
INSERT INTO Satisfaction_Scores (Event_ID, Participant_ID, Score) VALUES (2, 102, 0.9);
INSERT INTO Satisfaction_Scores (Event_ID, Participant_ID, Score) VALUES (2, 103, 0.8);
INSERT INTO Satisfaction_Scores (Event_ID, Participant_ID, Score) VALUES (3, 101, 0.6);
INSERT INTO Satisfaction_Scores (Event_ID, Participant_ID, Score) VALUES (3, 102, 0.85);
INSERT INTO Satisfaction_Scores (Event_ID, Participant_ID, Score) VALUES (3, 103, 0.95);

-- Realistic data for Event_Capacities
INSERT INTO Event_Capacities (Event_ID, Capacity) VALUES (1, 50);
INSERT INTO Event_Capacities (Event_ID, Capacity) VALUES (2, 100);
INSERT INTO Event_Capacities (Event_ID, Capacity) VALUES (3, 75);

-- Realistic data for Participants_in_Events
INSERT INTO Participants_in_Events (Event_ID, Participant_ID) VALUES (1, 101);
INSERT INTO Participants_in_Events (Event_ID, Participant_ID) VALUES (1, 102);
INSERT INTO Participants_in_Events (Event_ID, Participant_ID) VALUES (1, 103);
INSERT INTO Participants_in_Events (Event_ID, Participant_ID) VALUES (2, 101);
INSERT INTO Participants_in_Events (Event_ID, Participant_ID) VALUES (2, 102);
INSERT INTO Participants_in_Events (Event_ID, Participant_ID) VALUES (2, 103);
INSERT INTO Participants_in_Events (Event_ID, Participant_ID) VALUES (3, 101);
INSERT INTO Participants_in_Events (Event_ID, Participant_ID) VALUES (3, 102);
INSERT INTO Participants_in_Events (Event_ID, Participant_ID) VALUES (3, 103);
```
