The Branch and Bound (B&B) method is a widely-used approach to solve Integer Linear Programming (ILP) problems, where some or all of the variables are required to be integers. It systematically explores possible solutions, dividing the problem into smaller subproblems (branching) and eliminating portions of the solution space that cannot contain the optimal solution (bounding). Let me explain the method in detail:

Steps in the Branch and Bound Method

1. Initial Linear Programming Relaxation (LP Relaxation):

   • Start by solving the relaxed version of the problem, where integer constraints are ignored. This provides an optimal solution for the relaxed problem.
   • If the solution is integer, it's feasible for the ILP, and the process ends.

2. Branching:

   • If the solution from the relaxed problem is not integer, branching is performed:
     • Divide the feasible region into smaller regions by introducing additional constraints. For example:
       • If $x_1 = 1.5$ (non-integer), create two subproblems:
         1. $x_1 \leq 1$
         2. $x_1 \geq 2$
   • This creates a tree structure, where each node represents a subproblem.

3. Bounding:

   • For each subproblem (node), solve the relaxed problem again.
   • If the optimal value of a subproblem is worse than the best integer solution found so far, discard (fathom) that branch (it cannot improve the solution).

4. Fathoming:

   • A subproblem is considered fathomed (or solved) if:
     • The relaxed solution is integer and better than the current best solution.
     • The subproblem is infeasible (no solution exists).
     • The objective value of the relaxed solution is worse than the current best integer solution.

5. Optimality:

   • The process continues until all branches are either fathomed or explored.
   • The best integer solution found during the process is declared the optimal solution.

Graphical Illustration in the Question

From the uploaded diagrams:

• The original relaxed problem (LP0) has a fractional solution $(x_1 = 2, x_2 = 1.5)$ with $z = 32.5$.
• Branching is performed based on the integer constraints:
  • At LP1, $x_2 \leq 1$: Solution becomes $x_1 = 2, x_2 = 1$, yielding $z = 29$ (an integer solution).
  • At LP2, $x_2 \geq 2$: A fractional solution $x_1 = 1.5, x_2 = 2$ is found with $z = 30.5$, so further branching occurs.
• After exploring all branches, $x^* = (2, 1)$ with $z^* = 29$ is determined to be the best solution.

Key Notes:

• Each subproblem is explored systematically using the tree structure.
• Branching decisions and bounding criteria ensure that only promising solutions are pursued, minimizing computation.

Would you like further explanation of any step or help with a similar example?

Related Questions:

1. What are LP relaxation and how does it help in integer programming?
2. How do we determine which variable to branch on in the B&B method?
3. What happens if there are multiple optimal integer solutions?
4. Can the B&B method handle constraints that are non-linear?
5. How does the B&B method differ from other optimization techniques like dynamic programming?

Tip:

When constructing the tree diagram, ensure clarity by labeling nodes with subproblem constraints and objective values to avoid confusion.