Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 607 https://internationalpubls.com Hybrid Minkowski-Based Fuzzy AHP with MCDM Optimization for Green Supply Chain Transportation Hasim Khan Department of Mathematics, College of Science, Jazan University, P.O. Box 114 Jazan, 45142, Kingdom of Saudi Arabia hhanif@jazanu.edu.sa Article History: Received: 02-10-2024 Revised: 30-11-2024 Accepted: 08-12-2024 Abstract: In this research, a Hybrid Minkowski Consistency Operator is introduced for the application of Green Supply Chain (GSC) transportation problem based on Fuzzy Analytic Hierarchy Process (AHP)-Multi Criteria Decision Making (MCDM) techniques. Consequently, the model primarily focuses on enhancing efficiency in transportation processes within green supply chain logistics while minimizing its environmental impact. This is enabled by taking into account these multiple criteria often in conflict with one another: cost, emission, and delivery time. It combines fuzzy logic with the Hybrid Minkowski Consistency Operator for managing uncertainties that may usually come up in a real-world supply chain system, hence making the decisions more precise. This research demonstrates the proposed model's potential to produce optimal and sustainable logistics solutions, a real-case study is employed for validation. In addition, besides this optimization of transport, the paper discusses a hybrid multi- objective optimization methodology that involves Ant Colony Optimization (ANT) and Firefly Algorithm (FA). This hybrid optimization enhances overall performance in this model and provides an overall framework whereby all existing challenges are solved and a foundation for future developments is laid regarding green supply chain management. Keywords: Hybrid Minkowski Consistency, Fuzzy AHP, Green Supply Chain Management (GSCM), Transportation Problem, MCDM 1. Introduction Reducing environmental effects without sacrificing operational efficiency has been a primary goal for firms in today's globalized and ecological marketplace, according to GSCM [1]. The most noteworthy revelations on the intricacies of supply networks concerned transportation, which encompasses Delivery time, carbon emission, and multi-objective cost [2]. There is unprecedented pressure on decision-makers due to ever-tighter surroundings and sustainability goals, which necessitates finding strategies to balance conflicting agendas [3]. A general term for a collection of methods that identify the best course of action when it is difficult to evaluate options by several intricate and contradictory criteria is MCDM [4]. Since the 1960s, MCDM has become a more recognized branch of study that works with the development of mathematical and computational tools to help decision-makers select the optimal course of action in complicated situations [5]. The use of MCDM approaches increased rapidly after the 1990s. Many MCDM techniques have been developed over time; some examples are as follows: The techniques that have been employed are as follows: the PROMETHEE, the BWM, the TOPSIS, ELECTRE, the AHP, SAW, the DEMATEL, the ANP, the LINMAP [6], [7]. Green transportation planning involves inherent conflicts and uncertainties, which are effectively managed by combining fuzzy logic and mailto:hhanif@jazanu.edu.sa Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 608 https://internationalpubls.com MCDM methodologies [8]. One might list AHP as one of the most useful MCDM approaches. However conventional AHP lacks the adaptability needed to capture uncertainty in real-world issues [9]. In line with this, fuzzy AHP has emerged as a reliable method for incorporating imprecise data into decision-making procedures [10]. Thomas L. Saaty developed the AHP model, which is frequently used for rating, analyzing, prioritizing, and reviewing decision possibilities [11]. As a result, the AHP technique also divides problems into hierarchies according to the evaluations of those in charge. The challenges of the task are indicated by the number of tiers in a hierarchy [12]. Since Zadeh's explanation of OFS, they have been extensively employed in practically all study domains. Extensions to conventional fuzzy sets, such as HFS, IFS, T2FS, PFS, and NS, have been found in several investigations [13]. Pythagorean and Neutrosophic fuzzy sets combine to create three-dimensional SFS. AHP-based SFSs have been used in a few studies. It is suggested that green supply chain transportation issues be resolved using the Hybrid Distance-Based Aggregation Operator with Fuzzy AHP and MCDM Optimization that is provided here [14]. Fuzzy logic is combined with a hybrid distance-based technique to capture the uncertainties and interdependencies across parameters like cost, environmental impact, and delivery performance. This approach hasn't been matched up till now. When various criteria are taken into account throughout the process of making decisions, an optimal model is produced, which enables the creation of a suggested model to show the solution modeling of green transportation planning robustly and flexibly [15]. The remainder of the article will address the methodology, use of case studies, and possible drawbacks of this hybrid approach to provide a reliable and effective framework for transportation optimization in green supply chains. The following is the paper's contribution. 