Multi-criteria Group Decision-making Problems Research Articles

Comparative study between GRA and MEREC technique on an agricultural-based MCGDM problem in pentagonal neutrosophic environment.

In this research article, an improved Multi-criteria group decision-making (MCGDM) strategy has been developed in pentagonal neutrosophic environment incorporating grey relational analysis and method on the removal effects of criteria (MEREC) techniques to address the relative advantages and disadvantages of these aspects in MCGDM. The aim of the study is to improve MCGDM technique which can capture the underlying uncertainties in robust way and can produce consistent results in a more rigorous way. Here, the conception of Hamming distance between two pentagonal neutrosophic number (PNN)s is introduced and the weighted arithmetic and geometric averaging operators in PNN arena are deployed to craft our computational technique more progressive and robust. An agriculture-based numerical problem is illustrated to demonstrate the ranking results of the alternatives by both of the techniques. After evaluating the problem by two aggregation operators, it is found that "plantation crop" is the best alternative under certain circumstances. Lastly, the sensitivity investigation is performed which reveals that withthe appliance of arithmetic and geometric aggregation operators the best ranked alternative preserves its position by both of the ranking methods, which definitely exhibit the consistency and robustness of our executed methodology.

An approach to hesitant fuzzy linguistic multiple criteria group decision making with uncertain criteria weights considering incomparability between alternatives

In multi-criteria group decision-making (MCGDM) problems, the consensus of experts is taken as the result, and it is common that the result may include incomparability between alternatives. The longest ranking of alternatives that attains consensus and allows the incomparability between alternatives is called a maximum consensus sequence (MCS). In decision-making problems related to emerging industry, it is hard to accurately obtain criteria weights. In this case, how to weight different criteria values and individual opinions to mine an MCS is worth studying. This article applies the hesitant fuzzy linguistic ORESTE method to obtain the ranking of alternatives, allowing for uncertain criteria weights and incomparability between alternatives. Then, a compromise-or-not mechanism based on uncertain criteria weights, which can derive experts’ attitudes towards collective result, is developed to tackle the consensus issue. Two kinds of conflicting items about the incomparability in the collective result and divergent opinions of uncompromising experts are identified. Through information exchange among experts regarding the conflicting items, an MCS can be obtained. The proposed method provides an inspiration for MCGDM in emerging industries.

A new social network driven consensus reaching process for multi-criteria group decision making with probabilistic linguistic information

Consensus reaching process (CRP) is very important for multi-criteria group decision making (MCGDM). Recently, social network driven CRP methods have been deeply investigated. However, these methods rarely considered knowledge levels of DMs. In fact, knowledge levels of DMs can decide the correctness of final decision result which play an important role in CRP. Therefore, this paper proposes knowledge degree to measure knowledge level of DM and develops a new social network driven CRP method for probabilistic linguistic MCGDM problems. Based on the similarity degree and knowledge degree, similarity degree-based social network (SDSN) and trust relationship-based social network (TRSN) are constructed successively. According to the constructed SDSN and TRSN, the proposed CRP method consists of two stages. In the first stage, pair-wise DMs in complementary social network adjust their evaluations. In the second stage, the evaluations of DMs who contribute less to group consensus are modified. To ensure that the group consensus index (GCI) is non-decreasing in the above two stages, the constructed programming models not only improve similarity degrees between DMs-to-adjust and reference DMs but also consider similarity degrees between DMs-to-adjust and other DMs. Finally, an investment selection example and comparison analyses demonstrate the applicability and advantages of the proposed method.

Linguistic Single-Valued Neutrosophic Multi-Criteria Group Decision Making Based on Personalized Individual Semantics and Consensus

In practical linguistic multi-criteria group decision-making (MCGDM) problems, words may indicate different meanings for various decision makers (DMs), and a high level of group consensus indicates that most of the group members are satisfied with the final solution. This study aims at developing a novel framework that considers the personalized individual semantics (PISs) and group consensus of DMs to tackle linguistic single-valued neutrosophic MCGDM problems. First, a novel discrimination measure for linguistic single-valued neutrosophic numbers (LSVNNs) is proposed, based on which a discrimination-based optimization model is built to assign personalized numerical scales (PNSs). Second, an extended consensus-based optimization model is constructed to identify the weights of DMs considering the group consensus. Then, the overall evaluations of all the alternatives are obtained based on the LSVNN aggregation operator to identify the ranking of alternatives. Finally, the results of the illustrative example, sensitivity and comparative analysis are presented to verify the feasibility and effectiveness of the proposed method.

