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Integer Linear Programming (ILP) is a specialized branch of mathematical optimization that deals with problems in which some or all decision variables are required to take integer values. Unlike linear programming, where variables are allowed to take fractional or continuous values, ILP imposes integer constraints on the variables, which makes the problem significantly more complex to solve. This restriction arises because in many real-world problems, it is not feasible to implement fractional solutions. For example, when assigning workers to shifts, scheduling aircraft, or deciding on the number of machines to be installed, the decision variables can only take whole number values. Thus, integer linear programming provides a framework for solving problems that demand discrete decision-making. Formally, an integer linear programming problem can be described as follows: maximize or minimize a linear objective function subject to a set of linear constraints, with the additional restriction that some or all variables must take integer values. When all variables are restricted to be integers, the problem is known as a pure integer programming problem. If only some of the variables are restricted to be integers and others can remain continuous, the problem is called a mixed integer linear programming problem. An even stricter version, where variables can only take binary values (0 or 1), is known as a binary integer programming problem. These types are commonly used in optimization scenarios such as capital budgeting, network design, facility location, and production planning. The solution techniques for integer _____ __________ _______ ____ ____ __________ __________.
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