Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 621 https://internationalpubls.com Applying a Fuzzy Ordering Approach in Transportation Problems with Decagonal Intuitionistic Fuzzy Numbers a Nathiya K, b Balasubramanian KR, c Gunasekar T, d Ramesh R, e Seenivasan M a Research Scholar, Department of Mathematics, H.H. The Rajah’s College, (Affiliated to Bharathidasan University), Pudukkottai, Tamil Nadu, India. Email: nathiyagk30@gmail.com b Department of Mathematics, H.H. The Rajah’s College, (Affiliated to Bharathidasan University), Pudukkottai, Tamil Nadu, India. Email: balamohitha@gmail.com C Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Chennai, Tamil Nadu, India, Email: tguna84@gmail.com d Department of Mathematics, Arignar Anna Government Arts College, Musiri, Tamil Nadu, India, Email: rameshsanju123@gmail.com e Mathematics Wings - CDOE, Annamalai University, Annamalai Nagar, Tamil Nadu, India, Email: emseeni@yahoo.com Article History: Received: 20-04-2024 Revised: 10-06-2024 Accepted: 24-06-2024 Abstract: In this study, the paper delves into precision challenges within traditional transportation problem solutions, which rigidly define cost, supply, and demand. Acknowledging the inherent vagueness in real world contexts, the research explores the efficacy of intuitive fuzzy sets as a potent tool. Organized into four distinct sections, this work utilizes decagonal intuitionistic fuzzy numbers for managing supply and demand, while upholding conventional approaches for cost considerations. Employing a fuzzy ordering method, optimal solutions are derived by adjusting the configuration of decagonal intuitive fuzzy numbers across each segment. Through a comparative analysis, the Study identifies the most effective solution, with initial sections addressing balanced geometric intuitionistic fuzzy transportation problems and the final part focusing on unbalanced scenarios, specifically emphasizing supply and demand complexities. Keywords: Fuzzy Set; Intuitionistic; Ranking; Transportation Problem; Numerical Analysis. 2020AMS subject classifications: 94D05; 03F55; 62F07; 90C08; 62L86. 1. Introduction Transportation problems, a foundational aspect of operations research, are pivotal in optimizing the distribution of goods among diverse sources and destinations. Traditional solutions to transportation problems often assume deterministic parameters, overlooking the inherent uncertainties in real-world scenarios. Sharma and Taha gave the basic concepts of Operations research ((12; 16)). In recent years, the integration of fuzzy set theory has provided a robust framework for capturing and modeling this uncertainty ((19)). This introduction aims to explore fundamental