Cubic Bipolar Fuzzy VIKOR and ELECTRE-II Algorithms for E ffi cient Freight Transportation in Industry 4.0

: The theory of cubic bipolar fuzzy sets (CBFSs) is a robust approach for dealing with vagueness and bipolarity in real-life circumstances. This theory provides a hybrid machine learning paradigm that can accurately describe two-sided contrasting features for medical diagnosis.


Introduction
Multi criteria group decision making (MCGDM) is an optimization approach in which a group of decision makers (DMs) unanimously choose a best choice from a collection of feasible choices under heterogeneous criterion. However, due to the inadequate data and inherent human judgments, this process entails ambiguous and vague information. The DMs often face difficulties to address two-sided contrasting features of bipolar information. Classical techniques are unable to determine the best option in the presence of vagueness and bipolarity. The abstraction of fuzzy set (FS) was first designed by Zadeh [1] to address vagueness and since been FS theory has been constructively adopted to solve a extensive range of real-world issues. An intuitionistic fuzzy set (IFS) [3,4] is a robust model to assign membership and non-membership grades in decision making problems. Table  1 lists numerous fuzzy set extensions that are helpful for grasping the concept of FS theory and its extension. Zadeh [2] Interval grading IFS Atanassov [3,4] µ, ν ∈ [0, 1], µ + ν ≤ 1 Pythagorean fuzzy set (PFS) Yager [5,6] µ, ν ∈ [0, 1], µ 2 + ν 2 ≤ 1 q-Rung orthopair fuzzy set (q-ROFS) Yager [7] µ, ν ∈ [0, 1], µ q + ν q ≤ 1, q ≥ 1 Bipolar fuzzy set (BFS) Zhang [8,9] Positive grading µ + ∈ [0, 1] and negative grading µ − ∈ [−1, 0] Cubic set (CS) Jun et al. [10] Hybrid model of FS and IVFS. Cubic bipolar fuzzy set (CBFS) Riaz and Tehrim [11] Hybrid model of CS and BFS Many researchers have employed these models successfully in recent decades. All of these models were developed as a response to the imperative need to manage the instability intrinsic in the challenges that arise in the actual world. The discipline of multi-criteria decision making (MCDM) offers a robust approach to the DMs with assistance in the process of determining the appropriate action to take. In addition to this, it guarantees that the appropriate thought is given to two sided conflicting aspects of the problem in hand. The minimal components of MCDM are at least one or more decision makers, two distinct options, and two essential criterion.
A significant amount of research effort has been put into the expansion and refinement of ELECTRE methods, and these approaches have been deployed in a variety of real world applications. Various activities, also known as alternatives, are evaluated based on the suitable qualitative or quantitative scales associated with the criterion. On a more individual level, they frequently offer appraisals that are at odds with one another. The most significant benefit that it provides is the incorporation of the viewpoints of a number of DMs who share their professional competence in the fields that are pertinent to the discussion. As a direct consequence of this, this feature could end up being a more dependable and useful option. The following should be on a short list of traditional MCDM techniques: SIR [12], LAM [13], AHP [14], VIKOR [15], TOPSIS [16], and ELECTRE [17], which is the subject of this study. The ELECTRE strategy is a collection of outranking techniques that works best when there are many competing solutions and competing criteria [18]. In general, algorithms that are based on outranking perform out an inter-comparison of different options in a way that is systematic and is based on each criteria. The concordance and discordance are computed by contrasting the results obtained by evaluating every possible combination of possibilities in terms of a number of different factors. The concordance set illustrates the components (or criteria) that provide credence to the claim that one choice is superior to another. The discordance set, on the other hand, illustrates the feature (or standards) that run counter to the judgement that one option is more desirable than the other. Both the concordance and discordance sets are used to identify the comparison between the proposed relations, which are then included into the production of a more convincing proposition. In the years after the introduction of the ELECTRE strategy, other variations and additional outranking techniques have been developed [19]. Each one relates to details regarding the nature of the central problem. Almost every element of daily life, including the selection of environmentally friendly items, financial management, and power projects, has seen extensive usage of the ELECTRE methods for MCDM. The ELECTRE-I strategy is appropriate when there are many options in the problem because it can establish a partial prioritising by choosing a promising alternatives set. The ELECTRE-I method fails to establish a preference ordering of activities, whereas the ELECTRE-II strategy succeeds by rating the alternatives [20]. Some extensions of ELECTRE technique are expressed in Table 2. Group decision making m-PFL-ELECTRE-I Adeel et al. [25] Analysis of salaries Crisp-ELECTRE-I Benayoun et al. [26] Aircraft problem Fuzzy-ELECTRE-I Sevkli et al. [27] Supplier selection Fuzzy-ELECTRE-I Rouyendegh and Erkan [28] Academic staff selection HF-ELECTRE-I Chen et al. [29] Project management IF-ELECTRE-I Rouyendegh [30] Best plant selection BF-ELECTRE-I Akram et al. [31] Medical diagnosis ELECTRE Wang and Triantaphyllou [32] Ranking irregularities PF-ELECTRE Akram et al. [33] Healthcare management The two forms of integrated outranking links are established by ELECTRE-II by taking into account a set of concordance and discordance threshold values [34]. The graphs of strong and weak outranking show both in a visual way. Numerous ranking issues have been successfully solved using the ELECTRE-II method and its notable variations [35][36][37][38]. IF-ELECTRE II proposed by Victor and Rekha [39] to better understand gender inequality in society. Nimra and Riaz proposed CBF-topological structure, CBF-TOPSIS and CBF-ELECTRE-I to deal with CBF information and applied it decision analysis [40,41]. Riaz and Tehrim [42,43] proposed CBF-AOs for ranking index of alternative in MCGDM problems. Zhan et al. [44] developed PF-TOPSIS and Gwak et al. [45] proposed clustering analysis algorithm. Recently, Farid et al. [46] introduced the abstraction of information aggregation (IA) dynamic decision-making with T-spherical fuzzy sets for hierarchical medical diagnosis.

