In this research, we develop two procedures to determine the common weights relative to the performance indices across all units of organization. The first one is to determine the common weights by searching the benchmark unit in the organization. One virtual benchmark is defined as units with efficiency score 1.0 and all units are asked to approach the virtual benchmark as close as possible. The units with zero gaps to the virtual benchmark are the real benchmark. In the structure of data envelopment analysis, the determination of common weights in this research means that the organization determines the favorable weight to maximize the organization efficiency. The obtained common weights can assist the organization managers in generating the individual efficiency score for all units and the corresponding ranking problem can be addressed by comparing with the scores. However, in the first procedure, sometimes there is existing some units with the equivalent efficiency score 1.0 due to the constraint that none of DMUs’ efficiency scores is allowed to exceed 1.0. It possibly leads to the obstruction of efficiency development. In order to avoid the bias in measurement due to the upper bound of efficiency, we develop the second procedure to determine one compromise common set of weights by eliminating the restriction with upper bound 1.0 in efficiency score. It leads to the more complete ranking without the repeatable efficiency scores.
Several interesting subjects for the further development of this research are discussed.
Besides the scenarios of benchmark chasing and neutral compromise, risk avoidance owns the highest potential for the management. Risk avoidance focuses the prediction of possible and potential UOAs with the worst performance and provides the improvement plan in advance.
The excellent risk avoidance always saves a possible significant lost for organizations.
In this research, the common set of weights is applied to all UOAs under different scenarios and the performance indices are assumed given. Some methods for the selection of performance indices will help this research to possess reliable assessment outcomes.
Statistical approaches and other methods such as analytic hieratical process (AHP) [37] and analytic network process (ANP) [38] would help to select appropriate combinations.
We used non-negative data for the numerical examples of the procedures proposed. One should examine the applicability of the proposed procedures to the other data types, such as negative data, probabilistic data, fuzzy data, ordinal data, and interval data in determining the common set of weights.
In numerical example 3, Table 14 shows that five interval limitations were set for both the proportional virtual output and proportional virtual input, respectively. In our particular numerical example, we observed that the rankings of the UOAs possess the robustness under the considerable amount of combinations. In fact, how to determine the amount of interval limitations for obtaining the ranking robustness is a critical issue. One would observe the interaction between the setting of boundary intervals and the rankings by observing more and more combinations. Generally, the rule for setting the interval limitations is straightforward;
the lower bounds are in increasing order while the upper bounds are in decreasing order.
Section 4 provides the analysis similar to Least Square Method (LSM). While there are existing multiple dependent variables, this model may provide the corresponding analysis.
The slope of the diagonal line is also another issue. A non-linear programming model can search for the optimal slope so that the total gap is further minimized.
References
[1] Charnes A, Cooper WW, Rhodes E. Measuring the efficiency of decision making units.
European Journal of Operational Research 1978;2:429-444.
[2] Cooper WW, Tone K. Measures of inefficiency in data envelopment analysis and stochastic frontier estimation. European Journal of Operational Research 1997;99(1):72- 88.
[3] Andersen P, Petersen NC. A procedure for ranking efficient units in data envelopment analysis. Management Science 1993;39(10):1261-1264.
[4] Hashimoto A. A ranked voting system using a DEA/AR exclusion model: A note.
European Journal of Operational Research 1997;97:600-604.
[5] Sueyoshi T. Data envelopment analysis non-parametric ranking test and index measurement: Slack-adjusted DEA and an application to Japanese agriculture cooperatives. Omega International Journal of Management Science 1999;27:315-326.
[6] Thrall RM. Duality, classification and slacks in data envelopment analysis. The Annals of Operations Research 1996;66:109-138.
[7] Zhu J. Robustness of the efficient decision-making units in data envelopment analysis.
European Journal of Operational Research 1996;90:451-460.
[8] Dula JH, Hickman BL. Effects of excluding the column being scored from the DEA envelopment LP technology mayrix. Journal of the Operational Research Society 1997;48:1001-1012.
[9] Seiford LM, Zhu J. Infeasibility of super-efficiency data envelopment analysis models.
INFOR 1999;37(2):174-187.
[10] Mehrabian S, Alirezaee MR, Jahanshahloo GR. A complete efficiency ranking of decision making units in data envelopment analysis. Computational Optimization and Applications 1999;14:261-266.
[11] Cook W, Kress M, Seiford L. Prioritization models for frontier decision making units in DEA. European Journal of Operational Research 1992;59:319-323.
