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Here we present a few examples of how real benefits are achieved by rigorously applying mathematical programming and optimization on real-world problems.
Optimal Schedules
Optimization is a powerful tool for determining workable schedules that are in accordance with certain requirements. Examples include
- Put together a lesson plan taking into account teacher availability and teacher preferences such as morning/afternoon sessions, preferred classes, or days without lessons. At the same time restricted lab availability is taken into account, gaps in teachers' schedules are minimized, and all required subjects are taught for each class.
- Create a seminar schedule for multiple locations, topics, and audiences. The requirements to be met are instructor availability in terms of skills/topic coverage, instructor willingness to travel certain distances and to give a certain number of courses, meet customer requirements (offer specific courses within certain time periods in certain regions), and to minimize travel cost.
- In a specific implementation we managed to maximize the number of offered training courses for high school teachers across three U.S. states while observing constraints such as travel restrictions, trainer skills and availability, and course demand and frequency. The computed plans do not conflict with business and trainer constraints while the effort to come up with a workable plan was reduced from three days to an hour.
The problem was formulated as a mixed-integer linear program (MILP) and included approx. 75 locations, 100 instructors, 25 time slots, and 20 different courses. Depending on the specific constraints, the resulting schedule contains about 200-250 scheduled courses during the planning period of one quarter.
- In a specific implementation we managed to maximize the number of offered training courses for high school teachers across three U.S. states while observing constraints such as travel restrictions, trainer skills and availability, and course demand and frequency. The computed plans do not conflict with business and trainer constraints while the effort to come up with a workable plan was reduced from three days to an hour.
- Optimize utilization of beds in a hospital depending on total number of beds, allocations to certain departments, expected duration of the individual stays in the hospital, required safety quantities, etc. The benefits of optimization include maximized utilization of available beds, minimized waiting time, minimized bottlenecks, and increased visibility into the hospital supply chain.
Other Applications
Apart from optimizing schedules there is an endless list of other applications where using applied mathematics and in particular optimization, greatly improves profitability. Beyond more common areas like route planning, location determination in a logistics network, and commission structures in a multi-level marketing environment, here are examples of successful projects we have done:
- Medical diagnosis: we successfully contributed to a lung cancer detection method based on breath tests. Advanced applied mathematics involving genetic algorithms help in the involved weighted digital analysis of measurement data.
- Web-based optimal chemical formulation and blending: Upon entering a set of desired chemical and physical properties of the end product on a web page the application computes the optimal chemical formulation in terms of meeting the specifications while minimizing cost and returns the result in the user's web browser. The benefits for sales and marketing include more accurate quotes within significantly shorter response times.
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