Iteration final - PROBLEM_DESCRIPTION
Sequence: 5
Timestamp: 2025-07-27 23:14:46

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: Visits[StuID, ResID] - number of visits each student makes to each restaurant (integer)
- Operational parameters align with expected linear objective: maximize total_satisfaction = ∑(Rating[ResID] × Visits[StuID, ResID])
- Business configuration includes: The budget constraint for each student (used for Used as a constraint bound in the optimization model)
- Business logic formulas to express in natural language: Calculates the total spent by a student at a restaurant (calculation method for Used to calculate total spending in the 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": "restaurant_1",
  "iteration": 1,
  "business_context": "Optimize the allocation of student visits to restaurants to maximize the total satisfaction score based on restaurant ratings, while considering budget constraints.",
  "optimization_problem_description": "Determine the optimal number of visits each student should make to each restaurant to maximize the total satisfaction score, subject to budget constraints on the total amount spent by each student.",
  "optimization_formulation": {
    "objective": "maximize total_satisfaction = \u2211(Rating[ResID] \u00d7 Visits[StuID, ResID])",
    "decision_variables": "Visits[StuID, ResID] - number of visits each student makes to each restaurant (integer)",
    "constraints": [
      "\u2211(Cost_Per_Visit[StuID, ResID] \u00d7 Visits[StuID, ResID]) \u2264 Budget[StuID] for all StuID",
      "Visits[StuID, ResID] \u2265 0 for all StuID, ResID"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "Rating[ResID]": {
        "currently_mapped_to": "Restaurant.Rating",
        "mapping_adequacy": "good",
        "description": "Satisfaction score contribution from visiting a restaurant"
      }
    },
    "constraint_bounds": {
      "Budget[StuID]": {
        "currently_mapped_to": "Student_Budget.Budget",
        "mapping_adequacy": "good",
        "description": "Budget constraint for each student"
      }
    },
    "decision_variables": {
      "Visits[StuID, ResID]": {
        "currently_mapped_to": "Visits_Restaurant.Visits",
        "mapping_adequacy": "good",
        "description": "Number of visits a student makes to a restaurant",
        "variable_type": "integer"
      }
    }
  },
  "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 tables for budget constraints and cost per visit, modifying existing tables to include necessary columns, and updating configuration logic for scalar parameters and formulas.

CREATE TABLE Restaurant (
  ResID INTEGER,
  Rating FLOAT
);

CREATE TABLE Student_Budget (
  StuID INTEGER,
  Budget INTEGER
);

CREATE TABLE Cost_Per_Visit (
  StuID INTEGER,
  ResID INTEGER,
  Cost FLOAT
);

CREATE TABLE Visits_Restaurant (
  StuID INTEGER,
  ResID INTEGER,
  Visits INTEGER
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical student budgets and restaurant costs, ensuring a variety of scenarios for optimization. Ratings were chosen to reflect a range of restaurant quality.

-- Realistic data for Restaurant
INSERT INTO Restaurant (ResID, Rating) VALUES (1, 4.5);
INSERT INTO Restaurant (ResID, Rating) VALUES (2, 3.8);
INSERT INTO Restaurant (ResID, Rating) VALUES (3, 5.0);

-- Realistic data for Student_Budget
INSERT INTO Student_Budget (StuID, Budget) VALUES (101, 120);
INSERT INTO Student_Budget (StuID, Budget) VALUES (102, 180);
INSERT INTO Student_Budget (StuID, Budget) VALUES (103, 150);

-- Realistic data for Cost_Per_Visit
INSERT INTO Cost_Per_Visit (StuID, ResID, Cost) VALUES (101, 1, 12.0);
INSERT INTO Cost_Per_Visit (StuID, ResID, Cost) VALUES (102, 2, 15.0);
INSERT INTO Cost_Per_Visit (StuID, ResID, Cost) VALUES (103, 3, 20.0);