concepts in transportation problems, fuzzy sets, and their multifaceted applications, drawing insights from various relevant research papers. The seminal work of Zadeh in 1965 introduced fuzzy set theory as a means to handle imprecise and Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 622 https://internationalpubls.com uncertain information ((19)). Fuzzy sets generalize classical set theory, allowing elements to belong to a set to varying degrees, mirroring the inherent ambiguity in human reasoning. Transportation problems involve decision-making in uncertain environments, where the utilization of fuzzy sets has significantly enhanced traditional optimization techniques. Atanassov. K.T introduced intuitionistic Fuzzy Sets, a substantial extension within Fuzzy sets and systems ((2)). burillo et. al discussed with intuitionistic Fuzzy Sets ((4)). Annie M. S Christi has made substantial contributions by applying fuzzy sets to transportation problems. Her research explores the use of pentagonal intuitionistic fuzzy numbers to solve transportation problems, employing ranking techniques and Russell’s method ((5; 6)). Syamala et. al discussed some Fuzzy and Intuitionistic Fuzzy concepts ((1; 13; 14; 15)). By incorporating intuitionistic fuzzy numbers, which extend traditional fuzzy sets by considering both the degree of membership and non- membership, the model becomes more adept at handling imprecise data. Building upon Christi’s work, Mohideen et al. expanded the scope to octagon fuzzy numbers and introduced α-cut and ranking techniques to solve fuzzy transportation problems ((11)). This research further exemplifies the adaptability of fuzzy set theory to different geometric configurations, demonstrating its applicability in diverse scenarios. Felix et. al used octagonal fuzzy numbers ((7)). The exploration of generalized hexagonal and octagonal fuzzy numbers in transportation problems is presented by Ghadle and Pathade ((8; 9)). Beaula et. al discussed also fuzzy transportation problems ((3)).Their work introduces a ranking method as a means to find optimal solutions, showcasing the versatility of fuzzy sets in addressing various problem formulations. The incorporation of generalized fuzzy numbers allows for a broader representation of uncertainty, contributing to a more realistic modeling of transportation systems. Thamaraiselvi and Santhi investigate the application of hexagonal intuitionistic fuzzy numbers in solving transportation problems ((17; 18)). Their work emphasizes the importance of intuitionistic fuzzy sets in capturing uncertainty and vagueness, providing a more nuanced approach to modeling real-world decision- making processes. Li used ratio