Iteration final - PROBLEM_DESCRIPTION
Sequence: 7
Timestamp: 2025-07-25 22:31:07

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 2), 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: assign[student_id, dorm_id] ∈ {0, 1} (binary decision variable indicating whether student is assigned to dorm)
- Operational parameters align with expected linear objective: minimize ∑(distance[student_id, dorm_id] * assign[student_id, dorm_id])
- 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": "dorm_1",
  "iteration": 2,
  "business_context": "Optimize dormitory assignments to minimize the total distance students travel from their home cities to their assigned dorms, while respecting dorm capacity and gender constraints.",
  "optimization_problem_description": "Minimize the total distance traveled by students from their home cities to their assigned dorms, subject to dorm capacity limits and gender matching constraints.",
  "optimization_formulation": {
    "objective": "minimize \u2211(distance[student_id, dorm_id] * assign[student_id, dorm_id])",
    "decision_variables": "assign[student_id, dorm_id] \u2208 {0, 1} (binary decision variable indicating whether student is assigned to dorm)",
    "constraints": [
      "\u2211(assign[student_id, dorm_id]) = 1 for all student_id (each student is assigned to exactly one dorm)",
      "\u2211(assign[student_id, dorm_id]) \u2264 student_capacity[dorm_id] for all dorm_id (dorm capacity is not exceeded)",
      "assign[student_id, dorm_id] = 0 if gender[student_id] \u2260 gender[dorm_id] (gender constraints are respected)"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "distance[student_id, dorm_id]": {
        "currently_mapped_to": "DistanceMatrix.distance",
        "mapping_adequacy": "good",
        "description": "Distance from student's home city to dorm"
      }
    },
    "constraint_bounds": {
      "student_capacity[dorm_id]": {
        "currently_mapped_to": "Dorm.student_capacity",
        "mapping_adequacy": "good",
        "description": "Maximum number of students a dorm can accommodate"
      },
      "gender[student_id]": {
        "currently_mapped_to": "GenderInfo.gender",
        "mapping_adequacy": "good",
        "description": "Gender of student"
      },
      "gender[dorm_id]": {
        "currently_mapped_to": "Dorm.gender",
        "mapping_adequacy": "good",
        "description": "Gender constraint for dorm"
      }
    },
    "decision_variables": {
      "assign[student_id, dorm_id]": {
        "currently_mapped_to": "Assignment.assign",
        "mapping_adequacy": "good",
        "description": "Binary variable indicating whether student is assigned to dorm",
        "variable_type": "binary"
      }
    }
  },
  "missing_optimization_requirements": [],
  "iteration_status": {
    "complete": true,
    "confidence": "high",
    "next_focus": "Ready for convergence"
  }
}

FINAL DATABASE SCHEMA:
```sql
-- Iteration 2 Database Schema
-- Objective: Added Assignment table for decision variables, updated data dictionary, and ensured all optimization mappings are complete.

CREATE TABLE DistanceMatrix (
  student_id INTEGER,
  dorm_id INTEGER,
  distance FLOAT
);

CREATE TABLE GenderInfo (
  student_id INTEGER,
  dorm_id INTEGER,
  gender STRING
);

CREATE TABLE Dorm (
  dorm_id INTEGER,
  student_capacity INTEGER,
  gender STRING
);

CREATE TABLE Assignment (
  student_id INTEGER,
  dorm_id INTEGER,
  assign BOOLEAN
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic dormitory scenarios, considering typical student populations, dorm capacities, and geographical distances. Gender constraints and assignment logic were respected to ensure feasibility.

-- Realistic data for DistanceMatrix
INSERT INTO DistanceMatrix (student_id, dorm_id, distance) VALUES (1, 1, 12.5);
INSERT INTO DistanceMatrix (student_id, dorm_id, distance) VALUES (1, 2, 18.3);
INSERT INTO DistanceMatrix (student_id, dorm_id, distance) VALUES (2, 1, 15.0);
INSERT INTO DistanceMatrix (student_id, dorm_id, distance) VALUES (2, 2, 10.2);
INSERT INTO DistanceMatrix (student_id, dorm_id, distance) VALUES (3, 1, 20.1);
INSERT INTO DistanceMatrix (student_id, dorm_id, distance) VALUES (3, 2, 14.7);

-- Realistic data for GenderInfo
INSERT INTO GenderInfo (student_id, dorm_id, gender) VALUES (1, 1, 'Male');
INSERT INTO GenderInfo (student_id, dorm_id, gender) VALUES (2, 2, 'Female');
INSERT INTO GenderInfo (student_id, dorm_id, gender) VALUES (3, 1, 'Male');

-- Realistic data for Dorm
INSERT INTO Dorm (dorm_id, student_capacity, gender) VALUES (1, 100, 'Male');
INSERT INTO Dorm (dorm_id, student_capacity, gender) VALUES (2, 150, 'Female');

-- Realistic data for Assignment
INSERT INTO Assignment (student_id, dorm_id, assign) VALUES (1, 1, True);
INSERT INTO Assignment (student_id, dorm_id, assign) VALUES (2, 2, True);
INSERT INTO Assignment (student_id, dorm_id, assign) VALUES (3, 1, True);


