## 4. Mathematical Optimization Formulation

#### Decision Variables
- Let \( x_{ij} \) be a binary decision variable where \( x_{ij} = 1 \) if submission \( j \) is accepted for workshop \( i \), and \( x_{ij} = 0 \) otherwise.

#### Objective Function
Maximize the total quality score of accepted submissions:
\[ \text{Maximize } \sum_{i} \sum_{j} \text{Scores}_j \times x_{ij} \]

Data Source Verification:
- Coefficient \(\text{Scores}_j\) comes from `submission.Scores`.

#### Constraints
1. **Workshop Capacity Constraints**: Ensure that the total number of accepted submissions for each workshop does not exceed its capacity.
   \[
   \sum_{j} x_{ij} \leq \text{capacity}_i \quad \forall i
   \]

   Data Source Verification:
   - Coefficient \(\text{capacity}_i\) comes from `VenueCapacity.capacity`.

2. **Minimum Submissions Constraints**: Ensure that each workshop has at least the minimum required number of accepted submissions.
   \[
   \sum_{j} x_{ij} \geq \text{min\_submissions}_i \quad \forall i
   \]

   Data Source Verification:
   - Coefficient \(\text{min\_submissions}_i\) comes from `WorkshopSubmissions.min_submissions`.

3. **Binary Decision Variables**: Ensure that each decision variable is binary.
   \[
   x_{ij} \in \{0, 1\} \quad \forall i, j
   \]

This formulation provides a complete linear optimization model that can be solved using standard linear programming or mixed-integer programming solvers. The objective function and constraints are derived directly from the provided data, ensuring alignment with the business problem's requirements.

## 5. Gurobipy Implementation

```python
# Complete GUROBIPY implementation - Retry Attempt 4

import gurobipy as gp
from gurobipy import GRB

def optimize_workshop_paper():
    # 1. MODEL & DATA SETUP
    model = gp.Model("workshop_paper")
    
    # Data from the database
    scores = [8.5, 6.0, 9.2, 7.3, 5.5]
    capacities = {1: 50, 2: 100, 3: 150}
    min_submissions = {1: 3, 2: 5, 3: 7}
    
    # Number of submissions
    n_submissions = len(scores)
    n_workshops = len(capacities)
    
    # CRITICAL: Validate array lengths before loops
    assert len(scores) == n_submissions, "Array length mismatch for scores"
    
    # 2. VARIABLES
    # Binary decision variables for acceptance of submissions
    x = model.addVars(n_workshops, n_submissions, vtype=GRB.BINARY, name="x")
    
    # 3. OBJECTIVE FUNCTION
    # Maximize the total quality score of accepted submissions
    model.setObjective(gp.quicksum(scores[j] * x[i, j] for i in range(n_workshops) for j in range(n_submissions)), GRB.MAXIMIZE)
    
    # 4. CONSTRAINTS - CORRECT SYNTAX PATTERNS
    
    # Workshop Capacity Constraints
    for i in range(n_workshops):
        model.addConstr(gp.quicksum(x[i, j] for j in range(n_submissions)) <= capacities[i+1], name=f"capacity_{i+1}")
    
    # Minimum Submissions Constraints
    for i in range(n_workshops):
        model.addConstr(gp.quicksum(x[i, j] for j in range(n_submissions)) >= min_submissions[i+1], name=f"min_submissions_{i+1}")
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in range(n_workshops):
            for j in range(n_submissions):
                if x[i, j].x > 1e-6:
                    print(f"x[{i},{j}] = {x[i, j].x:.3f}")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")
    
    return model

# Run the optimization
optimize_workshop_paper()
```

### Execution Results
**Status**: INFEASIBLE
**Error**: Problem is infeasible
**Analysis**: Gurobipy consistently reported the problem as infeasible across multiple retry attempts, indicating a potential issue with the problem formulation or data constraints.

## 6. DOCplex Implementation

```python
# Complete DOCPLEX implementation - Retry Attempt 4

from docplex.mp.model import Model

def optimize_workshop_submissions():
    # 1. MODEL & DATA SETUP
    mdl = Model(name="workshop_paper")
    
    # Data from the database
    scores = [8.5, 6.0, 9.2, 7.3, 5.5]  # Submission scores
    capacities = [50, 100, 150]  # Venue capacities for each workshop
    min_submissions = [3, 5, 7]  # Minimum submissions required for each workshop
    
    # Validate array lengths to prevent IndexError
    n_submissions = len(scores)
    n_workshops = len(capacities)
    assert len(min_submissions) == n_workshops, "Array length mismatch"
    
    # Safe indexing
    safe_range_submissions = range(n_submissions)
    safe_range_workshops = range(n_workshops)
    
    # 2. VARIABLES
    # Binary decision variables for acceptance of submissions
    x = {(i, j): mdl.binary_var(name=f"x_{i}_{j}") for i in safe_range_workshops for j in safe_range_submissions}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize the total quality score of accepted submissions
    objective = mdl.sum(scores[j] * x[i, j] for i in safe_range_workshops for j in safe_range_submissions)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS
    # Workshop Capacity Constraints
    for i in safe_range_workshops:
        mdl.add_constraint(mdl.sum(x[i, j] for j in safe_range_submissions) <= capacities[i], ctname=f"capacity_{i}")
    
    # Minimum Submissions Constraints
    for i in safe_range_workshops:
        mdl.add_constraint(mdl.sum(x[i, j] for j in safe_range_submissions) >= min_submissions[i], ctname=f"min_submissions_{i}")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for i in safe_range_workshops:
            for j in safe_range_submissions:
                if solution.get_value(x[i, j]) > 0.5:  # Check if submission is accepted
                    print(f"Submission {j} accepted for Workshop {i}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")

    return mdl

# Execute the optimization
optimize_workshop_submissions()
```

### Execution Results
**Status**: INFEASIBLE
**Error**: Problem is infeasible
**Analysis**: DOCplex also reported the problem as infeasible, aligning with the results from other solvers, suggesting a systemic issue rather than a solver-specific problem.

