Iteration final - PROBLEM_DESCRIPTION
Sequence: 5
Timestamp: 2025-07-28 00:20:44

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: x_ij: binary variable indicating if host j is assigned to party i
- Operational parameters align with expected linear objective: minimize ∑(1 × x_ij) where x_ij is a binary variable indicating if host j is assigned to party i
- Business configuration includes: Maximum number of hosts that can be assigned to a party (used for Used as a constraint bound 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": "party_host",
  "iteration": 1,
  "business_context": "A company organizes themed parties and aims to optimize the allocation of hosts to parties to minimize the total number of hosts required while ensuring each party has at least one main host in charge and does not exceed the maximum number of hosts allowed per party.",
  "optimization_problem_description": "Minimize the total number of hosts assigned to parties while ensuring each party has at least one main host and does not exceed the maximum number of hosts allowed per party.",
  "optimization_formulation": {
    "objective": "minimize \u2211(1 \u00d7 x_ij) where x_ij is a binary variable indicating if host j is assigned to party i",
    "decision_variables": "x_ij: binary variable indicating if host j is assigned to party i",
    "constraints": [
      "\u2211(x_ij for all j) >= 1 for each party i",
      "\u2211(x_ij for all j) <= max_hosts_per_party for each party i"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "1[x_ij]": {
        "currently_mapped_to": "objective_coefficients.coefficient_value",
        "mapping_adequacy": "good",
        "description": "Coefficient for each host assignment in the objective function"
      }
    },
    "constraint_bounds": {
      "min_hosts_per_party[i]": {
        "currently_mapped_to": "business_configuration_logic.key",
        "mapping_adequacy": "good",
        "description": "Ensures each party has at least one host"
      },
      "max_hosts_per_party[i]": {
        "currently_mapped_to": "business_configuration_logic.max_hosts_per_party",
        "mapping_adequacy": "good",
        "description": "Ensures no party exceeds the maximum number of hosts"
      }
    },
    "decision_variables": {
      "x_ij": {
        "currently_mapped_to": "party_host_assignment.is_assigned",
        "mapping_adequacy": "good",
        "description": "Binary decision variable for host assignment",
        "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 creating new tables for decision variables and objective coefficients, modifying existing tables to fill mapping gaps, and updating business configuration logic for scalar parameters and formulas.

CREATE TABLE party_host_assignment (
  party_id INTEGER,
  host_id INTEGER,
  is_assigned BOOLEAN
);

CREATE TABLE objective_coefficients (
  coefficient_id INTEGER,
  coefficient_value FLOAT
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical party hosting scenarios, ensuring each party has a reasonable number of hosts and that the constraints are neither too tight nor too loose.

-- Realistic data for party_host_assignment
INSERT INTO party_host_assignment (party_id, host_id, is_assigned) VALUES (1, 101, True);
INSERT INTO party_host_assignment (party_id, host_id, is_assigned) VALUES (1, 102, True);
INSERT INTO party_host_assignment (party_id, host_id, is_assigned) VALUES (2, 103, True);
INSERT INTO party_host_assignment (party_id, host_id, is_assigned) VALUES (2, 104, True);
INSERT INTO party_host_assignment (party_id, host_id, is_assigned) VALUES (3, 105, True);

-- Realistic data for objective_coefficients
INSERT INTO objective_coefficients (coefficient_id, coefficient_value) VALUES (1, 1.0);
INSERT INTO objective_coefficients (coefficient_id, coefficient_value) VALUES (2, 1.0);
INSERT INTO objective_coefficients (coefficient_id, coefficient_value) VALUES (3, 1.0);


```

DATA DICTIONARY:
{
  "tables": {
    "party_host_assignment": {
      "business_purpose": "Tracks host assignments to parties",
      "optimization_role": "decision_variables",
      "columns": {
        "party_id": {
          "data_type": "INTEGER",
          "business_meaning": "Identifier for the party",
          "optimization_purpose": "Links host assignments to specific parties",
          "sample_values": "1, 2, 3"
        },
        "host_id": {
          "data_type": "INTEGER",
          "business_meaning": "Identifier for the host",
          "optimization_purpose": "Links assignments to specific hosts",
          "sample_values": "101, 102, 103"
        },
        "is_assigned": {
          "data_type": "BOOLEAN",
          "business_meaning": "Indicates if a host is assigned to a party",
          "optimization_purpose": "Binary decision variable for host assignment",
          "sample_values": "true, false"
        }
      }
    },
    "objective_coefficients": {
      "business_purpose": "Stores coefficients for the optimization objective function",
      "optimization_role": "objective_coefficients",
      "columns": {
        "coefficient_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for the coefficient",
          "optimization_purpose": "Identifies specific coefficients in the objective function",
          "sample_values": "1, 2, 3"
        },
        "coefficient_value": {
          "data_type": "FLOAT",
          "business_meaning": "Value of the coefficient",
          "optimization_purpose": "Used in the calculation of the objective function",
          "sample_values": "1.0"
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "max_hosts_per_party": {
    "data_type": "INTEGER",
    "business_meaning": "Maximum number of hosts that can be assigned to a party",
    "optimization_role": "Used as a constraint bound in optimization model",
    "configuration_type": "scalar_parameter",
    "value": 5,
    "business_justification": "A maximum of 5 hosts per party is realistic for managing different aspects without overcrowding."
  }
}

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: party_host

## 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: binary variable indicating if host j is assigned to party i
- Operational parameters align with expected linear objective: minimize ∑(1 × x_ij) where x_ij is a binary variable indicating if host j is assigned to party i
- Business configuration includes: Maximum number of hosts that can be assigned to a party (used for Used as a constraint bound 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: minimize
- Metric to optimize: minimize ∑(1 × x_ij) where x_ij is a binary variable indicating if host j is assigned to party i
- 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 for all j) >= 1 for each party i', '∑(x_ij for all j) <= max_hosts_per_party for each party i'] (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 new tables for decision variables and objective coefficients, modifying existing tables to fill mapping gaps, and updating business configuration logic for scalar parameters and formulas.

CREATE TABLE party_host_assignment (
  party_id INTEGER,
  host_id INTEGER,
  is_assigned BOOLEAN
);

CREATE TABLE objective_coefficients (
  coefficient_id INTEGER,
  coefficient_value 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 party hosting scenarios, ensuring each party has a reasonable number of hosts and that the constraints are neither too tight nor too loose.

-- Realistic data for party_host_assignment
INSERT INTO party_host_assignment (party_id, host_id, is_assigned) VALUES (1, 101, True);
INSERT INTO party_host_assignment (party_id, host_id, is_assigned) VALUES (1, 102, True);
INSERT INTO party_host_assignment (party_id, host_id, is_assigned) VALUES (2, 103, True);
INSERT INTO party_host_assignment (party_id, host_id, is_assigned) VALUES (2, 104, True);
INSERT INTO party_host_assignment (party_id, host_id, is_assigned) VALUES (3, 105, True);

-- Realistic data for objective_coefficients
INSERT INTO objective_coefficients (coefficient_id, coefficient_value) VALUES (1, 1.0);
INSERT INTO objective_coefficients (coefficient_id, coefficient_value) VALUES (2, 1.0);
INSERT INTO objective_coefficients (coefficient_id, coefficient_value) VALUES (3, 1.0);


