Iteration final - PROBLEM_DESCRIPTION
Sequence: 5
Timestamp: 2025-07-25 22:29:28

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[accelerator_id, browser_id] ∈ {0, 1} (binary variable indicating if accelerator_id is compatible with browser_id)
- Operational parameters align with expected linear objective: maximize ∑(market_share[browser_id] × x[accelerator_id, browser_id])
- Business configuration includes: Maximum number of accelerators that can be selected (used for Constraint bound for the total number of accelerators)
- Business logic formulas to express in natural language: Formula to calculate compatibility score between accelerators and browsers (calculation method for Used in decision variable calculations)
- 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": "browser_web",
  "iteration": 1,
  "business_context": "A company aims to optimize the selection of web client accelerators to maximize compatibility with browsers based on market share, ensuring that the chosen accelerators are compatible with the most widely used browsers.",
  "optimization_problem_description": "Maximize the total market share of browsers compatible with selected web client accelerators, subject to constraints on the number of accelerators that can be selected and ensuring compatibility with at least one browser per accelerator.",
  "optimization_formulation": {
    "objective": "maximize \u2211(market_share[browser_id] \u00d7 x[accelerator_id, browser_id])",
    "decision_variables": "x[accelerator_id, browser_id] \u2208 {0, 1} (binary variable indicating if accelerator_id is compatible with browser_id)",
    "constraints": [
      "\u2211(x[accelerator_id, browser_id]) \u2264 max_accelerators (total number of selected accelerators cannot exceed max_accelerators)",
      "\u2211(x[accelerator_id, browser_id]) \u2265 1 for each accelerator_id (each selected accelerator must be compatible with at least one browser)"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "market_share[browser_id]": {
        "currently_mapped_to": "browser.market_share",
        "mapping_adequacy": "good",
        "description": "Market share of each browser used as coefficient in the objective function"
      }
    },
    "constraint_bounds": {
      "max_accelerators": {
        "currently_mapped_to": "business_configuration_logic.max_accelerators",
        "mapping_adequacy": "good",
        "description": "Maximum number of accelerators that can be selected"
      }
    },
    "decision_variables": {
      "x[accelerator_id, browser_id]": {
        "currently_mapped_to": "accelerator_compatible_browser.compatibility_score",
        "mapping_adequacy": "partial",
        "description": "Binary decision variable indicating compatibility between accelerator and browser",
        "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 adding a max_accelerators parameter to business_configuration_logic.json, enhancing the accelerator_compatible_browser table with additional attributes, and ensuring all tables meet the 3-row minimum rule. Configuration logic updates include scalar parameters and formulas for optimization constraints and metrics.

CREATE TABLE browser (
  browser_id INTEGER,
  market_share FLOAT
);

CREATE TABLE accelerator_compatible_browser (
  accelerator_id INTEGER,
  browser_id INTEGER,
  compatibility_score FLOAT,
  last_updated_date DATE
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic market share data for browsers, compatibility scores reflecting real-world scenarios, and a maximum number of accelerators that aligns with business constraints.

-- Realistic data for browser
INSERT INTO browser (browser_id, market_share) VALUES (1, 0.35);
INSERT INTO browser (browser_id, market_share) VALUES (2, 0.25);
INSERT INTO browser (browser_id, market_share) VALUES (3, 0.15);

