Benchmark Selection of DEA on Two-Construct Output Measures 資料包絡分析在雙構面產出指標的標竿評選

1. Benchmark Selection of DEA on Two-Construct Output Measures資料包絡分析在雙構面產出指標的標竿評選 Dong-Shang Chang Department of Business Administration National Central University, TAIWAN

2. OUTLINE • PROBLEM AND MOTIVATION • PREVIOUS WORK • TWO-OBJECTIVE DEA (TODEA) • CONCLUDING REMARKS

3. 1. PROBLEM AND MOTIVATION Output 1 Input 1 Output 2 Input 2 • Data Envelopment Analysis (DEA) • A linear-programming-based technique for evaluating the efficiency • DEA can provide benchmarks for inefficient DMUs DMUj (j = 1, …, n) … … Input m Output s

4. Return of Equity Profitability Construct Construct Return of Asset Return of Sales 1. PROBLEM AND MOTIVATION • “Two-Construct” or “Two-Objective” Output Measures • “Operating and Quality Constructs” • “Operating and Profitability Constructs” • An unobservable or latent concept that cannot be measured directly but can be represented or measured by one or more variables, i.e. indicators (Hair et al., 2006) .

5. Two-Objective Output Measures 1. PROBLEM AND MOTIVATION Two-ObjectiveInput Measures Input 1 Output 1 Input 2 Output 2 … … Input 1 Output 1 Input f Output p DMUj (j = 1, …, n) DMUj (j = 1, …, n) Input 2 Output2 … … Input 1 Output 1 Input m Output s Input 2 Output 2 … … Input g Output q Two distinct objectives are normally mutually exclusive

6. TCDEA E-TCDEA 1. PROBLEM AND MOTIVATION Desirable outputs Operating expenses Inputs Outputs Undesirable outputs Capital expenditures Ecological issues Electricity generation plants; Coal mines; Gas industry; Petroleum industry Profitability Inputs Non-ecological issues Complications of medical operations in healthcare; Tax payment in commerce Sustainability

7. 1. PROBLEM AND MOTIVATION • Appropriate Benchmarks • Performance of benchmarks should not be dominated by that of the corresponding inefficient DMUs under either objective • DEA may provide inappropriate benchmarks for inefficient DMUs when the output measures are derived from two distinct objectives A Hypothetical Banking Case The branch A might be identified as the benchmark for the branch B

8. 1. PROBLEM AND MOTIVATION • Imposing weight restrictions on the output measures • weight restriction methods (Thompson et al., 1986 ) • Two distinct objectives are normally mutually exclusive • This predicament explains the difficulty in subjectively giving an unbiased tradeoff between the output measures

9. 1. PROBLEM AND MOTIVATION • Without involving tradeoff information between two objectives in solving procedures • Quality-Adjusted DEA (Q-DEA)(Sherman and Zhu, 2006) • Two-Model DEA (TM-DEA)(Shimshak and Lenard, 2007) • The limitations of TM-DEA • Value judgments or a prior information is involved into the solving procedures • DMUs remaining in the final evaluation may be part of the original ones • Performance of benchmark DMUs may be dominated by that of the corresponding inefficient DMUs on another objective

10. Research Purpose • To seek the appropriate benchmarks in the presence of two-objective output measures

11. TM-DEA Q-DEA Efficiency-profitability matrix A Profit Sleeper Star Quality Rating Quality Efficiency Minimal Acceptable Level Minimal Acceptable Level B Cows Dog Operating Efficiency Operating Efficiency Operating Efficiency 2. PREVIOUS WORKS • Efficiency-Profitability Matrix • Boussofiane, et al. (1991); Camanho and Dyson (1999) • Quality-Adjusted DEA (Q-DEA) • Sherman and Zhu (2006) • Two-Model DEA (TM-DEA) • Shimshak and Lenard (2007); Shimshak et al. (2009) • Multiple Objective DEA (MODEA) • Shimshak et al. (2009) • Assurance Regions (AR-I) • Thompson et al. (1986); Shimshak et al. (2009)

12. 2. PREVIOUS WORK • POTENTIAL LIMITATIONS OF TM-DEA • The TM-DEA requires establishing the minimal acceptable levels of operating efficiency and quality efficiency in order to exclude inappropriate benchmark DMUs from analysis. • After removing inappropriate benchmark DMUs, the adjusted DMUs in the final evaluation may only remain part of the original ones. • If the minimal acceptable level is not given high enough (i.e., close to 100%), the selected operating (or quality) benchmark might be dominated by the corresponding inefficient DMUs on quality (or operating) objective.

13. EXAMPLE • Table 1 considers 10 DMUs, each having two inputs derived from objective P and two inputs derived from objective Q and a single output of one. • The TM-DEA measures P efficiency and Q efficiency in separate DEA models. P inputs are used to measure P efficiency; Q inputs are used to measure Q efficiency; measuring P and Q efficiencies uses the common single output. Table 1 A numerical example

14. POTENTIAL LIMITATIONS OF TM-DEA –cont. • For qualifying P benchmarks, the minimal acceptable level of Q efficiency score was set as 0.8. • For qualifying Qbenchmarks, the minimal acceptable level of Pefficiency score was set as 0.7.. • Clearly, the limitation (1) of TM-DEA is that decision makers require subjectively giving the minimal acceptable level.

