Optimal Allocation of Resources Using the Ideal-Solutions

Document Type : research paper


1 Department of Mathematics, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Isfahan, Iran.

2 Department of Mathematics, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Isfahan, Iran


This paper proposes a new method based on the ideal input vector to estimate inputs of a given decision making unit (DMU) when some or all of its outputs are increased to maintain its current efficiency level. In other words, this paper studied the following question: How much would be the increase in the inputs of the DMU if the decision maker increases certain outputs to a particular unit in which the DMU maintains its current efficiency level? In this study, unlike other proposed methods, the above question was addressed using just the single-objective linear programming (LP) problems. The problem of estimation of inputs was investigated based on the non-radial models. Necessary and sufficient conditions are proposed for estimation of inputs using just the single-objective LP problems. In addition, the level of deficiency (if any exists) in each of the output components is specified. An example with real data is presented to illustrate our proposed method.


Article Title [فارسی]

تخصیص بهینه منابع با بکارگیری جوابهای ایده آل

Authors [فارسی]

  • سعید قبادی 1
  • سعید جهانگیری 2
1 گروه ریاضی، واحد خمینی شهر، دانشگاه آزاد اسلامی، خمینی شهر، اصفهان، ایران
2 گروه ریاضی، واحد خمینی شهر، ، دانشگاه آزاد اسلامی، خمینی شهر، اصفهان، ایران
Abstract [فارسی]

این مقاله یک روش جدید بر اساس بردار ورودی ایده آل برای تخمین ورودی ها تحت حفظ اندازه کارایی از یک واحد تصمیم گیرنده وقتی که برخی یا همه خروجی های آن افزایش یافته است، پیشنهاد می دهد. بعبارت دیگر، این مقاله سوال زیر را مطالعه کرده است: تحت حفظ کارایی، به چه میزانی می بایستی ورودی های یک واحد تصمیم گیرنده افزایش یابد در شرایطی که برخی یا همه خروجی های آن افزایش داده شده باشد؟ در روش ارایه شده در این مقاله، برخلاف روش های پیشنهاد شده دیگر، سوال فوق فقط بر پایه مسایل برنامه ریزی خطی تک هدفی پاسخ داده شده است. مساله تخمین ورودی ها بر پایه مدل غیر شعاعی راسل پیشرفته مورد بررسی قرار گرفته است. شرایط لازم و کافی برای تخمین ورودی ها بر پایه برنامه ریزی خطی پیشنهاد گردیده است. بعلاوه، اگر کمبودی در هر یک از مولفه های خروجی واحد تصمیم گیرنده وجود داشته باشد شناسایی می شود. یک مثال با داده های واقعی برای توضیح از روش پیشنهادی ارایه شده است.

Keywords [فارسی]

