In sectors like production, health services, and logistics, decision-makers often face scenarios where limited time, capacity, or resources prevent the execution of all tasks. Precisely defining what to do and in what order is crucial for positive results. Therefore, having tools that support these critical decisions is paramount.
In education specifically, this challenge was highlighted during the pandemic. When schools and universities had to rapidly reorganize teaching plans and prioritize content, they struggled significantly. This revealed a crucial problem: a lack of clear tools or defined frameworks for making vital decisions about what content to teach and what to omit.
At Usach, Dr. Óscar C. Vásquez (Industrial Engineering) is leading a Fondecyt Regular project that offers a solution using mathematics and engineering. His work extends the classic knapsack problem into the Knapsack Problem with Scheduled Items (KPsi). This optimization model selects items of varying value and weight that fit into a limited-capacity knapsack in a specific sequence to maximize the overall benefit.
Dr. Vásquez explains: "As a teacher in a crisis, you're often forced to decide what to teach and when with limited time and resources. It's like shopping: you choose what to buy, but you also have to arrange the products in your bag. You can't put eggs at the bottom and heavy items on top, or they'll break. It matters what you choose and in what sequence. That's the problem with the knapsack with sequenced items."
The project's goal is to translate this everyday dilemma into mathematical language. Based on this logic, the research formulates optimization models that incorporate both the selection and sequence of elements in a decision. It then applies various algorithms to solve the problem efficiently. Essentially, the aim is to create a system that can explore different organizational possibilities, like puzzle pieces, to determine which ones should be chosen and in what order to maximize efficiency.
Viable Solutions in a Reasonable Time
In education, this problem translates into planning what content to teach and when within a subject or curriculum, recognizing that learning impact often hinges on the presented sequence. Therefore, the solution must not only involve selecting the right elements and sequence, but also finding viable results quickly, especially when time and resources are scarce.
Dr. Vásquez states, "This presents a new challenge: it's not just about selecting and sequencing actions, but about finding a solution that's truly applicable in practice with the available resources." He adds that this practicality is the core motivation, stemming from real-world fields like education or health, where "we not only have to choose what to do, but also how to sequence those decisions so that they really work."
Faced with this challenge, the research incorporates resolution methods that allow uncertainty to be explored without evaluating every possibility, thus offering good-quality answers in less time. These methods, instead of pursuing the absolute optimal solution, apply practical rules to quickly find sufficiently good and efficient options. This approach is crucial when time or resources are severely limited.
Dr. Vásquez offers a healthcare example: "In a hospital, it's one thing to select and sequence patients for treatment, but another entirely to find a solution that integrates this with the hospital's actual resources." He clarifies the core challenge: "It lies in using technology to make these decisions truly useful for people. This means formalizing a problem mathematically, capturing the practical rules people use, and then drawing conclusions that genuinely benefit society."
The four-year project is structured into distinct stages, starting with problem characterization and progressing through the implementation of resolution methods and result validation. Crucially, it includes the training of undergraduate and graduate students and collaboration with national and international researchers. This approach aims to establish new research lines applied primarily to education, with potential expansion into sectors like health.
Dr. Óscar C. Vásquez concludes: "Our goal isn't to replace decision-makers, but to support them. The mathematical models and resolution methods we develop act as a tool that complements their experience, helping them make better decisions in complex scenarios, especially in a sector as noble as education." He adds, "With this, we are not only making a direct contribution to society, we are training good professionals with an ethical and sustainable approach, who will help promote the country's development."