The linear ordering problem revisited

Abstract

The Linear Ordering Problem is a popular combinatorial optimisation problem which has been extensively addressed in the literature. However, in spite of its popularity, little is known about the characteristics of this problem. This paper studies a procedure to extract static information from an instance of the problem, and proposes a method to incorporate the obtained knowledge in order to improve the performance of local searchbased algorithms. The procedure introduced identifies the positions where the indexes cannot generate local optima for the insert neighbourhood, and thus global optima solutions. This information is then used to propose a restricted insert neighbourhood that discards the insert operations which move indexes to positions where optimal solutions are not generated.In order to measure the efficiency of the proposed restricted insert neighbourhood system, two state-of-the-art algorithms for the LOP that include local search procedures have been modified. Conducted experiments confirm that the restricted versions of the algorithms outperform the classical designs systematically. The statistical test included in the experimentation reports significant differences in all the cases, which validates the efficiency of our proposal.

An Algebraic Differential Evolution for the Linear Ordering Problem

“…According to [6] and the more recent [2], MA and ILS are to be considered the state-of-the-art algorithms for LOP.…”

Extending distance-based ranking models in estimation of distribution algorithms

“…The instances of the TSP were downloaded from TSPLIB [41], and the instances of the QAP and PFSP were obtained from the Taillard's Benchmark [42]. As regards the LOP, the smallest 15 instances were obtained from the LOLIB benchmark, and the rest were artificially generated as specified in [32] 2 .…”

“…A particular feature of this problem worth noting is that the contribution of an index i to the objective function depends on the previous and posterior sets of indexes, but not on their relative ordering [32].…”

“…Given a matrix B = [b ij ] n⇥n of numerical entries, the Linear Ordering Problem (LOP) [31], [32] consists of finding a simultaneous permutation of the rows and columns of B such that the sum of the entries above the main diagonal is maximised (or equivalently, the sum of the entries below the main diagonal is minimised). The equation below formalises the LOP function:…”

A Note on the Boltzmann Distribution and the Linear Ordering Problem

“…, n} positions. However, it does not change for any ordering of the items within the subsets [1]. In the following section, we demonstrate that such a notion of independence between the items is interpreted as L-decomposability of the associated Boltzmann distribution.…”