-- Realistic data for accelerator_compatible_browser
INSERT INTO accelerator_compatible_browser (accelerator_id, browser_id, compatibility_score, last_updated_date) VALUES (1, 1, 0.9, 2023-10-01);
INSERT INTO accelerator_compatible_browser (accelerator_id, browser_id, compatibility_score, last_updated_date) VALUES (1, 2, 0.8, 2023-10-01);
INSERT INTO accelerator_compatible_browser (accelerator_id, browser_id, compatibility_score, last_updated_date) VALUES (2, 1, 0.85, 2023-10-02);
INSERT INTO accelerator_compatible_browser (accelerator_id, browser_id, compatibility_score, last_updated_date) VALUES (2, 3, 0.7, 2023-10-02);
INSERT INTO accelerator_compatible_browser (accelerator_id, browser_id, compatibility_score, last_updated_date) VALUES (3, 2, 0.75, 2023-10-03);
INSERT INTO accelerator_compatible_browser (accelerator_id, browser_id, compatibility_score, last_updated_date) VALUES (3, 3, 0.65, 2023-10-03);


```

DATA DICTIONARY:
{
  "tables": {
    "browser": {
      "business_purpose": "Stores browser market share data for optimization",
      "optimization_role": "objective_coefficients",
      "columns": {
        "browser_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each browser",
          "optimization_purpose": "Index for objective coefficients",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "market_share": {
          "data_type": "FLOAT",
          "business_meaning": "Market share percentage of the browser",
          "optimization_purpose": "Coefficient in the objective function",
          "sample_values": [
            0.25,
            0.3,
            0.2
          ]
        }
      }
    },
    "accelerator_compatible_browser": {
      "business_purpose": "Stores compatibility data between accelerators and browsers",
      "optimization_role": "decision_variables",
      "columns": {
        "accelerator_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each accelerator",
          "optimization_purpose": "Index for decision variables",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "browser_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each browser",
          "optimization_purpose": "Index for decision variables",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "compatibility_score": {
          "data_type": "FLOAT",
          "business_meaning": "Score indicating compatibility between accelerator and browser",
          "optimization_purpose": "Used in decision variable calculations",
          "sample_values": [
            0.8,
            0.9,
            0.7
          ]
        },
        "last_updated_date": {
          "data_type": "DATE",
          "business_meaning": "Date when the compatibility data was last updated",
          "optimization_purpose": "Ensures data freshness",
          "sample_values": [
            "2023-10-01",
            "2023-10-02",
            "2023-10-03"
          ]
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "max_accelerators": {
    "data_type": "INTEGER",
    "business_meaning": "Maximum number of accelerators that can be selected",
    "optimization_role": "Constraint bound for the total number of accelerators",
    "configuration_type": "scalar_parameter",
    "value": 3,
    "business_justification": "A realistic limit on the number of accelerators the company can feasibly implement and maintain."
  },
  "compatibility_score_formula": {
    "data_type": "STRING",
    "business_meaning": "Formula to calculate compatibility score between accelerators and browsers",
    "optimization_role": "Used in decision variable calculations",
    "configuration_type": "business_logic_formula",
    "formula_expression": "(market_share * compatibility_weight) / total_weight"
  }
}

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: browser_web

## 1. Problem Context and Goals

### Context  
[Regenerate business context that naturally aligns with LINEAR optimization formulation. Ensure:]
- Business decisions match expected decision variables: x[accelerator_id, browser_id] ∈ {0, 1} (binary variable indicating if accelerator_id is compatible with browser_id)
- Operational parameters align with expected linear objective: maximize ∑(market_share[browser_id] × x[accelerator_id, browser_id])
- Business configuration includes: Maximum number of accelerators that can be selected (used for Constraint bound for the total number of accelerators)
- Business logic formulas to express in natural language: Formula to calculate compatibility score between accelerators and browsers (calculation method for Used in decision variable calculations)
- 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 ∑(market_share[browser_id] × x[accelerator_id, browser_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: ['∑(x[accelerator_id, browser_id]) ≤ max_accelerators (total number of selected accelerators cannot exceed max_accelerators)', '∑(x[accelerator_id, browser_id]) ≥ 1 for each accelerator_id (each selected accelerator must be compatible with at least one browser)'] (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 adding a max_accelerators parameter to business_configuration_logic.json, enhancing the accelerator_compatible_browser table with additional attributes, and ensuring all tables meet the 3-row minimum rule. Configuration logic updates include scalar parameters and formulas for optimization constraints and metrics.

CREATE TABLE browser (
  browser_id INTEGER,
  market_share FLOAT
);

CREATE TABLE accelerator_compatible_browser (
  accelerator_id INTEGER,
  browser_id INTEGER,
  compatibility_score FLOAT,
  last_updated_date DATE
);


```

### 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 market share data for browsers, compatibility scores reflecting real-world scenarios, and a maximum number of accelerators that aligns with business constraints.

-- Realistic data for browser
INSERT INTO browser (browser_id, market_share) VALUES (1, 0.35);
INSERT INTO browser (browser_id, market_share) VALUES (2, 0.25);
INSERT INTO browser (browser_id, market_share) VALUES (3, 0.15);

