Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 765 https://internationalpubls.com A Simplified Approach to Solving Balanced Generalized Quadratic Fuzzy Transportation Problems M.Venkatachalapathy1, K.Abdul Razak2, M.Ayyappan3 ,V.S.Mathu Suresh4, S.N.Subhathra5 and B.Usha6 1 School of Sciences, Division of Mathematics, SRM Institute of Science and Technology, Tiruchirappalli Campus, SRM Nagar, Irungalur, Trichirappalli, Tamil Nadu, India-621105 2Department of Mathematics, K.Ramakrishnan College of Engineering (Autonomous),Samayapuram,Tiruchirappalli, Tamil Nadu, India-621112 3Department of Mathematics, Francis Xavier Engineering College (Autonomous), Tirunelveli, Tamil Nadu, India-627003 4Department of Mathematics, Rohini College of Engineering and Technology, Palkulam, Kanyakumari, Tamil Nadu, India-629401 5Department of Mathematics, Excel Engineering College (Autonomous), Komarapalayam-637 303, Tamil Nadu,India 6Department of Mathematics, Kongu Engineering College, Perundurai-638 060, Tamil Nadu, India venkatachalapathymaths@gmail.com,arrazak76@gmail.com,ayyappanmath72@gmail.com, sreeuthrad@gmail.com,subhathrasn.eec@excelcolleges.com,usha_b.sh@kongu.edu Article History: Received: 28-10-2024 Revised:12-11-2024 Accepted:19-12-2024 Abstract: The transportation model is a specialized linear programming problem aimed at optimizing the shipment of goods from several sources to different locations. The primary objective is to minimize total transportation costs while ensuring that the supply constraints at each source and the demand requirements at each destination, represented as quadratic fuzzy numbers, are satisfied. This paper introduces a novel approach for solving generalized quadratic fuzzy transportation problems using the Piecewise Quadratic Fuzzy Vogel’s Approximation Method (PQFVAM) and the Piecewise Quadratic Fuzzy Modified Distribution Method (PQFMODI)The proposed method is simple, relying on basic arithmetic and logical calculations with generalized quadratic fuzzy numbers, making it efficient and practical for real-world applications. Keywords: Transportation Problem; Generalized Quadratic Fuzzy Number (GQFN); Piecewise Quadratic Fuzzy Vogel’s Approximation Method (PQFVAM); Piecewise Quadratic Fuzzy Modified Distribution Method (PQFMODI); 1. Introduction In practical decision-making scenarios, data are often incomplete or approximate. To address this, Zadeh [27] introduced fuzzy set theory in 1965, providing a framework for handling uncertainty. In 1987, Dubois and Prade [11] further defined a fuzzy number as a subset of the real line, enabling the mathematical modeling of linguistic variables and uncertain environments. Over time, numerous researchers have contributed to advancing fuzzy transportation problem-solving techniques. Bortolan and Degani explored various ranking methods for ordering fuzzy subsets. Using left and right spreads at particular α-levels, Abbasbandy and Hajjari [1] suggested a ranking method for trapezoidal fuzzy integers. With an emphasis on trapezoidal fuzzy numbers, Venkatachalapathy M. and A. Edward Samuel [25] presented an alternate approach to fuzzy transportation problems utilizing ranking functions. Amit Kumar and Amarpreet Kaur [2] created a novel method for Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 766 https://internationalpubls.com generalized trapezoidal fuzzy numbers in transportation-related issues. Ali Ebrahim [4] and Adrian Ban [3] focused on estimating and simplifying fuzzy numbers for real-world use. Amit Kumar et al. [5] introduced the RM method for ranking generalized trapezoidal fuzzy numbers. Other significant contributions include strategies for handling multi-objective fuzzy transportation problems, such as those proposed by Ammar E.E. and Youness E.A. [6], and solving two-stage problems with symmetric trapezoidal fuzzy numbers by Annie Christi M.S. and Shoba Kumari K. Advancements in Transportation Problem Methods, Hitchcock [15] was among the pioneers in studying the distribution of goods from multiple sources to various destinations. Dharani K. and Selvi D. [11] applied MATLAB to solve intuitionistic fuzzy transportation problems using ranking methods. Edward Samuel A. and Venkatachalapathy M. [13] enhanced the Vogel’s Approximation Method for fuzzy transportation problems. Halgurd Namiq Azeez and Abdulqader Othman Hamad Ameen [16] addressed stochastic transportation problems with fuzzy probability distributions using MATLAB. Jain K.K. et al. [17] employed lexicographic goal programming for multi-objective