Iteration final - PROBLEM_DESCRIPTION
Sequence: 9
Timestamp: 2025-07-25 22:36:07

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 3), 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_full_time[shop_id]: number of full-time employees to hire for shop_id (integer), x_part_time[shop_id]: number of part-time employees to hire for shop_id (integer), y[employee_id]: binary decision to award bonus to employee_id (binary)
- Operational parameters align with expected linear objective: minimize ∑(cost_full_time[shop_id] × x_full_time[shop_id] + cost_part_time[shop_id] × x_part_time[shop_id])
- Business configuration includes: Cost per full-time employee for each shop (used for Objective coefficient for full-time hiring cost), Cost per part-time employee for each shop (used for Objective coefficient for part-time hiring cost)
- 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": "employee_hire_evaluation",
  "iteration": 3,
  "business_context": "A retail chain aims to optimize employee hiring and allocation across shops to minimize total hiring costs while ensuring each shop has sufficient staff and employees are fairly evaluated based on their bonuses. The optimization problem is linear, focusing on minimizing costs and meeting staffing requirements.",
  "optimization_problem_description": "Minimize the total hiring costs by deciding the number of full-time and part-time employees to hire for each shop, ensuring that each shop meets its staffing requirements and that employees receive bonuses based on their evaluations. The problem is formulated as a linear optimization problem with linear constraints.",
  "optimization_formulation": {
    "objective": "minimize \u2211(cost_full_time[shop_id] \u00d7 x_full_time[shop_id] + cost_part_time[shop_id] \u00d7 x_part_time[shop_id])",
    "decision_variables": "x_full_time[shop_id]: number of full-time employees to hire for shop_id (integer), x_part_time[shop_id]: number of part-time employees to hire for shop_id (integer), y[employee_id]: binary decision to award bonus to employee_id (binary)",
    "constraints": "x_full_time[shop_id] + x_part_time[shop_id] \u2265 min_staff[shop_id] for all shop_id, x_full_time[shop_id] \u2264 max_full_time[shop_id] for all shop_id, x_part_time[shop_id] \u2264 max_part_time[shop_id] for all shop_id"
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "cost_full_time[shop_id]": {
        "currently_mapped_to": "employee_costs.cost_full_time",
        "mapping_adequacy": "good",
        "description": "Cost per full-time employee for each shop"
      },
      "cost_part_time[shop_id]": {
        "currently_mapped_to": "employee_costs.cost_part_time",
        "mapping_adequacy": "good",
        "description": "Cost per part-time employee for each shop"
      }
    },
    "constraint_bounds": {
      "min_staff[shop_id]": {
        "currently_mapped_to": "shop_staff_requirements.min_staff",
        "mapping_adequacy": "good",
        "description": "Minimum staff required for each shop"
      },
      "max_full_time[shop_id]": {
        "currently_mapped_to": "shop_staff_requirements.max_full_time",
        "mapping_adequacy": "good",
        "description": "Maximum full-time employees allowed for each shop"
      },
      "max_part_time[shop_id]": {
        "currently_mapped_to": "shop_staff_requirements.max_part_time",
        "mapping_adequacy": "good",
        "description": "Maximum part-time employees allowed for each shop"
      }
    },
    "decision_variables": {
      "x_full_time[shop_id]": {
        "currently_mapped_to": "employee_hiring_decisions.x_full_time",
        "mapping_adequacy": "good",
        "description": "Number of full-time employees to hire for each shop",
        "variable_type": "integer"
      },
      "x_part_time[shop_id]": {
        "currently_mapped_to": "employee_hiring_decisions.x_part_time",
        "mapping_adequacy": "good",
        "description": "Number of part-time employees to hire for each shop",
        "variable_type": "integer"
      },
      "y[employee_id]": {
        "currently_mapped_to": "employee_bonus_decisions.y",
        "mapping_adequacy": "good",
        "description": "Binary decision to award bonus to each employee",
        "variable_type": "binary"
      }
    }
  },
  "missing_optimization_requirements": [],
  "iteration_status": {
    "complete": true,
    "confidence": "high",
    "next_focus": "Ready for convergence"
  }
}

