One of my colleagues was on a master’s men’s swim team that set a world record in the 400-meter medley. He pointed out how the rules generate the complexities of creating the best team. The team members’ ages must sum to at least 200 years. Assuming each person’s time slows at an accelerating rate, you might think that minimizing the relay team’s time would require having each member be age 50. Yet my colleague is 53, but he is an exceptionally fine swimmer, especially at freestyle, which he swam in the meet. Having him on the team doesn’t slow the total time very much (he’s never the best among masters in any event); but including him allows the team also to include a 47-year-old who has several individual world records.
The programming problem becomes even more complex because each swimmer’s time slows at a different rate in different events. Since teams need time to practice, constructing the optimal team requires taking into account the sum of ages constraint, the current time of each potential team member in each relay leg, and the rate at which times are slowing for each person in each event. (HT: MS)
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“you might think that minimizing the relay team’s time would require having each member be age 50.”
No one might think that. Other than Professor Hamermesh, of course.
This kind of problem is known (in academic IT circles at least) as “NP-Complete”, in that adding one extra swimmer to the talent pool makes the calculation more complex at an exponential rate .
Without going into the gory details of what NP-Complete means, it’s the kind of thing that modern desktop computers can do very quickly, assuming you have someone able to spend a couple of hours coding, rather than solve “by hand” in a spreadsheet.
Let’s make life more complex
- do some swimmers in the target group swim better at certain times of the day (and this is important) relative to each other? If so, you have to optimise based on when the event will be swum.
- do some swimmers perform more consistently than others?
- are there some combinations of swimmers where there is such personal antipathy that one or more might be tempted to perform sub-optimally to deny the other a share in any glory?
- is the nature of swimming at this level such that training together makes a difference? (I suspect less so than for most team sports, but I don’t have any data.) If so, what is the response curve for such training, since you’d then need to optimise for not just the selected people but date of announcement of the team.
I suspect that the hard part is actually gathering the data you need to determine the inputs.
This is a nice example of trying to maximize a certain output under a constraint, just like firms try to maximize their productivity and their profits with a limited amount of resources at hand. In this situation, one tries to minimize the time required to complete the relay race by selecting individuals to perform different parts of the race, while being constrained by the age requirement. With careful analisys one should be able to find the optimal combination, but in larger-scale situations like the operation of a firm there are so many variables that it becomes hard to control. Another example from the sports realm could be managing a football team, which involves finding the ideal position for each player as well as teams of players that work well with each other.
Now do it again with the C.A.F.E. standards.
Simply get a 150 year old Turtle named Crush on the roster.
Or, you could just stage a set of qualifying races….
An objective function and a constraint are not sufficient conditions for a problem to be economics.
Just in case you were trying to take over operations research, or some other related field.
You don’t at any point mention if this team has a maximum or required number of members — if not, the problem is trivial.