IHJPAS. 36(1)2023 318 This work is licensed under a Creative Commons Attribution 4.0 International License Fuzzy Transportation Model Technique to Determine the Minimum Total Cost Using a Novel Ranking Function Rasha Jalal Mitlif rasha.j.mitlif@uotechnology.edu.iq Mathematics and Computer Applications, Department of Applied Sciences, University of Technology, Baghdad, Iraq. Abstract The transportation problem (TP) is employed in many different situations, such as scheduling, performance, spending, plant placement, inventory control, and employee scheduling. When all variables, including supply, demand, and unit transportation costs (TC), are precisely known, effective solutions to the transportation problem can be provided. However, understanding how to investigate the transportation problem in an uncertain environment is essential. Additionally, businesses and organizations should seek the most economical and environmentally friendly forms of transportation, considering the significance of environmental issues and strict environmental legislation. This research employs a novel ranking function to solve the transportation problem (TP), where fuzzy triangular numbers represent the fuzzy demand and supply (DAS). The fuzzy model is transformed and compressed to a crisp model (CM), and the results are compared using the northwest corner method and the least cost method. In addition, a numerical example of the fuzzy transportation model (FTM) is shown. Keywords: Fuzzy transportation model, Fuzzy set, Ranking function, Triangular fuzzy number, Membership function. 1. Introduction In today's economy, where there is intense competition, businesses must find new ways to produce and offer value to customers. Finding the best ways and times to deliver goods to clients in the quantities they want efficiently is getting more and more challenging. Around the world, some practical problems are typically solved by focusing on the transportation problem. In the manufacturing sector, the transportation issue is significant, as are other issues. Fuzzy transportation (FT) tries to find the lowest transportation costs (TC) of particular commodities via a capacitated network when the supply and demand of nodes and the capacity and cost of edges are represented as fuzzy integers. Many researchers have studied the fuzzy transportation problem (FTP). In a fuzzy environment, Maheswari and Ganesan provided a simple technique for solving the FTP in which TC, sources of doi.org/10.30526/36.1.2991 Article history: Received 27 Augest 2022, Accepted 24 October 2022, Published in January 2023. Ibn Al-Haitham Journal for Pure and Applied Sciences Journal homepage: jih.uobaghdad.edu.iq https://creativecommons.org/licenses/by/4.0/ mailto:rasha.j.mitlif@uotechnology.edu.iq IHJPAS. 