ranking method ((10)). Beyond transportation-specific applications, fuzzy set theory, as introduced by Zadeh and further elaborated by researchers like Zimmermann ((20)), has found applications in various disciplines. Fuzzy set theory proves beneficial in situations where available information is inherently imprecise or vague. In addressing transportation problems, effective solutions often hinge on precise specifications of costs, supply, and demand. Intuitionistic fuzzy sets emerge as a potent tool for handling the inherent vagueness in such scenarios. The subsequent sections of this paper delve into exploring the application of decagonal intuitive fuzzy numbers in transportation problem formulations. These sections employ a proposed ranking method to determine optimal solutions across various methodological approaches, conducting comparative analyses to identify favorable solutions. While the initial sections focus on balanced decagonal intuitionistic Fuzzy Transportation Problems, the concluding section addresses unbalanced transportation problems, particularly involving supply and demand, culminating the paper’s investigation. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 623 https://internationalpubls.com | ∈ 2. Preliminaries Definition 2.1. Let X be a nonempty set. A fuzzy set à of X is defined as à = {< x, µÃ(x) > /x ∈ X}. Where µÃ(x) is called membership function, which maps each element of x to [0,1]. Definition 2.2. Given a non-empty set X , an Intuitionistic Fuzzy Set ÃI de- fined over X is described by its elements in the form < x, µÃI (x), θÃI (x) >, where x belongs to X . Here, µÃI (x) and θÃI (x) denote the membership and non- membership functions, respectively. Both functions, µÃI and θÃI , are mappings from X to the interval [0, 1] and must adhere to the constraint 0 ≤ µÃI (x) + θÃI (x) ≤ 1 for every x ∈ X . Definition 2.3. A fuzzy number, represents a range of possible values rather than a single precise value. It is characterized by a connected set of values on the real line R, where each value within this set is assigned a weight, known as its membership function, ranging from 0 to 1. For a fuzzy number to be valid: 1. There must be at least one value x ∈ R for which the membership function µÃ(x) equals 2. The membership function µÃ(x) must exhibit piecewise continuity, ensuring the smooth transition between different values within the set. Definition 2.4. An Intuitionistic Fuzzy Subset ÃI defined as < x, µÃI (x), θÃI (x) > x X on the real line R is termed an Intuitionistic Fuzzy Number if it satisfies the following conditions, there exists a value m in the real line R such that µÃI (m) = 1, and simultaneously, θÃI (m) = 0. 