Iteration final - PROBLEM_DESCRIPTION
Sequence: 7
Timestamp: 2025-07-27 23:20:55

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: assignment[i,j] where i is actor_id and j is musical_id, binary
- Operational parameters align with expected linear objective: minimize total_age = ∑(age[i] * assignment[i,j])
- 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": "musical",
  "iteration": 2,
  "business_context": "A theater company is optimizing the casting of actors for various musicals to minimize the total age of the cast while ensuring each musical has a complete cast. Each actor can only be assigned to one musical.",
  "optimization_problem_description": "Assign actors to musicals such that the total age of all actors assigned is minimized. Each musical requires a specific number of actors, and each actor can only be assigned to one musical.",
  "optimization_formulation": {
    "objective": "minimize total_age = \u2211(age[i] * assignment[i,j])",
    "decision_variables": "assignment[i,j] where i is actor_id and j is musical_id, binary",
    "constraints": [
      "\u2211(assignment[i,j]) = required_actors[j] for each musical j",
      "\u2211(assignment[i,j] for all j) \u2264 1 for each actor i"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "age[i]": {
        "currently_mapped_to": "actor_details.age",
        "mapping_adequacy": "good",
        "description": "Age of actor i used as the coefficient in the objective function"
      }
    },
    "constraint_bounds": {
      "required_actors[j]": {
        "currently_mapped_to": "musical_requirements.required_actors",
        "mapping_adequacy": "good",
        "description": "Number of actors required for musical j"
      }
    },
    "decision_variables": {
      "assignment[i,j]": {
        "currently_mapped_to": "actor_musical_assignment.assignment",
        "mapping_adequacy": "good",
        "description": "Binary variable indicating if actor i is assigned to musical j",
        "variable_type": "binary"
      }
    }
  },
  "missing_optimization_requirements": [],
  "iteration_status": {
    "complete": true,
    "confidence": "high",
    "next_focus": "Ready for convergence"
  }
}

FINAL DATABASE SCHEMA:
```sql
-- Iteration 2 Database Schema
-- Objective: Incorporated actor ages into the schema to complete the linear formulation and updated business configuration logic for scalar parameters.

CREATE TABLE musical_requirements (
  musical_id INTEGER,
  required_actors INTEGER
);

CREATE TABLE actor_musical_assignment (
  actor_id INTEGER,
  musical_id INTEGER,
  assignment BOOLEAN
);

CREATE TABLE actor_details (
  actor_id INTEGER,
  age INTEGER
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical age ranges for actors and realistic casting requirements for musicals, ensuring a diverse age distribution and varied musical requirements.

-- Realistic data for musical_requirements
INSERT INTO musical_requirements (musical_id, required_actors) VALUES (1, 3);
INSERT INTO musical_requirements (musical_id, required_actors) VALUES (2, 4);
INSERT INTO musical_requirements (musical_id, required_actors) VALUES (3, 5);

-- Realistic data for actor_musical_assignment
INSERT INTO actor_musical_assignment (actor_id, musical_id, assignment) VALUES (101, 1, True);
INSERT INTO actor_musical_assignment (actor_id, musical_id, assignment) VALUES (102, 2, True);
INSERT INTO actor_musical_assignment (actor_id, musical_id, assignment) VALUES (103, 3, True);
INSERT INTO actor_musical_assignment (actor_id, musical_id, assignment) VALUES (104, 2, False);
INSERT INTO actor_musical_assignment (actor_id, musical_id, assignment) VALUES (105, 3, False);

-- Realistic data for actor_details
INSERT INTO actor_details (actor_id, age) VALUES (101, 25);
INSERT INTO actor_details (actor_id, age) VALUES (102, 30);
INSERT INTO actor_details (actor_id, age) VALUES (103, 35);


