Iteration final - PROBLEM_DESCRIPTION
Sequence: 5
Timestamp: 2025-07-25 22:45:44

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: assignment[technician_id, repair_ID] ∈ {0, 1} (binary decision variable indicating if a technician is assigned to a repair)
- Operational parameters align with expected linear objective: minimize ∑(repair_time[technician_id, repair_ID] × assignment[technician_id, repair_ID])
- Business configuration includes: maximum number of repairs a technician can handle (used for constraint bound), minimum priority level for machine repairs (used for constraint bound)
- Business logic formulas to express in natural language: calculation of repair time based on technician and machine complexity (calculation method for objective coefficient)
- 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": "machine_repair",
  "iteration": 1,
  "business_context": "A repair shop aims to optimize the assignment of technicians to machine repairs to minimize the total repair time while considering technician expertise and machine priority.",
  "optimization_problem_description": "Minimize the total repair time by assigning technicians to machine repairs efficiently. Constraints include ensuring each repair is assigned to one technician, each technician is not overloaded, and high-priority machines are repaired first.",
  "optimization_formulation": {
    "objective": "minimize \u2211(repair_time[technician_id, repair_ID] \u00d7 assignment[technician_id, repair_ID])",
    "decision_variables": "assignment[technician_id, repair_ID] \u2208 {0, 1} (binary decision variable indicating if a technician is assigned to a repair)",
    "constraints": [
      "\u2211(assignment[technician_id, repair_ID] for all technicians) = 1 for each repair_ID (each repair is assigned to one technician)",
      "\u2211(assignment[technician_id, repair_ID] for all repairs) \u2264 max_repairs[technician_id] for each technician_id (technician capacity constraint)",
      "priority[repair_ID] \u2265 min_priority for high-priority repairs (priority constraint)"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "repair_time[technician_id, repair_ID]": {
        "currently_mapped_to": "repair_time.repair_time",
        "mapping_adequacy": "good",
        "description": "time taken by a technician to complete a repair"
      }
    },
    "constraint_bounds": {
      "max_repairs[technician_id]": {
        "currently_mapped_to": "technician_capacity.max_repairs",
        "mapping_adequacy": "good",
        "description": "maximum number of repairs a technician can handle"
      },
      "min_priority": {
        "currently_mapped_to": "business_configuration_logic.min_priority",
        "mapping_adequacy": "good",
        "description": "minimum priority level for machine repairs"
      }
    },
    "decision_variables": {
      "assignment[technician_id, repair_ID]": {
        "currently_mapped_to": "repair_assignment.technician_id, repair_assignment.repair_ID",
        "mapping_adequacy": "good",
        "description": "binary decision variable indicating if a technician is assigned to a repair",
        "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 repair time, technician capacity, and machine priority. Configuration logic updates include scalar parameters for max repairs per technician and min priority, and a formula for repair time calculation.

CREATE TABLE repair_time (
  technician_id INTEGER,
  repair_ID INTEGER,
  repair_time FLOAT
);

CREATE TABLE technician_capacity (
  technician_id INTEGER,
  max_repairs INTEGER
);

CREATE TABLE machine_priority (
  Machine_ID INTEGER,
  priority INTEGER
);

CREATE TABLE repair_assignment (
  technician_id INTEGER,
  repair_ID INTEGER
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic business scenarios, considering technician expertise, machine complexity, and repair shop capacity.

-- Realistic data for repair_time
INSERT INTO repair_time (technician_id, repair_ID, repair_time) VALUES (1, 101, 2.5);
INSERT INTO repair_time (technician_id, repair_ID, repair_time) VALUES (2, 102, 3.0);
INSERT INTO repair_time (technician_id, repair_ID, repair_time) VALUES (3, 103, 4.0);

-- Realistic data for technician_capacity
INSERT INTO technician_capacity (technician_id, max_repairs) VALUES (1, 5);
INSERT INTO technician_capacity (technician_id, max_repairs) VALUES (2, 4);
INSERT INTO technician_capacity (technician_id, max_repairs) VALUES (3, 3);

-- Realistic data for machine_priority
INSERT INTO machine_priority (Machine_ID, priority) VALUES (201, 1);
INSERT INTO machine_priority (Machine_ID, priority) VALUES (202, 2);
INSERT INTO machine_priority (Machine_ID, priority) VALUES (203, 3);