-- Realistic data for Visits_Restaurant
INSERT INTO Visits_Restaurant (StuID, ResID, Visits) VALUES (101, 1, 3);
INSERT INTO Visits_Restaurant (StuID, ResID, Visits) VALUES (102, 2, 4);
INSERT INTO Visits_Restaurant (StuID, ResID, Visits) VALUES (103, 3, 2);


```

DATA DICTIONARY:
{
  "tables": {
    "Restaurant": {
      "business_purpose": "Stores restaurant information including ratings",
      "optimization_role": "objective_coefficients",
      "columns": {
        "ResID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each restaurant",
          "optimization_purpose": "Index for restaurant ratings",
          "sample_values": "1, 2, 3"
        },
        "Rating": {
          "data_type": "FLOAT",
          "business_meaning": "Satisfaction score contribution from visiting a restaurant",
          "optimization_purpose": "Coefficient in the objective function",
          "sample_values": "4.5, 3.8, 5.0"
        }
      }
    },
    "Student_Budget": {
      "business_purpose": "Stores budget constraints for each student",
      "optimization_role": "constraint_bounds",
      "columns": {
        "StuID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each student",
          "optimization_purpose": "Index for budget constraints",
          "sample_values": "101, 102, 103"
        },
        "Budget": {
          "data_type": "INTEGER",
          "business_meaning": "Budget constraint for each student",
          "optimization_purpose": "Constraint bound in the optimization model",
          "sample_values": "100, 150, 200"
        }
      }
    },
    "Cost_Per_Visit": {
      "business_purpose": "Stores cost per visit for each student-restaurant pair",
      "optimization_role": "business_data",
      "columns": {
        "StuID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each student",
          "optimization_purpose": "Index for cost data",
          "sample_values": "101, 102, 103"
        },
        "ResID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each restaurant",
          "optimization_purpose": "Index for cost data",
          "sample_values": "1, 2, 3"
        },
        "Cost": {
          "data_type": "FLOAT",
          "business_meaning": "Cost per visit for a student at a restaurant",
          "optimization_purpose": "Used in cost calculations for constraints",
          "sample_values": "10.0, 15.0, 20.0"
        }
      }
    },
    "Visits_Restaurant": {
      "business_purpose": "Tracks the number of visits each student makes to each restaurant",
      "optimization_role": "decision_variables",
      "columns": {
        "StuID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each student",
          "optimization_purpose": "Index for decision variables",
          "sample_values": "101, 102, 103"
        },
        "ResID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each restaurant",
          "optimization_purpose": "Index for decision variables",
          "sample_values": "1, 2, 3"
        },
        "Visits": {
          "data_type": "INTEGER",
          "business_meaning": "Number of visits a student makes to a restaurant",
          "optimization_purpose": "Decision variable in the optimization model",
          "sample_values": "2, 3, 1"
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "Budget[StuID]": {
    "data_type": "INTEGER",
    "business_meaning": "The budget constraint for each student",
    "optimization_role": "Used as a constraint bound in the optimization model",
    "configuration_type": "scalar_parameter",
    "value": 150,
    "business_justification": "Reflects an average budget for students, allowing for multiple visits"
  },
  "Cost_Per_Visit_Formula": {
    "data_type": "STRING",
    "business_meaning": "Calculates the total spent by a student at a restaurant",
    "optimization_role": "Used to calculate total spending in the optimization model",
    "configuration_type": "business_logic_formula",
    "formula_expression": "Spent[StuID, ResID] = Cost_Per_Visit[StuID, ResID] * Visits[StuID, ResID]"
  }
}

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: restaurant_1

## 1. Problem Context and Goals

### Context  
[Regenerate business context that naturally aligns with LINEAR optimization formulation. Ensure:]
- Business decisions match expected decision variables: Visits[StuID, ResID] - number of visits each student makes to each restaurant (integer)
- Operational parameters align with expected linear objective: maximize total_satisfaction = ∑(Rating[ResID] × Visits[StuID, ResID])
- Business configuration includes: The budget constraint for each student (used for Used as a constraint bound in the optimization model)
- Business logic formulas to express in natural language: Calculates the total spent by a student at a restaurant (calculation method for Used to calculate total spending in the 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: maximize
- Metric to optimize: maximize total_satisfaction = ∑(Rating[ResID] × Visits[StuID, ResID])
- 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: ['∑(Cost_Per_Visit[StuID, ResID] × Visits[StuID, ResID]) ≤ Budget[StuID] for all StuID', 'Visits[StuID, ResID] ≥ 0 for all StuID, ResID'] (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 tables for budget constraints and cost per visit, modifying existing tables to include necessary columns, and updating configuration logic for scalar parameters and formulas.

CREATE TABLE Restaurant (
  ResID INTEGER,
  Rating FLOAT
);

CREATE TABLE Student_Budget (
  StuID INTEGER,
  Budget INTEGER
);

CREATE TABLE Cost_Per_Visit (
  StuID INTEGER,
  ResID INTEGER,
  Cost FLOAT
);

CREATE TABLE Visits_Restaurant (
  StuID INTEGER,
  ResID INTEGER,
  Visits 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 student budgets and restaurant costs, ensuring a variety of scenarios for optimization. Ratings were chosen to reflect a range of restaurant quality.

-- Realistic data for Restaurant
INSERT INTO Restaurant (ResID, Rating) VALUES (1, 4.5);
INSERT INTO Restaurant (ResID, Rating) VALUES (2, 3.8);
INSERT INTO Restaurant (ResID, Rating) VALUES (3, 5.0);

-- Realistic data for Student_Budget
INSERT INTO Student_Budget (StuID, Budget) VALUES (101, 120);
INSERT INTO Student_Budget (StuID, Budget) VALUES (102, 180);
INSERT INTO Student_Budget (StuID, Budget) VALUES (103, 150);

-- Realistic data for Cost_Per_Visit
INSERT INTO Cost_Per_Visit (StuID, ResID, Cost) VALUES (101, 1, 12.0);
INSERT INTO Cost_Per_Visit (StuID, ResID, Cost) VALUES (102, 2, 15.0);
INSERT INTO Cost_Per_Visit (StuID, ResID, Cost) VALUES (103, 3, 20.0);

-- Realistic data for Visits_Restaurant
INSERT INTO Visits_Restaurant (StuID, ResID, Visits) VALUES (101, 1, 3);
INSERT INTO Visits_Restaurant (StuID, ResID, Visits) VALUES (102, 2, 4);
INSERT INTO Visits_Restaurant (StuID, ResID, Visits) VALUES (103, 3, 2);


