Iteration final - PROBLEM_DESCRIPTION
Sequence: 5
Timestamp: 2025-07-27 23:59:04

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: resource_allocation.amount[Team_ID] (continuous)
- Operational parameters align with expected linear objective: maximize ∑(basketball_match.All_Games_Percent[Team_ID] × resource_allocation.amount[Team_ID])
- Business configuration includes: total resources available for allocation across all teams (used for constraint bound for total resources), minimum resources that must be allocated to each team (used for constraint bound for minimum allocation), maximum resources that can be allocated to each team (used for constraint bound for maximum allocation)
- 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": "university_basketball",
  "iteration": 1,
  "business_context": "A university basketball league aims to optimize resource allocation to maximize team performance, measured by win percentage, while adhering to resource constraints.",
  "optimization_problem_description": "The objective is to maximize the overall performance of basketball teams by optimally allocating resources such as training hours, coaching staff, and budget. The performance is measured by the win percentage in all games. Constraints include limits on total resources available and minimum and maximum resource allocation requirements for each team.",
  "optimization_formulation": {
    "objective": "maximize \u2211(basketball_match.All_Games_Percent[Team_ID] \u00d7 resource_allocation.amount[Team_ID])",
    "decision_variables": "resource_allocation.amount[Team_ID] (continuous)",
    "constraints": [
      "\u2211(resource_allocation.amount[Team_ID]) \u2264 business_configuration_logic.Total_Resources_Available",
      "resource_allocation.amount[Team_ID] \u2265 resource_constraints.Minimum_Allocation[Team_ID]",
      "resource_allocation.amount[Team_ID] \u2264 resource_constraints.Maximum_Allocation[Team_ID]"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "All_Games_Percent[Team_ID]": {
        "currently_mapped_to": "basketball_match.All_Games_Percent",
        "mapping_adequacy": "good",
        "description": "win percentage of each team in all games"
      }
    },
    "constraint_bounds": {
      "Total_Resources_Available": {
        "currently_mapped_to": "business_configuration_logic.Total_Resources_Available",
        "mapping_adequacy": "good",
        "description": "total resources available for allocation across all teams"
      },
      "Minimum_Allocation[Team_ID]": {
        "currently_mapped_to": "resource_constraints.Minimum_Allocation",
        "mapping_adequacy": "good",
        "description": "minimum resources that must be allocated to the team"
      },
      "Maximum_Allocation[Team_ID]": {
        "currently_mapped_to": "resource_constraints.Maximum_Allocation",
        "mapping_adequacy": "good",
        "description": "maximum resources that can be allocated to the team"
      }
    },
    "decision_variables": {
      "amount[Team_ID]": {
        "currently_mapped_to": "resource_allocation.amount",
        "mapping_adequacy": "good",
        "description": "amount of resources allocated to the team",
        "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 new tables for resource allocation constraints and decision variables, and updating existing tables to fill mapping gaps. Configuration logic updates include adding scalar parameters for total resources and resource allocation limits.

CREATE TABLE basketball_match (
  All_Games_Percent FLOAT,
  Resource_Allocation FLOAT
);

CREATE TABLE resource_allocation (
  Team_ID INTEGER,
  amount FLOAT
);

CREATE TABLE resource_constraints (
  Team_ID INTEGER,
  Minimum_Allocation FLOAT,
  Maximum_Allocation FLOAT
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical resource allocation scenarios in university sports, ensuring that constraints are neither too tight nor too loose, allowing for meaningful optimization.

-- Realistic data for basketball_match
INSERT INTO basketball_match (All_Games_Percent, Resource_Allocation) VALUES (0.75, 110);
INSERT INTO basketball_match (All_Games_Percent, Resource_Allocation) VALUES (0.6, 140);
INSERT INTO basketball_match (All_Games_Percent, Resource_Allocation) VALUES (0.85, 130);

-- Realistic data for resource_allocation
INSERT INTO resource_allocation (Team_ID, amount) VALUES (1, 110);
INSERT INTO resource_allocation (Team_ID, amount) VALUES (2, 140);
INSERT INTO resource_allocation (Team_ID, amount) VALUES (3, 130);

