Admasu Tadesse *1 , Sirkumar Acharya 2, Berhanu Belay3 1 Department of Mathematics, Hawassa university, Ethiopia 2 Department of Mathematics, KIIT University, Bhubaneswar, India 3Department of Mathematics, Debretabor University, Ethiopia KEYWORDS: Multi-Objective Programming; Triangular Fuzzy Number; Fuzzy Transportation Problem; Fuzzy Decision Variables; Ranking Function; Fuzzy Programming Method. ABSTRACT The aim this study is presenting the solution methodology of multi- objective fuzzy transportation problem with fuzzy decision vari- ables, where all the input parameters and decisions variables of the programming problems are assumed to be triangular fuzzy num- ber and triangular fuzzy decision variables respectively. More- over the objectives under considerations are minimization of cost of transportation and minimization of shipping time under fuzzy environment. The fuzziness of the objective functions and the fuzzy constraints of the programming problem are defuzzified us- ing the ranking function and the equality property between two fuzzy numbers, respectively. The consequent crisp multi-objective fuzzy transportation problem is tackled by employing fuzzy math- ematical programming approach. Finally fuzzy decision is made after solving the resultant mathematical programming problem us- ing LINGO(Schrage and LINDO Systems (1997)) software. Illus- trative numerical example is presented in support of the proposed methodology. the transportation of a product man- ufactured at different plants ( supply origins) to a number of different warehouses (demand destinations). Transportation problems were well known as a basic network problem in its classical category. The formulation and dis- cussion of transportation model was intro- duced by Hitchcock (1941). Classical TP models and techniques have been effectively applied to problems with a well-defined or accurately known parameters for many years. The coefficient parameters of the majority of TP models are considered *Corresponding author: East African Journal of Biophysical and Computational Sciences Journal homepage : https://journals.hu.edu.et/hu-journals/index.php/eajbcs College of Natural & Com H pu aw tat as ion sa U al n Sc ive ie r nc sit e y s Year 2021 Volume xx No xx Email: admasut@hu.edu.et Transportation problem(TP) is a particular class of linear programming, which is associ- ated with day-to-day activities in our real life and mainly deals with logistics. It helps in solving problems on distribution and trans- portation of resources from one place to an- other. The goods are transported from a set of sources (e.g., factory) to a set of desti- nations (e.g., warehouse) to meet the spe- cific r equirements, Das et al. (2016). Inother words, transportation problems deal with _______________________________ https://dx.doi.org/10.4314/eajbcs.v4i2.4S East Afr. J. Biophys. Comput. Sci. (2023), Vol. 4, Issue. 