1 Ratio-Based Efficiency Analysis Antti Punkka and Ahti Salo Systems Analysis Laboratory Aalto University School of Science P.O. Box 11100, 00076 Aalto.

Slides:



Advertisements
Similar presentations
Teknillinen korkeakoulu Systeemianalyysin laboratorio 1 Graduate school seminar Rank-Based DEA-Efficiency Analysis Samuli Leppänen Systems.
Advertisements

Developing the Strategic Research Agenda (SRA) for the Forest-Based Sector Technology Platform (FTP) RPM-Analysis Ahti Salo, Totti Könnölä and Ville Brummer.
Minimum Clique Partition Problem with Constrained Weight for Interval Graphs Jianping Li Department of Mathematics Yunnan University Jointed by M.X. Chen.
C&O 355 Lecture 23 N. Harvey TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A A A A A.
LINEAR PROGRAMMING (LP)
Approximation Algorithms Chapter 14: Rounding Applied to Set Cover.
Computing Kemeny and Slater Rankings Vincent Conitzer (Joint work with Andrew Davenport and Jayant Kalagnanam at IBM Research.)
Schedule On Thursdays we will be here in SOS180 for: – (today) – – Homework 1 is on the web, due to next Friday (17: ).
Preference Elicitation Partial-revelation VCG mechanism for Combinatorial Auctions and Eliciting Non-price Preferences in Combinatorial Auctions.
1 Helsinki University of Technology Systems Analysis Laboratory Robust Portfolio Modeling for Scenario-Based Project Appraisal Juuso Liesiö, Pekka Mild.
Helsinki University of Technology Systems Analysis Laboratory RPM – Robust Portfolio Modeling for Project Selection Pekka Mild, Juuso Liesiö and Ahti Salo.
Helsinki University of Technology Systems Analysis Laboratory RICHER – A Method for Exploiting Incomplete Ordinal Information in Value Trees Antti Punkka.
1 Helsinki University of Technology Systems Analysis Laboratory Multi-Criteria Capital Budgeting with Incomplete Preference Information Pekka Mild, Juuso.
CMOS Circuit Design for Minimum Dynamic Power and Highest Speed Tezaswi Raja, Dept. of ECE, Rutgers University Vishwani D. Agrawal, Dept. of ECE, Auburn.
An Optimization Approach to Improving Collections of Shape Maps Andy Nguyen, Mirela Ben-Chen, Katarzyna Welnicka, Yinyu Ye, Leonidas Guibas Computer Science.
Linear Programming Unit 2, Lesson 4 10/13.
Computational Methods for Management and Economics Carla Gomes
S ystems Analysis Laboratory Helsinki University of Technology A Preference Programming Approach to Make the Even Swaps Method Even Easier Jyri Mustajoki.
Data Envelopment Analysis (DEA). Which Unit is most productive? DMU = decision making unit DMU labor hrs. #cust
Helsinki University of Technology Systems Analysis Laboratory A Portfolio Model for the Allocation of Resources to Standardization Activities Antti Toppila,
Stevenson and Ozgur First Edition Introduction to Management Science with Spreadsheets McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies,
S ystems Analysis Laboratory Helsinki University of Technology Using Intervals for Global Sensitivity and Worst Case Analyses in Multiattribute Value Trees.
1 Helsinki University of Technology Systems Analysis Laboratory Robust Portfolio Selection in Multiattribute Capital Budgeting Pekka Mild and Ahti Salo.
Distributed Constraint Optimization Michal Jakob Agent Technology Center, Dept. of Computer Science and Engineering, FEE, Czech Technical University A4M33MAS.
Helsinki University of Technology Systems Analysis Laboratory Ahti Salo and Antti Punkka Systems Analysis Laboratory Helsinki University of Technology.
1 Helsinki University of Technology Systems Analysis Laboratory Robust Portfolio Modeling in the Development of National Research Priorities Ville Brummer.
