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

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[StuID, ResID] ∈ {0, 1} (binary decision variable indicating whether student StuID visits restaurant ResID)
- Operational parameters align with expected linear objective: maximize ∑(satisfaction_score[StuID, ResID] * x[StuID, ResID])
- Business configuration includes: Total budget for student spending (used for Total spending constraint)
- Business logic formulas to express in natural language: Satisfaction score based on time spent and restaurant rating (calculation method for Objective coefficient calculation)
- 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": "A university aims to optimize student allocation to restaurants based on preferences and spending habits, maximizing satisfaction while keeping total spending within a predefined budget. Satisfaction is modeled as a linear function of restaurant ratings and time spent.",
  "optimization_problem_description": "Maximize student satisfaction by allocating students to preferred restaurants, ensuring total spending does not exceed the budget and restaurant capacities are not exceeded. The problem is formulated as a linear optimization model.",
  "optimization_formulation": {
    "objective": "maximize \u2211(satisfaction_score[StuID, ResID] * x[StuID, ResID])",
    "decision_variables": "x[StuID, ResID] \u2208 {0, 1} (binary decision variable indicating whether student StuID visits restaurant ResID)",
    "constraints": [
      "\u2211(spending[StuID, ResID] * x[StuID, ResID]) \u2264 Budget (total spending constraint)",
      "\u2211(x[StuID, ResID]) \u2264 capacity[ResID] for all ResID (restaurant capacity constraint)",
      "\u2211(x[StuID, ResID]) = 1 for all StuID (each student visits exactly one restaurant)"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "satisfaction_score[StuID, ResID]": {
        "currently_mapped_to": "Satisfaction_Scores.score",
        "mapping_adequacy": "good",
        "description": "Satisfaction score for student StuID at restaurant ResID"
      }
    },
    "constraint_bounds": {
      "Budget": {
        "currently_mapped_to": "business_configuration_logic.Budget",
        "mapping_adequacy": "good",
        "description": "Total budget for student spending"
      },
      "capacity[ResID]": {
        "currently_mapped_to": "Restaurant_Capacities.capacity",
        "mapping_adequacy": "good",
        "description": "Maximum capacity of restaurant ResID"
      }
    },
    "decision_variables": {
      "x[StuID, ResID]": {
        "currently_mapped_to": "Visits_Restaurant.visited",
        "mapping_adequacy": "good",
        "description": "Binary decision variable indicating whether student StuID visits restaurant ResID",
        "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 tables for satisfaction scores and restaurant capacities, modifying the Visits_Restaurant table to better map decision variables, and adding business configuration logic for budget and satisfaction score calculation.

CREATE TABLE Satisfaction_Scores (
  StuID INTEGER,
  ResID INTEGER,
  score FLOAT
);

CREATE TABLE Restaurant_Capacities (
  ResID INTEGER,
  capacity INTEGER
);

CREATE TABLE Visits_Restaurant (
  StuID INTEGER,
  ResID INTEGER,
  visited BOOLEAN
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic university restaurant scenarios, considering typical student spending, restaurant capacities, and satisfaction metrics derived from time spent and restaurant ratings.

-- Realistic data for Satisfaction_Scores
INSERT INTO Satisfaction_Scores (StuID, ResID, score) VALUES (1, 1, 4.5);
INSERT INTO Satisfaction_Scores (StuID, ResID, score) VALUES (1, 2, 3.0);
INSERT INTO Satisfaction_Scores (StuID, ResID, score) VALUES (2, 1, 3.8);

-- Realistic data for Restaurant_Capacities
INSERT INTO Restaurant_Capacities (ResID, capacity) VALUES (1, 50);
INSERT INTO Restaurant_Capacities (ResID, capacity) VALUES (2, 75);
INSERT INTO Restaurant_Capacities (ResID, capacity) VALUES (3, 100);