Motivation
The following summarizes the main facts that guided this research work.
1. In comparison with ELECTRE-I methodology, the ELECTRE-II method generates a preferential ranking of the options available. The ability to provide a set of concordance and discordance  [62] Bibliometric analysis MCDM in shipping industry 4.0 Krstic et al. [63] Comprehensive distance based reverse logistics technologies in ranking (COBRA) method Industry 4.0 Yavuz et al. [64] HFS linguistic model Evaluation of alternative-fuel vehicles Farid and Riaz [65] Prioritized interactive Evaluation of efficient aggregation operators autonomous vehicles Gružauskas et al. [66] Optimization cost effective Minimizing the trade-off performance with autonomous vehicles Gerhátová et al. [67] Bibliometric analysis Implementation industry 4.0 Railway transport Qahtan et al. [68] q-ROF rough sets model Sustainable shipping transportation industry Zhu et al. [69] CO2 emissions future scenario Road freight transportation simulation Callefi et al. [70] A multi-method study Road freight transportation Yilmaz [71] IF-VIKOR method Evaluating industry 4.0 barriers Bravo and Vidal [72] Optimization models Freight transportation function in supply chain barrier values is one of its features, and it allows for two distinct kinds of engrained outranking relations to be implemented (weak and strong outranking relations). These embedded outranking interactions are shown in a clear and concise manner by the weak and strong outranking graphs, which are then used in order to derive decision outcomes. This demonstrates that we have a major edge over our competitors, which we intend to put to good use in this scenario.
2. As problems get more complicated, MCGDM may be able to offer more dependable and persuasive answers. Because it effectively makes use of expert opinions from those who are knowledgeable about the opposing sides of the underlying issue.
3. CBFS increases the space that positive and negative grades are permitted to occupy along with IVFS and FS. For the purposes of MCDA, conflicting viewpoints can be accommodated more effectively.
4. As a result, we draw the conclusion that the ELECTRE-II method combined with CBFS information will result in a MCGDM method that clearly outperforms previous approaches.

Propounded techniques
In this section, we will discuss some propounded MCDM techniques to deal CBF data.