2, 43-53 43 INTRODUCTION Fuzzy Programming Approach to Solve Multi-Objective Fully Fuzzy Transportation Problem Research article to be single real number which is accurately known. However, such assumptions are not suitable for dealing with a number of issues that occur in real life because many of the parameters are imprecise and vague as a re- sult of both natural and anthropogenic ef- fects. This motivates to formulate the TP models in uncertain environment. One of the uncertain environments is fuzzy environ-ment. Moreover in most of the literatures, authors assumed the decision parameters as fuzzy numbers while the decision variables as crisp ones. Since the variables are crisp, the solution obtained as the crisp, which is a real number. The solution is exact value in the fuzzy programming problems with fuzzy parameters. The fuzzy aspect of the decision is partly lost in this case so, it is reasonable and important to consider fuzzy mathemat- ical programming problem with fuzzy deci- sion variables. Fuzzy transportation prob- lems involving fuzzy decision variables are considered in this study. Kaur and Kumar (2012) studied a special type of fuzzy TP by assuming that a decision maker is uncertain about the precise values of transportation cost only, where the transportation cost is represented by generalized trapezoidal fuzzy numbers. In their proposed work all the sup- ply and demand of products are crisp pa- rameters that means there is no uncertainty about the supply and demand of the prod- uct. According to the explanation of Ku- mar and Kaur (2011), there may be factors which imposes the occurrences of fuzziness in TP. Some of these are, the decision maker has not enough information about the unit transportation cost of transportation opera- tion and thus the transportation cost uncer- tain, there may be some sort of vagueness with respect to the demand of a newly intro- duced product to the market and there exists uncertainty about the product availability at a source or supplier because of time factor. Many authors introduced tools to solve TP. Since transportation problem (TP) is spe- cial case of linear programming (LP) prob- lem, one straightforward approach is to ap- ply the existing LP techniques to the fuzzy TP. These techniques are discussed by many researchers namely; Buckley (1988), Buckley (1990), Mitlif (2016), Ebrahimnejad (2013), Ebrahimnejad (2015), etc. However, some these techniques only give crisp solutions, which represent a compromise in terms of fuzzy data. The traditional view on TP is mainly con- cerned with distributing any homogeneous product from a group of supply centers, called sources, to any group of receiving cen- ters, called destinations, in such a way as to minimize the single objective total trans- portation cost, where the transportation cost per unit product is constant regardless of the amount transported, but most of the time in real-life situation, the TPs are not designed as single objective function. The TP that deals with multiple-objective func- tions is called a multi-objective transporta- tion problem (MOTP). The MOTP is a special type of multi-objective linear pro- gramming problem in which objective func- tions conflict with each other. Further- more, objective functions are frequently in conflict, thus there is no one best (global optimum) solution, but rather a group of equally good (non-dominated) alternatives known as pareto optimal (PO) solutions. In the framework of multi-objective pro- gramming problems, numerous scholars from a wide variety of academic disciplines dis- cussed their work. Recently, multi-objective problems have been proposed by researchers such as Sayyah et al. (2019); Sahih et al. (2021); Sosa and Dhodiya (2021); Geshni- ani et al. (2020), and others. Researchers namely; Acharya et al. (2014) and Dutta et al. (2016) introduced