An Intuitionistic Fuzzy Consensus WASPAS Method for Assessment of Open-Source Software Learning Management Systems

Ineffective evaluation of open-source software learning management system (OSS-LMS) packages can negatively impact organizational effectiveness. Clients may struggle to select the best OSS-LMS package from a wide range of options, leading to a complex multi-criteria group decision-making (MCGDM) problem. This evaluates OSS-LMS packages based on several criteria like usability, functionality, e-learning standards, reliability, activity tracking, course development, assessment, backup and recovery, error reporting, efficiency, operating system compatibility, computer-managed instruction, authentication, authorization, troubleshooting, maintenance, upgrading, and scalability. Handling uncertain data is a vital aspect of OSS-LMS package evaluation. To tackle MCGDM issues, this study presents a consensus weighted sum product (c-WASPAS) method which is applied to an educational OSS-LMS package selection problem to evaluate four OSS-LMS packages, namely ATutor, eFront, Moodle, and Sakai. The findings indicate that the priority order of alternatives is Moodle > Sakai > eFront > ATutor and, therefore, MOODLE is the best OSS-LMS package for the case study. A sensitivity analysis of criteria weights is also conducted, as well as a comparative study, to demonstrate the effectiveness of the proposed method. It is essential to note that proper OSS-LMS package evaluation is crucial to avoid negative impacts on organizational performance. By addressing MCGDM issues and dealing with uncertain information, the c-WASPAS method presented in this study can assist clients in selecting the most appropriate OSS-LMS package from multiple alternatives. The findings of this study can benefit educational institutions and other organizations that rely on OSS-LMS packages to run their operations.

Complex spherical fuzzy TOPSIS based on entropy

Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is a traditional approach, which is quite powerful since it shows the best and worst alternatives at the same time, for solving multi-criteria group decision-making (MCGDM) problems. On the other hand, the theory of complex spherical fuzzy set (CSFS) has a noticeable advantage over the other set theories which process the uncertain data since this theory includes the abstinence part of uncertain information in addition to satisfaction and dissatisfaction in two-dimensional data. This paper proposes to merge the feasible and remarkable structure of the CSFS with the excellent tendencies of the traditional TOPSIS where the information about the weights of both the decision-makers (DMs) and criteria are completely unknown. So, we establish a novel TOPSIS method under the complex spherical fuzzy environment by calculating the weights of both the DMs and criteria objectively with the novel entropy measure function that we constructed for this method. Furthermore, we give a numerical example to explain the proposed method step by step. Finally, we compare the solutions to the current problem with the various existing MCGDM methods to provide the capabilities and validity of the presented method. We also give a sensitivity analysis by changing the entropy to demonstrate the robustness of the weights of the criteria under the prevailing entropy measure function.

Spherical Fuzzy Choquet Integral-Based VIKOR Method for Multi-Criteria Group Decision-Making Problems

Inter-dependency among the decision criteria and difficulty of handling the information in form of “yes,” “abstain,” “no,” and “refusal” are two important issues to be addressed in multicriteria group decision-making (MCGDM) problems in the environment of uncertainty. Recently, spherical fuzzy set (SFS) has gained the attention of the researchers for MCGDM problems due to their capability of handling decision makers’ preferences in the form “yes,” “abstain,” “no,” and “refusal” more efficiently than picture fuzzy set. Choquet integral operator has an edge over traditional aggregation operator in the modeling of interaction among the preferences of decision makers and decision criteria and therefore, we define Choquet integral operator for SFS. The main objective of this study is to define the spherical fuzzy Choquet integral (SFCI) operator and to extend the VIsekriterijumska optimizacija I Kmpromisno Resenje (VIKOR) for spherical fuzzy environment. In this study we define spherical fuzzy Choquet average, spherical fuzzy Choquet geometric and spherical fuzzy Choquet integral distance (SFCID) operators and propose SFCI-based VIKOR method for MCGDM problems. Proposed SFCI-based VIKOR method is applied on warehouse selection problems and results are compared with existing MCGDM methods.

Multicriteria group decision making for COVID-19 testing facility based on picture cubic fuzzy aggregation information.

