Application of Combinatorial Programming to a Class of All-Zero-One Integer Programming Problems

Abstract

Problem-solving procedures based on the methods of combinatorial programming are presented for solving a class of integer programming problems in which all elements are zero or one. All of the procedures seek first a feasible solution and then successively better and better feasible solutions until ultimately one is discovered which is shown to be optimal. By representing the problem elements in a binary computer as bits in a word and employing logical "and" and "or" operations in the problem-solving process, a number of problems involving several hundred integer variables have been solved in a matter of seconds.

Scheduling Ocean Transportation of Crude Oil

“…A lexicographic in-situ sort is employed to express the SPP in block-staircase form (e.g., Christofides and Korman 1975, Garfinkel and Newhauser 1969, Pierce 1968. Rows may also be reordered by length (e.g., Marsten 1974).…”

The set covering problem : a group theoretic approach

“…Pierce [14] created a purely enumerative procedure to solve the partitioning problem; according to comparative testing made at MJ.T., it seemed to perform better than group theory for problems with few (less than 50) rows and several hundred columns; in these cases. Pierce's method was faster than the LP solution itself !…”

Une nouvelle méthode pour le problème de partitionnement fondée sur une évaluation par excès de la solution

“…Nous avons montré dans [1] (1,2,3,6,7,10), (1,4,5,6,8,11), (2,5,7,8,9,12), (3,4,9,10,11,12), (5,6,7,10,11,12) où (1,2,3,4,8,9), par exemple, correspond à la contrainte x A + x 2 + x 3 + x 4 + x 8 + x 9 -1.…”

“…Le problème de partitionnement PP : où A = (a ij ) est une m xn matrice donnée et telle que a tj = 0 ou 1; e, un m xl vecteur dont tous les éléments sont égaux à 1; c, un lxn vecteur à coefficients positifs connus; x, le vecteur nxl des inconnues xy, est un problème très contraint qui se résout bien par énumération implicite [10,6]. Cependant ces auteurs développent d'importantes arborescences.…”