problems. In engineering and management sciences, fuzzy mathematical models were developed by A. Kaufmann and M.M. Gupta [18]. The Vogel's Approximation Method for transportation problems was enhanced by Korukoğlu S. and Balli S. [19]. To get an initial fundamental workable solution for transportation problems, Yusuf Sahin and Kenan Karagul [20] presented a unique approximation method. Genetic algorithms were used by F.T. Lin [21] to solve transportation problems using fuzzy coefficients. T.S. Liou and M.J. Wang [22] created a ranking method for integral-valued fuzzy numbers. Anuradha D. and Pandian P. [23] presented a novel method for resolving solid transportation issues. Multi-objective solid transportation issues were examined by Sifaoui T. and Aïder M. [24] in relation to sustainable development and adaptations of Vogel's Approximation Method for transportation problem solving. Fuzzy numbers provide a flexible representation for uncertain cost parameters, which often cannot be accurately described as random variables. As demonstrated by Zimmermann [29], solutions from fuzzy linear programming consistently deliver efficient results. This has inspired the development of numerous fuzzy optimization methods for transportation problems, including the contributions by Samuel, Venkatachalapathy M. [14], and A. Jayaraja [26], who refined the Modified Vogel’s Approximation Method (MVAM) for solving problems involving generalized trapezoidal and orthogonal fuzzy numbers. By leveraging these advancements, this study proposes a robust and efficient solution framework tailored to generalized quadratic fuzzy transportation problems. , Additionally, a study on finding optimal solutions for orthogonal fuzzy numbers using the Modified Vogel's Approximation Method (MVAM) was presented. The concept of GQFNs has gained traction as researchers sought advanced methods for capturing fuzzy uncertainty in more complex systems. Early research on fuzzy numbers focused on triangular and trapezoidal shapes due to their simplicity. However, as applications grew in complexity (e.g., in engineering, economics, decision-making, and machine learning), the need for more generalized forms, such as GQFNs, became apparent. Studies have demonstrated the effectiveness of GQFNs in modelling nonlinear uncertainties. Applications in fields like financial forecasting, medical diagnostics, and risk assessment have shown that quadratic membership functions can capture subtle variations in data. Comparisons between GQFNs and traditional linear fuzzy numbers reveal that GQFNs often provide better precision and model fit for complex, real-world data. This paper aims to enhance the balanced generalized quadratic fuzzy transportation problem by introducing modifications that yield an initial basic feasible solution, contributing a refined approach to managing fuzzy data in transportation models. This paper introduces a straight forward and easily understandable method for solving balanced generalized quadratic fuzzy transportation problems. The proposed approach is illustrated with suitable numerical examples and detailed algorithms. The Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 767 https://internationalpubls.com structure of the paper is as follows Section 2, Preliminaries and basic definitions. Sections 3 and 4: Proposed algorithms and numerical examples of balanced generalized piecewise quadratic fuzzy transportation problems, utilizing the PQFVAM and the PQFMODI to formulate transportation problems mathematically and obtain optimal solutions. The conclusion is discussed in section 5. 2. Preliminaries 2.1 Formation of Crisp Balanced Transportation Problems A Fuzzy balanced transportation problem can be represented into tabular form as shown by Fuzzy Cost per unit Distributed Destination 1 2 3 . . n Supply iS ~ 1 11 ~ C 12 ~ C 13 ~ C . . n C 1 ~ 1 ~ S 2 21 ~ C 22 ~ C 23 ~ C . . n C 2 ~ 2 ~ S 3 31 ~ C 32 ~ C 33 ~ C . . n C 3 ~ 3 ~ S . . . . . . . . . . . . . . m 1 ~ m C 2 ~ m C 3 ~ m C . . mn C ~ m S ~ Demand jD ~ 1 ~ D 2 ~ D 3 ~ D 4 ~ D Table .1 Formation of a Fuzzy cost transportation problem where, (i) m represents the number of sources. (ii) n represents the number of destinations. (iii) iS ~ Represents the i th source and jD ~ represents demand. 