1) In this research, we theoretically suggested a new consistency formula in the fuzzy AHP methodology is called Minkowski consistency, which is founded on the Minkowski distance. 2) The proposed method offers an improved outcome by applying it to a green supply chain challenge based on a transportation issue. 3) The application of a Green Supply Chain is demonstrated as a transportation problem to minimize the transportation cost, emission of carbon footprints, and delivery time. 2. Related Work Demircan and Yetilmezsov, 2023 [16] evaluated four distinct intelligent disposal of waste solutions making use of a mixed fuzzy MCDM method to solve the shortage in research. The efficiency of the suggested approach substitutes was evaluated using research and fifteen sub-criteria set up into four primary groups. Fuzzy AHP, or fuzzy analytic hierarchy procedure, was paired with a fuzzy preference for order by resemblance method to generate fuzzy TOPSIS, which offered the most effective response. Demir et.al., 2023 [17] developed a two-stage fuzzy set for use in MCDM investigations about transportation. The proposed strategy is a special tactic that expands on Pythagorean fuzzy sets and combines the AHP and VIKOR approaches. In this way, the suggested method makes use of databases from the assessment of three train lines in Antalya, one of the biggest cities in Turkey. One Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 609 https://internationalpubls.com popular MCDM method has the FAHP, which weights thirteen sub-factors in addition to four primary criteria. AHP has been supported by IVPFNs. Du et.al., 2021 [18] addressed the TSP, a modified ACO algorithm called AHACOβ€”which is recommended to rely on an adaptive heuristic element. To enhance the algorithm's performance, the AHACO proposes three primary enhancements. First, a city classification technique called k-means is presented. To increase population diversity and enhance the algorithm's search capabilities, the AHACO offers various mobility techniques for various city classes. To further fine-tune the solution, a changed 2-opt local optimizer is suggested. To prevent the algorithm from stagnating, a mechanism for exiting the local optimum is finally included. Fidanova et.al., 2020 [19] discovered an algorithm based on the Ant Colony Optimization approach that has been suggested for passenger flow and transit modeling. The problem has been referred to as multi-objective optimization. Minimal transportation time and minimum cost are the two optimization goals. There have been some hazy elements in it. The price has been regarded as equal when it falls within a predetermined range. In the same way as the initial journey time. The percentage of passengers who have preferred employing a train over a bus due to factors like price, time, or preference has the target. Calik, 2021 [20] developed a unique approach to group decision-making that uses Industry 4.0 components to combine AHP and TOPSIS approaches in a Pythagorean fuzzy environment to choose the top green vendor. The suggested method states that many experts' opinions are expressed using linguistic phrases generated by PFN. The criteria weights have been determined using the interval- valued Pythagorean Fuzzy AHP approach. To rank and choose the best fit, the Pythagorean Fuzzy TOPSIS method has been applied, taking into consideration the distances between the providers. A genuine case study on an agricultural machinery and equipment company has been included at the conclusion to demonstrate the efficacy and precision of the suggested selection procedure. Tripathi et.al., 2021 [21] revealed eleven factors that fall under three categories: environmental, geographic, and socioeconomic. The study’s purpose has been regulated as to whether the MCDA approach would be best for choosing new hospital locations. AHP and FAHP have two MCDA techniques employed here. This study also presented an MCDA approach based on GIS. A comparison was made between the outcomes of the AHP and FAHP techniques. Abdullah et.al., 2023 [22] identified 21 social, economic, and technical factors that have been taken into attention when developing the best site development for Indonesian nuclear