I have a situation involving two independent variables, each of which has constraints, and a measure derived from the ratio of the variables. I am trying to wrap my head around how to derive the solution space for the measure but the non-linearity is making my head hurt.
The details are, let X and Y be two independent variables with the constraints X >= A > 0 and 0 < Y <= B. The measure is M = CX/Y where C is a positive constant. Given these constraints, can anyone point me to a method for tackling this?
I should have been clearer, the variables are physical values and will always be positive, and while there is no direct dependence between X and Y, in this case no “acceptable” value of Y is possible near 0 since that would result in values of X that are less than the lower bound A. I should have specified the boundary conditions for Y as 0 < B[sub]min[/sub] <= Y <= B[sub]max[/sub]
My question is not how do I maximize the value of the derived measure M, but from the constraints on the variables how can I map the possible values of M in terms of the constraints.
There is an intuitive way and a mathematical way to do this. Lets start with the mathematical way:
M=CX/Y we can rewrite this as Y=CX/M. So 0 < CX/M <=B. Since C X and Y are all positive, M must be positiive, so we can multiply by M/C. 0< X <= MB/C. Meanwhile we know that X>= A so
0 <A <= X <= MB/C, so in particular A<=MB/C so M >= CA/B
Now the intuitive way. M gets larger as X gets larger and Y gets smaller. So the lowest value for M will be at the smallest X and the largest Y namely CA/B, meanwhile X can get arbitrarily large and Y arbitrarily small so there is no upper limit to the size of M.
For more complicated non-linear equations, for which the associations aren’t so obvious, the standard mathematical way of approaching the problem is with the calculus of variations. But that requires understanding multivariate calculus.
If X and Y are free to vary independently between Ymin & Ymax and Xmin and Xmax respectively, then clearly M can vary from the lowest value of CXmin/Ymax to the highest of CXmax/Ymin and any value in between.
Now, if, as I suspect, there’s some complicated relationship between the allowed values of X and Y, then it’s possible M can’t get quite that low or that high, but those two values are certainly outer bounds for what it can be. And to go any farther, mathmatically, you’ll need to mathematically describe how X and Y relate.
Thanks to Buck Godot and Quercus for responding, very helpful.
I don’t mean to be vague about X and Y, but there are intellectual property issues that prevent me from being much more forthcoming. What I can say is that this involves a type of training where participants are presented with a series of stimuli to which they respond. X is the percentage of correct responses and Y is the time required to respond. So trying to derive a mathematical relationship between X and Y is not really possible, but there is a minimum possible time, given the nature and number of stimuli presented, and the scores X can range from 0 to 100. To make matters more complicated, there is also a consistency criteria so the values of X and Y are actually averages for a number of consecutive trials. We have been using a metric where the participant must be able to respond with an average score greater than or equal to a particular value and an average time less than or equal to another particular value to “pass”.
A proposal has been made to use a new measure of accurate speed, the M described above. My task is to examine the relationship between the new measure and the existing passing constraints to see if M is compatible. M has proven useful already in examining the progress that the participants exhibit but I got bogged down trying to see how M could be used to replace the previous criteria. I resorted to a numeric stimulation where I generate random sets of X and Y values close to the passing constraint values and test the average values of them against the current criteria, and against a sequence of “reasonable” values for M. What I find is that while there is never perfect agreement, there is a value of M that results in a minimum disagreement. I have run the stimulation under 3 different programming languages to try to account for possible limitations of their random number generators and the results agree across all platforms.
I will think about your responses and see where I can take them.