A slacks-based model for dynamic data envelopment analysis

Previous Article
Pricing options on investment project expansions under commodity price uncertainty
JIMO Home
This Issue
Next Article
Pricing vulnerable options under a Markov-modulated jump-diffusion model with fire sales

January 2019, 15(1): 275-291. doi: 10.3934/jimo.2018043

A slacks-based model for dynamic data envelopment analysis

Mohammad Afzalinejad ^, and Zahra Abbasi

Department of Mathematics, Tafresh University, Tafresh, 3951879611, Iran

* Corresponding author: Mohammad Afzalinejad

Received May 2017 Revised October 2017 Published April 2018

Dynamic Data Envelopment Analysis (DDEA) deals with efficiency analysis of decision making units in time dependent situations. A finite number of time periods and some carry-over activities between each two consecutive periods are assumed in DDEA. There are many models in DEA for efficiency evaluation of decision making units over time periods. One important class of dynamic models is the class of slacks-based models. By using a numerical example we show that some slacks-based DDEA models, especially ones proposed by Tone and Tsutsui, suffer from efficiency overestimation. A new dynamic slacks-based DEA model is proposed to overcome the deficiencies of the available slacks-based models. The model proposed in this paper is capable of revealing all sources of inefficiencies and providing more discrimination between decision making units. The theoretical and practical examinations demonstrate the merits of the new model.

Keywords: Data envelopment analysis, dynamic efficiency, DDEA, slacks-based model, carry-over activity.

Mathematics Subject Classification: Primary: 90C05, 90B50; Secondary: 91B06.

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

References:

[1]	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. Google Scholar
[2]	W. W. Cooper, L. M. Seiford and K. Tone, Data Envelopment Analysis: A Comprehensive Text with Models, Applications, References and DEA-solver Software, 2^nd edition, Springer-Verlag, New York, 2007. doi: 10.1007/978-0-387-45283-8. Google Scholar
[3]	D. K. Despotis, D. Sotiros and G. Koronakos, A network DEA approach for series multi-stage processes, Omega, 61 (2016), 35-48. doi: 10.1016/j.omega.2015.07.005. Google Scholar
[4]	A. Emrouznejad and E. Thanassoulis, A mathematical model for dynamic efficiency using data envelopment analysis, Applied Mathematics and Computation, 160 (2005), 363-378. doi: 10.1016/j.amc.2003.09.026. Google Scholar
[5]	R. Färe, S. Grosskopf, C. A. K. Lovell and C. Pasurka, Multilateral productivity comparisons when some outputs are undesirable: A nonparametric approach, The Review of Economics and Statistics, 71 (1989), 90-98. doi: 10.2307/1928055. Google Scholar
[6]	R. Färe and S. Grosskopf, Intertemporal Production Frontiers: With Dynamic DEA, Springer-Verlag, Netherlands, 1996. doi: 10.1007/978-94-009-1816-0. Google Scholar
[7]	R. Färe, S. Grosskopf, M. Norris and Z. Zhang, Productivity growth, technical progress, and efficiency change in industrialized countries, The American Economic Review, 84 (1994), 66-83. Google Scholar
[8]	H. Fukuyama and W. L. Weber, Measuring Japanese bank performance: A dynamic network DEA approach, Journal of Productivity Analysis, 44 (2015), 249-264. doi: 10.1007/s11123-014-0403-1. Google Scholar
[9]	S. Hung, D. He and W. Lu, Evaluating the dynamic performances of business groups from the carry-over perspective: A case study of Taiwan's semiconductor industry, Omega, 46 (2014), 1-10. doi: 10.1016/j.omega.2014.01.003. Google Scholar
[10]	C. Kao, Dynamic data envelopment analysis: A relational analysis, European Journal of Operational Research, 227 (2013), 325-330. doi: 10.1016/j.ejor.2012.12.012. Google Scholar
[11]	C. Kao and S. Hwang, Efficiency decomposition in two-stage data envelopment analysis: An application to non-life insurance companies in Taiwan, European Journal of Operational Research, 185 (2008), 418-429. doi: 10.1016/j.ejor.2006.11.041. Google Scholar
[12]	G. A. Klopp, The Analysis of the Efficiency of Production System with Multiple Inputs and Outputs, Ph. D thesis, Industrial and System Engineering College, University of Illinois in Chicago, 1985.Google Scholar
[13]	L. Liang, W. D. Cook and J. Zhu, DEA models for two-stage processes: Game approach and efficiency decomposition, Naval Research Logistics, 55 (2008), 643-653. doi: 10.1002/nav.20308. Google Scholar
[14]	P. Moreno and S. Lozano, Super SBI Dynamic Network DEA approach to measuring efficiency in the provision of public services, International Transactions in Operational Research, 25 (2018), 715-735. doi: 10.1111/itor.12257. Google Scholar
[15]	J. Nemoto and M. Goto, Dynamic data envelopment analysis: modeling intertemporal behavior of a firm in the presence of productive inefficiencies, Economic Letters, 64 (1999), 51-56. doi: 10.1016/S0165-1765(99)00070-1. Google Scholar
[16]	J. Nemoto and M. Goto, Measuring dynamic efficiency in production: an application of data envelopment analysis to Japanese electric utilities, Journal of Productivity Analysis, 19 (2003), 191-210. doi: 10.1023/A:1022805500570. Google Scholar
[17]	H. Omrani and E. Soltanzadeh, Dynamic DEA models with network structure: An application for Iranian airlines, Journal of Air Transport Management, 57 (2016), 52-61. doi: 10.1016/j.jairtraman.2016.07.014. Google Scholar
[18]	K. S. Park and K. Park, Measurement of multiperiod aggregative efficiency, European Journal of Operational Research, 193 (2009), 567-580. doi: 10.1016/j.ejor.2007.11.028. Google Scholar
[19]	H. Scheel, Undesirable outputs in efficiency valuations, European Journal of Operational Research, 132 (2001), 400-410. doi: 10.1016/S0377-2217(00)00160-0. Google Scholar
[20]	J. K. Sengupta, A dynamic efficiency model using data envelopment analysis, International Journal of Production Economics, 62 (1999), 209-218. doi: 10.1016/S0925-5273(98)00244-8. Google Scholar
[21]	M. Shafiee, M. Sangi and M. Ghaderi, Bank performance evaluation using dynamic DEA: A slacks-based measure approach, Journal of Data Envelopment Analysis and Decision Science, 2013 (2013), 1-12. doi: 10.5899/2013/dea-00026. Google Scholar
[22]	M. Soleimani-damaneh, An effective computational attempt in DDEA, Applied Mathematical Modelling, 33 (2009), 3943-3948. doi: 10.1016/j.apm.2009.01.013. Google Scholar
[23]	T. Sueyoshi and K. Sekitani, Returns to scale in dynamic DEA, European Journal of Operational Research, 161 (2005), 536-544. doi: 10.1016/j.ejor.2003.08.055. Google Scholar
[24]	K. Tone, A slacks-based measure of efficiency in data envelopment analysis, European Journal of Operational Research, 130 (2001), 498-509. doi: 10.1016/S0377-2217(99)00407-5. Google Scholar
[25]	K. Tone and M. Tsutsui, Dynamic DEA: A slacks-based measure approach, Omega, 38 (2010), 145-156. doi: 10.1016/j.omega.2009.07.003. Google Scholar
[26]	K. Tone and M. Tsutsui, Dynamic DEA with network structure: A slacks-based measure approach, Omega, 42 (2014), 124-131. doi: 10.1016/j.omega.2013.04.002. Google Scholar

show all references

References:

[1]	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. Google Scholar
[2]	W. W. Cooper, L. M. Seiford and K. Tone, Data Envelopment Analysis: A Comprehensive Text with Models, Applications, References and DEA-solver Software, 2^nd edition, Springer-Verlag, New York, 2007. doi: 10.1007/978-0-387-45283-8. Google Scholar
[3]	D. K. Despotis, D. Sotiros and G. Koronakos, A network DEA approach for series multi-stage processes, Omega, 61 (2016), 35-48. doi: 10.1016/j.omega.2015.07.005. Google Scholar
[4]	A. Emrouznejad and E. Thanassoulis, A mathematical model for dynamic efficiency using data envelopment analysis, Applied Mathematics and Computation, 160 (2005), 363-378. doi: 10.1016/j.amc.2003.09.026. Google Scholar
[5]	R. Färe, S. Grosskopf, C. A. K. Lovell and C. Pasurka, Multilateral productivity comparisons when some outputs are undesirable: A nonparametric approach, The Review of Economics and Statistics, 71 (1989), 90-98. doi: 10.2307/1928055. Google Scholar
[6]	R. Färe and S. Grosskopf, Intertemporal Production Frontiers: With Dynamic DEA, Springer-Verlag, Netherlands, 1996. doi: 10.1007/978-94-009-1816-0. Google Scholar
[7]	R. Färe, S. Grosskopf, M. Norris and Z. Zhang, Productivity growth, technical progress, and efficiency change in industrialized countries, The American Economic Review, 84 (1994), 66-83. Google Scholar
[8]	H. Fukuyama and W. L. Weber, Measuring Japanese bank performance: A dynamic network DEA approach, Journal of Productivity Analysis, 44 (2015), 249-264. doi: 10.1007/s11123-014-0403-1. Google Scholar
[9]	S. Hung, D. He and W. Lu, Evaluating the dynamic performances of business groups from the carry-over perspective: A case study of Taiwan's semiconductor industry, Omega, 46 (2014), 1-10. doi: 10.1016/j.omega.2014.01.003. Google Scholar
[10]	C. Kao, Dynamic data envelopment analysis: A relational analysis, European Journal of Operational Research, 227 (2013), 325-330. doi: 10.1016/j.ejor.2012.12.012. Google Scholar
[11]	C. Kao and S. Hwang, Efficiency decomposition in two-stage data envelopment analysis: An application to non-life insurance companies in Taiwan, European Journal of Operational Research, 185 (2008), 418-429. doi: 10.1016/j.ejor.2006.11.041. Google Scholar
[12]	G. A. Klopp, The Analysis of the Efficiency of Production System with Multiple Inputs and Outputs, Ph. D thesis, Industrial and System Engineering College, University of Illinois in Chicago, 1985.Google Scholar
[13]	L. Liang, W. D. Cook and J. Zhu, DEA models for two-stage processes: Game approach and efficiency decomposition, Naval Research Logistics, 55 (2008), 643-653. doi: 10.1002/nav.20308. Google Scholar
[14]	P. Moreno and S. Lozano, Super SBI Dynamic Network DEA approach to measuring efficiency in the provision of public services, International Transactions in Operational Research, 25 (2018), 715-735. doi: 10.1111/itor.12257. Google Scholar
[15]	J. Nemoto and M. Goto, Dynamic data envelopment analysis: modeling intertemporal behavior of a firm in the presence of productive inefficiencies, Economic Letters, 64 (1999), 51-56. doi: 10.1016/S0165-1765(99)00070-1. Google Scholar
[16]	J. Nemoto and M. Goto, Measuring dynamic efficiency in production: an application of data envelopment analysis to Japanese electric utilities, Journal of Productivity Analysis, 19 (2003), 191-210. doi: 10.1023/A:1022805500570. Google Scholar
[17]	H. Omrani and E. Soltanzadeh, Dynamic DEA models with network structure: An application for Iranian airlines, Journal of Air Transport Management, 57 (2016), 52-61. doi: 10.1016/j.jairtraman.2016.07.014. Google Scholar
[18]	K. S. Park and K. Park, Measurement of multiperiod aggregative efficiency, European Journal of Operational Research, 193 (2009), 567-580. doi: 10.1016/j.ejor.2007.11.028. Google Scholar
[19]	H. Scheel, Undesirable outputs in efficiency valuations, European Journal of Operational Research, 132 (2001), 400-410. doi: 10.1016/S0377-2217(00)00160-0. Google Scholar
[20]	J. K. Sengupta, A dynamic efficiency model using data envelopment analysis, International Journal of Production Economics, 62 (1999), 209-218. doi: 10.1016/S0925-5273(98)00244-8. Google Scholar
[21]	M. Shafiee, M. Sangi and M. Ghaderi, Bank performance evaluation using dynamic DEA: A slacks-based measure approach, Journal of Data Envelopment Analysis and Decision Science, 2013 (2013), 1-12. doi: 10.5899/2013/dea-00026. Google Scholar
[22]	M. Soleimani-damaneh, An effective computational attempt in DDEA, Applied Mathematical Modelling, 33 (2009), 3943-3948. doi: 10.1016/j.apm.2009.01.013. Google Scholar
[23]	T. Sueyoshi and K. Sekitani, Returns to scale in dynamic DEA, European Journal of Operational Research, 161 (2005), 536-544. doi: 10.1016/j.ejor.2003.08.055. Google Scholar
[24]	K. Tone, A slacks-based measure of efficiency in data envelopment analysis, European Journal of Operational Research, 130 (2001), 498-509. doi: 10.1016/S0377-2217(99)00407-5. Google Scholar
[25]	K. Tone and M. Tsutsui, Dynamic DEA: A slacks-based measure approach, Omega, 38 (2010), 145-156. doi: 10.1016/j.omega.2009.07.003. Google Scholar
[26]	K. Tone and M. Tsutsui, Dynamic DEA with network structure: A slacks-based measure approach, Omega, 42 (2014), 124-131. doi: 10.1016/j.omega.2013.04.002. Google Scholar