MOT problem in stochastic environment. Chakraborty and Chakraborty (2010) discussed cost-time min- imization TP, where the demand, supply and transportation cost per unit of the quantities are fuzzy. Nomani et al. (2017) introduced a weighted goal programming to solve multi- objective transportation problems with crisp parameters. They used weighted approach based on goal programming to obtain com- promise solutions. Roy et al. (2018) pro- East Afr. J. Biophys. Comput. Sci. (2023), Vol. 4, Issue. 2, 43-53 44 posed multi-objective transportation prob- lem (MOTP) under intuitionistic fuzzy en- vironment. They have assumed transporta- tion cost, the supply and the demand param- eters as a intuitionistic fuzzy numbers. Jalil et al. (2017) proposed a solution approach for obtaining compromise optimal solution of fully fuzzy(all the parameters and deci- sion variables are fuzzy) multi-objective solid transportation problems. In their proposed problem they used ranking function for the defuzzification of fuzzy objective function and the property of equality between fuzzy numbers for the defuzzification of fuzzy con- straints. El Sayed and Abo-Sinna (2021); Moges et al. (2023); Malik and Gupta (2022); Niksirat (2022), etc. are the works done un- der fuzzy environment. As is mentioned above (Paragraph 2), in most of the literature, authors regarded the decision variables as being crisp while the de- cision parameters are assumed to be fuzzy. This assumptions leads to crisp decisions which is illogical. A fuzzy decision multi- objective fully fuzzy transportation prob- lem proposed in this study. Ranking func- tion and equality between two triangular fuzzy numbers are employed for defuzzifica- tion purpose and finally the equivalent crisp multi-objective model is solved fuzzy pro- gramming method. The paper is organized as follows: following the introduction, basic preliminaries are pre- sented in Sect. 2. The mathematical model of multi-objective fuzzy TP is presented in Sect. 3. Solution procedures are provided in 4. Numerical examples are provided in support of the proposed method in Sect. 5. Finally, Conclusion is provided in Sect. 6 followed by supportive references. Definition: [Roy et al. (2018)]: A tri-angular fuzzy number ã is denoted by (ap, a, ao), where ap, a, ao are real numbers. The membership function (µã(x)) of ã is given below: µã(x) =  0, x ≤ ap x−ap a−ap , ap ≤ x ≤ a ao−x ao−a , a ≤ x ≤ ao 0, otherwise Note: The point ’a’ is the core value of tri- angular fuzzy number Ã, where µã(a) =1 ap and ao are the lower and upper bounds of support of triangular fuzzy number à re- spectively. Definition : [Roy et al. (2018)]: Let ã = (ap, a, ao) and b̃ = (bp, b, bo) be two triangular fuzzy numbers ,then (i) (ap, a, ao)⊕(bp, b, bo) = (ap+ap, a+b, ao+ bo) (ii) k(ap, a, ao) = (kap, ka, kao), k ≥ 0 (iii) (ap, a, ao)⊗ (bp, b, bo) = (apbp, ab, aobo, if ap ≥ 0 and bp ≥ 0 Definition : [Ebrahimnejad (2017)]: Let ã = (ap, a, ao) and b̃ = (bp, b, bo) be two tri- angular fuzzy numbers ,then (i) ã = b̃ iff ap = bp, a=b and ao = bo (ii) ã = (ap, a, ao) ≥ 0 iff ap ≥ 0 Definition :Ebrahimnejad (2017)]: Fuzzy TP is said to be balanced transporta- tion problem when total supply from all the sources is equal to the total demand in all destinations. Definition : [Kumar et al. (2011)]: A ranking function is a function R:F (R) → R, where F(R) is a set