-- Realistic data for resource_constraints
INSERT INTO resource_constraints (Team_ID, Minimum_Allocation, Maximum_Allocation) VALUES (1, 50, 200);
INSERT INTO resource_constraints (Team_ID, Minimum_Allocation, Maximum_Allocation) VALUES (2, 60, 180);
INSERT INTO resource_constraints (Team_ID, Minimum_Allocation, Maximum_Allocation) VALUES (3, 70, 190);


```

DATA DICTIONARY:
{
  "tables": {
    "basketball_match": {
      "business_purpose": "stores match results and team performance metrics",
      "optimization_role": "objective_coefficients",
      "columns": {
        "All_Games_Percent": {
          "data_type": "FLOAT",
          "business_meaning": "win percentage of each team in all games",
          "optimization_purpose": "coefficient in the objective function",
          "sample_values": "0.75, 0.60, 0.85"
        },
        "Resource_Allocation": {
          "data_type": "FLOAT",
          "business_meaning": "amount of resources allocated to the team",
          "optimization_purpose": "decision variable for resource allocation",
          "sample_values": "100, 150, 120"
        }
      }
    },
    "resource_allocation": {
      "business_purpose": "stores the amount of resources allocated to each team",
      "optimization_role": "decision_variables",
      "columns": {
        "Team_ID": {
          "data_type": "INTEGER",
          "business_meaning": "unique identifier for each team",
          "optimization_purpose": "index for resource allocation",
          "sample_values": "1, 2, 3"
        },
        "amount": {
          "data_type": "FLOAT",
          "business_meaning": "amount of resources allocated to the team",
          "optimization_purpose": "decision variable for resource allocation",
          "sample_values": "100, 150, 120"
        }
      }
    },
    "resource_constraints": {
      "business_purpose": "stores resource allocation constraints for each team",
      "optimization_role": "constraint_bounds",
      "columns": {
        "Team_ID": {
          "data_type": "INTEGER",
          "business_meaning": "unique identifier for each team",
          "optimization_purpose": "index for constraints",
          "sample_values": "1, 2, 3"
        },
        "Minimum_Allocation": {
          "data_type": "FLOAT",
          "business_meaning": "minimum resources that must be allocated to the team",
          "optimization_purpose": "constraint bound for minimum allocation",
          "sample_values": "50, 60, 70"
        },
        "Maximum_Allocation": {
          "data_type": "FLOAT",
          "business_meaning": "maximum resources that can be allocated to the team",
          "optimization_purpose": "constraint bound for maximum allocation",
          "sample_values": "200, 180, 190"
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "Total_Resources_Available": {
    "data_type": "INTEGER",
    "business_meaning": "total resources available for allocation across all teams",
    "optimization_role": "constraint bound for total resources",
    "configuration_type": "scalar_parameter",
    "value": 1000,
    "business_justification": "This value reflects a realistic total budget for resource allocation across all teams, ensuring that the league can operate within its financial means."
  },
  "Minimum_Resource_Allocation": {
    "data_type": "INTEGER",
    "business_meaning": "minimum resources that must be allocated to each team",
    "optimization_role": "constraint bound for minimum allocation",
    "configuration_type": "scalar_parameter",
    "value": 50,
    "business_justification": "Ensures that each team receives a baseline level of resources necessary to participate in the league."
  },
  "Maximum_Resource_Allocation": {
    "data_type": "INTEGER",
    "business_meaning": "maximum resources that can be allocated to each team",
    "optimization_role": "constraint bound for maximum allocation",
    "configuration_type": "scalar_parameter",
    "value": 200,
    "business_justification": "Prevents any single team from monopolizing resources, promoting fair competition."
  }
}

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: university_basketball

## 1. Problem Context and Goals

### Context  
[Regenerate business context that naturally aligns with LINEAR optimization formulation. Ensure:]
- Business decisions match expected decision variables: resource_allocation.amount[Team_ID] (continuous)
- Operational parameters align with expected linear objective: maximize ∑(basketball_match.All_Games_Percent[Team_ID] × resource_allocation.amount[Team_ID])
- Business configuration includes: total resources available for allocation across all teams (used for constraint bound for total resources), minimum resources that must be allocated to each team (used for constraint bound for minimum allocation), maximum resources that can be allocated to each team (used for constraint bound for maximum allocation)
- 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 ∑(basketball_match.All_Games_Percent[Team_ID] × resource_allocation.amount[Team_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: ['∑(resource_allocation.amount[Team_ID]) ≤ business_configuration_logic.Total_Resources_Available', 'resource_allocation.amount[Team_ID] ≥ resource_constraints.Minimum_Allocation[Team_ID]', 'resource_allocation.amount[Team_ID] ≤ resource_constraints.Maximum_Allocation[Team_ID]'] (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 resource allocation constraints and decision variables, and updating existing tables to fill mapping gaps. Configuration logic updates include adding scalar parameters for total resources and resource allocation limits.

CREATE TABLE basketball_match (
  All_Games_Percent FLOAT,
  Resource_Allocation FLOAT
);

CREATE TABLE resource_allocation (
  Team_ID INTEGER,
  amount FLOAT
);

CREATE TABLE resource_constraints (
  Team_ID INTEGER,
  Minimum_Allocation FLOAT,
  Maximum_Allocation 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 typical resource allocation scenarios in university sports, ensuring that constraints are neither too tight nor too loose, allowing for meaningful optimization.

-- Realistic data for basketball_match
INSERT INTO basketball_match (All_Games_Percent, Resource_Allocation) VALUES (0.75, 110);
INSERT INTO basketball_match (All_Games_Percent, Resource_Allocation) VALUES (0.6, 140);
INSERT INTO basketball_match (All_Games_Percent, Resource_Allocation) VALUES (0.85, 130);

-- Realistic data for resource_allocation
INSERT INTO resource_allocation (Team_ID, amount) VALUES (1, 110);
INSERT INTO resource_allocation (Team_ID, amount) VALUES (2, 140);
INSERT INTO resource_allocation (Team_ID, amount) VALUES (3, 130);

-- Realistic data for resource_constraints
INSERT INTO resource_constraints (Team_ID, Minimum_Allocation, Maximum_Allocation) VALUES (1, 50, 200);
INSERT INTO resource_constraints (Team_ID, Minimum_Allocation, Maximum_Allocation) VALUES (2, 60, 180);
INSERT INTO resource_constraints (Team_ID, Minimum_Allocation, Maximum_Allocation) VALUES (3, 70, 190);