-- Realistic data for repair_assignment
INSERT INTO repair_assignment (technician_id, repair_ID) VALUES (1, 101);
INSERT INTO repair_assignment (technician_id, repair_ID) VALUES (2, 102);
INSERT INTO repair_assignment (technician_id, repair_ID) VALUES (3, 103);


```

DATA DICTIONARY:
{
  "tables": {
    "repair_time": {
      "business_purpose": "time taken by a technician to complete a repair",
      "optimization_role": "objective_coefficients",
      "columns": {
        "technician_id": {
          "data_type": "INTEGER",
          "business_meaning": "identifier for the technician",
          "optimization_purpose": "links to technician in repair assignment",
          "sample_values": "1, 2, 3"
        },
        "repair_ID": {
          "data_type": "INTEGER",
          "business_meaning": "identifier for the repair",
          "optimization_purpose": "links to repair in repair assignment",
          "sample_values": "101, 102, 103"
        },
        "repair_time": {
          "data_type": "FLOAT",
          "business_meaning": "time taken to complete the repair",
          "optimization_purpose": "coefficient in the objective function",
          "sample_values": "2.5, 3.0, 4.0"
        }
      }
    },
    "technician_capacity": {
      "business_purpose": "maximum number of repairs a technician can handle",
      "optimization_role": "constraint_bounds",
      "columns": {
        "technician_id": {
          "data_type": "INTEGER",
          "business_meaning": "identifier for the technician",
          "optimization_purpose": "links to technician in repair assignment",
          "sample_values": "1, 2, 3"
        },
        "max_repairs": {
          "data_type": "INTEGER",
          "business_meaning": "maximum number of repairs the technician can handle",
          "optimization_purpose": "bound in the constraint",
          "sample_values": "5, 6, 7"
        }
      }
    },
    "machine_priority": {
      "business_purpose": "priority level for machine repairs",
      "optimization_role": "constraint_bounds",
      "columns": {
        "Machine_ID": {
          "data_type": "INTEGER",
          "business_meaning": "identifier for the machine",
          "optimization_purpose": "links to repair in repair assignment",
          "sample_values": "201, 202, 203"
        },
        "priority": {
          "data_type": "INTEGER",
          "business_meaning": "priority level of the machine",
          "optimization_purpose": "bound in the constraint",
          "sample_values": "1, 2, 3"
        }
      }
    },
    "repair_assignment": {
      "business_purpose": "assignment of technicians to repairs",
      "optimization_role": "decision_variables",
      "columns": {
        "technician_id": {
          "data_type": "INTEGER",
          "business_meaning": "identifier for the technician",
          "optimization_purpose": "decision variable in the optimization model",
          "sample_values": "1, 2, 3"
        },
        "repair_ID": {
          "data_type": "INTEGER",
          "business_meaning": "identifier for the repair",
          "optimization_purpose": "decision variable in the optimization model",
          "sample_values": "101, 102, 103"
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "max_repairs_per_technician": {
    "data_type": "INTEGER",
    "business_meaning": "maximum number of repairs a technician can handle",
    "optimization_role": "constraint bound",
    "configuration_type": "scalar_parameter",
    "value": 5,
    "business_justification": "This value ensures that technicians are not overloaded while maintaining efficient repair operations."
  },
  "min_priority": {
    "data_type": "INTEGER",
    "business_meaning": "minimum priority level for machine repairs",
    "optimization_role": "constraint bound",
    "configuration_type": "scalar_parameter",