```

DATA DICTIONARY:
{
  "tables": {
    "musical_requirements": {
      "business_purpose": "Defines the number of actors required for each musical",
      "optimization_role": "constraint_bounds",
      "columns": {
        "musical_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each musical",
          "optimization_purpose": "Links to musicals for constraint application",
          "sample_values": "1, 2, 3"
        },
        "required_actors": {
          "data_type": "INTEGER",
          "business_meaning": "Number of actors required for the musical",
          "optimization_purpose": "Constraint bound for actor assignment",
          "sample_values": "3, 4, 5"
        }
      }
    },
    "actor_musical_assignment": {
      "business_purpose": "Tracks which actors are assigned to which musicals",
      "optimization_role": "decision_variables",
      "columns": {
        "actor_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each actor",
          "optimization_purpose": "Links to actors for decision variable",
          "sample_values": "101, 102, 103"
        },
        "musical_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each musical",
          "optimization_purpose": "Links to musicals for decision variable",
          "sample_values": "1, 2, 3"
        },
        "assignment": {
          "data_type": "BOOLEAN",
          "business_meaning": "Indicates if the actor is assigned to the musical",
          "optimization_purpose": "Binary decision variable for optimization",
          "sample_values": "true, false"
        }
      }
    },
    "actor_details": {
      "business_purpose": "Stores details about actors including their age",
      "optimization_role": "objective_coefficients",
      "columns": {
        "actor_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each actor",
          "optimization_purpose": "Links to actors for objective coefficient",
          "sample_values": "101, 102, 103"
        },
        "age": {
          "data_type": "INTEGER",
          "business_meaning": "Age of the actor",
          "optimization_purpose": "Objective coefficient for minimizing total age",
          "sample_values": "25, 30, 35"
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:


TASK: Create structured markdown documentation for SECTIONS 1-3 ONLY (Problem Description).

EXACT MARKDOWN STRUCTURE TO FOLLOW:

# Complete Optimization Problem and Solution: musical

## 1. Problem Context and Goals

### Context  
[Regenerate business context that naturally aligns with LINEAR optimization formulation. Ensure:]
- Business decisions match expected decision variables: assignment[i,j] where i is actor_id and j is musical_id, binary
- Operational parameters align with expected linear objective: minimize total_age = ∑(age[i] * assignment[i,j])
- 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 total_age = ∑(age[i] * assignment[i,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: ['∑(assignment[i,j]) = required_actors[j] for each musical j', '∑(assignment[i,j] for all j) ≤ 1 for each actor 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 2 Database Schema
-- Objective: Incorporated actor ages into the schema to complete the linear formulation and updated business configuration logic for scalar parameters.

CREATE TABLE musical_requirements (
  musical_id INTEGER,
  required_actors INTEGER
);

CREATE TABLE actor_musical_assignment (
  actor_id INTEGER,
  musical_id INTEGER,
  assignment BOOLEAN
);

CREATE TABLE actor_details (
  actor_id INTEGER,
  age 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 typical age ranges for actors and realistic casting requirements for musicals, ensuring a diverse age distribution and varied musical requirements.

-- Realistic data for musical_requirements
INSERT INTO musical_requirements (musical_id, required_actors) VALUES (1, 3);
INSERT INTO musical_requirements (musical_id, required_actors) VALUES (2, 4);
INSERT INTO musical_requirements (musical_id, required_actors) VALUES (3, 5);

-- Realistic data for actor_musical_assignment
INSERT INTO actor_musical_assignment (actor_id, musical_id, assignment) VALUES (101, 1, True);
INSERT INTO actor_musical_assignment (actor_id, musical_id, assignment) VALUES (102, 2, True);
INSERT INTO actor_musical_assignment (actor_id, musical_id, assignment) VALUES (103, 3, True);
INSERT INTO actor_musical_assignment (actor_id, musical_id, assignment) VALUES (104, 2, False);
INSERT INTO actor_musical_assignment (actor_id, musical_id, assignment) VALUES (105, 3, False);

