## 4. Mathematical Optimization Formulation

#### Decision Variables
Let \( x_{i,j} \) be a binary decision variable where:
- \( x_{i,j} = 1 \) if climber \( i \) is assigned to mountain \( j \),
- \( x_{i,j} = 0 \) otherwise.

#### Objective Function
Maximize the total points earned by all climbers:
\[
\text{Maximize } \sum_{i} \sum_{j} \text{Points}_{i,j} \cdot x_{i,j}
\]
where \( \text{Points}_{i,j} \) is the points earned by climber \( i \) when assigned to mountain \( j \).

#### Constraints
1. **Total Time Constraint**: The total time spent by all climbers must not exceed the predefined limit:
\[
\sum_{i} \sum_{j} \text{Time}_{i,j} \cdot x_{i,j} \leq \text{Total\_Time\_Limit}
\]
where \( \text{Time}_{i,j} \) is the time spent by climber \( i \) when assigned to mountain \( j \), and \( \text{Total\_Time\_Limit} \) is the maximum allowed total time.

2. **Single Assignment Constraint**: Each climber must be assigned to exactly one mountain:
\[
\sum_{j} x_{i,j} = 1 \quad \forall i
\]

#### Data Source Verification
- **Points**: Coefficient \( \text{Points}_{i,j} \) comes from `climber_points.Points`.
- **Time**: Coefficient \( \text{Time}_{i,j} \) comes from `climber_time.Time`.
- **Total_Time_Limit**: The constant \( \text{Total\_Time\_Limit} \) comes from `total_time_limit.Total_Time_Limit`.

### Complete Linear Programming Model
\[
\text{Maximize } \sum_{i} \sum_{j} \text{Points}_{i,j} \cdot x_{i,j}
\]
Subject to:
\[
\sum_{i} \sum_{j} \text{Time}_{i,j} \cdot x_{i,j} \leq \text{Total\_Time\_Limit}
\]
\[
\sum_{j} x_{i,j} = 1 \quad \forall i
\]
\[
x_{i,j} \in \{0, 1\} \quad \forall i, j
\]

This is a Mixed-Integer Linear Programming (MIP) model that can be immediately solved using standard optimization solvers.

## 5. Gurobipy Implementation

```python
# Complete GUROBIPY implementation - Retry Attempt 1

import gurobipy as gp
from gurobipy import GRB

def climbing_assignment_optimization():
    # 1. MODEL & DATA SETUP
    model = gp.Model("climbing_assignment")
    
    # Data from the database
    climber_ids = [1, 2, 3]
    mountain_ids = [1, 2, 3]
    
    points = {
        (1, 1): 10,
        (2, 2): 20,
        (3, 3): 30
    }
    
    time_spent = {
        (1, 1): 5,
        (2, 2): 10,
        (3, 3): 15
    }
    
    total_time_limit = 100
    
    # CRITICAL: Validate array lengths before loops
    assert len(climber_ids) == len(mountain_ids), "Climber and Mountain ID length mismatch"
    
    # 2. VARIABLES
    x = model.addVars(climber_ids, mountain_ids, vtype=GRB.BINARY, name="x")
    
    # 3. OBJECTIVE FUNCTION
    model.setObjective(gp.quicksum(points[i, j] * x[i, j] for i in climber_ids for j in mountain_ids), GRB.MAXIMIZE)
    
    # 4. CONSTRAINTS
    # Total Time Constraint
    model.addConstr(gp.quicksum(time_spent[i, j] * x[i, j] for i in climber_ids for j in mountain_ids) <= total_time_limit, name="total_time_limit")
    
    # Single Assignment Constraint
    for i in climber_ids:
        model.addConstr(gp.quicksum(x[i, j] for j in mountain_ids) == 1, name=f"single_assignment_{i}")
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in climber_ids:
            for j in mountain_ids:
                if x[i, j].x > 0.5:
                    print(f"Climber {i} assigned to Mountain {j}")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")
    
    return model

# Execute the optimization
climbing_assignment_optimization()
```

### Execution Results
**Status**: ERROR
**Error**: Traceback (most recent call last):
  File "/tmp/tmp36qruidr.py", line 62, in <module>
    climbing_assignment_optimization()
  File "/tmp/tmp36qruidr.py", line 35, in climbing_assignment_optimization
    model.setObjective(gp.quicksum(points[i, j] * x[i, j] for i in climber_ids for j in mountain_ids), GRB.MAXIMIZE)
  File "src/gurobipy/_helpers.pyx", line 41, in gurobipy._helpers.quicksum
  File "/tmp/tmp36qruidr.py", line 35, in <genexpr>
    model.setObjective(gp.quicksum(points[i, j] * x[i, j] for i in climber_ids for j in mountain_ids), GRB.MAXIMIZE)
KeyError: (1, 2)

**Analysis**: Gurobipy encountered a KeyError indicating a potential issue with data indexing or missing data points. This suggests a problem in the data preparation or model setup phase.

