Iteration final - PROBLEM_DESCRIPTION
Sequence: 5
Timestamp: 2025-07-27 22:36:48

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[i,j] are binary variables indicating whether film i is distributed in market j
- Operational parameters align with expected linear objective: maximize total_gross_revenue = sum(Low_Estimate[i,j] * x[i,j])
- Business configuration includes: Maximum budget allowed for film distribution (used for Constraint in optimization model)
- Business logic formulas to express in natural language: Preference score for distributing a film in a market (calculation method for Used to prioritize film-market pairs)
- 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": "film_rank",
  "iteration": 1,
  "business_context": "A film distribution company aims to maximize its total gross revenue from films across different markets. Each film has estimated revenue ranges in different markets, and the company needs to decide which films to distribute in which markets to maximize revenue while considering market-specific constraints such as budget limits.",
  "optimization_problem_description": "Maximize the total gross revenue from distributing films across various markets, considering the estimated revenue ranges for each film-market pair and adhering to budget constraints.",
  "optimization_formulation": {
    "objective": "maximize total_gross_revenue = sum(Low_Estimate[i,j] * x[i,j])",
    "decision_variables": "x[i,j] are binary variables indicating whether film i is distributed in market j",
    "constraints": [
      "sum(Low_Estimate[i,j] * x[i,j]) <= budget_limit",
      "x[i,j] \u2208 {0, 1} for all i, j"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "Low_Estimate[i,j]": {
        "currently_mapped_to": "film_market_estimation.Low_Estimate",
        "mapping_adequacy": "good",
        "description": "Estimated lower bound of revenue for film-market pair"
      }
    },
    "constraint_bounds": {
      "budget_limit": {
        "currently_mapped_to": "business_configuration_logic.budget_limit",
        "mapping_adequacy": "good",
        "description": "Maximum budget allowed for film distribution"
      }
    },
    "decision_variables": {
      "x[i,j]": {
        "currently_mapped_to": "film_market_decision.x",
        "mapping_adequacy": "good",
        "description": "Decision variable indicating if film i is distributed in market j",
        "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 a new table for decision variables, modifying existing tables to ensure all optimization requirements are met, and updating business configuration logic for scalar parameters and formulas.

CREATE TABLE film_market_estimation (
  film_id INTEGER,
  market_id INTEGER,
  Low_Estimate FLOAT,
  High_Estimate FLOAT
);

CREATE TABLE film_market_decision (
  film_id INTEGER,
  market_id INTEGER,
  x BOOLEAN
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical revenue estimates for film distribution in various markets, ensuring a mix of high and low estimates to reflect market variability. Budget limits were set to allow for strategic decision-making without trivializing the optimization problem.

-- Realistic data for film_market_estimation
INSERT INTO film_market_estimation (film_id, market_id, Low_Estimate, High_Estimate) VALUES (1, 101, 12000.0, 18000.0);
INSERT INTO film_market_estimation (film_id, market_id, Low_Estimate, High_Estimate) VALUES (1, 102, 15000.0, 22000.0);
INSERT INTO film_market_estimation (film_id, market_id, Low_Estimate, High_Estimate) VALUES (2, 101, 8000.0, 12000.0);
INSERT INTO film_market_estimation (film_id, market_id, Low_Estimate, High_Estimate) VALUES (2, 103, 20000.0, 30000.0);
INSERT INTO film_market_estimation (film_id, market_id, Low_Estimate, High_Estimate) VALUES (3, 102, 25000.0, 35000.0);

