Can a Branch and Bound algorithm solve all instances of SALBP-1 efficiently?

Abstract

Following the publication by Sewell and Jacobson of a Branch and Bound algorithm which has solved all existing benchmarks for the simple assembly line balancing problem of type 1 (SALBP-1) in less than 1 second per problem, this paper investigates the possible existence of harder instances and how they can be designed. SALBP-1 is a generalization of the bin packing problem. We demonstrate how a hard bin packing case can be generated. We prove that, in the worst case, Branch and Bound algorithms with Lower Bounds computed in polynomial time are unable to solve even moderate sized examples of the problem in appropriate time limit. Branch and Bound is an excellent approach based on decomposition and enumeration. Intelligent techniques were elaborated to avoid a complete enumeration. Nevertheless, as demonstrated here, it is always possible to find cases where a Branch and Bound will necessitate a complete enumeration.