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A new algorithm for fuzzy transportation problems with trapezoidal fuzzy numbers under fuzzy circumstances

Journal of Fuzzy Extension and Applications
Journal of Fuzzy Extension and Applications

The document presents a new algorithm for solving fully fuzzy transportation problems with trapezoidal fuzzy numbers. Transportation problems aim to minimize transportation costs while meeting supply and demand constraints. Existing methods often provide precise solutions rather than fuzzy solutions that account for uncertainty in costs, supplies, and demands. The proposed two-step method converts the fuzzy transportation problem into two interval transportation problems, whose optimal solutions provide the optimal fuzzy solution to the original problem. This allows the method to consider all parameters as fuzzy numbers and provide a fuzzy optimal solution and cost without negative components.

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Corresponding Author: fsegmath@gmail.com http://dx.doi.org/10.22105/jfea.2021.287198.1148 E-ISSN: 2717-3453 | P-ISSN: 2783-1442 | Abstract 1 | Introduction Transportation problem is an important network structured linear programming problem that arises in several contexts and has deservedly received a great deal of attention in the literature. The central concept in this problem is to find the least total transportation cost of a commodity in order to satisfy demands at destinations using available supplies at origins. Transportation problem can be used for a wide variety of situations such as scheduling, production, investment, plant location, inventory control, employment scheduling and many others. Journal of Fuzzy Extension and Applications www.journal-fea.com J. Fuzzy. Ext. Appl. Vol. 2, No. 3 (2021) 204–225. Paper Type: Research Paper The New Algorithm for Fully Fuzzy Transportation Problem by Trapezoidal Fuzzy Number (A Generalization of Triangular Fuzzy Number) Ladji Kané1,* , Moctar Diakité1, Souleymane Kané1, Hawa Bado1, Moussa Konaté1, Koura Traoré1 1 Faculty of Economics and Management, University of Social Sciences and Management of Bamako (USSGB), Quartier du Fleuve Rue 310, Porte 238, Mali; fsegmath@gmail.com; moctardia@yahoo.fr; garalo2018@gmail.com; badohawa@yahoo.fr; konatefseg@gmail.com. Citation: Kané, L., Diakité, M., Kané, S., Bado, H., Konaté, M., & Traoré, K. (2021). The new algorithm for fully fuzzy transportation problem by trapezoidal fuzzy number (a generalization of triangular fuzzy number). Journal of fuzzy extension and application, 2 (3), 204-225. Accept: 29/07/2021 Revised: 17/06/2021 Reviewed: 01/06/2021 Received: 20/05/2021 Transportation problem is an important network structured linear programming problem that arises in several contexts and has deservedly received a great deal of attention in the literature. The central concept in this problem is to find the least total transportation cost of a commodity in order to satisfy demands at destinations using available supplies at origins in a crisp environment. In real life situations, the decision maker may not be sure about the precise values of the coefficients belonging to the transportation problem. The aim of this paper is to introduce a formulation of fully fuzzy transportation problem involving trapezoidal fuzzy numbers for the transportation costs and values of supplies and demands. We propose a two-step method for solving fuzzy transportation problem where all of the parameters are represented by triangular fuzzy numbers i.e. two interval transportation problems. Since the proposed approach is based on classical approach it is very easy to understand and to apply on real life transportation problems for the decision makers. To illustrate the proposed approach four application examples are solved. The results show that the proposed method is simpler and computationally more efficient than existing methods in the literature. Keywords: Fully fuzzy linear programming, Transportation problem, Trapezoidal fuzzy numbers, fuzzy numbers. Licensee Journal of Fuzzy Extension and Applications. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons. org/licenses/by/4.0).
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205 A new algorithm for fuzzy transportation problems with trapezoidal fuzzy numbers under fuzzy circumstances In general, transportation problems are solved with the assumptions that the transportation costs and values of supplies and demands are specified in a precise way i.e., in crisp environment. However, in many cases the decision maker has no crisp information about the coefficients belonging to the transportation problem. If the nature of the information is vague, that is, if it has some lack of precision, the corresponding coefficients or elements defining the problem can be formulated by means of fuzzy sets, and thus fuzzy transportation problems arise. Several researchers have carried out investigations on fuzzy transportation problem. Zimmermann [4] developed Zimmermann's fuzzy linear programming into several fuzzy optimization methods for solving the transportation problems. ÓhÉigeartaigh [5] proposed an algorithm for solving transportation problems where the supplies and demands are fuzzy sets with linear or triangular membership functions. Chanas et al. [6] investigated the transportation problem with fuzzy supplies and demands and solved them via the parametric programming technique. Their method provided solution which simultaneously satisfies the constraints and the goal to a maximal degree. In addition, Chanas et al. [7] formulated the classical, interval and fuzzy transportation problem and discussed the methods for solution for the fuzzy transportation problem. Kuchta [8] discussed the type of transportation problems with fuzzy cost coefficients and converted the problem into a bicriterial transportation problem with crisp objective function. Their method only gives crisp solutions based on efficient solutions of the converted problems. Jiménez and Verdegay [9], [10] investigated the fuzzy solid transportation problem in which supplies, demands and conveyance capacities are represented by trapezoidal fuzzy numbers and applied a parametric approach for finding the fuzzy solution. Liu and Kao [11] developed a procedure, based on extension principle to derive the fuzzy objective value of fuzzy transportation problem, in that the cost coefficients and the supply and demand quantities are fuzzy numbers. Gani and Razak [12] presented a two-stage cost minimizing fuzzy transportation problem in which supplies and demands are as trapezoidal fuzzy numbers and used a parametric approach for finding a fuzzy solution with the aim of minimizing the sum of the transportation costs in the two stages. Li et al. [13] proposed a new method based on goal programming for solving fuzzy transportation problem with fuzzy costs. Lin [14] used genetic algorithm for solving transportation problems with fuzzy coefficients. Dinagar and Palanivel [15] investigated fuzzy transportation problem, with the help of trapezoidal fuzzy numbers and applied fuzzy modified distribution method to obtain the optimal solution in terms of fuzzy numbers. Pandian and Natarajan [16] introduced a new algorithm namely, fuzzy zero-point method for finding fuzzy optimal solution for such fuzzy transportation problem in which the transportation cost, supply and demand are represented by trapezoidal fuzzy numbers. Kumar and Kaur [17] proposed a new method based on fuzzy linear programming problem for finding the optimal solution of fuzzy transportation problem. Gupta et al. [18] proposed a new method named as Mehar's method, to find the exact fuzzy optimal solution of fully fuzzy multi-objective transportation problems. Ebrahimnejad [19] applied a fuzzy bounded dual algorithm for solving bounded transportation problems with fuzzy supplies and demands. Shanmugasundari and Ganesan [20] developed the fuzzy version of Vogel's and MODI methods for obtaining the fuzzy initial basic feasible solution and fuzzy optimal feasible solution, respectively, without converting them into classical transportation problem. Also, [21] Chandran and Kandaswamy [21] proposed an algorithm to find an optimal solution of a fuzzy transportation problem, where supply, demand and cost coefficients all are fuzzy numbers. Their algorithm provides decision maker with an effective solution in comparison to existing methods. Ebrahimnejad [22] using an example showed that their method will not always lead to a fuzzy optimal solution. Moreover, Kumar and Kaur [23] pointed out the limitations and shortcomings of the existing methods for solving fuzzy solid transportation problem and to overcome these limitations and shortcomings proposed a new method to find the fuzzy optimal solution of unbalanced fuzzy solid transportation problems. In addition, Ebrahimnejad [24] proposed a two-step method for solving fuzzy transportation problem where all of the parameters are represented by non-negative triangular fuzzy numbers. Some researchers applied generalized fuzzy numbers for solving transportation problems. Kumar and Kaur [25] proposed a new method based on ranking function for solving fuzzy transportation problem by assuming that transportation cost, supply and demand of the commodity are represented by generalized trapezoidal fuzzy numbers. After that, Kaur and Kumar [26] introduced a similar algorithm for solving a special type of fuzzy
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206 Kané et al. |J. Fuzzy. Ext. Appl. 2(3) (2021) 204-225 transportation problem by assuming that a decision maker is uncertain about the precise values of transportation cost only but there is no uncertainty about the supply and demand of the product. Ebrahimnejad [27] demonstrated that once the ranking function is chosen, the fuzzy transportation problem introduced by Kaur and Kumar [26] is converted into crisp one, which is easily solved by the standard transportation algorithms. The contributions of the present study are summarized as follows: (a) in the Eq. (15) under consideration, all of the parameters, such as the transportation costs, supplies and demands are considered as fuzzy numbers. (b) According to the proposed approach, the Eq. (15) is converted into two interval transportation problems Eq. (16) and Eq. (17). The integration of the optimal solution of the two sub-problems provides the optimal solution of the Eq. (15). (c) In contrast to most existing approaches, which provide a precise solution, the proposed method provides a fuzzy optimal solution. (d) In contrast to existing methods that include negative parts in the obtained fuzzy optimal solution and fuzzy optimal cost, the proposed method provides a fuzzy optimal solution and optimal cost. (e) Similarly, to the competing methods in the literature, the proposed method is applicable for all types of trapezoidal fuzzy numbers. (f) The complexity of computation is greatly reduced compared with commonly used existing methods in the literature. The rest of this paper is organized as follows: In Section 2, we recall the definitions of interval number linear programming, interval numbers and fully fuzzy transportation problem. In Section 3, a new method is proposed for obtaining the fuzzy optimal solution of the Eq. (15). The advantages of the proposed method are discussed in Section 4. Two application examples are provided to illustrate the effectiveness of the proposed method in Section 5, and a comparative study in Section 6. Finally, concluding remarks are presented in Section 7. 2| Materials and Methods In this section, some basic definitions, arithmetic operations for closed Intervals numbers and of linear programming problems involving interval numbers are presented [28]. 2.1| A New Interval Arithmetic In this section, some arithmetic operations for two intervals are presented [28]. Let   1 4 1 4 1 4 a a ,a :a a , a ,a           R R be the set of all proper intervals. We shall use the terms “interval” and “interval number” interchangeably. The mid-point and width (or half-width) of an interval number 1 4 a a ,a        are defined as   4 1 a a m a 2   and   4 1 a a w a 2   . The interval number a can also be expressed in terms of its midpoint and width as For any two intervals     1 4 a a ,a m a ,w a           and     1 4 b b ,b m b ,w b           , the arithmetic operations on a and b are defined as:                    4 1 4 1 1 4 a a a a a a ,a m a ,w a , . 2 2 (1)               a b m a m b ,w a w b . (2)               a b m a m b ,w a w b . (3)
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207 A new algorithm for fuzzy transportation problems with trapezoidal fuzzy numbers under fuzzy circumstances 2.2| A New Interval Arithmetic for Trapezoidal Fuzzy Numbers via Intervals Numbers The aim of this section is to present some notations, notions and results which are of useful in our further consideration. A number   1 2 3 4 a a ,a ,a ,a  % (where 1 2 3 4 a a a a    ) is said to be a trapezoidal fuzzy number if its membership function is given by [1]-[3]: Assume that       1 2 3 4 14 23 1 4 2 3 a a ,a ,a ,a a |a a ,a | a ,a                % , and       1 2 3 4 14 23 1 4 2 3 b b ,b ,b ,b b |b b ,b | b ,b                % are two trapezoidal fuzzy numbers. For any two trapezoidal fuzzy numbers   14 23 a a |a  % and   14 23 b b |b  % , the arithmetic operations on a % and b % are defined as: 2.3| Formulation of a Transportation Problems Involving Interval Numbers We consider the Transportation Problem involving Interval numbers as follows [28]:                       αm a ,αw a   if  α 0, αa αm a , αw a   if  α 0. (4)                                                                                1 1 1 1 4 1 m a m b w a w b ,m a w b m b w a ifa 0, b 0, a b m a m b m a w b ,m b w a w b w a ifa 0, b 0, m a m b w a w b ,m b w a m a w b ifa 0, b 0. (5)                         % 1 1 2 2 1 a 4 3 4 3 4 x a , a x a , a a μ x x a , a x a . a a (6)                                     % % 14 23 14 23 1 4 1 4 2 3 2 3 a b a |a b |b a ,a b , b | a ,a b , b . (7)                                    % % 14 23 14 23 1 4 1 4 2 3 2 3 ab a |a b |b a ,a b , b | a ,a b , b (8)
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208 Kané et al. |J. Fuzzy. Ext. Appl. 2(3) (2021) 204-225 where 1 4 ij ij ij c c ,c        , 1 4 i i i a a ,a        , 1 4 j j j b b ,b        are non-negative interval numbers and 1 4 ij ij ij x x ,x        are unrestricted interval numbers. Objective function transformation. Transformation of constraints.         n n ij i ij ij i i j 1 j 1 x a m x ,w x m a ,w a           for i 1, 2, ,m   and         m m ij j ij ij j j i 1 i 1 x b m x ,w x m b ,w b           for j 1,2, ,n   . Now we can say that                                    n m ij ij j 1i 1 n ij i j 1 m ij j i 1 MinZ x c x Subjecttotheconstraints x a ,fori 1,2, ,m , x b , for j 1,2, ,n. (9)                                   n m n m n m 1 4 1 4 ij ij ij ij ij ij ij ij ij ij j 1 i 1 j 1 i 1 j 1 i 1 Z x c x c ,c x ,x m c x ,w c x , Where                                            1 ij ij ij ij ij 1 4 ij ij ij ij ij ij ij ij 4 ij ij ij ij ij m c m x w c w x if x 0, m c x m c m x w c m x if x 0andx 0, m c m x w c w x if x 0. And (10)                                            1 ij ij ij ij ij 1 4 ij ij ij ij ij ij ij ij 4 ij ij ij ij ij m c w x w c m x if x 0, w c x m c w x w c w x if x 0andx 0, m c w x w c m x if x 0. (11)
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209 A new algorithm for fuzzy transportation problems with trapezoidal fuzzy numbers under fuzzy circumstances is equivalent to Optimal Solution (9) according to the choice of the decision maker:         * 1 4 ij ij ij ij ij ij ij x x ,x m x w x ,m x w x                   where     n ij i j 1 w x w a    , i 1, 2, ,m   . Remark. I. n ij i j 1 x a    if and only if     n ij i j 1 m x m a    and     n ij i j 1 w x w a    for i 1, 2, ,m   . II. n ij i j 1 x a    if and only if     n ij i j 1 m x m a    and     n ij i j 1 w x w a    for i 1, 2, ,m   . III. m ij j i 1 x b    if and only if     m ij j i 1 m x m b    and     m ij j i 1 w x w b    for j 1,2, ,n   . IV. m ij j i 1 x b    if and only if     m ij j i 1 m x m b    and     m ij j i 1 w x w b    for j 1,2, ,n   . 2.4| Formulation of a Fully Fuzzy Transportation Problem The fuzzy linear programming formulation of a fully fuzzy transportation problem can be written as follows as follows [2], [3]:                                                                   n m ij ij ij ij j 1 i 1 n ij ij i i j 1 m ij ij j j i 1 MinZ x m c x ,w c x Subjecttotheconstraints m x ,w x m a ,w a ,i 1,2, ,m, m x ,w x m b ,w b , j 1,2, ,n. (12)                                                     n m ij ij j 1 i 1 n ij i j 1 m ij j i 1 m MinZ x m c x Subjecttotheconstraints m x m a ,fori 1,2, ,m, m x m b , for j 1,2, ,n. (13)
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210 Kané et al. |J. Fuzzy. Ext. Appl. 2(3) (2021) 204-225 with m n i j i 1 j 1 a b     % % where ij x % are unrestricted fuzzy numbers and ij c % , i a % and j b % are non-negatives fuzzy numbers. 3| Main Results In this section, we will describe our method of solving. 3.1| Transportation Problem with Trapezoidal Fuzzy Numbers For all the rest of this paper, we will consider the following transportation problem with trapezoidal fuzzy numbers as follows: where   1 2 3 4 ij ij ij ij ij c c ,c ,c ,c ,  %   1 2 3 4 j ij ij ij ij x x ,x ,x ,x  % ,   1 2 3 4 j j j j j b b ,b ,b ,b  % and   1 2 3 4 i i i i i a a ,a ,a ,a  % are trapezoidal fuzzy numbers with     14 23 1 4 2 3 ij ij ij ij ij ij ij x x |x x ,x | x ,x               % where 14 1 4 ij ij ij x x ,x        and 23 2 3 ij ij ij x x ,x        ,     14 23 1 4 2 3 ij ij ij ij ij ij ij c c |c c ,c | c ,c               % where 14 1 4 ij ij ij c c ,c        and 23 2 3 ij ij ij c c ,c ,            14 23 1 4 2 3 j j j j j j j b b |b b ,b | b ,b               % where 14 1 4 j j j b b ,b        and 23 2 3 j j j b b ,b        and     14 23 1 4 2 3 i i i i i i i a a |a a ,a | a ,a               % where 14 1 4 i i i a a ,a        and 23 2 3 i i i a a ,a        . For all the rest of this paper, we will consider the following transportation problems involving interval numbers 14 ij x , 14 ij c , 14 i a and 14 j b as follows:                                     % % % % % % % % m n ij ij i 1 j 1 n ij i j 1 m ij j i 1 MinZ x c x Subjecttotheconstraints x a , fori 1,2, ,m x b , for j 1,2, ,n (14)                                     % % % % % % % % m n ij ij i 1 j 1 n ij i j 1 m ij j i 1 MinZ x c x Subjecttotheconstraints x a ,for i 1,2, ,m, x b , for j 1,2, ,n. (15)
