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

2 n Decision making unit k (DMU k ) defined by –N outputs y k = (y 1k,...,y Nk ) –M inputs x k = (x 1k,...,x Mk ) n Efficiency Ratio to model the efficiency of a DMU n Non-negative weights u n (v m ) measure the relative values of outputs (relative costs of inputs) –E.g., if u 3 =1, u 4 =2, then 2 units of output 3 is as valuable as 1 unit of output 4 Efficiency Ratio

3 Efficiency Ratio in CCR-DEA n Efficient DMUs maximize Efficiency Ratio for some weights n Efficiency score of DMU k is computed with weights that maximize [min l=1,...,K E k /E l ] = E k /E * –Efficiency with other weights not communicated –These weights depend on what DMUs are considered  the order of two DMUs’ scores can depend on other DMUs –Comparisons only with the most efficient DMU »Not necessarily a realistic benchmark for ’very inefficient’ DMUs 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 * =0.98 N=2, M=1

4 Ratio-Based Efficiency Analysis (REA) n Given a pair of DMUs, is the first DMU more efficient than the second for all feasible weights? →Dominance relation n What are the best and worst possible rankings of a DMU over all feasible weights? →Ranking intervals n Considering all feasible weights, how efficient a DMU is compared to the most (least) efficient of a benchmark group? →Efficiency bounds

5 Incomplete preference information and feasible weights n Feasible weights fulfill (possible) preference statements about the values of outputs and inputs –cf. Assurance regions in DEA literature –In the absense of preference information, all non-negative u ≠0, v ≠0 are feasible n Statements correspond to linear constraints on the weights –”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, –Several methods for weight elicitation exist n If preference statements are collected to matrices A u and A v, the feasible weight sets are

6 Pairwise dominance relation (1/2) n DMU k dominates DMU l iff (i) its Efficiency Ratio is at least as high as that of DMU l for all feasible weights (ii) its Efficiency Ratio 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 –Also the inefficient DMU 2 is non- dominated n Dominance between two DMUs does not depend on other DMUs u 1 =1/3 u 2 =2/3 u 1 =1/2 u 2 =1/2 E1E1 E2E2 E3E3 E4E4 E*E* E

7 Pairwise dominance relation (2/2) n Dominance graph displays dominance structure of several DMUs – – –A DMU does not dominate itself n Additional preference information can only establish new dominance relations, not break existing ones »An exception: if A dominates B and E A = E B for some feasible weights, then it is possible that E A = E B throughout the smaller feasible region –Statement 5u 1 ≥ 4u 2 leads to new relations 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

8 Ranking intervals n For any (u,v), the DMUs can be ranked based on Efficiency Ratios →The minimum ranking of DMU k →The maximum ranking of DMU k n Properties –Provide a holistic view of the Efficiency Ratios at a glance –Compare all DMUs against all other DMUs –Addition / removal of a DMU changes the rankings by at most 1 –Show how ’good’ and ’bad’ DMUs can be –Additional preference information cannot 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

9 Efficiency bounds n How efficient is a DMU compared to –... the most efficient DMU, DMU * ? –... the least efficient DMU, DMU 0 ? n Select a benchmark group –Other DMUs: ”How efficient is DMU 1 compared to the most efficient one of the other DMUs?” [0.75,1.18] –A subset of DMUs: ”How much more efficient can DMU 1 be than DMU 3 ?” 43% n Additional preference information cannot 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 *

10 Computation of dominance relations (1/2) n How to determine whether DMU k dominates DMU l (S u,S v ) is open, not bounded, and the objective function non-linear... How to solve the optimization problem?

11 Option to normalize n Let (u,v)  (S u, S v ), c u,c v > 0 and consider DMUs k and l. Then, –Weights stay feasible: u  S u  c u u  S u for any positive c u »Similarly for v n For each (u,v), choose c u (u,v) and c v (u,v) e.g. so that –The Efficiency Ratio of DMU * is equal to 1 (cf. DEA) –The output (input) value of DMU k is equal to 1 –The output value of DMU k is equal to the input value of DMU k  E k =1 –The output (input) weights sum up to 1

12 Computation of dominance relations (2/2) n Normalize so that –The input value of DMU k =1 –The output value of DMU l is equal to its input value  Set of decision variables u,v is now bounded + closed by linear constraints Objective function is linear n Minimize LP; if the minimum >1, k dominates l <1, k does not dominate l =1, maximize the same objective function; if the maximum is >1, then k dominates l

13 Computation of ranking intervals and efficiency bounds n Minimum (best) rankings for DMU k 1.For every other DMU, define a binary variable z l so that z l = 1 if E l > E k 2.Choose a suitable normalization to get linear constraints 3.The minimum ranking is 1 + min Σ l z l over (S u,S v ) –An MILP; model for maximum rankings very similar n Efficiency bounds compared to DMU * –Maximum with LP similar to the computation of DEA scores –Minimum 1.Minimize the linear model used for the computation of dominance relations against all DMUs in the benchmark group 2.The smallest of these is the minimum –Models for bounds compared to DMU 0 very similar (LPs)

14 Example: Efficiency analysis of TKK’s departments (2007) n University departments consume inputs to produce outputs –Data from TKK’s reporting system –2 inputs, 44 outputs n Preferences from 7 members of the Resources Committee –Each member responded to elicitation questions, which yielded crisp weights –The feasible weights = all possible convex combinations of these 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

15 A D, F, H B C, E G I J K L Efficiency compared to DMU * Ranking intervals Dominance relations n Departments A, J and L are 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 ratio is 71% n Efficiency intervals of D, F and H overlap with those of B and G –Yet, B and G are more efficient for all feasible weights

16 Specification of performance targets n How much must Department D increase its Efficiency Ratio 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 much should to be non- dominated? –88,18% –Computation: LP models A D, F, H B C, E G I J K L

17 Conclusion n REA uses all feasible weights to compare DMUs –Dominance relations compare DMUs pairwise –Ranking intervals show which rankings are attainable for the DMUs –Efficiency bounds extend DEA-Efficiency scores n Input/output weights can be constrained with incomplete preference information –More information  narrower intervals, more dominance relations n Linear formulations allow ’realistically large’ analyses –Tens, even hundreds of DMUs depending on # of outputs and inputs n Submitted manuscript A. Salo, A. Punkka: Ranking Intervals and Dominance Relations for Ratio-Based Efficiency Analysis –Available at