-- Realistic data for film_market_decision
INSERT INTO film_market_decision (film_id, market_id, x) VALUES (1, 101, True);
INSERT INTO film_market_decision (film_id, market_id, x) VALUES (1, 102, False);
INSERT INTO film_market_decision (film_id, market_id, x) VALUES (2, 101, False);
INSERT INTO film_market_decision (film_id, market_id, x) VALUES (2, 103, True);
INSERT INTO film_market_decision (film_id, market_id, x) VALUES (3, 102, True);


```

DATA DICTIONARY:
{
  "tables": {
    "film_market_estimation": {
      "business_purpose": "Estimates revenue for film-market pairs",
      "optimization_role": "objective_coefficients/constraint_bounds",
      "columns": {
        "film_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each film",
          "optimization_purpose": "Identifies films in optimization",
          "sample_values": "1, 2, 3"
        },
        "market_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each market",
          "optimization_purpose": "Identifies markets in optimization",
          "sample_values": "101, 102, 103"
        },
        "Low_Estimate": {
          "data_type": "FLOAT",
          "business_meaning": "Estimated lower bound of revenue",
          "optimization_purpose": "Objective coefficient",
          "sample_values": "10000.0, 20000.0, 30000.0"
        },
        "High_Estimate": {
          "data_type": "FLOAT",
          "business_meaning": "Estimated upper bound of revenue",
          "optimization_purpose": "Constraint bound",
          "sample_values": "15000.0, 25000.0, 35000.0"
        }
      }
    },
    "film_market_decision": {
      "business_purpose": "Stores decision variables for film-market distribution",
      "optimization_role": "decision_variables",
      "columns": {
        "film_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each film",
          "optimization_purpose": "Identifies films in optimization",
          "sample_values": "1, 2, 3"
        },
        "market_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each market",
          "optimization_purpose": "Identifies markets in optimization",
          "sample_values": "101, 102, 103"
        },
        "x": {
          "data_type": "BOOLEAN",
          "business_meaning": "Decision variable for film distribution",
          "optimization_purpose": "Indicates if film is distributed in market",
          "sample_values": "true, false"
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "budget_limit": {
    "data_type": "INTEGER",
    "business_meaning": "Maximum budget allowed for film distribution",
    "optimization_role": "Constraint in optimization model",
    "configuration_type": "scalar_parameter",
    "value": 50000,
    "business_justification": "The budget limit is set to challenge the optimization process, requiring strategic selection of film-market pairs to maximize revenue within constraints."
  },
  "film_market_preference": {
    "data_type": "STRING",
    "business_meaning": "Preference score for distributing a film in a market",
    "optimization_role": "Used to prioritize film-market pairs",
    "configuration_type": "business_logic_formula",
    "formula_expression": "preference_score = market_popularity * film_rating"
  }
}

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: film_rank

## 1. Problem Context and Goals

### Context  
[Regenerate business context that naturally aligns with LINEAR optimization formulation. Ensure:]
- Business decisions match expected decision variables: x[i,j] are binary variables indicating whether film i is distributed in market j
- Operational parameters align with expected linear objective: maximize total_gross_revenue = sum(Low_Estimate[i,j] * x[i,j])
- Business configuration includes: Maximum budget allowed for film distribution (used for Constraint in optimization model)
- Business logic formulas to express in natural language: Preference score for distributing a film in a market (calculation method for Used to prioritize film-market pairs)
- 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 total_gross_revenue = sum(Low_Estimate[i,j] * x[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: ['sum(Low_Estimate[i,j] * x[i,j]) <= budget_limit', 'x[i,j] ∈ {0, 1} for all i, j'] (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 a new table for decision variables, modifying existing tables to ensure all optimization requirements are met, and updating business configuration logic for scalar parameters and formulas.

CREATE TABLE film_market_estimation (
  film_id INTEGER,
  market_id INTEGER,
  Low_Estimate FLOAT,
  High_Estimate FLOAT
);

CREATE TABLE film_market_decision (
  film_id INTEGER,
  market_id INTEGER,
  x BOOLEAN
);


```

### 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 revenue estimates for film distribution in various markets, ensuring a mix of high and low estimates to reflect market variability. Budget limits were set to allow for strategic decision-making without trivializing the optimization problem.

-- Realistic data for film_market_estimation
INSERT INTO film_market_estimation (film_id, market_id, Low_Estimate, High_Estimate) VALUES (1, 101, 12000.0, 18000.0);
INSERT INTO film_market_estimation (film_id, market_id, Low_Estimate, High_Estimate) VALUES (1, 102, 15000.0, 22000.0);
INSERT INTO film_market_estimation (film_id, market_id, Low_Estimate, High_Estimate) VALUES (2, 101, 8000.0, 12000.0);
INSERT INTO film_market_estimation (film_id, market_id, Low_Estimate, High_Estimate) VALUES (2, 103, 20000.0, 30000.0);
INSERT INTO film_market_estimation (film_id, market_id, Low_Estimate, High_Estimate) VALUES (3, 102, 25000.0, 35000.0);

