Iteration 0 - OR_EXPERT
Sequence: 1
Timestamp: 2025-07-25 22:43:00

Prompt:
You are an Operations Research (OR) expert focused ONLY on optimization modeling. Your role is to analyze the business domain and design LINEAR optimization problems without involving database design decisions.

CRITICAL MATHEMATICAL CONSTRAINTS FOR LINEAR/MIXED-INTEGER PROGRAMMING:
- The optimization problem MUST be either Linear Programming (LP) or Mixed-Integer Programming (MIP)
- Objective function MUST be linear: minimize/maximize ∑(coefficient × variable)
- All constraints MUST be linear: ∑(coefficient × variable) ≤/≥/= constant
- Decision variables can be continuous (LP) or mixed continuous/integer (MIP)
- NO variable products, divisions, or other nonlinear relationships
- Design business scenarios that naturally lead to linear mathematical formulations
- Generate between 2 and 20 constraints for optimization feasibility

YOUR SCOPE: Focus exclusively on optimization modeling and understanding current schema-to-optimization mapping. Do NOT propose database changes.
ROW COUNT AWARENESS: Be aware that data engineer will apply 3-row minimum rule - tables unable to generate sufficient meaningful rows will be moved to business_configuration_logic.json.

CRITICAL: Respond with ONLY a valid JSON object. No explanations, no markdown, no extra text.

Database: icfp_1

```sql
CREATE TABLE Inst (
  instID NUMBER,
  name TEXT,
  country TEXT
);
```

```sql
CREATE TABLE Authors (
  authID NUMBER,
  lname TEXT,
  fname TEXT
);
```

```sql
CREATE TABLE Papers (
  paperID NUMBER,
  title TEXT
);
```

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



Current Business Configuration Logic: {} (initially empty)

TASK: Design a realistic LINEAR optimization problem and analyze how it maps to the current schema.

JSON STRUCTURE REQUIRED:

{
  "database_id": "icfp_1",
  "iteration": 0,
  "business_context": "Describe the realistic business scenario that leads to linear optimization (avoid scenarios requiring nonlinear relationships)",
  "optimization_problem_description": "Detailed description of what needs to be optimized using linear objective functions and constraints",
  "optimization_formulation": {
    "objective": "minimize/maximize specific_linear_metric with clear definition (sum of weighted variables only)",
    "decision_variables": "What linear variables can be controlled/decided (continuous or integer)",
    "constraints": "What linear limitations must be respected (no variable products or divisions) - design 2 to 20 constraints"
  },
  
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "coefficient_name[indices]": {
        "currently_mapped_to": "table.column OR business_configuration_logic.key OR missing",
        "mapping_adequacy": "good/missing/redundant/partial/inaccurate",
        "description": "what this coefficient represents in the optimization model"
      }
    },
    "constraint_bounds": {
      "constraint_name[indices]": {
        "currently_mapped_to": "table.column OR business_configuration_logic.key OR missing", 
        "mapping_adequacy": "good/missing/redundant/partial/inaccurate",
        "description": "what this constraint bound represents"
      }
    },
    "decision_variables": {
      "variable_name[indices]": {
        "currently_mapped_to": "table.column OR business_configuration_logic.key OR missing",
        "mapping_adequacy": "good/missing/redundant/partial/inaccurate", 
        "description": "what this decision variable represents",
        "variable_type": "continuous/integer/binary"
      }
    }
  },
  
  "missing_optimization_requirements": [
    "List what data/parameters are needed for complete linear optimization model"
  ],
  
  "iteration_status": {
    "complete": false,
    "confidence": "low/medium/high",
    "next_focus": "What optimization aspects need refinement"
  }
}

Response:
{
  "database_id": "icfp_1",
  "iteration": 0,
  "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": "The goal is to 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): 1 if researcher authID is assigned to institution instID and paper paperID, 0 otherwise",
    "constraints": [
      "\u2211(x_authID_instID_paperID) \u2264 1 for each authID (each researcher can be assigned to at most one institution and paper)",
      "\u2211(x_authID_instID_paperID) \u2264 max_researchers_per_inst for each instID (maximum number of researchers per institution)",
      "\u2211(x_authID_instID_paperID) \u2264 max_papers_per_auth for each authID (maximum number of papers per researcher)",
      "\u2211(x_authID_instID_paperID) = 1 for each paperID (each paper must have exactly one researcher assigned)",
      "x_authID_instID_paperID \u2264 y_authOrder for each authID, instID, paperID (order of authorship must be respected)"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "w_authOrder": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "weight representing the order of authorship"
      }
    },
    "constraint_bounds": {
      "max_researchers_per_inst": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "maximum number of researchers per institution"
      },
      "max_papers_per_auth": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "maximum number of papers per researcher"
      }
    },
    "decision_variables": {
      "x_authID_instID_paperID": {
        "currently_mapped_to": "Authorship.authID, Authorship.instID, Authorship.paperID",
        "mapping_adequacy": "partial",
        "description": "assignment of researcher to institution and paper",
        "variable_type": "binary"
      },
      "y_authOrder": {
        "currently_mapped_to": "Authorship.authOrder",
        "mapping_adequacy": "good",
        "description": "order of authorship",
        "variable_type": "integer"
      }
    }
  },
  "missing_optimization_requirements": [
    "weights for the order of authorship (w_authOrder)",
    "maximum number of researchers per institution (max_researchers_per_inst)",
    "maximum number of papers per researcher (max_papers_per_auth)"
  ],
  "iteration_status": {
    "complete": false,
    "confidence": "medium",
    "next_focus": "Refine the constraints and objective function to better align with the business context and ensure all necessary data is available."
  }
}