FINAL DATABASE SCHEMA:
```sql
-- Iteration 3 Database Schema
-- Objective: Schema changes include creating a new table for objective coefficients (cost_full_time and cost_part_time) and updating business configuration logic to ensure all optimization requirements are mapped. Existing tables and configuration logic are reviewed for completeness and alignment with OR expert's analysis.

CREATE TABLE shop_staff_requirements (
  shop_id INTEGER,
  min_staff INTEGER,
  max_full_time INTEGER,
  max_part_time INTEGER
);

CREATE TABLE employee_bonus_decisions (
  employee_id INTEGER,
  y BOOLEAN
);

CREATE TABLE employee_hiring_decisions (
  shop_id INTEGER,
  x_full_time INTEGER,
  x_part_time INTEGER
);

CREATE TABLE employee_costs (
  shop_id INTEGER,
  cost_full_time INTEGER,
  cost_part_time INTEGER
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 3 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on industry standards for retail staffing and costs, ensuring realistic and meaningful data for optimization.

-- Realistic data for shop_staff_requirements
INSERT INTO shop_staff_requirements (shop_id, min_staff, max_full_time, max_part_time) VALUES (1, 5, 10, 8);
INSERT INTO shop_staff_requirements (shop_id, min_staff, max_full_time, max_part_time) VALUES (2, 6, 12, 10);
INSERT INTO shop_staff_requirements (shop_id, min_staff, max_full_time, max_part_time) VALUES (3, 7, 15, 12);

-- Realistic data for employee_bonus_decisions
INSERT INTO employee_bonus_decisions (employee_id, y) VALUES (101, True);
INSERT INTO employee_bonus_decisions (employee_id, y) VALUES (102, False);
INSERT INTO employee_bonus_decisions (employee_id, y) VALUES (103, True);

-- Realistic data for employee_hiring_decisions
INSERT INTO employee_hiring_decisions (shop_id, x_full_time, x_part_time) VALUES (1, 2, 3);
INSERT INTO employee_hiring_decisions (shop_id, x_full_time, x_part_time) VALUES (2, 3, 4);
INSERT INTO employee_hiring_decisions (shop_id, x_full_time, x_part_time) VALUES (3, 4, 5);

-- Realistic data for employee_costs
INSERT INTO employee_costs (shop_id, cost_full_time, cost_part_time) VALUES (1, 1500, 800);
INSERT INTO employee_costs (shop_id, cost_full_time, cost_part_time) VALUES (2, 1600, 850);
INSERT INTO employee_costs (shop_id, cost_full_time, cost_part_time) VALUES (3, 1700, 900);