The information aggregation of cubic fuzzy numbers and picture fuzzy numbers have played an important role in decision making. This paper introduces a novel approach to address the problem of testing facility of COVID-19 under picture cubic fuzzy environment. As the picture cubic fuzzy set is a generalized fuzzy structure to handle more uncertainty and ambiguity in decision making problems We discuss its various properties. Based on geometric aggregation operators and Hamacher operations, we introduce some Hamacher geometric aggregation operators under picture cubic fuzzy information. Namely, picture cubic fuzzy Hamacher weighted geometric aggregation operator, picture cubic fuzzy Hamacher hybrid geometric operator, picture cubic fuzzy Hamacher order weighted geometric aggregation operator. Discuss some properties of the defined operators. To verify the importance of the proposed operators, develop multicriteria group decision making (MCGDM) algorithm under picture cubic fuzzy environment and apply this strategy for the selection of an authentic laboratory for COVID-19 test. Further to validate the supremacy of our proposed operators, we present a comparative analysis with pre-existing aggregation operators. Results show that the proposed technique is more effective and suitable for MCGDM problems.

Sustainable supply chain partner selection and order allocation: A hybrid fuzzy PL-TODIM based MCGDM approach

Sustainability, as a trend of social development and the embodiment of corporate social responsibility, has begun to receive more attention. To achieve this goal, sustainable supplier selection (SSS) and order allocation (OA) are seen as the crucial activities in corporate management. In the process of SSS, the psychological behavior of decision-makers (DMs) could play a critical role in the evaluation results. Therefore, introducing it into the decision-making process may lead to decision in line with the actual situation. In the uncertain multi-criteria group decision-making (MCGDM) problem described by probability linguistic term sets (PLTS), the DMs can evaluate the criteria of each supplier based on his own preference and hesitation, which is useful to avoid the loss of information. For this reason, this study develops a novel multi-criteria group decision-making combined with fuzzy multi-objective optimization (MCGDM-FMOO) model for SSS/OA problems by considering the triple bottom line (TBL) in which includes economic, environmental and social factors. The proposed method includes four stages. (1) the best-worst method (BWM) and entropy weight method are utilized to assign the weights of criteria to obtain the comprehensive weight. According to the output weights, the an acronym for interactive and multi-criteria decision-making in Portugese (TODIM) approach is applied to rank the suppliers under PLTS environment; (2) a FMOO model that can effectively deal with uncertainties and dynamic nature of parameter is formulated for allocating optimal order quantities; (3) two novel approaches are utilized to solve the FMOO model in order to obtain the richer Pareto frontier; and (4) the final OA solution is achieved by technique for order preference by similarity to ideal solution (TOPSIS) method. Finally, the validity and practicability of proposed MCGDM-FMOO model are verified by an example and comparative analysis with other classical MCGDM methods.

A novel consensus reaching method for MCGDM in social network with intuitionistic grey linguistic numbers

PurposeThe existing consensus reaching mechanisms ignore the influence of social triangle structure on the decision-makers’ (DMs') weights, and the consensus reaching process (CRP) fails to fully reflect the DMs' subjectivity and can be time consuming and costly. To solve these issues, a novel CRP for multi-criteria group decision-making (MCGDM) problems with intuitionistic grey linguistic numbers (IGLNs) is proposed in this paper.Design/methodology/approachFirst, a weight calculation method is proposed by analysing the triangle structure of DMs' social network and scale of adjacent nodes. Then, a consensus degree index based on three-level polygon area is defined and applied to identify the inconsistent DMs. Finally, the feedback mechanism based on particle swarm optimisation (PSO) algorithm under grey linguistic environment is developed, where subjective trust relationships in social network is utilised to determine the adjustment coefficient.FindingsThe advantages of the proposed method are highlighted by two practical applications of the evaluation of tunnel construction method and the selection of a hotel for the centralised isolation. Comparision analysis and numerical simulation are performed to reveal the effectiveness and applicability of the method.Practical implicationsThe proposed model can not only reflect the effect of triangle structure in social network on DMs' weights, but also reduce the time and cost of decision-making.Originality/valueThe main contribution of this paper is to propose a new MCGDM model based on intuitionistic grey linguistic numbers, which can handle the problem of inconsistency of information more effectively.