2.2 Formulation of Generalized Quadratic Fuzzy Linear Programming Model The transportation problem is represented by the generalized quadratic fuzzy linear programming model, which is provided as Min  == = n j jiji m i xcZ 11 ~ . Subject to  =  n j iji Sx 1 ~ ; mi ,......,2,1=  =  m i jji Dx 1 ~ ; nj ,......,2,1= 0jix . for all i and j . In mathematical terms the above problem can be expressed as finding a set of jix ’s, mi ,......,2,1= ; nj ,......,2,1= to Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 768 https://internationalpubls.com Min  == = n j jiji m i xcZ 11 ~ . Subject to  = = n j iji Sx 1 ~ ; mi ,......,2,1=  = = m i jji Dx 1 ~ ; nj ,......,2,1= 0jix . for all i and j . 2.3 Generalized Quadratic Fuzzy Numbers (GQFNs) Generalized Quadratic Fuzzy Numbers (GQFNs) extend the concept of fuzzy numbers, particularly useful in scenarios with uncertain or imprecise data. Unlike traditional fuzzy numbers, which are linear, GQFNs introduce a quadratic (parabolic) membership function that can handle more complex data structures, offering greater flexibility and precision in representing uncertainty. 2.4 Basic Definition of Generalized Quadratic Fuzzy Numbers A membership function that specifies the level of belongingness of every element in a universe is commonly used to express a fuzzy number. For GQFNs, the membership function is quadratic, which means it includes a parabolic function in its construction. Membership Function: For a GQFN, the membership function μ(x) is defined as a quadratic function, generally of the form: where l,m,n,r represent the parameters that define the fuzzy number's "support" and "core" values. The quadratic expression ensures a parabolic shape, allowing more flexible representation for uncertainty and variability. This parabolic nature of the membership function in GQFNs allows for a more accurate depiction of uncertainties compared to linear membership functions. 2.5 Applications of Generalized Quadratic Fuzzy Numbers 2.5.1 Engineering: GQFNs are used in control systems and optimization, where systems may have complex, uncertain inputs that are better captured with quadratic models. 2.5.2 Decision-Making and Risk Analysis: GQFNs provide a refined approach in risk assessment, where the uncertain or ambiguous nature of risk factors can be better quantified. 2.5.3 Medical Diagnosis: In medical fields, where patient data may not be clear-cut, GQFNs offer an enhanced representation of uncertainty in diagnostic values, enhancing interpretability. 2.5.4 Economics and Finance: In financial modelling, GQFNs help in dealing with market uncertainties, providing more robust predictions by capturing nonlinear variations.       +−+− = otherwise rnmlxedxbcxa x 0 ,,,~) ~ ( ~2)~(~ )(~ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 769 https://internationalpubls.com 2.6 Definition: Piecewise Quadratic Fuzzy Numbers In essence, a piecewise quadratic fuzzy number A is a fuzzy number that ),,,,( ~ 54321 lllllA= is represented by the membership function. 2 1 1 22 2 1 2 3 2 32 3 2 2 3 3 42 4 3 2 5 4 52 5 4 1 ( ) 2( ) 1 ( ) 1 2( ) 1 ( ) ( ) 1 2( ) 1 ( ) 2( ) 0 A x l for l x l l l x l for l x l l l x x l for l x l l l x l for l x l l l otherwise   −   −   − +   −   = − +   −  −   −    The piecewise quadratic fuzzy number is a bell shaped curve symmetric about the line 3x l= , possess a supporting interval  1 5,l l .Moreover, 3 1 5 1 ( ) 2 l l l= + and 3 2 4 3l l l l− = − . The  -cut for 1 2  = between the points 2 4( , )l l and they are called cross over points. The interval of confidence at level  is given to be  1 3 1 5 5 3( ) , ( )A l l l l l l  = + − − − . 2.7 Arithmetic Operations The following presents these arithmetic operations as they apply to PQFN.Let 1 2 3 4 5( , , , , )A l l l l l= and 1 2 3 4 5( , , , , )B m m m m m= are two piecewise quadratic fuzzy numbers, Then the arithmetic operations on A and B are given as follows: (i) Addition : 1 1 2 2 3 3 4 4 5 5( , , , , )A B l m l m l m l m l m+ = + + + + + (ii) Subtraction : 1 5 2 4 3 3 4 2 5 1( , , , , )A B l m l m l m l m l m− = − − − − − (iii) Scalar Multiplication 1 2 3 4 5 5 4 3 2 1 ( , , , , ) 0 ( , , , , ) 0 A l l l l l for A l l l l l for               =  =  (iv) Multiplication : 5 1 1 5 4 2 2 4 3 3 2 2 4 4 1 1 5 5 1 1 1 1. ( ( ), ( ), , ( ), ( )) 2 2 2 2 A B l m l m l m l m l m l m l m l m l m= + + + + Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 770 https://internationalpubls.com (v) Division 3 51 2 4 1 5 2 4 3 2 4 1 5 22 2 2 , , , , l ll l lA B m m m m m m m m m   =   + + + +  (If all m’s are non-zero) 5 34 2 1 1 5 2 4 3 2 4 1 5 2 2 2 2 , , , , a aa a aA B b b b b b b b b b   =   + + + +  (If B is negative and all im ’s are non-zero) 3. Algorithms 3.1. Algorithm for Solving a Balanced or Unbalanced Transportation Problems by using Piecewise Quadratic VAM