Figure 1. $DMU_{b}$ is dominated by $DMU_{a}$ and is not efficient.

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2. Comparison of rank order (input-oriented).

Figure Options

Download full-size image

Download as PowerPoint slide

Table 1. Data of 10 bank branches

DMUs	Average monthly salaries			Operating expense			Total loans			Net profit			Loan Losses
DMUs	t=1	t=2	t=3	t=1	t=2	t=3	t=1	t=2	t=3	t=1	t=2	t=3	t=1	t=2	t=3
DMU1	2.828	2.705	3.775	27.55	35.25	50.43	40.01	49.85	54.38	57.95	58.85	66.64	12.41	7.88	7.4
DMU2	5.667	5.825	7.657	84.5	122	105.5	282.9	297.6	322.5	94.18	87.29	111.6	41.34	34.95	28.64
DMU3	6.23	6.32	8.899	183.6	159.5	170.8	184.5	191.4	188.4	103.7	120.5	121.6	28.44	22.71	21.41
DMU4	5.577	5.532	7.552	122.7	94.48	94.97	195.9	200.5	202.2	58.98	58.42	58.25	22.8	25.68	26.69
DMU5	3.864	4.526	5.72	57.19	38.43	40.27	106.2	102.9	98.36	32.41	42.5	48.91	8.51	6.25	8.93
DMU6	4.696	4.601	6.196	72.07	2.64	3.41	175.5	176.1	190.7	60.7	58.88	47.68	10.35	11.89	10.22
DMU7	3.582	3.108	4.221	21.83	21.3	29.76	21.56	24.38	28.28	18.68	19.17	19.42	1.91	1.24	2.02
DMU8	5.395	5.522	7.139	63.85	56.14	49	133	147.1	156.8	76.77	99.79	100.9	30.49	21.06	18.07
DMU9	7.761	7.522	10.746	27.93	34.4	31.14	872.9	815.4	803.3	314.7	312.8	31.21	80.96	119.5	115.5
DMU10	3.748	3.593	5.138	59.99	96.5	60.43	113.7	121.6	122.9	72.64	84.51	81.45	7.33	3.28	13.53