of fuzzy numbers defined on set of real numbers, which maps each fuzzy number into the real line, where a nat- ural order exists. Let ã = (ap, a, ao) be a tri- angular fuzzy number, then R(ã) = ap+2a+ao 4 Definition : [Hasan et al. (2015)]: Multi-Objective Optimization Problem (MOOP) involves more than one objective function that are to be minimized or maxi- mized. Answer of MOOP is the set of solu- tions that define t he b est t radeoff between competing objectives. The following is gen- eral mathematical Form of MOOP: East Afr. J. Biophys. Comput. Sci. (2023), Vol. 4, Issue. 2, 43-53 45 BASIC PRELIMINARIES max /min Zm(x),m = 1, 2, 3, ...,M (2.1) s.t. gj(x) ≥ 0, j = 1, 2, ..., J (2.2) hk(x) ≥ 0, k = 1, 2, ..., K (2.3) xL ≤xi ≤ xU (2.4) The mathematical model for MOFTP with fuzzy decision variables is represented as: min : Z̃k ≈ m∑ i=1 n∑ j=1 ((ckij) pxp ij, c k ijxij, (c k ij) oxo ij), k ∈ {1, 2...K} (3.1) subject to n∑ j=1 (xp ij, xij, x o ij) ≈ (api , ai, a o i ), i ∈ {1, 2, 3, ...,m} (3.2) m∑ i=1 (xp ij, xij, x o ij) ≈ (bpj , bj, b o j), j ∈ {1, 2, 3, ..., n} (3.3) x̃ij ⪰ 0, i ∈ {1, 2, 3, ...,m}; j ∈ {1, 2, 3, ..., n} (3.4) where, i. the fuzzy total availability and fuzzy total demand are assumed to be equal(balanced fuzzy transportation problem), ′m′ is total number of sup- ply points and ’n’ is total number of destination points, ii. ãi=(api , ai, a o i ) is the fuzzy availability of the commodity at ith origin and as- sumed to be triangular fuzzy number, iii. b̃j=(bpj , bj, b o j) is the fuzzy requirement of the commodity at jth destination and assumed to be triangular fuzzy number, iv. c̃ij=(cpi j, cij, c o i j) is the fuzzy cost coef- ficient involved with fuzzy variables in the objective function from ith origin to jth destination, which is also assumed to be triangular fuzzy number, v. x̃ij=(xp ij, xij, x o ij) is the fuzzy quantity that should be transported from ith ori- gin to jth destination and assumed to be triangular fuzzy decision variables. Crisp equivalent of multi- objective fuzzy transporta- tion problem Since FMP model from (3.1) to (3.4) can- not be solved directly, so ranking function and the property of equality between the fuzzy numbers are respectively applied on the fuzzy objective functions and fuzzy con- straints models of MOFTP. The resultant crisp equivalent is obtained as: min : Zk = 1 4 m∑ i=1 n∑ j=1 ((cpij) kxp ij+2ckijxij+(coij) kxo ij), k = 1, 2.....K (3.5) subject to n∑ j=1 xp ij = api , i ∈ {1, 2, 3, ...,m}; (3.6) n∑ j=1 xij = ai, i ∈ {1, 2, 3, ...,m} (3.7) East Afr. J. Biophys. Comput. Sci. (2023), Vol. 4, Issue. 2, 43-53 46 MATHEMATICAL MODEL n∑ j=1 xo ij = aoi , i ∈ {1, 2, 3, ...,m} (3.8) m∑ i=1 xp ij = bpj , j ∈ {1, 2, 3, ..., n} (3.9) m∑ i=1 xij = bj, j ∈ {1, 2, 3, ..., n} (3.10) m∑ i=1 xo ij = boj , j ∈ {1, 2, 3, ..., n} (3.11) xo ij−xij ≥ 0, i ∈ {1, 2, 3, ...,m}, j ∈ {1, 2, 3, ..., n} (3.12) xij−xp ij ≥ 0, i ∈ {1, 2, 3, ...,m}, j ∈ {1, 2, 3, ..., n} (3.13) xp ij ≥ 0, i ∈ {1, 2, 3, ...,m}, j ∈ {1, 2, 3, ..., n} (3.14) Solution procedure Now we use the fuzzy programming pro- gramming technique to solve the crisp multi- objective programming problem of 3.5 to 3.14. The solution procedures based on the fuzzy programming method is detailed be- low. Step 1: Find the ideal solutions x1, x2, ..., xk by picking one objective function at a time and leaving the other objective functions. Step 2: Construct a pay-matrix with the help of individual best solutions found by the above step. Using table 1 es- timate the bounds of zk (k=1,2,3...,K) from the Payoff matrix. Step 3: For every objective function zk (k=1,2,3...,K), we formulate member- ship function using any one of the fol- lowing techniques of maximization or minimization: Table 1: Payoff matrix z1(x) z2(x) . . . zK(x) x(1) z1(x (1)) z2(x (1)) . . . zK(x (1)) x(2) z1(x (2)) z2(x (2)) . . . zK(x (2)) . . . . . . . . . . . . . . . . . . . . . x(K) z1(x (K)) z2(x (K)) . . . .zK(x (K)) Step 4 Case 1: Membership function is formulated in the case of maximization problem as: µzk(x) =  0, if zk ≤ lb−k zk−lb−k ub∗−lb−k , if lb−k ≤ zk ≤ ub∗k 1, if zk ≥ ub∗k where lb−k denotes the worst lower bound of zk and ub∗k denotes the best upper bound of zk Case 2: For minimization problem the mem- bership function is formulated as: µzk(x) =  0, if zk ≥ ub−k ub−−zk ub−−lb∗k , if lb∗k ≤ zk ≤ ub−k 1, if zk ≤ lb∗k Where ub−k denotes the worst upper bound of zk and lb∗k denotes best lower bound of zk. Case 1 : Apply the augmented variable, λ with max-min operator to formulate a crisp single objective mixed integer programming problem as: max : λ (3.15) subject to µzk(x) ≥ λ, k = 1, 2, ...K (3.16) n∑ j=1 xp ij = api , i ∈ {1, 2, 3, ...,m} (3.17) East Afr. J. Biophys. Comput. Sci. (2023), Vol. 4, Issue. 2, 43-53 47 n∑ j=1 xij = ai, i ∈ {1, 2, 3, ...,m} (3.18) n∑ j=1 xo ij = aoi , i ∈ {1, 2, 3, ...,m} (3.19) m∑ i=1 xp ij = bpj , j ∈ {1, 2, 3, ..., n} (3.20) m∑ i=1 xij = bj, j ∈ {1, 2, 3, ..., n} (3.21) m∑ i=1 xo ij = boj , j ∈ {1, 2, 3, ..., n} (3.22) xo ij−xij ≥ 0, i ∈ {1, 2, 3, ...,m}, j ∈ {1, 2, 3, ..., n} (3.23) xij−xp ij ≥ 0, i ∈ {1, 2, 3, ...,m}, j ∈ {1, 2, 3, ..., n} (3.24) xp ij ≥ 0, i ∈ {1, 2, 3, ...,m}, j ∈ {1, 2, 3, ..., n} (3.25) 0 ≤ λ ≤ 1 (3.26) Case 2 : Apply the augmented variable, λ with min-max operator to formulate a crisp single objective mixed integer programming problem as: min : λ (3.27) subject to µzk(x) ≤ λ, k ∈ {1, 2, 3, ....K} (3.28) n∑ j=1 xp ij = api , i ∈ {1, 2, 3...m} (3.29) n∑ j=1 xij = ai, i ∈ {1, 2, 3...m} (3.30) n∑ j=1 xo ij = aoi , i ∈ {1, 2, 3...m} (3.31) m∑ i=1 xp ij = bpj , j ∈ {1, 2, 3...n} (3.32) m∑ i=1 xij = bj, j ∈ {1, 2, 3...n} (3.33) m∑ i=1 xo ij = boj , j ∈ {1, 2, 3...n} (3.34) xo ij−xij ≥ 0, i ∈ {1, 2, 3...m}; j ∈ {1, 2, 3...n} (3.35) xij−xp ij ≥ 0, i ∈ {1, 2, 3...m}; j ∈ {1, 2, 3...n} (3.36) xp ij ≥ 0, i ∈ {1, 2, 3...m}; j ∈ {1, 2, 3...n} (3.37) 0 ≤ λ ≤ 1 (3.38) Step 5 At the end, the equivalent single ob- jective MP model is solved by using appropriate techniques or existing soft- ware. The obtained PO solutions sub- stituted back to original fuzzy objec- tive function, as the result we can find fuzzy optimal values of each fuzzy ob- jective functions. Numerical example In this part, application numerical example is solved using the provided approach, and the conclusions drawn from the results are discussed in further detail. A firm has two sources O 1 and O 2 and three destinations D1 ,D2 and D3. The fuzzy sup- ply of the commodity from O1 and O2 are (75, 95, 125) and (45, 65, 95), respectively. Request of fuzzy demanded product at D1 ,D2 and D3 are (35, 45, 55), (25, 35, 45) and (60, 80, 110), respectively. The com- pany wants to determine the fuzzy quan- tity of the commodity that should be trans- ported from each origin to each destination so that