doi: 10.3934/naco.2022006
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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. 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. 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. 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. 

show all references

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. 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. 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. 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. 

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
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
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 $
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 $
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 $
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 $
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
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
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
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
[1]

Laurent Denis, Anis Matoussi, Jing Zhang. The obstacle problem for quasilinear stochastic PDEs with non-homogeneous operator. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5185-5202. doi: 10.3934/dcds.2015.35.5185

[2]

Corentin Audiard. On the non-homogeneous boundary value problem for Schrödinger equations. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 3861-3884. doi: 10.3934/dcds.2013.33.3861

[3]

Mahdi Mahdiloo, Abdollah Noorizadeh, Reza Farzipoor Saen. Developing a new data envelopment analysis model for customer value analysis. Journal of Industrial and Management Optimization, 2011, 7 (3) : 531-558. doi: 10.3934/jimo.2011.7.531

[4]

Cheng-Kai Hu, Fung-Bao Liu, Hong-Ming Chen, Cheng-Feng Hu. Network data envelopment analysis with fuzzy non-discretionary factors. Journal of Industrial and Management Optimization, 2021, 17 (4) : 1795-1807. doi: 10.3934/jimo.2020046

[5]

Mohammad Afzalinejad, Zahra Abbasi. A slacks-based model for dynamic data envelopment analysis. Journal of Industrial and Management Optimization, 2019, 15 (1) : 275-291. doi: 10.3934/jimo.2018043

[6]

Giuseppe Maria Coclite, Lorenzo di Ruvo. A note on the non-homogeneous initial boundary problem for an Ostrovsky-Hunter type equation. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3357-3389. doi: 10.3934/dcdss.2020236

[7]

Shenghao Li, Min Chen, Bing-Yu Zhang. A non-homogeneous boundary value problem of the sixth order Boussinesq equation in a quarter plane. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2505-2525. doi: 10.3934/dcds.2018104

[8]

Virginia Giorno, Serena Spina. On the return process with refractoriness for a non-homogeneous Ornstein-Uhlenbeck neuronal model. Mathematical Biosciences & Engineering, 2014, 11 (2) : 285-302. doi: 10.3934/mbe.2014.11.285

[9]

Habibe Zare Haghighi, Sajad Adeli, Farhad Hosseinzadeh Lotfi, Gholam Reza Jahanshahloo. Revenue congestion: An application of data envelopment analysis. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1311-1322. doi: 10.3934/jimo.2016.12.1311

[10]

Alain Miranville, Costică Moroşanu. Analysis of an iterative scheme of fractional steps type associated to the nonlinear phase-field equation with non-homogeneous dynamic boundary conditions. Discrete and Continuous Dynamical Systems - S, 2016, 9 (2) : 537-556. doi: 10.3934/dcdss.2016011

[11]

Shitao Liu, Roberto Triggiani. Recovering damping and potential coefficients for an inverse non-homogeneous second-order hyperbolic problem via a localized Neumann boundary trace. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5217-5252. doi: 10.3934/dcds.2013.33.5217

[12]

Pooja Bansal, Aparna Mehra. Integrated dynamic interval data envelopment analysis in the presence of integer and negative data. Journal of Industrial and Management Optimization, 2022, 18 (2) : 1339-1363. doi: 10.3934/jimo.2021023

[13]

Narges Torabi Golsefid, Maziar Salahi. Second order cone programming formulation of the fixed cost allocation in DEA based on Nash bargaining game. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021032

[14]

Imam Wijaya, Hirofumi Notsu. Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1197-1212. doi: 10.3934/dcdss.2020234

[15]

Demetris Hadjiloucas. Stochastic matrix-valued cocycles and non-homogeneous Markov chains. Discrete and Continuous Dynamical Systems, 2007, 17 (4) : 731-738. doi: 10.3934/dcds.2007.17.731

[16]

Christine Chambers, Nassif Ghoussoub. Deformation from symmetry and multiplicity of solutions in non-homogeneous problems. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 267-281. doi: 10.3934/dcds.2002.8.267

[17]

Aníbal Rodríguez-Bernal, Robert Willie. Singular large diffusivity and spatial homogenization in a non homogeneous linear parabolic problem. Discrete and Continuous Dynamical Systems - B, 2005, 5 (2) : 385-410. doi: 10.3934/dcdsb.2005.5.385

[18]

Cheng-Kai Hu, Fung-Bao Liu, Cheng-Feng Hu. Efficiency measures in fuzzy data envelopment analysis with common weights. Journal of Industrial and Management Optimization, 2017, 13 (1) : 237-249. doi: 10.3934/jimo.2016014

[19]

Hasan Hosseini-Nasab, Vahid Ettehadi. Development of opened-network data envelopment analysis models under uncertainty. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022027

[20]

Pooja Bansal. Sequential Malmquist-Luenberger productivity index for interval data envelopment analysis. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022058

 Impact Factor: 

Article outline

Figures and Tables

[Back to Top]