power plants (NPPs). These factors have included the operating costs, transmission network, geology, geotechnics, seismology, economic impact, environment, meteorology, proximity to hazardous facilities, population density, hydrology, cooling water, topography, evacuation route, proximity to wetland, transportation network, security, the impact of tourism, legal considerations, public acceptance, historical sites, and land ownership. Rouyendegh and Savalan, 2022 [23] have shown how the complexity of agricultural production has increased the need for more sophisticated agricultural production techniques (APTs). Using organic management techniques, these lacks downsides consequences associated with conventional and genetically engineered production, organic agriculture seeks to safeguard the environment while also enhancing consumer pleasure. In the meantime, goods made by genetic engineering and conventional methods are more affordable. This challenge serves as the dataset for the causal agent to demonstrate Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 610 https://internationalpubls.com the superiority of the suggested fuzzy MCDM hybrid model. The evaluation approach has increased the ratio of input data to output data because the challenge contains a lot of contradictory quantitative and qualitative criteria. It has been crucial to regulate agricultural productivity comprehensively as a result. Nguyen et.al., 2022 [24] suggested integrating SF-AHP, the SF-WASPAS, and DEA to have been capable of finding a maintainable supplier for Vietnam's steel manufacturing sector. This research considers both quantitative and qualitative criteria through expert interviews and a comprehensive literature analysis. DEA has been used in the initial step to verify high-efficiency suppliers according to several quantitative standards. In the second phase, these suppliers were assessed further based on qualitative standards like social, environmental, and economic aspects. While the SF-WASPAS was used to find maintainable suppliers, the SF-AHP was used to determine the importance of the criterion. Nguyen et.al., 2022 [25] advised methods include the PF-AHP and PF-CoCoSo. Benefit expectations (BEs) have been ranked according to AI adoption by PF-CoCoSo, whereas PF-AHP establishes the important element weighting criteria. The accuracy of the suggested technique in tackling the issue of AI technology adoption is attested to by the study's findings and comparisons between the output of the recommended methodology and other MCDM methodologies. These contributions, which offer methods and recommendations for analyzing AI applications, will be helpful to practitioners and researchers in the field. 3. The Fuzzy AHP Method The final result of integrating the method of AHP with a fuzzy concept method is fuzzy AHP. For example, selecting a new park's site. In pairs, assess every criterion about of the objective. For instance, how significant are the costs? and access. The 1–9 scale displayed in Table 1 is usually employed for this. 9 implies that one criterion is most important, whereas 1 signifies two equally important factors. Table 1: Noteworthy nine-point intensity scale Scale Value Vocal Judgment 1 Equal Importance 3 Moderate Importance 5 Strong Importance 7 Very Strong Importance 9 Extreme Importance 2, 4, 6, 8 Intermediate Values The fuzzy AHP value was calculated using the TFN. In linguistics, the fuzzy theory of associations, or TFN, is used to explain how people impartially evaluate one another. Hierarchical sets with several membership levels are called fuzzy sets. The weight from the AHP scale could be transformed into a fuzzy number using a triangle fuzzy number. Using the values q, r, and s which have the most potential but the lowest probability - the TFN has been found. The TFN and its grade are shown in Table 2. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 611 https://internationalpubls.com Table 2: The TFN and its grade Grade TFN (1,1,1) Equivalence level (1,1,3) Lower level (1,3,5) Average level (3,5,7) Better level (5,7,9) Excellence level 4. Methodology The current paper reflects the concepts of transportation problems based on fuzzy AHP. Consider source 𝑆𝑖 βˆ€ 𝑖 = 1,2, … , 𝑛 and demand 𝐷𝑗 βˆ€ 𝑗 = 1,2, … , π‘š for the transportation. Hence the objective function for the transportation is given below. min 𝑍 = βˆ‘(𝑀𝑇𝐢 . 𝑇𝐢 + 𝑀𝐢𝐹 . 𝐢𝐹 + 𝑀𝐷𝑇. 𝐷𝑇) (1) Subject to 𝑇𝐢 = βˆ‘ βˆ‘ 𝑐𝑖𝑗 π‘š 𝑗=𝑖 𝑛 𝑖=1 (2) 𝐢𝐹 = βˆ‘ βˆ‘ 𝑒𝑖𝑗 π‘š 𝑗=𝑖 𝑛 𝑖=1 (3) 𝐷𝑇 = βˆ‘ βˆ‘ 𝑑𝑖𝑗 π‘š 𝑗=𝑖 𝑛 𝑖=1 , π‘₯𝑖𝑗 β‰₯ 0 βˆ€ 𝑖, 𝑗 (4) Here, 𝑇𝐢 denotes the transportation cost, 𝐢𝐹 represents the emission of carbon footprint for the transportation and 𝐷𝑇 specifies the delivery time taken to reach the destination. The notations of the transportation problem are as follows. 