Optimization problem (with inequality and number type constraints)

[Nerj]

Hello everyone! I'm trying to sort out resources for a gig I'm working on and realized that I've got an optimization problem on my hands. The constraints are a little unique compared to what I'm familiar with, so I turn to you all in hopes that someone might be able to help me.


Variables: x, y
Equation: 11xy = 10
Constraints: y >= 1 and is a natural number

Given the above, I need to maximize x and y. Does anyone have any ideas on how to approach this problem?

 

[ksdhart2]

Well, it's extremely unclear to me what you mean by "maximize x and y," but the restriction \(\displaystyle y \in \left\{\mathbb{N} \backslash \left\{0\right\}\right\}\) suggests a good place to start would be to simply try some cases. Suppose y = 1. What does that make the value of x? Next, suppose that y = 2. What does that make the value of x? You'll now have to decide, using whatever unknown criteria you have, if this is a more desirable outcome than the case where y = 1. After that, suppose y = 3 and evaluate that as well. By this point, you should definitely be seeing a pattern emerging, and so you should be able to determine the values which "maximize x and y" or else determine that such a maximization is impossible.

 

[JeffM]

Hello everyone! I'm trying to sort out resources for a gig I'm working on and realized that I've got an optimization problem on my hands. The constraints are a little unique compared to what I'm familiar with, so I turn to you all in hopes that someone might be able to help me.


Variables: x, y
Equation: 11xy = 10
Constraints: y >= 1 and is a natural number

Given the above, I need to maximize x and y. Does anyone have any ideas on how to approach this problem?

It should be intuitively obvious that, if y is greater than 1, y is positive and so therefore must x be. Moreover, it should be equally obvious that, the larger you make x, the smaller you make y, and the larger you make x. So it is impossible to maximize both simultaneously. And finally, there only four natural numbers that can be considered: 1, 2, 5, or 10.

 

[Nerj]

Well, it's extremely unclear to me what you mean by "maximize x and y," but the restriction \(\displaystyle y \in \left\{\mathbb{N} \backslash \left\{0\right\}\right\}\) suggests a good place to start would be to simply try some cases. Suppose y = 1. What does that make the value of x? Next, suppose that y = 2. What does that make the value of x? You'll now have to decide, using whatever unknown criteria you have, if this is a more desirable outcome than the case where y = 1. After that, suppose y = 3 and evaluate that as well. By this point, you should definitely be seeing a pattern emerging, and so you should be able to determine the values which "maximize x and y" or else determine that such a maximization is impossible.

Thanks! This really helped. Was able to figure out what to do. Many thanks, and happy math-ing.