the total fuzzy transportation cost is minimum with minimum transfer time. For i=1,2, j=1,2,3., let the fuzzy transportation cost for unit quantity of the commodity from ith source to jth destinations be c̃ij , the fuzzy transportation time is also considered as t̃ij and x̃ij represents the allocations(or amounts), which is non negative triangular fuzzy real variable. The theoretical data on fuzzy cost of transportation and fuzzy deliv- ery time, is given in the table 2 below. East Afr. J. Biophys. Comput. Sci. (2023), Vol. 4, Issue. 2, 43-53 48 Table 2: Fuzzy transportation cost per unit(in Rupees) and fuzzy transportation time per unit(in minute) D1 D2 D3 c̃1j(O1) (15,25,35) (55, 65,85) (85,95,105) c̃2j(O2) (65,75,85) (80,90,110) (30,40,50) t̃1j(O1) (4, 6, 8) (6, 8,10) (7,9,11) t̃2j(O2) (3,5,7) (5,7,9) (11,13,15) From the table 2 mathematical model for MOFTP with fuzzy decision variables becomes: min : z̃1 ≈ (15, 25, 35)⊗ x̃11 ⊕ (55, 65, 85)⊗ x̃12 ⊕ (85, 95, 105)⊗ x̃13 ⊕ (65, 75, 85)⊗ x̃21 ⊕(65, 75, 85)⊗ x̃21 ⊕ (80, 90, 110)⊗ x̃22 ⊕ (30, 40, 50)⊗ x̃23 (3.39) min : z̃2 ≈ (4, 6, 8)⊗ x̃11 ⊕ (6, 8, 10)⊗ x̃12 ⊕ (7, 9, 11)⊗ x̃13 ⊕ (3, 5, 7)⊗ x̃21 ⊕ (5, 7, 9)⊗ x̃22 ⊕(11, 13, 15)⊗ x̃23 (3.40) Subject to x̃11 ⊕ x̃12 ⊕ x̃13 ≈ (75, 95, 125) (3.41) x̃21 ⊕ x̃22 ⊕ x̃23 ≈ (45, 65, 95) (3.42) x̃11 ⊕ x̃21 ≈ (35, 45, 65) (3.43) x̃12 ⊕ x̃22 ≈ (25, 35, 45) (3.44) x̃13 ⊕ x̃23 ≈ (60, 80, 110) (3.45) x11, x̃12, x̃13 ⪰ 0 (3.46) x̃21, x̃22, x̃23 ⪰ 0 (3.47) defuzzification is done using the concept discussed in the subsection 3.1. Then the case(3.27-3.37) of evaluation of membership functions for the minimization (Max-min op- erator) is applied on the crisp MOTP along with all the procedures discussed above. The fuzzy programming method is applied on the aforementioned crisp equivalent MOTP to find the ideal solutions as detailed below x(1) = (xp 11, x11, x o 11, x p 12, x12, x o 12, x p 13, x13, x o 13, x p 21, x21, x o 21, x p 22, x22, x o 22x p 23, x23, x o 23)= (35,35,55,10,10,10,30,50,60,0,10,10,15,25,35,30,30,50) x(2) = (xp 11, x11, x o 11, x p 12, x12, x o 12, x p 13, x13, x o 13, x p 21, x21, x o 21, x p 22, x22, x o 22, x p 23, x23, x o 23)= (0,10,30,25,35,35,50,50,60,35,35,35,0,0,10,10,30,50) At the corresponding ideal points the val- ues of crisp objective functions are obtained and given as: z1=10737.5, z2=1440. After constructing a pay-of matrix 2, the bounds of the two objective functions are given by: 10737.5 ≤ z1(x) ≤ 11825 1440 ≤ z2(x) ≤ 1465. The the single objective crisp problem is ob- tained by formulating membership function. Thus the programming problem becomes: max : λ (3.48) East Afr. J. Biophys. Comput. Sci. (2023), Vol. 4, Issue. 2, 43-53 49 subject to z1 + 1387.5λ ≤ 11825 (3.49) z2 + 25λ ≤ 1465 (3.50) xp 11 + xp 12 + xp 13 = 75 (3.51) x11 + x12 + x13 = 95 (3.52) xo 11 + xo 12 + xo 13 = 125 (3.53) xp 21 + xp 22 + xp 23 = 45 (3.54) x21 + x22 + x23 = 65 (3.55) xo 21 + xo 22 + xo 23 = 95 (3.56) xp 11 + xp 21 = 35 (3.57) x11 + x21 = 45 (3.58) xo 11 + xo 21 = 65 (3.59) xp 12 + xp 22 = 25 (3.60) x12 + x22 = 35 (3.61) xo 12 + xo 22 = 45 (3.62) xp 13 + xp 23 = 60 (3.63) x13 + x23 = 80 (3.64) xo 13 + xo 23 = 110 (3.65) xo ij − xij ≥ 0 (3.66) xij − xp ij ≥ 0 (3.67) xp ij ≥ 0 (3.68) 0 ≤ λ ≤ 1 (3.69) By employing the LINGO software, the resultant equivalent crisp programming problem (3.48)-(3.69) is solved. The PO solutions and the agumated