Consensus reaching with dynamic expert credibility under Dempster-Shafer theory

Consensus building is crucial in multi-criteria group decision making (MCGDM) because a consensual decision outcome is beneficial for implementing the result. However, expert credibility and its dynamic variation are rarely considered in the consensus reaching process (CRP) for MCGDM problems. In reality, experts are not completely reliable, and their credibility changes dynamically during the CRP. Therefore, this paper proposes a novel CRP method based on dynamic expert credibility for GDM under Dempster-Shafer theory (DST). First, expert credibility is defined by measuring the extrinsic discrepancy and intrinsic ambiguity of decision information to quantify experts’ abilities to provide high-quality evaluations. Then, a novel consensus measure is proposed based on the inverse-square law to check whether the current consensus level satisfies the designated threshold. If not, a new feedback mechanism considering dynamic expert credibility is activated to promote group consensus. Furthermore, the attribute weights are comprehensively determined from both objective and subjective perspectives. Finally, an illustrative example and comparative analysis are presented to validate the proposed method.

A novel hybrid multi-criteria group decision-making approach with intuitionistic fuzzy sets to design reverse supply chains for COVID-19 medical waste recycling channels

The COVID-19 pandemic has led to exponential growth in COVID-19 medical waste (CMW) generation worldwide. This tremendous growth in CMW is a major transmission medium for COVID-19 virus and thus brings serious challenges to medical waste (MW) management. Designing an efficient and reliable CMW reverse supply chain in this situation can help to prevent epidemic spread. Nowadays, the assessment of CMW recycling channels has become a challenging mission for health-care institutions, especially in developing countries. It can be seen as a complex multi-criteria group decision-making (MCGDM) problem that requires the consideration of multiple conflicting tangible and intangible criteria. Nevertheless, few academics have been concerned about this issue. Moreover, current MCGDM methods have limited support for CMW recycling channel evaluation and they do not consider hospitals’ reverse supply chain strategy when evaluating. Thus, this study presents a novel MCGDM approach based on intuitionistic fuzzy sets (IFSs) and the VIKOR method for assessing the capacity of CWM recycling channels. According to the characteristics of CMW, processing flow and the TOE (Technology, Organization and Environment) theoretical framework, we established a new CMW recycling channel capacity evaluation index system which makes our proposed method more targeted and efficient. In the decision-making process, we integrate the best-worst method (BWM) and entropy to determine the decision makers (DMs) weighting in a more comprehensive way, considering both subjective and objective criteria, which was ignored by many MCGDM methods. A new aggregation operator called IFWA is proposed by us, considering the priority of DMs. Based on both the ranking of capacity and disposal charges, we then position the alternatives in the recycling channel priority index (RCPI) matrix constructed by us. According to this PCPI matrix and the reverse supply chain strategy of hospitals, a more reasonable CMW allocation strategy is determined and a more efficient CMW reverse supply chain is designed. Finally, a real case study from Wuhan was examined to illustrate the validation of our approach.

Type-2 fuzzy envelope of extended hesitant fuzzy linguistic term set: Application to multi-criteria group decision making

Multi-criteria group decision making (MCGDM) is a common activity in human-beings’ daily life. In real-world problems is common that decision makers prefer to use linguistic values instead of numerical values to express their opinions/preferences on the alternatives. Solving MCGDM problems under linguistic environment suffers difficulties such as loss of information during the computing with words (CWW) processes. Existing fuzzy encoding technology is still limited to single word, thus it is difficult to apply previous fuzzy encoding methodologies to carry out CWW processes when more complex linguistic information appears in MCGDM. To overcome these limitations, a novel MCGDM method has been proposed, in which extended hesitant fuzzy linguistic term sets (EHFLTS) is applied to decrease information loss in preference aggregation process. Novel type-1 and type-2 fuzzy envelopes of EHFLTS are proposed to decrease information loss in CWW processes. A new context-free grammar and the notion of rough comparative linguistic expression (RCLE) are introduced to supplement the computation with fuzzy envelopes of EHFLTS, and to increase the flexibility of preference elicitation. The proposed fuzzy encoding models and syntax extend the fuzzy perceptual computing technology from single words to EHFLTS and RCLE. Finally, the proposed MCGDM method which adopts fuzzy envelopes of EHFLTS is applied in a cross-border e-commerce selection situation.

