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1 Ratio-Based Efficiency Analysis Antti Punkka and Ahti Salo Systems Analysis Laboratory Aalto University School of Science P.O. Box 11100, Aalto Finland

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2 n Efficiency Ratio of DMU k, k = 1,...,K n 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”: u doctoral ≥ 2u master’s, u doctoral ≤ 7u master’s, n Feasible weights (u,v) fulfill these linear constraints –Without preference statements, all non-negative u ≠0, v ≠0 are feasible Efficiency Ratio and preference statements

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3 Efficiency Ratio in CCR-DEA n Efficient DMUs maximize Efficiency Ratio with some (u,v) –For any (u,v), let E * (u,v) = max {E 1 (u,v),...,E K (u,v)} n Efficiency score of DMU k is max u,v [E k (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 n DMU 1 and DMU 3 are efficient –If DMU 5 is included, then DMU 2 becomes more efficient than DMU 3 in terms of efficiency score E1E1 E2E2 E3E3 E4E4 E*E* E 1 / E * =1 E E 4 / E * =0.82 u1u1 E5E5 E 3 / E * =1 E 3 / E * = outputs, 1 input

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4 New results for Ratio-Based Efficiency Analysis (REA) n 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 over all feasible weights? →Ranking intervals 3.Considering all feasible weights, how efficient is a DMU compared to the most (or the least) efficient DMU of a benchmark group? →Efficiency bounds n Can be computed in presence of preference statements about the relative values of outputs and inputs

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5 Dominance relation (1/2) n DMU k dominates DMU l iff its Efficiency Ratio is (i) at least as high as that of DMU l for all feasible weights (ii) is higher than that of DMU l for some feasible weights n Example: 2 outputs, 1 input –Feasible weights such that 2u 1 ≥ u 2 ≥ u 1 –DMU 3 and DMU 2 dominate DMU 4 –CCR-DEA-inefficient DMU 2 is non- dominated, too n Computation: LP models u 1 =1/3 u 2 =2/3 u 1 =1/2 u 2 =1/2 E1E1 E2E2 E3E3 E4E4 E*E* E

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6 Dominance relation (2/2) n A graph shows dominance relations –Transitive: –Asymmetric: no DMU dominates itself and n Additional preference statements can lead to new relations –Relation ”A dominates B” still holds, unless E A = E B throughout the revised weight set –Statement 5u 1 ≥ 4u 2 leads to new relations n 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 u1=4u25u1=4u2 u 1 =1/3 u 2 =2/3 u 1 =1/2 u 2 =1/2 E1E1 E2E2 E3E3 E4E4 E*E* E

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7 Ranking intervals n For any (u,v), the DMUs can be ranked based on Efficiency Ratios →DMUs’ minimum and maximum rankings n 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 DMU 1 DMU 3 DMU 2 DMU 4 ranking 1 ranking 2 ranking 3 ranking 4 u 1 =1/3 u 2 =2/3 u 1 =1/2 u 2 =1/2 E1E1 E2E2 E3E3 E4E4 E*E* E DMU 4 ranked 4 th 3 rd

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8 Efficiency bounds n Select a benchmark group and compare against its most or least efficient DMU with all feasible weights –”How efficient is DMU 1 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, DMU 0 ?” n Computation: LP models n Additional preference statements do not widen the intervals u 1 =1/3 u 2 =2/3 u 1 =1/2 u 2 =1/2 E1E1 E2E2 E3E3 E4E4 E*E* E 1 / E * =1 E 1 / E * =0.75 E 3 / E * = 1 E 3 / E * = 0.7 E E 4 / E * =0.82 E 2 / E * =0.98 E 4 / E * =0.6 E 2 / E * =0.85 E 4 / E 0 =1.07 u 1 =1/3 u 2 =2/3 u 1 =1/2 u 2 =1/2 E1E1 E2E2 E3E3 E4E4 E*E* E0E0 E 1 / E 0 =1.67 E 1 / E 0 =1 E 3 / E 0 = 1.17 E E 4 / E 0 =1 E 3 / E 0 = 1.33 E 2 / E 0 =1.1 E 2 / E 0 =1.42 Compared to DMU 0 E 1 [1.00,1.67]E 0 E 2 [1.10,1.42]E 0 E 3 [1.17,1.33]E 0 E 4 [1.00,1.07]E 0 Compared to DMU * E 1 [0.75,1.00]E * E 2 [0.85,0.98]E * E 3 [0.70,1.00]E * E 4 [0.60,0.82]E *

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9 Example: Efficiency analysis of TKK’s departments n 2 inputs and 44 outputs describe the 12 university dept’s –Data from TKK’s reporting system n 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 Department x 1 (Budget funding) y 1 (Master’s Theses) y 2 (Dissertations) y 3 (Int’l publications) x 2 (Project funding) TKK = Helsinki University of Technology. As of , TKK is part of the Aalto University

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10 A D, F, H B C, E G I J K L Efficiency bounds compared to DMU * Ranking intervals Dominance relations n 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, E A /E * is only 57 % »For K, the smallest such ratio is 71% n 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

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11 Specification of performance targets: examples n 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 n How big an increase to be non-dominated? »88,18% –Computation: LP models A D, F, H B C, E G I J K L

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12 Conclusion n 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 n Admits preference statements –Helps exclude use of extreme weights –More information narrower intervals, more dominance relations n A. Salo, A. Punkka (2011): Ranking Intervals and Dominance Relations for Ratio-Based Efficiency Analysis, Management Science 57(1), pp

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