## 7. Pyomo Implementation

```python
# Complete PYOMO implementation - Retry Attempt 4

import pyomo.environ as pyo
from pyomo.opt import SolverFactory

def conference_optimization():
    """Pyomo model for optimizing conference workshop submissions"""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Assuming 5 submissions and 3 workshops based on the provided data
    n_submissions = 5
    n_workshops = 3
    
    # Scores for each submission
    scores = [8.5, 6.0, 9.2, 7.3, 5.5]
    
    # Workshop capacities
    capacities = {1: 50, 2: 100, 3: 150}
    
    # Minimum submissions required for each workshop
    min_submissions = {1: 3, 2: 5, 3: 7}
    
    # CRITICAL: Validate array lengths before indexing
    assert len(scores) == n_submissions, "Array length mismatch for scores"
    
    # 3. SETS
    model.Submissions = pyo.RangeSet(1, n_submissions)
    model.Workshops = pyo.RangeSet(1, n_workshops)
    
    # 4. PARAMETERS
    model.Scores = pyo.Param(model.Submissions, initialize={i+1: scores[i] for i in range(n_submissions)})
    model.Capacity = pyo.Param(model.Workshops, initialize=capacities)
    model.MinSubmissions = pyo.Param(model.Workshops, initialize=min_submissions)
    
    # 5. VARIABLES
    model.x = pyo.Var(model.Workshops, model.Submissions, within=pyo.Binary)
    
    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.Scores[j] * model.x[i, j] for i in model.Workshops for j in model.Submissions)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.maximize)
    
    # 7. CONSTRAINTS
    
    # Workshop Capacity Constraints
    def capacity_rule(model, i):
        return sum(model.x[i, j] for j in model.Submissions) <= model.Capacity[i]
    model.capacity_constraint = pyo.Constraint(model.Workshops, rule=capacity_rule)
    
    # Minimum Submissions Constraints
    def min_submissions_rule(model, i):
        return sum(model.x[i, j] for j in model.Submissions) >= model.MinSubmissions[i]
    model.min_submissions_constraint = pyo.Constraint(model.Workshops, rule=min_submissions_rule)
    
    # 8. SOLVING WITH GUROBI
    solver = SolverFactory('gurobi')
    
    # Optional: Set solver options
    solver.options['TimeLimit'] = 300  # 5 minutes
    solver.options['MIPGap'] = 0.01    # 1% gap
    
    # Solve the model
    results = solver.solve(model, tee=True)
    
    # 9. RESULT PROCESSING
    # Check solver status
    if results.solver.termination_condition == pyo.TerminationCondition.optimal:
        print("Optimal solution found!")
        print(f"Optimal value: {pyo.value(model.objective)}")
        
        # Extract variable values
        print("\nAccepted Submissions:")
        for i in model.Workshops:
            for j in model.Submissions:
                if pyo.value(model.x[i, j]) > 0.5:  # Binary variable, check if it's 1
                    print(f"Workshop {i}, Submission {j} accepted")
        
    elif results.solver.termination_condition == pyo.TerminationCondition.infeasible:
        print("Problem is infeasible")
    elif results.solver.termination_condition == pyo.TerminationCondition.unbounded:
        print("Problem is unbounded")
    else:
        print(f"Solver terminated with condition: {results.solver.termination_condition}")
    
    return model

# Execute the optimization
conference_optimization()
```

### Execution Results
**Status**: INFEASIBLE
**Error**: Problem is infeasible
**Analysis**: Pyomo's infeasibility result is consistent with the other solvers, reinforcing the likelihood of an issue with the constraints or data inputs.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | INFEASIBLE | N/A | 0.21s | N/A | 4 |
| Docplex | INFEASIBLE | N/A | 1.23s | N/A | 4 |
| Pyomo | INFEASIBLE | N/A | 1.43s | N/A | 4 |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex, pyomo
**Solver Retry Summary**: gurobipy: 4 attempts, docplex: 4 attempts, pyomo: 4 attempts

### Final Recommendation
**Confidence Level**: LOW
**Preferred Solver(s)**: multiple
**Reasoning**: All solvers consistently reported infeasibility, suggesting the issue lies with the problem setup rather than the solvers themselves.

### Business Interpretation
**Overall Strategy**: The current problem formulation cannot be solved as it stands due to infeasibility, indicating that the constraints or data inputs need to be revisited.
**Objective Value Meaning**: The objective value cannot be interpreted as the problem is infeasible. The goal was to maximize the total quality score of accepted submissions.
**Resource Allocation Summary**: No resources can be allocated as the problem constraints are not satisfiable with the current data.
**Implementation Recommendations**: Review and verify the input data for capacity and minimum submissions. Check for logical consistency in constraints and ensure that they do not conflict.