-- Realistic data for Visits_Restaurant
INSERT INTO Visits_Restaurant (StuID, ResID, visited) VALUES (1, 1, True);
INSERT INTO Visits_Restaurant (StuID, ResID, visited) VALUES (2, 2, True);
INSERT INTO Visits_Restaurant (StuID, ResID, visited) VALUES (3, 3, True);


```

DATA DICTIONARY:
{
  "tables": {
    "Satisfaction_Scores": {
      "business_purpose": "Satisfaction score based on time spent and restaurant rating",
      "optimization_role": "objective_coefficients",
      "columns": {
        "StuID": {
          "data_type": "INTEGER",
          "business_meaning": "Student ID",
          "optimization_purpose": "Identifies the student",
          "sample_values": "1, 2, 3"
        },
        "ResID": {
          "data_type": "INTEGER",
          "business_meaning": "Restaurant ID",
          "optimization_purpose": "Identifies the restaurant",
          "sample_values": "1, 2, 3"
        },
        "score": {
          "data_type": "FLOAT",
          "business_meaning": "Satisfaction score",
          "optimization_purpose": "Used in the objective function",
          "sample_values": "4.5, 3.8, 5.0"
        }
      }
    },
    "Restaurant_Capacities": {
      "business_purpose": "Maximum number of students a restaurant can accommodate",
      "optimization_role": "constraint_bounds",
      "columns": {
        "ResID": {
          "data_type": "INTEGER",
          "business_meaning": "Restaurant ID",
          "optimization_purpose": "Identifies the restaurant",
          "sample_values": "1, 2, 3"
        },
        "capacity": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum capacity",
          "optimization_purpose": "Used in the capacity constraint",
          "sample_values": "50, 75, 100"
        }
      }
    },
    "Visits_Restaurant": {
      "business_purpose": "Allocation of students to restaurants",
      "optimization_role": "decision_variables",
      "columns": {
        "StuID": {
          "data_type": "INTEGER",
          "business_meaning": "Student ID",
          "optimization_purpose": "Identifies the student",
          "sample_values": "1, 2, 3"
        },
        "ResID": {
          "data_type": "INTEGER",
          "business_meaning": "Restaurant ID",
          "optimization_purpose": "Identifies the restaurant",
          "sample_values": "1, 2, 3"
        },
        "visited": {
          "data_type": "BOOLEAN",
          "business_meaning": "Whether the student visited the restaurant",
          "optimization_purpose": "Binary decision variable",
          "sample_values": "true, false"
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "Budget": {
    "data_type": "INTEGER",
    "business_meaning": "Total budget for student spending",
    "optimization_role": "Total spending constraint",
    "configuration_type": "scalar_parameter",
    "value": 15000,
    "business_justification": "A budget of $15,000 is realistic for a university to allocate for student spending across multiple restaurants."
  },
  "Satisfaction_Score_Formula": {
    "data_type": "STRING",
    "business_meaning": "Satisfaction score based on time spent and restaurant rating",
    "optimization_role": "Objective coefficient calculation",
    "configuration_type": "business_logic_formula",
    "formula_expression": "time_spent * restaurant_rating"
  }
}

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: x[StuID, ResID] ∈ {0, 1} (binary decision variable indicating whether student StuID visits restaurant ResID)
- Operational parameters align with expected linear objective: maximize ∑(satisfaction_score[StuID, ResID] * x[StuID, ResID])
- Business configuration includes: Total budget for student spending (used for Total spending constraint)
- Business logic formulas to express in natural language: Satisfaction score based on time spent and restaurant rating (calculation method for Objective coefficient calculation)
- 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 ∑(satisfaction_score[StuID, ResID] * x[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: ['∑(spending[StuID, ResID] * x[StuID, ResID]) ≤ Budget (total spending constraint)', '∑(x[StuID, ResID]) ≤ capacity[ResID] for all ResID (restaurant capacity constraint)', '∑(x[StuID, ResID]) = 1 for all StuID (each student visits exactly one restaurant)'] (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 satisfaction scores and restaurant capacities, modifying the Visits_Restaurant table to better map decision variables, and adding business configuration logic for budget and satisfaction score calculation.

CREATE TABLE Satisfaction_Scores (
  StuID INTEGER,
  ResID INTEGER,
  score FLOAT
);

CREATE TABLE Restaurant_Capacities (
  ResID INTEGER,
  capacity INTEGER
);

CREATE TABLE Visits_Restaurant (
  StuID INTEGER,
  ResID INTEGER,
  visited BOOLEAN
);


```

### 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 university restaurant scenarios, considering typical student spending, restaurant capacities, and satisfaction metrics derived from time spent and restaurant ratings.

-- Realistic data for Satisfaction_Scores
INSERT INTO Satisfaction_Scores (StuID, ResID, score) VALUES (1, 1, 4.5);
INSERT INTO Satisfaction_Scores (StuID, ResID, score) VALUES (1, 2, 3.0);
INSERT INTO Satisfaction_Scores (StuID, ResID, score) VALUES (2, 1, 3.8);

-- Realistic data for Restaurant_Capacities
INSERT INTO Restaurant_Capacities (ResID, capacity) VALUES (1, 50);
INSERT INTO Restaurant_Capacities (ResID, capacity) VALUES (2, 75);
INSERT INTO Restaurant_Capacities (ResID, capacity) VALUES (3, 100);

