# What is the optimal small-sample experimental design for multiple treatments and costly trials?

I have an experimental outcome in three outcome variables, y1, y2, y3. My objective function is:

a * ln(y1 - y1*) + b * ln(y2 - y2*) + ln(y3 - y3*)

for a, b, c, y1, y2, y3 >0. The starred variables represent the minimum acceptable value for each outcome variable.

The outcome is the effect of five treatments, each of which is continuously variable, plus a random factor. I can set the value of each control variable, but trials are time consuming and costly, and need to be performed sequentially. Trials that pass one of lower acceptable values are potentially very, very costly.

I have a very rough idea of the location of the optimum. The response curves are generally unimodal for each variable, but not necessarily symmetric. The form of interaction between them is unknown, but I have reason to believe that there are interactions, and that they are probably not terribly complicated.

I have reason to believe that simultaneous halving or doubling of all the treatments at once would hit one or more of the minimum acceptable values, though this need not be true of any individual treatment -- it might be optimal to zero out one treatment while increasing others to compensate.

Given these constraints, I would like to design a sensible experimental protocol. What sort of designs should I consider? Is there a good text explaining alternatives for problems of this sort? My statistical background is in econometrics, so my training in multivariate controlled experimental protocols is pretty limited. I am not sure if i should treat this as a problem in search or in estimation of a response surface

Also, if anyone is aware of a literature on determining optimal dosing in multi-drug regimes – for efficacy, not toxicity, especially for the small-sample or n-of-one case – a pointer toward it would be particularly appreciated.

This sounds like a response surface design would do, you should lok into the book Box & Draper Do you know, more or less, the relevant range for the five continuous variables? Can they be controlled independently of each other, or will some "corners" of the covariate space be cut off? (That might happen because, say, you know that high values of $$x_1$$ and $$x_2$$ simultaneously will destroy the product).