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211 A new algorithm for fuzzy transportation problems with trapezoidal fuzzy numbers under fuzzy circumstances And the transportation problems involving interval numbers 23 ij x , 23 ij c , 23 i a and 23 j b as follows: 3.2| Formulation of a Transportation Problem involving Midpoint Thanks to the new interval arithmetic and Eq. (16), we can write the following transportation problem involving midpoint ( 14 ij x , 14 ij c , 14 i a and 14 j b ) [28] as follows: where   4 1 ij ij 14 ij c c m c 2   ,   4 1 14 i i i a a m a 2   ,   4 1 j j 14 j b b m b 2   and   4 1 14 i i i a a w a 2   . Thanks to the new interval arithmetic and Eq. (17), we can write the following transportation problem involving midpoint ( 23 ij x , 23 ij c , 23 i a and 23 j b ) [28] as follows:                                     m n 14 14 14 14 ij ij i 1 j 1 n 14 14 ij i j 1 m 14 14 ij j i 1 MinZ x c x Subjecttotheconstraints x a ,for i 1,2, ,m, x b ,for j 1,2, ,n. (16)                                    m n 23 23 23 23 ij ij i 1 j 1 n 23 23 ij i j 1 m 23 23 ij j i 1 MinZ x c x Subjecttotheconstraints x a ,for i 1,2, ,m, x b ,for j 1,2, ,n. (17)                                           m n 14 14 14 14 ij ij i 1 j 1 n 14 14 ij i j 1 14 14 ij j i 1 m MinZ x m c x Subjecttotheconstraints x m a ,for i 1,2, ,m, x m b ,for j 1,2, ,n. (18)
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212 Kané et al. |J. Fuzzy. Ext. Appl. 2(3) (2021) 204-225 where   3 2 ij ij 23 ij c c m c 2   ,   3 2 23 i i i a a m a 2   ,   3 2 j j 23 j b b m b 2   and   i 3 i 2 23 i a a w a 2   . Thanks to the new interval arithmetic, we can write the following proposition [28]: Proposition 1. If 14 *1 *4 ij ij ij x x ,x        is an optimal solution to the Eq. (16) and 23 *2 *3 ij ij ij x x ,x        is an optimal solution to the Eq. (17), then   * * ij m n x x   % % is an optimal solution to the Eq. (15) with       * 14 23 *1 *4 *2 *3 *1 *2 *3 *4 ij ij ij ij ij ij ij ij ij ij ij x x |x x ,x | x ,x x ,x ,x ,x                % . 3.3| The Steps of Our Computational Method The steps of our method for solving the fully fuzzy transportation problem involving trapezoidal fuzzy numbers as follows. 3.3.1| Solution procedure for transportation problem with trapezoidal fuzzy numbers Step 1. Construct the fuzzy transportation Problem (15), and then convert it into a balanced one if it is not. Else, go to Step 2. Step 2. Solving Eq. (17) via Eq. (19). Determine 3 ij 2 x =     *2 *3 23 23 23 23 ij ij ij ij ij ij x ,x x w x ,x w x                  by applying the following conditions: I.   23 ij w x 0  if and only if 23 ij x 0  and II.     23 ij 23 23 ij i x 0 w x w a    with     23 23 ip ik w x w x  if 4 ip 4 ik c c  for i 1, ,m   . III. Go to Step 3. Step 3. Solving Eq. (16) via Eq. (18). Determine 4 ij 1 x =     *1 *4 ij ij ij ij ij i 14 14 14 14 j x ,x x w x ,x w x                  for i 1, ,m   . Considering the following cases: Case 1. If     n 14 23 i ij ij i i 3 1 j 1 2 4 E x x w a w a       , then 14 ij x =     *1 *4 ij ij ij ij ij i 14 14 14 14 j x ,x x w x ,x w x                  with     4 ij 1 i 14 j i x 0 14 w x w a    . Else, go to Case 2.                                           m n 23 23 23 23 ij ij i 1 j 1 n 23 23 ij i j 1 23 23 ij j i 1 m MinZ x m c x Subjecttotheconstraints x m a ,fori 1,2, ,m, x m b ,for j 1,2, ,n. (19)
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213 A new algorithm for fuzzy transportation problems with trapezoidal fuzzy numbers under fuzzy circumstances Case 2. If     n 14 23 i ij ij i i 3 1 j 1 2 4 E x x w a w a       , then 14 ij x =     *1 *4 ij ij ij ij ij i 14 14 14 14 j x ,x x w x ,x w x                  with     14 14 2 23 3 ij ij ij ij w x x x w x    and go to Step 4. Step 4. The Optimal Solution (15) according to the choice of the decision maker is:        m n 1 2 3 4 *1 *2 *3 *4 *1 *2 *3 *4 ij ij ij ij ij ij ij ij j 1 i 1 Min Z x c ,c ,c ,c x ,x ,x ,x Z ,Z ,Z ,Z       % % where       * 14 23 *1 *4 *2 *3 *1 *2 *3 *4 ij ij ij ij ij ij ij ij ij ij ij x x |x x ,x | x ,x x ,x ,x ,x                % . 3.3.2| Solution procedure for transportation problem with triangular fuzzy numbers The steps of our method for solving the fully fuzzy transportation problem involving triangular fuzzy numbers as follows: The fuzzy number   * *1 *2 *3 *4 ij ij ij ij ij x x ,x ,x ,x  % is said to be triangular if and only if *2 *3 ij ij x x  . Then   *3 *2 ij ij 23 ij x x w x 0 2    . Therefore   14 14 23 ij ij ij w x x x   and       * *1 *2 *3 *4 *1 *2 *2 *4 *1 *2 *4 ij ij ij ij ij ij ij ij ij ij ij ij x x ,x ,x ,x x ,x ,x ,x x ,x ,x    % . In this article, consider fuzzy triangular numbers by:       * 2 13 2 *1 *3 *1 *2 *3 ij ij ij ij ij ij ij ij ij x x |x x | x ,x x ,x ,x          % , where     *1 *3 13 13 13 13 ij ij ij ij ij ij x ,x x w x ,x w x                  . Step 1. Construct the fuzzy transportation Problem (15), and then convert it into a balanced one if it is not. Else, go to Step 2. Step 2. Solving Eq. (17) via Eq. (19). Determine 3 ij 2 x = *2 *3 ij ij x ,x        *2 *2 ij ij x ,x        *2 ij x , for i 1, ,m   and go to Step 3. Step 3. Solving Eq. (16) via Eq. (18). Determine 3 ij 1 x =     *1 *3 ij ij ij ij ij i 13 13 13 13 j x ,x x w x ,x w x                  for i 1, ,m   . Considering the following cases: Case 1. If   n 13 i ij ij 2 3 1 1 i j E x x w a      , then 3 ij 1 x =     *1 *3 ij ij ij ij ij i 13 13 13 13 j x ,x x w x ,x w x                  with     3 ij 1 i 13 j i x 0 13 w x w a    . Else, go to Case 2. Case 2. If   13 n i ij i 13 i j 1 2 j E x x w a      , then 3 ij 1 x =     *1 *3 ij ij ij ij ij i 13 13 13 13 j x ,x x w x ,x w x                  with   ij i 13 13 j 2 j i w x x x   and go to Step 4. Step 4. The Optimal Solution (15) according to the choice of the decision maker is:
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214 Kané et al. |J. Fuzzy. Ext. Appl. 2(3) (2021) 204-225        m n 1 2 3 *1 *2 *3 *1 *2 *3 ij ij ij ij ij ij j 1 i 1 Min Z x c ,c ,c x ,x ,x Z ,Z ,Z       % % with       * 2 13 2 *1 *3 *1 *2 *3 ij ij ij ij ij ij ij ij ij x x |x x | x ,x x ,x ,x          % . 4| Advantages of the Proposed Method Let us explore the main advantages of the proposed method:  The new proposed method is applicable to all Eq. (15) where ij x % are unrestricted triangular or trapezoidal fuzzy numbers and ij c % , i a % and j b % are non-negatives triangular or trapezoidal fuzzy numbers.  The proposed technique does not use the goal and parametric approaches which are difficult to apply in real life situations.  By applying the proposed approach for finding the fuzzy optimal solution, there is no need of much knowledge of fuzzy linear programming technique, Zimmerman approach and crisp linear programming which are difficult to learn for a new decision maker.  The proposed method to solve Eq. (15) is based on traditional transportation algorithms. Thus, the existing and easily available software can be used for the same. However, the existing method [1], [2], [3], [11], [29] to solve Eq. (15) should be implemented into a programming language.  To solve the Eq. (15) by using the existing method [1], [2], [3], [11], [29], there is need to use arithmetic operations of generalized fuzzy numbers. While, if the proposed technique is used for the same then there is need to use arithmetic operations of real numbers. This proves that it is much easy to apply the proposed method as compared to the existing method [1], [2], [3], [11], [29].  Moreover, it is possible to assume a generic ranking index for comparing the fuzzy numbers involved in the Eq. (15), in such a way that each time in which the decision maker wants to solve the FFTP under consideration (s)he can choose (or propose) the ranking index that best suits the Eq. (15). 5| Numerical Illustration In this section, to illustrate the new method proposed, and the existing fully fuzzy transportation problem due to [1], [2], [3], [11], [29], [30], presented in Examples 1-4, are solved by the proposed method. Example 1. [3]. Step 1. Construct the fuzzy transportation Problem (15), and then convert it into a balanced one if it is not. Table 1. Eq. (15) in trapezoidal unbalanced form. 1 R 2 R 3 R 3 R %i a Supply ( ) A   10,10,10,10   50,50,50,50   80,80,80,80   0,0,0,0   70,90,90,100 B   60,70,80,90   60,60,60,60   20,20,20,20   0,0,0,0   40,60,70,80 C   0,0,0,0   0,0,0,0   0,0,0,0   0,0,0,0   0,0,10,50 Demand ( % j b )   30,40,50,70   20,30,40,50   40,50,50,80   20,30,30,30     % % m n i j i 1 j 1 a b
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215 A new algorithm for fuzzy transportation problems with trapezoidal fuzzy numbers under fuzzy circumstances Step 2. Solving Eq. (17) via Eq. (19).   23 23 23 23 23 23 23 23 11 12 13 21 22 23 MinZ x 25x 65x 100x 75x 90x 40x       subject to the constraints 23 23 23 23 11 12 13 14 x x x x 90     , 23 23 23 23 21 22 23 24 x x x x 65     , 23 23 23 23 31 32 33 34 x x x x 5     , 23 23 23 11 21 31 x x x 45    , 23 23 23 12 22 32 x x x 35    , 23 23 23 13 23 33 x x x 50    and 23 23 23 14 24 34 x x x 30    . Optimal solution is: 23 11 x 45  , 23 12 x 30  , 23 13 x 0  , 23 14 x 15  , 23 21 x 0  , 23 22 x 0  , 23 23 x 50  , 23 24 x 15  , 23 31 x 0  , 23 32 x 5  , 23 33 x 0  and 23 34 x 0  . Furthermore   23 1 w a 0  ,   23 2 w a 5  and   23 3 w a 5  . For i 1  , then     23 1 j 23 23 1j 1 x 0 w x w a 0     with         23 23 23 23 11 12 13 14 w x w x w x w x 0     . We get 23 11 x 45,45        , 23 12 x 30,30        , 23 13 x 0  and 23 14 x 15,15        . For i 2  , then     23 2 j 23 23 2 j 2 x 0 w x w a 5     with     23 23 21 22 w x w x 0   ,   23 23 w x 1  and   23 24 w x 4  . We get 23 21 x 0,0        , 23 22 x 0,0        , 23 23 x 49,51        and 23 24 x 11,19        . For i 3  , then     23 3 j 23 23 3 j 3 x 0 w x w a 5     with   23 31 w x 0  ,   23 32 w x 5  ,   23 33 w x 0  and   23 34 w x 0  . We get 23 31 x 0  , 23 32 x 0,10        , 23 33 x 0  and 23 34 x 0  . Step 3. Solving Eq. (16) via Eq. (18). We get   14 14 14 14 14 14 14 14 11 12 13 21 22 23 MinZ x 25x 70x 100x 75x 95x 40x       subject to the constraints 14 14 14 14 11 12 13 14 x x x x 85     , 14 14 14 14 21 22 23 24 x x x x 60     , 14 14 14 14 31 32 33 34 x x x x 25     , 14 14 14 11 21 31 x x x 50    , 14 14 14 12 22 32 x x x 35    , 14 14 14 13 23 33 x x x 60    and 14 14 14 14 24 34 x x x 25    . Optimal solution is: 14 11 x 50  , 14 12 x 10  , 14 13 x 0  , 14 14 x 25  , 14 21 x 0  , 14 22 x 0  , 14 23 x 60  , 14 24 x 0  , 14 31 x 0  , 14 32 x 25  , 14 33 x 0  and 14 34 x 0  . Furthermore   14 1 w a 15  ,   14 2 w a 20  and   14 3 w a 25  For i 1  , we have     4 14 23 23 14 1 1j 1j 1 1 j 1 E x x w a 50 45 10 30 0 0 25 15 0 35 15 w a                  , then     14 32 14 23 1j 1j 1j 1j w x x x w x    . We have   14 11 w x 5,    14 12 w x 20  ,   14 13 w x 0  and   14 14 w x 10  . We get 14 11 x 45,55        , 14 12 x 10,30         , 14 13 x 0  and 14 14 x 15,35        . For i 2  , we have
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216 Kané et al. |J. Fuzzy. Ext. Appl. 2(3) (2021) 204-225     4 14 23 23 14 2 2 j 2 j 2 2 j 1 E x x w a 0 0 0 0 60 50 0 15 5 30 20 w a                  , then     14 14 23 23 2 j 2 j 2 j 2 j w x x x w x    . We have     14 14 21 22 w x w x 0   ,   23 23 w x 11  and   14 24 w x 19  . We get 14 21 x 0,0        , 14 22 x 0,0        , 14 23 x 49,71        and 14 24 x 19,19         . For i 3  , we have     4 14 23 23 14 3 3 j 3 j 3 3 j 1 E x x w a 0 0 25 5 0 0 0 0 5 25 25 w a                  , then     14 3 j 14 14 3 j 3 x 0 w x w a 25     . We have   14 31 w x 0  ,   14 32 w x 25  ,   14 33 w x 0  and   14 34 w x 0  We get 14 31 x 0  , 14 32 x 0,50        , 14 33 x 0  and 14 34 x 0  . Step 4. The optimal solution according to the choice of the decision maker is:          m n 1 2 3 4 *1 *2 *3 *4 *1 *2 *3 *4 ij ij ij ij ij ij ij ij j 1 i 1 Min Z x c ,c ,c ,c x ,x ,x ,x Z ,Z ,Z ,Z 930,2930,2970,3470        % % and       * 14 23 *1 *4 *2 *3 *1 *2 *3 *4 ij ij ij ij ij ij ij ij ij ij ij x x |x x ,x | x ,x x ,x ,x ,x                % with   * 11 x 45, 45, 45,55  % ,   * 12 x 10, 30, 30, 30   % , * 13 x 0  % % ,   * 14 x 15, 15, 15, 35  % , * 21 x 0  % % , * 22 x 0  % % ,   * 23 x 49, 49, 51, 71  % ,   * 24 x 19, 11, 19, 19   % , * 31 x 0  % % ,   * 32 x 0,0,10,50  % , * 33 x 0  % % and * 34 x 0  % % . Example 2. [29]. Step 1. Construct the fuzzy transportation Problem (15), and then convert it into a balanced one if it is not. Table 2. Eq. (15) in trapezoidal balanced form. 1 R 2 R 3 R Supply %i a A   1,4,9,19   1,2,5,9   2,5,8,18   1,5,7,9 B   8,9,12,26   3,5,8,12   7,9,13,28   4,7,8,10 C   11,12,20,27   0,5,10,15   4,5,8,11   4,5,8,11 Demand % j b   3,5,8,12 (4, 8, 9, 10) (2, 4, 6, 8)     % % m n i j i 1 j 1 a b
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217 A new algorithm for fuzzy transportation problems with trapezoidal fuzzy numbers under fuzzy circumstances Step 2. Solving Eq. (17) via Eq. (19). Optimal solution is: 23 11 x 6  , 23 12 x 0  , 23 13 x 0  , 23 21 1 x 2  , 23 22 x 7  , 23 23 x 0  , 23 31 x 0  , 23 32 3 x 2  and 14 33 x 5  . Furthermore   23 1 2 w a 1 2   ,   23 2 w a 2 1  ,   23 3 w a 2 3  . For i 1  , then     3 1 j 2 1j 23 23 x 0 1 w x w a 1     with   23 11 w x 1  . We get 3 11 2 5,7 x        , 3 12 2 0,0 x        and 23 13 x 0,0        . For i 2  , then     3 2 j 2 2 j 23 x 0 23 2 1 w x w a 2     with   23 21 1 w x 4  and   23 22 1 w x 4  . We get 23 21 1 3 , 4 4 x            , 23 22 27 29 , 4 4 x            and 23 23 x 0,0        . For i 3  , then     3 3 j 2 3 j 23 x 0 23 3 3 w x w a 2     with   23 32 w x 2 1  and   23 33 w x 1  . We get 23 31 0,0 x        , 23 32 1,2 x        and 23 33 x 4,6        . Step 3. Solving Eq. (16) via Eq. (18). Optimal solution is: 14 11 x 5  , 14 12 x 0  and 14 13 x 0  , 14 21 5 x 2  , 14 22 9 x 2  , 14 23 x 0  , 14 31 x 0  , 14 32 5 x 2  and 14 33 x 5  . Furthermore   14 1 w a 4  ,   14 2 w a 3  ,   14 3 w a 2 7  . For i 1  , we have     23 n 14 1 23 1 1j 1j 1 1 j 1 4 E x x w a 5 6 0 0 0 0 1 2 4 w a                , then     4 1 j 1 1j 14 14 x 0 1 w x w a 4     with   14 11 w x 4  . We get 4 11 1 1,9 x        , 4 12 1 0,0 x        and 14 13 x 0,0        . For i 2  , we have     n 14 23 2 2 j 2 j 2 2 j 2 1 1 3 4 5 1 9 1 E x x w a 7 0 0 5 w a 3 2 2 2 2                , then     14 23 2 14 3 2 j 2 j 2 j 2 j w x x x w x    with   14 21 9 w x 4  and   14 22 11 w x 4  . We get 14 21 1 19 , 4 4 x            , 14 22 7 29 , 4 4 x            and 14 23 x 0,0        . For i 3  , we have     n 14 23 3 3 j 3 j 23 14 3 3 j 1 5 3 3 5 E x x w a 5 5 0 0 w a 2 2 2 2 2 7                , then     14 3 j 14 14 3 j 3 x 0 7 w x w a 2     with   14 32 3 w x 2  and   14 33 w x 2  . We get 14 31 0,0 x        , 14 32 1,4 x        and 14 33 x 3,7        . Step 4. The optimal solution according to the choice of the decision maker is:
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218 Kané et al. |J. Fuzzy. Ext. Appl. 2(3) (2021) 204-225       * 14 23 *1 *4 *2 *3 *1 *2 *3 *4 ij ij ij ij ij ij ij ij ij ij ij x x |x x ,x | x ,x x ,x ,x ,x                % .   81 1037 Min Z x ,81, 173, 4 2            % % with   * 11 x 1, 5, 7 ,9  % , * 12 x 0  % % , * 13 x 0  % % , * 21 1 1 3 19 x , , , 4 4 4 4            % , * 22 7 27 29 29 x , , , 4 4 4 4            % , * 23 x 0  % % , * 31 x 0  % % ,   * 32 x 1,1,2,4  % and   * 33 x 3,4,6,7  % . Example 3. [29]. Step 1. Construct the fuzzy transportation Problem (15), and then convert it into a balanced one if it is not. Table 3. Eq. (15) in triangular balanced form. Step 2. Solving Eq. (17) via Eq. (19). We get   2 2 2 2 2 2 11 12 21 22 MinZ x 31x 19x 39x 10x     subject to the constraints 2 2 11 12 x x 201   , 2 2 21 22 x x 99   , 2 2 11 21 x x 150   and 2 2 12 22 x x 150   . Optimal solution is: 2 11 x 150  , 2 12 x 51  , 2 21 x 0  and 2 22 x 99  . Step 3. Solving Eq. (16) via Eq. (18). We have   13 13 13 13 13 13 11 12 21 22 MinZ x 28x 22x 42x 10x     subject to the constraints 13 13 11 12 x x 198   , 13 13 21 22 x x 102   , 13 13 11 21 x x 150   and 13 13 12 22 x x 150   . Optimal solution is: 13 11 x 150  , 13 12 x 48  , 13 21 x 0  and 13 22 x 102  . Furthermore   13 1 w a 48  ,   13 2 w a 52  . For i 1  , we have   2 13 2 13 1 1j 1j 1 j 1 E x x 150 150 48 51 3 48 w a            , then     3 1 j 1 1j 13 x 0 13 1 w x w a 48     with   13 11 w x 10  and   13 12 w x 38  . We get 13 11 x 140,160        and 13 12 x 10,86        . 1 R 2 R Supply ( %i a ) A   22,31,34   15,19,29   150,201,246 B   30,39,54   8,10,12   50,99,154 Demand ( % j b )   100,150,200   100,150,200     % % m n i j i 1 j 1 a b
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219 A new algorithm for fuzzy transportation problems with trapezoidal fuzzy numbers under fuzzy circumstances For i 2  , we have   2 13 2 13 2 2 j 2 j 2 j 1 E x x 0 0 102 99 3 52 w a            , then     3 2 j 1 2 j 13 x 0 13 2 w x w a 52     with   13 21 w x 0  and   13 22 w x 52  . We get 13 21 x 0,0        and 13 22 x 50,154        . Step 4. The optimal solution according to the choice of the decision maker is     m n 1 2 3 ij ij ij ij i 1 j 1 MinZ x c ,c ,c x      % % % with       * 2 13 2 *1 *3 *1 *2 *3 ij ij ij ij ij ij ij ij ij x x |x x | x ,x x ,x ,x          % . We have Min     * Z 6609| 3630, 9782 3630,6609, 9782         % where   * 11 x 140,150, 160  % ,   * 12 x 10,51,86  % , * 21 x 0  % % and   * 22 x 50,99,154  % . 5.1| Interpretation of Results We will now interpret the minimum total fuzzy transportation cost obtained in Example 3. by using the proposed methods presented in Section 3. Similarly, the obtained fuzzy optimal solution will also be interpreted. By using the methods proposed the minimum total fuzzy transportation cost is   3630,6609, 9782 , which can be physically interpreted as follows:  The least amount of the minimum total transportation cost is 3630 .  The most possible amount of minimum total transportation cost is 6609 .  The greatest amount of the minimum total transportation cost is 9782 i.e., the minimum total transportation cost will always be greater than 3630 and less than6609 , and the highest chances are that the minimum total transportation cost will be9782 . Example 4. [1]-[3]. Step 1. Construct the fuzzy transportation Problem (15), and then convert it into a balanced one if it is not. Step 2. Solving Eq. (17) via Eq. (19). We get   2 2 2 2 2 2 2 2 2 2 11 12 13 14 21 22 23 24 MinZ x 10x 22x 10x 20x 15x 20x 12x 8x        2 2 2 2 31 32 33 34 20x 12x 10x 15x     subject to the constraints 2 2 2 2 11 12 13 14 x x x x 8     , 2 2 2 2 21 22 23 24 x x x x 14     , 2 2 2 2 31 32 33 34 x x x x 12     , 2 2 2 11 21 31 x x x 7    , 2 2 2 12 22 32 x x x 10    , 2 2 2 13 23 33 x x x 8    and 2 2 2 14 24 34 x x x 9    .
