Dr. Konur has recently started working out and he is working out with a trainer in the Centre (a gym in Rolla, MO). The trainer, Matt the Mean, always come up with a work-out plan for the training sessions and he enjoys making Dr. Konur suffer, moreover, he laughs right at Dr. Konur's face while doing that. For this last session, he is considering activities from different categories. Each activity under each category has a different calorie burning rate for one minute spent doing that activity. Specifically, there are three categories of activities: UPs, Cardio, and Pain-givers. Each category has three activities. The data for each activity under different categories are summarized in the table below.
Dr. Konur and Matt the Mean has a work-out session of 60 plus or minus 5 minutes, that is, the work-out sessions should be between 55 and 65 minutes. Matt the Mean wants to tire Dr. Konur as much as he can during a session, so he wants to prepare a work-out plan to maximize the calories burnt. In preparing a work-out plan, Matt the Mean should determine which categories to include in the plan, and how much time to allocate to each activity under the included categories. Each category has a 3-minute get-ready time, where Matt the Mean, shows how the activities will be executed and Dr. Konur psychologically prepares himself for the upcoming pain. If get-ready-time is not done, activities under that category cannot be done. Matt the Mean has some restrictions above the work-out plan:
- He cannot allocate more than 40 minutes under the same category. That is, the sum of the times allocated to the activities under the same category plus the get-ready time for that catefory cannot be more than 40 minutes.
- He cannot allocate more than 15 minutes to any activity under any category.
- If a category is included in the plan, he should allocate at least 20 minutes, in total, to the activities under that category.
- He has to include at least two categories in the plan. Mathematically formulate Matt the Mean's work-out plan preparation problem as a mixed-integer-linear programming model. To do so, define your decision variables, objective and express your objective function and constraints in terms of your decision variables, and combine everything to get the final formulation.