Table Options

Download as excel

DMUs	Average monthly salaries			Operating expense			Total loans			Net profit			Loan Losses
DMUs	t=1	t=2	t=3	t=1	t=2	t=3	t=1	t=2	t=3	t=1	t=2	t=3	t=1	t=2	t=3
DMU1	2.828	2.705	3.775	27.55	35.25	50.43	40.01	49.85	54.38	57.95	58.85	66.64	12.41	7.88	7.4
DMU2	5.667	5.825	7.657	84.5	122	105.5	282.9	297.6	322.5	94.18	87.29	111.6	41.34	34.95	28.64
DMU3	6.23	6.32	8.899	183.6	159.5	170.8	184.5	191.4	188.4	103.7	120.5	121.6	28.44	22.71	21.41
DMU4	5.577	5.532	7.552	122.7	94.48	94.97	195.9	200.5	202.2	58.98	58.42	58.25	22.8	25.68	26.69
DMU5	3.864	4.526	5.72	57.19	38.43	40.27	106.2	102.9	98.36	32.41	42.5	48.91	8.51	6.25	8.93
DMU6	4.696	4.601	6.196	72.07	2.64	3.41	175.5	176.1	190.7	60.7	58.88	47.68	10.35	11.89	10.22
DMU7	3.582	3.108	4.221	21.83	21.3	29.76	21.56	24.38	28.28	18.68	19.17	19.42	1.91	1.24	2.02
DMU8	5.395	5.522	7.139	63.85	56.14	49	133	147.1	156.8	76.77	99.79	100.9	30.49	21.06	18.07
DMU9	7.761	7.522	10.746	27.93	34.4	31.14	872.9	815.4	803.3	314.7	312.8	31.21	80.96	119.5	115.5
DMU10	3.748	3.593	5.138	59.99	96.5	60.43	113.7	121.6	122.9	72.64	84.51	81.45	7.33	3.28	13.53

Table 2. Comparison of the efficiency scores resulted from Tone and Tsutsui's model and the proposed model.

	Overall input-oriented efficiency		Overall output-oriented efficiency			Non-oriented combined efficiency
DMUs	$ \theta^*_{o}(\text{TT})$	$ \theta^{*}_{o}$	$ \tau^{*}_{o}(\text{TT})$	$ \varphi^{*}_{o}$	Model (16) with TT objective: $ \varphi^{*}_{o}(\text{TT})$	$ \theta^{}_{o}(\text{TT})\times\tau^{}_{o}(TT)$	$ \theta^{}_{o}\times\varphi^{}_{o}$
DMU1	0.9194	0.8792	0.6771	0.7587	0.6771	0.6225	0.6671
DMU2	1	0.7040	1	0.8458	0.7492	1	0.5954
DMU3	0.6521	0.4735	0.7618	0.7959	0.7230	0.4968	0.3761
DMU4	0.5133	0.6773	0.5840	0.7117	0.5456	0.2998	0.4821
DMU5	0.7648	0.7614	0.7916	0.8100	0.7286	0.6054	0.6168
DMU6	1	1	1	1	1	1	1
DMU7	0.8854	0.6421	0.9409	0.8979	0.8700	0.8331	0.5766
DMU8	0.7765	0.7020	0.7482	0.7766	0.6841	0.5810	0.5451
DMU9	1	1	1	1	1	1	1
DMU10	1	0.5660	0.1	0.6979	0.9977	1	0.3950

Table Options

Download as excel

	Overall input-oriented efficiency		Overall output-oriented efficiency			Non-oriented combined efficiency
DMUs	$ \theta^*_{o}(\text{TT})$	$ \theta^{*}_{o}$	$ \tau^{*}_{o}(\text{TT})$	$ \varphi^{*}_{o}$	Model (16) with TT objective: $ \varphi^{*}_{o}(\text{TT})$	$ \theta^{}_{o}(\text{TT})\times\tau^{}_{o}(TT)$	$ \theta^{}_{o}\times\varphi^{}_{o}$
DMU1	0.9194	0.8792	0.6771	0.7587	0.6771	0.6225	0.6671
DMU2	1	0.7040	1	0.8458	0.7492	1	0.5954
DMU3	0.6521	0.4735	0.7618	0.7959	0.7230	0.4968	0.3761
DMU4	0.5133	0.6773	0.5840	0.7117	0.5456	0.2998	0.4821
DMU5	0.7648	0.7614	0.7916	0.8100	0.7286	0.6054	0.6168
DMU6	1	1	1	1	1	1	1
DMU7	0.8854	0.6421	0.9409	0.8979	0.8700	0.8331	0.5766
DMU8	0.7765	0.7020	0.7482	0.7766	0.6841	0.5810	0.5451
DMU9	1	1	1	1	1	1	1
DMU10	1	0.5660	0.1	0.6979	0.9977	1	0.3950

