Comparing Multi-Objective Local Search Algorithms for the Beam Angle Selection Problem

In intensity-modulated radiation therapy, treatment planners aim to irradiate the tumour according to a medical prescription while sparing surrounding organs at risk as much as possible. Although this problem is inherently a multi-objective optimisation (MO) problem, most of the models in the literature are single-objective ones. For this reason, a large number of single-objective algorithms have been proposed in the literature to solve such single-objective models rather than multi-objective ones. Further, a difficulty that one has to face when solving the MO version of the problem is that the algorithms take too long before converging to a set of (approximately) non-dominated points. In this paper, we propose and compare three different strategies, namely random PLS (rPLS), judgement-function-guided PLS (jPLS) and neighbour-first PLS (nPLS), to accelerate a previously proposed Pareto local search (PLS) algorithm to solve the beam angle selection problem in IMRT. A distinctive feature of these strategies when compared to the PLS algorithms in the literature is that they do not evaluate their entire neighbourhood before performing the dominance analysis. The rPLS algorithm randomly chooses the next non-dominated solution in the archive and it is used as a baseline for the other implemented algorithms. The jPLS algorithm first chooses the non-dominated solution in the archive that has the best objective function value. Finally, the nPLS algorithm first chooses the solutions that are within the neighbourhood of the current solution. All these strategies prevent us from evaluating a large set of BACs, without any major impairment in the obtained solutions’ quality. We apply our algorithms to a prostate case and compare the obtained results to those obtained by the PLS from the literature. The results show that algorithms proposed in this paper reach a similar performance than PLS and require fewer function evaluations.

Pythagorean Cubic fuzzy Hamacher aggregation operators and their application in green supply selection problem

<abstract><p>The green chain supplier selection process plays a major role in the environmental decision for the efficient and effective supply chain management. Therefore, the aim of this paper is to develop a mechanism for decision making on green chain supplier problem. First, we define the Hamacher operational law for Pythagorean cubic fuzzy numbers (PCFNs) and study their fundamental properties. Based on the Hamacher operation law of PCFNs, we defined Pythagorean cubic fuzzy aggregation operators by using Hamacher t-norm and t-conorm. Further, we develop a series of Pythagorean cubic fuzzy Hamacher weighted averaging (PCFHWA), Pythagorean cubic fuzzy Hamacher order weighted averaging (PCFHOWA) Pythagorean Cubic fuzzy Hamacher hybrid averaging (PCFHHA), Pythagorean Cubic fuzzy Hamacher weighted Geometric (PCFHWG), Pythagorean Cubic fuzzy Hamacher order weighted Geometric (PCFHOWG), and Pythagorean Cubic fuzzy Hamacher hybrid geometric (PCFHHA) operators. Furthermore, we apply these aggregation operators of Pythagorean Cubic fuzzy numbers to the decision making problem for green supplier selection. We construct an algorithm for the group decision making by using aggregation operators and score function. The proposed decision making method applies to green chain supplier selection problem and find the best green supplier for green supply chain management. The proposed method compared with other group decision techniques under Pythagorean cubic fuzzy information. From the comparison and sensitivity analysis, we concluded that our proposed method is more generalized and effective method.</p></abstract>