1. µÃI is a continuous function from R→[0,1] 2. 0≤µÃI(x)+θÃI(x)≤1 for all x∈ X. The membership and non-membership functions of ÃI are in the following form µÃI(x)= { 0 𝑓𝑜𝑟 − ∞ < 𝑥 < 𝜗1 𝑓(𝑥) 𝑓𝑜𝑟 𝜗1 ≤ 𝑥 < 𝜗2 1 𝑓𝑜𝑟 𝑥 = 𝜗2 𝑔(𝑥) 𝑓𝑜𝑟𝜗2 ≤ 𝑥 < 𝜗3 0 𝑓𝑜𝑟𝜗3 ≤ 𝑥 < ∞ ⬚ θA˜I (x) = { 1 𝑓𝑜𝑟 − ∞ < 𝑥 < 𝜗1 𝑓′(𝑥)𝑓𝑜𝑟𝜗1 0 ≤ 𝑥 < 𝜗2 0 𝑓𝑜𝑟𝑥 = 𝜗2 𝑔′(𝑥)𝑓𝑜𝑟𝜗2 ≤ 𝑥 < 𝜗3 0 1 𝑓𝑜𝑟𝜗3 0 ≤ 𝑥 < ∞ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 624 https://internationalpubls.com Where, ϑ1, ϑ2, ϑ3, ϑ1’, ϑ2’, and ϑ3’ are real numbers. f, f’, g, g’ are functions from R → [0, 1]. f and g 0 are strictly increasing functions and g and f 0 are strictly decreasing functions with the conditions 0 ≤ f(x) + f ‘(x) ≤ 1 and 0 ≤ g(x) + g’(x) ≤ 1. Definition 2.5. A Decagonal Fuzzy Number, is represented by D˜ = (ϑ1, ϑ2, ϑ3, ϑ4, ϑ5, ϑ6, ϑ7, ϑ8, ϑ9, ϑ10). Here, ϑ1, ϑ2, ϑ3, ϑ4, ϑ5, ϑ6, ϑ7, ϑ8, ϑ9, and ϑ10 are real numbers. The membership function µD˜ (x) characterizing this Generalized Decagonal Fuzzy Number is defined as follows, 𝜇�̃�I(𝑥) = { p ( x − ϑ1 ϑ2 − ϑ1 )𝑓𝑜𝑟ϑ1 < 𝑥 < ϑ2 𝑝 + (𝑞 − 𝑝) ( x − ϑ2 ϑ3 − ϑ2 ) 𝑓𝑜𝑟𝜗2 ≤ 𝑥 < 𝜗3 𝑞 + (𝑟 − 𝑞) ( x − ϑ3 ϑ4 − ϑ3 ) 𝑓𝑜𝑟𝜗3 ≤ 𝑥 < 𝜗4 𝑟 + (1 − 𝑟) ( x − ϑ4 ϑ5 − ϑ4 ) 𝑓𝑜𝑟𝜗4 ≤ 𝑥 < 𝜗5 1 𝑓𝑜𝑟ϑ5 ≤ x < ϑ6 𝑟 + (1 − 𝑟) ( x − ϑ6 ϑ7 − ϑ6 ) 𝑓𝑜𝑟𝜗6 ≤ 𝑥 < 𝜗7 𝑛 + (1 − 𝑛) ( ϑ7 − x ϑ8 − ϑ7 ) 𝑓𝑜𝑟𝜗7 < 𝑥 ≤ 𝜗8 𝑚 + (𝑛 −𝑚) ( ϑ8 − x ϑ9 − ϑ8 ) 𝑓𝑜𝑟𝜗8 < 𝑥 ≤ 𝜗9 𝑚 ( ϑ9 − x ϑ10 − ϑ9 ) 𝑓𝑜𝑟𝜗9 < 𝑥 < 𝜗10 0 𝑓𝑜𝑟 x ≤ ϑ1, x ≤ ϑ10 where 0 < 𝑝 < 𝑞 < 𝑟 < 𝑛 < 𝑚1. Definition 2.6. A Decagonal Intuitionistic Fuzzy Number is specified by �̃�𝐷 𝐼 = (ϑ1, ϑ2, ϑ3, ϑ4, ϑ5, ϑ6, ϑ7, ϑ8, ϑ9, ϑ10), (ϑ1′, ϑ2′, ϑ3′, ϑ4′, ϑ5′, ϑ6′, ϑ7′, ϑ8′, ϑ9′, ϑ10′) Where 𝜗1, 𝜗2, 𝜗3, 𝜗4, 𝜗5, 𝜗6, 𝜗7, 𝜗8, 𝜗9, 𝜗10, ϑ1′, ϑ2′, ϑ3′, ϑ4′, ϑ5′, ϑ6′, ϑ7′, ϑ8′, ϑ9′ 𝑎𝑛𝑑 ϑ10′ are real numbers such thatϑ1′ ≤ 𝜗1 ≤ ϑ2′ ≤ 𝜗2 ≤ ϑ3′ ≤ 𝜗3 ≤ ϑ4′ ≤ 𝜗4 ≤ 𝜗5 ≤ 𝜗6 ≤ ϑ7′ ≤ 𝜗7 ≤ ϑ8′ ≤ 𝜗8 ≤ ϑ9′ ≤ 𝜗9 ≤ ϑ10′ ≤ ϑ10 and its membership and non membership functions are given below 𝜇�̃�𝐼(𝑥)µ is given below, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 625 https://internationalpubls.com 𝜇�̃�I(𝑥)= { p ( x−ϑ1 ϑ2−ϑ1 ) 𝑓𝑜𝑟ϑ1 < 𝑥 < ϑ2 𝑝 + (𝑞 − 𝑝) ( x−ϑ2 ϑ3−ϑ2 ) 𝑓𝑜𝑟𝜗2 ≤ 𝑥 < 𝜗3 𝑞 + (𝑟 − 𝑞) ( x−ϑ3 ϑ4−ϑ3 )𝑓𝑜𝑟𝜗3 ≤ 𝑥 < 𝜗4 𝑟 + (1 − 𝑟) ( x−ϑ4 ϑ5−ϑ4 ) 𝑓𝑜𝑟𝜗4 ≤ 𝑥 < 𝜗5 1 𝑓𝑜𝑟ϑ5 ≤ x < ϑ6 𝑟 + (1 − 𝑟) ( x−ϑ6 ϑ7−ϑ6 ) 𝑓𝑜𝑟𝜗6 ≤ 𝑥 < 𝜗7 𝑛 + (1 − 𝑛) ( ϑ7−x ϑ8−ϑ7 )𝑓𝑜𝑟𝜗7 < 𝑥 ≤ 𝜗8 𝑚 + (𝑛 − 𝑚) ( ϑ8−x ϑ9−ϑ8 ) 𝑓𝑜𝑟𝜗8 < 𝑥 ≤ 𝜗9 𝑚 ( ϑ9−x ϑ10−ϑ9 ) 𝑓𝑜𝑟𝜗9 < 𝑥 < 𝜗10 0 𝑓𝑜𝑟 x ≤ ϑ1, x ≤ ϑ10 𝜃�̃�I(𝑥) = { p ( x − ϑ1′ ϑ2′ − ϑ1′ ) 𝑓𝑜𝑟 ϑ1′ < 𝑥 < 𝜗2′ 𝑝 + (𝑞 − 𝑝) ( x − ϑ2′ ϑ3′ − ϑ2′ ) 𝑓𝑜𝑟𝜗2′ ≤ 𝑥 < 𝜗3′ 𝑞 + (𝑟 − 𝑞) ( x − ϑ3′ ϑ4′ − ϑ3′ ) 𝑓𝑜𝑟𝜗3′ ≤ 𝑥 < 𝜗4′ 𝑟 + (1 − 𝑟)( x − ϑ4′ ϑ5′ − ϑ4′ ) 𝑓𝑜𝑟𝜗4′ ≤ 𝑥 < 𝜗5′ 0 𝑓𝑜𝑟 ϑ5 ≤ x < 𝜗6 𝑟 + (1 − 𝑟) ( x − ϑ6 ϑ7′ − ϑ6 ) 𝑓𝑜𝑟𝜗6 ≤ 𝑥 < 𝜗7′ 𝑛 + (1 − 𝑛) ( ϑ7′ − x ϑ8′ − ϑ7′ ) 𝑓𝑜𝑟 𝜗7′ < 𝑥 ≤ 𝜗8′ 𝑚 + (𝑛 − 𝑚)( ϑ8′ − x ϑ9′ − ϑ8′ ) 