-- Realistic data for actor_details
INSERT INTO actor_details (actor_id, age) VALUES (101, 25);
INSERT INTO actor_details (actor_id, age) VALUES (102, 30);
INSERT INTO actor_details (actor_id, age) VALUES (103, 35);


```

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

Response:
# Complete Optimization Problem and Solution: musical

## 1. Problem Context and Goals

### Context  
The theater company is focused on optimizing the casting process for its musicals. The primary decision involves assigning actors to musicals, where each assignment is represented by a binary decision variable indicating whether a specific actor is cast in a particular musical. The objective is to minimize the total age of the cast, which is achieved by summing the ages of all actors assigned to musicals. Each musical has a predefined number of actors required, and each actor can only be assigned to one musical. This setup ensures that the optimization problem remains linear, as it involves straightforward summation and assignment without any nonlinear operations like multiplication or division of variables. The operational parameters, such as the age of actors and the number of actors required for each musical, are clearly defined and mapped to the relevant data sources. The business configuration includes these parameters to guide the optimization process effectively.

### Goals  
The primary goal of the optimization is to minimize the total age of the actors assigned to the musicals. This involves selecting actors in such a way that the sum of their ages is as low as possible while still meeting the casting requirements of each musical. Success in this optimization is measured by achieving the lowest possible total age, using the ages of actors as coefficients in the objective function. The goal is articulated in natural language to ensure clarity and alignment with the linear optimization framework.

## 2. Constraints    

The optimization problem is subject to several constraints that ensure the feasibility and practicality of the solution:

- Each musical must have exactly the number of actors it requires. This means that the sum of assignments for each musical should equal the number of actors needed for that musical.
- An actor can only be assigned to one musical. This constraint ensures that the sum of assignments for each actor across all musicals does not exceed one.

These constraints are expressed in business terms that naturally translate into linear mathematical forms, ensuring that the optimization remains within the linear framework.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 2 Database Schema
-- Objective: Incorporated actor ages into the schema to complete the linear formulation and updated business configuration logic for scalar parameters.

CREATE TABLE musical_requirements (
  musical_id INTEGER,
  required_actors INTEGER
);

CREATE TABLE actor_musical_assignment (
  actor_id INTEGER,
  musical_id INTEGER,
  assignment BOOLEAN
);

CREATE TABLE actor_details (
  actor_id INTEGER,
  age INTEGER
);
```

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

- **Musical Requirements Table**: This table defines the number of actors required for each musical. It plays a crucial role in setting the constraints for the optimization problem, ensuring that each musical has the necessary number of actors.
  - **musical_id**: Serves as a unique identifier for each musical, linking it to the required number of actors.
  - **required_actors**: Specifies the exact number of actors needed for each musical, forming the basis for the constraint on actor assignments.

- **Actor Musical Assignment Table**: This table tracks which actors are assigned to which musicals. It is central to the decision-making process in the optimization, representing the binary decision variables.
  - **actor_id**: A unique identifier for each actor, linking them to their potential assignments.
  - **musical_id**: Connects each assignment to a specific musical, ensuring that the assignment decisions are correctly applied.
  - **assignment**: A binary indicator showing whether an actor is assigned to a musical, forming the core of the decision variables in the optimization.

- **Actor Details Table**: This table stores information about the actors, including their ages, which are used as coefficients in the objective function.
  - **actor_id**: Links each actor to their details, ensuring that age information is correctly applied in the optimization.
  - **age**: Represents the age of each actor, used as a coefficient in the objective function to minimize the total age of the cast.

### Current Stored Values  
```sql
-- Iteration 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical age ranges for actors and realistic casting requirements for musicals, ensuring a diverse age distribution and varied musical requirements.

-- Realistic data for musical_requirements
INSERT INTO musical_requirements (musical_id, required_actors) VALUES (1, 3);
INSERT INTO musical_requirements (musical_id, required_actors) VALUES (2, 4);
INSERT INTO musical_requirements (musical_id, required_actors) VALUES (3, 5);

-- Realistic data for actor_musical_assignment
INSERT INTO actor_musical_assignment (actor_id, musical_id, assignment) VALUES (101, 1, True);
INSERT INTO actor_musical_assignment (actor_id, musical_id, assignment) VALUES (102, 2, True);
INSERT INTO actor_musical_assignment (actor_id, musical_id, assignment) VALUES (103, 3, True);
INSERT INTO actor_musical_assignment (actor_id, musical_id, assignment) VALUES (104, 2, False);
INSERT INTO actor_musical_assignment (actor_id, musical_id, assignment) VALUES (105, 3, False);

-- Realistic data for actor_details
INSERT INTO actor_details (actor_id, age) VALUES (101, 25);
INSERT INTO actor_details (actor_id, age) VALUES (102, 30);
INSERT INTO actor_details (actor_id, age) VALUES (103, 35);
```
