Iteration final - PROBLEM_DESCRIPTION
Sequence: 5
Timestamp: 2025-07-27 23:47:35

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[player_id, match_num] where x is a binary variable indicating if player_id is assigned to match_num
- Operational parameters align with expected linear objective: minimize total_travel_distance = ∑(distance[player_id, match_num] * x[player_id, match_num])
- Business configuration includes: Total available time for a player to play matches (used for Used as a constraint bound in optimization model)
- Business logic formulas to express in natural language: Formula to calculate travel distance between player and match location (calculation method for Used to determine objective coefficients 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": "wta_1",
  "iteration": 1,
  "business_context": "Optimize the scheduling of tennis matches to minimize the total travel distance for players while ensuring all matches are played within a tournament's duration.",
  "optimization_problem_description": "The goal is to minimize the total travel distance for players between matches in a tournament, considering the constraints of match scheduling and player availability.",
  "optimization_formulation": {
    "objective": "minimize total_travel_distance = \u2211(distance[player_id, match_num] * x[player_id, match_num])",
    "decision_variables": "x[player_id, match_num] where x is a binary variable indicating if player_id is assigned to match_num",
    "constraints": [
      "\u2211(x[player_id, match_num] * minutes[match_num]) <= available_time[player_id] for all player_id",
      "\u2211(x[player_id, match_num]) = 1 for all match_num"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "distance[player_id, match_num]": {
        "currently_mapped_to": "travel_distances.distance",
        "mapping_adequacy": "good",
        "description": "Travel distance for player to reach match location"
      }
    },
    "constraint_bounds": {
      "available_time[player_id]": {
        "currently_mapped_to": "player_availability.available_time",
        "mapping_adequacy": "good",
        "description": "Total available time for a player to play matches"
      },
      "minutes[match_num]": {
        "currently_mapped_to": "matches.minutes",
        "mapping_adequacy": "good",
        "description": "Duration of the match in minutes"
      }
    },
    "decision_variables": {
      "x[player_id, match_num]": {
        "currently_mapped_to": "matches.player_id",
        "mapping_adequacy": "good",
        "description": "Binary variable indicating if player_id is assigned to match_num",
        "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 travel distances and player availability, modifying existing tables for better mapping, and updating configuration logic for scalar parameters and formulas.

CREATE TABLE travel_distances (
  player_id INTEGER,
  match_num INTEGER,
  distance FLOAT
);

CREATE TABLE player_availability (
  player_id INTEGER,
  available_time INTEGER
);

CREATE TABLE matches (
  match_num INTEGER,
  minutes INTEGER,
  player_id INTEGER
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical tournament schedules, player travel patterns, and match durations to ensure a realistic and solvable optimization problem.

-- Realistic data for travel_distances
INSERT INTO travel_distances (player_id, match_num, distance) VALUES (1, 101, 12.5);
INSERT INTO travel_distances (player_id, match_num, distance) VALUES (2, 102, 18.0);
INSERT INTO travel_distances (player_id, match_num, distance) VALUES (3, 103, 8.0);

-- Realistic data for player_availability
INSERT INTO player_availability (player_id, available_time) VALUES (1, 150);
INSERT INTO player_availability (player_id, available_time) VALUES (2, 180);
INSERT INTO player_availability (player_id, available_time) VALUES (3, 120);

