For a simulation of the economy, I am trying to implement, I want to find an equation to represent how input resources impact the rate of production of a factory.
E.g, a factory wants an optimal amount of wood (the amount being v=10). When this level of wood is reached, the function returns 1, which will multiply with other production values to return 10 tables. However, if you have excess wood (say, 20 pieces), the factory doesn't have the capability to make 20 tables. Instead, it produces maybe only 15 or so. The specifics don't matter, but it needs to be less. To grease the wheels of the entire economy, the factory will produce tables even if no wood is provided (say, 1 table).
To implement this in a flexible way, I wanted to be able to control the y-intercept of the graph (to set the amount of production given 0 inputs), the point in which the optimal rate is met (the 10 wood), and the actual rate that is the optimal point (e.g, 1 or 1.5, etc.)
The two horizontal asymptotes can run wherever, but the center point needs to lie directly between the two of them.
A draft of what I need:
I have tried using the following equations, but as you can see at 11 wood it should be a rate < 1.1, but that isn't the case:
Variables for optimal amount, null rate, and optimal rate.
-sigmoid : 
- arctan :
The problem is, I think, that you need a maximum derivative of 1 that sits in the centre point, and for the neighboring gradients to be less than that. However, there always exist some y-intercept that will cause the centre point to get so bunched up that the gradient will exceed 1. I am not sure if this is the case for all possible equations though.
I am aware that I can modify the x-transformation until I get a gradient that's slanted enough for what I need. However, this won't be compatible with being able to specify the y-intercept as well.
A bottom asymptote may not necessarily be necessary, I only considered it to be so as I wanted the curve to "bottom out" as it approached 0. But if there is an alternative that will have a similar curve that I can spec to pass through a y-intercept than I am all ears.
What I want to know, is if what I want is even possible at all, what would work for a single equation, and if there isn't any, what would work on a piecewise equation.
Let me know if I need to clarify anything, or explain in further detail any section of the prompt.
Thank you.