# American Institute of Mathematical Sciences

doi: 10.3934/naco.2022006
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## A two-stage data envelopment analysis approach to solve extended transportation problem with non-homogenous costs

 1 Department of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran 2 Department of Mathematics, Faculty of Mathematical Science Computer, Kharazmi University, Tehran, Iran

*Corresponding author: Saeid_Mehrabian@khu.ac.ir

Received  March 2021 Revised  February 2022 Early access March 2022

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.

Citation: Ali Hadi, Saeid Mehrabian. A two-stage data envelopment analysis approach to solve extended transportation problem with non-homogenous costs. Numerical Algebra, Control and Optimization, doi: 10.3934/naco.2022006
##### References:
 [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. Chao, M. 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. Charnes, W. 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. Dantzig, Linear 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. Maity, D. Mardanya, S. 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. Maity, S. 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. Meng, B. Su, E. Thomson, D. 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. Midya, S. 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. Paradi, S. 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. Sadeghi, M. 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. Sudhakar, V. 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 Angiz, M. 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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##### References:
 [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. Chao, M. 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. Charnes, W. 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. Dantzig, Linear 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. Maity, D. Mardanya, S. 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. Maity, S. 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. Meng, B. Su, E. Thomson, D. 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. Midya, S. 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. Paradi, S. 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. Sadeghi, M. 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. Sudhakar, V. 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 Angiz, M. 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.
Extended transportation problem
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
 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
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$
 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$
Composite efficiency
 I J K AB $e_{AI}=0.6682 $$e_{BI}=0.7507 e_{AJ}=0.9175$$ e_{BJ}=0.7305$ $e_{AK}=0.7296 $$e_{BK}=0.7507 C e_{CI}=0.9461 e_{CJ}=0.6657 e_{CK}=0.8653 D e_{DI}=1,0000 e_{DJ}=0.7805 e_{DK}=1,0000 E e_{EI}=1,0000 e_{EJ}=0.7963 e_{EK}=1,0000 F e_{FI}=1,0000 e_{FJ}=1,0000 e_{FK}=0.8971 G e_{GI}=0.7281 e_{GJ}=0.8285 e_{GK}=0.8287 H e_{HI}=1,0000 e_{HJ}=1,0000 e_{HK}=0.9597  I J K AB e_{AI}=0.6682$$ 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 $e_{HI}=1,0000$ $e_{HJ}=1,0000$ $e_{HK}=0.9597$
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
 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
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
 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
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