-- Realistic data for matches
INSERT INTO matches (match_num, minutes, player_id) VALUES (101, 90, 1);
INSERT INTO matches (match_num, minutes, player_id) VALUES (102, 120, 2);
INSERT INTO matches (match_num, minutes, player_id) VALUES (103, 60, 3);


```

DATA DICTIONARY:
{
  "tables": {
    "travel_distances": {
      "business_purpose": "Stores travel distances for players between match locations",
      "optimization_role": "objective_coefficients",
      "columns": {
        "player_id": {
          "data_type": "INTEGER",
          "business_meaning": "Identifier for the player",
          "optimization_purpose": "Used to index travel distances",
          "sample_values": "1, 2, 3"
        },
        "match_num": {
          "data_type": "INTEGER",
          "business_meaning": "Identifier for the match",
          "optimization_purpose": "Used to index travel distances",
          "sample_values": "101, 102, 103"
        },
        "distance": {
          "data_type": "FLOAT",
          "business_meaning": "Travel distance for player to reach match location",
          "optimization_purpose": "Coefficient in objective function",
          "sample_values": "10.5, 20.0, 15.3"
        }
      }
    },
    "player_availability": {
      "business_purpose": "Stores available time for each player to play matches",
      "optimization_role": "constraint_bounds",
      "columns": {
        "player_id": {
          "data_type": "INTEGER",
          "business_meaning": "Identifier for the player",
          "optimization_purpose": "Used to index available time",
          "sample_values": "1, 2, 3"
        },
        "available_time": {
          "data_type": "INTEGER",
          "business_meaning": "Total available time for a player to play matches",
          "optimization_purpose": "Constraint bound in optimization model",
          "sample_values": "120, 150, 180"
        }
      }
    },
    "matches": {
      "business_purpose": "Stores match details including duration and player assignments",
      "optimization_role": "decision_variables",
      "columns": {
        "match_num": {
          "data_type": "INTEGER",
          "business_meaning": "Identifier for the match",
          "optimization_purpose": "Used to index matches",
          "sample_values": "101, 102, 103"
        },
        "minutes": {
          "data_type": "INTEGER",
          "business_meaning": "Duration of the match in minutes",
          "optimization_purpose": "Constraint bound in optimization model",
          "sample_values": "90, 120, 60"
        },
        "player_id": {
          "data_type": "INTEGER",
          "business_meaning": "Identifier for the player assigned to the match",
          "optimization_purpose": "Decision variable in optimization model",
          "sample_values": "1, 2, 3"
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "available_time": {
    "data_type": "INTEGER",
    "business_meaning": "Total available time for a player to play matches",
    "optimization_role": "Used as a constraint bound in optimization model",
    "configuration_type": "scalar_parameter",
    "value": 150,
    "business_justification": "150 minutes is a realistic average available time for players in a tournament setting."
  },
  "distance_formula": {
    "data_type": "STRING",
    "business_meaning": "Formula to calculate travel distance between player and match location",
    "optimization_role": "Used to determine objective coefficients in optimization model",
    "configuration_type": "business_logic_formula",
    "formula_expression": "distance[player_id, match_num] = calculate_distance(player_location, match_location)"
  }
}

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: wta_1

## 1. Problem Context and Goals

### Context  
[Regenerate business context that naturally aligns with LINEAR optimization formulation. Ensure:]
- Business decisions match expected decision variables: x[player_id, match_num] where x is a binary variable indicating if player_id is assigned to match_num
- Operational parameters align with expected linear objective: minimize total_travel_distance = ∑(distance[player_id, match_num] * x[player_id, match_num])
- Business configuration includes: Total available time for a player to play matches (used for Used as a constraint bound in optimization model)
- Business logic formulas to express in natural language: Formula to calculate travel distance between player and match location (calculation method for Used to determine objective coefficients 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 total_travel_distance = ∑(distance[player_id, match_num] * x[player_id, match_num])
- 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[player_id, match_num] * minutes[match_num]) <= available_time[player_id] for all player_id', '∑(x[player_id, match_num]) = 1 for all match_num'] (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 travel distances and player availability, modifying existing tables for better mapping, and updating configuration logic for scalar parameters and formulas.

CREATE TABLE travel_distances (
  player_id INTEGER,
  match_num INTEGER,
  distance FLOAT
);

CREATE TABLE player_availability (
  player_id INTEGER,
  available_time INTEGER
);

CREATE TABLE matches (
  match_num INTEGER,
  minutes INTEGER,
  player_id 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 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical tournament schedules, player travel patterns, and match durations to ensure a realistic and solvable optimization problem.

-- Realistic data for travel_distances
INSERT INTO travel_distances (player_id, match_num, distance) VALUES (1, 101, 12.5);
INSERT INTO travel_distances (player_id, match_num, distance) VALUES (2, 102, 18.0);
INSERT INTO travel_distances (player_id, match_num, distance) VALUES (3, 103, 8.0);

-- Realistic data for player_availability
INSERT INTO player_availability (player_id, available_time) VALUES (1, 150);
INSERT INTO player_availability (player_id, available_time) VALUES (2, 180);
INSERT INTO player_availability (player_id, available_time) VALUES (3, 120);

-- Realistic data for matches
INSERT INTO matches (match_num, minutes, player_id) VALUES (101, 90, 1);
INSERT INTO matches (match_num, minutes, player_id) VALUES (102, 120, 2);
INSERT INTO matches (match_num, minutes, player_id) VALUES (103, 60, 3);