𝑓𝑜𝑟 𝜗8′ < 𝑥 ≤ 𝜗9′ 𝑚 ( ϑ9′ − x ϑ10′ − ϑ9′ ) 𝑓𝑜𝑟 𝜗9′ < 𝑥 < 𝜗10′ 1 𝑓𝑜𝑟 x ≤ ϑ1′, x ≤ ϑ10′ 3. Proposed New Ordering of Decagonal Intuitionistic Fuzzy Numbers The ordering function of Decagonal Intuitionistic Fuzzy Number (OIFN) 𝐴′𝑂𝐶=(a1,a2,a3,a4,a5,a6,a7,a8,a9,a10) (a1 r, a2 r, a3 r, a4 r, a5 r, a6 r, a7 r, A8 r, a9 r, a10 r, a11 r, a12 r) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 626 https://internationalpubls.com defined as maps the set of all Fuzzy numbers to a set of real numbers defined as ′ R[AOC]=Max[Magµ(Aoc),Magθ(Aoc)]Where Solving Transportation Problem with Supply and Demand Values are Decagonal Intuitionistic Fuzzy Numbers using New Proposed Ordering Method 3.1 Numerical Example ConsiderSuppliesandDemandsaredecagonalIntuitionisticFuzzyNumber. B1 B2 B3 Supply A1 12.5 11.5 9.5 (2,4,5,6,7,8,9,1 (6,7,8,9,10,11,12,13,14,15) A2 10.5 6.5 13.5 (8,9,10,11,12,13,14,15,16,17) (4,6,7,8,9,10,12,13,15,16) A3 11.5 9.5 8.5 (4,6,7,8,9,10,11,12,13,14) (1,2,3,4,5,6,7,8,9,10) Demand (4,5,6,7,8,9,10,11,12,13) (6,7,8,9,10,11,12,13,14,15) (1,2,3,5,6,7,8,12,14,15) (0,1,2,3,4,7,9,10,12,13) (3,4,5,6,7,8,9,10,11,12) (8,9,10,11,12,13,14,15,16,17) ΣDemand=ΣSupply The problem is a balanced transportation problem. Using the proposed algorithm, the solution of the problem is as follows. Applying accuracy function on decagonal Intuitionistic Fuzzy Number [(1,2,3,5,6,7,8,10)(3,6,7,8,9,10,12,13)], we have R(Aoc)=10.5 Similarly applying for all the values, we have the following table after ordering 3.2 Applying VAM Method B1 B2 B3 Supply A1 12.5 11.5 9.5 [10.5] 10.5 A2 10.5 [7.25] 6.5 [7.25] 13.5 12.5 A3 11.5 [6.25] 9.5 8.5 [5] 6.25 Demand 11.5 7.25 13.5 Since the number of occupied cell m+n−1 = 5 and are also independent. There exist non-negative basic feasible solutions. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 627 https://internationalpubls.com The initial transportation cost is [(10.5×9.5)+(7.25×10.5)+(7.25×6.5)+(6.25×11.5)+(5×8.5)]=337.38. 3.3 Applying MODI Method Table corresponding to optimal solution is B1 B2 B3 Supply A1 12.5 11.5 9.5 [10.5] 10.5 A2 10.5 [7.25] 6.5 [7.25] 13.5 12.5 A3 11.5 [6.25] 9.5 8.5 [5] 9.25 Demand 11.5 7.25 13.5 Since all dij≥0 the solution in optimum and unique. The solution is given by x13=10.5,x21=7.25,x22=7.25,x31=6.25,x33=5.The optimal solution is =[(10.5×9.5)+(7.25×10.5)+(7.25×6.5)+(6.25 ×11.5)+(5×8.5)] =337.38 4. Solving Transportation Problem with Cost Values are Decagonal Intuitionistic Fuzzy Numbers using New Proposed Ordering Method 4.1 Numerical Example Consider Costs are Decagonal Intuitionistic Fuzzy Number. B1 B2 B3 Supply A1 (8,9,10,11,12,13,14,15,16,17) (6,7,8,9,10,11,12,13,14,15) (6,7,8,9,10,11,12,13,14,15) (1,2,3,4,5,6,7,8,9,10) (5,6,7,8,9,10,11,12,13,14,) (1,2,3,4,5,6,7,10,11,12) 