Representation of Double Lie-Groupoid

Subject Areas : Statistics

M. R. Farhangdoost ¹ (Department of Mathematics, College of Sciences, Shiraz University, P.O. Box 71457-44776, Shiraz, Iran.)
S. Merati ² (Department of Mathematics, College of Sciences, Shiraz University, P.O. Box 71457- 44776, Shiraz, Iran)

Received: 2021-02-05 Accepted : 2021-05-22 Published : 2022-06-22

Keywords: کلاف برداری دوگانه, نمایش, جبروار لی مضاعف, گروهوار لی مضاعف,

Abstract :

In this paper we introduce the bi-VB groupoid and representation of double Lie groupoids, using the concept of vector bundle object in the category of Lie groupoids or Lie groupoid object in the category of vector bundle. A bi-VB groupoid is a bi-vector bundle object in the category of Lie groupoids. By a bi-vector bundle, we mean that a manifold by two vector bundle structures over two manifolds. We study some properties of the representation of double Lie groupoid as a cochains and smooth groupoid cohomology. We can show that there exists a one to one corresponding between a representation of a double Lie groupoid and two continues degree one operator which the space of normalized cochains, satisfying graded Leibniz identity and vanished their square. We study some properties and some example of bi-VB groupoids. And then we show that any representation of double Lie groupoids induced a bi-VB groupoid structure on its action groupoid.

References:

[1] Charnes, A., Cooper, W.W., & Rhodes, E., (1978), Measuring the efficiency of decision making units, European journal of operational Research, 2, 429-444.

 [2] Wu, J., Chu, J.F., Sun, J.S., & Zhu, Q.Y., (2016), DEA cross efficiency evaluation based on Pareto improvement, European Journal of Operational Research,248 (2),571-579.

[3] Moeini, M., Karimi, B., and Khorram, E., (2015) Modelling, Computation and Optimization in Information Systems and Management Sciences. Springer International Publishing. 487-498.

[4] Zhou, Z., Lin, L., Xiao, H., Ma, C., Wu, S., (2017), Stochastic network DEA models for two-stage systems under the centralized control organization mechanism, Doi : http://dx.doi.org/10.1016/j.cie.2017.06.005.

[5] Yang, C. C., (2014), An enhanced DEA model for decomposition of technical efficiency in banking, Journal of the Annals of Operations Research,167-185.

[6] Emrouznejad, A., & Yang, G. L., (2018), A survey and analysis of the first 40 years of scholarly literature in DEA, Socio‐Economic Planning Sciences, 61, 4–8.

[7] Liu, S.T., (2018), A DEA ranking method based on cross efficiency intervals and signal-to-noise ratio, Annals of Operations Research, 261(1-2), 207-232.

[8] Sexton, T. R., Silkman, R. H., and Hogan, A. J., (1986), Data envelopment analysis: critique and extensions. In: Silkman R H (Ed.) Measuring Efficiency: An Assessment of data Envelopment Analysis, Jossey-Bass San Francisco, Pp: 73–105.

[9] Anderson, T.R.., Hollings Worth, K.B., & Inman, L. B., (2002), The fixed weighting nature of a cross evaluation model, Journal of Productivity Analysis, 18, 249-255.

[10] Lim, S., and Zhu, J., (2014), DEA cross-efficiency evaluation under variable returns to scale, Journal of the Operational Research Society,1-12.

[11] Cui, Q., & Li, Y., (2015), Evaluating energy efficiency for airlines: an application of VFB-DEA, Journal of Air Transport Management, 44, 34–41.

[12] Doyle, J. R., and Green, R. H., (1994), Efficiency and cross-efficiency in DEA: derivations, meanings and uses, Journal of the Operational Research Society, 45 (5): 567–578.

[13] Cook, W. D., and Zhu, J., (2014), DEA Cobb–Douglas frontier and cross-efficiency, Journal of the Operational Research Society, 65(2): 265–268.

[14] Wu, J., Chu, J.F., Sun, J.S., & Zhu, Q.Y., (2016), DEA cross efficiency evaluation based on Pareto improvement, European Journal of Operational Research,248 (2),571-579.

[15] Liu, S.T., (2018), A DEA ranking method based on cross efficiency intervals and signal-to-noise ratio, Annals of Operations Research, 261(1-2), 207-232.

[16] Kao, C., Liu, S. (2018), Cross efficiency measurement and decomposition in two basic network systems, Omega,1-10.

[17] Wu, J., Chu, J. F., Sun, J. S., Zhu, Q. Y., & Liang, L., (2016), Extended secondary goal models for weights selection in DEA cross-efficiency evaluation, Computers & Industrial Engineering, 93, 143-151.

[18] Ramón, N., Ruiz, J. L., and Sirvent, I., (2010), On the choice of weights profiles in cross efficiency evaluations, European Journal of Operational Research, 207 (3):1564-1572.

[19] Soltanifar, M., Shahghobadi, S., (2013), Selecting a benevolent secondary goal model in data an envelopment analysis cross-efficiency evaluation by a voting, Socio-Economic Planning Sciences,159-167.

[20] Thompson, R. G., Singleton, F. D., Thrall, R. M., and Smith, B. A., (1986), Comparative site evaluations for locating a high-energy physics lab in Texas, Interfaces 16 (6): 35–49.

 [21] Ramón, N., Ruiz, J. L., Sirvent, I., (2011), Reducing differences between profiles of weights: A ‘‘peer-restricted’’ cross-efficiency evaluation, Omega, 39 (6): 634–641.

[22] Jahanshahloo, G. R., Hosseinzadeh Lofti, F., Yafari, and Maddahi, R., (2011), Selecting symmetric weights as a secondary goal in DEA cross efficiency evaluation, Applied Mathematical Modelling, 35(1): 544–549.

[23] Wu, J., Chu, J. F., Sun, J. S., Zhu, Q. Y., & Liang, L., (2016), DEA cross-efficiency evaluation based on satisfaction degree: an application to technology, (2016).

[24] Charles, V., Cornillier, F., (2017), Value of the Stochastic Efficiency in Data Envelopment Analysis, Expert Systems With Applications, doi: 10.1016/j.eswa.2017.03.061.

[25] Cooper, W.W., Honghui, D., Huang, Z. & Li, S.X., (2004), Chance constrained programming approaches to congestion in stochastic data envelopment analysis, European Journal of Operational Research,155 (2), 487-501.

[26] Cooper, W.W., Deng, H., Huang, Z., & Li, S.X., (2002), Chance constrained programming approaches to technical efficiencies and inefficiencies in stochastic data envelopment analysis, Journal of the Operational Research Society, 53, 1347–1356.

[27] Khodabakhshi, M., (2009), Estimating most productive scale size with stochastic data in data envelopment analysis, Economic Modelling, doi: 10. 1016.j. econmod.2009.03.002.

[28] Zhou, Z., Lin, L., Xiao, H., Ma, C., Wu, S., (2017), Stochastic network DEA models for two-stage systems under the centralized control organization mechanism, Doi : http://dx.doi.org/10.1016/j.cie.2017.06.005