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A two-stage data envelopment analysis approach to solve extended transportation problem with non-homogenous costs

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  • The transportation problem is a particular type of linear programming problem in which the main objective is to minimize the cost. In marked contrast to the classical real-world transportation model, shipping supplies from one source to a destination cause several costs and benefits, each of which is incomparable to another. The extended transportation problem was first introduced in a study conducted by Amirteimoori [1]. In contrast, many important questions regarding the production possibility set, the place of costs, the benefits, and the essence of these costs were not fully addressed yet. Therefore, this paper focuses on transportation models that do not provide explicit output. This method is helpful because it is designed for a specific purpose: to send goods and supply-demand at the lowest cost and decision-maker; does not suffer from the confusion of costs and the various consequences of placing them costs and outputs. Furthermore, this model improves the contradiction between the essence of the problem and the input/output-oriented data envelopment analysis. In this paper, previous models that can not incorporate all the sources of inefficiency have been solved. We apply the slack-based measure(SBM) to calculate all identified inefficiency sources. A numerical example is considered to show the approach's applicability, as mentioned above, to actual life situations. As a result, the optimal costs achieved via the proposed method are more realistic and accurate by obtaining a more representative efficiency assessment. This example proved our proposed approach's efficiency, providing a more efficient solution by corporate all sources inefficiency and presenting efficient costs for each path.

    Mathematics Subject Classification: Primary: 90C05, 90C08; Secondary: 90B06.

    Citation:

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  • Figure 1.  Extended transportation problem

    Table 1.  The data for example

    I J K $ S_i $
    A (531, 3500,500) (431,380,600) (395, 3950,400) 10
    B (394, 2850,600) (418, 2395,700) (512, 2590,485) 13
    C (405,310,800) (512,409, 1000) (412,390, 1100) 11
    D (355,290,705) (493,385,617) (570,419,518) 7
    E (299,415,585) (398,512,490) (315,255,380) 9
    F (319,512,488) (464,215,305) (435,355,512) 9
    G (619,612,619) (490,510,505) (354,550,490) 4
    H (456,299,601) (394,512,432) (439,499,519) 6
    $ d_j $ 30 25 14
     | Show Table
    DownLoad: CSV

    Table 2.  Results for example

    I J K
    A $ \tilde{e}_{AI}=0.5743 $
    $ \bar{e}_{AI}=0.7988 $
    $ \tilde{e}_{AJ}=0.8475 $
    $ \bar{e}_{AJ}=1,0000 $
    $ \tilde{e}_{AK}=0.5743 $
    $ \bar{e}_{AK}=1,0000 $
    B $ \tilde{e}_{BI}=0.6009 $
    $ \bar{e}_{BI}=1,0000 $
    $ \tilde{e}_{BJ}=0.5754 $
    $ \bar{e}_{BJ}=1,0000 $
    $ \tilde{e}_{BK}=0.6009 $
    $ \bar{e}_{BK}=1,0000 $
    C $ \tilde{e}_{CI}=0.8978 $
    $ \bar{e}_{CI}=1,0000 $
    $ \tilde{e}_{CJ}=0.579 $
    $ \bar{e}_{CJ}=0.7830 $
    $ \tilde{e}_{CK}=0.8978 $
    $ \bar{e}_{CK}=0.8551 $
    D $ \tilde{e}_{DI}=1,0000 $
    $ \bar{e}_{DI}=1,0000 $
    $ \tilde{e}_{DJ}=0.6646 $
    $ \bar{e}_{DJ}=0.9454 $
    $ \tilde{e}_{DK}=1.3844 $
    $ \bar{e}_{DK}=1,0000 $
    E $ \tilde{e}_{EI}=1,0000 $
    $ \bar{e}_{EI}=1,0000 $
    $ \tilde{e}_{EJ}=0.9446 $
    $ \bar{e}_{EJ}=0.6886 $
    $ \tilde{e}_{EK}=1,0000 $
    $ \bar{e}_{EK}=1,0000 $
    F $ \tilde{e}_{FI}=1,0000 $
    $ \bar{e}_{FI}=1,0000 $
    $ \tilde{e}_{FJ}=1,0000 $
    $ \bar{e}_{FJ}=1,0000 $
    $ \tilde{e}_{FK}=1,0000 $
    $ \bar{e}_{FK}=0.8134 $
    G $ \tilde{e}_{GI}=0.7021 $
    $ \bar{e}_{GI}=1.2044 $
    $ \tilde{e}_{GJ}=1.189 $
    $ \bar{e}_{GJ}=0.7541 $
    $ \tilde{e}_{GK}=0.7021 $
    $ \bar{e}_{GK}=1,0000 $
    H $ \tilde{e}_{HI}= 1,0000 $
    $ \bar{e}_{HI}=1,0000 $
    $ \tilde{e}_{HJ}=1,0000 $
    $ \bar{e}_{HJ}=1,0000 $
    $ \tilde{e}_{HK}=1,0000 $
    $ \bar{e}_{HK}=0.9195 $
     | Show Table
    DownLoad: CSV