Table 3. The input-oriented period efficiency scores resulted from the TT model and the proposed model

DMUs	Tone and Tsutsui's model			The proposed model
DMUs	Period 1 efficiency	Period 2 efficiency	Period 3 efficiency	Period 1 efficiency	Period 2 efficiency	Period 3 efficiency
DMU1	0.7574	1	1	0.6364	1	1
DMU2	1	1	1	0.2769	0.8169	1
DMU3	0.5217	0.6987	0.7450	0.1962	0.4393	0.7247
DMU4	0.3887	0.6334	0.5324	0.1802	0.8321	1
DMU5	0.7047	0.8411	0.7477	0.5053	0.7744	1
DMU6	1	1	1	1	1	1
DMU7	0.6562	1	1	0.4236	0.5221	0.9991
DMU8	0.4524	0.8718	1	0.2319	0.8498	1
DMU9	1	1	1	1	1	1
DMU10	1	1	1	0.2602	0.8350	0.5901

Table Options

Download as excel

DMUs	Tone and Tsutsui's model			The proposed model
DMUs	Period 1 efficiency	Period 2 efficiency	Period 3 efficiency	Period 1 efficiency	Period 2 efficiency	Period 3 efficiency
DMU1	0.7574	1	1	0.6364	1	1
DMU2	1	1	1	0.2769	0.8169	1
DMU3	0.5217	0.6987	0.7450	0.1962	0.4393	0.7247
DMU4	0.3887	0.6334	0.5324	0.1802	0.8321	1
DMU5	0.7047	0.8411	0.7477	0.5053	0.7744	1
DMU6	1	1	1	1	1	1
DMU7	0.6562	1	1	0.4236	0.5221	0.9991
DMU8	0.4524	0.8718	1	0.2319	0.8498	1
DMU9	1	1	1	1	1	1
DMU10	1	1	1	0.2602	0.8350	0.5901