36(1)2023 319 supply, and demand at destinations were all represented by fuzzy pentagonal numbers (PFI). The problem of fuzzy transportation (FT) was solved without transforming to its corresponding crisp form, using a ranking methodology and a new pentagonal fuzzy arithmetic [1]. Christi and Kumari applied a robust ranking technique to discover the best fuzzy solution to the transportation problem [2]. To obtain the best solution, Mary and Sivasankari employed the direct method of FTP utilizing fuzzy hexagonal numbers (HFN) with alpha cut (AC) [3]. Purushothkumar et al. used the diagonal optimum technique approach to solve fully fuzzy transportation problems [4]. Several articles have introduced a new ranking function procedure by which the fuzzy number is converted into crisp [5-9]. The primary goal of this article is to propose a comparative method for solving FTP. The remainder of the paper is organized as follows: some fundamental definitions are provided in Section 2 and will be used throughout the article. The ranking function is presented for fuzzy numbers (FN) in Section 3. In Section 4, an algorithm is used to solve FTP. A numerical example is given in Section 5 to demonstrate the proposed strategy. Finally, in Section 6, the conclusions are provided. 2. Preliminaries notions Simple definitions relevant to this work are presented in this section: Definition1 [10]: Let ß̃ be the universal set. A fuzzy set in ß̃ is a set of ordered pairs, A= {ß̃, 𝜇ß̃(𝒢); 𝒢 𝜖 Ủ } , where 𝜇ß̃: Ủ →[0,1] is called the membership set. Definition 2 [11]: A triangular fuzzy number (TFN) �̃� is denoted as (𝓅 , 𝒹, ę), where 𝓅 , 𝒹, and ę are real numbers, and its membership function 𝜇ß̃ is as follows: 𝜇ß̃ = { 𝒢−𝓅 𝒹−𝓅 1 ę−𝒢 ę−𝒹 𝓅 ≤ 𝒢 ≤ 𝒹 𝒢 = 𝒹 𝒹 ≤ 𝒢 ≤ ę Definition 3 [12-13]: A ranking function of a FN R: F(Ủ) → Ủ is the set of all FN defined on Ủ:, which maps each FN into a real number. The properties of the RF are the following: (i) ß1̃ ≤ ß2̃ iff ℜ(ß1̃) ≤ ℜ(ß2̃) (ii) ß1̃ > ß2̃ iff ℜ(ß1̃) > ℜ(ß2̃) (iii) ß1̃ = ß2̃iff ℜ(ß1̃) = ℜ(ß2̃) A fuzzy RF meets ℜ (k ß1̃ +𝑙 ß2̃) = 𝑘 ℜ(ß1̃)+𝑙 ℜ(ß2̃) for every ß1̃, ß2̃∊ 𝐹(Ủ). 3. Suggested a Novel Ranking Function of the Fuzzy Transportation Model: The following is the membership function that is utilized: �̃�(ß̃) = { 𝒻(𝓈−ŗ) ℓ−ŗ 𝒻 𝒻(𝓉−𝓈) 𝓉−ℓ ŗ ≤ 𝓈 ≤ ℓ 𝓈 = ℓ ℓ ≤ 𝓈 ≤ 𝓉 IHJPAS. 36(1)2023 320 Now, utilizing the triangular membership function (TMF) is to find a novel ranking function as follows: 𝒻(𝓈 − ŗ) ℓ − ŗ = ʋ → 𝓈 = ŗ + 𝛼 𝒻 (ℓ − ŗ) = 𝑖𝑛𝑓 �̃� (ʋ) 𝒻(𝓉 − 𝓈) 𝓉 − ℓ = ʋ → 𝓈 = 𝓉 − 𝛼 𝒻 (𝓉 − ℓ) = 𝑠𝑢𝑝 �̃� (ʋ). Hence, the following ranking function 𝒥(�̃�) can be calculated: 𝒥(�̃�) = [ 1 2 ∫ ʋ10 2 𝛿 0 [ 𝑖𝑛𝑓 (�̃�( ʋ )) + 𝑠𝑢𝑝 (�̃�( ʋ )) ]𝑑ʋ ] ∫ ʋ10 2 𝛿 0 𝑑ʋ 𝒥(�̃�) = [ 1 2 ∫ ʋ10 2 𝛿 0 [ŗ + ʋ 𝛿 (ℓ − ŗ) + 𝓉 − ʋ 𝛿 (𝓉 − ℓ) ]𝑑ʋ ] ∫ ʋ10 2 𝛿 0 𝑑ʋ 𝒥(�̃�) = [ 1 2 ∫ ʋ10 2 𝛿 0 ŗ + ʋ11 2𝛿 (ℓ − ŗ) + ʋ10 2 𝓉 − ʋ11 2𝛿 (𝓉 − ℓ)]𝑑ʋ ] ∫ ʋ10 2 𝛿 0 𝑑ʋ 𝒥(�̃�) = [ 1 2 [ ʋ11 22 ŗ + ʋ12 24𝛿 (ℓ − ŗ) + ʋ11 22 𝓉 − ʋ12 24𝛿 (𝓉 − ℓ)]| 0 𝛿 ] ʋ11 22 | 0 𝛿 𝒥(�̃�) = [ 1 2 [ 𝛿11 22 ŗ + 𝛿12 24𝛿 (ℓ − ŗ) + 𝛿11 22 𝓉 − 𝛿12 24𝛿 (𝓉 − ℓ) ] 𝛿11 22 𝒥(�̃�) = [ 1 2 [ 𝛿11 22 ŗ + 𝛿12 24 (ℓ − ŗ) + 𝛿11 22 𝓉 − 𝛿12 24 (𝓉 − ℓ) ] ] 𝛿11 22 𝒥(�̃�) = [ 𝛿11 2 [ ŗ 22 + (ℓ − ŗ) 24 + 𝓉 22 − (𝓉 − ℓ) 24 ] ] 𝛿11 22 𝒥(�̃�) = [ 𝛿11 2 [ 24 ŗ + 22 ℓ − 22 ŗ 528 + 24 𝓉 − 22 𝓉 + 22 ℓ 528 ] ] 𝛿11 22 𝒥(�̃�) = [ 𝛿11 2 [ 24 ŗ + 22 ℓ − 22 ŗ + 24 𝓉 − 22 𝓉 + 22 ℓ 528 ] ] 𝛿11 22 IHJPAS. 