IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 16, Issue 6 Ser. II (Nov. – Dec. 2020), PP 26-35 www.iosrjournals.org DOI: 10.9790/5728-1606022635 www.iosrjournals.org 26 | Page Why does the classical Benders decomposition algorithm not work well for the b-Complementary Multisemigroup Dual Problem? Eleazar Madriz, Yuri Tavares dos Passos, Lucas Carvalho Souza CETEC-UFRB,Cruz das Almas, BA, Brazil Abstract We present an adaptation of the classical algorithm of the decomposition of Bender to the b-complementary multisemigroup dual problem. Despite this decomposition has been shown in literature as a good tool for dealing with high dimensional mixed-integer linear programming, that is not the case for the presented one in this paper, which is better to be solved by the simplex algorithm without partitioning. We present results from computer experiments to show that conclusion. Keywords: Multisemigroup; Complementary; Duality; Benders Decomposition. --------------------------------------------------------------------------------------------------------------------------------------- Date of Submission: 13-11-2020 Date of Acceptance: 29-11-2020 --------------------------------------------------------------------------------------------------------------------------------------- I. Introduction Aráoz in 1973 defined the Semigroup Problem (SP), characterizes the polyhedra, and shows the relation between the minimal system of linear inequality of the polyhedra and extreme points and rays. Aráoz and Johnson, in 1982, presented the polyhedra of a multivalued additive system problem. A particular case of multivalued additive systems is the b-complementary Multisemigroups (b-CMS). A b-CMS is an associative, an abelian, a b-consistent, and a b-complementary additive system. Madriz, in 2016, constructed the dual problem associated with a b-CMS problem, extending the duality result of semigroup by Johnson in 1980. Madriz's work is based on the theorem presented by Ar´aoz and Johnson in 1989, where they determine that, given a base of the subadditive cone, it is possible to establish a system of equations and inequalities that define the polyhedron associated with a multivalued associative additive system. Aráoz and Johnson in 1982 defined the finite b-complementary multisemigroup as an associative, commutative, consistent, and complementary additive multivalued system and they showed the following characterization of the faces of the convex hull of multisemigroup solutions. Let be an b-complementary multisemigroup and . The master corner polyhedron is defined as for some fixed right-hand side element and denotes the cardinality of the set and is the set of integer numbers. Let be the subadditive cone associated with (see [1]), Aráoz and Johnson in [2] showed the following characterization of the . Theorem 1.1 (Theorem 3.8 in [2])} Let (L,E) be a base of . The following system defined a In general, given a finite b-complementary multisemigroup A , the b-complementary multisemigroup master problem is defined as