To solve a problem using the **Branch and Bound** method, we typically follow these steps:

### Problem Statement:
Let's say we have a general integer programming problem that we want to solve using the Branch and Bound method:

**Objective:**  
\[ \text{Maximize or Minimize } z = c_1x_1 + c_2x_2 + \dots + c_nx_n \]

**Subject to:**  
\[ a_{11}x_1 + a_{12}x_2 + \dots + a_{1n}x_n \leq b_1 \]  
\[ a_{21}x_1 + a_{22}x_2 + \dots + a_{2n}x_n \leq b_2 \]  
\[ \vdots \]  
\[ a_{m1}x_1 + a_{m2}x_2 + \dots + a_{mn}x_n \leq b_m \]  
\[ x_1, x_2, \dots, x_n \geq 0 \text{ and integer} \]

### Steps in Branch and Bound Method:

1. **Relaxation:**
   Solve the linear programming relaxation of the problem by ignoring the integer constraints (i.e., allow \( x_i \) to take any non-negative real value). This gives an upper bound for a maximization problem or a lower bound for a minimization problem.

2. **Branching:**
   Identify the first variable that has a non-integer value in the relaxed solution. Create two new subproblems (branches) by adding constraints that force this variable to take integer values:
   - In one branch, add a constraint \( x_k \leq \text{floor}(x_k) \).
   - In the other branch, add a constraint \( x_k \geq \text{ceil}(x_k) \).

3. **Bounding:**
   For each branch, solve the linear programming relaxation. If the solution is worse than the best known feasible integer solution, prune that branch. Otherwise, continue branching on that subproblem.

4. **Pruning:**
   If a subproblem yields an integer solution, compare it with the best known integer solution:
   - If it improves upon the best known solution, update the best solution.
   - If the solution does not improve or the relaxation's solution is worse, prune that branch.

5. **Termination:**
   The algorithm terminates when all branches have been pruned. The best known integer solution is the optimal solution to the original problem.

### Example:
Suppose we have a simple integer programming problem:

**Objective:**  
\[ \text{Maximize } z = 3x_1 + 2x_2 \]

**Subject to:**  
\[ x_1 + x_2 \leq 4 \]  
\[ 2x_1 + x_2 \leq 5 \]  
\[ x_1, x_2 \geq 0 \text{ and integer} \]

### Step-by-Step Solution:
1. **Relaxation:** Solve the relaxed problem (ignoring the integer constraint):

   \[ z = 3x_1 + 2x_2 \]

   Subject to:

   \[ x_1 + x_2 \leq 4 \]
   \[ 2x_1 + x_2 \leq 5 \]

   \(\Rightarrow x_1 = 1.5, x_2 = 2.5\)  
   \( z

Explore how to solve a problem using the Branch and Bound method in integer programming. This detailed guide covers steps including relaxation, branching, bounding, pruning, and termination. Suitable for advanced college students and professionals.

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