Stochastic bank

Question 1

motivation for stochastic programming.

Answer 1

Uncertainty lead to models that behave somewhat differnetly. One might think that using expected value of parameters or values is good enough, but it can be shown that this often has unfortunate consequences.

There are two important concepts:
1) Expected value of perfect information (EVPI)
2) Value of the stochastic solution (VSS)

Both of these concepts illustrate that they give an edge compared to what we call the “expected value solution”.

In other words, there is a real value in doing it the stochastic way, which means that we are not fully optimized unless we account for uncertainty.

Question 2

in general, how can we describe our “situation” when we use stochastic programmign

Answer 2

We are considering some situation where we need to make a decision on a set of variables without the complete information on their effect. There will be a random event that would tell us the effect, but we do not know the outcome of the random event at the time when we solve the optimization problem.

Question 3

What dfferent decisions do we have?

Answer 3

Broadly speaking, we have:
1) First stage decisions
2) Second stage decisions

they differ in the sense that the first stage decisions are made on the basis of some probabilistic event.
The second stage decisions are easy to solve in the sense that they represent the best possible value given a certain event. At this point in time, we know the event’s outcome, and are able to set the values without uncertainty.

it is typical to say that:
We have a set of first stage decisions “x”, and a set of second stage decisions, or corrective actions “y” that needs to be taken after some random event. It is actually more common to refer to y as “y(s)” to highlight the fact that they depend on the specific scenario. y is a vector of actions/decisions that only make sense when in context of the specific scenario.
So, we get a set of y-variables for every scenario s in S.

Question 4

releationship between a scenario and constraints?

Answer 4

For each scenario, we have a set of variables AND a set of constraints on these variables.

Abstractly, we define the scenario specific problem as:

Q(x, randomEvent) = min {qy | Wy = h - Tx, y>=0}
Here, W is a fixed known resource matrix. It could values for things like “price” or “cost” etc. The important thing is that it is fixed.
The T matrix is random. why? it is the T matrix that actually hold all of the uncertainty. It is scenario specific. in the farmer example it holds values related to the yield of crops, which is directly tied to the first stage variables of the amount of acres of each crop.

Question 5

give the implicit form of a two-stage model. what is the benefit of this formulation?

Answer 5

min z = cx + L(x)
s.t.
Ax = b
x>=0

Represent the problem clearly as consisting of a regular part and a recourse part.

Question 6

what happens when we plan for “expected value” like in the vehicle routing problem?

Answer 6

Planning with expected value creates a scenario where we add uncertainty into our considerations, but not in a good way. It sort of acts like “forgetting” that uncertainty exist. For instance, in VRP, we can end up believing that all is good for a single trip as long as the expected demand is at a level that satisfy a single trip. however, if the variance is large, we might very likely end up going for a trip to the depot and back.

Question 7

What is the better option (as compared to expected value variant)?

Answer 7

Plan for recourse actions. We solve a first stage problem and a second stage problem. Doing this allow us to initially determine a route, and then later make adjustments if necessary.