```

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

Response:
# Complete Optimization Problem and Solution: restaurant_1

## 1. Problem Context and Goals

### Context  
The objective is to optimize how students allocate their visits to various restaurants to achieve the highest possible satisfaction score. Each student has a specific budget that limits their total spending on restaurant visits. The decision involves determining the number of visits each student should make to each restaurant. The satisfaction score is derived from the restaurant's rating, which is multiplied by the number of visits. The budget constraint for each student is a critical factor, ensuring that the total cost of visits does not exceed their allocated budget. The cost per visit for each student-restaurant pair is used to calculate the total spending, ensuring that the optimization remains within linear constraints. This setup allows for precise decision-making, focusing on maximizing satisfaction while adhering to budgetary limits.

### Goals  
The primary goal is to maximize the total satisfaction score for all students. This is achieved by optimizing the number of visits each student makes to each restaurant, with the satisfaction score being the product of the restaurant's rating and the number of visits. The success of this optimization is measured by the total satisfaction score, which is directly influenced by the restaurant ratings and the number of visits, ensuring a clear linear relationship in the optimization process.

## 2. Constraints    

The optimization is subject to several constraints. Firstly, the total cost incurred by a student from visiting restaurants must not exceed their budget. This ensures that each student's spending remains within their financial limits. Additionally, the number of visits a student makes to any restaurant must be a non-negative integer, reflecting the realistic scenario where visits cannot be negative or fractional. These constraints are designed to maintain a linear relationship, ensuring the optimization problem remains solvable within the defined parameters.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include adding tables for budget constraints and cost per visit, modifying existing tables to include necessary columns, and updating configuration logic for scalar parameters and formulas.

CREATE TABLE Restaurant (
  ResID INTEGER,
  Rating FLOAT
);

CREATE TABLE Student_Budget (
  StuID INTEGER,
  Budget INTEGER
);

CREATE TABLE Cost_Per_Visit (
  StuID INTEGER,
  ResID INTEGER,
  Cost FLOAT
);

CREATE TABLE Visits_Restaurant (
  StuID INTEGER,
  ResID INTEGER,
  Visits INTEGER
);
```

### Data Dictionary  
- **Restaurant Table**: This table holds information about each restaurant, including a unique identifier and a rating that reflects the satisfaction score a student receives from visiting the restaurant. The rating serves as a coefficient in the optimization objective, influencing the total satisfaction score.

- **Student_Budget Table**: This table contains the budget constraints for each student, identified by a unique student ID. The budget represents the maximum amount a student can spend on restaurant visits, serving as a constraint bound in the optimization model.

- **Cost_Per_Visit Table**: This table records the cost associated with each visit for a student to a specific restaurant. It includes identifiers for both the student and the restaurant, and the cost per visit is used in calculating the total spending, ensuring it aligns with the budget constraints.

- **Visits_Restaurant Table**: This table tracks the number of visits each student makes to each restaurant. It includes identifiers for both the student and the restaurant, and the number of visits is the decision variable in the optimization model, directly affecting the total satisfaction score.

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical student budgets and restaurant costs, ensuring a variety of scenarios for optimization. Ratings were chosen to reflect a range of restaurant quality.

-- Realistic data for Restaurant
INSERT INTO Restaurant (ResID, Rating) VALUES (1, 4.5);
INSERT INTO Restaurant (ResID, Rating) VALUES (2, 3.8);
INSERT INTO Restaurant (ResID, Rating) VALUES (3, 5.0);

-- Realistic data for Student_Budget
INSERT INTO Student_Budget (StuID, Budget) VALUES (101, 120);
INSERT INTO Student_Budget (StuID, Budget) VALUES (102, 180);
INSERT INTO Student_Budget (StuID, Budget) VALUES (103, 150);

-- Realistic data for Cost_Per_Visit
INSERT INTO Cost_Per_Visit (StuID, ResID, Cost) VALUES (101, 1, 12.0);
INSERT INTO Cost_Per_Visit (StuID, ResID, Cost) VALUES (102, 2, 15.0);
INSERT INTO Cost_Per_Visit (StuID, ResID, Cost) VALUES (103, 3, 20.0);

-- Realistic data for Visits_Restaurant
INSERT INTO Visits_Restaurant (StuID, ResID, Visits) VALUES (101, 1, 3);
INSERT INTO Visits_Restaurant (StuID, ResID, Visits) VALUES (102, 2, 4);
INSERT INTO Visits_Restaurant (StuID, ResID, Visits) VALUES (103, 3, 2);
```