VIKOR method
This subsection details the fundamental framework of CBF-VIKOR, including its essential procedures, formulations, and terminology at the most fundamental level. The acronym VIKOR, which stands for Vlse Kriterijumska Optimizacija Kompromisno Resenje, is a Serbian term that refers to multiple optimization, conflict, and compromise factors. It is assumed that compromise is a suitable method for conflict resolution, the person making the choice appears to seek a solution that is as near to the ideal as is feasible, and the options are evaluated according to all of the signs. When determining the order of the available choices, VIKOR considers whatever compromise gets the closest to achieving the desired end result.
Step 1: Analyze the issue and set the group of DMs, alternatives and criterions.
Step 2: Get the decision matrix for each decision maker opinion for each alternative verse criteria.
Step 3: Generate(Calculate) the weights and normalized it.
Step 4: Aggregate the decision matrices by using formula.
Step 5: Evaluate the PIS and NIS by using formula: Step 6: Evaluate the "group utility" value S i , the "individual regret value" R i , and "compromise value" Q i by making use of listed formulas (3.2)-(3.4).
In order to select a compromise solution by majority vote, the value of the decision mechanism's coefficient K ∈ [0, 1]. The weight of the j th criterion, expressed as w j , indicates its relative importance.
Step 7: Consider your options carefully and come up with a compromise. Make three ranking lists S , R and Q. The alternative f will be deemed the compromise option if it scores highest in Q[.] and simultaneously meets the criteria: The flow chart of CBF-VIKOR algorithm is presented in Figure 1.

ELECTRE-II method
This section entails the formulation and fundamental terms as well as the main structure of CBF-ELECTRE-II.

Basic data
Analyze the issue and set the group of DMs, alternatives and criterions.

CBF-concordance, CBF-disconcordance and CBF-indifference matrices
The argument that ρm is at least as excellent as ρn is used to establish the outranking relation between any two alternatives. Two indices, the concordance index and the disconcordance index, are used in outranking-based approaches. These indices outline the factors that are both in favor of and against an outranking problem.

CBF-concordance sets
The list of criteria for concordance includes subscripts that highlight the alternatives (ρ m , ρ n ) where m, n = 1, 2, ..., 6. The CBF-concordance set is categorized into eight sets (B ψ k ) expressed in Equations (3.5)-(3.12) if P l 1 ≥ P l 2 and P u 1 ≥ P u 2 .

CBF-indifference set
It is possible that both the alternatives ρ m and ρ n will have the same accuracy degree and score degree, i.e. they will be equally indifferent to one another. The CBF-indifference set B = mn is defined as follows in order to represent this difference relation:

CBF-concordance matrix
The concordance indices, denoted by ψ mn in the range of [0,1], are used to form the CBFconcordance matrix. Equation (3.22) is used to calculate the index ψ mn , where η ′ b ∈ [0, 1] are the normalized weights related to the b th criteria.
where ω i are the respective weights assigned to the CBF-concordance sets specified by the experts. The Concondance matrix is given in Table 5. Table 5. Concondance matrix

CBF-disconcordance matrix
The disconcordance indices δ mn ∈ [0, 1] make up the CBF-disconcordance matrix ∆ = (δ mn ) stimess . The indices δ mn express how strongly one choice outranks another (ρ n over ρ m ). In other words, the evaluation of ρ m is poorer than ρ n the greater the value of δ mn . Equation (3.23) is used to get the index δ mn .

Ranking of alternatives
By creating two embedding relations-strong and weak outranking, denoted as O S and O W -the ELECTRE-II approach allows for the preference ordering of alternatives. These outranking were created by combining elemental memberships that were concordant and discordant. The threshold values ψ − , ψ 0 & ψ + represent three strictly rising degrees of concordance, or low, average, and high levels are integers. Additionally, δ 0 & δ + must represent strictly decreasing levels of disconcordance, such as average and low levels. Outranking of alternatives is shown in Table 6.

Exploration of outranking graphs
The two embedded outranking relationships are taken into consideration throughout the ranking process via the ELECTRE-II approach. For the strong outranking connection O S , draw the strong Table 6. Outranking of alternatives outranking graph G S = (V S , E S ), and for the weak outranking relationship OW, draw the weak outranking graph G W = (V W , E W ). The collection of directed arcs between the two alternatives, E S and E W , respectively, indicate the outranking in accordance with the principles listed in Table ??. The first step is to construct a forward ordering (φ ′ ) and a reverse ordering (φ"), with an average ordering (φ) serving as the final ranking. Forward ordering φ ′ Consider the set of vertices V S = ρ 1 , ρ 2 , ..., ρ 6 and T (V) be the subset of V S , the following is a breakdown of the phases involved in the forward ordering process: 1. First, identify the no-precedent and incoming arrow vertices of the strong outranking graph G S .
Put these vertices together into a set represented by H(V).. c Give an alternative ρ k the order v as follows: φ ′ (rho k ) = v∀F (V).

Find the arcs from E W with both endpoints from
d By determining V(v + 1) = V(v) − F (v) and deleting all arcs from graphs G S & G W , coming from the alternatives in sets F (v), the forward-ranked alternatives are eliminated from the system. All the options are rated if V(v + 1) = ∅. Set v = v + 1 and proceed to step 2 if V(v + 1) ∅.