```

DATA DICTIONARY:
{
  "tables": {
    "shop_staff_requirements": {
      "business_purpose": "Staffing requirements for each shop",
      "optimization_role": "constraint_bounds",
      "columns": {
        "shop_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each shop",
          "optimization_purpose": "Index for shop-specific constraints",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "min_staff": {
          "data_type": "INTEGER",
          "business_meaning": "Minimum staff required for the shop",
          "optimization_purpose": "Lower bound for staffing constraint",
          "sample_values": [
            5,
            6,
            7
          ]
        },
        "max_full_time": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum full-time employees allowed for the shop",
          "optimization_purpose": "Upper bound for full-time hiring constraint",
          "sample_values": [
            10,
            12,
            15
          ]
        },
        "max_part_time": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum part-time employees allowed for the shop",
          "optimization_purpose": "Upper bound for part-time hiring constraint",
          "sample_values": [
            8,
            10,
            12
          ]
        }
      }
    },
    "employee_bonus_decisions": {
      "business_purpose": "Decisions to award bonuses to employees",
      "optimization_role": "decision_variables",
      "columns": {
        "employee_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each employee",
          "optimization_purpose": "Index for bonus decision variable",
          "sample_values": [
            101,
            102,
            103
          ]
        },
        "y": {
          "data_type": "BOOLEAN",
          "business_meaning": "Binary decision to award bonus",
          "optimization_purpose": "Decision variable for bonus allocation",
          "sample_values": [
            true,
            false,
            true
          ]
        }
      }
    },
    "employee_hiring_decisions": {
      "business_purpose": "Decisions on the number of full-time and part-time employees to hire for each shop",
      "optimization_role": "decision_variables",
      "columns": {
        "shop_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each shop",
          "optimization_purpose": "Index for hiring decision variables",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "x_full_time": {
          "data_type": "INTEGER",
          "business_meaning": "Number of full-time employees to hire for the shop",
          "optimization_purpose": "Decision variable for full-time hiring",
          "sample_values": [
            2,
            3,
            4
          ]
        },
        "x_part_time": {
          "data_type": "INTEGER",
          "business_meaning": "Number of part-time employees to hire for the shop",
          "optimization_purpose": "Decision variable for part-time hiring",
          "sample_values": [
            3,
            4,
            5
          ]
        }
      }
    },
    "employee_costs": {
      "business_purpose": "Cost per full-time and part-time employee for each shop",
      "optimization_role": "objective_coefficients",
      "columns": {
        "shop_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each shop",
          "optimization_purpose": "Index for shop-specific costs",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "cost_full_time": {
          "data_type": "INTEGER",
          "business_meaning": "Cost per full-time employee for the shop",
          "optimization_purpose": "Objective coefficient for full-time hiring cost",
          "sample_values": [
            1500,
            1600,
            1700
          ]
        },
        "cost_part_time": {
          "data_type": "INTEGER",
          "business_meaning": "Cost per part-time employee for the shop",
          "optimization_purpose": "Objective coefficient for part-time hiring cost",
          "sample_values": [
            800,
            850,
            900
          ]
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "cost_full_time": {
    "data_type": "INTEGER",
    "business_meaning": "Cost per full-time employee for each shop",
    "optimization_role": "Objective coefficient for full-time hiring cost",
    "configuration_type": "scalar_parameter",
    "value": 1500,
    "business_justification": "Average cost for a full-time employee in the retail industry."
  },
  "cost_part_time": {
    "data_type": "INTEGER",
    "business_meaning": "Cost per part-time employee for each shop",
    "optimization_role": "Objective coefficient for part-time hiring cost",
    "configuration_type": "scalar_parameter",
    "value": 800,
    "business_justification": "Average cost for a part-time employee in the retail industry."
  }
}

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: employee_hire_evaluation

## 1. Problem Context and Goals

### Context  
[Regenerate business context that naturally aligns with LINEAR optimization formulation. Ensure:]
- Business decisions match expected decision variables: x_full_time[shop_id]: number of full-time employees to hire for shop_id (integer), x_part_time[shop_id]: number of part-time employees to hire for shop_id (integer), y[employee_id]: binary decision to award bonus to employee_id (binary)
- Operational parameters align with expected linear objective: minimize ∑(cost_full_time[shop_id] × x_full_time[shop_id] + cost_part_time[shop_id] × x_part_time[shop_id])
- Business configuration includes: Cost per full-time employee for each shop (used for Objective coefficient for full-time hiring cost), Cost per part-time employee for each shop (used for Objective coefficient for part-time hiring cost)
- 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 ∑(cost_full_time[shop_id] × x_full_time[shop_id] + cost_part_time[shop_id] × x_part_time[shop_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: x_full_time[shop_id] + x_part_time[shop_id] ≥ min_staff[shop_id] for all shop_id, x_full_time[shop_id] ≤ max_full_time[shop_id] for all shop_id, x_part_time[shop_id] ≤ max_part_time[shop_id] for all shop_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 3 Database Schema
-- Objective: Schema changes include creating a new table for objective coefficients (cost_full_time and cost_part_time) and updating business configuration logic to ensure all optimization requirements are mapped. Existing tables and configuration logic are reviewed for completeness and alignment with OR expert's analysis.