    Table 3.  Composite efficiency

    I J K
    A
    B
    $ e_{AI}=0.6682 $
    $ e_{BI}=0.7507 $
    $ e_{AJ}=0.9175 $
    $ e_{BJ}=0.7305 $
    $ e_{AK}=0.7296 $
    $ e_{BK}=0.7507 $
    C $ e_{CI}=0.9461 $ $ e_{CJ}=0.6657 $ $ e_{CK}=0.8653 $
    D $ e_{DI}=1,0000 $ $ e_{DJ}=0.7805 $ $ e_{DK}=1,0000 $
    E $ e_{EI}=1,0000 $ $ e_{EJ}=0.7963 $ $ e_{EK}=1,0000 $
    F $ e_{FI}=1,0000 $ $ e_{FJ}=1,0000 $ $ e_{FK}=0.8971 $
    G $ e_{GI}=0.7281 $ $ e_{GJ}=0.8285 $ $ e_{GK}=0.8287 $
    H $ e_{HI}=1,0000 $ $ e_{HJ}=1,0000 $ $ e_{HK}=0.9597 $
     | Show Table
    DownLoad: CSV

    Table 4.  Results comparing

    Arc Destination aspect Source aspect Average SBM-Without-Input
    AI 0.5743 0.7988 0.6865 0.6682
    AJ 0.8475 1 0.9237 0.9175
    AK 0.5743 1 0.7871 0.7296
    BI 0.6009 1 0.8004 0.7507
    BJ 0.5754 1 0.7877 0.7305
    BK 0.6009 1 0.8004 0.7507
    CI 0.8978 1 0.9489 0.9461
    CJ 0.579 0.783 0.681 0.6657
    CK 0.8978 0.8351 0.8664 0.8653
    DI 1 1 1 1
    DJ 0.6646 0.9454 0.805 0.7805
    DK 1 1 1 1
    EI 1 1 1 1
    EJ 0.9446 0.6883 0.8164 0.7963
    EK 1 1 1 1
    FI 1 1 1 1
    FJ 1 1 1 1
    FK 1 0.8134 0.9067 0.8971
    GI 0.7021 0.7541 0.7281 0.7272
    GJ 0.6575 1 0.8287 0.7934
    GK 0.7021 1 0.851 0.825
    HI 1 1 1 1
    HJ 1 1 1 1
    HK 1 0.9195 0.9597 0.9581
    Mean 0.825783 0.939067 0.882404 0.866746
    STD 0.182328 0.099468 0.108758 0.122179
     | Show Table
    DownLoad: CSV

