# Can't find the right value function

This is a problem I cannot solve because I don't see why the athlete doesn't choose the time allocation all at once...

Problem:

You are a professional athlete preparing a triathlon. For each sport, your score is: $$S = 500\sqrt{h/H}$$, where $$h$$ is the number of hours of training and $$H=(100, 75, 50)$$ the number of hours required to have a perfect score of 500. You will train at least 25 hours for each sport, but have a total amount of 150 hours of training to spend. You decide it is more efficient to spend a whole day on each sport at a time (15 hours). Therefore, $$h$$ is a multiple of 15 for each sport. Solve the problem as a dynamic program. Determine the state variable (with its transition law) and decision epoch, the reward function, write out the optimization problem and solve it for the right time investment strategy.

Where I am:

I think the state variable is $$h_t$$ the vector of hours already spent on the 3 sports. I am not sure how to write down the transition law though... or the optimization problem.

Many thanks for any help!

• I don't see any dynamics in the problem. You may want to ask your teacher for a clarification. Mar 31, 2021 at 16:10
• Thanks for your reply! He will introduce dynamics at a later stage but still wants us to solve it as a dynamic model with 6 periods... There may be no point here but I'm nonetheless struggling with defining the model, in particular the choice variable $h_{t+1} = h_t + ???$ Mar 31, 2021 at 17:03