```

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

Response:
# Complete Optimization Problem and Solution: wta_1

## 1. Problem Context and Goals

### Context  
The objective is to optimize the scheduling of tennis matches in a tournament to minimize the total travel distance for players. Each player is assigned to matches, and the decision variable is whether a player is assigned to a specific match. The operational parameters include minimizing the total travel distance, which is calculated by summing the travel distances for each player to their assigned matches. The business configuration includes the total available time for a player to participate in matches, which serves as a constraint in the optimization model. The travel distance between a player's location and the match location is determined using a predefined formula, ensuring that the calculation method aligns with the objective coefficients in the optimization model. The focus is on precise operational decision-making that leads to linear formulations, avoiding any nonlinear relationships. Resource limitations are expressed as linear constraints, and specific operational parameters are mapped to expected coefficient sources. The business configuration parameters, including scalar parameters and business logic formulas, are integral to the model.

### Goals  
The primary goal is to minimize the total travel distance for players participating in the tournament. This involves optimizing the assignment of players to matches such that the sum of the travel distances is minimized. Success is measured by the reduction in total travel distance, aligning with the expected coefficient sources. The optimization goal is clearly defined in natural language, focusing on minimizing the travel distance without using mathematical formulas or symbolic notation.

## 2. Constraints    

The constraints are designed to ensure that the optimization problem remains linear and solvable. The first constraint ensures that the total time a player spends in matches does not exceed their available time. This is expressed as the sum of the time spent in each match for a player being less than or equal to their available time. The second constraint ensures that each match has exactly one player assigned to it. These constraints are described in business terms, naturally leading to linear mathematical forms without involving variable products or divisions.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating new tables for travel distances and player availability, modifying existing tables for better mapping, and updating configuration logic for scalar parameters and formulas.

CREATE TABLE travel_distances (
  player_id INTEGER,
  match_num INTEGER,
  distance FLOAT
);

CREATE TABLE player_availability (
  player_id INTEGER,
  available_time INTEGER
);

CREATE TABLE matches (
  match_num INTEGER,
  minutes INTEGER,
  player_id INTEGER
);
```

### Data Dictionary  
The data dictionary provides a comprehensive mapping of tables and columns to their business purposes and optimization roles. 

- **Travel Distances Table**: This table stores the travel distances for players between match locations. It plays a critical role in determining the objective coefficients for the optimization model. Each entry includes a player identifier, a match identifier, and the travel distance, which is used to calculate the total travel distance.

- **Player Availability Table**: This table records the total available time each player has to participate in matches. It serves as a constraint bound in the optimization model, ensuring that players do not exceed their available time when assigned to matches.

- **Matches Table**: This table contains details about each match, including its duration and the player assigned to it. The match duration is used as a constraint bound, while the player assignment is a 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 tournament schedules, player travel patterns, and match durations to ensure a realistic and solvable optimization problem.

-- Realistic data for travel_distances
INSERT INTO travel_distances (player_id, match_num, distance) VALUES (1, 101, 12.5);
INSERT INTO travel_distances (player_id, match_num, distance) VALUES (2, 102, 18.0);
INSERT INTO travel_distances (player_id, match_num, distance) VALUES (3, 103, 8.0);

-- Realistic data for player_availability
INSERT INTO player_availability (player_id, available_time) VALUES (1, 150);
INSERT INTO player_availability (player_id, available_time) VALUES (2, 180);
INSERT INTO player_availability (player_id, available_time) VALUES (3, 120);

-- Realistic data for matches
INSERT INTO matches (match_num, minutes, player_id) VALUES (101, 90, 1);
INSERT INTO matches (match_num, minutes, player_id) VALUES (102, 120, 2);
INSERT INTO matches (match_num, minutes, player_id) VALUES (103, 60, 3);
```
