# Decision making with respect to utility function

I am currently working on a small project targeted towards predicting survival times (red, green functions) of certain engine parts. The ultimate goal is to decide what part would be the best choice with respect to some utility function (purple). Currently, I am deciding based on the expected survival time, however, this decision criterion does not take into account the underlying utility. However, when deciding based on the expected utility of each predicted survival function, my predictions get worse. I think this is primarily due to the fact that my utility function is highly non-smooth. In the example below, the expected utility for the green function would be higher than the expected utility of the red curve, however, it centers a lot of probability mass close to the critical region of the utility function, hence is more likely to achieve no utility at all if the predicted survival functions are somewhat uncertain. Therefore, I would like to find some decision criterion that factors the above into account. Regarding this, my question are:

1. Can I find some relaxation (surrogate) of my utility function (e.g. concave, smooth function) that assigns less utility near the critical point? How would I approach this?
2. I thought about making decisions wrt. risk aversion measures, however, did not find any measure that worked well in my experiments. Is there any risk measure that would adequately penalize survival functions with high probability near the critical point?
3. Do you have any suggestions with taking this utility function into account?