Iteration final - PROBLEM_DESCRIPTION
Sequence: 5
Timestamp: 2025-07-25 22:41:41

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: Allocation_ij: continuous variable representing the proportion of film i allocated to market j
- Operational parameters align with expected linear objective: maximize ∑(Average_Revenue_ij × Allocation_ij) where Average_Revenue_ij is the average revenue estimate for film i in market j and Allocation_ij is the proportion of film i allocated to market j
- Business configuration includes: Maximum proportion of a film that can be allocated to any market (used for Used to set constraint bounds)
- Business logic formulas to express in natural language: Average revenue estimate for film i in market j (calculation method for Used to calculate objective coefficients)
- 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 studio aims to maximize its total gross revenue by strategically allocating its films to different markets based on average revenue estimates.",
  "optimization_problem_description": "The studio wants to decide the proportion of each film to allocate to each market to maximize total gross revenue, using average revenue estimates for each film-market pair and ensuring that the total allocation for each film does not exceed 100%.",
  "optimization_formulation": {
    "objective": "maximize \u2211(Average_Revenue_ij \u00d7 Allocation_ij) where Average_Revenue_ij is the average revenue estimate for film i in market j and Allocation_ij is the proportion of film i allocated to market j",
    "decision_variables": "Allocation_ij: continuous variable representing the proportion of film i allocated to market j",
    "constraints": "\u2211(Allocation_ij) \u2264 1 for each film i, Allocation_ij \u2265 0 for all i, j"
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "Average_Revenue_ij": {
        "currently_mapped_to": "film_market_revenue_estimate.average_revenue",
        "mapping_adequacy": "good",
        "description": "Average revenue estimate for film i in market j"
      }
    },
    "constraint_bounds": {
      "Allocation_Constraint_i": {
        "currently_mapped_to": "business_configuration_logic.allocation_constraint_bound",
        "mapping_adequacy": "good",
        "description": "Maximum proportion of a film that can be allocated to any market"
      }
    },
    "decision_variables": {
      "Allocation_ij": {
        "currently_mapped_to": "film_market_allocation.allocation_proportion",
        "mapping_adequacy": "good",
        "description": "Proportion of film i allocated to market j",
        "variable_type": "continuous"
      }
    }
  },
  "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 tables for film-market allocations and revenue estimates, modifying existing tables to better map optimization requirements, and adding business configuration logic for scalar parameters and formulas.

CREATE TABLE film_market_allocation (
  allocation_proportion FLOAT
);

CREATE TABLE film_market_revenue_estimate (
  low_estimate FLOAT,
  high_estimate FLOAT,
  average_revenue FLOAT
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on industry standards for film revenue estimates and allocation proportions, ensuring they are realistic and align with the business context of maximizing gross revenue.

-- Realistic data for film_market_allocation
INSERT INTO film_market_allocation (allocation_proportion) VALUES (0.3);
INSERT INTO film_market_allocation (allocation_proportion) VALUES (0.5);
INSERT INTO film_market_allocation (allocation_proportion) VALUES (0.2);

