Stochastic optimization (finding the best solution when you have randomness) is a tricky topic. As an example, think about picking the fastest route for driving home from work. Depending on the traffic, different paths might be fastest. It might make sense at first to say "I want to pick the path that will be the fastest today." But when that outcome has uncertainty, it is usually impossible to answer until after you actually have driven home.
There are a number of different ways people handle this issue in optimization. For driving home, it probably makes sense to pick the route with the lowest expected transit time. Over the years of driving that path, some days might take longer than you'd like, but in the long run you'll come out ahead. Other times when you have a dinner-appointment, it might make sense to pick the route that has the lowest probability of taking more than 25 minutes so you are not late. If you were managing the electric grid and avoiding power outages, you will account for the uncertainty in a different kind of way.
As you build a model of the world, different objectives will compress or amplify the effects of uncertainty. Frequently when you are driving somewhere, there will be several routes that have basically the same expected transit time. But the likely worst-case (say, average of the worst 5%) of driving times will often be very different across routes. In the first case, the uncertainties are a relatively small issue because they all get averaged away. In the latter case, extreme cases have a much bigger effect, and therefore the uncertainty will too.
Since models are just attempts to represent reality in a useful way, which model to use for a stochastic problem will depend a lot on your best guess of the costs of uncertainty. If you use an expected-value objective but care a lot about the worst-case tails, you are going to have a bad time when you implement your solution. On the other hand, if you optimize for the worst case for a low-stakes situation like how much inventory to order for a promotion, your company probably will not stay in business too long.
While I have been focusing on either worst-case or expected value as an objective, there are countless ways to design your stochastic optimization model. The short version is that in the field, we are typically trying to reduce the random outcome of our decisions to a single number which allows us to pick the "optimal" solution. While there are ways to optimize over "multiple objectives," they still tend to focus on either subjective decision making or weighting the objectives to obtain a single number.
I welcome comments either on the blog or directly to me. I'm getting this topic ready for a short talk, and I think this will be only my second talk on optimization to a room full of not-optimization people.
There are a number of different ways people handle this issue in optimization. For driving home, it probably makes sense to pick the route with the lowest expected transit time. Over the years of driving that path, some days might take longer than you'd like, but in the long run you'll come out ahead. Other times when you have a dinner-appointment, it might make sense to pick the route that has the lowest probability of taking more than 25 minutes so you are not late. If you were managing the electric grid and avoiding power outages, you will account for the uncertainty in a different kind of way.
As you build a model of the world, different objectives will compress or amplify the effects of uncertainty. Frequently when you are driving somewhere, there will be several routes that have basically the same expected transit time. But the likely worst-case (say, average of the worst 5%) of driving times will often be very different across routes. In the first case, the uncertainties are a relatively small issue because they all get averaged away. In the latter case, extreme cases have a much bigger effect, and therefore the uncertainty will too.
Since models are just attempts to represent reality in a useful way, which model to use for a stochastic problem will depend a lot on your best guess of the costs of uncertainty. If you use an expected-value objective but care a lot about the worst-case tails, you are going to have a bad time when you implement your solution. On the other hand, if you optimize for the worst case for a low-stakes situation like how much inventory to order for a promotion, your company probably will not stay in business too long.
While I have been focusing on either worst-case or expected value as an objective, there are countless ways to design your stochastic optimization model. The short version is that in the field, we are typically trying to reduce the random outcome of our decisions to a single number which allows us to pick the "optimal" solution. While there are ways to optimize over "multiple objectives," they still tend to focus on either subjective decision making or weighting the objectives to obtain a single number.
I welcome comments either on the blog or directly to me. I'm getting this topic ready for a short talk, and I think this will be only my second talk on optimization to a room full of not-optimization people.