1) 𝑆𝑖 are the suppliers (source). 2) 𝐷𝑗 are the customers (destination). 3) π‘₯𝑖𝑗 be the amount of goods transported (capacity) from the supplier 𝑖 to destination 𝑗. 4) 𝑐𝑖𝑗 be the cost of transportation per unit of goods from the supplier 𝑖 to destination 𝑗. 5) 𝑒𝑖𝑗 be the emission of carbon footprint per unit of goods transported from the supplier 𝑖 to destination 𝑗. 6) 𝑑𝑖𝑗 be the delivery time per unit of goods transported from the supplier 𝑖 to destination 𝑗. 7) 𝑀𝑇𝐢 is the weight for transportation costs. 8) 𝑀𝐢𝐹 is the weight of carbon footprints. 9) 𝑀𝐷𝑇 is the weight for delivery time. Step 1: It requires a decision matrix to be set up first, representing a hierarchy of criteria pertinent to the problem. Criteria 𝑆𝑖 βˆ€ 𝑖 = 1,2, … , 𝑛 refer to different sources or criteria to be analyzed within the problem. On the one side, alternatives 𝐷𝑗 βˆ€ 𝑗 = 1,2, … , π‘š refer to options or decisions that need to be ranked or decided. The decision matrix shows how each option 𝐷𝑗 fares against every criterion 𝑆𝑖; hence, it provides the ground for pairwise comparisons. Step 2: Decision takers establish fuzzy pairwise contrast matrices corresponding to the following criterion utilizing linguistic values for TFNs. The fuzzy pairwise comparison matrix reflects subjective judgments by the decision-makers, encapsulating uncertainty and imprecision in human judgment. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 612 https://internationalpubls.com Step 3: Given the availability of the fuzzy pairwise comparison matrix, then the next step involves normalizing every element of that matrix, with each of the fuzzy elements divided by the column sum in the matrix to which it belongs. After this, a normalized fuzzy matrix will have been obtained, where each element will be a relative weight. Step 4: By the centroid method, we convert the normalized fuzzy matrix into crisped values through the process of defuzzification. Thus, the defuzzified value of each criterion is obtained. The result is a crisp value for each criterion, which facilitates comparison and ranking of the alternatives. πΆπ‘’π‘›π‘‘π‘Ÿπ‘œπ‘–π‘‘ π‘“π‘œπ‘Ÿπ‘šπ‘’π‘™π‘Ž = π‘₯1+π‘₯2+π‘₯3 3 (5) Step 5: Next, sum up the defuzzified values obtained from the pairwise comparison. Then, we calculate the final weight by separating each defuzzified value by its sum of the defuzzified values corresponding to its rows. Thus, the fuzzy weight of the criteria is obtained whose sum is one. Step 6: By contemplating the random distribution of the decision makers as [03, 0.4, 0.3]. To determine the criteria's ultimate weight, we shall multiply the random distribution of the decision makers by each element of fuzzy weight. Thus, results from the final weight of the criteria which has the total one. Step 7: Proposed validation It is more important to check the consistency in fuzzy AHP. For consistency checking, we have proposed a new formula using Minkowski distance that will provide a successful result. Step 7.1: Firstly, change the fuzzy pairwise comparison matrix into hard values using the centroid formula of equation (5). Let us denote the crisp values as 𝐴. The general Minkowski distance is given by 𝐷𝑝(𝐴, 𝐡) = (βˆ‘ (π‘₯𝑖𝑗 βˆ’ 𝑦𝑖𝑗) 𝑝𝑛 𝑖,𝑗=1 ) 𝑝 . In the proposed methodology we shall modify the Minkowski distance as 𝑃 = 1 and fixing π‘₯𝑖𝑗 as the Centre matrix element. Thus, the newly proposed Minkowski consistency ratio is as follows. π‘€π‘–π‘›π‘˜π‘œπ‘€π‘ π‘˜π‘– 𝐢𝑅 = 𝑠𝑖𝑛 ( βˆ‘ (π‘₯π‘–π‘—βˆ’π‘¦π‘–π‘—)𝑛 𝑖,𝑗=1 det 𝐴 ) (6) Thus, 0.1 is the maximum CR value. Hence the newly proposed Minkowski consistency ratio reveals the superficial consistency of the fuzzy AHP. 4.1 Application of the study A vital component of the economy and daily life is public transportation. Different forms of public transportation provide various benefits. The transportation problem is one of the optimization problems, which requires the transportation of goods from various sources to different destinations in the most efficient way. A green supply chain is one in which the supply chain incorporates environmentally friendly practices in its operation, reducing ecological impact along all steps from production to delivery. In focus will come such issues as sustainability, waste minimization, and reduction of carbon footprints, while sustaining efficiency and profitability. We consider an application of green supply chain-based transportation problems using Fuzzy AHP. A company Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 613 https://internationalpubls.com needs to ship goods from a collection of suppliers to a collection of destinations. Let the suppliers be 𝑆𝑖 βˆ€ 𝑖 = 1,2,3. The destinations of the suppliers are named as customers and denoted as 𝐷𝑗 βˆ€ 𝑗 = 1,2,3,4. The company aims to minimize the total transportation cost, emission