13.5 A2 (5,6,7,8,9,10,11,12,13,14) (1,2,3,4,5,6,7,8,9,10) (1,2,3,4,5,6,7,8,9,10) (0,1,2,3,4,5,6,7,8,9) (9,10,11,12,13,14,15,16,1 7,18) (3,4,5,6,7,8,9,10,11,12) 11.5 A3 (6,7,8,9,10,11,12,13,14,15) (3,4,5,6,7,8,9,10,11,12) (4,5,6,7,8,9,10,11,13,15) (1,2,3,5,6,7,8,10,11,13) (3,4,5,6,7,8,9,10,11,12) (1,2,3,4,5,6,7,8,9,10) 7.25 Demand 12.5 9.25 10.5 Σ Demand=ΣSupply The problem is a balanced transportation problem. Using the proposed algorithm, the solution of the problem is as follows. Applying accuracy function on decagonal Intuitionistic Fuzzy Number (8,9,10,11,12,13,14,15,16,17)(6,7,8,9,10,12,13,14,15),we have R(Aoc)=12.5 Similarly applying for all the values, we have the following table after ranking Reduced Table B1 B2 B3 Supply A1 12.5 10.5 9.5 13.5 A2 10.5 6.5 13.5 11.5 A3 11.5 9.5 8.5 7.25 Demand 12.5 9.25 10.5 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 628 https://internationalpubls.com − 4.2 Applying VAM Method Table corresponding to initial basic feasible solution is B1 B2 B3 Supply A1 12.5 [5] 10.5 9.5 [10.5] 13.5 A2 10.5 [4.25] 6.5 [9.25] 13.5 11.5 A3 11.5 [7.25] 9.5 8.5 7.25 Demand 12.5 9.25 10.5 Since the number of occupied cell m +n-1 = 5 and are also independent. There exist non-negative basic feasible solutions. The initial transportation cost is [(5×12.5)+(9.5×10.5)+(4.25×10.5)+(9.25×6.5)+(7.25×11.5)]=350.38 4.3 Applying MODI Method Table corresponding to optimal solution is B1 B2 B3 Supply A1 12.5 [5] 10.5 9.5 [10.5] 13.5 A2 10.5 [4.25] 6.5 [9.25] 13.5 11.5 A3 11.5 [7.25] 9.5 8.5 7.25 Demand 12.5 9.25 10.5 Since all dij≥0 the solution in optimum and unique. The solution is given by x11=5, x13=10.5, x21=4.25, x22=9.25, x31=7.25. The optimal solution is [(5×12.5)+(9.5×10.5)+(4.25×10.5)+(9.25×6.5)+(7.25×11.5)]=350.38. 5. Solving Transportation Problem with Cost, Sup- ply and Demand Values are Decagonal Intuitionistic Fuzzy Numbers using New Proposed Ordering Method 5.1 Numerical Example Consider Costs, Supplies and Demands are Decagonal Intuitionistic Fuzzy Number. B1 B2 B3 Supply A1 (2,3,4,5,6,7,8,9,10,11) (1,2,3,4,5,6,7,8,9,10) (4,5,6,7,8,9,10,11,12,12) (1,2,3,4,5,6,7,8,9,10) (6,7,8,9,10,11,12,13,14,15) (4,5,6,7,8,9,10,11,12,13) (3,4,5,6,7,8,9,10,11,12) (9,10,11,12,13,14,15,16,17,18) A2 (5, 6, 7, 8, 9, 10, 11, 12, 13, 14) (1,2,3,4,5,6,7,10,11,12) (9,10, 11, 12, 13, 14,15, 16, 17, 18) (4,5,6,7,8,9,10,11,12,13) (6,7, 8, 9, 10, 12, 13, 14, 15, 16) (2,3,4,5,6,7,S,9) (3,4,5,6,7,8,9,10,11,12) (6,7, 8, 9, 10, 11, 12, 13, 14, 15) A3 (5, 6, 7, 8, 9, 10, 11, 12, 13, 14) (1,2,3,4,5,6,7,8,9,10) (7, 8, 9, 10,11, 12, 13,14) (6,7, 8, 9, 10, 12, 13, 14, 15, 16) (5, 6, 7, 8, 9, 10, 11, 12, 13, 