Step 1: Formulate the Transportation Problem Formulate the transportation problem in matrix form: Represent the problem as a matrix, where rows correspond to supply sources and columns correspond to demand destinations. Each element in the matrix represents the transportation cost between a source and a destination. Check if the problem is balanced: (i) Balanced problem: Total supply equals total demand. (ii) Unbalanced problem: Total supply does not equal total demand. If unbalanced, introduce a dummy source or dummy destination (as needed) to balance the problem. The dummy source/ destination will have a cost of zero for all cells involving the dummy. Step 2: Calculate Penalties Calculate penalties for each row and column: For each row and column, find the difference between the smallest and the second smallest cost in that row or column.If two or more costs are identical and the smallest, the penalty is zero for that row or column. Step 3: Select the Highest Penalty Row or Column Identify the row or column with the highest penalty: Look for the row or column with the largest penalty value. If multiple rows or columns have the same penalty, proceed with the following tiebreakers, Select the row or column with the smallest unit cost.If there's still a tie, choose the row or column where the largest allocation can be made (i.e., the one with the highest remaining supply or demand).Allocate to the lowest-cost cell: In the row or column with the highest penalty, allocate as much as possible to the cell with the lowest cost. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 771 https://internationalpubls.com Step 4: Update the Problem Remove satisfied rows or columns: Once a row or column’s supply or demand has been completely allocated, remove it from the matrix (or mark it as satisfied). Step 5: Repeat Until All Constraints are satisfied Repeat steps 2 and 3: Continue calculating penalties and allocating resources until all supply and demand constraints are fully satisfied. Step 6: Obtain the Initial Basic Feasible Solution Final solution: The result of the allocation process in Step 5 gives you the initial basic feasible solution for the transportation problem. This solution is feasible, but it may not necessarily be optimal. 3.2 Numerical Example A beverage company operates three bottling plants located across a region, with each plant having the following daily production capacities: Plant 1, Plant 2, and Plant 3. The company needs to supply three distribution hubs Hub X, Hub Y, and Hub Z each with specific daily beverage requirements. The transportation cost for shipping one million liters of beverage from each plant to each distribution hub is provided in the table below (in hundreds of rupees). The company aims to determine the optimal shipment quantities from each bottling plant to each distribution hub in order to minimize the total transportation cost, which is represented as a generalized quadratic fuzzy number. The goal is to determine the Initial Basic Feasible Solution (IBFS) for this transportation problem. Plant 1 Plant 2 Plant 3 Supply Hub X (3,5,6,7,9) (7,9,10,11,13) (11,13,14,15,17) (47,49,50,51,53) Hub Y (9,11,12,13,15) (16,18,19,20,22) (18,20,21,22,24) (47,49,50,51,53) Hub Z (12,14,15,16,18) (11,13,14,15,17) (14,16,17,18,20) (47,49,50,51,53) Demand (27,29,30,31,33) (37,39,40,41,43) (52,54,55,56,58) Table 2. Generalized Quadratic Fuzzy Transportation table Check for Balance: Fuzzy Total Supply =(47,49,50,51,53)+(47,49,50,51,53)+(47,49,50,51,53)=(141,147,150,153,159) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 772 https://internationalpubls.com Fuzzy Total Demand =(27,29,30,31,33)+(37,39,40,41,43)+(52,54,55,56,58) =(116,122,125,128,134) Since Fuzzy Total Supply ≠ Fuzzy Total Demand, It is called unbalanced. Step 1: Add a Dummy Column, To balance the problem, a dummy Column is added with costs of (0,0,0,0,0) and demand equal to the excess supply: Step 2: Formulate the Piecewise Quadratic Fuzzy Transportation Problem in Matrix Form The transportation matrix for costs (in generalized quadratic fuzzy numbers) is: Plant 1 Plant 2 Plant 3 Plant 4 Supply Penalty Hub X (3,5,6,7,9) (7,9,10,11,13) (11,13,14,15,17) (0,0,0,0,0) (47,49,50,51,53) (3,5,6,7,9) Hub Y (9,11,12,13,15) (16,18,19,20,22) (18,20,21,22,24) (0,0,0,0,0) (47,49,50,51,53) (9,11,12,13,15) Hub Z (12,14,15,16,18) (11,13,14,15,17) (14,16,17,18,20) (0,0,0,0,0) (47,49,50,51,53) (12,13,14,15,17) Demand (27,29,30,31,33) (37,39,40,41,43) (52,54,55,56,58) (7,19,25,31,43) P en al ty (0,4,6,8,12) (-2,2,4,6,10) (-3,1,3,5,9) (0,0,0,0,0) Table 3. Balanced Generalized Quadratic Fuzzy Transportation table Step 3: Calculate Penalties and Allocate Supplies Using the Piecewise Quadratic Fuzzy Vogel's Approximation Method (PQFVAM): To solve the problem, The difference between the smallest and next smallest transportation expenses is used to calculate the penalties for each row and column. After determining which row or column has the largest penalty, the cell with the lowest cost receives the maximum allowable allocation. The supply and demand are modified in accordance with each allocation. This process is repeated until all supply and demand constraints are satisfied. The optimal allocation for the generalized quadratic fuzzy transportation problem is determined. (3,5,6,7,9) (37,39,40,41,43) (7,9,10,11,13) (4,8,10,12,16) (11,13,14,15,17) (0,0,0,0,0) (27,29,30,31,33) (9,11,12,13,15) (16,18,19,20,22) (14,18,20,22,26) (18,20,21,22,24) (0,0,0,0,0) (12,14,15,16,18) (11,13,14,15,17) (4,18,25,32,46) (14,16,17,18,20) (7,19,25,31,43) (0,0,0,0,0) Table 4. Optimal Assignment Generalized Quadratic Fuzzy Transportation table Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 773 https://internationalpubls.com The total cost of the solution is calculated by multiplying the transportation cost per unit in each basic cell by the corresponding number of units allocated, and then summing the results across all allocated cells. = (7,19,25,31,43)(0,0,0,0,0) + (27,29,30,31,33)(9,11,12,13,15)+ (37, 39, 40, 41, 43)(7,9,10,11,13) + (14,18,20,22,26)(18,20,21,22,24)+ (4, 18, 25, 32, 46)(14,16,17,18,20) + (4,8,10,12,16)(11,13,14,15,17) = (1628, 1732, 1745, 1758, 1862) 3.2.1 Membership Function The PQFN is a bell shaped curve symmetric about the line x=1745, possess a supporting interval [1628, 1862]. Moreover, 17453 =l and 13= 3 2 4 3l l l l− = − . The  -cut for 1 2  = between the points (1732, 1758).The interval of confidence at level  is given to be  5.1803,5.1686=A .              − + − + −  − = otherwise xfor x xfor x xfor x xfor x x A 0 18621758 21632 )1862( 175817451 338 )1745( 174517321 338 )1732( 17321628 21632 )1628( )( 2 2 2 2 ~ % MATLAB Code to Plot Symmetric Bell-Shaped Fuzzy Curve Computing the Valuation of Piecewise Quadratic Fuzzy Number Figure 1. Symmetric Bell shaped fuzzy curve with  -cut Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 774 https://internationalpubls.com 4. Piecewise Quadratic Fuzzy Modified Distribution Method (PQFMODI) The PQFMODI is an effective technique for verifying the optimality of the initial feasible solution in transportation problems. 4.1 Algorithm The PQFMODI, also known as the u−v method, offers an efficient approach to finding the minimum-cost solution for transportation problems. Unlike the Stepping Stone Method, which requires evaluating costs for all unoccupied cells by drawing closed paths, the PQFMODI method only needs one closed path for the unoccupied cell with the highest opportunity cost. Step 1 : Determine Initial Basic Feasible Solution Start by finding an initial feasible solution using Piecewise Quadratic Fuzzy Vogel’s Approximation Method. Step 2: Calculate Dual Variables ui and vj Determine the values of the dual variables ui and vj for each row and column. Step 3: Compute Opportunity Costs Calculate the opportunity cost for each unoccupied cell. Step4: Evaluate Opportunity Costs (i) If all unoccupied cells have opportunity costs that are positive or zero, the solution is optimal. (ii)If any unoccupied cell has a negative opportunity cost, the solution is not optimal, and further cost reduction is possible. Step 5: Select Cell with Smallest Negative Opportunity Cost Choose the unoccupied cell with the most negative opportunity cost for the next iteration. Step 6: Draw Closed Path for Selected Cell Draw a closed path or loop starting from the selected cell. Right-angle turns are allowed only at occupied cells and the initially selected unoccupied cell. Step 7: Assign Signs on Path Cells Mark the corner cells of the closed path with alternating plus and minus signs, starting with a plus sign on the cell being evaluated. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 775 https://internationalpubls.com Step 8: Determine Units to Ship Determine the maximum number of units that can be shipped to the selected unoccupied cell. This is based on the smallest value at the cells marked with a minus sign. Add this quantity to cells marked with a plus and subtract it from cells marked with a minus to adjust the allocations. The unoccupied cell becomes occupied. Step 9: Repeat Until Optimal Solution is Achieved Continue this process until all opportunity costs for unoccupied cells are positive or zero, indicating an optimal solution. 