-- Realistic data for accelerator_compatible_browser
INSERT INTO accelerator_compatible_browser (accelerator_id, browser_id, compatibility_score, last_updated_date) VALUES (1, 1, 0.9, 2023-10-01);
INSERT INTO accelerator_compatible_browser (accelerator_id, browser_id, compatibility_score, last_updated_date) VALUES (1, 2, 0.8, 2023-10-01);
INSERT INTO accelerator_compatible_browser (accelerator_id, browser_id, compatibility_score, last_updated_date) VALUES (2, 1, 0.85, 2023-10-02);
INSERT INTO accelerator_compatible_browser (accelerator_id, browser_id, compatibility_score, last_updated_date) VALUES (2, 3, 0.7, 2023-10-02);
INSERT INTO accelerator_compatible_browser (accelerator_id, browser_id, compatibility_score, last_updated_date) VALUES (3, 2, 0.75, 2023-10-03);
INSERT INTO accelerator_compatible_browser (accelerator_id, browser_id, compatibility_score, last_updated_date) VALUES (3, 3, 0.65, 2023-10-03);


```

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

Response:
# Complete Optimization Problem and Solution: browser_web

## 1. Problem Context and Goals

### Context  
A company is focused on optimizing the selection of web client accelerators to ensure maximum compatibility with the most widely used browsers. The decision-making process involves determining which accelerators are compatible with which browsers, with the goal of maximizing the overall market share of compatible browsers. The company has a limited number of accelerators it can feasibly implement and maintain, as defined by a maximum number of accelerators that can be selected. This limit is a key operational constraint.  

The compatibility between accelerators and browsers is evaluated using a compatibility score, which is calculated based on the market share of the browser and a predefined weighting formula. This score ensures that the selection process prioritizes browsers with higher market shares, aligning with the company’s goal of maximizing user coverage.  

The business configuration includes a scalar parameter that defines the maximum number of accelerators that can be selected, currently set to three. Additionally, the compatibility score formula incorporates the market share of the browser and a compatibility weight, divided by a total weight, to determine the final score. This formula ensures that the decision variables are calculated in a way that supports linear optimization.  

### Goals  
The primary goal of this optimization problem is to maximize the total market share of browsers that are compatible with the selected accelerators. This is achieved by selecting accelerators that are compatible with the most widely used browsers, as measured by their market share. Success is measured by the extent to which the selected accelerators cover the highest possible market share of browsers, ensuring broad user compatibility.  

## 2. Constraints  

The optimization problem is subject to two key constraints:  
1. **Maximum Accelerators Constraint**: The total number of accelerators selected cannot exceed the predefined maximum number of accelerators, which is currently set to three. This ensures that the company does not exceed its operational capacity for implementing and maintaining accelerators.  
2. **Minimum Compatibility Constraint**: Each selected accelerator must be compatible with at least one browser. This ensures that no accelerator is selected without providing compatibility benefits to at least one browser.  

These constraints are designed to align with the company’s operational limitations and ensure that the optimization process remains feasible and practical.  

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include adding a max_accelerators parameter to business_configuration_logic.json, enhancing the accelerator_compatible_browser table with additional attributes, and ensuring all tables meet the 3-row minimum rule. Configuration logic updates include scalar parameters and formulas for optimization constraints and metrics.

CREATE TABLE browser (
  browser_id INTEGER,
  market_share FLOAT
);

CREATE TABLE accelerator_compatible_browser (
  accelerator_id INTEGER,
  browser_id INTEGER,
  compatibility_score FLOAT,
  last_updated_date DATE
);
```

### Data Dictionary  
- **browser**:  
  - **Business Purpose**: Stores browser market share data, which is critical for determining the optimization objective.  
  - **Optimization Role**: Provides the coefficients for the objective function, ensuring that the optimization process prioritizes browsers with higher market shares.  
  - **Columns**:  
    - **browser_id**: Unique identifier for each browser, used to index the objective coefficients.  
    - **market_share**: Percentage of the market share for each browser, used as the coefficient in the objective function.  

- **accelerator_compatible_browser**:  
  - **Business Purpose**: Stores compatibility data between accelerators and browsers, which is essential for determining the decision variables.  
  - **Optimization Role**: Provides the data used to calculate the decision variables, ensuring that the optimization process selects accelerators that are compatible with the most widely used browsers.  
  - **Columns**:  
    - **accelerator_id**: Unique identifier for each accelerator, used to index the decision variables.  
    - **browser_id**: Unique identifier for each browser, used to index the decision variables.  
    - **compatibility_score**: Score indicating the compatibility between an accelerator and a browser, used in the calculation of the decision variables.  
    - **last_updated_date**: Ensures that the compatibility data is current and reliable.  

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic market share data for browsers, compatibility scores reflecting real-world scenarios, and a maximum number of accelerators that aligns with business constraints.

-- Realistic data for browser
INSERT INTO browser (browser_id, market_share) VALUES (1, 0.35);
INSERT INTO browser (browser_id, market_share) VALUES (2, 0.25);
INSERT INTO browser (browser_id, market_share) VALUES (3, 0.15);

-- Realistic data for accelerator_compatible_browser
INSERT INTO accelerator_compatible_browser (accelerator_id, browser_id, compatibility_score, last_updated_date) VALUES (1, 1, 0.9, 2023-10-01);
INSERT INTO accelerator_compatible_browser (accelerator_id, browser_id, compatibility_score, last_updated_date) VALUES (1, 2, 0.8, 2023-10-01);
INSERT INTO accelerator_compatible_browser (accelerator_id, browser_id, compatibility_score, last_updated_date) VALUES (2, 1, 0.85, 2023-10-02);
INSERT INTO accelerator_compatible_browser (accelerator_id, browser_id, compatibility_score, last_updated_date) VALUES (2, 3, 0.7, 2023-10-02);
INSERT INTO accelerator_compatible_browser (accelerator_id, browser_id, compatibility_score, last_updated_date) VALUES (3, 2, 0.75, 2023-10-03);
INSERT INTO accelerator_compatible_browser (accelerator_id, browser_id, compatibility_score, last_updated_date) VALUES (3, 3, 0.65, 2023-10-03);
```