of carbon footprints of the transportation, and the delivery time taken by the supplier to reach the customer. The capacity of the suppliers 𝑆𝑖 βˆ€ 𝑖 = 1,2,3 are 70, 100, and, 90 respectively. The measuring units of the transportation cost, emission of carbon footprints, and delivery time are to be rupees, kilograms, and hours respectively. Table 4 depicts the dataset of the suppliers in the company. Table 4: Dataset π’„π’Šπ’‹ (rupees) π’†π’Šπ’‹ (kg) π’•π’Šπ’‹ (hrs) π‘«πŸ π‘«πŸ π‘«πŸ‘ π‘«πŸ’ π‘«πŸ π‘«πŸ π‘«πŸ‘ π‘«πŸ’ π‘«πŸ π‘«πŸ π‘«πŸ‘ π‘«πŸ’ π‘ΊπŸ 5 4 6 7 4 6 8 7 3 4 5 4 π‘ΊπŸ 8 5 7 6 9 5 6 8 6 3 4 7 π‘ΊπŸ‘ 6 9 5 8 4 7 3 6 5 6 5 4 As aspects of the decision maker, we obtain a pairwise comparison matrix based on the TFN. Table 5 contains the tabulation of the pairwise comparison. Table 5: Pairwise comparison π’„π’Šπ’‹ π’†π’Šπ’‹ π’•π’Šπ’‹ π’„π’Šπ’‹ (1,1,1) (2,3,4) (1/5,1/4,1/3) π’†π’Šπ’‹ (1/6,1/5,1/4) (1,1,1) (3,4,5) π’•π’Šπ’‹ (5,6,7) (1/7,1/6,1/5) (1,1,1) Each fuzzy element can be divided by the total of the corresponding column elements to form a normalized matrix, which can be found in Table 6 as the pairwise comparison matrix. Table 6: Normalized matrix π’„π’Šπ’‹ π’†π’Šπ’‹ π’•π’Šπ’‹ π’„π’Šπ’‹ (0.162,0.138,0.12) (0.636,0.720,0.769) (0.047,0.047,0.052) π’†π’Šπ’‹ (0.027,0.027,0.030) (0.318,0.240,0.192) (0.714,0.761,0.789) π’•π’Šπ’‹ (0.811,0.83,0.848) (0.045,0.039,0.038) (0.238,0.190,0.157) Using the centroid formula as mentioned in equation (5), we get the defuzzified value of each criterion and thus tabulated in Table 7. Table 7: Defuzzified values π’„π’Šπ’‹ π’†π’Šπ’‹ π’•π’Šπ’‹ sum π’„π’Šπ’‹ 0.14 0.708 0.048 0.896 π’†π’Šπ’‹ 0.028 0.25 0.754 1.032 π’•π’Šπ’‹ 0.830 0.038 0.195 1.063 To obtain the fuzzy weight in Table 8, we regularize them by separating each defuzzified value by all its corresponding sum of the rows. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 614 https://internationalpubls.com Table 8: Fuzzy weight π’„π’Šπ’‹ π’†π’Šπ’‹ π’•π’Šπ’‹ π’„π’Šπ’‹ 0.156 0.790 0.053 π’†π’Šπ’‹ 0.027 0.242 0.730 π’•π’Šπ’‹ 0.780 0.036 0.188 The graph that compares fuzzy weights of the criteria is shown in Figure 1. Figure 1: Fuzzy weights According to step 6 to find the final weight of the criteria the random distribution [0.3 0.4 0.3] is multiplied by the elements of Table 8. The method is as follows. [0.3 0.4 0.3] Γ— [ 0.156 0.790 0.053 0.027 0.242 0.730 0.780 0.036 0.188 ] = 0.2916 0.3446 0.3643 As a result, the criteria's ultimate weight is reached. The weight for the transportation cost is 0.2916. The weight for the carbon footprints is 0.3446. The weight for the delivery time is 0.3646. From equations (2), (3) and (4), we have 𝑇𝐢 = βˆ‘ βˆ‘ 𝑐𝑖𝑗 4 𝑗=𝑖 3 𝑖=1 = 5 + 4 + 6 + 7 + 8 + 5 + 7 + 6 + 6 + 9 + 5 + 8 = 76. 𝐢𝐹 = βˆ‘ βˆ‘ 𝑒𝑖𝑗 4 𝑗=𝑖 3 𝑖=1 = 4 + 6 + 8 + 7 + 9 + 5 + 6 + 8 + 4 + 7 + 3 + 6 = 73. 𝐷𝑇 = βˆ‘ βˆ‘ 𝑑𝑖𝑗 4 𝑗=𝑖 3 𝑖=1 = 3 + 4 + 5 + 4 + 6 + 3 + 4 + 7 + 5 + 6 + 5 + 4 = 56. Hence equation (1) becomes min 𝑍 = βˆ‘ (𝑀𝑇𝐢 . 𝑇𝐢 + 𝑀𝐢𝐹 . 𝐢𝐹 + 𝑀𝐷𝑇 . 𝐷𝑇) = 0.2916 Γ— 76 + 0.3446 Γ— 73 + 0.3643 Γ— 56 = 22.1616 + 25.1558 + 20.4008 = 67.7182. Hence the minimum transportation cost of the company is 67.7182, respectively. To find the consistency for the required fuzzy AHP, we convert the values of fuzzy pairwise comparison into crisp value matrix 𝐴. Thus, the crisp value matrix 𝐴 = [ 1 3 0.26 0.21 1 4 6 0.169 1 ]. calculating the determinant of 𝐴, 𝑑𝑒𝑑 𝐴 = 1 Γ— | 1 4 0.169 1 | βˆ’ 3 Γ— | 0.21 4 6 1 | + 0.26 Γ— | 0.21 1 6 0.169 | = 70.143. βˆ‘ (π‘₯𝑖𝑗 βˆ’ 𝑦𝑖𝑗) 𝑛 𝑖,𝑗=1 = βˆ‘ (π‘₯22 βˆ’ 𝑦𝑖𝑗) 3 𝑖,𝑗=1 = 0 + 2 + 0.74 + 0.6241 + 0 + 3 + 5 + 0.690561 + 0 = 12.054661. We use the new proposed Minkowski consistency ratio to validate the fuzzy AHP problem. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 615 https://internationalpubls.com π‘€π‘–π‘›π‘˜π‘œπ‘€π‘ π‘˜π‘– 𝐢𝑅 = 𝑠𝑖𝑛 ( βˆ‘ (π‘₯𝑖𝑗 βˆ’ 𝑦𝑖𝑗)𝑛 𝑖,𝑗=1 det 𝐴 ) = 𝑠𝑖𝑛 ( 12.054661 70.143 ) = 𝑠𝑖𝑛(0.17185) = 0.002999 < 0.1. As the consequence is less than 0.1, the matrix is considered consistent, and the required fuzzy AHP yields superficial outcomes. 5. Ant Colony Optimization Swarm intelligence, which derives inspiration from nature, represents one of the most efficient methods for solving optimization problems in an array of situations. Ant Colony Optimization (ACO) is a stochastic optimization technique that mimics the behaviours of real ants as they forage for food. It was created by Dorigo. By collaborating and transferring data, it calculates the quickest path an ant colony takes for reaching sources of food. Because they happen to be going along a common roadway, the ants are following each other. This is caused by the reality that, when they adhered to the path, every worker ant left a substance known as pheromone. After establishing the potency of the pheromone, the remaining ants pursue the route