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220 Kané et al. |J. Fuzzy. Ext. Appl. 2(3) (2021) 204-225 Optimal solution is: 2 11 x 7  , 2 12 x 0  , 2 13 x 1  , 2 14 x 0  , 2 21 x 0  , 2 22 x 0  , 2 23 x 5  , 2 24 x 9  , 2 31 x 0  , 2 32 x 10  , 2 33 x 2  and 2 34 x 0  . Table 4. Eq. (15) in triangular balanced form (in U.S. dollar). Step 3. Solving Eq. (16) via Eq. (18). We get   13 13 13 13 13 13 13 13 11 12 13 14 21 22 MinZ x 9.4x 22.2x 9.3x 20.4x 15x 20.1x       13 13 13 13 13 13 23 24 31 32 33 34 11.5x 7.4x 19.7 x 11.3x 9.3x 15x       subject to the constraints 13 13 13 13 11 12 13 14 x x x x 8     , 13 13 13 13 21 22 23 24 x x x x 14     , 13 13 13 13 31 32 33 34 x x x x 12     , 13 13 13 11 21 31 x x x 7    , 13 13 13 12 22 32 x x x 10    , 13 13 13 13 23 33 x x x 8    and 13 13 13 14 24 34 x x x 9    . Optimal solution is: 13 11 x 7  , 13 12 x 0  , 13 13 x 1  , 13 14 x 0  , 13 21 x 0  , 13 22 x 0  , 13 23 x 5  , 13 24 x 9  , 13 31 x 0  , 13 32 x 10  , 13 33 x 2  and 13 34 x 0  . We have   13 1 4 w a 5  ,   13 2 w a 2  and   13 3 9 w a 5  . Furthermore:  13 11 34 36 x , 5 5            , 13 12 x 0,0        , 13 13 2 8 x , 5 5            and 13 14 x 0,0        where   13 11 1 w x 5  ,   13 12 w x 0  ,   13 13 3 w x 5  and   13 14 w x 0  .  13 21 x 0,0        , 13 22 x 0,0        , 13 23 9 11 x , 2 2            and 13 24 15 21 x , 2 2            where   13 21 w x 0  ,   13 22 w x 0  ,   13 23 1 w x 2  and   13 24 3 w x 2  . Source Destination Supply (𝒂 ̃𝒊) (000 dozen bottles) Taichung Chiayi Kaohsiung Taipei Changhua   $8, $10, $10.8   $20.4, $22, $24   $8,$10,$10.6   $18.8, $20, $22   7.2,8,8.8 Touliu   $14, $15, $16   $18.2, $20, $22   $10, $12, $13   $6, $8, $8.8   12,14,16 Hsinchu   $18.4, $20, $21   $9.6, $12, $13   $7.8, $10, $10.8   $14, $15, $16   10.2,12,13.8 Demand ( j b % ) (000 dozen bottles)   6.2,7,7.8   8.9,10,11.1   6.5,8,9.5   7.8,9,10.2 ∑𝑎̃𝑖 𝑚 𝑖=1 = ∑𝑏̃𝑗 𝑛 𝑗=1
18
221 A new algorithm for fuzzy transportation problems with trapezoidal fuzzy numbers under fuzzy circumstances  13 31 x 0,0        , 13 32 47 53 x , 5 5            , 13 33 4 16 x , 5 5            and 13 34 x 0,0        with   13 31 w x 0  ,   13 32 3 w x 5  ,   13 33 6 w x 5  and   13 34 w x 0  . Step 4. The optimal solution according to the choice of the decision maker is The value of the objective function:        m n 1 2 3 *1 *2 *3 *1 *2 *3 ij ij ij ij ij ij j 1 i 1 Min Z x c ,c ,c x ,x ,x Z ,Z ,Z       % % with       * 2 13 2 *1 *3 *1 *2 *3 ij ij ij ij ij ij ij ij ij x x |x x | x ,x x ,x ,x          % . We have Min * 12204 21549 12204 21549 Z 352$| $, $ $,352$, $ 50 50 50 50                                   % where * 11 34 36 x , 7 , 5 5            % , * 12 x 0  % % , * 13 2 8 x , 1, 5 5            % , * 14 x 0  % % , * 21 x 0  % % , * 22 x 0  % % , * 23 9 11 x , 5, 2 2            % , * 24 15 21 x , 9, 2 2            % , * 31 x 0  % % , * 32 47 53 x , 10, 5 5            % , * 33 4 16 x , 2, 5 5            % and * 34 x 0  % % . 5.2| Interpretation of Results We will now interpret the minimum total fuzzy transportation cost obtained in Example 4 by using the proposed methods presented in Section 3. Similarly, the obtained fuzzy optimal solution will also be interpreted. By using the methods proposed the minimum total fuzzy transportation cost is 12204 21549 $,352$, $ 50 50           , which can be physically interpreted as follows:  The least amount of the minimum total transportation cost is 12204 $ 50 .  The most possible amount of minimum total transportation cost is 352$ .  The greatest amount of the minimum total transportation cost is 21549 $ 50 i.e., the minimum total transportation cost will always be greater than 12204 $ 50 and less than 352$ , and the highest chances are that the minimum total transportation cost will be 21549 $ 50 .
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222 Kané et al. |J. Fuzzy. Ext. Appl. 2(3) (2021) 204-225 6| A Comparative Study Author's of [3], [11], [29] have proposed a method to find the crisp optimal solution of such fuzzy transportation problems in which all the parameters are represented by triangular or trapezoidal fuzzy numbers. Then, they have used their new method proposed to find the crisp optimal solution of a real- life fuzzy transportation problem. However, it is often better to find a fuzzy optimal solution than a crisp optimal solution. In this section we will therefore show how in the problem considered by [3], [11], [29], we can obtain a fuzzy optimal solution of the same real-life problem using the new method proposed. 7| Concluding Remarks and Future Research Directions 7.1| Concluding Remarks These days a number of researchers have shown interest in the area of fuzzy transportation problems and various attempts have been made to study the solution of these problems. In this paper, to overcome the shortcomings of the existing methods we introduced a new formulation of transportation problem involving trapezoidal (or triangular) fuzzy numbers for the transportation costs and values of supplies and demands. We propose a fuzzy linear programming approach for solving trapezoidal (or triangular) fuzzy numbers transportation problem based on the converting into two interval transportation problems Eq. (16) and Eq. (17). To show the advantages of the proposed methods over existing methods, some fuzzy transportation problems, may or may not be solved by the existing methods, are solved by using the proposed methods and it is shown that it is better to use the proposed methods as compared to the existing methods for solving the transportation problems. From both theoretical and algorithmic considerations, and examples solved in this paper, it can be noticed that some shortcomings of the methods for solving the fuzzy transportation problems known from the literature can be resolved by using the new methods proposed in Section 5. 7.2| Future Research Directions Finally, we feel that, there are many other points of research and should be studied later on interval numbers or fuzzy numbers. Some of these points are below: We will consider the following transportation Problems (14) with fuzzy numbers as follows: Example Minimum Total Fuzzy Transportation Cost [3] [29] Method Proposed in This Paper 1   2100,2900,3500,4200   930,2930,2970,3470 2   31,80,199,460           81 1037 ,81,173, 4 2 3   3630,6609,9782 4   238.44,347.8,428.9           12204 21549 ,352, 50 50
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223 A new algorithm for fuzzy transportation problems with trapezoidal fuzzy numbers under fuzzy circumstances   n m ij ij j 1i 1 MinZ x c x     % % % % subject to the constraints n ij i j 1 x a   % % and m ij j i 1 x b    % % where   1 2 t ij ij ij ij c c ,c , ,c   % ,   1 2 t ij ij ij ij x x ,x , ,x   % ,   1 2 t i i i i a a ,a , ,a   % and   1 2 t j j j j b b ,b , ,b   % with 1 t  N . Let pq p q ij ij ij x x ,x        where p q  and 1 p,q  N . The same applies to pq ij c , pq i a and pq j b . I. Solution procedure for classical Transportation Problem   t 1  :   1 ij ij x x .  II. Solution procedure for Transportation Problem with Interval numbers   t 2  : 12 1 2 ij ij ij x x ,x .        III. Solution procedure for Transportation Problem with Triangular fuzzy numbers   t 3  :     1 2 3 2 13 ij ij ij ij ij ij x x ,x ,x x ,x .   % IV. Solution procedure for Transportation Problem with Trapezoidal fuzzy numbers   t 4  :     1 2 3 4 23 14 ij ij ij ij ij ij ij x x ,x ,x ,x x ,x .   % V. Solution procedure for Transportation Problem with Pentagonal fuzzy numbers   t 5  :     1 2 3 4 5 3 24 15 ij ij ij ij ij ij ij ij ij x x , x , x , x , x x , x , x .   % VI. Solution procedure for Transportation Problem with Hexagonal fuzzy numbers   t 6  :     1 2 3 4 5 6 34 25 16 ij ij ij ij ij ij ij ij ij ij x x ,x ,x ,x ,x ,x x ,x ,x .   % VII. Solution procedure for Transportation Problem with Heptagonal fuzzy numbers   t 7      1 2 3 4 5 6 7 4 35 26 17 ij ij ij ij ij ij ij ij ij ij ij ij : x x ,x ,x ,x ,x ,x ,x x ,x ,x ,x .   % VIII. Solution procedure for Transportation Problem with Octagonal fuzzy numbers   t 8      1 2 3 4 5 6 7 8 45 36 27 18 ij ij ij ij ij ij ij ij ij ij ij ij ij : x x ,x ,x ,x ,x ,x ,x ,x x ,x ,x ,x .   % IX. Solution procedure for Transportation Problem with Nonagonal fuzzy numbers   t 9  :     1 2 3 4 5 6 7 8 9 5 46 37 28 19 ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij x x ,x ,x ,x ,x ,x ,x ,x ,x x ,x ,x ,x ,x .   % X. Solution procedure for Transportation Problem with Decagonal fuzzy numbers   t 10  :     1 2 3 4 5 6 7 8 9 10 56 47 38 29 110 ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij x x ,x ,x ,x ,x ,x ,x ,x ,x ,x x ,x ,x ,x ,x .   % XI. Solution procedure for Transportation Problem with Hendecagonal fuzzy numbers   t 11  :     1 2 3 4 5 6 7 8 9 10 11 6 57 48 39 210 111 ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij x x ,x ,x ,x ,x ,x ,x ,x ,x ,x ,x x ,x ,x ,x ,x ,x .   % XII. Solution procedure for Transportation Problem with Dodecagonal fuzzy numbers   t 12      1 2 3 4 5 6 7 8 9 10 11 12 67 58 49 310 211 112 ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij : x x ,x ,x ,x ,x ,x ,x ,x ,x ,x ,x ,x x ,x ,x ,x ,x ,x .   % Acknowledgements The authors are very grateful to the anonymous referees for their valuable comments and suggestions to improve the paper in the present form. Conflicts of interest The authors declare no conflicts of interest.
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224 Kané et al. |J. Fuzzy. Ext. Appl. 2(3) (2021) 204-225 References Kaur, J., & Kumar, A. (2016). An introduction to fuzzy linear programming problems theory, methods and applications. Springer. Nasseri, S. H., Ebrahimnejad, A., & Cao, B. Y. (2019). Fuzzy linear programming. In Fuzzy linear programming: solution techniques and applications (pp. 39-61). Springer, Cham. Kaur, A., Kacprzyk, J., Kumar, A., & Analyst, C. F. (2020). Fuzzy transportation and transshipment problems. Springer International Publishing. Zimmermann, H. J. (1978). Fuzzy programming and linear programming with several objective functions. Fuzzy sets and systems, 1(1), 45-55. ÓhÉigeartaigh, M. (1982). A fuzzy transportation algorithm. Fuzzy sets and systems, 8(3), 235-243. Chanas, S., Kołodziejczyk, W., & Machaj, A. (1984). A fuzzy approach to the transportation problem. Fuzzy sets and systems, 13(3), 211-221. Chanas, S., Delgado, M., Verdegay, J. L., & Vila, M. A. (1993). Interval and fuzzy extensions of classical transportation problems. Transportation planning and technology, 17(2), 203-218. Kuchta, D. (1996). A concept of the optimal solution of the transportation problem with fuzzy cost coefficients. Fuzzy sets and systems, 82(299), 299-305. Jiménez, F., & Verdegay, J. L. (1998). Uncertain solid transportation problems. Fuzzy sets and systems, 100(1-3), 45-57. Jiménez, F., & Verdegay, J. L. (1999). Solving fuzzy solid transportation problems by an evolutionary algorithm based parametric approach. European journal of operational research, 117(3), 485-510. Liu, S. T., & Kao, C. (2004). Solving fuzzy transportation problems based on extension principle. European journal of operational research, 153(3), 661-674. Gani, A. N., & Razak, K. A. (2006). Two stage fuzzy transportation problem. J. Phys. Sci. 10, 63–69. http://inet.vidyasagar.ac.in:8080/jspui/handle/123456789/720 Li, L., Huang, Z., Da, Q., & Hu, J. (2008, May). A new method based on goal programming for solving transportation problem with fuzzy cost. 2008 international symposiums on information processing (pp. 3- 8). IEEE. Lin, F. T. (2009, August). Solving the transportation problem with fuzzy coefficients using genetic algorithms. 2009 IEEE international conference on fuzzy systems (pp. 1468-1473). IEEE. Dinagar, D. S., & Palanivel, K. (2009). The transportation problem in fuzzy environment. International journal of algorithms, computing and mathematics, 2(3), 65-71. Pandian, P., & Natarajan, G. (2010). A new algorithm for finding a fuzzy optimal solution for fuzzy transportation problems. Applied mathematical sciences, 4(2), 79-90. Kumar, A., & Kaur, A. (2011). Application of linear programming for solving fuzzy transportation problems. Journal of applied mathematics & informatics, 29(3_4), 831-846. Gupta, A., Kumar, A., & Kaur, A. (2012). Mehar’s method to find exact fuzzy optimal solution of unbalanced fully fuzzy multi-objective transportation problems. Optimization letters, 6(8), 1737-1751. Ebrahimnejad, A. (2015). A duality approach for solving bounded linear programming problems with fuzzy variables based on ranking functions and its application in bounded transportation problems. International journal of systems science, 46(11), 2048-2060. Shanmugasundari, M., & Ganesan, K. (2013). A novel approach for the fuzzy optimal solution of fuzzy transportation problem. Transportation, 3(1), 1416-1424. Chandran, S., & Kandaswamy, G. (2016). A fuzzy approach to transport optimization problem. Optimization and engineering, 17(4), 965-980. Ebrahimnejad, A. (2016). Note on “A fuzzy approach to transport optimization problem”. Optimization and engineering, 17(4), 981-985. Kumar, A., & Kaur, A. (2014). Optimal way of selecting cities and conveyances for supplying coal in uncertain environment. Sadhana, 39(1), 165-187. Ebrahimnejad, A. (2015). An improved approach for solving fuzzy transportation problem with triangular fuzzy numbers. Journal of intelligent & fuzzy systems, 29(2), 963-974. Kaur, A., & Kumar, A. (2011). A new method for solving fuzzy transportation problems using ranking function. Applied mathematical modelling, 35(12), 5652-5661.
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Corresponding Author: fsegmath@gmail.com http://dx.doi.org/10.22105/jfea.2021.287198.1148 E-ISSN: 2717-3453 | P-ISSN: 2783-1442 | Abstract 1 | Introduction Transportation problem is an important network structured linear programming problem that arises in several contexts and has deservedly received a great deal of attention in the literature. The central concept in this problem is to find the least total transportation cost of a commodity in order to satisfy demands at destinations using available supplies at origins. Transportation problem can be used for a wide variety of situations such as scheduling, production, investment, plant location, inventory control, employment scheduling and many others. Journal of Fuzzy Extension and Applications www.journal-fea.com J. Fuzzy. Ext. Appl. Vol. 2, No. 3 (2021) 204–225. Paper Type: Research Paper The New Algorithm for Fully Fuzzy Transportation Problem by Trapezoidal Fuzzy Number (A Generalization of Triangular Fuzzy Number) Ladji Kané1,* , Moctar Diakité1, Souleymane Kané1, Hawa Bado1, Moussa Konaté1, Koura Traoré1 1 Faculty of Economics and Management, University of Social Sciences and Management of Bamako (USSGB), Quartier du Fleuve Rue 310, Porte 238, Mali; fsegmath@gmail.com; moctardia@yahoo.fr; garalo2018@gmail.com; badohawa@yahoo.fr; konatefseg@gmail.com. Citation: Kané, L., Diakité, M., Kané, S., Bado, H., Konaté, M., & Traoré, K. (2021). The new algorithm for fully fuzzy transportation problem by trapezoidal fuzzy number (a generalization of triangular fuzzy number). Journal of fuzzy extension and application, 2 (3), 204-225. Accept: 29/07/2021 Revised: 17/06/2021 Reviewed: 01/06/2021 Received: 20/05/2021 Transportation problem is an important network structured linear programming problem that arises in several contexts and has deservedly received a great deal of attention in the literature. The central concept in this problem is to find the least total transportation cost of a commodity in order to satisfy demands at destinations using available supplies at origins in a crisp environment. In real life situations, the decision maker may not be sure about the precise values of the coefficients belonging to the transportation problem. The aim of this paper is to introduce a formulation of fully fuzzy transportation problem involving trapezoidal fuzzy numbers for the transportation costs and values of supplies and demands. We propose a two-step method for solving fuzzy transportation problem where all of the parameters are represented by triangular fuzzy numbers i.e. two interval transportation problems. Since the proposed approach is based on classical approach it is very easy to understand and to apply on real life transportation problems for the decision makers. To illustrate the proposed approach four application examples are solved. The results show that the proposed method is simpler and computationally more efficient than existing methods in the literature. Keywords: Fully fuzzy linear programming, Transportation problem, Trapezoidal fuzzy numbers, fuzzy numbers. Licensee Journal of Fuzzy Extension and Applications. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons. org/licenses/by/4.0).