    Table 5.  Optimal costs

    I J K
    A 0 10 0
    B 10 0 3
    C 11 0 0
    D 0 0 7
    E 9 0 0
    F 0 9 0
    G 0 0 4
    H 0 6 0
     | Show Table
    DownLoad: CSV
  • [1] A. Amirteimoori, An extended transportation problem: a DEA based approach, Central European Journal of Operations Research, 19 (2011), 513-521.  doi: 10.1007/s10100-010-0140-0.
    [2] A. Amirteimoori, An extended shortest path problem: A data envelopment anlysis approach, Applied Mathmatics Letters, 25 (2012), 1839-1843.  doi: 10.1016/j.aml.2012.02.042.
    [3] M. S. Bazaraa, J. J. Jarvis and H. D. Sherali, Linear Programming and Network Flows, Willey, New York, 2011.
    [4] C. M. ChaoM. M. Yu and M. C. Chen, Measuring the performance of financial holding companies, The Service Industries Journal, 30 (2010), 811-829.  doi: 10.1080/02642060701849857.
    [5] A. CharnesW. W. Cooper and E. Rhodes, Measuring the efficiency of decision making units, European Journal of Operational Research, 2 (1978), 429-444.  doi: 10.1016/0377-2217(78)90138-8.
    [6] L. H. Chen and H. W. Lu, Responses and comments to "A comment on "An extended assignment problem considering multiple inputs and outputs"", Appl. Math. Model., 32 (2008), 2463-2466.  doi: 10.1016/j.apm.2007.09.029.
    [7] G. DantzigLinear Programming and Extensions, Princeton University Press, 1963. 
    [8] K. Djordjevi'c, Evaluation of energy-environment effciency of European transport sectors: Non-radial DEA and TOPSIS approach, Energies, 12 (2019), 1-27. 
    [9] F. Hitchcock, The distribution of a product from several sources to numerous localities, J. Math. Phys., 20 (1941), 224-230.  doi: 10.1002/sapm1941201224.
    [10] L. Kantorovich, Mathematical methods of organizingand planning production, Manag. Sci., 6 (1960), 336-422.  doi: 10.1287/mnsc.6.4.366.
    [11] G. MaityD. MardanyaS. K. Roy and G. W. Weber, A new approach for solving dual-hesitant fuzzy transportation problem with restrictions, Indian Academy of Sciences, 75 (2018), 44-75.  doi: 10.1007/s12046-018-1045-1.
    [12] G. MaityS. K. Roy and J. L. Verdegay, Analyzing multimodal transportation problem and its application, Neural Computing and Applications, 32 (2020), 2243-2256.  doi: 10.1007/s00521-019-04393-5.
    [13] F. MengB. SuE. ThomsonD. Zhou and P. Zhou, Measuring China's regional energy and carbon emission effciency with DEA models: A survey, Appl. Energy, 183 (2016), 1-21.  doi: 10.1016/j.apenergy.2016.08.158.
    [14] S. MidyaS. K. Roy and Vincent F. Yu, Intuitionistic fuzzy multi-stage multi-objective fixed-charge solid transportation problem in a green supply chain, International Journal of Machine Learning and Cybernetics, 12 (2021), 699-717.  doi: 10.1007/s13042-020-01197-1.
    [15] P. Pandian and G. Natrajan, An optimal more-for-less solution to fuzzy transportation problems with mixed constraints, Applied Mathematical Sciences, 4 (2010), 1405-1415. 
    [16] J. C. ParadiS. Rouatt and H. Zhu, Two-stage evaluation of bank branch efficiency using data envelopment analysis, Omega, 39 (2011), 99-109.  doi: 10.1016/j.omega.2010.04.002.
    [17] M. A. Saati, Generalized dealing problems with fuzzy differential costs with the help of DEA, ACECR Journals, 18 (2008), 1-10. 
    [18] J. SadeghiM. Ghiyasi and A. Dehnokhalaji, Resource allcoaction and target setting based on virtual profit improvement, Numerical Algebra, Control and Optimization, 10 (2020), 127-142.  doi: 10.3934/naco.2019043.
    [19] A. SudhakarV. J. N. Arunsankar and T. Karpagam, A new approach for finding an optimal solution for transportation problems, European Journal of Scientific Research, (2020), 254-257. 
    [20] Z. M. Tao and J. P. Xu, A class of rough multiple objective programming and its application to solid transportation problem, Inf. Sci., 188 (2012), 215-235.  doi: 10.1016/j.ins.2011.11.022.
    [21] K. Tone, A slacks-based measure of efficiency in data envelopment analysis, European Journal of Operational Research, 130 (2002), 498-509.  doi: 10.1016/S0377-2217(99)00407-5.
    [22] L. M. Zarafat AngizM. S. Saati and M. Mokhtaran, An alternative approach to assignment problem with non-homogeneous costs using common set of weights in DEA, Far East J. Appl. Math., 10 (2003), 29-39. 
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