Notations

 

I = number of the first stage decision variables, i=1,...,I  index of variable;

K = number of the second stage decision variables, k=1,...,K  index of variable;

= vector of the first stage decision variables;

= vector of the second stage decision variables;

= matrix of the first stage linear constraints;

= vectors of lower and upper bounds for the first stage linear constraints;

= lower and upper bounds for the first stage variables, i=1,...,I;

= cost vectors for the second stage objective;

= matrices of the second stage linear constraints;

= lower and upper bounds for the second stage variables, k=1,...,K;

= probability of scenario ;

= vectors of lower and upper bounds for the second stage linear constraints for scenario ;

 = recourse function for scenario :

subject to

linear constraints for the second stage

,

bounds on the second stage decision variables

;

= Average of Recourse function over scenarios.

Optimization Problem

minimizing Average of Recourse function

               (CS.1)

subject to

linear constraints on the first stage decision variables

                       (CS.2)

bounds on the first stage decision variables

       (CS.3)