205 A new algorithm for fuzzy transportation problems with trapezoidal fuzzy numbers under fuzzy circumstances In general, transportation problems are solved with the assumptions that the transportation costs and values of supplies and demands are specified in a precise way i.e., in crisp environment. However, in many cases the decision maker has no crisp information about the coefficients belonging to the transportation problem. If the nature of the information is vague, that is, if it has some lack of precision, the corresponding coefficients or elements defining the problem can be formulated by means of fuzzy sets, and thus fuzzy transportation problems arise. Several researchers have carried out investigations on fuzzy transportation problem. Zimmermann [4] developed Zimmermann's fuzzy linear programming into several fuzzy optimization methods for solving the transportation problems. ÓhÉigeartaigh [5] proposed an algorithm for solving transportation problems where the supplies and demands are fuzzy sets with linear or triangular membership functions. Chanas et al. [6] investigated the transportation problem with fuzzy supplies and demands and solved them via the parametric programming technique. Their method provided solution which simultaneously satisfies the constraints and the goal to a maximal degree. In addition, Chanas et al. [7] formulated the classical, interval and fuzzy transportation problem and discussed the methods for solution for the fuzzy transportation problem. Kuchta [8] discussed the type of transportation problems with fuzzy cost coefficients and converted the problem into a bicriterial transportation problem with crisp objective function. Their method only gives crisp solutions based on efficient solutions of the converted problems. Jiménez and Verdegay [9], [10] investigated the fuzzy solid transportation problem in which supplies, demands and conveyance capacities are represented by trapezoidal fuzzy numbers and applied a parametric approach for finding the fuzzy solution. Liu and Kao [11] developed a procedure, based on extension principle to derive the fuzzy objective value of fuzzy transportation problem, in that the cost coefficients and the supply and demand quantities are fuzzy numbers. Gani and Razak [12] presented a two-stage cost minimizing fuzzy transportation problem in which supplies and demands are as trapezoidal fuzzy numbers and used a parametric approach for finding a fuzzy solution with the aim of minimizing the sum of the transportation costs in the two stages. Li et al. [13] proposed a new method based on goal programming for solving fuzzy transportation problem with fuzzy costs. Lin [14] used genetic algorithm for solving transportation problems with fuzzy coefficients. Dinagar and Palanivel [15] investigated fuzzy transportation problem, with the help of trapezoidal fuzzy numbers and applied fuzzy modified distribution method to obtain the optimal solution in terms of fuzzy numbers. Pandian and Natarajan [16] introduced a new algorithm namely, fuzzy zero-point method for finding fuzzy optimal solution for such fuzzy transportation problem in which the transportation cost, supply and demand are represented by trapezoidal fuzzy numbers. Kumar and Kaur [17] proposed a new method based on fuzzy linear programming problem for finding the optimal solution of fuzzy transportation problem. Gupta et al. [18] proposed a new method named as Mehar's method, to find the exact fuzzy optimal solution of fully fuzzy multi-objective transportation problems. Ebrahimnejad [19] applied a fuzzy bounded dual algorithm for solving bounded transportation problems with fuzzy supplies and demands. Shanmugasundari and Ganesan [20] developed the fuzzy version of Vogel's and MODI methods for obtaining the fuzzy initial basic feasible solution and fuzzy optimal feasible solution, respectively, without converting them into classical transportation problem. Also, [21] Chandran and Kandaswamy [21] proposed an algorithm to find an optimal solution of a fuzzy transportation problem, where supply, demand and cost coefficients all are fuzzy numbers. Their algorithm provides decision maker with an effective solution in comparison to existing methods. Ebrahimnejad [22] using an example showed that their method will not always lead to a fuzzy optimal solution. Moreover, Kumar and Kaur [23] pointed out the limitations and shortcomings of the existing methods for solving fuzzy solid transportation problem and to overcome these limitations and shortcomings proposed a new method to find the fuzzy optimal solution of unbalanced fuzzy solid transportation problems. In addition, Ebrahimnejad [24] proposed a two-step method for solving fuzzy transportation problem where all of the parameters are represented by non-negative triangular fuzzy numbers. Some researchers applied generalized fuzzy numbers for solving transportation problems. Kumar and Kaur [25] proposed a new method based on ranking function for solving fuzzy transportation problem by assuming that transportation cost, supply and demand of the commodity are represented by generalized trapezoidal fuzzy numbers. After that, Kaur and Kumar [26] introduced a similar algorithm for solving a special type of fuzzy
206 Kané et al. |J. Fuzzy. Ext. Appl. 2(3) (2021) 204-225 transportation problem by assuming that a decision maker is uncertain about the precise values of transportation cost only but there is no uncertainty about the supply and demand of the product. Ebrahimnejad [27] demonstrated that once the ranking function is chosen, the fuzzy transportation problem introduced by Kaur and Kumar [26] is converted into crisp one, which is easily solved by the standard transportation algorithms. The contributions of the present study are summarized as follows: (a) in the Eq. (15) under consideration, all of the parameters, such as the transportation costs, supplies and demands are considered as fuzzy numbers. (b) According to the proposed approach, the Eq. (15) is converted into two interval transportation problems Eq. (16) and Eq. (17). The integration of the optimal solution of the two sub-problems provides the optimal solution of the Eq. (15). (c) In contrast to most existing approaches, which provide a precise solution, the proposed method provides a fuzzy optimal solution. (d) In contrast to existing methods that include negative parts in the obtained fuzzy optimal solution and fuzzy optimal cost, the proposed method provides a fuzzy optimal solution and optimal cost. (e) Similarly, to the competing methods in the literature, the proposed method is applicable for all types of trapezoidal fuzzy numbers. (f) The complexity of computation is greatly reduced compared with commonly used existing methods in the literature. The rest of this paper is organized as follows: In Section 2, we recall the definitions of interval number linear programming, interval numbers and fully fuzzy transportation problem. In Section 3, a new method is proposed for obtaining the fuzzy optimal solution of the Eq. (15). The advantages of the proposed method are discussed in Section 4. Two application examples are provided to illustrate the effectiveness of the proposed method in Section 5, and a comparative study in Section 6. Finally, concluding remarks are presented in Section 7. 2| Materials and Methods In this section, some basic definitions, arithmetic operations for closed Intervals numbers and of linear programming problems involving interval numbers are presented [28]. 2.1| A New Interval Arithmetic In this section, some arithmetic operations for two intervals are presented [28]. Let   1 4 1 4 1 4 a a ,a :a a , a ,a           R R be the set of all proper intervals. We shall use the terms “interval” and “interval number” interchangeably. The mid-point and width (or half-width) of an interval number 1 4 a a ,a        are defined as   4 1 a a m a 2   and   4 1 a a w a 2   . The interval number a can also be expressed in terms of its midpoint and width as For any two intervals     1 4 a a ,a m a ,w a           and     1 4 b b ,b m b ,w b           , the arithmetic operations on a and b are defined as:                    4 1 4 1 1 4 a a a a a a ,a m a ,w a , . 2 2 (1)               a b m a m b ,w a w b . (2)               a b m a m b ,w a w b . (3)
207 A new algorithm for fuzzy transportation problems with trapezoidal fuzzy numbers under fuzzy circumstances 2.2| A New Interval Arithmetic for Trapezoidal Fuzzy Numbers via Intervals Numbers The aim of this section is to present some notations, notions and results which are of useful in our further consideration. A number   1 2 3 4 a a ,a ,a ,a  % (where 1 2 3 4 a a a a    ) is said to be a trapezoidal fuzzy number if its membership function is given by [1]-[3]: Assume that       1 2 3 4 14 23 1 4 2 3 a a ,a ,a ,a a |a a ,a | a ,a                % , and       1 2 3 4 14 23 1 4 2 3 b b ,b ,b ,b b |b b ,b | b ,b                % are two trapezoidal fuzzy numbers. For any two trapezoidal fuzzy numbers   14 23 a a |a  % and   14 23 b b |b  % , the arithmetic operations on a % and b % are defined as: 2.3| Formulation of a Transportation Problems Involving Interval Numbers We consider the Transportation Problem involving Interval numbers as follows [28]:                       αm a ,αw a   if  α 0, αa αm a , αw a   if  α 0. (4)                                                                                1 1 1 1 4 1 m a m b w a w b ,m a w b m b w a ifa 0, b 0, a b m a m b m a w b ,m b w a w b w a ifa 0, b 0, m a m b w a w b ,m b w a m a w b ifa 0, b 0. (5)                         % 1 1 2 2 1 a 4 3 4 3 4 x a , a x a , a a μ x x a , a x a . a a (6)                                     % % 14 23 14 23 1 4 1 4 2 3 2 3 a b a |a b |b a ,a b , b | a ,a b , b . (7)                                    % % 14 23 14 23 1 4 1 4 2 3 2 3 ab a |a b |b a ,a b , b | a ,a b , b (8)
208 Kané et al. |J. Fuzzy. Ext. Appl. 2(3) (2021) 204-225 where 1 4 ij ij ij c c ,c        , 1 4 i i i a a ,a        , 1 4 j j j b b ,b        are non-negative interval numbers and 1 4 ij ij ij x x ,x        are unrestricted interval numbers. Objective function transformation. Transformation of constraints.         n n ij i ij ij i i j 1 j 1 x a m x ,w x m a ,w a           for i 1, 2, ,m   and         m m ij j ij ij j j i 1 i 1 x b m x ,w x m b ,w b           for j 1,2, ,n   . Now we can say that                                    n m ij ij j 1i 1 n ij i j 1 m ij j i 1 MinZ x c x Subjecttotheconstraints x a ,fori 1,2, ,m , x b , for j 1,2, ,n. (9)                                   n m n m n m 1 4 1 4 ij ij ij ij ij ij ij ij ij ij j 1 i 1 j 1 i 1 j 1 i 1 Z x c x c ,c x ,x m c x ,w c x , Where                                            1 ij ij ij ij ij 1 4 ij ij ij ij ij ij ij ij 4 ij ij ij ij ij m c m x w c w x if x 0, m c x m c m x w c m x if x 0andx 0, m c m x w c w x if x 0. And (10)                                            1 ij ij ij ij ij 1 4 ij ij ij ij ij ij ij ij 4 ij ij ij ij ij m c w x w c m x if x 0, w c x m c w x w c w x if x 0andx 0, m c w x w c m x if x 0. (11)
209 A new algorithm for fuzzy transportation problems with trapezoidal fuzzy numbers under fuzzy circumstances is equivalent to Optimal Solution (9) according to the choice of the decision maker:         * 1 4 ij ij ij ij ij ij ij x x ,x m x w x ,m x w x                   where     n ij i j 1 w x w a    , i 1, 2, ,m   . Remark. I. n ij i j 1 x a    if and only if     n ij i j 1 m x m a    and     n ij i j 1 w x w a    for i 1, 2, ,m   . II. n ij i j 1 x a    if and only if     n ij i j 1 m x m a    and     n ij i j 1 w x w a    for i 1, 2, ,m   . III. m ij j i 1 x b    if and only if     m ij j i 1 m x m b    and     m ij j i 1 w x w b    for j 1,2, ,n   . IV. m ij j i 1 x b    if and only if     m ij j i 1 m x m b    and     m ij j i 1 w x w b    for j 1,2, ,n   . 2.4| Formulation of a Fully Fuzzy Transportation Problem The fuzzy linear programming formulation of a fully fuzzy transportation problem can be written as follows as follows [2], [3]:                                                                   n m ij ij ij ij j 1 i 1 n ij ij i i j 1 m ij ij j j i 1 MinZ x m c x ,w c x Subjecttotheconstraints m x ,w x m a ,w a ,i 1,2, ,m, m x ,w x m b ,w b , j 1,2, ,n. (12)                                                     n m ij ij j 1 i 1 n ij i j 1 m ij j i 1 m MinZ x m c x Subjecttotheconstraints m x m a ,fori 1,2, ,m, m x m b , for j 1,2, ,n. (13)
210 Kané et al. |J. Fuzzy. Ext. Appl. 2(3) (2021) 204-225 with m n i j i 1 j 1 a b     % % where ij x % are unrestricted fuzzy numbers and ij c % , i a % and j b % are non-negatives fuzzy numbers. 3| Main Results In this section, we will describe our method of solving. 3.1| Transportation Problem with Trapezoidal Fuzzy Numbers For all the rest of this paper, we will consider the following transportation problem with trapezoidal fuzzy numbers as follows: where   1 2 3 4 ij ij ij ij ij c c ,c ,c ,c ,  %   1 2 3 4 j ij ij ij ij x x ,x ,x ,x  % ,   1 2 3 4 j j j j j b b ,b ,b ,b  % and   1 2 3 4 i i i i i a a ,a ,a ,a  % are trapezoidal fuzzy numbers with     14 23 1 4 2 3 ij ij ij ij ij ij ij x x |x x ,x | x ,x               % where 14 1 4 ij ij ij x x ,x        and 23 2 3 ij ij ij x x ,x        ,     14 23 1 4 2 3 ij ij ij ij ij ij ij c c |c c ,c | c ,c               % where 14 1 4 ij ij ij c c ,c        and 23 2 3 ij ij ij c c ,c ,            14 23 1 4 2 3 j j j j j j j b b |b b ,b | b ,b               % where 14 1 4 j j j b b ,b        and 23 2 3 j j j b b ,b        and     14 23 1 4 2 3 i i i i i i i a a |a a ,a | a ,a               % where 14 1 4 i i i a a ,a        and 23 2 3 i i i a a ,a        . For all the rest of this paper, we will consider the following transportation problems involving interval numbers 14 ij x , 14 ij c , 14 i a and 14 j b as follows:                                     % % % % % % % % m n ij ij i 1 j 1 n ij i j 1 m ij j i 1 MinZ x c x Subjecttotheconstraints x a , fori 1,2, ,m x b , for j 1,2, ,n (14)                                     % % % % % % % % m n ij ij i 1 j 1 n ij i j 1 m ij j i 1 MinZ x c x Subjecttotheconstraints x a ,for i 1,2, ,m, x b , for j 1,2, ,n. (15)
211 A new algorithm for fuzzy transportation problems with trapezoidal fuzzy numbers under fuzzy circumstances And the transportation problems involving interval numbers 23 ij x , 23 ij c , 23 i a and 23 j b as follows: 3.2| Formulation of a Transportation Problem involving Midpoint Thanks to the new interval arithmetic and Eq. (16), we can write the following transportation problem involving midpoint ( 14 ij x , 14 ij c , 14 i a and 14 j b ) [28] as follows: where   4 1 ij ij 14 ij c c m c 2   ,   4 1 14 i i i a a m a 2   ,   4 1 j j 14 j b b m b 2   and   4 1 14 i i i a a w a 2   . Thanks to the new interval arithmetic and Eq. (17), we can write the following transportation problem involving midpoint ( 23 ij x , 23 ij c , 23 i a and 23 j b ) [28] as follows:                                     m n 14 14 14 14 ij ij i 1 j 1 n 14 14 ij i j 1 m 14 14 ij j i 1 MinZ x c x Subjecttotheconstraints x a ,for i 1,2, ,m, x b ,for j 1,2, ,n. (16)                                    m n 23 23 23 23 ij ij i 1 j 1 n 23 23 ij i j 1 m 23 23 ij j i 1 MinZ x c x Subjecttotheconstraints x a ,for i 1,2, ,m, x b ,for j 1,2, ,n. (17)                                           m n 14 14 14 14 ij ij i 1 j 1 n 14 14 ij i j 1 14 14 ij j i 1 m MinZ x m c x Subjecttotheconstraints x m a ,for i 1,2, ,m, x m b ,for j 1,2, ,n. (18)
212 Kané et al. |J. Fuzzy. Ext. Appl. 2(3) (2021) 204-225 where   3 2 ij ij 23 ij c c m c 2   ,   3 2 23 i i i a a m a 2   ,   3 2 j j 23 j b b m b 2   and   i 3 i 2 23 i a a w a 2   . Thanks to the new interval arithmetic, we can write the following proposition [28]: Proposition 1. If 14 *1 *4 ij ij ij x x ,x        is an optimal solution to the Eq. (16) and 23 *2 *3 ij ij ij x x ,x        is an optimal solution to the Eq. (17), then   * * ij m n x x   % % is an optimal solution to the Eq. (15) with       * 14 23 *1 *4 *2 *3 *1 *2 *3 *4 ij ij ij ij ij ij ij ij ij ij ij x x |x x ,x | x ,x x ,x ,x ,x                % . 3.3| The Steps of Our Computational Method The steps of our method for solving the fully fuzzy transportation problem involving trapezoidal fuzzy numbers as follows. 3.3.1| Solution procedure for transportation problem with trapezoidal fuzzy numbers Step 1. Construct the fuzzy transportation Problem (15), and then convert it into a balanced one if it is not. Else, go to Step 2. Step 2. Solving Eq. (17) via Eq. (19). Determine 3 ij 2 x =     *2 *3 23 23 23 23 ij ij ij ij ij ij x ,x x w x ,x w x                  by applying the following conditions: I.   23 ij w x 0  if and only if 23 ij x 0  and II.     23 ij 23 23 ij i x 0 w x w a    with     23 23 ip ik w x w x  if 4 ip 4 ik c c  for i 1, ,m   . III. Go to Step 3. Step 3. Solving Eq. (16) via Eq. (18). Determine 4 ij 1 x =     *1 *4 ij ij ij ij ij i 14 14 14 14 j x ,x x w x ,x w x                  for i 1, ,m   . Considering the following cases: Case 1. If     n 14 23 i ij ij i i 3 1 j 1 2 4 E x x w a w a       , then 14 ij x =     *1 *4 ij ij ij ij ij i 14 14 14 14 j x ,x x w x ,x w x                  with     4 ij 1 i 14 j i x 0 14 w x w a    . Else, go to Case 2.                                           m n 23 23 23 23 ij ij i 1 j 1 n 23 23 ij i j 1 23 23 ij j i 1 m MinZ x m c x Subjecttotheconstraints x m a ,fori 1,2, ,m, x m b ,for j 1,2, ,n. (19)
