The PROMETHEE is a significant method for evaluating alternatives with respect to criteria in multi-criteria decision-making problems. It is characterized by many types of preference functions that are used for assigning the differences between alternatives in judgements. This paper proposes a preference of green suppliers using the PROMETHEE under the usual criterion preference functions. Comparable results are presented to check the effect of different preference functions on the final preference. Seven economical and environmental criteria, four suppliers and five decision makers were the main structures in the green supplier selection problem. Data were collected via personal communication with decision makers using five-point Likert scale. The algorithm of PROMETHEE under usual criterion function was implemented, and the results show that supplier A₁ is the most preferred alternative. Comparative results also show that supplier A₁ is the most preferred alternative despite the difference in preference functions used.

Combinatorial optimization problems are often considered NP-hard problems in the field of decision science and the industrial revolution. As a successful transformation to tackle complex dimensional problems, metaheuristic algorithms have been implemented in a wide area of combinatorial optimization problems. Metaheuristic algorithms have been evolved and modified with respect to the problem nature since it was recommended for the first time. As there is a growing interest in incorporating necessary methods to develop metaheuristics, there is a need to rediscover the recent advancement of metaheuristics in combinatorial optimization. From the authors’ point of view, there is still a lack of comprehensive surveys on current research directions. Therefore, a substantial part of this paper is devoted to analyzing and discussing the modern age metaheuristic algorithms that gained popular use in mostly cited combinatorial optimization problems such as vehicle routing problems, traveling salesman problems, and supply chain network design problems. A survey of seven different metaheuristic algorithms (which are proposed after 2000) for combinatorial optimization problems is carried out in this study, apart from conventional metaheuristics like simulated annealing, particle swarm optimization, and tabu search. These metaheuristics have been filtered through some key factors like easy parameter handling, the scope of hybridization as well as performance efficiency. In this study, a concise description of the framework of the selected algorithm is included. Finally, a technical analysis of the recent trends of implementation is discussed, along with the impacts of algorithm modification on performance, constraint handling strategy, the handling of multi-objective situations using hybridization, and future research opportunities.

The fuzzy decision-making trial and evaluation laboratory (fuzzy DEMATEL) has been used to solve various multi-criteria group decision-making problems where triangular fuzzy numbers are utilized in defining decision makers’ linguistic judgements. Most of the fuzzy DEMATEL modifications are built from linguistic variables based on fuzzy sets. Recent literature suggests that Pythagorean fuzzy sets (PFS) can offer a better alternative particularly when fuzzy sets have some extent of limitations in handling vagueness and uncertainty. This paper proposes a modification fuzzy DEMATEL characterized by PFS for linguistic variables. Differently from the typical fuzzy DEMATEL which directly utilizes triangular fuzzy numbers with a single membership, this modification introduces membership and non-membership of PFS to enhance judgements in the group decision-making environment. The proposed method has a number of attractive features. It includes linguistic variables, expert’s weights, and score function, in which all of these features are expressed by PFS. The proposed modification is applied to a case of solid waste management (SWM) where ten criteria are considered in assessment. Six experts in SWM were invited to provide linguistic judgments with respect to the criteria, and the eleven-step computational procedure of the proposed method was implemented without losing the general structure of the DEMATEL method. The results unveiled that four criteria are identified as ‘cause group’ and six criteria are identified as ‘effect group’ in SWM. The grouping of criteria would help policy makers in identifying the criteria that could enhance the efficiency of SWM.

This paper aims to resolve the issue of the ranking of the fuzzy numbers in decision analysis, artificial intelligence, and optimization. In the literature, many ideas have been established for the ranking of the fuzzy numbers, and those ideas have some restrictions and limitations. We propose a method based on spherical fuzzy numbers (SFNs) for ranking to overcome the existing restrictions. Further, we investigate the basic properties of SFNs, compare the idea of spherical fuzzy set with the picture fuzzy set, and establish some distance operators, namely spherical fuzzy distance-weighted averaging (SFDWA), spherical fuzzy distance order-weighted averaging (SFDOWA), and spherical fuzzy distance order-weighted average weighted averaging (SFDOWA WA) operators with the attribute weights’ information incompletely described. Further, we design an algorithm to solve decision analysis problems. Finally, to validate the usage and applicability of the established procedure, we assume the child development influence environmental factors problem as a practical application.