Aggregation of Priorities in Multi-Criteria Decision Analysis (MCDA): Connecting Decision Spaces in the Cognitive Space

In Multi-Criteria Decision Analysis, the well-known weighted sum method for aggregating normalised relative priorities ignores the unit of scale that may vary across the criteria and thus causes rank reversals. A new aggregation rule that explicitly includes the norms of priority vectors is derived and shown as a remedy for it. An algorithmic procedure is presented to demonstrate how it can as well be used in the Analytic Hierarchy Process when norms of priority vectors are not readily available. Also, recursion relations connecting two decision spaces with added or deleted alternatives give an opportunity to extend the idea of connectivity to a new concept of cognitive space. Expanded analytic modelling embracing multiple decision spaces or scenarios may assist in detecting deficiencies in analytic models and also grasping the big picture in decision making.

Exploring a Promising Region and Enhancing Decision Space Diversity for Multimodal Multi-Objective Optimization

During the past decade,research efforts have been gradually directed to the widely existing yet less noticed multimodal multi-objective optimization problems(MMOPs)in the multi-objective optimization community.Recently,researchers have begun to investigate enhancing the decision space diversity and preserving valuable dominated solutions to overcome the shortage caused by a preference for objective space convergence.However,many existing methods still have limitations,such as giving unduly high priorities to convergence and insufficient ability to enhance decision space diversity.To overcome these shortcomings,this article aims to explore a promising region(PR)and enhance the decision space diversity for handling MMOPs.Unlike traditional methods,we propose the use of non-dominated solutions to determine a limited region in the PR in the decision space,where the Pareto sets(PSs)are included,and explore this region to assist in solving MMOPs.Furthermore,we develop a novel neighbor distance measure that is more suitable for the complex geometry of PSs in the decision space than the crowding distance.Based on the above methods,we propose a novel dual-population-based coevolutionary algorithm.Experimental studies on three benchmark test suites demonstrates that our proposed methods can achieve promising performance and versatility on different MMOPs.The effectiveness of the proposed neighbor distance has also been justified through comparisons with crowding distance methods.

Zoning Search With Adaptive Resource Allocating Method for Balanced and Imbalanced Multimodal Multi-Objective Optimization

Maintaining population diversity is an important task in the multimodal multi-objective optimization.Although the zoning search(ZS)can improve the diversity in the decision space,assigning the same computational costs to each search subspace may be wasteful when computational resources are limited,especially on imbalanced problems.To alleviate the above-mentioned issue,a zoning search with adaptive resource allocating(ZS-ARA)method is proposed in the current study.In the proposed ZS-ARA,the entire search space is divided into many subspaces to preserve the diversity in the decision space and to reduce the problem complexity.Moreover,the computational resources can be automatically allocated among all the subspaces.The ZS-ARA is compared with seven algorithms on two different types of multimodal multi-objective problems(MMOPs),namely,balanced and imbalanced MMOPs.The results indicate that,similarly to the ZS,the ZS-ARA achieves high performance with the balanced MMOPs.Also,it can greatly assist a“regular”algorithm in improving its performance on the imbalanced MMOPs,and is capable of allocating the limited computational resources dynamically.

Turiyam Set a Fourth Dimension Data Representation

Recently most of the authors focused on dealing with uncertainty. To deal with that algebra of fuzzy sets is introduced in three-way fuzzy space. It gave way to characterize the attributes in true, false or uncertain regions. It is found in chemistry also as an electron, proton and neutron. The problem arises when the attributes are beyond the three-dimensions. One of suitable examples is winning, draw or loss of a match cannot define the win draw or loss of series. It is measured in international relationships also like USA-China, USA-Russia, India-Pakistan or Israel-Philistine. Some countries support the USA, some countries support China, some countries are neutral in hidden consciousness to fulfill their agenda. It is also measured in democratic countries like India where some people vote in favor of a party, some people vote against the given party, some people may be absent or walkout whereas some people vote NOTA. This last dimension is called as Turiyam or fourth dimension. This last dimension is influenced from Yin-Yang theory of Chinese Taoism as well as Sanskrit Yamma-Yammi and its complement. The Turiya means measuring each dimension in silent mode. There is less attention paid towards the necessity of fourth dimensions and its algebra. This paper will discuss gimbal lock of three-way fuzzy space using the Tuiyam set and its algebra.

A Dynamic Resource Allocation Strategy with Reinforcement Learning for Multimodal Multi-objective Optimization

Many isolation approaches, such as zoning search, have been proposed to preserve the diversity in the decision space of multimodal multi-objective optimization(MMO). However, these approaches allocate the same computing resources for subspaces with different difficulties and evolution states. In order to solve this issue, this paper proposes a dynamic resource allocation strategy(DRAS)with reinforcement learning for multimodal multi-objective optimization problems(MMOPs). In DRAS, relative contribution and improvement are utilized to define the aptitude of subspaces, which can capture the potentials of subspaces accurately. Moreover, the reinforcement learning method is used to dynamically allocate computing resources for each subspace. In addition, the proposed DRAS is applied to zoning searches. Experimental results demonstrate that DRAS can effectively assist zoning search in finding more and better distributed equivalent Pareto optimal solutions in the decision space.