36(1)2023 321 𝒥(�̃�) = [ 𝜆11 2 [ 2 ŗ + 44 ℓ + 2 𝓉 528 ] ] 𝛿11 22 𝒥(�̃�) = [[ 2 ŗ + 44 ℓ + 2 𝓉 528 ] ] 𝛿11 22 The following is the proposed formula for a class of valuation ranking functions: 𝒥(�̃�) = 2 ŗ + 44 ℓ + 2 𝓉 48 4. Algorithm Fuzzy Transportation Model Technique to Determine the Minimum Total Cost using a Novel Ranking Function In this section, a novel RF is used to discover the best solution for the FTP. The RF of each FN in the fuzzy problem under consideration is used and resulted in an equivalent crisp linear programming problem (CLPP), which is solved using two methods performed on each mathematical operation on FN. In contrast, in our proposed method, crisp numbers perform all arithmetic operations. The following are the steps in the suggested method: Step1. Triangular Fuzzy Transportation Problems (TFTP) with fuzzy demand, supply, and cost were considered. The parameters are in triangular fuzzy number expressions. Step2. The FT problems were converted to crisp value programming problems (CVPP) by the new RF 𝒥(�̃�) = 2 ŗ + 44 ℓ + 2 𝓉 48 Step3. Transportation problem is tested to be balanced or unbalanced. Step4. The crisp transportation model (CTM) is solved using NWC and LCM methods. Step5. The two methods are compared to find the optimal solution (OS). Step6. An example is given to test the optimality. Step7. The chosen FTP example1 is solved utilizing all the provided approaches, indicating that it is possible to choose the optimal method among the NWC and LCM. Example1: Consider a transportation problem (TP) with parameters as triangular fuzzy numbers. Table 1.Fuzzy Transportation Problem 𝑭𝑩𝟏 𝑭𝑩𝟐 𝑭𝑩𝟑 Fuzzy Supply 𝑭𝑨𝟏 [14,16,18] [56,58,60] [44,46,48] [66,68,70] 𝑭𝑨𝟐 [82,84,86] [30,32,34] [14,16,18] [38,40,42] 𝑭𝑨𝟑 [86,88,90] [26,28,30] [68,70,72] [86,88,90] Fuzzy Demand [38,40,42] [66,68,70] [86,88,90] IHJPAS. 36(1)2023 322 The given fuzzy problem (FP) can be converted into a crisp value problem (CVP) applying the new ranking 𝒥(�̃�) = 2 ŗ + 44 ℓ + 2 𝓉 48 Table 2. Crisp Transportation Problem (new ranking function) 𝐵1 𝐵2 𝐵3 Supply 𝐴1 16 58 46 68 𝐴2 84 32 16 40 𝐴3 88 28 70 88 Demand 40 68 88 Next, the transportation problem (TP) is checked proving to be balanced, as the sum of supply and demand is equal to 196. The problem is solved using 1) is shown in Table 3: Table 3. The North west corner method 𝑩𝟏 𝑩𝟐 𝑩𝟑 Supply 𝑨𝟏 16 58 46 68 𝑨𝟐 84 32 16 40 𝑨𝟑 88 28 70 88 Demand 40 68 88 The total cost is found to be equal to 9704 (= 16*40+58*28+32*40+16*0+70*88) 2) Least cost method procedure is shown in Table 4: Table 4. The procedure of Least cost method 𝑩𝟏 𝑩𝟐 𝑩𝟑 Supply 𝑨𝟏 16 58 46 68 𝑨𝟐 84 32 16 40 𝑨𝟑 88 28 70 88 Demand 40 68 88 40 40 0 28 88 88 40 20 88 28 40 68 IHJPAS. 36(1)2023 323 The total cost is found to be equal to = 16*40+46*28+16*40+28*68+70*20= 5872 The two methods are compared as shown in Table 5. Table 5. Total cost for North West Corner and Least Cost Methods No Method Transportation Cost 1 North West Corner Method 9704 2 Least Cost Method 5872 The total cost was calculated for each method of transportation model according to Table 5. The value describes the total cost of decision-makers in the transportation model. Generally, it is chosen to be a monotonically decreasing function concerning the transportation model, which means the lower the total cost value, the stronger the relationship between the compared results. The minimum determined the fuzzy probabilistic preference relation between the two methods. Obtain the comparison result obtained with the total