Iteration 1 - OR_EXPERT_REFINEMENT
Sequence: 3
Timestamp: 2025-07-25 22:43:00

Prompt:
You are an Operations Research (OR) expert in iteration 1 of an alternating optimization process. The algorithm alternates between OR expert analysis and data engineering implementation until convergence.

CRITICAL MATHEMATICAL CONSTRAINTS FOR LINEAR/MIXED-INTEGER PROGRAMMING:
- The optimization problem MUST remain 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
- If previous iteration introduced nonlinear elements, redesign as linear formulation
- Maintain between 2 and 20 constraints for optimization feasibility

YOUR SCOPE: Focus exclusively on optimization modeling and mapping analysis. Do NOT propose database changes.
ROW COUNT AWARENESS: Understand that data engineer applies 3-row minimum rule - insufficient table data gets moved to business_configuration_logic.json.


DATA AVAILABILITY CHECK: 
Before listing missing requirements, verify:
- Check current schema for required data columns
- Check business configuration logic for required parameters  
- Only list as "missing" if data is truly unavailable
- If all mappings are "good", missing_requirements should be []

CONSISTENCY RULES:
- IF all mapping_adequacy == "good" THEN missing_optimization_requirements = []
- IF missing_optimization_requirements = [] THEN complete CAN be true
- IF complete == true THEN confidence should be "high"

SELF-CHECK: Before responding, verify:
1. Does current schema contain the data I claim is missing?
2. Are my mapping assessments consistent with missing requirements?
3. Is my complete status consistent with missing requirements?

MAPPING COMPLETENESS CHECK: Ensure logical consistency between:
- All objective coefficients mapped with adequacy evaluation
- All constraint bounds mapped with adequacy evaluation  
- All decision variables mapped with adequacy evaluation
- Missing requirements list matches inadequate mappings only


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



CURRENT STATE (iteration 0):
{
  "iteration": 1,
  "converged": false,
  "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": "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.",
  "objective": "maximize \u2211(w_authOrder * x_authID_instID_paperID)",
  "table_count": 0,
  "key_changes": [
    "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."
  ],
  "math_consistency": "high",
  "next_iteration_focus": "Refine the constraints and objective function to better align with the business context and ensure all necessary data is available.",
  "mapping_adequacy_summary": "needs_improvement"
}

CURRENT 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 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
          ]
        }
      }
    }
  }
}


CURRENT BUSINESS CONFIGURATION LOGIC:
{
  "w_authOrder": {
    "sample_value": 1.0,
    "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"
  },
  "max_researchers_per_inst": {
    "sample_value": 5,
    "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"
  },
  "max_papers_per_auth": {
    "sample_value": 3,
    "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"
  }
}


TASK: Refine the optimization problem formulation by analyzing current data schema mapping and identifying requirements while maintaining LINEAR structure.

JSON STRUCTURE REQUIRED:

{
  "database_id": "icfp_1",
  "iteration": 1,
  "business_context": "Updated realistic business scenario description that supports linear optimization",
  "optimization_problem_description": "Refined description of LINEAR optimization problem", 
  "optimization_formulation": {
    "objective": "refined linear minimize/maximize with mathematical precision (sum of weighted variables only)",
    "decision_variables": "clearly defined controllable linear variables (continuous or integer)",
    "constraints": "mathematically precise LINEAR constraint definitions (no variable products or divisions) - maintain 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 still needed for linear formulation (should be [] if all mappings are good)"
  ],
  
  "iteration_status": {
    "complete": true/false based on linear formulation completeness,
    "confidence": "updated confidence level",
    "next_focus": "What needs refinement for linear optimization OR 'Ready for convergence'"
  }
}

Response:
{
  "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"
  }
}
