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Ratio-Based Efficiency Analysis

Ratio-Based Efficiency Analysis. Antti Punkka and Ahti Salo Systems Analysis Laboratory Aalto University School of Science P.O. Box 11100, 00076 Aalto Finland antti.punkka@tkk.fi, ahti.salo@tkk.fi. Efficiency Ratio and preference statements. Efficiency Ratio of DMU k , k = 1 ,...,K

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Ratio-Based Efficiency Analysis

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  1. Ratio-Based Efficiency Analysis Antti Punkka and Ahti SaloSystems Analysis LaboratoryAalto University School of ScienceP.O. Box 11100, 00076 AaltoFinlandantti.punkka@tkk.fi, ahti.salo@tkk.fi

  2. Efficiency Ratio and preference statements • Efficiency Ratio of DMUk, k = 1,...,K • Possible preference statements constrain the relative values of outputs and inputs • Linear constraints on output and input weights, cf. assurance regions of type I • ”A doctoral thesis is at least as valuable as 2 master’s theses, but not more valuable than 7 master’s theses”: udoctoral≥ 2umaster’s,udoctoral≤ 7umaster’s, • Feasible weights (u,v) fulfill these linear constraints • Without preference statements, all non-negative u≠0, v≠0 are feasible

  3. Efficiency Ratio in CCR-DEA • Efficient DMUs maximize Efficiency Ratio with some (u,v) • For any (u,v), let E*(u,v) = max {E1(u,v),...,EK(u,v)} • Efficiency score of DMUk is maxu,v [Ek(u,v)/E*(u,v)] • Based on comparisons with one weights, with one DMU • Order of two DMUs’ efficiency scores can depend on what other DMUs are considered • Does not show how ’bad’ a DMU can be • Efficiency score of an efficient DMU is 1 • DMU1 and DMU3are efficient • If DMU5 is included, then DMU2 becomes more efficient than DMU3 in terms of efficiency score E E5 E3 / E*=1 E1 / E*=1 E3 / E*=0.98 E1 E* E2 E3 E4 / E*=0.82 E4 2 outputs,1 input u1

  4. New results for Ratio-Based Efficiency Analysis (REA) • All results are based on comparing DMUs’ Efficiency Ratios 1. Given a pair of DMUs, is the first DMU more efficient than the second for all feasible weights? • Dominance relations 2. What are the best and worst possible efficiency rankings of a DMU overall feasible weights? • Ranking intervals 3. Considering all feasible weights, how efficientis a DMU compared to the most (or the least) efficient DMU of a benchmark group? • Efficiency bounds • Can be computed in presence of preference statements about the relative values of outputs and inputs

  5. Dominance relation (1/2) • DMUk dominates DMUl iff its Efficiency Ratio is (i) at least as high as that of DMUlfor all feasible weights (ii) is higher than that of DMUlfor some feasible weights • Example: 2 outputs, 1 input • Feasible weights such that 2u1 ≥u2≥u1 • DMU3 and DMU2 dominate DMU4 • CCR-DEA-inefficient DMU2 is non-dominated,too • Computation: LP models E E1 E* E2 E3 E4 u1=1/3 u2=2/3 u1=1/2 u2=1/2

  6. Dominance relation (2/2) 1 2 3 3 • A graph shows dominance relations • Transitive: • Asymmetric: no DMU dominates itself and • Additional preference statements can lead to new relations • Relation ”A dominates B” still holds, unless EA = EB throughout the revised weight set • Statement 5u1≥ 4u2 leads to new relations • Dominance vs. CCR-DEA-efficiency • Efficient DMUs are non-dominated • A dominates B⇔B is inefficient among {A,B} • Dominance between two DMUs does not depend on other DMUs 2 4 1 4 E E1 E* E2 E3 E4 5u1= 4u2 u1=1/3 u2=2/3 u1=1/2 u2=1/2

