Download presentation

Presentation is loading. Please wait.

Published byBrodie Hoffman Modified over 2 years ago

1
1 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

2
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

3
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 * =0.98 2 outputs, 1 input

4
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

5
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

6
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 12 4 3 1 2 4 3 5u1=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

7
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

8
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 *

9
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 1.1.2010, TKK is part of the Aalto University

10
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

11
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

12
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. 200-214

Similar presentations

OK

1 Helsinki University of Technology Systems Analysis Laboratory Rank-Based Sensitivity Analysis of Multiattribute Value Models Antti Punkka and Ahti Salo.

1 Helsinki University of Technology Systems Analysis Laboratory Rank-Based Sensitivity Analysis of Multiattribute Value Models Antti Punkka and Ahti Salo.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on db2 introduction to statistics Ppt on introduction to product management Multifunction display ppt on tv Ppt on regional rural banks in india Sense organs for kids ppt on batteries Ppt on indian politics quotes Free download ppt on palm vein technology Ppt on five monuments of india Ppt on building management system Ppt on question tags rules