Question 2: A) Explain integrality constraints in IP. OR B) How do mixed integer problems differ from pure integer problems?
Answer (1)
In mathematical programming, Integer Programming (IP) is a type of optimization problem where some or all of the decision variables are required to take on integer values. Here’s a detailed explanation of integrality constraints and how mixed integer problems differ from pure integer problems:
A) Explaining Integrality Constraints in IP:
Integer programming involves constraints that require some variables to have integer values. These are called integrality constraints. The presence of these constraints makes the problem more complex and computationally difficult to solve as compared to linear programming.
To illustrate, if you're optimizing a production schedule, you might have a constraint stating the number of products produced must be a whole number since you can't produce a fraction of an item.
The formal expression, in a simple integer programming problem, might look like this:
Maximize z = c 1 x 1 + c 2 x 2 + … + c n x n
Subject to:
a 11 x 1 + a 12 x 2 + … + a 1 n x n ≤ b 1
a 21 x 1 + a 22 x 2 + … + a 2 n x n ≤ b 2
x i ∈ Z
Where:
z is the objective function.
x i are the decision variables, which are required to be integers.
a ij and b i are coefficients.
B) How Mixed Integer Problems Differ from Pure Integer Problems:
Pure Integer Problems:
In a pure integer programming problem, all decision variables are required to take integer values. This constraint can often make the problem much harder to solve since the solution space is only a subset of the continuous space.
Mixed Integer Problems (MIP):
In a mixed integer programming problem, only some of the decision variables are restricted to integer values, while others can take on non-integer values. This flexibility can make MIPs easier to solve than pure integer problems because the continuous variables allow for linear relaxation techniques to be applied, helping to find solutions more efficiently.
In summary, integrality constraints are central to IP, as they force decision variables to take on integer values. Mixed integer programming offers a middle ground where only some variables are integers, making it a versatile tool that balances complexity and solvability.