213 A new algorithm for fuzzy transportation problems with trapezoidal fuzzy numbers under fuzzy circumstances Case 2. If     n 14 23 i ij ij i i 3 1 j 1 2 4 E x x w a w a       , then 14 ij x =     *1 *4 ij ij ij ij ij i 14 14 14 14 j x ,x x w x ,x w x                  with     14 14 2 23 3 ij ij ij ij w x x x w x    and go to Step 4. Step 4. The Optimal Solution (15) according to the choice of the decision maker is:        m n 1 2 3 4 *1 *2 *3 *4 *1 *2 *3 *4 ij ij ij ij ij ij ij ij j 1 i 1 Min Z x c ,c ,c ,c x ,x ,x ,x Z ,Z ,Z ,Z       % % where       * 14 23 *1 *4 *2 *3 *1 *2 *3 *4 ij ij ij ij ij ij ij ij ij ij ij x x |x x ,x | x ,x x ,x ,x ,x                % . 3.3.2| Solution procedure for transportation problem with triangular fuzzy numbers The steps of our method for solving the fully fuzzy transportation problem involving triangular fuzzy numbers as follows: The fuzzy number   * *1 *2 *3 *4 ij ij ij ij ij x x ,x ,x ,x  % is said to be triangular if and only if *2 *3 ij ij x x  . Then   *3 *2 ij ij 23 ij x x w x 0 2    . Therefore   14 14 23 ij ij ij w x x x   and       * *1 *2 *3 *4 *1 *2 *2 *4 *1 *2 *4 ij ij ij ij ij ij ij ij ij ij ij ij x x ,x ,x ,x x ,x ,x ,x x ,x ,x    % . In this article, consider fuzzy triangular numbers by:       * 2 13 2 *1 *3 *1 *2 *3 ij ij ij ij ij ij ij ij ij x x |x x | x ,x x ,x ,x          % , where     *1 *3 13 13 13 13 ij ij ij ij ij ij x ,x x w x ,x w x                  . Step 1. Construct the fuzzy transportation Problem (15), and then convert it into a balanced one if it is not. Else, go to Step 2. Step 2. Solving Eq. (17) via Eq. (19). Determine 3 ij 2 x = *2 *3 ij ij x ,x        *2 *2 ij ij x ,x        *2 ij x , for i 1, ,m   and go to Step 3. Step 3. Solving Eq. (16) via Eq. (18). Determine 3 ij 1 x =     *1 *3 ij ij ij ij ij i 13 13 13 13 j x ,x x w x ,x w x                  for i 1, ,m   . Considering the following cases: Case 1. If   n 13 i ij ij 2 3 1 1 i j E x x w a      , then 3 ij 1 x =     *1 *3 ij ij ij ij ij i 13 13 13 13 j x ,x x w x ,x w x                  with     3 ij 1 i 13 j i x 0 13 w x w a    . Else, go to Case 2. Case 2. If   13 n i ij i 13 i j 1 2 j E x x w a      , then 3 ij 1 x =     *1 *3 ij ij ij ij ij i 13 13 13 13 j x ,x x w x ,x w x                  with   ij i 13 13 j 2 j i w x x x   and go to Step 4. Step 4. The Optimal Solution (15) according to the choice of the decision maker is:
214 Kané et al. |J. Fuzzy. Ext. Appl. 2(3) (2021) 204-225        m n 1 2 3 *1 *2 *3 *1 *2 *3 ij ij ij ij ij ij j 1 i 1 Min Z x c ,c ,c x ,x ,x Z ,Z ,Z       % % with       * 2 13 2 *1 *3 *1 *2 *3 ij ij ij ij ij ij ij ij ij x x |x x | x ,x x ,x ,x          % . 4| Advantages of the Proposed Method Let us explore the main advantages of the proposed method:  The new proposed method is applicable to all Eq. (15) where ij x % are unrestricted triangular or trapezoidal fuzzy numbers and ij c % , i a % and j b % are non-negatives triangular or trapezoidal fuzzy numbers.  The proposed technique does not use the goal and parametric approaches which are difficult to apply in real life situations.  By applying the proposed approach for finding the fuzzy optimal solution, there is no need of much knowledge of fuzzy linear programming technique, Zimmerman approach and crisp linear programming which are difficult to learn for a new decision maker.  The proposed method to solve Eq. (15) is based on traditional transportation algorithms. Thus, the existing and easily available software can be used for the same. However, the existing method [1], [2], [3], [11], [29] to solve Eq. (15) should be implemented into a programming language.  To solve the Eq. (15) by using the existing method [1], [2], [3], [11], [29], there is need to use arithmetic operations of generalized fuzzy numbers. While, if the proposed technique is used for the same then there is need to use arithmetic operations of real numbers. This proves that it is much easy to apply the proposed method as compared to the existing method [1], [2], [3], [11], [29].  Moreover, it is possible to assume a generic ranking index for comparing the fuzzy numbers involved in the Eq. (15), in such a way that each time in which the decision maker wants to solve the FFTP under consideration (s)he can choose (or propose) the ranking index that best suits the Eq. (15). 5| Numerical Illustration In this section, to illustrate the new method proposed, and the existing fully fuzzy transportation problem due to [1], [2], [3], [11], [29], [30], presented in Examples 1-4, are solved by the proposed method. Example 1. [3]. Step 1. Construct the fuzzy transportation Problem (15), and then convert it into a balanced one if it is not. Table 1. Eq. (15) in trapezoidal unbalanced form. 1 R 2 R 3 R 3 R %i a Supply ( ) A   10,10,10,10   50,50,50,50   80,80,80,80   0,0,0,0   70,90,90,100 B   60,70,80,90   60,60,60,60   20,20,20,20   0,0,0,0   40,60,70,80 C   0,0,0,0   0,0,0,0   0,0,0,0   0,0,0,0   0,0,10,50 Demand ( % j b )   30,40,50,70   20,30,40,50   40,50,50,80   20,30,30,30     % % m n i j i 1 j 1 a b
215 A new algorithm for fuzzy transportation problems with trapezoidal fuzzy numbers under fuzzy circumstances Step 2. Solving Eq. (17) via Eq. (19).   23 23 23 23 23 23 23 23 11 12 13 21 22 23 MinZ x 25x 65x 100x 75x 90x 40x       subject to the constraints 23 23 23 23 11 12 13 14 x x x x 90     , 23 23 23 23 21 22 23 24 x x x x 65     , 23 23 23 23 31 32 33 34 x x x x 5     , 23 23 23 11 21 31 x x x 45    , 23 23 23 12 22 32 x x x 35    , 23 23 23 13 23 33 x x x 50    and 23 23 23 14 24 34 x x x 30    . Optimal solution is: 23 11 x 45  , 23 12 x 30  , 23 13 x 0  , 23 14 x 15  , 23 21 x 0  , 23 22 x 0  , 23 23 x 50  , 23 24 x 15  , 23 31 x 0  , 23 32 x 5  , 23 33 x 0  and 23 34 x 0  . Furthermore   23 1 w a 0  ,   23 2 w a 5  and   23 3 w a 5  . For i 1  , then     23 1 j 23 23 1j 1 x 0 w x w a 0     with         23 23 23 23 11 12 13 14 w x w x w x w x 0     . We get 23 11 x 45,45        , 23 12 x 30,30        , 23 13 x 0  and 23 14 x 15,15        . For i 2  , then     23 2 j 23 23 2 j 2 x 0 w x w a 5     with     23 23 21 22 w x w x 0   ,   23 23 w x 1  and   23 24 w x 4  . We get 23 21 x 0,0        , 23 22 x 0,0        , 23 23 x 49,51        and 23 24 x 11,19        . For i 3  , then     23 3 j 23 23 3 j 3 x 0 w x w a 5     with   23 31 w x 0  ,   23 32 w x 5  ,   23 33 w x 0  and   23 34 w x 0  . We get 23 31 x 0  , 23 32 x 0,10        , 23 33 x 0  and 23 34 x 0  . Step 3. Solving Eq. (16) via Eq. (18). We get   14 14 14 14 14 14 14 14 11 12 13 21 22 23 MinZ x 25x 70x 100x 75x 95x 40x       subject to the constraints 14 14 14 14 11 12 13 14 x x x x 85     , 14 14 14 14 21 22 23 24 x x x x 60     , 14 14 14 14 31 32 33 34 x x x x 25     , 14 14 14 11 21 31 x x x 50    , 14 14 14 12 22 32 x x x 35    , 14 14 14 13 23 33 x x x 60    and 14 14 14 14 24 34 x x x 25    . Optimal solution is: 14 11 x 50  , 14 12 x 10  , 14 13 x 0  , 14 14 x 25  , 14 21 x 0  , 14 22 x 0  , 14 23 x 60  , 14 24 x 0  , 14 31 x 0  , 14 32 x 25  , 14 33 x 0  and 14 34 x 0  . Furthermore   14 1 w a 15  ,   14 2 w a 20  and   14 3 w a 25  For i 1  , we have     4 14 23 23 14 1 1j 1j 1 1 j 1 E x x w a 50 45 10 30 0 0 25 15 0 35 15 w a                  , then     14 32 14 23 1j 1j 1j 1j w x x x w x    . We have   14 11 w x 5,    14 12 w x 20  ,   14 13 w x 0  and   14 14 w x 10  . We get 14 11 x 45,55        , 14 12 x 10,30         , 14 13 x 0  and 14 14 x 15,35        . For i 2  , we have
216 Kané et al. |J. Fuzzy. Ext. Appl. 2(3) (2021) 204-225     4 14 23 23 14 2 2 j 2 j 2 2 j 1 E x x w a 0 0 0 0 60 50 0 15 5 30 20 w a                  , then     14 14 23 23 2 j 2 j 2 j 2 j w x x x w x    . We have     14 14 21 22 w x w x 0   ,   23 23 w x 11  and   14 24 w x 19  . We get 14 21 x 0,0        , 14 22 x 0,0        , 14 23 x 49,71        and 14 24 x 19,19         . For i 3  , we have     4 14 23 23 14 3 3 j 3 j 3 3 j 1 E x x w a 0 0 25 5 0 0 0 0 5 25 25 w a                  , then     14 3 j 14 14 3 j 3 x 0 w x w a 25     . We have   14 31 w x 0  ,   14 32 w x 25  ,   14 33 w x 0  and   14 34 w x 0  We get 14 31 x 0  , 14 32 x 0,50        , 14 33 x 0  and 14 34 x 0  . Step 4. The optimal solution according to the choice of the decision maker is:          m n 1 2 3 4 *1 *2 *3 *4 *1 *2 *3 *4 ij ij ij ij ij ij ij ij j 1 i 1 Min Z x c ,c ,c ,c x ,x ,x ,x Z ,Z ,Z ,Z 930,2930,2970,3470        % % and       * 14 23 *1 *4 *2 *3 *1 *2 *3 *4 ij ij ij ij ij ij ij ij ij ij ij x x |x x ,x | x ,x x ,x ,x ,x                % with   * 11 x 45, 45, 45,55  % ,   * 12 x 10, 30, 30, 30   % , * 13 x 0  % % ,   * 14 x 15, 15, 15, 35  % , * 21 x 0  % % , * 22 x 0  % % ,   * 23 x 49, 49, 51, 71  % ,   * 24 x 19, 11, 19, 19   % , * 31 x 0  % % ,   * 32 x 0,0,10,50  % , * 33 x 0  % % and * 34 x 0  % % . Example 2. [29]. Step 1. Construct the fuzzy transportation Problem (15), and then convert it into a balanced one if it is not. Table 2. Eq. (15) in trapezoidal balanced form. 1 R 2 R 3 R Supply %i a A   1,4,9,19   1,2,5,9   2,5,8,18   1,5,7,9 B   8,9,12,26   3,5,8,12   7,9,13,28   4,7,8,10 C   11,12,20,27   0,5,10,15   4,5,8,11   4,5,8,11 Demand % j b   3,5,8,12 (4, 8, 9, 10) (2, 4, 6, 8)     % % m n i j i 1 j 1 a b
217 A new algorithm for fuzzy transportation problems with trapezoidal fuzzy numbers under fuzzy circumstances Step 2. Solving Eq. (17) via Eq. (19). Optimal solution is: 23 11 x 6  , 23 12 x 0  , 23 13 x 0  , 23 21 1 x 2  , 23 22 x 7  , 23 23 x 0  , 23 31 x 0  , 23 32 3 x 2  and 14 33 x 5  . Furthermore   23 1 2 w a 1 2   ,   23 2 w a 2 1  ,   23 3 w a 2 3  . For i 1  , then     3 1 j 2 1j 23 23 x 0 1 w x w a 1     with   23 11 w x 1  . We get 3 11 2 5,7 x        , 3 12 2 0,0 x        and 23 13 x 0,0        . For i 2  , then     3 2 j 2 2 j 23 x 0 23 2 1 w x w a 2     with   23 21 1 w x 4  and   23 22 1 w x 4  . We get 23 21 1 3 , 4 4 x            , 23 22 27 29 , 4 4 x            and 23 23 x 0,0        . For i 3  , then     3 3 j 2 3 j 23 x 0 23 3 3 w x w a 2     with   23 32 w x 2 1  and   23 33 w x 1  . We get 23 31 0,0 x        , 23 32 1,2 x        and 23 33 x 4,6        . Step 3. Solving Eq. (16) via Eq. (18). Optimal solution is: 14 11 x 5  , 14 12 x 0  and 14 13 x 0  , 14 21 5 x 2  , 14 22 9 x 2  , 14 23 x 0  , 14 31 x 0  , 14 32 5 x 2  and 14 33 x 5  . Furthermore   14 1 w a 4  ,   14 2 w a 3  ,   14 3 w a 2 7  . For i 1  , we have     23 n 14 1 23 1 1j 1j 1 1 j 1 4 E x x w a 5 6 0 0 0 0 1 2 4 w a                , then     4 1 j 1 1j 14 14 x 0 1 w x w a 4     with   14 11 w x 4  . We get 4 11 1 1,9 x        , 4 12 1 0,0 x        and 14 13 x 0,0        . For i 2  , we have     n 14 23 2 2 j 2 j 2 2 j 2 1 1 3 4 5 1 9 1 E x x w a 7 0 0 5 w a 3 2 2 2 2                , then     14 23 2 14 3 2 j 2 j 2 j 2 j w x x x w x    with   14 21 9 w x 4  and   14 22 11 w x 4  . We get 14 21 1 19 , 4 4 x            , 14 22 7 29 , 4 4 x            and 14 23 x 0,0        . For i 3  , we have     n 14 23 3 3 j 3 j 23 14 3 3 j 1 5 3 3 5 E x x w a 5 5 0 0 w a 2 2 2 2 2 7                , then     14 3 j 14 14 3 j 3 x 0 7 w x w a 2     with   14 32 3 w x 2  and   14 33 w x 2  . We get 14 31 0,0 x        , 14 32 1,4 x        and 14 33 x 3,7        . Step 4. The optimal solution according to the choice of the decision maker is:
218 Kané et al. |J. Fuzzy. Ext. Appl. 2(3) (2021) 204-225       * 14 23 *1 *4 *2 *3 *1 *2 *3 *4 ij ij ij ij ij ij ij ij ij ij ij x x |x x ,x | x ,x x ,x ,x ,x                % .   81 1037 Min Z x ,81, 173, 4 2            % % with   * 11 x 1, 5, 7 ,9  % , * 12 x 0  % % , * 13 x 0  % % , * 21 1 1 3 19 x , , , 4 4 4 4            % , * 22 7 27 29 29 x , , , 4 4 4 4            % , * 23 x 0  % % , * 31 x 0  % % ,   * 32 x 1,1,2,4  % and   * 33 x 3,4,6,7  % . Example 3. [29]. Step 1. Construct the fuzzy transportation Problem (15), and then convert it into a balanced one if it is not. Table 3. Eq. (15) in triangular balanced form. Step 2. Solving Eq. (17) via Eq. (19). We get   2 2 2 2 2 2 11 12 21 22 MinZ x 31x 19x 39x 10x     subject to the constraints 2 2 11 12 x x 201   , 2 2 21 22 x x 99   , 2 2 11 21 x x 150   and 2 2 12 22 x x 150   . Optimal solution is: 2 11 x 150  , 2 12 x 51  , 2 21 x 0  and 2 22 x 99  . Step 3. Solving Eq. (16) via Eq. (18). We have   13 13 13 13 13 13 11 12 21 22 MinZ x 28x 22x 42x 10x     subject to the constraints 13 13 11 12 x x 198   , 13 13 21 22 x x 102   , 13 13 11 21 x x 150   and 13 13 12 22 x x 150   . Optimal solution is: 13 11 x 150  , 13 12 x 48  , 13 21 x 0  and 13 22 x 102  . Furthermore   13 1 w a 48  ,   13 2 w a 52  . For i 1  , we have   2 13 2 13 1 1j 1j 1 j 1 E x x 150 150 48 51 3 48 w a            , then     3 1 j 1 1j 13 x 0 13 1 w x w a 48     with   13 11 w x 10  and   13 12 w x 38  . We get 13 11 x 140,160        and 13 12 x 10,86        . 1 R 2 R Supply ( %i a ) A   22,31,34   15,19,29   150,201,246 B   30,39,54   8,10,12   50,99,154 Demand ( % j b )   100,150,200   100,150,200     % % m n i j i 1 j 1 a b
219 A new algorithm for fuzzy transportation problems with trapezoidal fuzzy numbers under fuzzy circumstances For i 2  , we have   2 13 2 13 2 2 j 2 j 2 j 1 E x x 0 0 102 99 3 52 w a            , then     3 2 j 1 2 j 13 x 0 13 2 w x w a 52     with   13 21 w x 0  and   13 22 w x 52  . We get 13 21 x 0,0        and 13 22 x 50,154        . Step 4. The optimal solution according to the choice of the decision maker is     m n 1 2 3 ij ij ij ij i 1 j 1 MinZ x c ,c ,c x      % % % with       * 2 13 2 *1 *3 *1 *2 *3 ij ij ij ij ij ij ij ij ij x x |x x | x ,x x ,x ,x          % . We have Min     * Z 6609| 3630, 9782 3630,6609, 9782         % where   * 11 x 140,150, 160  % ,   * 12 x 10,51,86  % , * 21 x 0  % % and   * 22 x 50,99,154  % . 5.1| Interpretation of Results We will now interpret the minimum total fuzzy transportation cost obtained in Example 3. by using the proposed methods presented in Section 3. Similarly, the obtained fuzzy optimal solution will also be interpreted. By using the methods proposed the minimum total fuzzy transportation cost is   3630,6609, 9782 , which can be physically interpreted as follows:  The least amount of the minimum total transportation cost is 3630 .  The most possible amount of minimum total transportation cost is 6609 .  The greatest amount of the minimum total transportation cost is 9782 i.e., the minimum total transportation cost will always be greater than 3630 and less than6609 , and the highest chances are that the minimum total transportation cost will be9782 . Example 4. [1]-[3]. Step 1. Construct the fuzzy transportation Problem (15), and then convert it into a balanced one if it is not. Step 2. Solving Eq. (17) via Eq. (19). We get   2 2 2 2 2 2 2 2 2 2 11 12 13 14 21 22 23 24 MinZ x 10x 22x 10x 20x 15x 20x 12x 8x        2 2 2 2 31 32 33 34 20x 12x 10x 15x     subject to the constraints 2 2 2 2 11 12 13 14 x x x x 8     , 2 2 2 2 21 22 23 24 x x x x 14     , 2 2 2 2 31 32 33 34 x x x x 12     , 2 2 2 11 21 31 x x x 7    , 2 2 2 12 22 32 x x x 10    , 2 2 2 13 23 33 x x x 8    and 2 2 2 14 24 34 x x x 9    .
220 Kané et al. |J. Fuzzy. Ext. Appl. 2(3) (2021) 204-225 Optimal solution is: 2 11 x 7  , 2 12 x 0  , 2 13 x 1  , 2 14 x 0  , 2 21 x 0  , 2 22 x 0  , 2 23 x 5  , 2 24 x 9  , 2 31 x 0  , 2 32 x 10  , 2 33 x 2  and 2 34 x 0  . Table 4. Eq. (15) in triangular balanced form (in U.S. dollar). Step 3. Solving Eq. (16) via Eq. (18). We get   13 13 13 13 13 13 13 13 11 12 13 14 21 22 MinZ x 9.4x 22.2x 9.3x 20.4x 15x 20.1x       13 13 13 13 13 13 23 24 31 32 33 34 11.5x 7.4x 19.7 x 11.3x 9.3x 15x       subject to the constraints 13 13 13 13 11 12 13 14 x x x x 8     , 13 13 13 13 21 22 23 24 x x x x 14     , 13 13 13 13 31 32 33 34 x x x x 12     , 13 13 13 11 21 31 x x x 7    , 13 13 13 12 22 32 x x x 10    , 13 13 13 13 23 33 x x x 8    and 13 13 13 14 24 34 x x x 9    . Optimal solution is: 13 11 x 7  , 13 12 x 0  , 13 13 x 1  , 13 14 x 0  , 13 21 x 0  , 13 22 x 0  , 13 23 x 5  , 13 24 x 9  , 13 31 x 0  , 13 32 x 10  , 13 33 x 2  and 13 34 x 0  . We have   13 1 4 w a 5  ,   13 2 w a 2  and   13 3 9 w a 5  . Furthermore:  13 11 34 36 x , 5 5            , 13 12 x 0,0        , 13 13 2 8 x , 5 5            and 13 14 x 0,0        where   13 11 1 w x 5  ,   13 12 w x 0  ,   13 13 3 w x 5  and   13 14 w x 0  .  13 21 x 0,0        , 13 22 x 0,0        , 13 23 9 11 x , 2 2            and 13 24 15 21 x , 2 2            where   13 21 w x 0  ,   13 22 w x 0  ,   13 23 1 w x 2  and   13 24 3 w x 2  . Source Destination Supply (𝒂 ̃𝒊) (000 dozen bottles) Taichung Chiayi Kaohsiung Taipei Changhua   $8, $10, $10.8   $20.4, $22, $24   $8,$10,$10.6   $18.8, $20, $22   7.2,8,8.8 Touliu   $14, $15, $16   $18.2, $20, $22   $10, $12, $13   $6, $8, $8.8   12,14,16 Hsinchu   $18.4, $20, $21   $9.6, $12, $13   $7.8, $10, $10.8   $14, $15, $16   10.2,12,13.8 Demand ( j b % ) (000 dozen bottles)   6.2,7,7.8   8.9,10,11.1   6.5,8,9.5   7.8,9,10.2 ∑𝑎̃𝑖 𝑚 𝑖=1 = ∑𝑏̃𝑗 𝑛 𝑗=1
221 A new algorithm for fuzzy transportation problems with trapezoidal fuzzy numbers under fuzzy circumstances  13 31 x 0,0        , 13 32 47 53 x , 5 5            , 13 33 4 16 x , 5 5            and 13 34 x 0,0        with   13 31 w x 0  ,   13 32 3 w x 5  ,   13 33 6 w x 5  and   13 34 w x 0  . Step 4. The optimal solution according to the choice of the decision maker is The value of the objective function:        m n 1 2 3 *1 *2 *3 *1 *2 *3 ij ij ij ij ij ij j 1 i 1 Min Z x c ,c ,c x ,x ,x Z ,Z ,Z       % % with       * 2 13 2 *1 *3 *1 *2 *3 ij ij ij ij ij ij ij ij ij x x |x x | x ,x x ,x ,x          % . We have Min * 12204 21549 12204 21549 Z 352$| $, $ $,352$, $ 50 50 50 50                                   % where * 11 34 36 x , 7 , 5 5            % , * 12 x 0  % % , * 13 2 8 x , 1, 5 5            % , * 14 x 0  % % , * 21 x 0  % % , * 22 x 0  % % , * 23 9 11 x , 5, 2 2            % , * 24 15 21 x , 9, 2 2            % , * 31 x 0  % % , * 32 47 53 x , 10, 5 5            % , * 33 4 16 x , 2, 5 5            % and * 34 x 0  % % . 5.2| Interpretation of Results We will now interpret the minimum total fuzzy transportation cost obtained in Example 4 by using the proposed methods presented in Section 3. Similarly, the obtained fuzzy optimal solution will also be interpreted. By using the methods proposed the minimum total fuzzy transportation cost is 12204 21549 $,352$, $ 50 50           , which can be physically interpreted as follows:  The least amount of the minimum total transportation cost is 12204 $ 50 .  The most possible amount of minimum total transportation cost is 352$ .  The greatest amount of the minimum total transportation cost is 21549 $ 50 i.e., the minimum total transportation cost will always be greater than 12204 $ 50 and less than 352$ , and the highest chances are that the minimum total transportation cost will be 21549 $ 50 .