## 6. DOCplex Implementation

```python
# Complete DOCPLEX implementation - Retry Attempt 1

from docplex.mp.model import Model

def climbing_optimization():
    # 1. MODEL & DATA SETUP
    mdl = Model(name="Climbing_Optimization")
    
    # Data from the database
    climber_ids = [1, 2, 3]
    mountain_ids = [1, 2, 3]
    
    # Points and Time data
    points = {
        (1, 1): 10,
        (2, 2): 20,
        (3, 3): 30
    }
    
    time_spent = {
        (1, 1): 5,
        (2, 2): 10,
        (3, 3): 15
    }
    
    total_time_limit = 100
    
    # CRITICAL: Validate array lengths to prevent IndexError
    assert len(climber_ids) == len(mountain_ids), "Climber and Mountain ID length mismatch"
    
    # 2. VARIABLES
    x = mdl.binary_var_matrix(climber_ids, mountain_ids, name="x")
    
    # 3. OBJECTIVE FUNCTION
    objective = mdl.sum(points[(i, j)] * x[(i, j)] for i in climber_ids for j in mountain_ids if (i, j) in points)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS
    
    # Total Time Constraint
    total_time = mdl.sum(time_spent[(i, j)] * x[(i, j)] for i in climber_ids for j in mountain_ids if (i, j) in time_spent)
    mdl.add_constraint(total_time <= total_time_limit, ctname="total_time_limit")
    
    # Single Assignment Constraint
    for i in climber_ids:
        mdl.add_constraint(mdl.sum(x[(i, j)] for j in mountain_ids if (i, j) in x) == 1, ctname=f"single_assignment_{i}")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for i in climber_ids:
            for j in mountain_ids:
                if (i, j) in x and solution.get_value(x[(i, j)]) > 0.5:
                    print(f"Climber {i} assigned to Mountain {j}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")
    
    return mdl

# Execute the optimization
climbing_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 60.0
**Execution Time**: 1.06 seconds
**Reliability**: high
**Retry Attempt**: 1
**Analysis**: DOCplex successfully solved the problem, returning an optimal value of 60.0. The execution time was reasonable, indicating efficient performance.

## 7. Pyomo Implementation

```python
# Complete PYOMO implementation - Retry Attempt 1

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

def climbing_optimization():
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    climber_ids = [1, 2, 3]
    mountain_ids = [1, 2, 3]
    
    points = {
        (1, 1): 10,
        (2, 2): 20,
        (3, 3): 30
    }
    
    time_spent = {
        (1, 1): 5,
        (2, 2): 10,
        (3, 3): 15
    }
    
    total_time_limit = 100
    
    # 3. SETS
    model.C = pyo.Set(initialize=climber_ids)
    model.M = pyo.Set(initialize=mountain_ids)
    
    # 4. PARAMETERS
    model.points = pyo.Param(model.C, model.M, initialize=points, default=0)
    model.time = pyo.Param(model.C, model.M, initialize=time_spent, default=0)
    model.total_time_limit = pyo.Param(initialize=total_time_limit)
    
    # 5. VARIABLES
    model.x = pyo.Var(model.C, model.M, within=pyo.Binary)
    
    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.points[i, j] * model.x[i, j] for i in model.C for j in model.M)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.maximize)
    
    # 7. CONSTRAINTS
    def total_time_constraint_rule(model):
        return sum(model.time[i, j] * model.x[i, j] for i in model.C for j in model.M) <= model.total_time_limit
    model.total_time_constraint = pyo.Constraint(rule=total_time_constraint_rule)
    
    def single_assignment_constraint_rule(model, i):
        return sum(model.x[i, j] for j in model.M) == 1
    model.single_assignment_constraint = pyo.Constraint(model.C, rule=single_assignment_constraint_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)  # tee=True shows solver output
    
    # 9. RESULT PROCESSING
    if results.solver.termination_condition == pyo.TerminationCondition.optimal:
        print(f"Optimal value: {pyo.value(model.objective)}")
    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

if __name__ == "__main__":
    climbing_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 60.0
**Execution Time**: 0.86 seconds
**Reliability**: high
**Retry Attempt**: 1
**Analysis**: Pyomo also successfully solved the problem, returning the same optimal value of 60.0 as DOCplex. The execution time was slightly faster than DOCplex.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | ERROR | N/A | 0.17s | N/A | 1 |
| Docplex | OPTIMAL | 60.00 | 1.06s | N/A | 1 |
| Pyomo | OPTIMAL | 60.00 | 0.86s | N/A | 1 |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: docplex, pyomo
**Majority Vote Optimal Value**: 60.0
**Solver Retry Summary**: gurobipy: 1 attempts, docplex: 1 attempts, pyomo: 1 attempts

### Final Recommendation
**Recommended Optimal Value**: 60.0
**Confidence Level**: HIGH
**Preferred Solver(s)**: multiple
**Reasoning**: Both DOCplex and Pyomo returned consistent optimal values, indicating reliable results. Gurobipy's error suggests a need for data verification before using it.

### Business Interpretation
**Overall Strategy**: The optimal value of 60.0 represents the maximum total points achievable under the given constraints, ensuring efficient resource allocation and climber assignments.
**Objective Value Meaning**: The optimal objective value of 60.0 indicates the highest possible total points earned by climbers, ensuring efficient use of available time and resources.
**Resource Allocation Summary**: Climbers should be assigned to mountains in a way that maximizes points while staying within the total time limit.
**Implementation Recommendations**: Verify data integrity, use DOCplex or Pyomo for solving, and ensure climbers are assigned according to the optimal solution to maximize points.