Reverse ordering φ"
The processes involved in reversing the ordering of φ" may be illustrated as: a By flipping the arc directions E S in G S and E W in G W , you may get the mirror image of the outranking relations.
b Utilize the mean of the above-mentioned inverted graphs to derive an ordering, ϖ(ρk), and proceed as described in the preceding φ ′ forward ordering demonstration.

Average ordering φ
Establish the average ordering φ as follows: The flow chart of CBF ELECTRE-II algorithm is shown in Figure 2.

MCGDM application
Road freight transportation (RFT) in Industry 4.0 enables various smart features of business promotion with end-to-end (E2E) visibility, digitization and undoubtedly supply chain operations as well as tracking, control and trustworthy logistics recognition. Nowadays, the growing challenges of sustainable planning and decision management in RFT have put pressure on steakholders of industry 4.0. Managing the cybersecurity and risk factors of autonomous vehicles is more complex in logistics industry because supply chain requires sustainability, accuracy and cost efficiency. The selection of autonomous FRT companies depend on various types of preferences and constraints including, distance, nature of goods, size and volume of goods, flexibility of various modes of transportation, priorities, and cost.
The manufacturing industry and everyday living both rely substantially on the ability to transport things. It produces materials for both industrial and domestic use, making it an indispensable part of modern society. However, autonomous vehicles have a major effect on the environment as a whole. The promotion of ecologically friendly technology, autonomous vehicles facilitated close monitoring of transportation patterns. Artificial Intelligence (AI) and Machine Learning (ML) have promoted and assisted the industry 4.0. Supply chain and logistics are greatly regulated by advances in AI approaches. By optimizing and streamlining an extensive range of business operations, this modern technology helps organizations save both time and money. Building efficient and trustworthy modes AI/ML tools has increased the interest of investors in investing more on FRT.
Overall, defining and optimizing any end-to-end supply chain method from an environmental viewpoint requires attention to major existing concerns. 1. Balance supply and demand according to market need. Supply of needed products as well as not keeping superfluous products for maximum profit of all business partners. 2. Improve quality and reduce pollution. Significantly reduce carbon emissions. 3. Autonomous vehicle network for freight transportation. Smart storage of liquid, gas, solid, and dry products. 4. Sustainable environmental resources. Efficient performance with reliability. 5. Recycling and reuse of waste material. 6. Cost effective performance with stimulate clean energy resources. 7. Adoption of ML/AI tools. E2E visibility with tracking systems. Use of robotics to minimize labor cost.

Numerical example
Consider the problem of capital investment in the ranking of efficient FRT companies. Assume there are six options ℘ 1 , ℘ 2 , ℘ 3 , ℘ 4 , ℘ 5 , ℘ 6 of FRT companies that business experts must evaluate based on four criterion ζ 1 , ζ 2 , ζ 3 , ζ 4 , where ζ 1 = Smart technologies, cloud computing, robotics, networking ζ 2 = E2E visibility, tracking systems, ζ 3 = Automation, cost efficiency, save time and money, ζ 4 = Reliability, sustainability, clean environment, Three DMs are called to put their expert opinions for MCGDM framework. Freight transportation selection criteria is given in Figure 3.  Table 7, decision matrices are expressed in Table 8 and fuzzy values of alternative v/s criterions id listed in Table 9.   Table 8. Decision matrices Table 9. Fuzzy values of alternative v/s criterions

CBF-VIKOR technique
Using formula (4.1) We have PIS and NIS as given in Table 10 and distance between alternatives and positive ideal solution is expressed in Table 11.  Table 11. Distance between alternatives and positive ideal solution Fix κ = 0.5. The utility, regret and compromise values are listed in Table 12. The final ranking is drawn as

Concordance matrix
The Concordance matrices is presented in Table 13. Indifference matrix is given in Table 14.
The concordance membership grades matrix is given in Table 15.

Disconcordance matrix
The disconcordance matrices is expressed in Table 16 and 17.
Strong relation based graph is expressed in Figure 4, strong relation based mirror graph is given in Figure 5, weak relation based graph is expressed in Figure 6 and Weak relation based mirror graph is given in Figure 7.
The final ranking is The optimum choice is ℘ 3 .