15. POTENTIAL LIMITATIONS OF TM-DEA • Table 2 shows the limitation (2) of TM-DEA because only half of the original DMUs are remained in the final evaluation. Table 2 Benchmarking for objective P by TM-DEAa,b aPES: P efficiency score; QES: Q efficiency score; RS: reference set; R: Removed. bDMUs achieving P efficiency of 1 but Q efficiency below 0.8 are removed from analysis. cThe reference set is given to DMUs for objective P.

16. POTENTIAL LIMITATIONS OF TM-DEA Table 3 Benchmarking for objective Q by TM-DEAa,b Table 3 shows the potential limitation (3) because, in this case, the TM-DEA selected DMU E as the benchmark DMU for DMUs B and C for objective Q, even though the performance of DMU E was dominated by that of DMUs B and C in terms of objective P. aPES: P efficiency score; QES: Q efficiency score; RS: reference set; R: Removed. bDMUs achieving Q efficiency of 1 but P efficiency below 0.7 are removed from analysis. cThe reference set is given to DMUs for objective Q. dThe DMUs remained in the second round are partitioned into various efficiency levels for objective Q and P, respectively.

17. 3. TWO-OBJECTIVE DEA (TODEA) • The TODEA is executed by a Two-Stagesolution process: • Stage 1: Stratify Efficiency For each objective, the stratification DEA model is employed to classify all DMUs into different categories of efficiency level. • Stage 2: Cross-Evaluation The DMUs within dominated categories are hierarchical evaluated by another objective..

18. Level 2 For objective P ΘA = ΘB ΘC = ΘB Level 3 For objective Q ΘA > ΘB ΘC > ΘB Level 1 For objective P ΘD > ΘF ΘG = ΘF For objective Q ΘD > ΘF ΘG > ΘF Stage 1 Stage 2 (I) A H E D P input 2 Q input 2 > I > B F J A B C G C D P input 1 Q input 1 (a) Stratify all DMUs for objective P (b) Evaluate A, B, C, and D by objective Q Stage 2 (II) J Stage 2 (III) F F D Q input 2 Q input 2 D B B A A I H C C E E G G Q input 1 Q input 1 (c) Evaluate E, F, and G for objective Q (d) Evaluate H, I, and J for objective Q

19. TODEA Algorithm (Two-objective Output Measures) • Let φ denote the whole set of the evaluated DMUs, e.g., φ = {Ωc,k| k = 1, 2} = {Ωc,1 ∪ Ωc,2}, φ = {Ωc,k | k = 1, 2, 3} = {Ωc,1 ∪ Ωc,2∪ Ωc,3}, and c = P or Q. The TODEA can be systematically formulated as an algorithm. When “output measures” are derived from objectives P and Q, the TODEA algorithm is as follows: • Step 1: Run the stratification DEA model to obtain ΩP,g (g = 1, …, G) using the output measures included in objective P and the common input measures and to attain ΩQ,h (h = 1, …, H) using theoutput measures included in objective Q and the common input measures. • Step 2: Select c = P or Q. If c = P (or Q), w = H (or G). Set j = 0. • Step 3: Let j = j + 1. If j > w, the algorithm stops. Otherwise, go to Step 4. • Step 4:φ = {Ωc,k| k = 1, …, j}. As c = P (or Q), evaluate φusing the output measures included in objective Q (or P) and the common input measures for seeking the benchmark DMUs forΩP (or Q),j. Go to Step 3.

20. TODEA Algorithm(Two-objective Input Measures) • When “input measures” are derived from objectives P and Q, the TODEA algorithm is as follows: • Step 1: Run the stratification DEA model to obtain ΩP,g (g = 1, …, G) using the input measures included in objective P and the common output measures and to attain ΩQ,h (h = 1, …, H) using the input measures included in objective Q and the common output measures. • Step 2: Select c = P or Q. If c = P (or Q), w = H (or G). Set j = 0. • Step 3: Let j = j + 1. If j > w, the algorithm stops. Otherwise, go to Step 4. • Step 4:φ = {Ωc,k | k = 1, …, j}. As c = P (or Q), evaluate φ using the input measures included in objective Q (or P) and the common output measures for seeking the benchmark DMUs for ΩP (or Q),j. Go to Step 3.

21. Demonstration of TODEA Table 4 Benchmarking for objectives Q and P by TODEAa aQEL: Q efficiency level; PES: P efficiency score; PEL: P efficiency level; QES: Q efficiency score; RS: reference set. bA superscript (or subscript) on each DMU and each benchmark indicates the specific efficiency level for objective P (or objective Q).

22. 4. CONCLUDING REMARKS • The proposed TODEA reveals the following features: • No value judgments or a prior information is involved into the solving procedures. • The benchmark for each inefficient DMU is found. • The performance of benchmarks is non-dominated. Reading Chang, D.S. and F.C. Yang (2010), “A New Benchmarking Method to Advance the Two-Model DEA Approach: Evidence from a Nursing Home Application,” INFOR , Vol. 48, No.2, pp. 77-88. (SCI, IF: 0.738, OPERATIONS RESEARCH & MANAGEMENT SCIENCE Ranking: 53/73)