In this paper a method to obtain a non-dominated point for the multi-objective transportation problem is presented. The superiority of this method over the other existing methods is that the presented non-dominated point is the closest solution to the ideal solution of that problem. The presented method does not need to have the ideal point and other parameters to find this solution. Also, the calculative load of this method is less than other methods in the literature

Bander, Abbas S.

Basirzadeh, Hadi

Morovati, Vahid

English

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Applications and Applied Mathematics: An International Journal (AAM) Volume 10 Issue 1 Article 32 6-2015 A Super Non-dominated Point for Multi-objective Transportation Problem Abbas S. Bander Shahid Chamran University Vahid Morovati Shahid Chamran University Hadi Basirzadeh Shahid Chamran University Follow this and additional works at: https://digitalcommons.pvamu.edu/aam  Part of the Numerical Analysis and Computation Commons Recommended Citation Bander, Abbas S.; Morovati, Vahid; and Basirzadeh, Hadi (2015). A Super Non-dominated Point for Multi-objective Transportation Problem, Applications and Applied Mathematics: An International Journal (AAM), Vol. 10, Iss. 1, Article 32. Available at: https://digitalcommons.pvamu.edu/aam/vol10/iss1/32 This Article is brought to you for free and open access by Digital Commons @PVAMU. It has been accepted for inclusion in Applications and Applied Mathematics: An International Journal (AAM) by an authorized editor of Digital Commons @PVAMU. For more information, please contact hvkoshy@pvamu.edu. 544   Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466  Vol. 10, Issue 1 (June 2015),  pp. 544 - 551 Applications and Applied Mathematics:  An International Journal (AAM)   A Super Non-dominated Point for Multi-objective    Transportation Problem   Abbas Sayadi Bander, Vahid Morovati, and Hadi Basirzadeh Department of Mathematics Faculty of Mathematical Sciences and Computer Shahid Chamran University Ahvaz 6135714463, Iran Abbas.sayadi@yahoo.com; v-morovati@phdstu.scu.ac.ir; basirzad@scu.ac.ir  Received: March 19, 2014;    Accepted: January 22, 2015  Abstract  In this paper a method to obtain a non-dominated point for the multi-objective transportation problem is presented. The superiority of this method over the other existing methods is that the presented non-dominated point is the closest solution to the ideal solution of that problem. The presented method does not need to have the ideal point and other parameters to find this solution. Also, the calculative load of this method is less than other methods in the literature.  Keywords: Multi-objective programming; Transportation problem  AMS-MSC 2010 No.: 90C29, 90C08    1.  Introduction  One of the undeniable problems in the real world is that of transportation. In general terms, the transportation model presents a plan with the least expenses to transfer goods from some places to some destinations. In a lot of practical problems the transportation formulation in single-objective form cannot be appropriate, because most of the problems include several objectives like minimizing the total cost, minimizing the total time etc. The transportation model was first developed by Hitchcock (1941). To obtain all of the non-dominated solutions of a multi-objective linear transportation problem an algorithm was introduced by Isermann (1979). Such problems can be solved by programming multi-objective linear techniques like parametric 1Bander et al.: A Super Non-dominated Point for Multi-objective Transportation ProblemPublished by Digital Commons @PVAMU, 2015AAM: Intern. J., Vol. 10, Issue 1 (June 2015)                                                                                                            545  method, the adjacent point method and the adjacent basic point method (see Gal (1975) and Zelenv (1974)). Martnez-Salazar (2014) solved a bi-objective transportation optimization problem using a heuristic algorithm. In Hongwei (2010) and Kundu (2013). Some methods to solve multi-objective transportation optimization problem were presented. It is an undeniable fact that among all of the non-dominated solutions the solution with the least distance from the ideal solution can be considered decision-maker, and in the second part of this paper a method is presented to calculate the closest solution to the ideal solution for multi-objective transportation problems. In the following, the convergence of the method is discussed. In the third part two  examples of the proposed method and the other existing methods are solved, the optimal solution and the calculative load of the methods are also compared. The results are provided in the final part.   2. Problem Formulation  The general form of a multi-objective transportation problem is as follows:                           ( )  ∑∑                                                                                                         ∑                               ∑                                                                        where the subscript on    and the superscript on      are used to identify the penalty criterion. Without loss of generality it may be assumed throughout this paper that      for all i,      for all j,        for all (i, j) and ∑    ∑           . Notice that the balance condition ∑   ∑           is both a necessary and sufficient condition for the existence of a feasible solution of problem   . Definition 2.1 (Ehrgott (2005)).     