CREATE TABLE shop_staff_requirements (
  shop_id INTEGER,
  min_staff INTEGER,
  max_full_time INTEGER,
  max_part_time INTEGER
);

CREATE TABLE employee_bonus_decisions (
  employee_id INTEGER,
  y BOOLEAN
);

CREATE TABLE employee_hiring_decisions (
  shop_id INTEGER,
  x_full_time INTEGER,
  x_part_time INTEGER
);

CREATE TABLE employee_costs (
  shop_id INTEGER,
  cost_full_time INTEGER,
  cost_part_time 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 3 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on industry standards for retail staffing and costs, ensuring realistic and meaningful data for optimization.

-- Realistic data for shop_staff_requirements
INSERT INTO shop_staff_requirements (shop_id, min_staff, max_full_time, max_part_time) VALUES (1, 5, 10, 8);
INSERT INTO shop_staff_requirements (shop_id, min_staff, max_full_time, max_part_time) VALUES (2, 6, 12, 10);
INSERT INTO shop_staff_requirements (shop_id, min_staff, max_full_time, max_part_time) VALUES (3, 7, 15, 12);

-- Realistic data for employee_bonus_decisions
INSERT INTO employee_bonus_decisions (employee_id, y) VALUES (101, True);
INSERT INTO employee_bonus_decisions (employee_id, y) VALUES (102, False);
INSERT INTO employee_bonus_decisions (employee_id, y) VALUES (103, True);

-- Realistic data for employee_hiring_decisions
INSERT INTO employee_hiring_decisions (shop_id, x_full_time, x_part_time) VALUES (1, 2, 3);
INSERT INTO employee_hiring_decisions (shop_id, x_full_time, x_part_time) VALUES (2, 3, 4);
INSERT INTO employee_hiring_decisions (shop_id, x_full_time, x_part_time) VALUES (3, 4, 5);

-- Realistic data for employee_costs
INSERT INTO employee_costs (shop_id, cost_full_time, cost_part_time) VALUES (1, 1500, 800);
INSERT INTO employee_costs (shop_id, cost_full_time, cost_part_time) VALUES (2, 1600, 850);
INSERT INTO employee_costs (shop_id, cost_full_time, cost_part_time) VALUES (3, 1700, 900);


```

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

Response:
# Complete Optimization Problem and Solution: employee_hire_evaluation

## 1. Problem Context and Goals

### Context  
A retail chain is focused on optimizing its employee hiring and allocation strategy across multiple shops to minimize total hiring costs while ensuring each shop meets its staffing requirements. The business decisions involve determining the number of full-time and part-time employees to hire for each shop, as well as deciding which employees should receive bonuses based on their performance evaluations.  

The cost of hiring a full-time employee varies by shop, with each shop having a specific cost per full-time employee. Similarly, the cost of hiring a part-time employee also varies by shop. These costs are critical inputs to the optimization problem, as they directly influence the total hiring expenses.  

The business configuration includes the following operational parameters:  
- **Cost per full-time employee for each shop**: This represents the expense associated with hiring a full-time employee at a specific shop.  
- **Cost per part-time employee for each shop**: This represents the expense associated with hiring a part-time employee at a specific shop.  

The optimization problem is designed to ensure that each shop has sufficient staff to meet its operational needs while adhering to limits on the number of full-time and part-time employees that can be hired. The problem avoids scenarios that would require nonlinear relationships, such as variable products or divisions, ensuring a linear formulation.  

### Goals  
The primary goal of this optimization problem is to minimize the total hiring costs across all shops. This is achieved by carefully deciding the number of full-time and part-time employees to hire for each shop, taking into account the specific costs associated with each type of hire.  

Success is measured by the ability to reduce total hiring expenses while ensuring that each shop meets its minimum staffing requirements and does not exceed the maximum allowed number of full-time or part-time employees. The optimization goal is directly aligned with the operational parameters, ensuring a clear and linear relationship between the decisions and the objective.  

## 2. Constraints  

The optimization problem is subject to the following constraints, which ensure that the staffing decisions align with business requirements:  

1. **Minimum Staffing Requirement**: For each shop, the total number of full-time and part-time employees hired must meet or exceed the minimum staffing requirement. This ensures that each shop has sufficient staff to operate effectively.  

2. **Maximum Full-Time Employees**: For each shop, the number of full-time employees hired must not exceed the maximum allowed number of full-time employees. This ensures compliance with business policies regarding full-time staffing levels.  

3. **Maximum Part-Time Employees**: For each shop, the number of part-time employees hired must not exceed the maximum allowed number of part-time employees. This ensures compliance with business policies regarding part-time staffing levels.  

These constraints are designed to naturally lead to linear mathematical forms, avoiding any complexity that would require nonlinear relationships.  

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 3 Database Schema
-- Objective: Schema changes include creating a new table for objective coefficients (cost_full_time and cost_part_time) and updating business configuration logic to ensure all optimization requirements are mapped. Existing tables and configuration logic are reviewed for completeness and alignment with OR expert's analysis.

CREATE TABLE shop_staff_requirements (
  shop_id INTEGER,
  min_staff INTEGER,
  max_full_time INTEGER,
  max_part_time INTEGER
);

CREATE TABLE employee_bonus_decisions (
  employee_id INTEGER,
  y BOOLEAN
);

CREATE TABLE employee_hiring_decisions (
  shop_id INTEGER,
  x_full_time INTEGER,
  x_part_time INTEGER
);

CREATE TABLE employee_costs (
  shop_id INTEGER,
  cost_full_time INTEGER,
  cost_part_time INTEGER
);
```

