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Ratio-Based Efficiency Analysis (REA)

Ratio-Based Efficiency Analysis (REA). Antti Punkka and Ahti Salo Systems Analysis Laboratory Aalto University School of Science and Technology P.O. Box 11100, 00076 Aalto Finland antti.punkka@tkk.fi, ahti.salo@tkk.fi. Efficiency Ratio. Decision making unit k (DMU k ) defined by

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Ratio-Based Efficiency Analysis (REA)

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

  2. Efficiency Ratio • Decision making unit k (DMUk) defined by • N outputs yk= (y1k,...,yNk) • M inputs xk= (x1k,...,xMk) • Efficiency Ratio to model the efficiency of a DMU • Non-negative weights un (vm) measure the relative values of outputs (relative costs of inputs) • E.g., if u3=1, u4=2, then 2 units of output 3 is as valuable as 1 unit of output 4

  3. Efficiency Ratio in CCR-DEA • Efficient DMUs maximize Efficiency Ratio for some weights • Efficiency score of DMUk is computed with weights that maximize [minl=1,...,KEk/El ] = Ek/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 • DMU1 and DMU3are efficient • If DMU5 is included, then DMU2 becomes more efficient than DMU3 in terms of Efficiency score E E3 / E*=1 E5 E1 / E*=1 E3 / E*=0.98 E1 E* E2 E3 E4 / E*=0.82 E4 N=2, M=1 u1

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

  5. Incomplete preference information and feasible weights • 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 • 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” • udoctoral≥ 2umaster’s,udoctoral≤ 7umaster’s, • Several methods for weight elicitation exist • If preference statements are collected to matrices Au and Av, the feasible weight sets are

  6. Pairwise dominance relation (1/2) • DMUk dominates DMUl iff (i) its Efficiency Ratio is at least as high as that of DMUlfor all feasible weights (ii) its Efficiency Ratio 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 • Also the inefficient DMU2 is non-dominated • Dominance between two DMUs does not depend on other DMUs E E1 E* E2 E3 E4 u1=1/3 u2=2/3 u1=1/2 u2=1/2

  7. Pairwise dominance relation (2/2) 1 2 3 3 • Dominance graph displays dominance structure of several DMUs • A DMU does not dominate itself • Additional preference information can only establish new dominance relations, not break existing ones • An exception: if A dominates B and EA = EB for some feasible weights, then it is possible that EA = EB throughout the smaller feasible region • Statement 5u1≥ 4u2 leads to new relations 2 4 1 4 E E1 E* E2 E3 E4 5u1= 4u2 u1=1/3 u2=2/3 u1=1/2 u2=1/2

  8. E Ranking intervals E1 • For any (u,v), the DMUs can be ranked based on Efficiency Ratios • The minimum ranking of DMUk • The maximum ranking of DMUk • 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 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

  9. 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 • How efficient is a DMU compared to • ... the most efficient DMU, DMU*? • ... the least efficient DMU, DMU0? • Select a benchmark group • Other DMUs: ”How efficient is DMU1 compared to the most efficient one of the other DMUs?” [0.75,1.18] • A subset of DMUs: ”How much more efficient can DMU1 be than DMU3?” 43% • Additional preference information cannot 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

  10. Computation of dominance relations (1/2) • How to determine whether DMUk dominates DMUl (Su,Sv) is open, not bounded, and the objective function non-linear... How to solve the optimization problem?

  11. Option to normalize • Let (u,v)  (Su, Sv), cu,cv > 0 and consider DMUs k and l. Then, • Weights stay feasible: uSu cuuSu for any positive cu • Similarly for v • For each (u,v), choose cu(u,v) and cv(u,v) e.g. so that • The Efficiency Ratio of DMU* is equal to 1 (cf. DEA) • The output (input) value of DMUk is equal to 1 • The output value of DMUk is equal to the input value of DMUk  Ek=1 • The output (input) weights sum up to 1

  12. Computation of dominance relations (2/2) • Normalize so that • The input value of DMUk=1 • The output value of DMUl is equal to its input value • Set of decision variables u,v is now bounded + closed by linear constraints Objective function is linear • 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 • Minimum (best) rankings for DMUk • For every other DMU, define a binary variable zl so that zl = 1 if El> Ek • Choose a suitable normalization to get linear constraints • The minimum ranking is 1 + min Σlzl over (Su,Sv) • An MILP; model for maximum rankings very similar • Efficiency bounds compared to DMU* • Maximum with LP similar to the computation of DEA scores • Minimum • Minimize the linear model used for the computation of dominance relations against all DMUs in the benchmark group • The smallest of these is the minimum • Models for bounds compared to DMU0 very similar (LPs)

  14. Example: Efficiency analysis of TKK’s departments (2007) • University departments consume inputs to produce outputs • Data from TKK’s reporting system • 2 inputs, 44 outputs • 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 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

  15. A J L K I C, E G B D, F, H Ranking intervals Efficiency compared to DMU* • Departments A, J and L are 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 ratio is 71% • 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 Dominance relations

  16. A J L K I C, E G B D, F, H Specification of performance targets • 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 • How much should to be non-dominated? • 88,18% • Computation: LP models

  17. Conclusion • 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 • Input/output weights can be constrained with incomplete preference information • More information  narrower intervals, more dominance relations • Linear formulations allow ’realistically large’ analyses • Tens, even hundreds of DMUs depending on # of outputs and inputs • Submitted manuscript A. Salo, A. Punkka: Ranking Intervals and Dominance Relations for Ratio-Based Efficiency Analysis • Available at http://www.sal.hut.fi/Publications/m-index.html

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