    "value": 2,
    "business_justification": "This value ensures that high and medium-priority machines are repaired first, aligning with business priorities."
  },
  "repair_time_formula": {
    "data_type": "STRING",
    "business_meaning": "calculation of repair time based on technician and machine complexity",
    "optimization_role": "objective coefficient",
    "configuration_type": "business_logic_formula",
    "formula_expression": "repair_time[technician_id, repair_ID] = base_time + (complexity_factor * machine_complexity)"
  }
}

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: machine_repair

## 1. Problem Context and Goals

### Context  
[Regenerate business context that naturally aligns with LINEAR optimization formulation. Ensure:]
- Business decisions match expected decision variables: assignment[technician_id, repair_ID] ∈ {0, 1} (binary decision variable indicating if a technician is assigned to a repair)
- Operational parameters align with expected linear objective: minimize ∑(repair_time[technician_id, repair_ID] × assignment[technician_id, repair_ID])
- Business configuration includes: maximum number of repairs a technician can handle (used for constraint bound), minimum priority level for machine repairs (used for constraint bound)
- Business logic formulas to express in natural language: calculation of repair time based on technician and machine complexity (calculation method for objective coefficient)
- 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 ∑(repair_time[technician_id, repair_ID] × assignment[technician_id, repair_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: ['∑(assignment[technician_id, repair_ID] for all technicians) = 1 for each repair_ID (each repair is assigned to one technician)', '∑(assignment[technician_id, repair_ID] for all repairs) ≤ max_repairs[technician_id] for each technician_id (technician capacity constraint)', 'priority[repair_ID] ≥ min_priority for high-priority repairs (priority constraint)'] (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 repair time, technician capacity, and machine priority. Configuration logic updates include scalar parameters for max repairs per technician and min priority, and a formula for repair time calculation.

CREATE TABLE repair_time (
  technician_id INTEGER,
  repair_ID INTEGER,
  repair_time FLOAT
);

CREATE TABLE technician_capacity (
  technician_id INTEGER,
  max_repairs INTEGER
);

CREATE TABLE machine_priority (
  Machine_ID INTEGER,
  priority INTEGER
);

CREATE TABLE repair_assignment (
  technician_id INTEGER,
  repair_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 realistic business scenarios, considering technician expertise, machine complexity, and repair shop capacity.

-- Realistic data for repair_time
INSERT INTO repair_time (technician_id, repair_ID, repair_time) VALUES (1, 101, 2.5);
INSERT INTO repair_time (technician_id, repair_ID, repair_time) VALUES (2, 102, 3.0);
INSERT INTO repair_time (technician_id, repair_ID, repair_time) VALUES (3, 103, 4.0);

-- Realistic data for technician_capacity
INSERT INTO technician_capacity (technician_id, max_repairs) VALUES (1, 5);
INSERT INTO technician_capacity (technician_id, max_repairs) VALUES (2, 4);
INSERT INTO technician_capacity (technician_id, max_repairs) VALUES (3, 3);

-- Realistic data for machine_priority
INSERT INTO machine_priority (Machine_ID, priority) VALUES (201, 1);
INSERT INTO machine_priority (Machine_ID, priority) VALUES (202, 2);
INSERT INTO machine_priority (Machine_ID, priority) VALUES (203, 3);

-- Realistic data for repair_assignment
INSERT INTO repair_assignment (technician_id, repair_ID) VALUES (1, 101);
INSERT INTO repair_assignment (technician_id, repair_ID) VALUES (2, 102);
INSERT INTO repair_assignment (technician_id, repair_ID) VALUES (3, 103);