-- Realistic data for Visits_Restaurant
INSERT INTO Visits_Restaurant (StuID, ResID, visited) VALUES (1, 1, True);
INSERT INTO Visits_Restaurant (StuID, ResID, visited) VALUES (2, 2, True);
INSERT INTO Visits_Restaurant (StuID, ResID, visited) VALUES (3, 3, True);


```

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

Response:
# Complete Optimization Problem and Solution: restaurant_1

## 1. Problem Context and Goals

### Context  
A university is focused on optimizing the allocation of students to restaurants based on their preferences and spending habits. The goal is to maximize student satisfaction while ensuring that the total spending across all students does not exceed a predefined budget. Student satisfaction is modeled as a linear function of restaurant ratings and the time spent at the restaurant. The university has a total budget of $15,000 allocated for student spending across all restaurants. Satisfaction scores are calculated by multiplying the time spent at a restaurant by its rating, ensuring a linear relationship that aligns with the optimization framework. The decision to allocate a student to a specific restaurant is represented as a binary choice, where each student is assigned to exactly one restaurant. The problem is designed to avoid any nonlinear relationships, such as variable products or divisions, ensuring a linear optimization formulation.

### Goals  
The primary goal of this optimization problem is to maximize the overall student satisfaction across all restaurant visits. This is achieved by strategically allocating students to restaurants in a way that their individual satisfaction scores are maximized. Success is measured by the sum of satisfaction scores for all student-restaurant pairs, ensuring that the allocation adheres to the university’s budget and respects the capacity limits of each restaurant. The optimization process ensures that each student is assigned to exactly one restaurant, and the total spending across all students does not exceed the allocated budget.

## 2. Constraints  

The optimization problem is subject to the following constraints:  
1. **Total Spending Constraint**: The total spending across all student-restaurant pairs must not exceed the university’s budget of $15,000. This ensures that the allocation remains financially feasible.  
2. **Restaurant Capacity Constraint**: The number of students assigned to each restaurant must not exceed its maximum capacity. This ensures that no restaurant is overbooked.  
3. **Student Assignment Constraint**: Each student must be assigned to exactly one restaurant. This ensures that all students are included in the allocation without duplication.  

These constraints are designed to align with linear mathematical relationships, avoiding any nonlinear complexities such as variable products or divisions.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating tables for satisfaction scores and restaurant capacities, modifying the Visits_Restaurant table to better map decision variables, and adding business configuration logic for budget and satisfaction score calculation.

CREATE TABLE Satisfaction_Scores (
  StuID INTEGER,
  ResID INTEGER,
  score FLOAT
);

CREATE TABLE Restaurant_Capacities (
  ResID INTEGER,
  capacity INTEGER
);

CREATE TABLE Visits_Restaurant (
  StuID INTEGER,
  ResID INTEGER,
  visited BOOLEAN
);
```

### Data Dictionary  
- **Satisfaction_Scores**:  
  - **Business Purpose**: Stores the satisfaction scores for each student-restaurant pair, calculated based on time spent and restaurant rating.  
  - **Optimization Role**: Provides the coefficients for the objective function, representing the satisfaction score for each student-restaurant pair.  
  - **Columns**:  
    - **StuID**: Identifies the student.  
    - **ResID**: Identifies the restaurant.  
    - **score**: Represents the satisfaction score, used in the objective function.  

- **Restaurant_Capacities**:  
  - **Business Purpose**: Stores the maximum number of students each restaurant can accommodate.  
  - **Optimization Role**: Provides the bounds for the restaurant capacity constraints.  
  - **Columns**:  
    - **ResID**: Identifies the restaurant.  
    - **capacity**: Represents the maximum capacity of the restaurant, used in the capacity constraint.  

- **Visits_Restaurant**:  
  - **Business Purpose**: Tracks the allocation of students to restaurants.  
  - **Optimization Role**: Represents the binary decision variables indicating whether a student visits a specific restaurant.  
  - **Columns**:  
    - **StuID**: Identifies the student.  
    - **ResID**: Identifies the restaurant.  
    - **visited**: A binary value indicating whether the student visits the restaurant, used as the decision variable.  

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic university restaurant scenarios, considering typical student spending, restaurant capacities, and satisfaction metrics derived from time spent and restaurant ratings.

-- Realistic data for Satisfaction_Scores
INSERT INTO Satisfaction_Scores (StuID, ResID, score) VALUES (1, 1, 4.5);
INSERT INTO Satisfaction_Scores (StuID, ResID, score) VALUES (1, 2, 3.0);
INSERT INTO Satisfaction_Scores (StuID, ResID, score) VALUES (2, 1, 3.8);

-- Realistic data for Restaurant_Capacities
INSERT INTO Restaurant_Capacities (ResID, capacity) VALUES (1, 50);
INSERT INTO Restaurant_Capacities (ResID, capacity) VALUES (2, 75);
INSERT INTO Restaurant_Capacities (ResID, capacity) VALUES (3, 100);

-- Realistic data for Visits_Restaurant
INSERT INTO Visits_Restaurant (StuID, ResID, visited) VALUES (1, 1, True);
INSERT INTO Visits_Restaurant (StuID, ResID, visited) VALUES (2, 2, True);
INSERT INTO Visits_Restaurant (StuID, ResID, visited) VALUES (3, 3, True);
```
