LU Decomposition in DEA with an Application to Hospitals

详细信息    查看全文

• 作者：Mehdi Toloo ; Rahele Jalili
• 关键词：Data envelopment analysis (DEA) ; Simplex algorithm ; LU decomposition ; Computational complexity
• 刊名：Computational Economics
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：47
• 期：3
• 页码：473-488
• 全文大小：713 KB
• 参考文献：Al-Kurdi, A., & Kincaid, R. D. (2006). LU decomposition with iterative refinement for solving sparse linear systems. Journal of Computational and Applied Mathematics, 185(2), 391–403.CrossRef
  Amin, G. R., & Toloo, M. (2004). A polynomial-time algorithm for finding \(\varepsilon \) in DEA models. Computer & Operation Research, 31, 803–805.CrossRef
  Banker, R. D., Charnes, A., & Cooper, W. W. (1984). Models for estimation of technical and scale inefficiencies in data envelopment analysis. Management Science, 30, 1078–1092.CrossRef
  Bartels, R., & Golub, G. (1969). The simplex method of linear programming using LU decomposition. Communications of the ACM, 12, 266–268.CrossRef
  Bazaraa, M. S., Jarvis, J. J., & Sherali, H. D. (2010). Linear programming and network flows (4th ed.). New Jersey: Wiley.
  Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measuring the efficiency of decision-making units. European Journal of Operational Research, 2, 429–444.CrossRef
  Charnes, A., Cooper, W. W., Golany, B., Seiford, L., & Stutz, J. (1985). Foundations of data envelopment analysis for Pareto–Koopmans efficient empirical production functions. Journal of Econometrics, 30, 91–107.CrossRef
  Cooper, W. W., Seiford, L. M., & Tone, K. (2007). Introduction to data envelopment analysis and its uses. New York: Springer.
  Ford, W. (2015). Numerical linear algebra with applications: Using MATLAB. New York: Elsevier.
  Gill, P. E., Murray, W., & Wright, M. H. (1991). Numerical linear algebra and optimization. Redwood City: Addison-Wesley.
  Gill, P. E., Golub, G. H., Murray, W., & Saunders, M. A. (1974). Methods for modifying matrix factorizations. Mathematics of Computation, 28, 505–535.CrossRef
  Gill, P. E., Murray, W., Saunders, M. A., & Wright, M. H. (1987). Maintaining LU factors of a general sparse matrix. Linear Algebra and its Application, 88(89), 239–270.CrossRef
  Hu, J., & Pan, P. (2008). An efficient approach to updating simplex multipliers in the simplex algorithm. Mathematical Programming, 114, 235–248. Springer.CrossRef
  Karska, E. E., & Dursun, M. (2014). An integrated supplier selection methodology incorporating QFD and DEA with imprecise data. Expert Systems with Applications, 41, 6995–7004.CrossRef
  Leleu, H., Moises, J., & Valdmanis, V. G. (2014). How do payer mix and technical inefficiency affect hospital profit? A weighted DEA approach. Operations Research for Health Care, 3(4), 231–237.CrossRef
  Mehrabian, S., Jahanshahloo, G. R., Alirezaee, M. R., & Amin, G. R. (2000). An assurance interval for the non-Archimedean epsilon in DEA models. Operations Research, 48(2), 3444–3347.CrossRef
  Millington, H. K., Lovell, J. E., & Lovell, C. A. K. (2015). A framework for guiding the management of urban stream health. Ecological Economics, 109, 222–233.CrossRef
  Mitropoulos, P., Talias, M. A., & Mitropoulos, I. (2015). Combining stochastic DEA with Bayesian analysis to obtain statistical properties of the efficiency scores: An application to Greek public hospitals. European Journal of Operational Research, 243(1), 302–311.CrossRef
  Mittal, R. C., & Al-Kurdi, A. (2002). LU-decomposition and numerical structure for solving large sparse nonsymmetric linear systems. Computers and Mathematics with Applications, 43, 131–155.
  Ping-Qi, P. (2014). Linear programming computation. Dordrecht: Springer.
  Saunders, M. A. (1976a). The complexity of LU updating in the simplex method. In R. S. Anderssen & R. P. Brent (Eds.), The complexity of computational problem solving (pp. 214–230). Queensland: Queensland University Press.
  Saunders, M. A. (1976b). A fast stable implementation of the simplex method using the Bartels-Golub updating. In J. R. Bunch & D. J. Rose (Eds.), Sparse matrix computations (pp. 213–226). New York, NY: Academic Press.
  Stoer, J., & Bulirsh, R. (2002). Introduction to numerical analysis (3rd ed.). New York: Springer.CrossRef
  Toloo, M., Masoumzadeh, A., & Barat, M. (2015). Finding an initial basic feasible solution for DEA models with an application on bank industry. Computational Economics, 245(2), 323–326.CrossRef
  Tone, K. (2001). A Slacks-based measure of efficiency in data envelopment analysis. European Journal of Operational Research, 130, 498–509.CrossRef
  Wu, S., Wang, C., & Zhang, G. (2015). Has China’s new health care reform improved efficiency at the provincial level? Evidence from a panel data of 31 Chinese provinces. Journal of Asian Public Policy, 8, 36–55. doi:10.​1080/​17516234.​2015.​1009399 .CrossRef
  
• 作者单位：Mehdi Toloo (1)
  Rahele Jalili (2)
  
  1. Faculty of Economics, Technical University of Ostrava, Sokolská tř. 33, Ostrava, 701 21, Czech Republic
  2. Department of Mathematics, Science and Research Branch, Islamic Azad University, Hesarak, Poonak, Tehran, 14778, Iran
  
• 刊物类别：Business and Economics
• 刊物主题：Economics
  Economic Theory
  
• 出版者：Springer Netherlands
• ISSN：1572-9974

文摘

A fundamental problem that usually appears in linear systems is to find a vector \(\mathbf{x}\) satisfying \(\mathbf{Bx}=\mathbf{b}\). This linear system is encountered in many research applications and more importantly, it is required to be solved in many contexts in applied mathematics. LU decomposition method, based on the Gaussian elimination, is particularly well suited for spars and large-scale problems. Linear programming (LP) is a mathematical method to obtain optimal solutions for a linear system that is more being considered in various fields of study in recent decades. The simplex algorithm is one of the mostly used mathematical techniques for solving LP problems. Data envelopment analysis (DEA) is a non-parametric approach based on linear programming to evaluate relative efficiency of decision making units (DMUs). The number of LP models that has to be solved in DEA is at least the same as the number of DMUs. Toloo et al. (Comput Econ 45(2):323–326, 2015) proposed an initial basic feasible solution for DEA models which practically reduces at least 50 % of the whole computations. The main contribution of this paper is in utlizing this solution to implement LU decomposition technique on the basic DEA models which is more accurate and numerically stable. It is shown that the number of computations in applying the Gaussian elimination method will be fairly reduced due to the special structure of basic DEA models. Potential uses are illustrated with applications to hospital data set.