Comparison analysis
In this study, we discovered a method of decision-making that combines the CBF model and the outranking ELECTRE-II methodology. A brief comparison of the CBF-ELECTRE-II with CBF-   Table 19. Strong relation    Table 20. Ranking TOPSIS method and CBF-ELECTRE-I method is given in Table 21 and Figure 8 to demonstrate the advantages of the proposed PF-ELECTRE-II.

Insights and limitations of CBF-ELECTRE-II method
The following are the main conclusions and restrictions of the suggested CBF-ELECTRE-II model:   Figure 8. Comparative analysis of ranking by some MCGDM methods 1. A CBFS is a robust model for handling bipolarity and fuzziness (NMGs). To specify the extent to which a given property does not belong, we use an NMG, which is defined by a negative interval and a negative number, and a PMG, which is defined by a positive interval and a positive number, to indicate the degree to which a given property does belong (or satisfaction level of its counter property). In order to expand the decision space available to DM while comparing candidates against predetermined criteria, CBF-ELECTRE-II makes use of the CBF model.

2.
A method for supporting group decisions is CBF-ELECTRE-II that adjusts the collective choice opinions to standard form using criteria related to cost and benefit-type standards, making it appropriate for all sorts of benefits of choice issues.
3. The criteria weights and DMs are derived by the simplest method for collecting opinions.
4. The ranking module of ELECTRE-II iteration processed of exploring outranking graphs is described in the finest and most understandable conceivable way, which enriches the viewing experience.

5.
To create a step-by-step knowledge for problem solving, a diagrammatic model of the group decision supporting system with two stages is offered.
6. When there are a variety of choices to choose from in the challenge, it is not always straightforward to provide the proper three kinds of concordance threshold values and two kinds of discordance threshold values.

Advantages and dominance of the proposed Method:
In the following section, we will examine the benefits and drawbacks of the proposed MCGDM methods. The proposed CBFS MCDGM are more accurate and reliable and cover drawbacks of existing methods.
1. Accuracy and supremacy: For a variety of input data types, the provided MCGDM frameworks are appropriate and applicable. The techniques are capable of managing uncertainties and ambiguities as well as resolving defects in the input data and shows high accuracy comparatively with other methods. As a hybrid structured set, the CBFSs can be used to collect information on a large scale under different criteria against each alternatives.

2.
Managing several criterion with efficiency: The decision support system problems involve several criteria and input data dependent on specific situations. The CBFSs that have been suggested are straightforward and uncomplicated, allowing for their seamless application in any situation involving different alternatives and criteria.
3. Superiority and flexibility. Our algorithms are characterized by its simplicity, flexibility, and superiority over other hybrid fuzzy sets and operators. Its high flexibility allows administrators to conduct comparative analysis at multiple levels, resulting in more optimal solutions. As a result of this study, a systematic approach to selecting the best algorithm from a list of algorithms. Our proposed method is less sensitive to input and output data variations, making it a valuable tool for managers who must deal with high levels of uncertainty and vagueness when evaluating options.

Conclusion
The absence of a ranking in judgments on real-world problems is effectively solved by the ELECTRE-II approach, an expanded version of the original ELECTRE method. When it is difficult to identify a single decision maker with sufficient training to accurately understand the entire problem and its constraints, decisions made using information from a group of DMs generate more dependable results. This study shows how ELECTRE-II and CBFS can be used in tandem when making group decisions. The CBF-ELECTRE-II model is presented in this article to armed with knowledge about CBF, tackle challenges in daily life. In a CBF context, the fundamental architecture of the ELECTRE-II model is defined, along with a detailed step-by-step process that has two fundamental steps: (i) group opinion aggregation, and (ii) ranking mechanism. To verify the efficacy of this method, it is utilized to the issue of RFT problem in industry 4.0. An existing method is then briefly juxtaposed with the choice outcomes. Our method is intended for application in any group decision-making environment, including but not limited to industrial engineering, the health sciences, corporate management, and similar fields. This key contributions of this study are listed below.
1. Modeling uncertain and sensitive information with cubic bipolar fuzzy hybrid model.

It describes benefits of utilizing CBFSs.
3. This investigated several significant concepts related to CBFSs.
4. An application of RFT in industry 4.0 is presented that leads to new MCGDM methods to seek reasonable decisions in a timely manner.
5. Such application enables organizations to quickly access required information and respond to inquiries or concerns promptly.
6. It also helps organizations to efficiently manage large amounts of data and information.

Conflicts of Interest:
The authors declare that they have no conflicts of interest.