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

- **shop_staff_requirements**:  
  - **shop_id**: Unique identifier for each shop.  
  - **min_staff**: Minimum number of employees required for the shop to operate effectively.  
  - **max_full_time**: Maximum number of full-time employees allowed for the shop.  
  - **max_part_time**: Maximum number of part-time employees allowed for the shop.  

- **employee_bonus_decisions**:  
  - **employee_id**: Unique identifier for each employee.  
  - **y**: Binary decision indicating whether a bonus is awarded to the employee.  

- **employee_hiring_decisions**:  
  - **shop_id**: Unique identifier for each shop.  
  - **x_full_time**: Number of full-time employees to hire for the shop.  
  - **x_part_time**: Number of part-time employees to hire for the shop.  

- **employee_costs**:  
  - **shop_id**: Unique identifier for each shop.  
  - **cost_full_time**: Cost per full-time employee for the shop.  
  - **cost_part_time**: Cost per part-time employee for the shop.  

### Current Stored Values  
```sql
-- Iteration 3 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on industry standards for retail staffing and costs, ensuring realistic and meaningful data for optimization.

-- Realistic data for shop_staff_requirements
INSERT INTO shop_staff_requirements (shop_id, min_staff, max_full_time, max_part_time) VALUES (1, 5, 10, 8);
INSERT INTO shop_staff_requirements (shop_id, min_staff, max_full_time, max_part_time) VALUES (2, 6, 12, 10);
INSERT INTO shop_staff_requirements (shop_id, min_staff, max_full_time, max_part_time) VALUES (3, 7, 15, 12);

-- Realistic data for employee_bonus_decisions
INSERT INTO employee_bonus_decisions (employee_id, y) VALUES (101, True);
INSERT INTO employee_bonus_decisions (employee_id, y) VALUES (102, False);
INSERT INTO employee_bonus_decisions (employee_id, y) VALUES (103, True);

-- Realistic data for employee_hiring_decisions
INSERT INTO employee_hiring_decisions (shop_id, x_full_time, x_part_time) VALUES (1, 2, 3);
INSERT INTO employee_hiring_decisions (shop_id, x_full_time, x_part_time) VALUES (2, 3, 4);
INSERT INTO employee_hiring_decisions (shop_id, x_full_time, x_part_time) VALUES (3, 4, 5);

-- Realistic data for employee_costs
INSERT INTO employee_costs (shop_id, cost_full_time, cost_part_time) VALUES (1, 1500, 800);
INSERT INTO employee_costs (shop_id, cost_full_time, cost_part_time) VALUES (2, 1600, 850);
INSERT INTO employee_costs (shop_id, cost_full_time, cost_part_time) VALUES (3, 1700, 900);
```