4.2 Degeneracy in Transportation Problems A transportation problem is considered degenerate when the number of positive allocations (occupied cells) is less than m+n−1, where m represents origins and n represents destinations. Degeneracy can occur in two stages: 1. Initial Solution: When setting up the initial feasible solution. 2. Testing Optimality: When evaluating the current solution for optimality. To resolve degeneracy, an artificial quantity is assigned to the unoccupied cell with the lowest transportation cost, helping to proceed with the calculations and maintain feasibility. 4.3 Numerical Examples with Piecewise Quadratic Fuzzy MODI method Plant 1 Plant 2 Plant 3 Supply Hub X (3,5,6,7,9) (7,9,10,11,13) (11,13,14,15,17) (47,49,50,51,53) Hub Y (9,11,12,13,15) (16,18,19,20,22) (18,20,21,22,24) (47,49,50,51,53) Hub Z (12,14,15,16,18) (11,13,14,15,17) (14,16,17,18,20) (47,49,50,51,53) Demand (27,29,30,31,33) (37,39,40,41,43) (52,54,55,56,58) Table 5. Generalized Quadratic Fuzzy Transportation table Step 1 Plant 1 Plant 2 Plant 3 Plant 4 Supply Hub X (3,5,6,7,9) (7,9,10,11,13) (11,13,14,15,17) (0,0,0,0,0) (47,49,50,51,53) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 776 https://internationalpubls.com Hub Y (9,11,12,13,15) (16,18,19,20,22) (18,20,21,22,24) (0,0,0,0,0) (47,49,50,51,53) Hub Z (12,14,15,16,18) (11,13,14,15,17) (14,16,17,18,20) (0,0,0,0,0) (47,49,50,51,53) Demand (27,29,30,31,33) (37,39,40,41,43) (52,54,55,56,58) (22,24,25,26,28) Table 6. Balanced Generalized Quadratic Fuzzy Transportation table Step 2, Initial Basic Feasible Solution Using PQFVAM . Let us use the PQFVAM to find the initial basic feasible solution. In the previous problems PQFVAM method is used to solve to this problem. 1v 2v 3v 4v 1u (3,5,6,7,9) (37,39,40,41,43) (4,8,10,12,16) (0,0,0,0,0) (7,9,10,11,13) (11,13,14,15,17) 2u (27,29,30,31,33) (16,18,19,20,22) (14,18,20,22,26) (0,0,0,0,0) (9,11,12,13,15) (18,20,21,22,24) 3u (12,14,15,16,18) (11,13,14,15,17) (4,18,25,32,46) (22,24,25,26,28) (14,16,17,18,20) (0,0,0,0,0) Table 7. First Optimal Assignment Generalized Quadratic Fuzzy Transportation table Now choose the smallest (most) negative value from opportunity cost is ( 16, 8, 4,0,8)− − − and draw a closed path. The following table shows the closed path (3,5,6,7,9) (37,39,40,41,43) (4,8,10,12,16) (0,0,0,0,0) (7,9,10,11,13) (11,13,14,15,17) (27,29,30,31,33) (16,18,19,20,22) (18,20,21,22,24) (0,0,0,0,0) (9,11,12,13,15) (18,20,21,22,24) -ve +ve (12,14,15,16,18) (11,13,14,15,17) (4,18,25,32,46) (22,24,25,26,28) (14,16,17,18,20) (0,0,0,0,0) +ve -ve Table 8. Closed Path Optimal Assignment Generalized Quadratic Fuzzy Transportation table Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 777 https://internationalpubls.com Select the smallest value with a negative location on the closed route from the following values (14, 18, 20, 22, 26). The quantity of units that can be transported to the entering cell is indicated by this amount. Then, deduct this amount from the cells with minus signs and add it to all the cells at the closed path's corner points that are indicated with plus signs. An empty cell becomes populated in this manner. Calculating iu and jv using i j iju v c+ = Now finding opportunity cost using ( )ij ij i jc u v = − + Next, create a closed path using the opportunity Fuzzy cost's smallest (most negative) value. Next, choose the smallest value with a negative location on the closed route from the range (2, 4, 5, 6, 8). This will show how many units can be transported to the cell that is entering. Once more, deduct this amount from the cells with minus signs and add it to all the cells at the closed path's corner points that are indicated with plus signs. By doing this, an empty cell is certain to be filled. The solution is now optimal as all of the opportunity costs are non-negative. 