  • تحلیل پوششی داده های معکوس(DEA)
  • برنامه ریزی خطی(LP)
  • تخصیص منابع
  • اندازه راسل پیشرفته(ERM)
  • جوابهای ایده آل
[1]   Charnes, A., Cooper, W. W., and Rhodes, E., Measuring the efficiency of decision making units, European Journal of Operational Research, 2 (1978) 429-444.
[2]   Banker, R. D., Charnes, A., and Cooper, W. W., Some models for estimating technical and scale efficiencies in data envelopment analysis, Management Science, 30 (1984) 1078-1092.
[3]   Cook W. D. and Seiford, L. M., Data envelopment analysis (DEA)-Thirty years on, European Journal of Operational Research, 192 (2009) 1-17.
[4]   Cooper, W. W., Seiford, L. M., and Tone, K., Data Envelopment Analysis: A Comprehensive Text With Models, Applications, References and DEA-Solver Software, Kluwer Academic Publisher (1999).
[5]   Hatami-Marbini, A., Emrouznejad, A., and Tavana, M., A taxonomy and review of the fuzzy data envelopment analysis literature: Two decades in the making, European Journal of Operational Research, 214 (2011) 457-472.
[6]  Zhang, X. S. and Cui, J. C. A project evaluation system in the state economic information system of china an operations research practice in public sectors, International Transactions in Operational Research, 6 (1999) 441-452.
[7]   Hadi-Vencheh, A. and Foroughi, A. A., A generalized DEA model for inputs/outputs estimation, Mathematical and Computer Modelling, 43 (2006) 447-457.
[8]  Hadi-Vencheh, A. and Foroughi, A. A., and Soleimani-damaneh, M., A DEA model for resource allocation, Economic Modelling, 25 (5) (2008) 983-993.
[9]   Jahanshahloo, G. R., Hosseinzadeh Lotfi, F., Shoja, N., Tohidi, G., and Razavyan, S., Input estimation and identification of extera inputs in inverse DEA models, Applied Mathematice and Computation, 156 (2004) 427-437.
[10]   Wei, Q. L., Zhang, J., and Zhang, X., An inverse DEA model for input/output estimate, European Journal of Operational Research, 121 (1) (2000) 151-163.
[11]   Yan, H., Wei, Q. L., and Hao, G., DEA models for resource reallocation and production input/output estimation, European Journal of Operational Research, 136 (2002) 19-31.
[12]  Jahanshahloo, G. R., Soleimani-damaneh, M., and ghobadi, S., Inverse DEA under inter-temporal dependence using multiple-objective programming, European Journal of Operational Research, 240 (2015) 447-456.
[13] Ghobadi, S., A Generalized DEA Model for Inputs (Outputs) Estimation under Inter-temporal Dependence, RAIRO-Operations Research, (In press). doi.org/10.1051/ro/201 8100.
[14] Ghobadi, S., Jahanshahloo, G. R., Hosseinzadeh Lotfi, F., and Rostamy- Malkhalifeh, M., Dynamic Inverse DEA in the Presence of Fuzzy Data, Advances in Environmental Biology, 8 (24) (2014) 139-151.
 [15]  Dong-Joon Lim, Inverse DEA with frontier changes for new target setting, European Journal of Operational Research, 254 (2016) 510-516
[16]  Emrouznejad, A., Amin, Gh. R., and Gattoufi, S., Modelling generalized firms’ restructuring using inverse DEA, Journal of Productivity Analysis, 48 (2017) 51-61.
[17]  Gattoufi, S., Amin, G. R., and Emrouznejad, E., A new inverse DEA method for merging banks, IMA J Management Math, 25 (2014) 73-87.
[18]  Ghobadi, S., Inputs and outputs estimation in inverse dea, Iranian Journal of Optimization, 9 (2) (2017) 119-129.
[19]  Ghobadi, S. and Jahangiri, S., Iverse DEA: Review, Extension and Application, International Journal of information Technology and Decision Making, 14 (4) (2015) 805-824.
[20]  Jahanshahloo, G. R., Hosseinzadeh Lotfi, F., Rostamy- Malkhalifeh, M., and ghobadi, S., Using Enhanced Russell Model to Solve Inverse Data Envelopment Analysis Problems, Hindawi Publishing Corporation, The Scientific World Journal, 2014) 1-10(
 [21] Joro, R., Korhonen, P., and Zionts, S., An interactive approach to improve estimates of value efficiency in data envelopment analysis, European Journal of Operations Research, 149 (2003) 688-699.
[22]  Lertworasirikul, S., Charnsethikul, P., and Fang, S. C., Inverse data envelopment analysis model to preserve relative efficiency values: The case of variable returns to scale, Computers and Industrial Engineering, 61 (2011) 1017-1023.
[23]  Li, X. and Cui, J., A comprehensive DEA approach for the resource alloca- tion problem based on scale economies classification, Journal of Systems Science and Complexity, 21 (2008) 540-557.
[24]  Li, X. and Cui, J., Inverse DEA model with considering returns to scale and elasticity, 11th International Symposium on Operations Research and its Applications in Engineering, Technology and Management, (2013) 100-104.
[25] Ghobadi, S., Inverse dea using enhanced russell measure in the presence of fuzzy data. Int. J. Industrial Mathematics 10 (2) (2018) 1-16.
[26]   Lin, H. T., An effficiency-driven approach for setting revenue target, Decision Support Systems 49 (2010) 311-317.
[27] Ghobadi, S., A dynamic DEA model for resource allocation. Int. J. of Mathematics in Operational Research (In press).
[28] Zenodin, E., Ghobadi, S., Merging decision-making units under inter-temporal dependence. IMA Journal of Management Mathematics 00 (2019) 1-28. doi.10.1093/I maman/dpz005.
[29] Amin, G. R., and Al-Muharrami, S., A new inverse data envelopment analysis model for mergers with negative data. IMA Journal of Management Mathematics 29 (2) (2016) 137–149.
[30] Amin, G. R., Emrouznejad, A., and Gattoufi, S., Minor and major consolidations in inverse DEA: Definition and determination. Computers & Industrial Engineering 103 (1) (2017) 193–200.
[31] Emrouznejad, A., Yang, G. L., and  Gattoufi, S., A novel inverse DEA model with application to allocate the CO2 emissions quota to different regions in Chinese manufacturing industries. Journal of the Operational Research Society 70 (7) (2018) 1-12.
[32]   Pastor, J. T., Ruiz, J. L., and Sirvent, I., An enhanced DEA Russell graph efficiency measure, European Journal of Operational Research, 115 (1999) 596-607.
[33]  Jahanshahloo, H., Nonradial model to measuring super efficiency and congestion, Ph.D Thesis, Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran (2013).