with the greatest intensity. They are trying to figure out the best route in this way. Traveling randomly is an expression employed to define an ant colony's effort to arrive at its objective first. During their return trip, the ants assess the strength of the pheromones and choose the route with the greatest concentration of those pheromones. It ought to be highlighted that pheromone dissipates with time; hence, the briefest roadway has more pheromone as it requires a shorter period to arrive there than the remaining routes. Because of this, almost every one of the ants would pick the most advantageous route since the shortest route would entice them through greater pheromone strength. The theoretical basis of ant behaviours can be described below: The kth ant at the location 𝑖 requires the subsequent probability formula as per equation (7) to move on to the next node, 𝑗. 𝑝𝑖𝑗 π‘˜ (𝑑) = { [πœπ‘–π‘—(𝑑)] 𝛼 [πœ‚π‘–π‘—(𝑑)] 𝛽 βˆ‘ [πœπ‘–π‘ (𝑑)]𝛼[πœ‚π‘–π‘ (𝑑)]𝛽 π‘ βˆˆπ½π‘˜(𝑖) , 𝑗 ∈ π½π‘˜(𝑖) 0, π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘€π‘–π‘ π‘’ (7) where Ξ± and Ξ² represent the relative importance of the pheromones and heuristic function, accordingly, πœπ‘–π‘— denotes the pheromone attention on the pathway between i and j, and πœ‚π‘–π‘— indicates the heuristic function, i.e., which corresponds to the reciprocal of the distance between the i and j positions. After ants finish their tour, the pheromone trial values are rationalized according to the equation (8). πœπ‘–π‘—(𝑑 + 1) = (1 βˆ’ 𝜌)[πœπ‘–π‘—(𝑑)] + βˆ‘ Ξ”πœπ‘–π‘— π‘˜π‘š π‘˜=1 (8) where 𝜌(0 < 𝜌 ≀ 1) implies the evaporation rate, Ξ”πœπ‘–π‘— π‘˜ signifies the extra pheromone left in the path (𝑖, 𝑗) by the k-th ant. Ξ”πœπ‘–π‘— π‘˜ signifies the extra pheromone left in the path (𝑖, 𝑗) by the k-th ant. Ξ”πœπ‘–π‘— π‘˜ could be considered through the equation (9). Ξ”πœπ‘–π‘— π‘˜ (𝑑) = { 𝑄 πΏπ‘˜ , 𝑖𝑓 π‘Žπ‘›π‘‘ π‘˜ π‘π‘Žπ‘ π‘ π‘’π‘  π‘‘β„Žπ‘’ π‘π‘Žπ‘‘β„Ž (𝑖, 𝑗) 0, π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘€π‘–π‘ π‘’ (9) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 616 https://internationalpubls.com where Ξ”πœπ‘–π‘— π‘˜ specifies the number of pheromones added by the kth ant, Q is the pheromone update constant, N is the total number of ants in the nest, and πΏπ‘˜ is the length of the path traveled by the kth ant. 5.1 Case Study The study ACO is put into an application of green supply chain. Here is a company which is named Eco Mobility. It is a manufacturing company of electric vehicles. With a motive of reducing environmental footprint, Eco Mobility is committed to this production for the adaption of Green Supply Chain management. Eco Mobility has many different suppliers that are spread across various locations. Eco Mobility has many suppliers who provide raw materials, which should be transferred to its main production facility. Each supplier provides components that are essential, such as batteries, electric motors, and control units. So, to maximize the supply chain of the corporation, the components of electric vehicles must be distributed and transported from the suppliers to its manufacturing plant. The key objective of the company is to decrease carbon emissions, fuel consumption, and delivery time. The distances covered between the routes chosen between suppliers and plant, time consumed, and environmental impact are not the same for each route. This makes the optimization problem complex. Eco Mobility strives to find the best routes with the ACO algorithm. Firstly, the transportation network can be represented as a graph in figure 2. In this model, each node represents a location, namely suppliers and the manufacturing plant. Figure 2: Transportation network graph Each edge in the graph represents every potential route between any two of the nodes. Each edge is associated with certain attributes, such as carbon emissions, fuel consumption, and delivery time. Let the alternatives be supplier 1 (S1), supplier 2 (S2), supplier 3 (S3), and the manufacturing plant (MP). The attributes are measured in certain units followed by distance in kilometers (km), carbon emission in grams of CO2 per km (g), fuel consumption in liters per kilometer (l/km), and the delivery time in hours (hrs.). Table 8 reflects the dataset of the company. Table 8: Dataset Routes Distance (km) carbon emission (g) Fuel consumption(l/km) Delivery time (hrs) S1-MP 100 50 0.2 1.5 S2-MP 150 60 0.25 2 S3-MP 200 70 0.3 3 S1-S2 90 45 0.22 1.8 S2-S3 125 55 0.18 1.2 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 617 https://internationalpubls.com The cost of the transportation problem depends on the route they follow. Thus, the cost of the route is calculated by the equation (10). π‘π‘œπ‘ π‘‘ π‘œπ‘“ π‘Ÿπ‘œπ‘’π‘‘π‘’ = π‘‘π‘–π‘ π‘‘π‘Žπ‘›π‘π‘’ Γ— 𝑓𝑒𝑒𝑙 π‘π‘œπ‘›π‘ π‘’π‘šπ‘π‘‘π‘–π‘œπ‘› + π‘π‘’π‘›π‘Žπ‘™π‘‘π‘¦ π‘’π‘šπ‘–π‘ π‘ π‘–π‘œπ‘› (10) Where penalty emission is proportional to emissions that are 0.001Γ—g CO2/km. Now we can calculate the total cost of each route in Table 9 using the equation (10) as follows. Table 9: Total cost Routes Distance (km) Fuel consumption(l/km) penalty emission Total cost S1-MP 100 0.2 5 25 S2-MP 150 0.25 9 46.5 S3-MP 200 0.3 14 74 S1-S2 90 0.22 4.05 23.85 S2-S3 125 0.18 6.86 29.36 The heuristic is inversely proportional to the cost and calculated by the equation (11) as follows. πœ‚π‘–π‘— = 1 π‘‘π‘œπ‘‘π‘Žπ‘™ π‘π‘œπ‘ π‘‘ π‘œπ‘“ π‘Ÿπ‘œπ‘’π‘‘π‘’ (11) We can evaluate the heuristic information for each route of the company’s transportation cost using equation (11) as follows. For the route S1-MP, πœ‚π‘–π‘— = 1 25 = 0.04 For the route S2-MP, πœ‚π‘–π‘— = 1 46.5 = 0.0215 For the route S3-MP, πœ‚π‘–π‘— = 1 74 = 0.01351 For the route S1-S2, πœ‚π‘–π‘— = 1 23.85 = 0.0419 For the route S2-S3, πœ‚π‘–π‘— = 1 29.36 = 0.0340 A supplier chooses a path based on the heuristic information (πœ‚π‘–π‘—) and pheromone level (τ𝑖𝑗). The probability 𝑝𝑖𝑗 for each route that a supplier selects the route 𝑖𝑗 from a starting node is calculated by equation (7). Let’s assume that the initial pheromone level τ𝑖𝑗 of each route to be 1. Assume the weight of the pheromone and heuristic values as 1 (that is 𝛼 = 1 and 𝛽 = 1). The possible routes from S1 are MP or S2. For the route S1-MP, 𝑃S1βˆ’MP = [𝜏S1βˆ’MP(𝑑)]𝛼[πœ‚S1βˆ’MP(𝑑)]𝛽 [𝜏S1βˆ’MP(𝑑)]𝛼[πœ‚S1βˆ’MP(𝑑)]𝛽 + [𝜏S1βˆ’S2(𝑑)]𝛼[πœ‚S1βˆ’S2(𝑑)]𝛽 = [1]1[0.04]1 [1]1[0.04]1 + [1]1[0.0419]1 = 0.04 0.04 + 0.0419 = 0.04 0.0819 β‰ˆ 0.4885 For the route S1-S2, 𝑃S1βˆ’S2 = [𝜏S1βˆ’S2(𝑑)]𝛼[πœ‚S1βˆ’S2(𝑑)]𝛽 [𝜏S1βˆ’MP(𝑑)]𝛼[πœ‚S1βˆ’MP(𝑑)]𝛽 + [𝜏S1βˆ’S2(𝑑)]𝛼[πœ‚S1βˆ’S2(𝑑)]𝛽 = [1]1[0.0419]1 [1]1[0.04]1 + [1]1[0.0419]1 = 0.0419 0.04 + 0.0419 = 0.0419 0.0819 β‰ˆ 05115 The next possible route from the node S2 is MP and S3. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 618 https://internationalpubls.com For the route S2-MP, 𝑃S2βˆ’MP = [𝜏S2βˆ’MP(𝑑)]𝛼[πœ‚S2βˆ’MP(𝑑)]𝛽 [𝜏S2βˆ’MP(𝑑)]𝛼[πœ‚S2βˆ’MP(𝑑)]𝛽 + [𝜏S2βˆ’S3(𝑑)]𝛼[πœ‚S2βˆ’S3(𝑑)]𝛽 = [1]1[0.0215]1 [1]1[0.0215]1 + [1]1[0.0340]1 = 0.0215 0.0215 + 0.0340 = 0.0215 0.0555 β‰ˆ 0.3874 For the route S2-S3, 𝑃S2βˆ’S3 = [𝜏S2βˆ’S3(𝑑)]𝛼[πœ‚S2βˆ’S3(𝑑)]𝛽 [𝜏S2βˆ’MP(𝑑)]𝛼[πœ‚S2βˆ’MP(𝑑)]𝛽 + [𝜏S2βˆ’S3(𝑑)]𝛼[πœ‚S2βˆ’S3(𝑑)]𝛽 = [1]1[0.0340]1 [1]1[0.0215]1 + [1]1[0.0340]1 = 0.0340 0.0215 + 0.0340 = 0.0340 0.0555 β‰ˆ 0.6126 The only possible route from S3 is MP. Thus, the probability is 𝑃S3βˆ’MP = 1. After the suppliers complete their paths, pheromone levels are updated. Consider the pheromone update constant 𝑄 = 100 and the evaporation rate 𝜌 = 0.1. For our convenience assume that the supply starts from S1 to MP that covers the distance 100 km. To calculate the additional pheromone level Ξ”πœπ‘–π‘— π‘˜ deposits of each route, we use equation (9). For the route S1-MP, Ξ”πœS1βˆ’MP π‘˜ = 𝑄 πΏπ‘˜ = 100 100 = 1 For the route S2-MP, Ξ”πœS2βˆ’MP π‘˜ = 𝑄 πΏπ‘˜ = 100 150 β‰ˆ 0.67 For the route S3-MP, Ξ”πœS3βˆ’MP π‘˜ = 𝑄 πΏπ‘˜ = 100 200 = 0.5 For the route S1-S2, Ξ”πœS1βˆ’S2 π‘˜ = 𝑄 πΏπ‘˜ = 100 90 β‰ˆ 1.11 For the route S2-S3, Ξ”πœS2βˆ’S3 π‘˜ = 𝑄 πΏπ‘˜ = 100 125 = 0.8 Now we calculate the update pheromone level πœπ‘–π‘—(𝑑 + 1) of each route as per the equation (8). For the route S1-MP, 𝜏S1βˆ’MP(𝑑 + 1) = (1 βˆ’ 0.1). 1 + 1 = 0.9 + 1 = 1.9 For the route S2-MP, 𝜏S2βˆ’MP(𝑑 + 1) = (1 βˆ’ 0.1). 1 + 0.67 = 0.9 + 0.67 = 1.57 For the route S3-MP, 𝜏S3βˆ’MP(𝑑 + 1) = (1 βˆ’ 0.1). 1 + 0.5 = 0.9 + 0.5 = 1.4 For the route S1-S2, 𝜏S1βˆ’S2(𝑑 + 1) = (1 βˆ’ 0.1). 1 + 1.11 = 0.9 + 1.11 = 2.01 For the route S2-S3, 𝜏S2βˆ’S3(𝑑 + 1) = (1 βˆ’ 0.1). 1 + 0.8 = 0.9 + 0.8 = 1.7 Hence the initial pheromone level, additional pheromone level and update pheromone level are tabulated in the table 10 as follows. Table 10: Pheromone level Routes π›•π’Šπ’‹(𝒕) πš«π‰π’Šπ’‹ π’Œ π‰π’Šπ’‹(𝒕 + 𝟏) S1-MP 1 1 1.9 S2-MP 1 0.67 1.57 S3-MP 1 0.5 1.4 S1-S2 1 1.11 2.01 S2-S3 1 0.8 1.7 After one cycle of the Ant Colony Optimization algorithm, the concentrations of pheromone on all routes have increased. The intensity of pheromone concentration is high on smaller routes such as Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 619 https://internationalpubls.com S1-S2 because it is normal considering the short distances that are involved hence making them more attractive in the optimization process. Iterating over many such iterations, the pheromone levels would continue to change; more suppliers will continue to trace higher pheromone levels' paths and converge upon routes that are optimized for consuming less fuel, fewer emissions of carbon, and the minimum delivery time. Considering the new level of pheromone, the developed routes, like S1 β†’ S2, are allocated an increasing attractiveness to the future suppliers. These will most likely be the most optimal routes in the green supply chain of Eco Mobility. 