Development of q-Rung Orthopair Trapezoidal Fuzzy Einstein Aggregation Operators and Their Application in MCGDM Problems

Compared to previous extensions, the q-rung orthopair fuzzy sets are superior to intuitionistic ones and Pythagorean ones because they allow decision-makers to use a more extensive domain to present judgment arguments. The purpose of this study is to explore the multicriteria group decision-making (MCGDM) problem with the q-rung orthopair trapezoidal fuzzy (q-ROTrF) context by employing Einstein t-conorms and t-norms. Firstly, some arithmetical operations for q-ROTrF numbers, such as Einstein-based sum, product, scalar multiplication, and exponentiation, are introduced based on Einstein t-conorms and t-norms. Then, Einstein operations-based averaging and geometric aggregation operators (AOs), viz., q-ROTrF Einstein weighted averaging and weighted geometric operators, are developed. Further, some prominent characteristics of the suggested operators are investigated. Then, based on defined AOs, a MCGDM model with q-ROTrF numbers is developed. In accordance with the proposed operators and the developed model, two numerical examples are illustrated. The impacts of the rung parameter on decision results are also analyzed in detail to reflect the suitability and supremacy of the developed approach.

Fermatean fuzzy ELECTRE multi-criteria group decision-making and most suitable biomedical material selection

ELECTRE is a family of multi-criteria decision analysis techniques, which has the ability to provide as much as possible precise and suitable set of actions or alternatives to the underlying problem by eliminating the alternatives, which are outranked by others. Group decision-making is an effective process to provide the most appropriate solution to real-world decision-making scenarios by considering and merging the expert opinions of multiple individuals on the problem. The aim of this study is to present an extended version of the ELECTRE I model called the Fermatean fuzzy ELECTRE I method for of multi-criteria group decision-making with Fermatean fuzzy human assessments. The method proposed in this study has the possibility to solve multi-criteria group decision-making problems by using the Fermatean fuzzy decision matrix obtained in Fermatean fuzzy number form in the evaluations made with the available alternatives based on expert opinions. First, the mathematical description of the multi-criteria group decision-making problem with Fermatean fuzzy information has been given. Then, the proposed Fermatean fuzzy ELECTRE I method to deal with the problem has been presented. After the determination of the relative importance degree of experts, the Fermatean fuzzy aggregated averaging operator is employed to merge the individual Fermatean fuzzy decision matrices produced by the experts into the aggregated Fermatean fuzzy decision matrix. Next, for pairwise comparison of available alternatives with respect to considered criteria, the concepts of Fermatean fuzzy strong, midrange, and weak concordance and discordance sets are based on the approach of score function and accuracy function defined for Fermatean fuzzy numbers. Afterward, Fermatean fuzzy concordance and discordance matrices are defined, constructed by concordance and discordance indices. Finally, Fermatean fuzzy effective concordance and discordance matrices are computed to obtain Fermatean fuzzy aggregated outranking matrix, indicating abstract information on dominations of suitable alternatives to the others. The proposed method will be used in material selection in distinct implementations, exclusively in biomedical applications where the prosthesis materials should have similar characteristics to human tissues. Since biomedical materials are used in various parts of the human body for many different purposes, in this study, material selection will be made using the method presented for the femoral component of the hip joint prosthesis for orthopedists and practitioners who will choose biomaterials.

Novel Distance Measures for Single-Valued Neutrosophic Fuzzy Sets and Their Applications to Multicriteria Group Decision-Making Problem

Single-valued neutrosophic sets are a hybrid of fuzzy sets that are used to represent uncertain, imprecise, partial, and inconsistent information in the actual world. The focus of this research is to develop two novel distance measures for single-valued neutrosophic fuzzy sets (SVNFSs). We introduced two new distance measures nameddηG,YanddζG,Yfor SVNFSs and apply these measures to different examples and also compare them with existing measures to show the validity of our proposed measures. Our results are reliable and useful for decision-making problems. We also proposed an algorithm for multicriteria group decision-making. Based on this algorithm, we find the ranking matrices using proposed distance measures. We also give an example to demonstrate the notion and concept of our algorithm.

