Considering correlation in stochastic programming

[tal...@hotmail.com]

In dealing with a production problem under demand uncertainty, I build
a model in the traditional form of stochastic programming where the
demand uncertainty is discretized into several scenarios, and are then
all scenarios for all products are permuted to generate all possible
combination scenarios between products. However, I think by doing so,
it implies that the correlation between demands of products is 0 and
therefore, it doesn't allow me to consider correlation between
products when I am generating demand scenarios. Does anyone know if
there is a way to put correlation into consideration when generating
scenarios? Thank you very much in advance!

[Horand....@googlemail.com]

Of course there is. The short answer is that you need to determine
the /joint/ distribution of the demands and generate your scenarios
accordingly. How did you get the univariate distributions in the first
place? If they came from historic data, then the correlation is
already built in. All yo need to do is to sample randomly (sounds like
this is what you would like to do) from the historic /periods/ and
pick the joint demands in each peiod you sampled.

[tal...@hotmail.com]

Hi, thank you very much for your reply. I am glad that there is a way
to do that, although I am not entirely clear about how this works.
Right now what I am doing is: For example, I have 2 products and I
discretize the distribution for each product into several
scenarios(e.g. 3), each with a probability p. so the demand
uncertainty will be represented by:
Product 1: d(1,1) with p(1,1)=0.3, d(1,2) with p(1,2)=0.4,d(1,3) with
p(1,3)=0.3
Product 2: d(2,1) with p(2,1)=0.3, d(2,2) with p(2,2)=0.4,d(2,3) with
p(2,3)=0.3
Then I will have totally 8 scenarios:

Scenario 1: d(1,1) & d(2,1) with probability p(1,1)*p(2,1)=0.09
Scenario 2: d(1,1) & d(2,2) with probability p(1,1)*p(2,2)=0.12
....
Scenario 8: d(1,3) & d(2,3) with probability p(1,3)*p(2,3)=0.09

by doing this, it implies the correlation between the two products is
0. Now assume that I know the correlation between product 1 and 2 is
t, how can I represent the demand uncertainty in this kind of format?
Or I have to use other form to do that?

Thank you very much! Looking forward to your reply.

[Horand....@googlemail.com]

Don't you mean 9 scenarios? Since you seem to be creating the
scenarios out of thin air, the easiest way is to proceed as follows:

Scenario 1:d(1,1) & d(2,1) with probability 0.3
Scenario 2:d(1,2) & d(2,2) with probability 0.4
Scenario 3:d(1,3) & d(2,3) with probability 0.3

These, of course, are perfectly correlated. If you want nonzero
corelation, you can try to form linear combinations of your scheme and
mine. There also is a vast lierature on scenario generation. In
particular, I would direct you to the work of Stein Wallace and co-
authors. This should be findable in google scholar.

[tal...@hotmail.com]

> authors. This should be findable in google scholar.- Hide quoted text -
>
> - Show quoted text -

Thank you very much again! It is very enlightful. I will check it out!