اندازه‌گیری کارایی سود کلی استوار با در نظر گرفتن عدم قطعیت در بردارهای قیمت ورودی و خروجی
محورهای موضوعی : آمارمحمدعلی رعایت پناه 1 , نازیلا آقایی 2
1 - گروه علوم ریاضی و کامپیوتر، دانشگاه خوارزمی، تهران، ایران
2 - گروه ریاضی، واحد اردبیل، دانشگاه آزاد اسلامی، اردبیل، ایران
کلید واژه: Data Envelopment Analysis, robust optimization, overall profit efficiency, uncertain data,
چکیده مقاله :
مدل کارایی سود کلی کلاسیک نیاز به اطلاعات دقیق از ورودی‌ها، خروجی‌ها و بردارهای قیمت ورودی و خروجی دارد. در حالی‌که در دنیای واقعی همه داده‌ها بطور دقیق در دسترس نمی‌باشد. در این حالت می‌توان از روش‌های تصادفی یا فازی برای محاسبه کارایی سود کلی استفاده نمود. در محاسبه کارایی سود کلی با این روش‌ها نیاز به اطلاعات بیشتری از داده‌ها از جمله تابع توزیع احتمال یا تابع عضویت داده‌ها می‌باشد، که در بعضی حالت‌ها ممکن است اطلاعات کافی برای تخمین این توابع وجود نداشته باشد و تنها دانش مربوط به پارامترها، تغییر آنها در یک فضای محدب بسته و کراندار است. لذا، در این مقاله با توجه به مدل عدم قطعیت بودجه‌ای در بهینه‌سازی استوار که قابل اعمال به مسایل بهینه‌سازی می‌باشد و نیز آنکه قابلیت تنظیم درجه محافظه کاری را دارد، مدل معادل استوار مساله محاسبه کارایی سود کلی با عدم قطعیت پارامتر بردار قیمت مطرح می‌گردد و سپس همتای استوار مدل برنامه‌ریزی خطی ارائه می‌شود. نتایج عددی نشان میدهند مقدار کارایی سود کلی واحدهای تصمیم گیرنده توسط مدل پیشنهادی در مقایسه با حالت خوش‌بینانه بیشتر است.
The classic overall profit needs precise information of inputs, outputs, inputs and outputs price vectors. In real word, all data are not certain. Therefore, in this case, stochastic and fuzzy methods use for measuring overall profit efficiency. These methods require more information about the data such as probability distribution function or data membership function, which in some cases may not have sufficient information to estimate them, and only we have knowledge about the parameters so that they change in a convex space that is closed and bounded. Therefore, in this paper, we consider a budget uncertainty model in the robust optimization problem that able to adjust the conservative degree. The robust model by the input and output price vectors is proposed to compute overall profit efficiency measure. To illustrate the application of the proposed method, a numerical example is presented and the results show that the robust overall efficiency of the decision making units is higher than the optimistic model.
[1] M. J. Farrell. The measurement of productive efficiency, Journal of the Royal Statistical Society, Series A, General 120 (3), 253–281, (1957).
[2] A. Charnes, W.W. Cooper, and E. Rodes. Measuring the efficiency of decision making units, European Journal of Operational Research, 2 (6), 429–444, (1978).
[3] R.D. Banker, A. Charens, and W.W. Cooper. Some models for estimating technical and scale inefficiencies in Data Envelopment Analysis, Management Science, 30, 1078–1092, (1984).
[4] A. Charnes, W.W. Cooper, B. Golany, L.M. Seiford, and J. Stutz. Foundations of data envelopment analysis and Pareto–Koopmans empirical production functions, Journal of Econometrics, 30, 91–107, (1985).
[5] K. Tone. A slacks-based measure of efficiency in data envelopment analysis, European Journal of Operational Research, 130, 498–509, (2001).
[6] H. Varian. The nonparametric approach to production analysis, Econometrical 54, 579-597.
[7] M. Toloo, N. Aghayi, and M. Rostamy-malkhalifeh, (2008). Measuring overall profit efficiency with interval data, Applied Mathematics and Computation, 201(1):640–649, (1988).
[8] Y-S. Wang, B. Xie, L. F. Shang, and W. H. Li. Measures to improve the performance of China’s thermal power industry in view of cost efficiency, Applied energy, 112, 1078-1086, (2013).
[9] H. Sakai and Y. Takahashi. Ten years after bus deregulation in Japan: An analysis of institutional changes and cost efficiency. Research in Transportation Economics, 39 (1), 215-225, (2013).
[10] N. Aghayi. Cost efficiency measurement with fuzzy data in DEA. Journal of Intelligent and Fuzzy Systems, 32, 409–420, (2017).
[11] A. Emrouznejad, M. Rostamy-Malkhalifeh, A. Hatami-Marbini, M. Tavana, and N. Aghayi. An overall profit Malmquist productivity index with fuzzy and interval data. Mathematical and Computer Modelling, 54 (11-12), 2827-2838, (2011).
[12] M. Rostamy-Malkhalifeh and N. Aghayi. Measuring overall profit efficiency with fuzzy data. Journal of Mathematical Extension, 5 (2), 73-90, (2011).
[13] M. Rostamy-Malkhalifeh and N. Aghayi. Two Ranking of Units on the Overall Profit Efficiency with Interval Data. Mathematics Scientific Journal, 8(2), 73–93, (2011).
[14] G. Cesaroni. Industry cost efficiency in data envelopment analysis. Socio-Economic Planning Sciences, 61, 37-43, (2018).
[15] S. Salehpour and N. Aghayi. The Most Revenue Efficiency with Price Uncertainty. International Journal of Data Envelopment Analysis, 3, 575-592, (2015).
[16] A.L. Soyster. Technical noteconvex programming with set-inclusive constraints and applications to inexact linear programming. Operations research, 21(5):1154–1157, (1973).
[17] A. Ben-Tal and A. Nemirovski. Robust convex optimization. Mathematics of Operations Research, 23, 769-805, (1998).
[18] A. Ben-Tal and A. Nemirovski. Robust solutions to uncertain programs. Operations Research Letters; 25; 1-13, (1999).
[19] A. Ben-Tal and A. Nemirovski. Robust solutions of linear programming problems contaminated with uncertain data. Mathematical Programming, 88, 411-424, (2000).
[20] D. Bertsimas and M. Sim. The price of robustness. Operations research, 52(1),35–53, (2004).
[21] L. El-Ghaoui and H. Lebret. Robust solutions to least-squares problems with uncertain data. SIAM Journal of Matrix Analysis and Applications, 18 (4), 1035-1064, (1997).
[22] L. El-Ghaoui, F. Oustry, and H. Lebret. Robust solutions to uncertain semidefinite programs. SIAM Journal on Optimization, 9, 33-52, (1998).
[23] N. Aghayi, M. Tavana, and M.A. Raayatpanah. Robust efficiency measurement with common set of weights under varying degrees of conservatism and data uncertainty, European Journal Industrial Engineering, 10, 385-405, (2016).
[24] N. Aghayi and B. Maleki. Efficiency Measurement of DMUs with Undesirable outputs under uncertainty based on the directional distance function: Application on Bank Industry. Energy, 112, 376-387, (2016).
[25] M. Asmild, J.C. Paradi, D.N. Reese, and F. Tam. Measuring overall efficiency and effectiveness using DEA, European Journal of Operational Research, 178, 305–321, (2007).
[26] W.W. Cooper, L.M. Seiford, and K. Tone. Data envelopment analysis: a comprehensive text with models, applications, references and DEA-solver software. Springer, (2006).