222 Kané et al. |J. Fuzzy. Ext. Appl. 2(3) (2021) 204-225 6| A Comparative Study Author's of [3], [11], [29] have proposed a method to find the crisp optimal solution of such fuzzy transportation problems in which all the parameters are represented by triangular or trapezoidal fuzzy numbers. Then, they have used their new method proposed to find the crisp optimal solution of a real- life fuzzy transportation problem. However, it is often better to find a fuzzy optimal solution than a crisp optimal solution. In this section we will therefore show how in the problem considered by [3], [11], [29], we can obtain a fuzzy optimal solution of the same real-life problem using the new method proposed. 7| Concluding Remarks and Future Research Directions 7.1| Concluding Remarks These days a number of researchers have shown interest in the area of fuzzy transportation problems and various attempts have been made to study the solution of these problems. In this paper, to overcome the shortcomings of the existing methods we introduced a new formulation of transportation problem involving trapezoidal (or triangular) fuzzy numbers for the transportation costs and values of supplies and demands. We propose a fuzzy linear programming approach for solving trapezoidal (or triangular) fuzzy numbers transportation problem based on the converting into two interval transportation problems Eq. (16) and Eq. (17). To show the advantages of the proposed methods over existing methods, some fuzzy transportation problems, may or may not be solved by the existing methods, are solved by using the proposed methods and it is shown that it is better to use the proposed methods as compared to the existing methods for solving the transportation problems. From both theoretical and algorithmic considerations, and examples solved in this paper, it can be noticed that some shortcomings of the methods for solving the fuzzy transportation problems known from the literature can be resolved by using the new methods proposed in Section 5. 7.2| Future Research Directions Finally, we feel that, there are many other points of research and should be studied later on interval numbers or fuzzy numbers. Some of these points are below: We will consider the following transportation Problems (14) with fuzzy numbers as follows: Example Minimum Total Fuzzy Transportation Cost [3] [29] Method Proposed in This Paper 1   2100,2900,3500,4200   930,2930,2970,3470 2   31,80,199,460           81 1037 ,81,173, 4 2 3   3630,6609,9782 4   238.44,347.8,428.9           12204 21549 ,352, 50 50
223 A new algorithm for fuzzy transportation problems with trapezoidal fuzzy numbers under fuzzy circumstances   n m ij ij j 1i 1 MinZ x c x     % % % % subject to the constraints n ij i j 1 x a   % % and m ij j i 1 x b    % % where   1 2 t ij ij ij ij c c ,c , ,c   % ,   1 2 t ij ij ij ij x x ,x , ,x   % ,   1 2 t i i i i a a ,a , ,a   % and   1 2 t j j j j b b ,b , ,b   % with 1 t  N . Let pq p q ij ij ij x x ,x        where p q  and 1 p,q  N . The same applies to pq ij c , pq i a and pq j b . I. Solution procedure for classical Transportation Problem   t 1  :   1 ij ij x x .  II. Solution procedure for Transportation Problem with Interval numbers   t 2  : 12 1 2 ij ij ij x x ,x .        III. Solution procedure for Transportation Problem with Triangular fuzzy numbers   t 3  :     1 2 3 2 13 ij ij ij ij ij ij x x ,x ,x x ,x .   % IV. Solution procedure for Transportation Problem with Trapezoidal fuzzy numbers   t 4  :     1 2 3 4 23 14 ij ij ij ij ij ij ij x x ,x ,x ,x x ,x .   % V. Solution procedure for Transportation Problem with Pentagonal fuzzy numbers   t 5  :     1 2 3 4 5 3 24 15 ij ij ij ij ij ij ij ij ij x x , x , x , x , x x , x , x .   % VI. Solution procedure for Transportation Problem with Hexagonal fuzzy numbers   t 6  :     1 2 3 4 5 6 34 25 16 ij ij ij ij ij ij ij ij ij ij x x ,x ,x ,x ,x ,x x ,x ,x .   % VII. Solution procedure for Transportation Problem with Heptagonal fuzzy numbers   t 7      1 2 3 4 5 6 7 4 35 26 17 ij ij ij ij ij ij ij ij ij ij ij ij : x x ,x ,x ,x ,x ,x ,x x ,x ,x ,x .   % VIII. Solution procedure for Transportation Problem with Octagonal fuzzy numbers   t 8      1 2 3 4 5 6 7 8 45 36 27 18 ij ij ij ij ij ij ij ij ij ij ij ij ij : x x ,x ,x ,x ,x ,x ,x ,x x ,x ,x ,x .   % IX. Solution procedure for Transportation Problem with Nonagonal fuzzy numbers   t 9  :     1 2 3 4 5 6 7 8 9 5 46 37 28 19 ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij x x ,x ,x ,x ,x ,x ,x ,x ,x x ,x ,x ,x ,x .   % X. Solution procedure for Transportation Problem with Decagonal fuzzy numbers   t 10  :     1 2 3 4 5 6 7 8 9 10 56 47 38 29 110 ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij x x ,x ,x ,x ,x ,x ,x ,x ,x ,x x ,x ,x ,x ,x .   % XI. Solution procedure for Transportation Problem with Hendecagonal fuzzy numbers   t 11  :     1 2 3 4 5 6 7 8 9 10 11 6 57 48 39 210 111 ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij x x ,x ,x ,x ,x ,x ,x ,x ,x ,x ,x x ,x ,x ,x ,x ,x .   % XII. Solution procedure for Transportation Problem with Dodecagonal fuzzy numbers   t 12      1 2 3 4 5 6 7 8 9 10 11 12 67 58 49 310 211 112 ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij : x x ,x ,x ,x ,x ,x ,x ,x ,x ,x ,x ,x x ,x ,x ,x ,x ,x .   % Acknowledgements The authors are very grateful to the anonymous referees for their valuable comments and suggestions to improve the paper in the present form. Conflicts of interest The authors declare no conflicts of interest.
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A new algorithm for fuzzy transportation problems with trapezoidal fuzzy numbers under fuzzy circumstances

  • 1. Corresponding Author: [email protected] http://dx.doi.org/10.22105/jfea.2021.287198.1148 E-ISSN: 2717-3453 | P-ISSN: 2783-1442 | Abstract 1 | Introduction Transportation problem is an important network structured linear programming problem that arises in several contexts and has deservedly received a great deal of attention in the literature. The central concept in this problem is to find the least total transportation cost of a commodity in order to satisfy demands at destinations using available supplies at origins. Transportation problem can be used for a wide variety of situations such as scheduling, production, investment, plant location, inventory control, employment scheduling and many others. Journal of Fuzzy Extension and Applications www.journal-fea.com J. Fuzzy. Ext. Appl. Vol. 2, No. 3 (2021) 204–225. Paper Type: Research Paper The New Algorithm for Fully Fuzzy Transportation Problem by Trapezoidal Fuzzy Number (A Generalization of Triangular Fuzzy Number) Ladji Kané1,* , Moctar Diakité1, Souleymane Kané1, Hawa Bado1, Moussa Konaté1, Koura Traoré1 1 Faculty of Economics and Management, University of Social Sciences and Management of Bamako (USSGB), Quartier du Fleuve Rue 310, Porte 238, Mali; [email protected]; [email protected]; [email protected]; [email protected]; [email protected]. Citation: Kané, L., Diakité, M., Kané, S., Bado, H., Konaté, M., & Traoré, K. (2021). The new algorithm for fully fuzzy transportation problem by trapezoidal fuzzy number (a generalization of triangular fuzzy number). Journal of fuzzy extension and application, 2 (3), 204-225. Accept: 29/07/2021 Revised: 17/06/2021 Reviewed: 01/06/2021 Received: 20/05/2021 Transportation problem is an important network structured linear programming problem that arises in several contexts and has deservedly received a great deal of attention in the literature. The central concept in this problem is to find the least total transportation cost of a commodity in order to satisfy demands at destinations using available supplies at origins in a crisp environment. In real life situations, the decision maker may not be sure about the precise values of the coefficients belonging to the transportation problem. The aim of this paper is to introduce a formulation of fully fuzzy transportation problem involving trapezoidal fuzzy numbers for the transportation costs and values of supplies and demands. We propose a two-step method for solving fuzzy transportation problem where all of the parameters are represented by triangular fuzzy numbers i.e. two interval transportation problems. Since the proposed approach is based on classical approach it is very easy to understand and to apply on real life transportation problems for the decision makers. To illustrate the proposed approach four application examples are solved. The results show that the proposed method is simpler and computationally more efficient than existing methods in the literature. Keywords: Fully fuzzy linear programming, Transportation problem, Trapezoidal fuzzy numbers, fuzzy numbers. Licensee Journal of Fuzzy Extension and Applications. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons. org/licenses/by/4.0).
  • 2. 205 A new algorithm for fuzzy transportation problems with trapezoidal fuzzy numbers under fuzzy circumstances In general, transportation problems are solved with the assumptions that the transportation costs and values of supplies and demands are specified in a precise way i.e., in crisp environment. However, in many cases the decision maker has no crisp information about the coefficients belonging to the transportation problem. If the nature of the information is vague, that is, if it has some lack of precision, the corresponding coefficients or elements defining the problem can be formulated by means of fuzzy sets, and thus fuzzy transportation problems arise. Several researchers have carried out investigations on fuzzy transportation problem. Zimmermann [4] developed Zimmermann's fuzzy linear programming into several fuzzy optimization methods for solving the transportation problems. ÓhÉigeartaigh [5] proposed an algorithm for solving transportation problems where the supplies and demands are fuzzy sets with linear or triangular membership functions. Chanas et al. [6] investigated the transportation problem with fuzzy supplies and demands and solved them via the parametric programming technique. Their method provided solution which simultaneously satisfies the constraints and the goal to a maximal degree. In addition, Chanas et al. [7] formulated the classical, interval and fuzzy transportation problem and discussed the methods for solution for the fuzzy transportation problem. Kuchta [8] discussed the type of transportation problems with fuzzy cost coefficients and converted the problem into a bicriterial transportation problem with crisp objective function. Their method only gives crisp solutions based on efficient solutions of the converted problems. Jiménez and Verdegay [9], [10] investigated the fuzzy solid transportation problem in which supplies, demands and conveyance capacities are represented by trapezoidal fuzzy numbers and applied a parametric approach for finding the fuzzy solution. Liu and Kao [11] developed a procedure, based on extension principle to derive the fuzzy objective value of fuzzy transportation problem, in that the cost coefficients and the supply and demand quantities are fuzzy numbers. Gani and Razak [12] presented a two-stage cost minimizing fuzzy transportation problem in which supplies and demands are as trapezoidal fuzzy numbers and used a parametric approach for finding a fuzzy solution with the aim of minimizing the sum of the transportation costs in the two stages. Li et al. [13] proposed a new method based on goal programming for solving fuzzy transportation problem with fuzzy costs. Lin [14] used genetic algorithm for solving transportation problems with fuzzy coefficients. Dinagar and Palanivel [15] investigated fuzzy transportation problem, with the help of trapezoidal fuzzy numbers and applied fuzzy modified distribution method to obtain the optimal solution in terms of fuzzy numbers. Pandian and Natarajan [16] introduced a new algorithm namely, fuzzy zero-point method for finding fuzzy optimal solution for such fuzzy transportation problem in which the transportation cost, supply and demand are represented by trapezoidal fuzzy numbers. Kumar and Kaur [17] proposed a new method based on fuzzy linear programming problem for finding the optimal solution of fuzzy transportation problem. Gupta et al. [18] proposed a new method named as Mehar's method, to find the exact fuzzy optimal solution of fully fuzzy multi-objective transportation problems. Ebrahimnejad [19] applied a fuzzy bounded dual algorithm for solving bounded transportation problems with fuzzy supplies and demands. Shanmugasundari and Ganesan [20] developed the fuzzy version of Vogel's and MODI methods for obtaining the fuzzy initial basic feasible solution and fuzzy optimal feasible solution, respectively, without converting them into classical transportation problem. Also, [21] Chandran and Kandaswamy [21] proposed an algorithm to find an optimal solution of a fuzzy transportation problem, where supply, demand and cost coefficients all are fuzzy numbers. Their algorithm provides decision maker with an effective solution in comparison to existing methods. Ebrahimnejad [22] using an example showed that their method will not always lead to a fuzzy optimal solution. Moreover, Kumar and Kaur [23] pointed out the limitations and shortcomings of the existing methods for solving fuzzy solid transportation problem and to overcome these limitations and shortcomings proposed a new method to find the fuzzy optimal solution of unbalanced fuzzy solid transportation problems. In addition, Ebrahimnejad [24] proposed a two-step method for solving fuzzy transportation problem where all of the parameters are represented by non-negative triangular fuzzy numbers. Some researchers applied generalized fuzzy numbers for solving transportation problems. Kumar and Kaur [25] proposed a new method based on ranking function for solving fuzzy transportation problem by assuming that transportation cost, supply and demand of the commodity are represented by generalized trapezoidal fuzzy numbers. After that, Kaur and Kumar [26] introduced a similar algorithm for solving a special type of fuzzy
  • 3. 206 Kané et al. |J. Fuzzy. Ext. Appl. 2(3) (2021) 204-225 transportation problem by assuming that a decision maker is uncertain about the precise values of transportation cost only but there is no uncertainty about the supply and demand of the product. Ebrahimnejad [27] demonstrated that once the ranking function is chosen, the fuzzy transportation problem introduced by Kaur and Kumar [26] is converted into crisp one, which is easily solved by the standard transportation algorithms. The contributions of the present study are summarized as follows: (a) in the Eq. (15) under consideration, all of the parameters, such as the transportation costs, supplies and demands are considered as fuzzy numbers. (b) According to the proposed approach, the Eq. (15) is converted into two interval transportation problems Eq. (16) and Eq. (17). The integration of the optimal solution of the two sub-problems provides the optimal solution of the Eq. (15). (c) In contrast to most existing approaches, which provide a precise solution, the proposed method provides a fuzzy optimal solution. (d) In contrast to existing methods that include negative parts in the obtained fuzzy optimal solution and fuzzy optimal cost, the proposed method provides a fuzzy optimal solution and optimal cost. (e) Similarly, to the competing methods in the literature, the proposed method is applicable for all types of trapezoidal fuzzy numbers. (f) The complexity of computation is greatly reduced compared with commonly used existing methods in the literature. The rest of this paper is organized as follows: In Section 2, we recall the definitions of interval number linear programming, interval numbers and fully fuzzy transportation problem. In Section 3, a new method is proposed for obtaining the fuzzy optimal solution of the Eq. (15). The advantages of the proposed method are discussed in Section 4. Two application examples are provided to illustrate the effectiveness of the proposed method in Section 5, and a comparative study in Section 6. Finally, concluding remarks are presented in Section 7. 2| Materials and Methods In this section, some basic definitions, arithmetic operations for closed Intervals numbers and of linear programming problems involving interval numbers are presented [28]. 2.1| A New Interval Arithmetic In this section, some arithmetic operations for two intervals are presented [28]. Let   1 4 1 4 1 4 a a ,a :a a , a ,a           R R be the set of all proper intervals. We shall use the terms “interval” and “interval number” interchangeably. The mid-point and width (or half-width) of an interval number 1 4 a a ,a        are defined as   4 1 a a m a 2   and   4 1 a a w a 2   . The interval number a can also be expressed in terms of its midpoint and width as For any two intervals     1 4 a a ,a m a ,w a           and     1 4 b b ,b m b ,w b           , the arithmetic operations on a and b are defined as:                    4 1 4 1 1 4 a a a a a a ,a m a ,w a , . 2 2 (1)               a b m a m b ,w a w b . (2)               a b m a m b ,w a w b . (3)
