Does two-stage stochastic programming involve 2 decision variables?

Question

I thought I was fairly confident in the formulation for “two-stage stochastic program with recourse” until I read the Wikipedia page.

I am used to seeing the problem formulated with one decision variable for the first stage, and another, different decision variable for the second stage (and the decision variable for the second stage is called “recourse decision variable”).

In fact, Wikipedia seems to imply this under the section “Two-stage problems”:

But, then, when I want to find out information about how to actually solve this, I look under the section later on in the document called Stochastic linear program (specifically, “Deterministic equivalent of a stochastic problem”). Now in that case it seems to be treating three decision variables x, y, and z.

Why are there three decision variables in the deterministic equivalent? I thought there would be only two, x, and y?

I found this to be a similar question: https://math.stackexchange.com/questions/2608742/what-is-the-role-of-the-recourse-variable-in-stochastic-programming — makansij, Jan 30 '18 at 07:10
It is early for me, but if I am correct it looks like they rename variables. In the first part $x$ are first stage and $y$ are second stage variables. In the DE formulation $x$ and $y$ are first stage and $z$ are second stage. The $x$ variable do not appear in the second stage whereas the $y$ variables appear both in the first stage and second stage (the $z$ variables only appear in the second stage).. — Erwin Kalvelagen, Jan 30 '18 at 11:09
I see, but since they are using vectors, as indicated by the transposing, I was thinking $x$ and $y$ would just be one vector (even if they are different variables), if they are both first stage variables. — makansij, Jan 30 '18 at 14:54
Also, I'm guessing the way you figured that $x$ and $y$ are for the first stage is because there is no probability multiplied by it, right? — makansij, Jan 30 '18 at 15:00
Stage 2 variables are indexed by the scenario. Stage 1 variables do not have a scenario index. — Erwin Kalvelagen, Jan 30 '18 at 16:37
PLease post a full-fledged answer (not only for me but also for others to see explanation) and i will accept. — makansij, Jan 31 '18 at 04:37
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score 1 · Accepted Answer · answered Jan 31 '18 at 18:22

The wikipedia page is not uniform in the naming of the variables. In the first part $x$ are first stage and $y$ are second stage variables. In the Deterministic Equalivalent (DE) formulation both $x$ and $y$ are first stage and $z$ are second stage. They make a distinction between $x$ and $y$ in how they appear in the second stage. The $x$ variable do not appear in the second stage whereas the $y$ variables appear both in the first stage and second stage. (The second stage variables $z$ only appear in the second stage).

You can recognize a stage 2 variable as being indexed by the scenario. Stage 1 variables do not have a scenario index.

Seems to me like that is some sloppy wiki-ing. The page lacks consistency — makansij, Feb 01 '18 at 04:03

Does two-stage stochastic programming involve 2 decision variables?

1 Answers1