  7. E Ranking intervals E1 • For any (u,v), the DMUs can be ranked based on Efficiency Ratios • DMUs’ minimum and maximum rankings • Properties • Addition / removal of a DMU changes the rankings by at most 1 • Show how ’good’ and ’bad’ DMUs can be • Minimum ranking of a CCR-DEA-efficient DMU is 1 • Computation: MILP models • K-1 binary variables • Additional preference statements do not widen the intervals E* E2 E3 E4 3rd DMU4 ranked 4th u1=1/3 u2=2/3 u1=1/2 u2=1/2 ranking 1 ranking 2 ranking 3 ranking 4 DMU1 DMU2 DMU3 DMU4

  8. E E3 / E* = 1 E1 / E*=1 E1 E2 / E*=0.98 E* E2 E2 / E*=0.85 E3 E E3 / E* = 0.7 E3 / E0 = 1.33 E1 / E*=0.75 E4 / E*=0.82 E4 E4 / E*=0.6 E1 / E0=1.67 E1 E* E2 E2 / E0=1.1 E2 / E0=1.42 E3 E3 / E0 = 1.17 E0 u1=1/3 u2=2/3 u1=1/2 u2=1/2 E1 / E0=1 E4 E4 / E0=1 u1=1/3 u2=2/3 u1=1/2 u2=1/2 Efficiency bounds • Select a benchmark group and compare against its most or least efficient DMU with all feasible weights • ”How efficient is DMU1 compared to the most efficient of other DMUs?” [0.75,1.18] • ”How efficient are the DMUs compared to • ... the most efficient of all DMUs, DMU*? • ... the least efficient of all DMUs, DMU0?” • Computation: LP models • Additional preference statements do not widen the intervals E4 / E0 =1.07 Compared to DMU* E1[0.75,1.00]E* E2[0.85,0.98]E* E3[0.70,1.00]E* E4[0.60,0.82]E* Compared to DMU0 E1[1.00,1.67]E0 E2[1.10,1.42]E0 E3[1.17,1.33]E0 E4[1.00,1.07]E0

  9. Example: Efficiency analysis of TKK’s departments • 2 inputs and 44 outputs describe the 12 university dept’s • Data from TKK’s reporting system • 7 Resources Committee members responded to preference elicitation questions which yielded crisp weightings • E.g. ”How many master’s theses are as valuable as a dissertation?” • Feasible weights modeled as all convex combinations of these 7 weightings y1 (Master’s Theses) Department x1 (Budget funding) y2 (Dissertations) x2 (Project funding) y3 (Int’l publications) TKK = Helsinki University of Technology. As of 1.1.2010, TKK is part of the Aalto University

  10. A J L K I C, E G B D, F, H Ranking intervals Efficiency bounds compared to DMU* • Dept’s A, J and L are CCR-DEA-efficient • But A can attain ranking 7 > 4, the worst ranking of K • For some feasible weights, EA/E* is only 57 % • For K, the smallest such ratio is 71% • Intervals set by Efficiency bounds of D, F and H overlap with those of B and G • Yet, B and G are more efficient for all feasible weights Dominance relations

  11. A J L K I C, E G B D, F, H Specification of performance targets: examples • How big a radial increase in its outputs must Department D make to be among the 6 most efficient departments • ... for some feasible weights? • 25,97 % • ... for all feasible weights? • 54,40 % • Computation: MILP models • How big an increase to be non-dominated? • 88,18% • Computation: LP models

  12. Conclusion • REA results use all feasible weights to compare DMUs • Dominance relations compare DMUs pairwise and provide a dominance structure for the DMUs • Ranking intervals show which efficiency rankings the DMUs can attain • Efficiency bounds extend CCR-DEA-Efficiency scores by allowing comparisons to the most or least efficient unit of any benchmark group • Computation with (MI)LP models allows comparing dozens of DMUs • Consistent with CCR-DEA results; they are obtained as special cases • Admits preference statements • Helps exclude use of extreme weights • More information  narrower intervals, more dominance relations • A. Salo, A. Punkka (2011): Ranking Intervals and Dominance Relations for Ratio-Based Efficiency Analysis, Management Science 57(1), pp. 200-214

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