# Improving efficiency by mathematical optimization

14.8.2017/Text: Jan Kronqvist

Our society is currently facing a variety of threats and challenges due to global warming and the depletion of our fossil fuel reserves. Many industries in our country are also facing challenges due to increased production costs and global competition. To reduce the environmental impact and simultaneously strengthen our industries’ competitiveness, it is of utter importance that all steps in the production are done as efficiently as possible. To achieve a maximal efficiency in production requires both optimal equipment for all tasks and an optimal utilization of all equipment. Mathematical optimization is a versatile tool that can play a crucial role in improving the efficiency in different industries. It might sound trivial, but significant reductions of harmful emissions can often be achieved by simply doing everything as efficient as possible.

What is mathematical optimization?

Optimization can simply be described as the search for the best possible solution to a given problem. Mathematical optimization uses a mathematical description of the problem, which enables the use of efficient techniques for analyzing and searching the best possible solution the problem.

There is a wide variety of applications, and mathematical optimization can for example be used for obtaining the optimal production schedule that minimizes waste and maximizes utilization of the equipment. Mathematical optimization has also been used to determine the optimal radiation treatment for cancer.

Why is it not more commonly used?

Because, unfortunately it is often very difficult to solve real world optimization problems. Optimization problems usually involve many decisions, and for every yes/no discussion the number of possible solutions doubles. This might not sound too bad, however, a problem containing 260 such decisions actually have more possible solutions than the number of atoms in our universe. Therefore, it is not possible to simply try out all possible solutions; instead we need a more sophisticated way to obtain the optimal solution.

My research

At the moment, there are efficient techniques for solving some specific type of optimization problems. For some type of problems it is even possible to obtain the optimal solution even if the problem contains more than a million decisions. However, many real-world optimization problems are still extremely difficult to solve. For such problems, it might be difficult to find the optimal solution even if the problem only contains a few decisions.

Most techniques that can successfully deal with such difficult optimization problems uses the same basic idea; instead of solving the difficult problem we solve an approximation of the problem. The approximated problem should be such that it has the same optimal solution as the difficult problem, which is the problem we are interested in. Furthermore, the approximated problem should also preferably be easy to solve. How to construct such approximations is still an open research question, which is also the focus of my PhD thesis.

At Carnegie Mellon University, there has been an ongoing research in this field since the early 90’s and many import results has been obtained. Staying at Carnegie Mellon as a visiting researcher, was an important experience for me. During the visit, I was able to get a deeper understanding of some of the most important mathematical theories in optimization. I also learned about several practical optimization applications in different industries. Being at Carnegie Mellon, was greatly beneficial for my research and I could collaborate with several other researchers in the field. Currently I am writing an article based on some of the results I obtained during my research visit, and some results will be presented at conference European Symposium on Computer-Aided Process Engineering.

To summarize, I learned a lot during my stay and got important results for my research. Furthermore, I got to know several researchers for future collaborations. The stay gave me new insights in my field of research and it was also a life experience that I will remember with joy.

Jan Kronqvist is a PhD student in process systems engineering at Åbo Akademi University. His research is focused on developing efficient techniques for finding the best possible solution to certain types of optimization problems. During the spring of 2017, he visited Carnegie Mellon University in Pittsburgh USA as a visiting scholar for six months. The research visit was partially funded by a grant from Tekniikan Edistämissäätiö given in 2016.

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