```

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

Response:
# Complete Optimization Problem and Solution: university_basketball

## 1. Problem Context and Goals

### Context  
The university basketball league is focused on optimizing the allocation of resources to its teams to enhance their performance. The primary decision involves determining the amount of resources, such as training hours, coaching staff, and budget, allocated to each team. This decision is represented by the continuous variable `resource_allocation.amount[Team_ID]`. The league's operational goal is to maximize the overall win percentage of the teams, which is directly influenced by the resources allocated. The win percentage for each team is captured in the `basketball_match.All_Games_Percent[Team_ID]`.

The league operates under specific resource constraints. The total resources available for allocation across all teams are limited, ensuring that the league remains within its financial means. Each team must receive a minimum level of resources to participate effectively, while no team should receive an excessive amount that could lead to unfair competition. These constraints are defined by the business configuration parameters: total resources available, minimum resources per team, and maximum resources per team.

### Goals  
The optimization goal is to maximize the overall performance of the basketball teams by strategically allocating resources. The metric for optimization is the sum of the product of each team's win percentage and the resources allocated to them. Success is measured by the increase in the league's overall win percentage, which is directly linked to the resources allocated to each team. The objective is to ensure that resources are distributed in a manner that maximizes this performance metric while adhering to the defined constraints.

## 2. Constraints    

The resource allocation process is governed by several linear constraints:

- The total amount of resources allocated to all teams must not exceed the total resources available. This ensures that the league operates within its budgetary limits.
- Each team must receive at least a minimum amount of resources to ensure they can compete effectively in the league.
- No team should receive more than the maximum allowable resources, preventing any single team from gaining an unfair advantage.

These constraints are designed to ensure a fair and efficient allocation of resources across all teams, promoting balanced competition and optimal performance.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating new tables for resource allocation constraints and decision variables, and updating existing tables to fill mapping gaps. Configuration logic updates include adding scalar parameters for total resources and resource allocation limits.

CREATE TABLE basketball_match (
  All_Games_Percent FLOAT,
  Resource_Allocation FLOAT
);

CREATE TABLE resource_allocation (
  Team_ID INTEGER,
  amount FLOAT
);

CREATE TABLE resource_constraints (
  Team_ID INTEGER,
  Minimum_Allocation FLOAT,
  Maximum_Allocation FLOAT
);
```

### Data Dictionary  
The data dictionary provides a business-oriented mapping of tables and columns to their purposes and roles in the optimization process:

- **basketball_match**: This table captures the performance metrics of each team, specifically their win percentage in all games. The column `All_Games_Percent` serves as a coefficient in the optimization objective, representing the impact of resource allocation on team performance.

- **resource_allocation**: This table records the amount of resources allocated to each team. The `Team_ID` column uniquely identifies each team, while the `amount` column represents the decision variable for resource allocation.

- **resource_constraints**: This table defines the constraints on resource allocation for each team. The `Team_ID` column identifies the team, while `Minimum_Allocation` and `Maximum_Allocation` specify the bounds for resource allocation, ensuring compliance with league rules.

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical resource allocation scenarios in university sports, ensuring that constraints are neither too tight nor too loose, allowing for meaningful optimization.

-- Realistic data for basketball_match
INSERT INTO basketball_match (All_Games_Percent, Resource_Allocation) VALUES (0.75, 110);
INSERT INTO basketball_match (All_Games_Percent, Resource_Allocation) VALUES (0.6, 140);
INSERT INTO basketball_match (All_Games_Percent, Resource_Allocation) VALUES (0.85, 130);

-- Realistic data for resource_allocation
INSERT INTO resource_allocation (Team_ID, amount) VALUES (1, 110);
INSERT INTO resource_allocation (Team_ID, amount) VALUES (2, 140);
INSERT INTO resource_allocation (Team_ID, amount) VALUES (3, 130);

-- Realistic data for resource_constraints
INSERT INTO resource_constraints (Team_ID, Minimum_Allocation, Maximum_Allocation) VALUES (1, 50, 200);
INSERT INTO resource_constraints (Team_ID, Minimum_Allocation, Maximum_Allocation) VALUES (2, 60, 180);
INSERT INTO resource_constraints (Team_ID, Minimum_Allocation, Maximum_Allocation) VALUES (3, 70, 190);
```