Novel entropy and distance measures for interval-valued intuitionistic fuzzy sets with application in multi-criteria group decision-making

Entropy and distance are the most important information-theoretic measures. These measures have found useful applications in different areas. In the present communication, we study the entropy and distance measures under an interval-valued intuitionistic fuzzy (IVIF) environment using an exponential function. First, it presents the novel exponential entropy and distance measures for interval-valued intuitionistic fuzzy sets (IVIFSs) with proof of their authenticity. A method is offered to solve multi-criteria group decision-making (MCGDM) problems in the IVIF environment based on the weighted exponential entropy measure. The performance of the proposed IVIF MCGDM method is shown by taking two case studies. Second, the success and strength of the proposed IVIF distance measures are demonstrated by comparing them with the existing ones. Further, the paper advances an approach to solve multi-attribute decision-making problems under the IVIF environment. Finally, it considers a real-world example to illustrate the applicability and authenticity of the proposed approach. In doing so, the proposed approach is compared with existing methods to exhibit its advantages. Thus, the proposed IVIF information measures and multi-criteria group decision-making method are more suitable to solve real-life decision-making problems.

Multicriteria q-Rung orthopair fuzzy decision analysis: a novel approach based on Archimedean aggregation operators with the confidence levels

The confidence levels can reduce the influence of the unreasonable evaluation value that was given by the decision-maker on the decision-making results. The Archimedean t-norm and t-conorm (ATS) also have many advantages for the processing of uncertain data. Under this environment, the confidence q-rung orthopair fuzzy aggregation operator based on ATS is one of the most successful extensions of confidence q-rung orthopair fuzzy numbers in which we decrease the deviation caused by the subjective perspective of the decision-maker in the multicriteria group decision-making problems. In this paper, we propose weighted, ordered weighted averaging aggregation operators and weighted, ordered weighted geometric aggregation operators based on ATS, respectively. Moreover, the properties and four specific forms associated with aggregation operators are also investigated. In this study, a novel MCGDM approach is introduced by using the proposed operator. A reasonable example is proposed and compared the results that are obtained by our operators and that in the existing literature, so as to verify the rationality and flexible of our method. From the study, we concluded that the proposed method can reduce the impact of extreme data, and make decision-making results more reasonable by considering the attitudes of decision-makers.

An extended VIKOR method based on particle swarm optimization and novel operations of probabilistic linguistic term sets for multicriteria group decision‐making problem

This paper proposes an extended VlseKriterijuska Optimizacija I Komoromisno Resenje (VIKOR) method based on the particle swarm optimization (PSO) algorithm for solving multicriteria group decision-making problems with probabilistic linguistic information. First, we define the novel operations of probabilistic linguistic term sets and then prove the corresponding properties. Second, we apply a modified PSO algorithm to the consensus reaching process to improve the collective consensus level. In the context of probabilistic linguistic information, each participant can be recognized as a particle moving towards the best position. The consensus level can be regarded as the objective function that is used to construct the fitness function. In the update function, the trust relationship and the similarity measure between experts are exploited to determine the adjustment coefficient. The new consensus model based on PSO can ensure that the ultimate evaluation achieves a high level of consensus. Afterward, we propose the extended VIKOR method to obtain the optimal solution, which not only avoids the loss of decision information, but also considers the separation of each alternative from the positive ideal solution and the negative ideal solution when criteria are interactive. The advantages of the proposed method are highlighted through a numerical example. Finally, we perform a comparative analysis and a sensitivity analysis to reveal the effectiveness and applicability of the method.

A New Idea to Evaluate Networking Problem and MCGDM Problem in Parametric Interval Valued Pythagorean Arena

In this article, the concept of parametric interval valued Pythagorean number (PIVPN) has been introduced, which is an extended version of Pythagorean number (PN). Here, a new score and accuracy function have been innovated in the PIVPN environment along with the De‐Pythagorean value concept. The new tool and techniques have been fruitfully applied to two realistic problems, namely the networking critical path model (CPM) problem and the multicriteria group decision making problem (MCGDM) problem. In order to solve the MCGDM problem, we have prepared Parametric Interval valued Pythagorean Weighted Arithmetic Mean Operator (PIVPWAMO) and Parametric Interval valued Pythagorean Weighted Geometric Mean Operator (PIVPWGMO) operator in PIVPN environment. Finally, sensitivity analysis and industrious comprehensive numerical simulations have been performed to identify the reliability, efficiency, and usefulness of this novel work. In this article, we have shown that PIVPNs are a more well‐organized representation to grip a real‐life problem, and they can handle inconsistent conditions in a better compatible way in comparison to the other existing methods.