variable λ obtained are given consecutively as: x∗=(31.66375,41.66375,61.66375,0,3.336248,3.33666248,43.33675,50,60, 3.336248 ,3.336248 ,3.336248 ,25,31.66375,41.66375,16.66375,30,50), λ=0.6668124. The com- promising fuzzy objective functions values are z̃1=(6875.31, 10341.77, 16075.04) and z̃2=(748.32, 1355.13, 2335). Figurative description of fuzzy optimal triangular cost and triangular transfer time are detailed in the figure below. Figure 1: Minimum fuzzy transportation cost East Afr. J. Biophys. Comput. Sci. (2023), Vol. 4, Issue. 2, 43-53 50 Figure 2: Minimum fuzzy transfer time The defuzzified crisp MOTP is solved by fuzzy programming method. Using Max-min operator, the resultant single objective crisp integer MP problem is coded into LINGO optimization package version 19.0 software and fuzzy PO solutions are obtained. The minimum total fuzzy cost of transportation and the minimum fuzzy transfer time are ob- tained and interpreted as follows: Fuzzy transportation cost and fuzzy transfer time z̃1=(6875.31, 10341.77, 16075.04) and z̃2=(748.32, 1355.13, 2335) respectively are calculated by back substitutions of PO solu- tions into fuzzy objective functions (3.1) of the the programming problem. It is found that the least amounts of min- imum total transportation cost and trans- fer time are 6875.31 and 748.32 units re- spectively. The the most possible amounts of minimum total transportation cost and transfer time are 10341.77 and 1355.13 units respectively. Furthermore, the greatest amounts of minimum total transportation cost and total transfer time are 16075.04 and 2335 units respectively. The classical transportation problem (TP) is primarily concerned with distributing any homogeneous product from a group of sup- ply centers, known as sources, to any group of receiving centers, known as destinations, in such a way that the single objective to- tal transportation cost is minimized, where all parameters are crisp (precisely defined). However, in many circumstances, the deci- sion maker lacks precise knowledge of the TP parameters. and the nature of the TPs are not designed as single objective func- tion. If the nature of the information is vague and the decision maker objectives preference are conflicting, t he c orresponding program- ming problem is fuzzy multi-objective pro- gramming problem, and thus fuzzy MOTP arises. In this paper, we have discussed a solution approach for solving TP, with more than one objective function by considering the presence of vagueness in the real life data of transportation problems, where all the pa- rameters and decision variables are consid- ered as triangular fuzzy numbers. Initially, the fuzzy objective functions are defuzzified by applying the ranking function for trian- East Afr. J. Biophys. Comput. Sci. (2023), Vol. 4, Issue. 2, 43-53 51 DISCUSSION CONCLUSION References Acharya S., Ranarahu N., Dash J.K. And Acharya M.M. 2014. Acharya,S.,etal.,2014. Computation of a multi-objective fuzzy stochastic transportation problem. International Journal of Fuzzy Computation and Modelling 1(2): 212–233. Buckley J. 1988. Possibilistic linear programming with triangular fuzzy numbers. Fuzzy sets Syst. 26 (1), 135–138. Buckley J.1990. Multi objective possibilistic linear programming. Fuzzy sets Syst. 35(1): 23–28. Chakraborty A. and Chakraborty M. 2010. 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