```

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

Response:
# Complete Optimization Problem and Solution: machine_repair

## 1. Problem Context and Goals

### Context  
A repair shop is tasked with efficiently assigning technicians to machine repairs to minimize the total repair time. Each technician has a specific expertise level, and each machine repair has a priority level that determines its urgency. The shop must ensure that each repair is assigned to exactly one technician, that no technician is overloaded beyond their capacity, and that high-priority repairs are addressed first.  

The repair time for each technician and repair combination is calculated based on the technician’s base repair time and the complexity of the machine being repaired. This calculation ensures that the repair time reflects both the technician’s skill and the machine’s difficulty.  

The shop operates under the following business configuration:  
- **Maximum Repairs per Technician**: Each technician can handle a maximum of 5 repairs at a time to prevent overloading and maintain efficiency.  
- **Minimum Priority Level**: Repairs with a priority level of 2 or higher are considered high-priority and must be addressed first.  

The decision to assign a technician to a repair is represented as a binary choice: either the technician is assigned (1) or not (0). This decision directly impacts the total repair time, which is the sum of the repair times for all assigned technician-repair pairs.  

### Goals  
The primary goal of this optimization is to minimize the total repair time across all assigned technician-repair pairs. This is achieved by efficiently assigning technicians to repairs while adhering to the constraints of technician capacity and repair priority. Success is measured by the reduction in total repair time, ensuring that high-priority repairs are completed promptly and that technicians are utilized effectively without exceeding their workload limits.  

## 2. Constraints  

The optimization must adhere to the following constraints:  
1. **Single Assignment per Repair**: Each repair must be assigned to exactly one technician. This ensures that all repairs are addressed without duplication or omission.  
2. **Technician Capacity Limit**: The total number of repairs assigned to a technician must not exceed their maximum capacity. This prevents overloading and maintains operational efficiency.  
3. **Priority Level Requirement**: Repairs with a priority level of 2 or higher must be addressed first. This ensures that high-priority machines are repaired promptly, aligning with business priorities.  

These constraints ensure that the optimization solution is both feasible and aligned with the repair shop’s operational requirements.  

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating new tables for repair time, technician capacity, and machine priority. Configuration logic updates include scalar parameters for max repairs per technician and min priority, and a formula for repair time calculation.

CREATE TABLE repair_time (
  technician_id INTEGER,
  repair_ID INTEGER,
  repair_time FLOAT
);

CREATE TABLE technician_capacity (
  technician_id INTEGER,
  max_repairs INTEGER
);

CREATE TABLE machine_priority (
  Machine_ID INTEGER,
  priority INTEGER
);

CREATE TABLE repair_assignment (
  technician_id INTEGER,
  repair_ID INTEGER
);
```

### Data Dictionary  
- **repair_time**:  
  - **Business Purpose**: Records the time taken by a technician to complete a specific repair.  
  - **Optimization Role**: Provides the coefficients for the objective function, representing the repair time for each technician-repair pair.  
  - **Columns**:  
    - `technician_id`: Identifier for the technician.  
    - `repair_ID`: Identifier for the repair.  
    - `repair_time`: Time taken to complete the repair.  

- **technician_capacity**:  
  - **Business Purpose**: Specifies the maximum number of repairs each technician can handle.  
  - **Optimization Role**: Provides the bounds for the technician capacity constraint.  
  - **Columns**:  
    - `technician_id`: Identifier for the technician.  
    - `max_repairs`: Maximum number of repairs the technician can handle.  

- **machine_priority**:  
  - **Business Purpose**: Indicates the priority level of each machine repair.  
  - **Optimization Role**: Provides the bounds for the priority constraint.  
  - **Columns**:  
    - `Machine_ID`: Identifier for the machine.  
    - `priority`: Priority level of the machine repair.  

- **repair_assignment**:  
  - **Business Purpose**: Tracks the assignment of technicians to repairs.  
  - **Optimization Role**: Represents the decision variables in the optimization model.  
  - **Columns**:  
    - `technician_id`: Identifier for the technician.  
    - `repair_ID`: Identifier for the repair.  

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic business scenarios, considering technician expertise, machine complexity, and repair shop capacity.

-- Realistic data for repair_time
INSERT INTO repair_time (technician_id, repair_ID, repair_time) VALUES (1, 101, 2.5);
INSERT INTO repair_time (technician_id, repair_ID, repair_time) VALUES (2, 102, 3.0);
INSERT INTO repair_time (technician_id, repair_ID, repair_time) VALUES (3, 103, 4.0);

-- Realistic data for technician_capacity
INSERT INTO technician_capacity (technician_id, max_repairs) VALUES (1, 5);
INSERT INTO technician_capacity (technician_id, max_repairs) VALUES (2, 4);
INSERT INTO technician_capacity (technician_id, max_repairs) VALUES (3, 3);

-- Realistic data for machine_priority
INSERT INTO machine_priority (Machine_ID, priority) VALUES (201, 1);
INSERT INTO machine_priority (Machine_ID, priority) VALUES (202, 2);
INSERT INTO machine_priority (Machine_ID, priority) VALUES (203, 3);

-- Realistic data for repair_assignment
INSERT INTO repair_assignment (technician_id, repair_ID) VALUES (1, 101);
INSERT INTO repair_assignment (technician_id, repair_ID) VALUES (2, 102);
INSERT INTO repair_assignment (technician_id, repair_ID) VALUES (3, 103);
```
