Knapsack Problems

At its core, a knapsack problem is a mathematical puzzle. It involves finding the most valuable combination of items to fit into a constrained container, often referred to as a knapsack.

The challenge lies in selecting items with the highest total value while ensuring that their combined size or weight does not exceed the knapsack’s capacity.

Although the concept sounds simplistic, knapsack problems have real-world applications that can make a big impact. For example:

1. Resource Allocation
   Imagine you’re an event planner organizing a conference with limited resources. You have a fixed budget and need to select the most impactful speakers, workshops, and activities within that constraint. The knapsack problem can help you determine the optimal combination that maximizes the value and impact of the event.
2. Inventory Management
   For businesses, efficient inventory management is crucial to minimize costs and maximize profits. When faced with limited warehouse space, the knapsack problem aids in choosing the inventory items that generate the highest revenue while fitting within the available storage capacity.
3. Travel Packing
   Packing for a vacation often involves making choices based on limited luggage space. Although the most basic knapsack problem only focuses on knapsack capacity and the capacity consumed by the individual items, you could take it to the next level—identifying the best items to carry, so you can optimize space while meeting your specific needs for the trip.

By applying knapsack problem solutions, we unlock several advantages in various domains:

1. Resource Optimization
   Knapsack problems, like many optimization problems, help you allocate your limited resources in the most efficient way possible. As a result, your business can minimize costs, maximize returns, and make informed decisions—especially when resources are scarce.
2. Time and Cost Savings
   By identifying the most valuable items to include in a limited space or considering the most impactful elements for an event, knapsack problem solutions save time and money. These solutions streamline decision-making processes and prevent wasted resources.
3. Improved Decision-Making
   Knapsack problems help you evaluate your choices in a systematic, data-driven way. They provide a framework for analyzing trade-offs, considering multiple factors, and selecting the best combination of items for a given objective.

The Defence Science and Technology Laboratory (Dstl), an executive agency of the United Kingdom’s Ministry of Defence, was facing a challenge. In the middle of complex operating environments, packers need to put materials and equipment into the right transport vehicles to ensure that the right items go to the right place at the right time—in the most efficient way possible.

Using a Gurobi-powered solution, created by Gurobi partner Decision Lab, Dstl was able to increase packing efficiency by 20% and provide logisticians with a solution in tens of seconds. The solution guides packers through the process in real-time, taking out the guesswork.

Read the Dstl case study to learn more about this real-world knapsack problem.