is an efficient solution for    if there is no other   belonged to the feasible region of problem    such that:  ( )    (  )         and   ( )    (  ) for some j. If *x  is an efficient solution for    then  (  )  (  (  )     (  )) is a non-dominated point for   . Definition 2.2 (Ehrgott (2005)). 2Applications and Applied Mathematics: An International Journal (AAM), Vol. 10 [2015], Iss. 1, Art. 32https://digitalcommons.pvamu.edu/aam/vol10/iss1/32546                                                                                                                                        Abbas Sayadi Bander et al.      (         ) is the ideal point of    if           ( )    for        .  Definition 2.3.     is a supper-efficient for    if 1)    is efficient 2)  (  ) has the least distance to the ideal point. If    is a supper efficient then  (  ) is called the super non-dominated point. In the following it is shown that a single-objective linear problem can be solved instead of a multi-objective problem    and the optimal solution which is obtained from the new problem is the super-efficient solution for problem   . Assume   ( )  ∑ ∑                 and consider the following problem:                           ( )    ( )    ( )      ( )                                                                   ∑                               ∑                                                                        Theorem 2.1.  Every optimal solution for problem    is an efficient solution for problem   .   Proof:   Assume that    is an optimal solution of   , but not the efficient solution of   , so, there is a feasible solution (the feasible region of both problems is identical) like   that dominates   , that is,   ( )    (  )         and   ( )    (  ) for some j, adding the side of the inequalities there is:   ( )      ( )    (  )      (  ), which is in contrast with    to optimal for   . So,    is an efficient point for   .  Definition 2.4.    If    (         ) is the ideal solution of problem   , the distance of each point of the objective  space from the ideal point (with the norm of L1) is considered as following: 3Bander et al.: A Super Non-dominated Point for Multi-objective Transportation ProblemPublished by Digital Commons @PVAMU, 2015AAM: Intern. J., Vol. 10, Issue 1 (June 2015)                                                                                                            547      ∑|      |      Theorem 2.2.   The problem of finding the super-efficient solution (the closest point to the ideal point for the multi-objective transportation problem) for the problem    is equal to finding the optimal solution for the problem   .  Proof:    If S is the feasible region of problem   , the problem of the closest solution to the ideal solution is considered as following:         ∑|      |                                              Because    is the ideal point of   , then for each     and for all i,         ( )  |      |                ∑(      )      ∑       ∑          Since the vector   is specified and obtained, then it can be assumed that     ∑         is a fixed value. So, for each     there is       {         ∑|      |  ∑            }  {                     ∑      }    This is the favorite result and the proof is complete.  3. Numerical example   Example 1.  Consider the following two-objective transportation problem: 4Applications and Applied Mathematics: An International Journal (AAM), Vol. 10 [2015], Iss. 1, Art. 32https://digitalcommons.pvamu.edu/aam/vol10/iss1/32548                                                                                                                                        Abbas Sayadi Bander et al.                                                                                                                                                                                                                                                                                                                                                                         ∑          ∑               ∑                                                                                ∑                ∑                ∑                ∑                                                                                                                                                                           ( )  Lushu et al. (2000) computed (163,190) (integer solution) or (163.33, 190.83) as the optimal compromise value of the objective vector (     ). Ringuest et al. (1978). Computed (156, 200) as the most preferred value of the objective vector (     ). Bit et al. (1992) obtained (160, 195) (integer solution) or (160.8591, 193.9260) as the optimal compromise value of the objective vector (     ).   For solving the problem above with the presented method this paper, we have to solve the following problem:                                                                                                                                                                                        ∑          ∑               ∑                                                                                ∑                ∑                ∑                ∑                                                                                                                                 The optimal solution *x is obtained:                    (                                               )                                                                                                                                                                                                                                                              By Theorem 2.1,  (  )  (  (  )   (  )  (       ) is a non-dominated point for the problem (1). By Definition 2.2, obtained    (       ) as the ideal point of the problem (1). 