Understanding the Bounded Knapsack problem

In many real-world applications, a software must make decisions under limited resources while also dealing with the limited availability of each option. In such situations, an item may not be restricted to a single choice but may also not be available an unlimited number of times. Instead, each item type may be selected multiple times up to a specified limit. Just ike the regular 0/1 knapsack problem, each option provides some benefit, but selecting it consumes part of the available budget and may prevent other options from being chosen. The goal, therefore, is to determine which combination of options produces the greatest overall value without exceeding the given constraint.

A fundamental problem in this space is the bounded knapsack problem, which extends the classic knapsack framework to scenarios where items come in limited quantities.