Application of the Branch and Bound Technique to Some Flow-Shop Scheduling Problems

Abstract

The branch-and-bound technique of Little, et al. and Land and Doig is presented and then applied to two flow-shop scheduling problems. Computational results for up to 9 jobs are given for the 2-machine problem when the objective is minimizing the mean completion time. This problem was previously untreated. Results for up to 10 jobs, including comparisons with other techniques, are given for the 3-machine problem when the objective is minimizing the makespan.

Efficient heuristic and optimal approaches for n/2/F/σCi scheduling problems

“…In this section, we develop formulae to compute the lower bounds. The first two lower bounds are similar to those of Ignall and Schrage [7]. Assuming that dk is a partial schedule containing k jobs, then its objective cost is a constantfk which can be computed by Eqs.…”

Dominance rules in combinatorial optimization problems

“…Among these methods, the "branch and bound" procedure has been popularized by Land and Doig [53] to solve Integer Linear Programming problems, by Little et al [54] to solve the traveling salesman problem, and by Ignall and Schrage [3] to solve scheduling flowshop problems. The branch and bound method has been examined and generalized by a lot of authors (see for example [55,21,56,57,58]).…”

“…In order to use the branch and bound method, it is only necessary to be able to describe the problem as a tree, in which each node represents a partial solution. In addition, it must be possible to fix, at each node, a lower bound on the objective function for all nodes that emanate from it [3]. Moreover, if dominance rules can be defined for identifying and eliminating active nodes leading to dominated solutions (because they are proved to be either not optimal or equivalent to other solutions 21 enumerated elsewhere in the search tree) this can dramatically improve the behavior of the branch and bound procedure [57].…”

“…We can find it in early papers by Manne [1], Levitan [2], Ignall and Schrage [3], Fox [4], Proshan and Bray [5], Weingartner and Ness [6] and Elmagharaby [7], although the term seems have taken several years to be established. Later, it was intensively used by Baker [8], Kohler and Steiglitz [9] and Ibaraki [10].…”

Probabilistic Analysis and Practical Algorithms for the Flow Shop Weighted Completion Time Problem

“…For instance, Ignall and Schrage (1965) first applied branch and bound to small size flow shop problems, while Krone and Steiglitz (1974) applied local search techniques. Kohler and Steiglitz (1975) combined these approaches to solve two-machine problems of up to 15 jobs to optimality, and of up to 50 jobs approximately.…”