-- Realistic data for film_market_revenue_estimate
INSERT INTO film_market_revenue_estimate (low_estimate, high_estimate, average_revenue) VALUES (200000.0, 600000.0, 400000.0);
INSERT INTO film_market_revenue_estimate (low_estimate, high_estimate, average_revenue) VALUES (300000.0, 800000.0, 550000.0);
INSERT INTO film_market_revenue_estimate (low_estimate, high_estimate, average_revenue) VALUES (100000.0, 400000.0, 250000.0);
INSERT INTO film_market_revenue_estimate (low_estimate, high_estimate, average_revenue) VALUES (250000.0, 700000.0, 475000.0);
INSERT INTO film_market_revenue_estimate (low_estimate, high_estimate, average_revenue) VALUES (150000.0, 500000.0, 325000.0);


```

DATA DICTIONARY:
{
  "tables": {
    "film_market_allocation": {
      "business_purpose": "Proportion of each film allocated to each market",
      "optimization_role": "decision_variables",
      "columns": {
        "allocation_proportion": {
          "data_type": "FLOAT",
          "business_meaning": "Proportion of film i allocated to market j",
          "optimization_purpose": "Decision variable in optimization formulation",
          "sample_values": "0.0 to 1.0"
        }
      }
    },
    "film_market_revenue_estimate": {
      "business_purpose": "Low and high revenue estimates for each film in each market",
      "optimization_role": "objective_coefficients",
      "columns": {
        "low_estimate": {
          "data_type": "FLOAT",
          "business_meaning": "Low revenue estimate for film i in market j",
          "optimization_purpose": "Used to calculate average revenue estimate",
          "sample_values": "100000.0 to 500000.0"
        },
        "high_estimate": {
          "data_type": "FLOAT",
          "business_meaning": "High revenue estimate for film i in market j",
          "optimization_purpose": "Used to calculate average revenue estimate",
          "sample_values": "500000.0 to 1000000.0"
        },
        "average_revenue": {
          "data_type": "FLOAT",
          "business_meaning": "Average revenue estimate for film i in market j",
          "optimization_purpose": "Objective coefficient in optimization formulation",
          "sample_values": "300000.0 to 750000.0"
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "revenue_estimation_formula": {
    "data_type": "STRING",
    "business_meaning": "Average revenue estimate for film i in market j",
    "optimization_role": "Used to calculate objective coefficients",
    "configuration_type": "business_logic_formula",
    "formula_expression": "(Low_Estimate_ij + High_Estimate_ij) / 2"
  },
  "allocation_constraint_bound": {
    "data_type": "FLOAT",
    "business_meaning": "Maximum proportion of a film that can be allocated to any market",
    "optimization_role": "Used to set constraint bounds",
    "configuration_type": "scalar_parameter",
    "value": 1.0,
    "business_justification": "Ensures that the total allocation for each film does not exceed 100%, maintaining feasibility in the optimization problem."
  }
}

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: Allocation_ij: continuous variable representing the proportion of film i allocated to market j
- Operational parameters align with expected linear objective: maximize ∑(Average_Revenue_ij × Allocation_ij) where Average_Revenue_ij is the average revenue estimate for film i in market j and Allocation_ij is the proportion of film i allocated to market j
- Business configuration includes: Maximum proportion of a film that can be allocated to any market (used for Used to set constraint bounds)
- Business logic formulas to express in natural language: Average revenue estimate for film i in market j (calculation method for Used to calculate objective coefficients)
- 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 ∑(Average_Revenue_ij × Allocation_ij) where Average_Revenue_ij is the average revenue estimate for film i in market j and Allocation_ij is the proportion of film i allocated to market 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: ∑(Allocation_ij) ≤ 1 for each film i, Allocation_ij ≥ 0 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 tables for film-market allocations and revenue estimates, modifying existing tables to better map optimization requirements, and adding business configuration logic for scalar parameters and formulas.

CREATE TABLE film_market_allocation (
  allocation_proportion FLOAT
);

CREATE TABLE film_market_revenue_estimate (
  low_estimate FLOAT,
  high_estimate FLOAT,
  average_revenue 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 industry standards for film revenue estimates and allocation proportions, ensuring they are realistic and align with the business context of maximizing gross revenue.

-- Realistic data for film_market_allocation
INSERT INTO film_market_allocation (allocation_proportion) VALUES (0.3);
INSERT INTO film_market_allocation (allocation_proportion) VALUES (0.5);
INSERT INTO film_market_allocation (allocation_proportion) VALUES (0.2);

-- Realistic data for film_market_revenue_estimate
INSERT INTO film_market_revenue_estimate (low_estimate, high_estimate, average_revenue) VALUES (200000.0, 600000.0, 400000.0);
INSERT INTO film_market_revenue_estimate (low_estimate, high_estimate, average_revenue) VALUES (300000.0, 800000.0, 550000.0);
INSERT INTO film_market_revenue_estimate (low_estimate, high_estimate, average_revenue) VALUES (100000.0, 400000.0, 250000.0);
INSERT INTO film_market_revenue_estimate (low_estimate, high_estimate, average_revenue) VALUES (250000.0, 700000.0, 475000.0);
INSERT INTO film_market_revenue_estimate (low_estimate, high_estimate, average_revenue) VALUES (150000.0, 500000.0, 325000.0);