```

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

Response:
# Complete Optimization Problem and Solution: party_host

## 1. Problem Context and Goals

### Context  
The company is focused on organizing themed parties and aims to efficiently allocate hosts to these events. The primary decision involves determining whether a specific host is assigned to a particular party. This decision is represented by a binary choice, where each host can either be assigned or not assigned to a party. The operational goal is to minimize the total number of hosts required across all parties. Each party must have at least one main host to ensure smooth operations, and no party should exceed a predefined maximum number of hosts. This maximum is set to ensure effective management and avoid overcrowding. The business configuration includes a scalar parameter that specifies the maximum number of hosts allowed per party, which is crucial for maintaining the balance between adequate staffing and resource efficiency.

### Goals  
The optimization goal is to minimize the total number of hosts assigned to parties. The metric for optimization is the total count of host assignments, with the aim of reducing this number while still meeting the operational requirements of each party. Success is measured by achieving the minimum possible number of host assignments, ensuring that each party is adequately staffed without exceeding the maximum host limit. The focus is on a linear optimization approach, where the objective is to streamline host allocation in a straightforward and efficient manner.

## 2. Constraints    

The constraints for this optimization problem are straightforward and align with the linear mathematical formulation. Each party must have at least one host assigned to it, ensuring that all parties are adequately staffed. Additionally, the number of hosts assigned to any party should not exceed the maximum number of hosts allowed, as specified in the business configuration. These constraints ensure that the staffing levels are both sufficient and efficient, adhering to the operational limits set by the company.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating new tables for decision variables and objective coefficients, modifying existing tables to fill mapping gaps, and updating business configuration logic for scalar parameters and formulas.

CREATE TABLE party_host_assignment (
  party_id INTEGER,
  host_id INTEGER,
  is_assigned BOOLEAN
);

CREATE TABLE objective_coefficients (
  coefficient_id INTEGER,
  coefficient_value FLOAT
);
```

### Data Dictionary  
The data dictionary provides a comprehensive overview of the tables and columns used in the optimization process, linking them to their business purposes and roles in the optimization model:

- **Party Host Assignment Table**: This table tracks which hosts are assigned to which parties. It includes:
  - **Party ID**: An identifier for each party, linking host assignments to specific events.
  - **Host ID**: An identifier for each host, ensuring that assignments are linked to the correct personnel.
  - **Is Assigned**: A binary indicator showing whether a host is assigned to a party, serving as the decision variable in the optimization model.

- **Objective Coefficients Table**: This table stores the coefficients used in the optimization objective function. It includes:
  - **Coefficient ID**: A unique identifier for each coefficient, ensuring clarity in the optimization process.
  - **Coefficient Value**: The value of the coefficient, used in calculating the objective function to minimize host assignments.

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical party hosting scenarios, ensuring each party has a reasonable number of hosts and that the constraints are neither too tight nor too loose.

-- Realistic data for party_host_assignment
INSERT INTO party_host_assignment (party_id, host_id, is_assigned) VALUES (1, 101, True);
INSERT INTO party_host_assignment (party_id, host_id, is_assigned) VALUES (1, 102, True);
INSERT INTO party_host_assignment (party_id, host_id, is_assigned) VALUES (2, 103, True);
INSERT INTO party_host_assignment (party_id, host_id, is_assigned) VALUES (2, 104, True);
INSERT INTO party_host_assignment (party_id, host_id, is_assigned) VALUES (3, 105, True);

-- Realistic data for objective_coefficients
INSERT INTO objective_coefficients (coefficient_id, coefficient_value) VALUES (1, 1.0);
INSERT INTO objective_coefficients (coefficient_id, coefficient_value) VALUES (2, 1.0);
INSERT INTO objective_coefficients (coefficient_id, coefficient_value) VALUES (3, 1.0);
```
