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. In the context of non-convex programming, two major types of special ordered sets (SOS) of variables have been identified and studied by researchers. SOS1 are sets of non-negative variables where, for each set, at most one of the variables can be non-zero in the final solution. The most common application of SOS1 is multiple choice programming (MCP) which can be found in the modeling of many integer programming problems in location, distribution, scheduling, etc. SOS2 requires that, for each set, at most two of the variables can be non-zero in the final solution and, if they are, they must be adjacent. SOS2 has been widely used in separable programming to model non-linear functions using sets of piece-wise linear functions. Bricker introduced an explicit reformulation technique for SOS in 1977. Lin and Bricker developed a reformulation and transformation technique (RTT) to implicitly compose the optimal Simplex tableau for MCP in 1991. They also elaborated upon it with a computational report in 1996. Without citing the work by Bricker, or that by Lin and Bricker, Ogryczak proposed an analogous reformulation technique called special ordered inequalities (SOI) for SOS in 1996. This note aims to elaborate upon the connection between SOI and RTT as the supplementary information for the future research in SOS.