```

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 studio is focused on maximizing its total gross revenue by strategically allocating its films to different markets. The studio must decide the proportion of each film to allocate to each market based on average revenue estimates. The average revenue for each film in each market is calculated as the midpoint between the low and high revenue estimates, ensuring a balanced and realistic projection. The studio has set a business rule that the total allocation for any single film across all markets cannot exceed 100%, ensuring feasible and practical distribution. This decision-making process is designed to be linear, avoiding complex relationships such as variable products or divisions, and relies on straightforward proportional allocations and revenue estimates.

### Goals  
The primary goal of the optimization is to maximize the studio's total gross revenue. This is achieved by determining the optimal proportion of each film to allocate to each market, weighted by the average revenue estimate for that film-market pair. Success is measured by the total revenue generated from these allocations, ensuring that the studio's resources are used efficiently and effectively. The optimization process is designed to be linear, focusing on proportional allocations and revenue estimates without introducing nonlinear complexities.

## 2. Constraints    

The optimization problem is subject to two key constraints. First, the total allocation for each film across all markets must not exceed 100%. This ensures that the studio does not overcommit its films to multiple markets beyond their availability. Second, the allocation proportion for each film in each market must be non-negative, ensuring that the allocations are realistic and feasible. These constraints are designed to be linear, avoiding any nonlinear relationships and ensuring that the problem remains straightforward and solvable using linear optimization techniques.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating tables for film-market allocations and revenue estimates, modifying existing tables to better map optimization requirements, and adding business configuration logic for scalar parameters and formulas.

CREATE TABLE film_market_allocation (
  allocation_proportion FLOAT
);

CREATE TABLE film_market_revenue_estimate (
  low_estimate FLOAT,
  high_estimate FLOAT,
  average_revenue FLOAT
);
```

### Data Dictionary  
- **film_market_allocation**: This table tracks the proportion of each film allocated to each market.  
  - **allocation_proportion**: Represents the proportion of a film allocated to a specific market. This is a continuous value ranging from 0 to 1, where 0 means no allocation and 1 means full allocation. It serves as the decision variable in the optimization problem.  

- **film_market_revenue_estimate**: This table contains revenue estimates for each film in each market.  
  - **low_estimate**: The minimum expected revenue for a film in a specific market.  
  - **high_estimate**: The maximum expected revenue for a film in a specific market.  
  - **average_revenue**: The midpoint between the low and high estimates, representing the average expected revenue. This value is used as the coefficient in the optimization objective to maximize total revenue.  

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on industry standards for film revenue estimates and allocation proportions, ensuring they are realistic and align with the business context of maximizing gross revenue.

-- Realistic data for film_market_allocation
INSERT INTO film_market_allocation (allocation_proportion) VALUES (0.3);
INSERT INTO film_market_allocation (allocation_proportion) VALUES (0.5);
INSERT INTO film_market_allocation (allocation_proportion) VALUES (0.2);

-- Realistic data for film_market_revenue_estimate
INSERT INTO film_market_revenue_estimate (low_estimate, high_estimate, average_revenue) VALUES (200000.0, 600000.0, 400000.0);
INSERT INTO film_market_revenue_estimate (low_estimate, high_estimate, average_revenue) VALUES (300000.0, 800000.0, 550000.0);
INSERT INTO film_market_revenue_estimate (low_estimate, high_estimate, average_revenue) VALUES (100000.0, 400000.0, 250000.0);
INSERT INTO film_market_revenue_estimate (low_estimate, high_estimate, average_revenue) VALUES (250000.0, 700000.0, 475000.0);
INSERT INTO film_market_revenue_estimate (low_estimate, high_estimate, average_revenue) VALUES (150000.0, 500000.0, 325000.0);
```