```

DATA DICTIONARY:
{
  "tables": {
    "DistanceMatrix": {
      "business_purpose": "Stores distances between student home cities and dorms",
      "optimization_role": "objective_coefficients",
      "columns": {
        "student_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for student",
          "optimization_purpose": "Links student to their home city",
          "sample_values": "1, 2, 3"
        },
        "dorm_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for dorm",
          "optimization_purpose": "Links dorm to its location",
          "sample_values": "1, 2, 3"
        },
        "distance": {
          "data_type": "FLOAT",
          "business_meaning": "Distance from student's home city to dorm",
          "optimization_purpose": "Used in objective function to minimize total distance",
          "sample_values": "10.5, 15.3, 20.1"
        }
      }
    },
    "GenderInfo": {
      "business_purpose": "Stores gender information for students and dorms",
      "optimization_role": "constraint_bounds",
      "columns": {
        "student_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for student",
          "optimization_purpose": "Links student to their gender",
          "sample_values": "1, 2, 3"
        },
        "dorm_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for dorm",
          "optimization_purpose": "Links dorm to its gender constraint",
          "sample_values": "1, 2, 3"
        },
        "gender": {
          "data_type": "STRING",
          "business_meaning": "Gender of student or dorm",
          "optimization_purpose": "Ensures gender constraints are respected",
          "sample_values": "Male, Female"
        }
      }
    },
    "Dorm": {
      "business_purpose": "Stores information about dorms",
      "optimization_role": "constraint_bounds",
      "columns": {
        "dorm_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for dorm",
          "optimization_purpose": "Links dorm to its capacity and gender",
          "sample_values": "1, 2, 3"
        },
        "student_capacity": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum number of students a dorm can accommodate",
          "optimization_purpose": "Used in capacity constraints",
          "sample_values": "100, 150, 200"
        },
        "gender": {
          "data_type": "STRING",
          "business_meaning": "Gender constraint for dorm",
          "optimization_purpose": "Ensures gender constraints are respected",
          "sample_values": "Male, Female"
        }
      }
    },
    "Assignment": {
      "business_purpose": "Stores binary assignment of students to dorms",
      "optimization_role": "decision_variables",
      "columns": {
        "student_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for student",
          "optimization_purpose": "Links student to their assignment",
          "sample_values": "1, 2, 3"
        },
        "dorm_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for dorm",
          "optimization_purpose": "Links dorm to its assignment",
          "sample_values": "1, 2, 3"
        },
        "assign": {
          "data_type": "BOOLEAN",
          "business_meaning": "Binary variable indicating whether student is assigned to dorm",
          "optimization_purpose": "Used in decision variables for optimization",
          "sample_values": "true, false"
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:


TASK: Create structured markdown documentation for SECTIONS 1-3 ONLY (Problem Description).

EXACT MARKDOWN STRUCTURE TO FOLLOW:

# Complete Optimization Problem and Solution: dorm_1

## 1. Problem Context and Goals

### Context  
[Regenerate business context that naturally aligns with LINEAR optimization formulation. Ensure:]
- Business decisions match expected decision variables: assign[student_id, dorm_id] ∈ {0, 1} (binary decision variable indicating whether student is assigned to dorm)
- Operational parameters align with expected linear objective: minimize ∑(distance[student_id, dorm_id] * assign[student_id, dorm_id])
- 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 ∑(distance[student_id, dorm_id] * assign[student_id, dorm_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: ['∑(assign[student_id, dorm_id]) = 1 for all student_id (each student is assigned to exactly one dorm)', '∑(assign[student_id, dorm_id]) ≤ student_capacity[dorm_id] for all dorm_id (dorm capacity is not exceeded)', 'assign[student_id, dorm_id] = 0 if gender[student_id] ≠ gender[dorm_id] (gender constraints are respected)'] (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 2 Database Schema
-- Objective: Added Assignment table for decision variables, updated data dictionary, and ensured all optimization mappings are complete.