  • 4. 207 A new algorithm for fuzzy transportation problems with trapezoidal fuzzy numbers under fuzzy circumstances 2.2| A New Interval Arithmetic for Trapezoidal Fuzzy Numbers via Intervals Numbers The aim of this section is to present some notations, notions and results which are of useful in our further consideration. A number   1 2 3 4 a a ,a ,a ,a  % (where 1 2 3 4 a a a a    ) is said to be a trapezoidal fuzzy number if its membership function is given by [1]-[3]: Assume that       1 2 3 4 14 23 1 4 2 3 a a ,a ,a ,a a |a a ,a | a ,a                % , and       1 2 3 4 14 23 1 4 2 3 b b ,b ,b ,b b |b b ,b | b ,b                % are two trapezoidal fuzzy numbers. For any two trapezoidal fuzzy numbers   14 23 a a |a  % and   14 23 b b |b  % , the arithmetic operations on a % and b % are defined as: 2.3| Formulation of a Transportation Problems Involving Interval Numbers We consider the Transportation Problem involving Interval numbers as follows [28]:                       αm a ,αw a   if  α 0, αa αm a , αw a   if  α 0. (4)                                                                                1 1 1 1 4 1 m a m b w a w b ,m a w b m b w a ifa 0, b 0, a b m a m b m a w b ,m b w a w b w a ifa 0, b 0, m a m b w a w b ,m b w a m a w b ifa 0, b 0. (5)                         % 1 1 2 2 1 a 4 3 4 3 4 x a , a x a , a a μ x x a , a x a . a a (6)                                     % % 14 23 14 23 1 4 1 4 2 3 2 3 a b a |a b |b a ,a b , b | a ,a b , b . (7)                                    % % 14 23 14 23 1 4 1 4 2 3 2 3 ab a |a b |b a ,a b , b | a ,a b , b (8)
  • 5. 208 Kané et al. |J. Fuzzy. Ext. Appl. 2(3) (2021) 204-225 where 1 4 ij ij ij c c ,c        , 1 4 i i i a a ,a        , 1 4 j j j b b ,b        are non-negative interval numbers and 1 4 ij ij ij x x ,x        are unrestricted interval numbers. Objective function transformation. Transformation of constraints.         n n ij i ij ij i i j 1 j 1 x a m x ,w x m a ,w a           for i 1, 2, ,m   and         m m ij j ij ij j j i 1 i 1 x b m x ,w x m b ,w b           for j 1,2, ,n   . Now we can say that                                    n m ij ij j 1i 1 n ij i j 1 m ij j i 1 MinZ x c x Subjecttotheconstraints x a ,fori 1,2, ,m , x b , for j 1,2, ,n. (9)                                   n m n m n m 1 4 1 4 ij ij ij ij ij ij ij ij ij ij j 1 i 1 j 1 i 1 j 1 i 1 Z x c x c ,c x ,x m c x ,w c x , Where                                            1 ij ij ij ij ij 1 4 ij ij ij ij ij ij ij ij 4 ij ij ij ij ij m c m x w c w x if x 0, m c x m c m x w c m x if x 0andx 0, m c m x w c w x if x 0. And (10)                                            1 ij ij ij ij ij 1 4 ij ij ij ij ij ij ij ij 4 ij ij ij ij ij m c w x w c m x if x 0, w c x m c w x w c w x if x 0andx 0, m c w x w c m x if x 0. (11)
  • 6. 209 A new algorithm for fuzzy transportation problems with trapezoidal fuzzy numbers under fuzzy circumstances is equivalent to Optimal Solution (9) according to the choice of the decision maker:         * 1 4 ij ij ij ij ij ij ij x x ,x m x w x ,m x w x                   where     n ij i j 1 w x w a    , i 1, 2, ,m   . Remark. I. n ij i j 1 x a    if and only if     n ij i j 1 m x m a    and     n ij i j 1 w x w a    for i 1, 2, ,m   . II. n ij i j 1 x a    if and only if     n ij i j 1 m x m a    and     n ij i j 1 w x w a    for i 1, 2, ,m   . III. m ij j i 1 x b    if and only if     m ij j i 1 m x m b    and     m ij j i 1 w x w b    for j 1,2, ,n   . IV. m ij j i 1 x b    if and only if     m ij j i 1 m x m b    and     m ij j i 1 w x w b    for j 1,2, ,n   . 2.4| Formulation of a Fully Fuzzy Transportation Problem The fuzzy linear programming formulation of a fully fuzzy transportation problem can be written as follows as follows [2], [3]:                                                                   n m ij ij ij ij j 1 i 1 n ij ij i i j 1 m ij ij j j i 1 MinZ x m c x ,w c x Subjecttotheconstraints m x ,w x m a ,w a ,i 1,2, ,m, m x ,w x m b ,w b , j 1,2, ,n. (12)                                                     n m ij ij j 1 i 1 n ij i j 1 m ij j i 1 m MinZ x m c x Subjecttotheconstraints m x m a ,fori 1,2, ,m, m x m b , for j 1,2, ,n. (13)
  • 7. 210 Kané et al. |J. Fuzzy. Ext. Appl. 2(3) (2021) 204-225 with m n i j i 1 j 1 a b     % % where ij x % are unrestricted fuzzy numbers and ij c % , i a % and j b % are non-negatives fuzzy numbers. 3| Main Results In this section, we will describe our method of solving. 3.1| Transportation Problem with Trapezoidal Fuzzy Numbers For all the rest of this paper, we will consider the following transportation problem with trapezoidal fuzzy numbers as follows: where   1 2 3 4 ij ij ij ij ij c c ,c ,c ,c ,  %   1 2 3 4 j ij ij ij ij x x ,x ,x ,x  % ,   1 2 3 4 j j j j j b b ,b ,b ,b  % and   1 2 3 4 i i i i i a a ,a ,a ,a  % are trapezoidal fuzzy numbers with     14 23 1 4 2 3 ij ij ij ij ij ij ij x x |x x ,x | x ,x               % where 14 1 4 ij ij ij x x ,x        and 23 2 3 ij ij ij x x ,x        ,     14 23 1 4 2 3 ij ij ij ij ij ij ij c c |c c ,c | c ,c               % where 14 1 4 ij ij ij c c ,c        and 23 2 3 ij ij ij c c ,c ,            14 23 1 4 2 3 j j j j j j j b b |b b ,b | b ,b               % where 14 1 4 j j j b b ,b        and 23 2 3 j j j b b ,b        and     14 23 1 4 2 3 i i i i i i i a a |a a ,a | a ,a               % where 14 1 4 i i i a a ,a        and 23 2 3 i i i a a ,a        . For all the rest of this paper, we will consider the following transportation problems involving interval numbers 14 ij x , 14 ij c , 14 i a and 14 j b as follows:                                     % % % % % % % % m n ij ij i 1 j 1 n ij i j 1 m ij j i 1 MinZ x c x Subjecttotheconstraints x a , fori 1,2, ,m x b , for j 1,2, ,n (14)                                     % % % % % % % % m n ij ij i 1 j 1 n ij i j 1 m ij j i 1 MinZ x c x Subjecttotheconstraints x a ,for i 1,2, ,m, x b , for j 1,2, ,n. (15)
  • 8. 211 A new algorithm for fuzzy transportation problems with trapezoidal fuzzy numbers under fuzzy circumstances And the transportation problems involving interval numbers 23 ij x , 23 ij c , 23 i a and 23 j b as follows: 3.2| Formulation of a Transportation Problem involving Midpoint Thanks to the new interval arithmetic and Eq. (16), we can write the following transportation problem involving midpoint ( 14 ij x , 14 ij c , 14 i a and 14 j b ) [28] as follows: where   4 1 ij ij 14 ij c c m c 2   ,   4 1 14 i i i a a m a 2   ,   4 1 j j 14 j b b m b 2   and   4 1 14 i i i a a w a 2   . Thanks to the new interval arithmetic and Eq. (17), we can write the following transportation problem involving midpoint ( 23 ij x , 23 ij c , 23 i a and 23 j b ) [28] as follows:                                     m n 14 14 14 14 ij ij i 1 j 1 n 14 14 ij i j 1 m 14 14 ij j i 1 MinZ x c x Subjecttotheconstraints x a ,for i 1,2, ,m, x b ,for j 1,2, ,n. (16)                                    m n 23 23 23 23 ij ij i 1 j 1 n 23 23 ij i j 1 m 23 23 ij j i 1 MinZ x c x Subjecttotheconstraints x a ,for i 1,2, ,m, x b ,for j 1,2, ,n. (17)                                           m n 14 14 14 14 ij ij i 1 j 1 n 14 14 ij i j 1 14 14 ij j i 1 m MinZ x m c x Subjecttotheconstraints x m a ,for i 1,2, ,m, x m b ,for j 1,2, ,n. (18)
  • 9. 212 Kané et al. |J. Fuzzy. Ext. Appl. 2(3) (2021) 204-225 where   3 2 ij ij 23 ij c c m c 2   ,   3 2 23 i i i a a m a 2   ,   3 2 j j 23 j b b m b 2   and   i 3 i 2 23 i a a w a 2   . Thanks to the new interval arithmetic, we can write the following proposition [28]: Proposition 1. If 14 *1 *4 ij ij ij x x ,x        is an optimal solution to the Eq. (16) and 23 *2 *3 ij ij ij x x ,x        is an optimal solution to the Eq. (17), then   * * ij m n x x   % % is an optimal solution to the Eq. (15) with       * 14 23 *1 *4 *2 *3 *1 *2 *3 *4 ij ij ij ij ij ij ij ij ij ij ij x x |x x ,x | x ,x x ,x ,x ,x                % . 3.3| The Steps of Our Computational Method The steps of our method for solving the fully fuzzy transportation problem involving trapezoidal fuzzy numbers as follows. 3.3.1| Solution procedure for transportation problem with trapezoidal fuzzy numbers Step 1. Construct the fuzzy transportation Problem (15), and then convert it into a balanced one if it is not. Else, go to Step 2. Step 2. Solving Eq. (17) via Eq. (19). Determine 3 ij 2 x =     *2 *3 23 23 23 23 ij ij ij ij ij ij x ,x x w x ,x w x                  by applying the following conditions: I.   23 ij w x 0  if and only if 23 ij x 0  and II.     23 ij 23 23 ij i x 0 w x w a    with     23 23 ip ik w x w x  if 4 ip 4 ik c c  for i 1, ,m   . III. Go to Step 3. Step 3. Solving Eq. (16) via Eq. (18). Determine 4 ij 1 x =     *1 *4 ij ij ij ij ij i 14 14 14 14 j x ,x x w x ,x w x                  for i 1, ,m   . Considering the following cases: Case 1. If     n 14 23 i ij ij i i 3 1 j 1 2 4 E x x w a w a       , then 14 ij x =     *1 *4 ij ij ij ij ij i 14 14 14 14 j x ,x x w x ,x w x                  with     4 ij 1 i 14 j i x 0 14 w x w a    . Else, go to Case 2.                                           m n 23 23 23 23 ij ij i 1 j 1 n 23 23 ij i j 1 23 23 ij j i 1 m MinZ x m c x Subjecttotheconstraints x m a ,fori 1,2, ,m, x m b ,for j 1,2, ,n. (19)
  • 10. 213 A new algorithm for fuzzy transportation problems with trapezoidal fuzzy numbers under fuzzy circumstances Case 2. If     n 14 23 i ij ij i i 3 1 j 1 2 4 E x x w a w a       , then 14 ij x =     *1 *4 ij ij ij ij ij i 14 14 14 14 j x ,x x w x ,x w x                  with     14 14 2 23 3 ij ij ij ij w x x x w x    and go to Step 4. Step 4. The Optimal Solution (15) according to the choice of the decision maker is:        m n 1 2 3 4 *1 *2 *3 *4 *1 *2 *3 *4 ij ij ij ij ij ij ij ij j 1 i 1 Min Z x c ,c ,c ,c x ,x ,x ,x Z ,Z ,Z ,Z       % % where       * 14 23 *1 *4 *2 *3 *1 *2 *3 *4 ij ij ij ij ij ij ij ij ij ij ij x x |x x ,x | x ,x x ,x ,x ,x                % . 3.3.2| Solution procedure for transportation problem with triangular fuzzy numbers The steps of our method for solving the fully fuzzy transportation problem involving triangular fuzzy numbers as follows: The fuzzy number   * *1 *2 *3 *4 ij ij ij ij ij x x ,x ,x ,x  % is said to be triangular if and only if *2 *3 ij ij x x  . Then   *3 *2 ij ij 23 ij x x w x 0 2    . Therefore   14 14 23 ij ij ij w x x x   and       * *1 *2 *3 *4 *1 *2 *2 *4 *1 *2 *4 ij ij ij ij ij ij ij ij ij ij ij ij x x ,x ,x ,x x ,x ,x ,x x ,x ,x    % . In this article, consider fuzzy triangular numbers by:       * 2 13 2 *1 *3 *1 *2 *3 ij ij ij ij ij ij ij ij ij x x |x x | x ,x x ,x ,x          % , where     *1 *3 13 13 13 13 ij ij ij ij ij ij x ,x x w x ,x w x                  . Step 1. Construct the fuzzy transportation Problem (15), and then convert it into a balanced one if it is not. Else, go to Step 2. Step 2. Solving Eq. (17) via Eq. (19). Determine 3 ij 2 x = *2 *3 ij ij x ,x        *2 *2 ij ij x ,x        *2 ij x , for i 1, ,m   and go to Step 3. Step 3. Solving Eq. (16) via Eq. (18). Determine 3 ij 1 x =     *1 *3 ij ij ij ij ij i 13 13 13 13 j x ,x x w x ,x w x                  for i 1, ,m   . Considering the following cases: Case 1. If   n 13 i ij ij 2 3 1 1 i j E x x w a      , then 3 ij 1 x =     *1 *3 ij ij ij ij ij i 13 13 13 13 j x ,x x w x ,x w x                  with     3 ij 1 i 13 j i x 0 13 w x w a    . Else, go to Case 2. Case 2. If   13 n i ij i 13 i j 1 2 j E x x w a      , then 3 ij 1 x =     *1 *3 ij ij ij ij ij i 13 13 13 13 j x ,x x w x ,x w x                  with   ij i 13 13 j 2 j i w x x x   and go to Step 4. Step 4. The Optimal Solution (15) according to the choice of the decision maker is:
  • 11. 214 Kané et al. |J. Fuzzy. Ext. Appl. 2(3) (2021) 204-225        m n 1 2 3 *1 *2 *3 *1 *2 *3 ij ij ij ij ij ij j 1 i 1 Min Z x c ,c ,c x ,x ,x Z ,Z ,Z       % % with       * 2 13 2 *1 *3 *1 *2 *3 ij ij ij ij ij ij ij ij ij x x |x x | x ,x x ,x ,x          % . 4| Advantages of the Proposed Method Let us explore the main advantages of the proposed method:  The new proposed method is applicable to all Eq. (15) where ij x % are unrestricted triangular or trapezoidal fuzzy numbers and ij c % , i a % and j b % are non-negatives triangular or trapezoidal fuzzy numbers.  The proposed technique does not use the goal and parametric approaches which are difficult to apply in real life situations.  By applying the proposed approach for finding the fuzzy optimal solution, there is no need of much knowledge of fuzzy linear programming technique, Zimmerman approach and crisp linear programming which are difficult to learn for a new decision maker.  The proposed method to solve Eq. (15) is based on traditional transportation algorithms. Thus, the existing and easily available software can be used for the same. However, the existing method [1], [2], [3], [11], [29] to solve Eq. (15) should be implemented into a programming language.  To solve the Eq. (15) by using the existing method [1], [2], [3], [11], [29], there is need to use arithmetic operations of generalized fuzzy numbers. While, if the proposed technique is used for the same then there is need to use arithmetic operations of real numbers. This proves that it is much easy to apply the proposed method as compared to the existing method [1], [2], [3], [11], [29].  Moreover, it is possible to assume a generic ranking index for comparing the fuzzy numbers involved in the Eq. (15), in such a way that each time in which the decision maker wants to solve the FFTP under consideration (s)he can choose (or propose) the ranking index that best suits the Eq. (15). 5| Numerical Illustration In this section, to illustrate the new method proposed, and the existing fully fuzzy transportation problem due to [1], [2], [3], [11], [29], [30], presented in Examples 1-4, are solved by the proposed method. Example 1. [3]. Step 1. Construct the fuzzy transportation Problem (15), and then convert it into a balanced one if it is not. Table 1. Eq. (15) in trapezoidal unbalanced form. 1 R 2 R 3 R 3 R %i a Supply ( ) A   10,10,10,10   50,50,50,50   80,80,80,80   0,0,0,0   70,90,90,100 B   60,70,80,90   60,60,60,60   20,20,20,20   0,0,0,0   40,60,70,80 C   0,0,0,0   0,0,0,0   0,0,0,0   0,0,0,0   0,0,10,50 Demand ( % j b )   30,40,50,70   20,30,40,50   40,50,50,80   20,30,30,30     % % m n i j i 1 j 1 a b
  • 12. 215 A new algorithm for fuzzy transportation problems with trapezoidal fuzzy numbers under fuzzy circumstances Step 2. Solving Eq. (17) via Eq. (19).   23 23 23 23 23 23 23 23 11 12 13 21 22 23 MinZ x 25x 65x 100x 75x 90x 40x       subject to the constraints 23 23 23 23 11 12 13 14 x x x x 90     , 23 23 23 23 21 22 23 24 x x x x 65     , 23 23 23 23 31 32 33 34 x x x x 5     , 23 23 23 11 21 31 x x x 45    , 23 23 23 12 22 32 x x x 35    , 23 23 23 13 23 33 x x x 50    and 23 23 23 14 24 34 x x x 30    . Optimal solution is: 23 11 x 45  , 23 12 x 30  , 23 13 x 0  , 23 14 x 15  , 23 21 x 0  , 23 22 x 0  , 23 23 x 50  , 23 24 x 15  , 23 31 x 0  , 23 32 x 5  , 23 33 x 0  and 23 34 x 0  . Furthermore   23 1 w a 0  ,   23 2 w a 5  and   23 3 w a 5  . For i 1  , then     23 1 j 23 23 1j 1 x 0 w x w a 0     with         23 23 23 23 11 12 13 14 w x w x w x w x 0     . We get 23 11 x 45,45        , 23 12 x 30,30        , 23 13 x 0  and 23 14 x 15,15        . For i 2  , then     23 2 j 23 23 2 j 2 x 0 w x w a 5     with     23 23 21 22 w x w x 0   ,   23 23 w x 1  and   23 24 w x 4  . We get 23 21 x 0,0        , 23 22 x 0,0        , 23 23 x 49,51        and 23 24 x 11,19        . For i 3  , then     23 3 j 23 23 3 j 3 x 0 w x w a 5     with   23 31 w x 0  ,   23 32 w x 5  ,   23 33 w x 0  and   23 34 w x 0  . We get 23 31 x 0  , 23 32 x 0,10        , 23 33 x 0  and 23 34 x 0  . Step 3. Solving Eq. (16) via Eq. (18). We get   14 14 14 14 14 14 14 14 11 12 13 21 22 23 MinZ x 25x 70x 100x 75x 95x 40x       subject to the constraints 14 14 14 14 11 12 13 14 x x x x 85     , 14 14 14 14 21 22 23 24 x x x x 60     , 14 14 14 14 31 32 33 34 x x x x 25     , 14 14 14 11 21 31 x x x 50    , 14 14 14 12 22 32 x x x 35    , 14 14 14 13 23 33 x x x 60    and 14 14 14 14 24 34 x x x 25    . Optimal solution is: 14 11 x 50  , 14 12 x 10  , 14 13 x 0  , 14 14 x 25  , 14 21 x 0  , 14 22 x 0  , 14 23 x 60  , 14 24 x 0  , 14 31 x 0  , 14 32 x 25  , 14 33 x 0  and 14 34 x 0  . Furthermore   14 1 w a 15  ,   14 2 w a 20  and   14 3 w a 25  For i 1  , we have     4 14 23 23 14 1 1j 1j 1 1 j 1 E x x w a 50 45 10 30 0 0 25 15 0 35 15 w a                  , then     14 32 14 23 1j 1j 1j 1j w x x x w x    . We have   14 11 w x 5,    14 12 w x 20  ,   14 13 w x 0  and   14 14 w x 10  . We get 14 11 x 45,55        , 14 12 x 10,30         , 14 13 x 0  and 14 14 x 15,35        . For i 2  , we have