1v 2v 3v 4v 1u (2,4,5,6,8) (37,39,40,41,43) (-4,2,5,8,14) (0,0,0,0,0) (3,5,6,7,9) (7,9,10,11,13) (11,13,14,15,17) 2u (19,23,25,27,31) (16,18,19,20,22) (18,20,21,22,24) (16,22,25,28,34) (9,11,12,13,15) (0,0,0,0,0) 3u (12,14,15,16,18) (11,13,14,15,17) (20,40,50,60,80) (0,0,0,0,0) (14,16,17,18,20) Table 9. Closed Path Optimal Assignment Generalized Quadratic Fuzzy Transportation table The minimum transportation cost is: = (2,4,5,6,8)(3,5,6,7,9)+(37,39,40,41,43)(7,9,10,11,13)+(-4,2,5,8,14)(11,13,14,15,17) +(19,23,25,27,31)(9,11,12,13,15)+(16,22,25,28,34)(0,0,0,0,0)+(20,40,50,60,80) (14,16,17,18,20) = (1497, 1633, 1650, 1667, 1803) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 778 https://internationalpubls.com 4.3.1 Membership Function The piecewise quadratic fuzzy number is a bell shaped curve symmetric about the line x=1745, possess a supporting interval [1497, 1803]. Moreover, 16503 =l and 17= 3 2 4 3l l l l− = − . The  -cut for 1 2  = between the points (1633, 1667) and they are called cross over points. The interval of confidence at level  is given to be  5.1726,5.1505=A . Computing the Valuation of membership function of Piecewise Quadratic Fuzzy Numbers                 − + − + −  − = otherwise xfor x xfor x xfor x xfor x x A 0 18031667 36992 2)1862( 166716501 578 2)1650( 165016331 578 2)1650( 16331497 36992 2)1497( )(~ % MATLAB Code to Plot Symmetric Bell-Shaped Fuzzy Curve Computing the Valuation of Piecewise Quadratic Fuzzy Number Figure.2. Symmetric Bell shaped fuzzy curve with  -cut The provided image displays a symmetric bell-shaped fuzzy number represented by a membership function μ(x), with x representing the transportation costs or some related variable. Let’s break down the explanation of the piecewise quadratic fuzzy numbers illustrated in the graph: Quadratic Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 779 https://internationalpubls.com functions ensure smoothness and gradual changes in the membership values.The symmetric bell- shaped curve effectively models uncertainty and variability in a system. By providing a range of possible values with associated likelihoods, it helps in robust optimization and better decision- making under uncertainty. 5. Conclusion The PQFMODI method offers a systematic approach through dual variables and opportunity costs, allowing quick identification of cost-saving reallocations. Meanwhile, the Stepping Stone method provides a deeper visual understanding of closed paths and feasible adjustments, which can be valuable in understanding the underlying allocation dynamics.For cases with degeneracy, both methods can handle it by introducing an artificial quantity, ensuring that the solution process remains feasible and accurate. Overall, these methods are essential tools in operations research and optimization, supporting decision-making in logistics, supply chain management, and other resource allocation applications by offering effective ways to minimize transportation costs while ensuring efficient resource distribution. this study tries to make a new method to solve transportation problem more efficiently. Compliance with Ethical Standards Acknowledgment The authors express their gratitude to the anonymous editor and reviewers for their time and effort in reviewing the manuscript. We deeply appreciate their valuable comments and suggestions, which have significantly improved the quality of the work. Conflicts of Interest The authors declare no conflicts of interest regarding the publication of this paper. All authors contributed equally to the research, reviewed, and approved the final manuscript. Special thanks are extended to the editor and anonymous referees for their constructive feedback. This research did not receive support from any funding sources. Research Involving Human Participants and/or Animals This article does not include any studies involving human participants or animals conducted by the authors. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 780 https://internationalpubls.com Funding Information This research was not supported by any specific grant from funding agencies in the public, commercial, or non-profit sectors. Ethical Approval No ethical approval was required for this research. Informed Consent All studies referred to in this paper are discussed in the Literature Review. References [1] Abbasbandy, S. & Hajjari, T., A new approach for the ranking of trapezoidal fuzzy numbers, Computers and Mathematics with Applications, 57, (2009), 413-419. [2] Amarpreet Kaur & Amit Kumar, A new approach for solving fuzzy transportation problems using generalized trapezoidal fuzzy numbers, Applied Soft Computing (2011), 1-34. [3] Adrian Ban 2008, ‘Approximation of fuzzy numbers by trapezoidal fuzzy numbers preserving the expected interval’, Fuzzy sets and systems, vol. 159, no. 11, pp. 1327-1344. [4] Ali Ebrahim, N.: A simplified new approach for solving fuzzy transportation problems with generalized trapezoidal fuzzy numbers. Appl. Soft Comput. 