14) (1,2,3,5,6,7,8,10,11,12) (1,2,3,5,6,7,8,13,14,15) (1,2,3,4,7,9,10,11,12,13) Demand (7,8, 9, 10, 11, 12, 13, 14, 15, 16) (3, 6, 7, 8, 9, 10, 12, 13, 14, 15) (2,4,6,7,8,9,10,11,13,15) (1,2,3,4,5,6,7,10,11,13) (1,2,3,5,6,7,8,10,11,12) (4,5,6,7,8,9,10,11,12,13) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 629 https://internationalpubls.com − × × × × × 5.2 Applying VAM Method Table corresponding to initial basic feasible solution is B1 B2 B3 Supply A1 [6.25]6.5 [9.25]8.5 11.5 13.5 A2 [8.25]9.5 13.5 [5.25]11.5 11.5 A3 10.5 12.5 [7.25]9.5 7.25 Demand 12.5 9.25 10.5 28.25 Since the number of occupied cell m+n-1=5 and are also independent. There exist non-negative basic feasible solutions. The initial transportation cost is [(6.25 6.5)+(9.25 8.5)+(8.25 9.5)+(5.25 10.5)+(7.25 9.5)]= 321.63. 5.3 Applying MODI Method Table corresponding to optimal solution is B1 B2 B3 Supply A1 [6.25]6.5 [9.25]8.5 (6)11.5 13.5 A2 [8.25]9.5 (4)13.5 [5.25]10.5 11.5 A3 (4)10.5 (2.5)12.5 [7.25]9.5 7.25 Demand 12.5 9.25 10.5 28.25 Since all dij≥0 the solution in optimum and unique. The solution is given by x11=4.25, x12=9.25,x21=8.25,x23=5.25,x33=7.25 The optimal solution is =[(6.25×6.5)+(9.25×8.5)+(8.25×9.5)+(5.25×10.5)+(7.25×9.5)] =321.63. 6. Solving Unbalanced Transportation Problem using Decagonal Intuitionistic Fuzzy Numbers 6.1 Numerical Example Dl D2 D3 Supply Sl (2,3,4,5,6,7,8,9,10,11) (4,5,6,7,8,9,10,11,12,23) (3,4,5,6,7,8,9,10,11,12) (1,2,3,4,5,6,7,8,9,10) (6,7, 8, 9, 10, 11, 12, 13, 14, 15) (3,4,5,6,7,8,9,10,11,12) (3,4,5,6,7,8,9,10,11,12) (8,9, 10, 11, 12, 13, 14, 15, 16,17) S2 (4,5,6,7,8,9,10,11,12,13) (1,2,3,4,5,6,7,10,11,12) (8,9, 10, 11, 12, 13, 14, 15, 16,17)1 (3,4,5,6,7,8,9,10,11,12) (3, 6, 7, 8, 9, 10, 12, 13, 14, 15) (2,3,4,5,6,7,8,9,10,11) (3,4,5,6,7,8,9,10,11,12) (6,7, 8, 9, 10, 11, 12, 13, 14, 15 S3 (5, 6, 7, 8, 9, 10, 11, 12, 13, 14) (1,2,3,4,5,6,7,8,9,10) (7,8, 9, 10, 11, 12, 13, 14, 15, 16) (3, 6, 7, 8, 9, 10, 12, 13, 14, 15) (4,5,6,7,8,9,10,11,12,13) (1,2,3,5,6,7,8,10,11,12) (1,2,3,5,6,7,8,10,11,12) (1,2,3,4,5,6,7,8,9,10) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 630 https://internationalpubls.com Demand (1,2,3,4,5,6,7,9,10,11) (3,4,5,6,7,8,10,12,13,14) (4,5,6,7,8,9,10,11,12,13) (1,2,3,4,5,6,7,8,9,10) (1,2,3,4,5,6,7,8,9,10) (1,2,3,4,5,6,7,8,9,10) ΣDemand/=ΣSupply The problem is unbalanced transportation problem, so convert the problem Into balanced transportation problem by introducing a dummy column Dl D2 D3 Dummy Supply Sl (2, 3, 4, 5, 6, 7,8, 9, 10, 11) (4, 5, 6, 7, 8, 9, 10, 11, 12, 23) (3, 4, 5, 6, 7, 8, 9, 10,11, 12) (1,2,3,4,5,6,7,8,9,10) (6,7, 8, 9, 10, 11, 12, 13, 14, 15) (3, 4, 5, 6, 7, 8, 9, 10,11, 12) (0,0,0,0,0,0,0,0) (0,0,0,0,0,0,0,0) (3, 