[12] Torgersen AM, Forsund FR, Kittelsen SAC. Slack-adjusted efficiency measures and ranking of efficient units. Journal of Productivity Analysis 1996;7:379-398.
[13] Bardhan I, Bowlin WF, Cooper WW, Sueyoshi T. Models for efficiency dominance in data envelopment analysis. Part Ι：Additive models and MED measures. Journal of the Operations Research Society of Japan 1996;39:322-332.
[14] Doyle JR, Green RH. Efficiency and cross-efficiency in DEA: derivatives, meanings and uses. Journal of the Operational Research Society 1994;45:567-578.
[15] Thompson RG, Singleton FD, Thrall RM, Smith BA. Comparative site evaluations for locating a high-energy physics lab in Texas. Interfaces 1986;16:35-49.
[16] Dyson RG, Thanassoulis E. Reducing weight flexibility in data envelopment analysis.
Journal of the Operational Research Society 1988;39(6):563-576.
[17] Charnes A, Cooper WW, Wei QL, Huang ZM. Cone ratio data envelopment analysis and multiobjective programming. International Journal of Systems Science 1989;20(7):1099- 1118.
[18] Charnes A, Cooper WW, Huang ZM, Sun DB. Polyhedral cone-ratio DEA models with an illustrative application to large commercial banks. Journal of Econometrics 1990;46(1-2):73-91.
[19] Wong YHB, Basely JE. Restricting weight flexibility in data envelopment analysis.
Journal of the Operational Research Society 1990;41:829-835.
[20] Sarrico CS, Dyson RG. Restricting virtual weights in data envelopment analysis.
European Journal of Operational Research 2004;159:17-34.
[21] Bernroider E, Stix V. A method using weight restrictions in data envelopment analysis for ranking and validity issues in decision making. Computers & Operations Research 2007;34:2637-2647.
[22] Estellita Lins MP, Moreira da Silva AC, Lovell CAK. Avoiding infeasibility in DEA models with weight restrictions. European Journal of Operational Research 2007;181:956-966.
[23] Alder N, Friedman L, Sinuany-Stern Z. Review of ranking methods in the data envelopment analysis context. European Journal of Operational Research 2002;140:249-265.
[24] Cook W, Roll Y, Kazakov A. A DEA model for measuring the relative efficiencies of highway maintenance patrols. INFOR 1990;28(2):113-124.
[25] Roll Y, Cook W, Golany B. Controlling factor weights in data envelopment analysis. IIE Transactions 1991;24:1-9.
[26] Cook WD, Kress M. Data envelopment model for aggregating preference ranking.
Management Science 1990;36(11):1302-1310.
[27] Cook WD, Kress M. A multiple criteria decision model with ordinal preference data.
European Journal of Operational Research 1991;54:191-198.
[28] Roll Y, Golany B. Alternate methods of treating factor weights in DEA. Omega 1993;21:99-109.
[29] Sinuany-Stern Z, Mehrez A, Barboy A. Academic department’s efficiency in DEA.
Computers & Operations Research 1994;21(5):543-556.
[30] Friedman L, Sinuany-Stern Z. Scaling units via the canonical correlation analysis and the data envelopment analysis. European Journal of Operational Research 1997;100(3):629- 637.
[31] Sinuany-Stern Z, Friedman L. DEA and the discriminant analysis of ratios for ranking units. European Journal of Operational Research 1998;111:470-478.
[32] Kao C, Hung HT. Data envelopment analysis with common weights: the compromise solution approach. Journal of the Operational Research Society 2005;56:1196-1203.
[33] Obata T, Ishii H. A method of discriminating efficient candidates with ranked voting data.
European Journal of Operational Research 2003;151:233-237.
[34] Liu F-HF, Chang HW, Peng HH. Ranking DEA efficient units with the most compromising common weights. Lecture Notes in Operations Research 2006;6:219-234.
[35] Liu F-HF, Lin SJ. Collective performance evaluation using centroid transformative effectiveness. Working paper, Department of Industrial Engineering Management, National Chiao Tung University, Taiwan, 2008.
[36] Ward JH. Hierarchical grouping to optimize an objective function. Journal of the American Statistical Association 1963; 58:236-244.
[37] Saaty TL. Decision Making for Leaders: The Analytic Hierarchy Process for Decisions in a Complex World. United States of America: RWS Publications, 1995.
[38] Saaty TL. Theory and Applications of the Analytic Network Process: Decision Making with Benefits, Opportunities, Costs, and Risks. United States of America: RWS Publications, 2005.