Iteration final - PROBLEM_DESCRIPTION
Sequence: 5
Timestamp: 2025-07-25 22:43:00

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_authID_instID_paperID: binary variable indicating whether researcher authID is assigned to institution instID for paper paperID
- Operational parameters align with expected linear objective: maximize ∑(w_authOrder * x_authID_instID_paperID)
- Business configuration includes: weight representing the order of authorship (used for used in the objective function to weight papers by authorship order), maximum number of researchers per institution (used for constraint on the number of researchers per institution), maximum number of papers per researcher (used for constraint on the number of papers per researcher)
- 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": "icfp_1",
  "iteration": 1,
  "business_context": "A research institution aims to optimize the allocation of its researchers across different projects to maximize the overall research output, measured by the number of papers published, while respecting constraints on the number of researchers per institution and the order of authorship.",
  "optimization_problem_description": "Maximize the total number of papers published by assigning researchers to institutions and projects in a way that respects the constraints on the number of researchers per institution and the order of authorship. The objective is to maximize the sum of papers published, weighted by the order of authorship.",
  "optimization_formulation": {
    "objective": "maximize \u2211(w_authOrder * x_authID_instID_paperID)",
    "decision_variables": "x_authID_instID_paperID: binary variable indicating whether researcher authID is assigned to institution instID for paper paperID",
    "constraints": [
      "\u2211(x_authID_instID_paperID) \u2264 max_researchers_per_inst for each instID",
      "\u2211(x_authID_instID_paperID) \u2264 max_papers_per_auth for each authID",
      "\u2211(x_authID_instID_paperID) = 1 for each paperID"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "w_authOrder": {
        "currently_mapped_to": "business_configuration_logic.w_authOrder",
        "mapping_adequacy": "good",
        "description": "weight representing the order of authorship"
      }
    },
    "constraint_bounds": {
      "max_researchers_per_inst": {
        "currently_mapped_to": "business_configuration_logic.max_researchers_per_inst",
        "mapping_adequacy": "good",
        "description": "maximum number of researchers per institution"
      },
      "max_papers_per_auth": {
        "currently_mapped_to": "business_configuration_logic.max_papers_per_auth",
        "mapping_adequacy": "good",
        "description": "maximum number of papers per researcher"
      }
    },
    "decision_variables": {
      "x_authID_instID_paperID": {
        "currently_mapped_to": "Authorship.authID, Authorship.instID, Authorship.paperID",
        "mapping_adequacy": "good",
        "description": "binary variable indicating whether researcher authID is assigned to institution instID for paper paperID",
        "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 adjustments include creating new tables for missing optimization data, modifying existing tables to better align with OR expert's mapping, and moving scalar parameters and formulas to business_configuration_logic.json.

CREATE TABLE Authorship (
  authID INTEGER,
  instID INTEGER,
  paperID INTEGER,
  authOrder INTEGER
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical research institution operations, ensuring realistic constraints and weights that align with the optimization objective of maximizing research output.

-- Realistic data for Authorship
INSERT INTO Authorship (authID, instID, paperID, authOrder) VALUES (1, 101, 201, 1);
INSERT INTO Authorship (authID, instID, paperID, authOrder) VALUES (2, 102, 202, 2);
INSERT INTO Authorship (authID, instID, paperID, authOrder) VALUES (3, 103, 203, 3);


```

DATA DICTIONARY:
{
  "tables": {
    "Authorship": {
      "business_purpose": "Represents the assignment of researchers to institutions and papers",
      "optimization_role": "decision_variables",
      "columns": {
        "authID": {
          "data_type": "INTEGER",
          "business_meaning": "Researcher ID",
          "optimization_purpose": "Identifier for the researcher",
          "sample_values": [
            1,
            2,
            3
          ]
        },
        "instID": {
          "data_type": "INTEGER",
          "business_meaning": "Institution ID",
          "optimization_purpose": "Identifier for the institution",
          "sample_values": [
            101,
            102,
            103
          ]
        },
        "paperID": {
          "data_type": "INTEGER",
          "business_meaning": "Paper ID",
          "optimization_purpose": "Identifier for the paper",
          "sample_values": [
            201,
            202,
            203
          ]
        },
        "authOrder": {
          "data_type": "INTEGER",
          "business_meaning": "Order of authorship",
          "optimization_purpose": "Determines the weight in the objective function",
          "sample_values": [
            1,
            2,
            3
          ]
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "w_authOrder": {
    "data_type": "FLOAT",
    "business_meaning": "weight representing the order of authorship",
    "optimization_role": "used in the objective function to weight papers by authorship order",
    "configuration_type": "scalar_parameter",
    "value": 0.8,
    "business_justification": "The weight of 0.8 for authorship order reflects the diminishing contribution of lower-order authors, which is a common practice in research evaluation."
  },
  "max_researchers_per_inst": {
    "data_type": "INTEGER",
    "business_meaning": "maximum number of researchers per institution",
    "optimization_role": "constraint on the number of researchers per institution",
    "configuration_type": "scalar_parameter",
    "value": 10,
    "business_justification": "A maximum of 10 researchers per institution is realistic, allowing for sufficient collaboration while maintaining manageable team sizes."
  },
  "max_papers_per_auth": {
    "data_type": "INTEGER",
    "business_meaning": "maximum number of papers per researcher",
    "optimization_role": "constraint on the number of papers per researcher",
    "configuration_type": "scalar_parameter",
    "value": 5,
    "business_justification": "A maximum of 5 papers per researcher ensures a balanced workload and prevents overcommitment, which is typical in research settings."
  }
}

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: icfp_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_authID_instID_paperID: binary variable indicating whether researcher authID is assigned to institution instID for paper paperID
- Operational parameters align with expected linear objective: maximize ∑(w_authOrder * x_authID_instID_paperID)
- Business configuration includes: weight representing the order of authorship (used for used in the objective function to weight papers by authorship order), maximum number of researchers per institution (used for constraint on the number of researchers per institution), maximum number of papers per researcher (used for constraint on the number of papers per researcher)
- 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 ∑(w_authOrder * x_authID_instID_paperID)
- 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_authID_instID_paperID) ≤ max_researchers_per_inst for each instID', '∑(x_authID_instID_paperID) ≤ max_papers_per_auth for each authID', '∑(x_authID_instID_paperID) = 1 for each paperID'] (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 adjustments include creating new tables for missing optimization data, modifying existing tables to better align with OR expert's mapping, and moving scalar parameters and formulas to business_configuration_logic.json.

CREATE TABLE Authorship (
  authID INTEGER,
  instID INTEGER,
  paperID INTEGER,
  authOrder INTEGER
);