4, 5, 6, 7, 8, 9, 10,11, 12) (8,9, 10, 11, 12, 13, 14,15, 16, 17) S2 (4, 5, 6, 7, 8, 9, 10, 11, 12, 13) (1, 2, 3, 4, 5, 6, 7, 10,11, 12) (8,9, 10, 11, 12, 13, 14,15, 16, 17)1 (3, 4, 5, 6, 7, 8, 9, 10,11, 12) (3, 6, 7, 8, 9, 10, 12, 13, 14, 15) (2, 3, 4, 5, 6, 7,8, 9, 10, 11) (0,0,0,0,0,0,0,0) (0,0,0,0,0,0,0,0) (3, 4, 5, 6, 7, 8, 9, 10,11, 12) (6,7, 8, 9, 10, 11, 12, 13, 14, 15 S3 (5, 6, 7, 8, 9, 10, 11, 12, 13, 14) (1,2,3,4,5,6,7,8,9,10) (7,8, 9, 10, 11, 12, 13, 14, 15, 16) (3, 6, 7, 8, 9, 10, 12, 13, 14, 15) (4, 5, 6, 7, 8, 9, 10, 11, 12, 13) (1, 2, 3, 5, 6, 7, 8, 10,11, 12) (0,0,0,0,0,0,0,0) (0,0,0,0,0,0,0,0) (1, 2, 3, 5, 6, 7, 8, 10,11, 12) (1,2,3,4,5,6,7,8,9,10) Demand (1, 2, 3, 4, 5, 6,7, 9, 10, 11) (3, 4, 5, 6, 7, 8, 10, 12, 13, 14) (4, 5, 6, 7, 8, 9, 10, 11, 12, 13) (1,2,3,4,5,6,7,8,9,10) (1,2,3,4,5,6,7,8,9,10) (1,2,3,4,5,6,7,8,9,10) (1, 2, 3, 5, 6, 7, 8, 10,11, 12) (4, 5, 6, 7, 8, 9, 10, 11, 13, 15) ΣDemand=ΣSupply Using the proposed algorithm, the solution of the problem is as follows. Ap- plying accuracy function on Decagonal Intuitionistic Fuzzy Number (2,3,4,5,6,7,8,9,10,11)(4,5,6,7,8,9,10,11,12,23),wehave R(Aoc)=8.75 Similarly applying for all the values, we have the following table after ordering Reduced Table: B1 B2 B3 Dummy Supply A1 6.5 8.5 11.5 0 13.5 A2 9.5 13.5 10.5 0 11.5 A3 10.5 12.5 9.5 0 7.25 Demand 8.75 9.5 5.5 10.5 6.2 Applying VAM Method Table corresponding to initial basic feasible solution is B1 B2 B3 Dummy Supply A1 [6]6.5 [9.5]8.5 11.5 0 13.5 A2 [4.75]9.5 13.5 [5.5]10.5 [5.25]0 11.5 A3 10.5 12.5 9.5 [7.25]0 7.25 Demand 8.75 9.5 5.5 10.5 Since the number of occupied cell m+n−1 = 6 and are also independent. There exists a non-negative basic feasible solution. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 631 https://internationalpubls.com The initial transportation cost is [(6×6.5)+(9.5×8.5)+(4.75×9.5)+(5.5×10.5)+(5.25×0)+(7.25×0)]= 222.63. 6.3 Applying MODI Method Table correspon ding to optimal solution is B1 B2 B3 Dummy Supply A1 [6]6.5 [9.5]8.5 (6)11.5 (5)0 13.5 A2 [4.75]9.5 (4)13.5 [5.5]10.5↓ [5.25]←0 11.5 A3 (3)10.5 (3)12.5 (−1)9.5→ [7.25]0↑ 7.25 Demand 8.75 9.5 5.5 10.5 Step−5: Again, find sign of each dij, the values are all positive (dij>0) then current basic feasible solution is optimal. B1 B2 B3 Dummy Supply A1 [6]6.5 [9.5]8.5 11.5 0 13.5 A2 [4.75]9.5 13.5 10.5 [8.75]0 11.5 A3 10.5 12.5 [5.5]9.5 [10.75]0 7.25 Demand 8.75 9.5 5.5 10.5 Since all dij≥0 the solution in optimum and unique. The solution is given by x11 = 6,x12 = 9.5,x21 = 4.75,x24 = 8.75,x33 = 5.5,x34=10.75. The optimal solution is = [(6×6.5)+(9.5×8.5)+(4.75×9.5)+(8.75×0) +(5.75×9.5)+(10.75×0)] = 219.5. 7. 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