A conic relaxation model for searching for the global optimum of network data envelopment analysis |
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| Affiliation: | 1. College of Auditing and Evaluation, Nanjing Audit University, Nanjing, Jiangsu Province, 211815, China;2. Schulich School of Business, York University, Toronto, Ontario M3J 1P3, Canada;3. Foisie Business School, Worcester Polytechnic Institute, Worcester, MA 01609, USA;1. Carey Business School, Johns Hopkins University, 100 International Drive, Baltimore, MD 21202, United States;2. Department of Information Systems and Business Analytics, Florida International University, Miami, FL 33199, United States;1. School of Business, Stevens Institute of Technology, 1 Castle Point Terrace, Hoboken, NJ 07030, USA;2. Lally School of Management, Rensselaer Polytechnic Institute, 110 8th Street, Pittsburgh Building, Troy, NY 12180, USA;3. Division of Economic and Risk Analysis, US Securities and Exchange Commission, 100 F St NE, Washington DC 20549, USA;4. Department of Electrical, Computer & Systems Engineering, Rensselaer Polytechnic Institute, Jonsson Engineering Center 6048, Troy, NY 12180, USA;1. Leiden University Mathematical Institute, Niels Bohrweg 1, 2333 CA, Leiden, NL, UK;2. Department of Management Science, Center for Transportation and Logistics, Lancaster University Management School, Bailrigg, Lancaster LA1 4YX, UK;1. Chair of Logistics and Quantitative Methods, Julius-Maximilians-Universität Würzburg, Sanderring 2, Würzburg 97070, Germany;2. MIT-Zaragoza International Logistics Program, Zaragoza Logistics Center, Edificio Náyade 5, C/ Bari 55 (PLAZA), Zaragoza 50197, Spain;1. School of Business, Pusan National University, Busan, Republic of Korea;2. Department of Information and Communication Engineering, DGIST, Daegu, Republic of Korea;1. Department of Industrial Engineering, University of Houston, 4800 Calhoun Road, Houston, TX 77204, USA;2. PROS Revenue Management, Houston, TX 77002, USA;3. Department of Radiation Physics, The University of Texas MD Anderson Cancer Center, Houston, TX 77030, USA |
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| Abstract: | Network data envelopment analysis (DEA) models the internal structures of decision-making units (DMUs). Unlike the standard DEA model, multiplier-based network DEA models are often highly non-linear and cannot be converted into linear programs. As such, obtaining a non-linear network DEA's global optimal solution is a challenge because it corresponds to a nonconvex optimization problem. In this paper, we introduce a conic relaxation model that searches for the global optimum to the general multiplier-based network DEA model. We reformulate the general network DEA models and relax the new models into second order cone programming (SOCP) problems. In comparison with linear relaxation models, which is potentially applicable to general network DEA structures, the conic relaxation model guarantees applicability in general network DEA, since McCormick envelopes involved are ensured to be finite. Furthermore, the conic relaxation model avoids unnecessary linear relaxations of some nonlinear constraints. It generates, in a more convenient manner, feasible approximations and tighter upper bounds on the global optimal overall efficiency. Compared with a line-parameter search method that has been applied to solve non-linear network DEA models, the conic relaxation model keeps track of the distances between the optimal overall efficiency and its approximations. As a result, it is able to determine whether a qualified approximation has been achieved or not, with the help of a branch and bound algorithm. Hence, our proposed approach can substantially reduce the computations involved. |
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