[1]	Cheng-Kai Hu, Fung-Bao Liu, Cheng-Feng Hu. Efficiency measures in fuzzy data envelopment analysis with common weights. Journal of Industrial & Management Optimization, 2017, 13 (1) : 237-249. doi: 10.3934/jimo.2016014
[2]	Dilek Günneç, Ezgi Demir. Fair-fixture: Minimizing carry-over effects in football leagues. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1565-1577. doi: 10.3934/jimo.2018110
[3]	Mahdi Mahdiloo, Abdollah Noorizadeh, Reza Farzipoor Saen. Developing a new data envelopment analysis model for customer value analysis. Journal of Industrial & Management Optimization, 2011, 7 (3) : 531-558. doi: 10.3934/jimo.2011.7.531
[4]	Habibe Zare Haghighi, Sajad Adeli, Farhad Hosseinzadeh Lotfi, Gholam Reza Jahanshahloo. Revenue congestion: An application of data envelopment analysis. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1311-1322. doi: 10.3934/jimo.2016.12.1311
[5]	Saber Saati, Adel Hatami-Marbini, Per J. Agrell, Madjid Tavana. A common set of weight approach using an ideal decision making unit in data envelopment analysis. Journal of Industrial & Management Optimization, 2012, 8 (3) : 623-637. doi: 10.3934/jimo.2012.8.623
[6]	Zheng Dai, I.G. Rosen, Chuming Wang, Nancy Barnett, Susan E. Luczak. Using drinking data and pharmacokinetic modeling to calibrate transport model and blind deconvolution based data analysis software for transdermal alcohol biosensors. Mathematical Biosciences & Engineering, 2016, 13 (5) : 911-934. doi: 10.3934/mbe.2016023
[7]	Angela Cadena, Adriana Marcucci, Juan F. Pérez, Hernando Durán, Hernando Mutis, Camilo Taútiva, Fernando Palacios. Efficiency analysis in electricity transmission utilities. Journal of Industrial & Management Optimization, 2009, 5 (2) : 253-274. doi: 10.3934/jimo.2009.5.253
[8]	Jingmei Zhou, Xiangmo Zhao, Xin Cheng, Zhigang Xu. Visualization analysis of traffic congestion based on floating car data. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1423-1433. doi: 10.3934/dcdss.2015.8.1423
[9]	Matthew O. Williams, Clarence W. Rowley, Ioannis G. Kevrekidis. A kernel-based method for data-driven koopman spectral analysis. Journal of Computational Dynamics, 2015, 2 (2) : 247-265. doi: 10.3934/jcd.2015005
[10]	Xinxin Tan, Shujuan Li, Sisi Liu, Zhiwei Zhao, Lisa Huang, Jiatai Gang. Dynamic simulation of a SEIQR-V epidemic model based on cellular automata. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 327-337. doi: 10.3934/naco.2015.5.327
[11]	Shi'an Wang, N. U. Ahmed. Optimum management of the network of city bus routes based on a stochastic dynamic model. Journal of Industrial & Management Optimization, 2019, 15 (2) : 619-631. doi: 10.3934/jimo.2018061
[12]	Wu Chanti, Qiu Youzhen. A nonlinear empirical analysis on influence factor of circulation efficiency. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 929-940. doi: 10.3934/dcdss.2019062
[13]	Jian Xiong, Yingwu Chen, Zhongbao Zhou. Resilience analysis for project scheduling with renewable resource constraint and uncertain activity durations. Journal of Industrial & Management Optimization, 2016, 12 (2) : 719-737. doi: 10.3934/jimo.2016.12.719
[14]	Konstantina Trivisa. Global existence and asymptotic analysis of solutions to a model for the dynamic combustion of compressible fluids. Conference Publications, 2003, 2003 (Special) : 852-863. doi: 10.3934/proc.2003.2003.852
[15]	Tyson Loudon, Stephen Pankavich. Mathematical analysis and dynamic active subspaces for a long term model of HIV. Mathematical Biosciences & Engineering, 2017, 14 (3) : 709-733. doi: 10.3934/mbe.2017040
[16]	Yan Zhang, Wanbiao Ma, Hai Yan, Yasuhiro Takeuchi. A dynamic model describing heterotrophic culture of chorella and its stability analysis. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1117-1133. doi: 10.3934/mbe.2011.8.1117
[17]	Sho Nanao, Hiroyuki Masuyama, Shoji Kasahara, Yutaka Takahashi. Queueing analysis of data block synchronization mechanism in peer-to-peer based video streaming system. Journal of Industrial & Management Optimization, 2011, 7 (3) : 699-716. doi: 10.3934/jimo.2011.7.699
[18]	Marie Levakova. Effect of spontaneous activity on stimulus detection in a simple neuronal model. Mathematical Biosciences & Engineering, 2016, 13 (3) : 551-568. doi: 10.3934/mbe.2016007
[19]	S. M. Crook, M. Dur-e-Ahmad, S. M. Baer. A model of activity-dependent changes in dendritic spine density and spine structure. Mathematical Biosciences & Engineering, 2007, 4 (4) : 617-631. doi: 10.3934/mbe.2007.4.617
[20]	Holly Gaff. Preliminary analysis of an agent-based model for a tick-borne disease. Mathematical Biosciences & Engineering, 2011, 8 (2) : 463-473. doi: 10.3934/mbe.2011.8.463

2018 Impact Factor: 1.025

Tools

Metrics

Tools

Article outline

Figures and Tables

2018 Impact Factor: 1.025

Tools

Figures and Tables

2018 Impact Factor: 1.025

American Institute of Mathematical Sciences