5Bander et al.: A Super Non-dominated Point for Multi-objective Transportation ProblemPublished by Digital Commons @PVAMU, 2015AAM: Intern. J., Vol. 10, Issue 1 (June 2015)                                                                                                            549  The calculated non-dominated points of the problem (1) by various other methods and their distance from the ideal point are presented in the following table:  Table 1. Result of Analysis for Example 1. Method    Lusha      Ringues       Bit Our method   (     )   (163,190) (156,200) (160,195) (176,175)   ∑|      |         43      46          45     41                    Table 1 shows that the obtained solution by the presented method has the minimum from the ideal point. Example 2.  In the following transportation objective problem obtained in Bit (1992), all of the non-dominated points and their distances to the ideal point have been collected in a table using a method known as Figueria.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 Solving both problems individually, the ideal solution was obtained (145,179). All of the non-dominated points of the problem which obtained by the Figueria method and the distance of these points to the ideal point are shown in the following table using the d  meter.  Regarding the Table 2, points of *              + are the non-dominated points which have the least distance to the ideal point. To obtain each of the non-dominated points using the Figueria method, complicated calculations are required, while using the proposed method solving a transportation linear programming problem which its target function is the sum of two given target functions, i.e.       and its feasible region is the feasible region of the two-objective problem, the solution of    (       ) is introduced as the optimized solution which is included in the points having the least distance to the ideal point. More surprising fact is that if the distance from the ideal point is being considered with the norm of    the    is the only point having the least distance from the ideal point. 6Applications and Applied Mathematics: An International Journal (AAM), Vol. 10 [2015], Iss. 1, Art. 32https://digitalcommons.pvamu.edu/aam/vol10/iss1/32550                                                                                                                                        Abbas Sayadi Bander et al.            Table 2. Result of Analysis for Example 2  Point   (  )   (  )   ∑ |      |     1x  145 215 36 2x  149 211 36 3x  153 207 36 4x  157 203 36 5x  161 199 36 6x  167 195 38 7x  173 191 40 8x  179 187 42 9x  185 183 39 10x  191 179 46   4. Conclusion  As in the case of a linear transportation programming problem a   non-dominated solution for the multi-objective transportation problem which has the least distance to the ideal point, can be obtained. Even through the method of solution may be relatively complicated and has a great deal of calculative load.  The ideal solution itself to problem (2.2) and any other parameters may not be needed. This represents an advantage.  Acknowledgment  The authors are grateful to the referrers for their helpful comments on some parts of the paper.  REFERENCES  Bit, AK, Biswal MP, Alam, SS. (1992). Fuzzy programming approach to multicriteria decision making transportation problem, Fuzzy Sets and Systems 50:35-41. 7Bander et al.: A Super Non-dominated Point for Multi-objective Transportation ProblemPublished by Digital Commons @PVAMU, 2015AAM: Intern. J., Vol. 10, Issue 1 (June 2015)                                                                                                            551  Ehrgott, M. (2005). Multicriteria Optimization, 2nd ed, Berlin. Eusebio, Augusto, Rui Figueria, Jose (2009). Finding non-dominated solutions in objective integer network flow problems, Comuter & Operation Research 36: 2554-2564.  Gal, T. (1975). Rim multi parametric linear programming. Management Sri. 2: 567-575. Hongwei, Zhang, Xiaoke, Cui, Shurong, Zou (2010). Multi-Objective Transportation Optimization based on Fmcica, ICIME, pp. 426-430. Iserman, HThe (1979). Enumeration of all efficient solutions for a linear multiple-objective transportation problem. Naval Res. Logist. Quart. 26(1): 123-39. Kundu, P., Kar, S. and Maiti, M. (2013). Multi-objective multi-item solid transportation problem in fuzzy environment, Appl. Math. Model. 37, 2028–2038. Li, Lushu, Lai, KK. (2000). A fuzzy approach to the multiobjective transportation problem. Computers & Operations Research 27: 43-57. Martnez-Salazar, I. A., Molina, J., Angel-Bello, F., Gomez, T., and Caballero, R. (2014). Solving a bi-objective transportation location routing problem by metaheuristic algorithms. European Journal of Operational Research, 234 (1): 25-36 Yu, P. L.  and Zeleny, M. (1974) The techniques of linear multiobjective programming. Revue Francaise Automatique, Inforrnatique et Recherche Operationelle 3: 5l-71. Ringuest J-L, Rinks, D.B. (1987). Interactive solutions for the linear multiobjective transportation problem, European Journal of Operations Research, 32:96-106.   8Applications and Applied Mathematics: An International Journal (AAM), Vol. 10 [2015], Iss. 1, Art. 32https://digitalcommons.pvamu.edu/aam/vol10/iss1/32

A Super Non-dominated Point for Multi-objective Transportation Problem

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A Super Non-dominated Point for Multi-objective Transportation Problem

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