  • 13. 216 Kané et al. |J. Fuzzy. Ext. Appl. 2(3) (2021) 204-225     4 14 23 23 14 2 2 j 2 j 2 2 j 1 E x x w a 0 0 0 0 60 50 0 15 5 30 20 w a                  , then     14 14 23 23 2 j 2 j 2 j 2 j w x x x w x    . We have     14 14 21 22 w x w x 0   ,   23 23 w x 11  and   14 24 w x 19  . We get 14 21 x 0,0        , 14 22 x 0,0        , 14 23 x 49,71        and 14 24 x 19,19         . For i 3  , we have     4 14 23 23 14 3 3 j 3 j 3 3 j 1 E x x w a 0 0 25 5 0 0 0 0 5 25 25 w a                  , then     14 3 j 14 14 3 j 3 x 0 w x w a 25     . We have   14 31 w x 0  ,   14 32 w x 25  ,   14 33 w x 0  and   14 34 w x 0  We get 14 31 x 0  , 14 32 x 0,50        , 14 33 x 0  and 14 34 x 0  . Step 4. The optimal solution according to the choice of the decision maker is:          m n 1 2 3 4 *1 *2 *3 *4 *1 *2 *3 *4 ij ij ij ij ij ij ij ij j 1 i 1 Min Z x c ,c ,c ,c x ,x ,x ,x Z ,Z ,Z ,Z 930,2930,2970,3470        % % and       * 14 23 *1 *4 *2 *3 *1 *2 *3 *4 ij ij ij ij ij ij ij ij ij ij ij x x |x x ,x | x ,x x ,x ,x ,x                % with   * 11 x 45, 45, 45,55  % ,   * 12 x 10, 30, 30, 30   % , * 13 x 0  % % ,   * 14 x 15, 15, 15, 35  % , * 21 x 0  % % , * 22 x 0  % % ,   * 23 x 49, 49, 51, 71  % ,   * 24 x 19, 11, 19, 19   % , * 31 x 0  % % ,   * 32 x 0,0,10,50  % , * 33 x 0  % % and * 34 x 0  % % . Example 2. [29]. Step 1. Construct the fuzzy transportation Problem (15), and then convert it into a balanced one if it is not. Table 2. Eq. (15) in trapezoidal balanced form. 1 R 2 R 3 R Supply %i a A   1,4,9,19   1,2,5,9   2,5,8,18   1,5,7,9 B   8,9,12,26   3,5,8,12   7,9,13,28   4,7,8,10 C   11,12,20,27   0,5,10,15   4,5,8,11   4,5,8,11 Demand % j b   3,5,8,12 (4, 8, 9, 10) (2, 4, 6, 8)     % % m n i j i 1 j 1 a b
  • 14. 217 A new algorithm for fuzzy transportation problems with trapezoidal fuzzy numbers under fuzzy circumstances Step 2. Solving Eq. (17) via Eq. (19). Optimal solution is: 23 11 x 6  , 23 12 x 0  , 23 13 x 0  , 23 21 1 x 2  , 23 22 x 7  , 23 23 x 0  , 23 31 x 0  , 23 32 3 x 2  and 14 33 x 5  . Furthermore   23 1 2 w a 1 2   ,   23 2 w a 2 1  ,   23 3 w a 2 3  . For i 1  , then     3 1 j 2 1j 23 23 x 0 1 w x w a 1     with   23 11 w x 1  . We get 3 11 2 5,7 x        , 3 12 2 0,0 x        and 23 13 x 0,0        . For i 2  , then     3 2 j 2 2 j 23 x 0 23 2 1 w x w a 2     with   23 21 1 w x 4  and   23 22 1 w x 4  . We get 23 21 1 3 , 4 4 x            , 23 22 27 29 , 4 4 x            and 23 23 x 0,0        . For i 3  , then     3 3 j 2 3 j 23 x 0 23 3 3 w x w a 2     with   23 32 w x 2 1  and   23 33 w x 1  . We get 23 31 0,0 x        , 23 32 1,2 x        and 23 33 x 4,6        . Step 3. Solving Eq. (16) via Eq. (18). Optimal solution is: 14 11 x 5  , 14 12 x 0  and 14 13 x 0  , 14 21 5 x 2  , 14 22 9 x 2  , 14 23 x 0  , 14 31 x 0  , 14 32 5 x 2  and 14 33 x 5  . Furthermore   14 1 w a 4  ,   14 2 w a 3  ,   14 3 w a 2 7  . For i 1  , we have     23 n 14 1 23 1 1j 1j 1 1 j 1 4 E x x w a 5 6 0 0 0 0 1 2 4 w a                , then     4 1 j 1 1j 14 14 x 0 1 w x w a 4     with   14 11 w x 4  . We get 4 11 1 1,9 x        , 4 12 1 0,0 x        and 14 13 x 0,0        . For i 2  , we have     n 14 23 2 2 j 2 j 2 2 j 2 1 1 3 4 5 1 9 1 E x x w a 7 0 0 5 w a 3 2 2 2 2                , then     14 23 2 14 3 2 j 2 j 2 j 2 j w x x x w x    with   14 21 9 w x 4  and   14 22 11 w x 4  . We get 14 21 1 19 , 4 4 x            , 14 22 7 29 , 4 4 x            and 14 23 x 0,0        . For i 3  , we have     n 14 23 3 3 j 3 j 23 14 3 3 j 1 5 3 3 5 E x x w a 5 5 0 0 w a 2 2 2 2 2 7                , then     14 3 j 14 14 3 j 3 x 0 7 w x w a 2     with   14 32 3 w x 2  and   14 33 w x 2  . We get 14 31 0,0 x        , 14 32 1,4 x        and 14 33 x 3,7        . Step 4. The optimal solution according to the choice of the decision maker is:
  • 15. 218 Kané et al. |J. Fuzzy. Ext. Appl. 2(3) (2021) 204-225       * 14 23 *1 *4 *2 *3 *1 *2 *3 *4 ij ij ij ij ij ij ij ij ij ij ij x x |x x ,x | x ,x x ,x ,x ,x                % .   81 1037 Min Z x ,81, 173, 4 2            % % with   * 11 x 1, 5, 7 ,9  % , * 12 x 0  % % , * 13 x 0  % % , * 21 1 1 3 19 x , , , 4 4 4 4            % , * 22 7 27 29 29 x , , , 4 4 4 4            % , * 23 x 0  % % , * 31 x 0  % % ,   * 32 x 1,1,2,4  % and   * 33 x 3,4,6,7  % . Example 3. [29]. Step 1. Construct the fuzzy transportation Problem (15), and then convert it into a balanced one if it is not. Table 3. Eq. (15) in triangular balanced form. Step 2. Solving Eq. (17) via Eq. (19). We get   2 2 2 2 2 2 11 12 21 22 MinZ x 31x 19x 39x 10x     subject to the constraints 2 2 11 12 x x 201   , 2 2 21 22 x x 99   , 2 2 11 21 x x 150   and 2 2 12 22 x x 150   . Optimal solution is: 2 11 x 150  , 2 12 x 51  , 2 21 x 0  and 2 22 x 99  . Step 3. Solving Eq. (16) via Eq. (18). We have   13 13 13 13 13 13 11 12 21 22 MinZ x 28x 22x 42x 10x     subject to the constraints 13 13 11 12 x x 198   , 13 13 21 22 x x 102   , 13 13 11 21 x x 150   and 13 13 12 22 x x 150   . Optimal solution is: 13 11 x 150  , 13 12 x 48  , 13 21 x 0  and 13 22 x 102  . Furthermore   13 1 w a 48  ,   13 2 w a 52  . For i 1  , we have   2 13 2 13 1 1j 1j 1 j 1 E x x 150 150 48 51 3 48 w a            , then     3 1 j 1 1j 13 x 0 13 1 w x w a 48     with   13 11 w x 10  and   13 12 w x 38  . We get 13 11 x 140,160        and 13 12 x 10,86        . 1 R 2 R Supply ( %i a ) A   22,31,34   15,19,29   150,201,246 B   30,39,54   8,10,12   50,99,154 Demand ( % j b )   100,150,200   100,150,200     % % m n i j i 1 j 1 a b
  • 16. 219 A new algorithm for fuzzy transportation problems with trapezoidal fuzzy numbers under fuzzy circumstances For i 2  , we have   2 13 2 13 2 2 j 2 j 2 j 1 E x x 0 0 102 99 3 52 w a            , then     3 2 j 1 2 j 13 x 0 13 2 w x w a 52     with   13 21 w x 0  and   13 22 w x 52  . We get 13 21 x 0,0        and 13 22 x 50,154        . Step 4. The optimal solution according to the choice of the decision maker is     m n 1 2 3 ij ij ij ij i 1 j 1 MinZ x c ,c ,c x      % % % with       * 2 13 2 *1 *3 *1 *2 *3 ij ij ij ij ij ij ij ij ij x x |x x | x ,x x ,x ,x          % . We have Min     * Z 6609| 3630, 9782 3630,6609, 9782         % where   * 11 x 140,150, 160  % ,   * 12 x 10,51,86  % , * 21 x 0  % % and   * 22 x 50,99,154  % . 5.1| Interpretation of Results We will now interpret the minimum total fuzzy transportation cost obtained in Example 3. by using the proposed methods presented in Section 3. Similarly, the obtained fuzzy optimal solution will also be interpreted. By using the methods proposed the minimum total fuzzy transportation cost is   3630,6609, 9782 , which can be physically interpreted as follows:  The least amount of the minimum total transportation cost is 3630 .  The most possible amount of minimum total transportation cost is 6609 .  The greatest amount of the minimum total transportation cost is 9782 i.e., the minimum total transportation cost will always be greater than 3630 and less than6609 , and the highest chances are that the minimum total transportation cost will be9782 . Example 4. [1]-[3]. Step 1. Construct the fuzzy transportation Problem (15), and then convert it into a balanced one if it is not. Step 2. Solving Eq. (17) via Eq. (19). We get   2 2 2 2 2 2 2 2 2 2 11 12 13 14 21 22 23 24 MinZ x 10x 22x 10x 20x 15x 20x 12x 8x        2 2 2 2 31 32 33 34 20x 12x 10x 15x     subject to the constraints 2 2 2 2 11 12 13 14 x x x x 8     , 2 2 2 2 21 22 23 24 x x x x 14     , 2 2 2 2 31 32 33 34 x x x x 12     , 2 2 2 11 21 31 x x x 7    , 2 2 2 12 22 32 x x x 10    , 2 2 2 13 23 33 x x x 8    and 2 2 2 14 24 34 x x x 9    .
  • 17. 220 Kané et al. |J. Fuzzy. Ext. Appl. 2(3) (2021) 204-225 Optimal solution is: 2 11 x 7  , 2 12 x 0  , 2 13 x 1  , 2 14 x 0  , 2 21 x 0  , 2 22 x 0  , 2 23 x 5  , 2 24 x 9  , 2 31 x 0  , 2 32 x 10  , 2 33 x 2  and 2 34 x 0  . Table 4. Eq. (15) in triangular balanced form (in U.S. dollar). Step 3. Solving Eq. (16) via Eq. (18). We get   13 13 13 13 13 13 13 13 11 12 13 14 21 22 MinZ x 9.4x 22.2x 9.3x 20.4x 15x 20.1x       13 13 13 13 13 13 23 24 31 32 33 34 11.5x 7.4x 19.7 x 11.3x 9.3x 15x       subject to the constraints 13 13 13 13 11 12 13 14 x x x x 8     , 13 13 13 13 21 22 23 24 x x x x 14     , 13 13 13 13 31 32 33 34 x x x x 12     , 13 13 13 11 21 31 x x x 7    , 13 13 13 12 22 32 x x x 10    , 13 13 13 13 23 33 x x x 8    and 13 13 13 14 24 34 x x x 9    . Optimal solution is: 13 11 x 7  , 13 12 x 0  , 13 13 x 1  , 13 14 x 0  , 13 21 x 0  , 13 22 x 0  , 13 23 x 5  , 13 24 x 9  , 13 31 x 0  , 13 32 x 10  , 13 33 x 2  and 13 34 x 0  . We have   13 1 4 w a 5  ,   13 2 w a 2  and   13 3 9 w a 5  . Furthermore:  13 11 34 36 x , 5 5            , 13 12 x 0,0        , 13 13 2 8 x , 5 5            and 13 14 x 0,0        where   13 11 1 w x 5  ,   13 12 w x 0  ,   13 13 3 w x 5  and   13 14 w x 0  .  13 21 x 0,0        , 13 22 x 0,0        , 13 23 9 11 x , 2 2            and 13 24 15 21 x , 2 2            where   13 21 w x 0  ,   13 22 w x 0  ,   13 23 1 w x 2  and   13 24 3 w x 2  . Source Destination Supply (𝒂 ̃𝒊) (000 dozen bottles) Taichung Chiayi Kaohsiung Taipei Changhua   $8, $10, $10.8   $20.4, $22, $24   $8,$10,$10.6   $18.8, $20, $22   7.2,8,8.8 Touliu   $14, $15, $16   $18.2, $20, $22   $10, $12, $13   $6, $8, $8.8   12,14,16 Hsinchu   $18.4, $20, $21   $9.6, $12, $13   $7.8, $10, $10.8   $14, $15, $16   10.2,12,13.8 Demand ( j b % ) (000 dozen bottles)   6.2,7,7.8   8.9,10,11.1   6.5,8,9.5   7.8,9,10.2 ∑𝑎̃𝑖 𝑚 𝑖=1 = ∑𝑏̃𝑗 𝑛 𝑗=1
  • 18. 221 A new algorithm for fuzzy transportation problems with trapezoidal fuzzy numbers under fuzzy circumstances  13 31 x 0,0        , 13 32 47 53 x , 5 5            , 13 33 4 16 x , 5 5            and 13 34 x 0,0        with   13 31 w x 0  ,   13 32 3 w x 5  ,   13 33 6 w x 5  and   13 34 w x 0  . Step 4. The optimal solution according to the choice of the decision maker is The value of the objective function:        m n 1 2 3 *1 *2 *3 *1 *2 *3 ij ij ij ij ij ij j 1 i 1 Min Z x c ,c ,c x ,x ,x Z ,Z ,Z       % % with       * 2 13 2 *1 *3 *1 *2 *3 ij ij ij ij ij ij ij ij ij x x |x x | x ,x x ,x ,x          % . We have Min * 12204 21549 12204 21549 Z 352$| $, $ $,352$, $ 50 50 50 50                                   % where * 11 34 36 x , 7 , 5 5            % , * 12 x 0  % % , * 13 2 8 x , 1, 5 5            % , * 14 x 0  % % , * 21 x 0  % % , * 22 x 0  % % , * 23 9 11 x , 5, 2 2            % , * 24 15 21 x , 9, 2 2            % , * 31 x 0  % % , * 32 47 53 x , 10, 5 5            % , * 33 4 16 x , 2, 5 5            % and * 34 x 0  % % . 5.2| Interpretation of Results We will now interpret the minimum total fuzzy transportation cost obtained in Example 4 by using the proposed methods presented in Section 3. Similarly, the obtained fuzzy optimal solution will also be interpreted. By using the methods proposed the minimum total fuzzy transportation cost is 12204 21549 $,352$, $ 50 50           , which can be physically interpreted as follows:  The least amount of the minimum total transportation cost is 12204 $ 50 .  The most possible amount of minimum total transportation cost is 352$ .  The greatest amount of the minimum total transportation cost is 21549 $ 50 i.e., the minimum total transportation cost will always be greater than 12204 $ 50 and less than 352$ , and the highest chances are that the minimum total transportation cost will be 21549 $ 50 .
  • 19. 222 Kané et al. |J. Fuzzy. Ext. Appl. 2(3) (2021) 204-225 6| A Comparative Study Author's of [3], [11], [29] have proposed a method to find the crisp optimal solution of such fuzzy transportation problems in which all the parameters are represented by triangular or trapezoidal fuzzy numbers. Then, they have used their new method proposed to find the crisp optimal solution of a real- life fuzzy transportation problem. However, it is often better to find a fuzzy optimal solution than a crisp optimal solution. In this section we will therefore show how in the problem considered by [3], [11], [29], we can obtain a fuzzy optimal solution of the same real-life problem using the new method proposed. 7| Concluding Remarks and Future Research Directions 7.1| Concluding Remarks These days a number of researchers have shown interest in the area of fuzzy transportation problems and various attempts have been made to study the solution of these problems. In this paper, to overcome the shortcomings of the existing methods we introduced a new formulation of transportation problem involving trapezoidal (or triangular) fuzzy numbers for the transportation costs and values of supplies and demands. We propose a fuzzy linear programming approach for solving trapezoidal (or triangular) fuzzy numbers transportation problem based on the converting into two interval transportation problems Eq. (16) and Eq. (17). To show the advantages of the proposed methods over existing methods, some fuzzy transportation problems, may or may not be solved by the existing methods, are solved by using the proposed methods and it is shown that it is better to use the proposed methods as compared to the existing methods for solving the transportation problems. From both theoretical and algorithmic considerations, and examples solved in this paper, it can be noticed that some shortcomings of the methods for solving the fuzzy transportation problems known from the literature can be resolved by using the new methods proposed in Section 5. 7.2| Future Research Directions Finally, we feel that, there are many other points of research and should be studied later on interval numbers or fuzzy numbers. Some of these points are below: We will consider the following transportation Problems (14) with fuzzy numbers as follows: Example Minimum Total Fuzzy Transportation Cost [3] [29] Method Proposed in This Paper 1   2100,2900,3500,4200   930,2930,2970,3470 2   31,80,199,460           81 1037 ,81,173, 4 2 3   3630,6609,9782 4   238.44,347.8,428.9           12204 21549 ,352, 50 50
  • 20. 223 A new algorithm for fuzzy transportation problems with trapezoidal fuzzy numbers under fuzzy circumstances   n m ij ij j 1i 1 MinZ x c x     % % % % subject to the constraints n ij i j 1 x a   % % and m ij j i 1 x b    % % where   1 2 t ij ij ij ij c c ,c , ,c   % ,   1 2 t ij ij ij ij x x ,x , ,x   % ,   1 2 t i i i i a a ,a , ,a   % and   1 2 t j j j j b b ,b , ,b   % with 1 t  N . Let pq p q ij ij ij x x ,x        where p q  and 1 p,q  N . The same applies to pq ij c , pq i a and pq j b . I. Solution procedure for classical Transportation Problem   t 1  :   1 ij ij x x .  II. Solution procedure for Transportation Problem with Interval numbers   t 2  : 12 1 2 ij ij ij x x ,x .        III. Solution procedure for Transportation Problem with Triangular fuzzy numbers   t 3  :     1 2 3 2 13 ij ij ij ij ij ij x x ,x ,x x ,x .   % IV. Solution procedure for Transportation Problem with Trapezoidal fuzzy numbers   t 4  :     1 2 3 4 23 14 ij ij ij ij ij ij ij x x ,x ,x ,x x ,x .   % V. Solution procedure for Transportation Problem with Pentagonal fuzzy numbers   t 5  :     1 2 3 4 5 3 24 15 ij ij ij ij ij ij ij ij ij x x , x , x , x , x x , x , x .   % VI. Solution procedure for Transportation Problem with Hexagonal fuzzy numbers   t 6  :     1 2 3 4 5 6 34 25 16 ij ij ij ij ij ij ij ij ij ij x x ,x ,x ,x ,x ,x x ,x ,x .   % VII. Solution procedure for Transportation Problem with Heptagonal fuzzy numbers   t 7      1 2 3 4 5 6 7 4 35 26 17 ij ij ij ij ij ij ij ij ij ij ij ij : x x ,x ,x ,x ,x ,x ,x x ,x ,x ,x .   % VIII. Solution procedure for Transportation Problem with Octagonal fuzzy numbers   t 8      1 2 3 4 5 6 7 8 45 36 27 18 ij ij ij ij ij ij ij ij ij ij ij ij ij : x x ,x ,x ,x ,x ,x ,x ,x x ,x ,x ,x .   % IX. Solution procedure for Transportation Problem with Nonagonal fuzzy numbers   t 9  :     1 2 3 4 5 6 7 8 9 5 46 37 28 19 ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij x x ,x ,x ,x ,x ,x ,x ,x ,x x ,x ,x ,x ,x .   % X. Solution procedure for Transportation Problem with Decagonal fuzzy numbers   t 10  :     1 2 3 4 5 6 7 8 9 10 56 47 38 29 110 ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij x x ,x ,x ,x ,x ,x ,x ,x ,x ,x x ,x ,x ,x ,x .   % XI. Solution procedure for Transportation Problem with Hendecagonal fuzzy numbers   t 11  :     1 2 3 4 5 6 7 8 9 10 11 6 57 48 39 210 111 ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij x x ,x ,x ,x ,x ,x ,x ,x ,x ,x ,x x ,x ,x ,x ,x ,x .   % XII. Solution procedure for Transportation Problem with Dodecagonal fuzzy numbers   t 12      1 2 3 4 5 6 7 8 9 10 11 12 67 58 49 310 211 112 ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij : x x ,x ,x ,x ,x ,x ,x ,x ,x ,x ,x ,x x ,x ,x ,x ,x ,x .   % Acknowledgements The authors are very grateful to the anonymous referees for their valuable comments and suggestions to improve the paper in the present form. Conflicts of interest The authors declare no conflicts of interest.
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