CREATE TABLE DistanceMatrix (
  student_id INTEGER,
  dorm_id INTEGER,
  distance FLOAT
);

CREATE TABLE GenderInfo (
  student_id INTEGER,
  dorm_id INTEGER,
  gender STRING
);

CREATE TABLE Dorm (
  dorm_id INTEGER,
  student_capacity INTEGER,
  gender STRING
);

CREATE TABLE Assignment (
  student_id INTEGER,
  dorm_id INTEGER,
  assign 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 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic dormitory scenarios, considering typical student populations, dorm capacities, and geographical distances. Gender constraints and assignment logic were respected to ensure feasibility.

-- Realistic data for DistanceMatrix
INSERT INTO DistanceMatrix (student_id, dorm_id, distance) VALUES (1, 1, 12.5);
INSERT INTO DistanceMatrix (student_id, dorm_id, distance) VALUES (1, 2, 18.3);
INSERT INTO DistanceMatrix (student_id, dorm_id, distance) VALUES (2, 1, 15.0);
INSERT INTO DistanceMatrix (student_id, dorm_id, distance) VALUES (2, 2, 10.2);
INSERT INTO DistanceMatrix (student_id, dorm_id, distance) VALUES (3, 1, 20.1);
INSERT INTO DistanceMatrix (student_id, dorm_id, distance) VALUES (3, 2, 14.7);

-- Realistic data for GenderInfo
INSERT INTO GenderInfo (student_id, dorm_id, gender) VALUES (1, 1, 'Male');
INSERT INTO GenderInfo (student_id, dorm_id, gender) VALUES (2, 2, 'Female');
INSERT INTO GenderInfo (student_id, dorm_id, gender) VALUES (3, 1, 'Male');

-- Realistic data for Dorm
INSERT INTO Dorm (dorm_id, student_capacity, gender) VALUES (1, 100, 'Male');
INSERT INTO Dorm (dorm_id, student_capacity, gender) VALUES (2, 150, 'Female');

-- Realistic data for Assignment
INSERT INTO Assignment (student_id, dorm_id, assign) VALUES (1, 1, True);
INSERT INTO Assignment (student_id, dorm_id, assign) VALUES (2, 2, True);
INSERT INTO Assignment (student_id, dorm_id, assign) VALUES (3, 1, True);


```

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

Response:
# Complete Optimization Problem and Solution: dorm_1

## 1. Problem Context and Goals

### Context  
The dormitory assignment problem involves optimizing the allocation of students to dormitories based on specific operational criteria. The primary business decision is whether to assign a student to a particular dorm, represented as a binary choice. The goal is to minimize the total distance students travel from their home cities to their assigned dorms. This distance is a key operational parameter, as it directly impacts the objective function. 

The problem must respect two critical constraints:  
1. **Dorm Capacity**: Each dorm has a maximum number of students it can accommodate, and this limit must not be exceeded.  
2. **Gender Matching**: Students can only be assigned to dorms that match their gender.  

The data used in this optimization includes the distance between each student's home city and each dorm, the capacity of each dorm, and the gender constraints for both students and dorms. These parameters are stored in the database and are essential for calculating the objective and enforcing constraints.

### Goals  
The optimization goal is to minimize the total distance traveled by all students to their assigned dorms. This is achieved by ensuring that each student is assigned to exactly one dorm, while respecting dorm capacities and gender constraints. Success is measured by the total distance metric, which is calculated as the sum of the distances for all assigned student-dorm pairs. The objective is linear, as it involves summing the product of binary assignment decisions and their corresponding distances.

## 2. Constraints  

The optimization problem is subject to the following constraints, described in business terms:  
1. **Single Assignment**: Each student must be assigned to exactly one dorm. This ensures that all students have a place to stay and no student is left unassigned.  
2. **Dorm Capacity**: The number of students assigned to a dorm cannot exceed its maximum capacity. This ensures that dorms are not overcrowded and can accommodate all assigned students comfortably.  
3. **Gender Matching**: A student can only be assigned to a dorm if the dorm's gender constraint matches the student's gender. This ensures that gender-specific dorm policies are respected.  

These constraints are linear in nature, as they involve sums of binary variables and comparisons with fixed values, without requiring any nonlinear relationships such as variable products or divisions.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 2 Database Schema
-- Objective: Added Assignment table for decision variables, updated data dictionary, and ensured all optimization mappings are complete.

CREATE TABLE DistanceMatrix (
  student_id INTEGER,
  dorm_id INTEGER,
  distance FLOAT
);

CREATE TABLE GenderInfo (
  student_id INTEGER,
  dorm_id INTEGER,
  gender STRING
);

CREATE TABLE Dorm (
  dorm_id INTEGER,
  student_capacity INTEGER,
  gender STRING
);

CREATE TABLE Assignment (
  student_id INTEGER,
  dorm_id INTEGER,
  assign BOOLEAN
);
```

