VNS Based MADM-Strategy Under Possibility Environment

Annals of Data Science https://doi.org/10.1007/s40745-022-00419-3 VNS Based MADM-Strategy Under Possibility Environment Bimal Shil1 · Prasenjit Sinha1 · Binod Chandra Tripathy2 · Suman Das2 Received: 18 January 2022 / Revised: 14 May 2022 / Accepted: 26 May 2022 © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2022 Abstract In this paper, we propose a Variable Neighborhood Search (VNS) algorithm based on Multi-AttributeDecision-Making (MADM) strategy under possibility environment. Further, we provide a numerical example to show the applicability and rationality of the proposed MADM strategy. Keywords MADM-Strategy · Variable neighborhood search · Possibility mean · Fuzzy set Mathematics Subject Classification 03E72 · 40A05 · 40F05 · 40G15 · 60B10 · 60A86 · 60E05 1 Introduction In 1965, Zadeh [1] grounded the notion of fuzzy set (FS) theory to deal with uncertainty events in the real world. Further, it has an ingressive amount of study for a different aspect. Interestingly, when felt that probability measure was unable to represent all fact of uncertainty theory, then possibility theory come into the image by Zadeh [2] in 1978. Thereafter, Dubois and Prade [3] introduced qualitative and quantitative approach to possibility theory in 1988. Kovalerchuk [4] in 2017 introduced the relationships between probability and possibility theories. B Bimal Shil Prasenjit Sinha Binod Chandra Tripathy ; Suman Das 1 Department of Statistics, Tripura University, Agartala, Tripura 799022, India 2 Department of Mathematics, Tripura University, Agartala, Tripura 799022, India 123 Annals of Data Science In 1997, Mladenovic and Hansen [5] proposed the Variable Neighborhood Search (VNS) algorithm, which is a framework for building heuristics based upon systematic changes of neighborhoods both in a descent phase, to find a local minimum, and in a perturbation phase, to escape from the corresponding valley. VNS algorithm represents a flexible framework for building heuristics for approximately solving combinatorial and non-linear continuous optimization problems. VNS search is the systematic change of neighborhood within a possible randomized local search algorithm that yields a simple and effective metaheuristic for combinatorial and global optimization. Contrary to the other metaheuristic based on local search methods. Rather than following a path, VNS explores more distant neighborhoods of the present solution, jumping from it to a new one and when an improvement is made. In this method, the solution’s beneficial qualities (e.g., many variables are already at their optimal value) are frequently preserved and exploited to find interesting surrounding solutions. Furthermore, to get from these adjacent solutions to local optima, a local search routine is used continuously. The variable neighborhood search algorithm is the stepwise change of neighborhood within the possible random variable. By using a distinct neighborhood sample as the value of the proposed function, it will move on to the next neighbour only when the value of the proposed function is slightly better than the first (or existing objective function) neighbour sample. Hansen and Mladenovic published Variable Neighborhood Search: Principles and Applications [6] in 2001. Hansen et al. investigated variable neighborhood search: methods and applications [7]. A VNS-algorithm heuristic has two parts: an improvement phase for potentially improving a given solution and a shaking phase for perhaps resolving local minima entrapment. The improvement phase and shaking method, as well as the neighborhood change step, are alternated until a predetermined stopping threshold is reached. VNS algorithm has successfully been applied in the field of design of experiment by finding the optimum allocation of experimental units with predictors into two treatment groups by Dash and Hore [8]. Later on, Hore [9] studied the VNS-algorithm to achieve an optimal allocation design for known covariates. At first, the concept of MAMD (Multiple Attribute Decision Making) was introduced by Hwang and Yoon [10] in their study of multiple attribute decision making methods and applications in 1981. In the field of fuzzy set theory, the MAMD was introduced by Chen, and Hwang [11] in 1992 on fuzzy multiple attribute decision making: methods and applications. After that, many authors around the globe contributed their work in this field [12–16]. Application of Neutrosophic Similarity Measures in Covid-19 [17] published by Das et al., and Separation Axioms on Spatial Topological Space and Spatial Data Analysis [18] studied by Das et al. This twenty-first century is called the era of information. Big data refers to datasets that are not only large, but also diverse and rapidly changing, making standard tools and procedures ineffective. Huge volumes of data have become available to decision makers in the digital age. Due to the increasing rise of such data, methods to handle and extract value and information from these datasets must be investigated and given. However, decision-makers must be able to extract useful information from a wide range of constantly changing data, including everyday transactions, customer experience, and social network data. Big data analytics, which is the application of advanced analytics techniques to large amounts of data, can give such value. An introduction to 123 Annals of Data Science business data mining [19] has been widely studied by Olson and Shi. The notion of Optimization based data mining [20] was introduced by Shi et al. Internet of things, real-time decision making and artificial intelligence [21] analyzed by Tien in the year 2017. Afterward, Advances in Big Data Analytics [22] were studied by Shi in 2022. In this paper, we introduce the concept of discrete possibility mean and variance. Then, in the possibility environment, we offer a MADM-strategy based on VNS. We also provide a numerical example to demonstrate the applicability and logic of our suggested MADM method. The remaining part of this article has been split into the following sections: In Sect. 2, we present some existing definitions and results that are relevant to the main results of this article. In Sect. 3, we introduce the notion of discrete possibility mean and variance under the possibility environment. In Sect. 4, we propose an MADM strategy based on VNS algorithm under the possibility environment. Section 5 deals with the validation of the proposed MADM strategy. In Sect. 6, a comparative study has been conducted to validate the results obtained from the proposed MADM strategy. Finally, in Sect. 7, wrap up the work presented in this article. 2 Preliminaries and Definitions In this section, we present some definitions and results those are relevant to the main results of this article. Definition 2.1. [2] Assume that Ẅ be a fixed set. Then N, a fuzzy set over Ẅ is defined as N = {(g, T N ( (...truncated)