qeqwqqweqweqeqeqeqwe The following concise notations are used to describe the main ideas of the proposed stochastic programming approach.
All the binary location decisionvariables regarding the forward processing facilities, collection facilities and hybrid process- ing facilities are included into vectors xt , yt and zt . All the binary linearization variables are includedinto vectors ut;t-1 , vt;t-1 and wt;t-1 . The vectors f t , f t , f t , f t , f t and f t stand for the corresponding cost coefﬁcient, respectively. The notation represents
1 2 3 45 6
the column vector where all the elements are equal to 1. All the continuous decision variables corresponding to the shipment of forward products and returned products areincluded into a vector qt and a vector gt , respectively. The vector ct and vector et represent the costs associated with the shipment of the forward products and the returned products,respectively. Deﬁne U as the set of all the possible scenarios of the uncertain parameters in the attempted problem and / 2 U stands for a par-
ticular scenario. The vectors dð/Þand sð/Þ correspond to the demand of forward products and the supply of returned prod- ucts, respectively, which are functions of a scenario /. The vectors ff t and fht correspond to the capacity forhandling forward products of forward processing facilities and hybrid processing facilities, respectively. The vectors rrt and rht correspond to the capacity for handling returned products ofcollection facilities and hybrid processing facilities, respectively. The matrices Dt , K t , Ht and Nt are appropriate matrices corresponding to the summations on the left-hand side of Eqs. (2)–(5),respectively. Thus, a two-stage stochastic programming model is further developed as follows:
ðTÞ min f ðx; y; z; u; v; wÞ¼
((f t ) xt þ (f t ) yt þ (f t ) zt þ (f t ) ut;t-1 þ (f t )...