```

### Data Dictionary  
[Create comprehensive business-oriented data dictionary mapping tables and columns to their business purposes and optimization roles - NOT technical database terms. Base this on the data_dictionary provided but rewrite in business language that clearly connects to the expected linear mathematical formulation]

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical research institution operations, ensuring realistic constraints and weights that align with the optimization objective of maximizing research output.

-- Realistic data for Authorship
INSERT INTO Authorship (authID, instID, paperID, authOrder) VALUES (1, 101, 201, 1);
INSERT INTO Authorship (authID, instID, paperID, authOrder) VALUES (2, 102, 202, 2);
INSERT INTO Authorship (authID, instID, paperID, authOrder) VALUES (3, 103, 203, 3);


```

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

Response:
# Complete Optimization Problem and Solution: icfp_1

## 1. Problem Context and Goals

### Context  
A research institution is focused on optimizing the allocation of its researchers across various projects to maximize the overall research output. The output is measured by the number of papers published, with a particular emphasis on the order of authorship to reflect the varying contributions of researchers. The institution must ensure that the allocation respects operational constraints, such as the maximum number of researchers that can be assigned to a single institution and the maximum number of papers a researcher can be involved in.  

The decision at hand is whether to assign a specific researcher to a particular institution for a given paper. This decision is represented by a binary indicator, where a value of 1 signifies an assignment and 0 signifies no assignment. The institution aims to maximize the total weighted research output, where the weight is determined by the order of authorship. For example, first authors are given a higher weight than subsequent authors, reflecting their greater contribution to the paper.  

The business configuration includes a weight for the order of authorship, which is used to prioritize papers based on the researcher's role. Additionally, there are constraints on the maximum number of researchers per institution and the maximum number of papers per researcher. These constraints ensure that the allocation remains feasible and aligns with the institution's operational capacity.  

### Goals  
The primary goal of this optimization problem is to maximize the total weighted research output. This is achieved by assigning researchers to institutions and papers in a way that respects the operational constraints. The weight for each paper is determined by the order of authorship, with higher weights assigned to papers where the researcher is a primary contributor. Success is measured by the total sum of these weighted assignments, ensuring that the institution's research output is optimized while adhering to the defined constraints.  

## 2. Constraints  

The optimization problem must adhere to the following constraints:  
1. **Maximum Researchers per Institution**: The total number of researchers assigned to any single institution cannot exceed the predefined maximum limit. This ensures that institutions are not overburdened with too many researchers.  
2. **Maximum Papers per Researcher**: The total number of papers assigned to any single researcher cannot exceed the predefined maximum limit. This ensures that researchers are not overcommitted and can maintain a balanced workload.  
3. **Single Assignment per Paper**: Each paper must be assigned to exactly one researcher-institution pair. This ensures that every paper is accounted for and that there are no overlaps or gaps in the allocation.  

These constraints are designed to ensure that the allocation of researchers to institutions and papers is both feasible and aligned with the institution's operational capacity.  

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema adjustments include creating new tables for missing optimization data, modifying existing tables to better align with OR expert's mapping, and moving scalar parameters and formulas to business_configuration_logic.json.

CREATE TABLE Authorship (
  authID INTEGER,
  instID INTEGER,
  paperID INTEGER,
  authOrder INTEGER
);
```

### Data Dictionary  
The **Authorship** table represents the assignment of researchers to institutions and papers. It plays a critical role in the optimization problem by providing the necessary data to make allocation decisions.  

- **authID**: Represents the unique identifier for each researcher. This is used to track which researcher is being assigned to a paper and institution.  
- **instID**: Represents the unique identifier for each institution. This is used to track which institution the researcher is being assigned to.  
- **paperID**: Represents the unique identifier for each paper. This is used to track which paper the researcher is being assigned to.  
- **authOrder**: Represents the order of authorship for the researcher on the paper. This is used to determine the weight of the paper in the objective function, with higher values indicating a greater contribution.  

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical research institution operations, ensuring realistic constraints and weights that align with the optimization objective of maximizing research output.

-- Realistic data for Authorship
INSERT INTO Authorship (authID, instID, paperID, authOrder) VALUES (1, 101, 201, 1);
INSERT INTO Authorship (authID, instID, paperID, authOrder) VALUES (2, 102, 202, 2);
INSERT INTO Authorship (authID, instID, paperID, authOrder) VALUES (3, 103, 203, 3);
```