6. Firefly Algorithm Three rules form the basis of the Firefly algorithm: β€’ Because they are all unisex, fireflies will always be drawn to one another regardless of gender. β€’ Their brightness determines how attractive they are; the dimmer will gravitate toward the brig hter one haphazard motion when they are evenly illuminated. β€’ Firefly brightness is correlated with the objective function's value. The relationship between the light intensity 𝐼 and the distance π‘Ÿ, is expressed as 𝐼 = 𝐼0 𝑒π‘₯𝑝(βˆ’π›Ύπ‘Ÿ), where 𝐼0 is the original light intensity, and 𝛾 is the absorption coefficient that ranges between 0 and 1. Attractiveness is given by 𝛽 = 𝛽0 exp (βˆ’π›Ύπ‘Ÿ), where 𝛽0 is the attractiveness when π‘Ÿ = 0. The distance in 𝑑-dimensions between two fireflies 𝑖 and 𝑗 at π‘₯𝑖 and π‘₯𝑗, respectively, is given by π‘Ÿπ‘–π‘— = β€–π‘₯𝑖 βˆ’ π‘₯𝑗‖ = βˆšβˆ‘ (π‘₯𝑖,π‘˜ βˆ’ π‘₯𝑗,π‘˜) 2𝑑 π‘˜=1 , and it is π‘Ÿπ‘–π‘— = √(π‘₯𝑖 βˆ’ π‘₯𝑗) 2 + (𝑦𝑖 βˆ’ 𝑦𝑗) 2 in two dimensions. The movement of firefly 𝑖 attracted to a brighter firefly 𝑗 is given by π‘₯𝑖 = π‘₯𝑖 + 𝛽0π‘’βˆ’π›Ύπ‘Ÿπ‘–π‘— 2 (π‘₯𝑗 βˆ’ π‘₯𝑖) + 𝛼 (π‘Ÿπ‘Žπ‘›π‘‘ βˆ’ 1 2 ), (12) where the following term is the outcome of randomization with a parameter for control 𝛼 between 0 and 1, and the second term is the consequence of attraction. An FF's flashing behaviour is employed to figure out an appropriate route for action when the robot meets with both static as well as moving things. For a robot to achieve the specific performance the need of the tasks that follow: identifying obstacles, avoidance of obstacles, conquering trap-like circumstances, preventing random excursions, and appropriate qualities should be recorded in a feedback-based function with objectives for route organizing. This enables the robot to produce an optimal path. Robots with sensors attached can determine their position in a new region and gain knowledge concerning their surroundings. In basic terms, navigation is a reduction issue or a route optimization issue. Considering the goal and the position of the obstacle, the function of objectives for the same could represent an appropriate parameter. Each time, a significantly stronger FF's location is determined, and the robot travels in an order of stages toward there it. In the scenario that the robot stumbles through an obstacle on its way toward accomplishing its current goal, the function calls are made itself. Based on the robot's sensor inputs, a set quantity of randomized fireflies appears around a challenge when a robot recognizes it. Each FF's distance from Euclid to the object being targeted and the obstacle is calculated. The robot Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 620 https://internationalpubls.com selects the most vibrant firefly (FF) and progresses toward it, maintaining its distance from impediments and an appropriate distance from its final location. The optimization of the path planning problem could be articulated as: 𝑓𝑖 = 𝐾1 1 min π‘œπ‘›βˆˆπ‘œπ‘ β€–π·π‘“π‘œβ€– + 𝐾2‖𝐷𝑓𝑔‖ (13) where 𝐾1 is the fitting parameter for the path safety and 𝐾2 defines the maximum and minimum path length of the navigation. π·π‘“π‘œ is the distance of the FF 𝑓𝑖 from the obstacle, and 𝐷𝑓𝑔 is its distance from the objective. The robot remains secure by circumventing the obstruction when 𝐾1 reaches its maximum value. The path length is maximized at the minimum value of 𝐾2 and minimized at the maximum value of 𝐾2. The optimum value of the function with objectives is established by correctly selecting the appropriate parameters, an objective that could be achieved through trial and error. 7. Conclusion The hybrid model proposed here solves the green supply chain transportation problem through the integrated use of the Hybrid Minkowski Consistency Operator, Fuzzy AHP, and MCDM techniques in a powerful and novel manner. The model has further proceeded into the multi-conflicting of these criteria of cost, emission, and delivery time to provide a more comprehensive and balanced optimization approach toward transportation efficiency in green supply chain management. It enabled the treatment of uncertainties, which has been quite common in a real-world supply chain, and reached more accurate and reliable decision-making processes with the utilization of fuzzy logic. In addition, the Hybrid Minkowski Consistency Operator has reinforced the model for complex criteria processing, while the multi-objective optimization technique combines ACO with FA, which further refined the overall performance. This hybrid optimization strategy strongly enhanced the model's capability to find a balance between competing objectives while furthering sustainability goals. 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