In Integer Programming, the decision variables are restricted to integer values. Therefore, a valid formulation includes the constraint x, y ∈ Z, which ensures that the variables take on integer values.

Linear programming relaxation involves solving a relaxed version of the Integer Programming problem, where the integer constraints are temporarily removed. The optimal solution to the relaxed problem provides a lower bound on the optimal objective value of the original Integer Programming problem.

Branching in branch-and-bound involves dividing the feasible region of the Integer Programming problem into smaller subregions. This is done by adding additional constraints that restrict the values of the decision variables, effectively creating new subproblems.

Cutting planes are additional constraints that are added to the linear programming relaxation of an Integer Programming problem. These constraints are designed to eliminate infeasible solutions and tighten the relaxation, resulting in a stronger lower bound on the optimal objective value.

The knapsack problem is a classic Integer Programming problem that involves selecting items from a set of items with different weights and values, subject to a weight constraint. Dynamic programming is a powerful technique for solving the knapsack problem and other similar Integer Programming problems.

To solve this problem, you can use a branch-and-bound algorithm or a dynamic programming approach. The optimal solution is x = 3, y = 2, which gives an objective value of z = 12.

A feasible solution to an Integer Programming problem is a set of values for the decision variables that satisfies all constraints of the problem, including the integer constraints. Finding a feasible solution is an important step in Integer Programming algorithms, as it provides an upper bound on the optimal objective value.

A mixed-integer programming problem is a type of Integer Programming problem where some of the decision variables are restricted to integer values while others are allowed to take on continuous values. The valid formulation for a mixed-integer programming problem includes constraints that specify which variables are integers and which are continuous.

A branch-and-cut algorithm is a type of branch-and-bound algorithm that incorporates cutting planes to strengthen the linear programming relaxation of an Integer Programming problem. By adding cutting planes, the algorithm tightens the relaxation and improves the lower bound on the optimal objective value.

To solve this problem, you can use a branch-and-bound algorithm or a dynamic programming approach. The optimal solution is x = 2, y = 3, which gives an objective value of z = 12.

A binary integer programming problem is a type of Integer Programming problem where the decision variables are restricted to binary values (0 or 1). The valid formulation for a binary integer programming problem includes constraints that specify that the variables can only take on binary values.

Lagrangian relaxation is a technique used in Integer Programming to decompose the problem into smaller subproblems. This is done by introducing a Lagrangian function that relaxes the integer constraints and allows the decision variables to take on continuous values. The subproblems are then solved independently, and the solutions are combined to obtain a solution to the original Integer Programming problem.

To solve this problem, you can use a branch-and-bound algorithm or a dynamic programming approach. The optimal solution is x = 2, y = 3, which gives an objective value of z = 12.

A set partitioning problem is a type of Integer Programming problem where the objective is to partition a set of elements into disjoint subsets, subject to certain constraints. The valid formulation for a set partitioning problem includes a constraint that ensures that each element is assigned to exactly one subset.