Iteration 0 - OR_EXPERT
Sequence: 1
Timestamp: 2025-07-27 22:35:15

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: shop_membership

```sql
CREATE TABLE member (
  Member_ID NUMBER,
  Card_Number TEXT,
  Name TEXT,
  Hometown TEXT,
  Level NUMBER
);
```

```sql
CREATE TABLE branch (
  Branch_ID NUMBER,
  Name TEXT,
  Open_year TEXT,
  Address_road TEXT,
  City TEXT,
  membership_amount TEXT
);
```

```sql
CREATE TABLE membership_register_branch (
  Member_ID NUMBER,
  Branch_ID TEXT,
  Register_Year TEXT
);
```

```sql
CREATE TABLE purchase (
  Member_ID NUMBER,
  Branch_ID TEXT,
  Year TEXT,
  Total_pounds 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": "shop_membership",
  "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": "shop_membership",
  "iteration": 0,
  "business_context": "A retail chain wants to optimize the allocation of membership marketing budget across different branches to maximize total purchases by members. The budget allocation should consider the membership levels and historical purchase data.",
  "optimization_problem_description": "The goal is to maximize the total purchases by members across all branches by optimally allocating a fixed marketing budget. The allocation should be based on historical purchase data and membership levels, ensuring that each branch receives a minimum budget.",
  "optimization_formulation": {
    "objective": "maximize total_purchases = \u2211(purchase_coefficient[branch_id] \u00d7 budget_allocation[branch_id])",
    "decision_variables": "budget_allocation[branch_id] (continuous)",
    "constraints": [
      "\u2211(budget_allocation[branch_id]) \u2264 total_budget",
      "budget_allocation[branch_id] \u2265 min_budget[branch_id] for all branch_id"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "purchase_coefficient[branch_id]": {
        "currently_mapped_to": "purchase.Total_pounds",
        "mapping_adequacy": "partial",
        "description": "Represents the historical purchase data for each branch, used to weigh the budget allocation"
      }
    },
    "constraint_bounds": {
      "total_budget": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "The total marketing budget available for allocation"
      },
      "min_budget[branch_id]": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "The minimum budget that must be allocated to each branch"
      }
    },
    "decision_variables": {
      "budget_allocation[branch_id]": {
        "currently_mapped_to": "missing",
        "mapping_adequacy": "missing",
        "description": "The amount of budget allocated to each branch",
        "variable_type": "continuous"
      }
    }
  },
  "missing_optimization_requirements": [
    "Total marketing budget (total_budget)",
    "Minimum budget allocation for each branch (min_budget[branch_id])"
  ],
  "iteration_status": {
    "complete": false,
    "confidence": "medium",
    "next_focus": "Determine the total marketing budget and minimum budget allocations for each branch"
  }
}
