Connecting special ordered inequalities and transformation and reformulation technique in multiple choice programming

An article entitled: “A Note on Modeling Multiple Choice Requirements for Simple Mixed Integer Programming Solvers” was published by Ogryczak (Comput. Oper. Res. 23 (1996) 199). In this article, Ogryczak proposed a reformulation technique called special ordered inequalities (SOI) to model the non-convex programming problems with special ordered sets (SOS) of variables. The SOI technique appears to be analogous to the reformulation technique introduced by Bricker (AIIE Trans. 9 (1977) 105) and is related to the reformulation and transformation technique (RTT) developed by Lin and Bricker (Eur. J. Oper. Res. 55(2) (1991) 228); Lin and Bricker (Eur. J. Oper. Res. 88 (1996) 182). Since none of this literature was cited in the references of Ogryczak (Comput. Oper. Res. 23 (1996) 199), we would like to use this note to differentiate SOI and RTT and to elaborate their connection.