cost relation, which is in the form of ranking the numbers. According to the values, the lower the deal, the better the transportation model's ranking will be. 5. Conclusion This research proposes a novel RF to determine the lowest total FTP cost. The cost of transportation was considered, as well as supply and demand. In nature, fuzzy triangular numbers are more realistic and general. A new classification process was used to change the FTP of the triple number to CTP. As shown in the comparison table (Table 5), the proposed least-cost method produced better results than the northwest corner method. This procedure is easy to understand and implement. References 1. Maheswari , P.U. ; Ganesan ,K., Solving fully fuzzy transportation problem using pentagonal fuzzy numbers , Journal of Physics: Conf. Series, 2018 ,1000(1), 1-8. 2. Annie Christi , M.S. and Kumari , Sh., Two Stage Fuzzy Transportation Problem Using Symmetric Trapezoidal Fuzzy Number, International Journal of Engineering Inventions, ,2015 , 4(1) , 07-10. 3. Arokia Mary ,A. ;Sivasankari, R., Direct Method of Fuzzy Transportation Problem Using Hexagonal Fuzzy Number with Alpha Cut, International Journal of Mathematics and its Applications, 2016 , 4(4),373-379. 4. Purushothkumar ,M. K., Ananathanarayanan , M. and Dhanasekar , S., FUZZY DIAGONAL OPTIMAL ALGORITHM TO SOLVE FULLY FUZZY TRANSPORTATION PROBLEMS, ARPN Journal of Engineering and Applied Sciences, 2019 ,14(9), 3450- 3454 . 5. Sudha ,L.;Shanmugapriya. R.;Rama, B. Fuzzy Transportation Problem Using Nanogonal Fuzzy Number, Jasc: Journal of Applied Science and Computations , 2019, VI ( III), 1100- 1105. 6. Bhopale ,M. B. , Solution to the Fuzzy Transportation Problem using a new Method of Ranking of Trapezoidal Fuzzy Numbers , International Journal of Engineering Development and Research, 2018, 6, 483- 486. 7. Anushya, B. ; Ramaand, B.; Sudha, L.Transportation ProblemUsing Intuitionistic Decagonal Fuzzy Number, International Journal of Research and Analytical Reviews, 2019 , 6, 271- 277. 8. Venkatachalapathy, M.;Samuel , A. E., An Alternative Method for Solving Fuzzy Transportation Problem using Ranking Function, International Journal of Applied Mathematical Sciences , 2016 ,9(1), 61-68. IHJPAS. 36(1)2023 324 9. Annie Christi , M. S.Solutions of Fuzzy Transportation Problem Using Best Candidates Method and Different Ranking Techniques, International Journal of Mathematical and Computational Sciences, 2017 ,11(4), 182- 187. 10.Jaikumar, K., New Approach to Solve Fully Fuzzy Transportation Problem, International Journal of Mathematics And its Applications, 2016, 4, 155-162. 11.Jayalakshmi, M. ; Pandian , P., A New method for finding an optimal fuzzy solution for fully fuzzy linear programming problems, International Journal of Engineering Research and Applications, 2012 , 2, 247-254. 12. Mitlif , R. J., A New Method for Solving Fully Fuzzy Multi-Objective Linear Programming Problems, Iraqi Journal of Science,2016 ,57(3C), 2307-2311. 13. Karyati , Wutsqa , D. U. ; Insani , N., Yager’s ranking method for solving the trapezoidal fuzzy number linear programming, International Conference on Mathematics, Science and Education , IOP Conf. Series: Journal of Physics, 2018,1-7 .