171–176 (2014). [5] Amit Kumar, Pushpinder Singh, Amarpreet Kaur & Parmpreet Kaur, 2010, RM Approach for Ranking of Generalized Trapezoidal fuzzy number, Fuzzy Information and Engineering, vol. 2, no. 1, pp. 37-47. [6] Ammar, EE & Youness, EA 2005, ‘Study on multi-objective transportation problem with fuzzy numbers’, Applied Mathematics and Computation, vol. 66, pp. 241-253. [7] Ameen, A. O. (2015). Improved Two-phase Solution Strategy for Multi objective Fuzzy Stochastic Linear Programming Problems with Uncertain Probability Distribution (1st ed.). Malaysia: Universiti Teknologi Malaysia. [8] Annie Christi, MS & Shoba Kumari, K 2015, ‘Two Stage Fuzzy Transportation Problem Using Symmetric Trapezoidal Fuzzy Number’, International journal of Engineering Invention, vol. 4, no. 11, pp. 07-10. [9] Balakrishnan, N. (1990) Modified Vogel’s Approximation Method for the Unbalanced Transportation Problem. Applied Mathematics Letters, 3, S11. http://dx.doi.org/10.1016/0893-9659(90)90003-t. [10] Dantzig, G.B. & M.N. Thapa, Springer: Linear Programming: 2: Theory and Extensions, Princeton University Press, New Jersey, 1963. [11] Dharani, K;Selvi, D. (2018). Solving intuitionistic fuzzy transportation problem with ranking method using MATLAB code. Applied Science and Computations, 5(12), 20-26. [12] Dubois, D. & Prade, H., Fuzzy set and systems theory and application. New [13] York, NY: Academic Press (1980). [14] Edward Samuel, A. and Venkatachalapathy, M. (2011) Modified Vogel’s Approximation Method for Fuzzy Transportation Problems. Applied Mathematical Sciences, 5, 1367-1372. [15] Edward Samuel, A. & Venkatachalapathy, M, A new procedure for solving the generalized trapezoidal fuzzy transportation problem, Advances in Fuzzy Sets and Systems, 12(2) (2012),111-125. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 781 https://internationalpubls.com [16] Hitchcock,F.L., The distribution of a product from several sources to Numerous localities, Journal of Mathematical Physics, 20 (1941) 224-230. [17] Halgurd Namiq Azeez , Abdulqader Othman Hamad Ameen, Solving Stochastic Transportation Electricity Problem With Fuzzy Information On Probability Distribution Using MATLAB Program, Science Journal of the University of Zakho, Vol. 12, No. 1, pp. 116 – 137, 2024. [18] Jain, K.K., Bhardwaj, R., Choudhary, S.: A Multi-Objective transportation problem solve by lexio graphic goal programming. Int. J. Recent Technol. Eng. (IJRTE) 1842–1846 (2019). [19] A.Kaufmann, M.M. Gupta, Fuzzy Mathematical Models in Engineering and Management Science, Amsterdam, The Netherland: Elsevier, (1988). [20] Korukoğlu, S. and Balli, S. (2011) An Improved Vogel’s Approximation Method for the Transportation Problem. Mathematical and Computational Applications, 16, 370-381. [21] Karagul, Kenan;Sahin, Yusuf. (2020). A novel approximation method to obtain initial basic feasible solution of transportation problem. Journal of King Saud University-Engineering Sciences, 32(3), 211-218. [22] F.T. Lin, Solving the Transportation Problem with Fuzzy Coefficients using Genetic Algorithms, Proceedings IEEE International Conference on Fuzzy Systems, (2009) 20-24. [23] T.S. Liou, M.J. Wang, Ranking fuzzy number with integral values, Fuzzy Sets and Systems 50 (1992) 247-255. [24] Pandian, P & Anuradha, D 2010, ‘A New Approach for Solving Solid Transportation Problems’, Applied Mathematical Sciences, vol. 4, no. 72, pp. 3603-3610. [25] Sifaoui, T., Aïder, M. (2022). A Multi-objective Solid Transportation Problem in Sustainable Development. In: Verdegay, J.L., Brito, J., Cruz, C. (eds) Computational Intelligence Methodologies Applied to Sustainable Development Goals. Studies in Computational Intelligence, vol 1036. Springer, Cham. [26] Venkatachalapathy.M. & Edward Samuel, A , An alternative method for solving fuzzy transportation problems using ranking functions, International Journal of Applied Mathematical Sciences, Volume 9, Number 1 (2016), pp. 61-68. [27] Venkatachalapathy.M, Jayaraja.A & A.E. Samuel, A study on solving octagonal fuzzy numbers using the modified Vogel’s approximation method, International Journal of Pure and Applied Mathematics, Volume 118, No. 6, 2018, 201-207. [28] Zadeh, L.A., Fuzzy sets, Information and Control, 8 (1965), 338-353. [29] Zena S. Mahdi,H. A. Wasi,Mushtak A. K. Shiker, Solving transportation problems by using modification to Vogel’s approximation method , AIP Conf. Proc. 2834, 080110 (2023). [30] Zimmermann, H.J., Fuzzy programming and linear programming with several objective functions, Fuzzy Sets and Systems, 1 (1978), 45-55. javascript:void(0) javascript:void(0) javascript:; javascript:; javascript:;