### Data Dictionary  
The following tables and columns are used in the optimization problem, with their business purposes and optimization roles clearly defined:  

- **DistanceMatrix**:  
  - **student_id**: Unique identifier for a student, linking them to their home city.  
  - **dorm_id**: Unique identifier for a dorm, linking it to its location.  
  - **distance**: The distance from a student's home city to a dorm, used in the objective function to minimize total travel distance.  

- **GenderInfo**:  
  - **student_id**: Unique identifier for a student, linking them to their gender.  
  - **dorm_id**: Unique identifier for a dorm, linking it to its gender constraint.  
  - **gender**: The gender of a student or the gender constraint of a dorm, used to enforce gender matching.  

- **Dorm**:  
  - **dorm_id**: Unique identifier for a dorm, linking it to its capacity and gender constraint.  
  - **student_capacity**: The maximum number of students a dorm can accommodate, used in capacity constraints.  
  - **gender**: The gender constraint for a dorm, used to enforce gender matching.  

- **Assignment**:  
  - **student_id**: Unique identifier for a student, linking them to their assignment.  
  - **dorm_id**: Unique identifier for a dorm, linking it to its assignment.  
  - **assign**: A binary variable indicating whether a student is assigned to a dorm, used as the decision variable in the optimization.  

### Current Stored Values  
```sql
-- Iteration 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic dormitory scenarios, considering typical student populations, dorm capacities, and geographical distances. Gender constraints and assignment logic were respected to ensure feasibility.

-- Realistic data for DistanceMatrix
INSERT INTO DistanceMatrix (student_id, dorm_id, distance) VALUES (1, 1, 12.5);
INSERT INTO DistanceMatrix (student_id, dorm_id, distance) VALUES (1, 2, 18.3);
INSERT INTO DistanceMatrix (student_id, dorm_id, distance) VALUES (2, 1, 15.0);
INSERT INTO DistanceMatrix (student_id, dorm_id, distance) VALUES (2, 2, 10.2);
INSERT INTO DistanceMatrix (student_id, dorm_id, distance) VALUES (3, 1, 20.1);
INSERT INTO DistanceMatrix (student_id, dorm_id, distance) VALUES (3, 2, 14.7);

-- Realistic data for GenderInfo
INSERT INTO GenderInfo (student_id, dorm_id, gender) VALUES (1, 1, 'Male');
INSERT INTO GenderInfo (student_id, dorm_id, gender) VALUES (2, 2, 'Female');
INSERT INTO GenderInfo (student_id, dorm_id, gender) VALUES (3, 1, 'Male');

-- Realistic data for Dorm
INSERT INTO Dorm (dorm_id, student_capacity, gender) VALUES (1, 100, 'Male');
INSERT INTO Dorm (dorm_id, student_capacity, gender) VALUES (2, 150, 'Female');

-- Realistic data for Assignment
INSERT INTO Assignment (student_id, dorm_id, assign) VALUES (1, 1, True);
INSERT INTO Assignment (student_id, dorm_id, assign) VALUES (2, 2, True);
INSERT INTO Assignment (student_id, dorm_id, assign) VALUES (3, 1, True);
```