-- Realistic data for film_market_decision
INSERT INTO film_market_decision (film_id, market_id, x) VALUES (1, 101, True);
INSERT INTO film_market_decision (film_id, market_id, x) VALUES (1, 102, False);
INSERT INTO film_market_decision (film_id, market_id, x) VALUES (2, 101, False);
INSERT INTO film_market_decision (film_id, market_id, x) VALUES (2, 103, True);
INSERT INTO film_market_decision (film_id, market_id, x) VALUES (3, 102, True);


```

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

Response:
# Complete Optimization Problem and Solution: film_rank

## 1. Problem Context and Goals

### Context  
A film distribution company is focused on maximizing its total gross revenue by strategically distributing films across various markets. The decision-making process involves determining whether to distribute each film in specific markets, represented by binary decisions. The company uses estimated revenue ranges for each film-market pair to guide these decisions. The primary objective is to maximize the total gross revenue, calculated as the sum of the lower revenue estimates for each film-market pair where the film is distributed. This decision-making process is subject to a maximum budget constraint, which limits the total expenditure on film distribution. Additionally, the company employs a preference score to prioritize film-market pairs, calculated based on market popularity and film rating. This approach ensures that the decision-making process remains linear and straightforward, avoiding complex nonlinear relationships.

### Goals  
The primary goal of the optimization process is to maximize the total gross revenue from film distribution. This involves selecting film-market pairs that contribute the most to revenue while adhering to budget constraints. The success of this optimization is measured by the total gross revenue achieved, which is calculated as the sum of the lower revenue estimates for the selected film-market pairs. The objective is clearly defined in linear terms, focusing on maximizing revenue within the given constraints.

## 2. Constraints    

The optimization process is governed by specific constraints that ensure the decision-making remains within practical and financial limits. The total revenue from the selected film-market pairs must not exceed the predefined budget limit. Additionally, each decision variable, representing whether a film is distributed in a market, is binary, meaning a film can either be distributed in a market or not. These constraints are designed to maintain a linear relationship, ensuring the optimization problem remains solvable using linear methods.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating a new table for decision variables, modifying existing tables to ensure all optimization requirements are met, and updating business configuration logic for scalar parameters and formulas.

CREATE TABLE film_market_estimation (
  film_id INTEGER,
  market_id INTEGER,
  Low_Estimate FLOAT,
  High_Estimate FLOAT
);

CREATE TABLE film_market_decision (
  film_id INTEGER,
  market_id INTEGER,
  x BOOLEAN
);
```

### Data Dictionary  
The data used in this optimization process is organized into two main tables, each serving a distinct purpose in the decision-making framework:

- **Film Market Estimation Table**: This table provides estimated revenue data for each film-market pair. It includes:
  - **Film ID**: A unique identifier for each film, used to track and manage film-specific data.
  - **Market ID**: A unique identifier for each market, facilitating market-specific analysis and decisions.
  - **Low Estimate**: The estimated lower bound of revenue for each film-market pair, serving as the objective coefficient in the optimization process.
  - **High Estimate**: The estimated upper bound of revenue, used as a reference for potential revenue but not directly in the optimization objective.

- **Film Market Decision Table**: This table records the decision variables for film distribution, indicating whether a film is distributed in a particular market. It includes:
  - **Film ID**: A unique identifier for each film, aligning with the estimation table for consistency.
  - **Market ID**: A unique identifier for each market, ensuring decisions are market-specific.
  - **Decision Variable (x)**: A binary indicator of whether a film is distributed in a market, forming the core decision variable in the optimization model.

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical revenue estimates for film distribution in various markets, ensuring a mix of high and low estimates to reflect market variability. Budget limits were set to allow for strategic decision-making without trivializing the optimization problem.

-- Realistic data for film_market_estimation
INSERT INTO film_market_estimation (film_id, market_id, Low_Estimate, High_Estimate) VALUES (1, 101, 12000.0, 18000.0);
INSERT INTO film_market_estimation (film_id, market_id, Low_Estimate, High_Estimate) VALUES (1, 102, 15000.0, 22000.0);
INSERT INTO film_market_estimation (film_id, market_id, Low_Estimate, High_Estimate) VALUES (2, 101, 8000.0, 12000.0);
INSERT INTO film_market_estimation (film_id, market_id, Low_Estimate, High_Estimate) VALUES (2, 103, 20000.0, 30000.0);
INSERT INTO film_market_estimation (film_id, market_id, Low_Estimate, High_Estimate) VALUES (3, 102, 25000.0, 35000.0);

-- Realistic data for film_market_decision
INSERT INTO film_market_decision (film_id, market_id, x) VALUES (1, 101, True);
INSERT INTO film_market_decision (film_id, market_id, x) VALUES (1, 102, False);
INSERT INTO film_market_decision (film_id, market_id, x) VALUES (2, 101, False);
INSERT INTO film_market_decision (film_id, market_id, x) VALUES (2, 103, True);
INSERT INTO film_market_decision (film_id, market_id, x) VALUES (3, 102, True);
```
