Finding the Optimum Analytically
In an optimization study, you want to find the best way to solve a problem.
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Getting Results with Simulation
The likelihood of failing to find the absolute best course of action in a study or experiment of a skills context is very high, because the subject matter is so complex. In simulation you run trials by varying individual variables while holding the other variables fixed. Since any of the variables could change during an actual performance of the task, you have to run enough trials to catch all the possible combinations. This could take a great deal of time.
Adding to this difficulty is the fact that behavioral environments are nonlinear. They may have many optimal solutions, just as a cluster of hills has many tops. Finding the best of the best can be very difficult, if not impossible, for all practical purposes.
Short of finding the absolute (global) optimum, one might go for less than best, but still better ways, unless new techniques are discovered to locate the best more efficiently. One possibility might be to use graphics to show the performance of the search mechanisms that control the integration. Glover, Kelly, and Laguna suggest this.
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Finding the Best, Analytically
In any formal study, you have to formulate the subject dynamics and find values of variables that yield the desired results. The equations define links that connect future states of a system with the present state. They tell you unequivocally, though not necessarily accurately, that, if a system is in state A, it will get to state B. You then plug in values of the variables for the state A, and out comes the answer, namely state B.
Unfortunately, the equations don't readily tell you what condition, X, the system must be in to be able to reach some desired state D. To find this unknown condition X, for D, you have to hunt for values of the variables that produce the answer D.
The optimization problem is even more complicated. We not only don't know X, but now we also don't know what state we are trying to reach. We only know it's the best that could happen, not what that state is, in fact.
Height of a Fly Ball
Suppose, for example, we wish to find the height attained by a fly ball as a function of starting upward speed. To simplify the illustration, assume the ball moves in a gravity-only environment (no resistance). The height attained depends only on the force of gravity, the starting upward speed, and the elapsed time since launch. So the vertical distance h of the ball is specified by the remarkable Newtonian equation of motion:
h = v0t - 1/2gt2
where v0 is the starting upward speed of the ball, g is the gravitational acceleration, and t is the elapsed time from launch. Treating gravity and speed as fixed parameters and solving for t analytically, we can find the two times t1 and t2 at which the ball attains a specific height h = D.
Maximum Height of a Fly Ball
Suppose we change the problem, slightly. This time, say our aim is to find the maximum height the ball can reach, given gravity and starting speed, as before. Since we're involved with the same fly ball, the dynamics remain unchanged and we can use the same equation for the height. Now, though, we need to find the time at which the ball is at its highest point. That is, we need to find the point at which the ball turns down. This is the highest point. Unfortunately, though, our equation doesn't help.
In order to solve for the time when the ball is at its greatest height, we have to know the height at that moment, because the elapsed time is expressed as a function of the height. But if we had the height, there wouldn't be a problem. We are therefore at an impasse. To find a solution, we must in some way change our perception of the problem.
One thing we can do is to bring into play another remarkable equation at our disposal. This is another formulation in Newtonian particle dynamics and reveals how particle velocity changes relative to its starting velocity. Applying it to the vertical component of motion of the ball, the speed v after an elapsed time t, is:
v = v0 - gt
where, again, v0 is the starting upward speed and g is the gravitational acceleration.
What makes this equation valuable is the fact that the vertical speed goes to zero at the top of the trajectory, which of course identifies the maximum height. Using this fact, we can now solve our problem analytically. By setting v equal to zero, we can easily find the time at which the maximum occurs. That is, we can write:
0 = v0 - gt,
or
t = v0/g.
Since we're given the starting speed and the gravitational acceleration, we can easily determine the time at which the maximum occurs. If we now substitute this value in the distance equation, we can find the maximum height directly. We start with:
h = v0t - 1/2gt2
and after the substitution we get:
h = v0(v0/g - 1/2g(v0/g)2
which is the same as:
h = (v0)2/g - (v0)2/2g = (v0)2/2g.
The analytical approach is definitely the method of choice in this example, because it yields our answer easily and accurately. But when equations get "messy," with things like air resistance, friction, or the like, they generally can't be solved analytically. Then the more likely prospect is to use the computer to conduct a sampling, or a trial and error search for the answer. In that case, the more appropriate approach is to develop a good search procedure.
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