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 RPM-Explorer - A Web-based Tool for Interactive Portfolio Decision Analysis Erkka Jalonen.
MIT and James Orlin © More Linear Programming Models.
Helsinki University of Technology Systems Analysis Laboratory Determining cost-effective portfolios of weapon systems Juuso Liesiö, Ahti Salo and Jussi.
Types of IP Models All-integer linear programs Mixed integer linear programs (MILP) Binary integer linear programs, mixed or all integer: some or all of.
1 Helsinki University of Technology Systems Analysis Laboratory INFORMS 2007 Seattle Efficiency and Sensitivity Analyses in the Evaluation of University.
3.4 Linear Programming p Optimization - Finding the minimum or maximum value of some quantity. Linear programming is a form of optimization where.
Helsinki University of Technology Systems Analysis Laboratory INFORMS Seattle 2007 Integrated Multi-Criteria Budgeting for Maintenance and Rehabilitation.
1 Helsinki University of Technology Systems Analysis Laboratory Selecting Forest Sites for Voluntary Conservation in Finland Antti Punkka and Ahti Salo.
S ystems Analysis Laboratory Helsinki University of Technology Practical dominance and process support in the Even Swaps method Jyri Mustajoki Raimo P.
A LINEAR PROGRAMMING PROBLEM HAS LINEAR OBJECTIVE FUNCTION AND LINEAR CONSTRAINT AND VARIABLES THAT ARE RESTRICTED TO NON-NEGATIVE VALUES. 1. -X 1 +2X.
1 Helsinki University of Technology Systems Analysis Laboratory Selecting Forest Sites for Voluntary Conservation with Robust Portfolio Modeling Antti.
Helsinki University of Technology Systems Analysis Laboratory Antti Punkka and Ahti Salo Systems Analysis Laboratory Helsinki University of Technology.
15.053Tuesday, April 9 Branch and Bound Handouts: Lecture Notes.
Prioritizing Failure Events in Fault Tree Analysis Using Interval-valued Probability Estimates PSAM ’11 and ESREL 2012, Antti Toppila and Ahti.
Helsinki University of Technology Systems Analysis Laboratory 1DAS workshop Ahti A. Salo and Raimo P. Hämäläinen Systems Analysis Laboratory Helsinki.
Constraints Feasible region Bounded/ unbound Vertices
1 Optimization Techniques Constrained Optimization by Linear Programming updated NTU SY-521-N SMU EMIS 5300/7300 Systems Analysis Methods Dr.
Helsinki University of Technology Systems Analysis Laboratory Incomplete Ordinal Information in Value Tree Analysis Antti Punkka and Ahti Salo Systems.
STATIC ANALYSIS OF UNCERTAIN STRUCTURES USING INTERVAL EIGENVALUE DECOMPOSITION Mehdi Modares Tufts University Robert L. Mullen Case Western Reserve University.
1 S ystems Analysis Laboratory Helsinki University of Technology Master’s Thesis Antti Punkka “ Uses of Ordinal Preference Information in Interactive Decision.
Schedule Reading material for DEA: F:\COURSES\UGRADS\INDR\INDR471\SHARE\reading material Homework 1 is due to tomorrow 17:00 ( ). Homework 2 will.
1 Ratio-Based Efficiency Analysis (REA) Antti Punkka and Ahti Salo Systems Analysis Laboratory Aalto University School of Science and Technology P.O. Box.
Recall: Consumer behavior Why are we interested? –New good in the market. What price should be charged? How much more for a premium brand? –Subsidy program:
Helsinki University of Technology Systems Analysis Laboratory EURO 2009, Bonn Supporting Infrastructure Maintenance Project Selection with Robust Portfolio.
3-5: Linear Programming. Learning Target I can solve linear programing problem.
Resource allocation and portfolio efficiency analysis Antti Toppila Systems Analysis Laboratory Aalto University School of Science and Technology P.O.
1 Helsinki University of Technology Systems Analysis Laboratory Standardization Portfolio Management for a Global Telecom Company Ville Brummer Systems.
Retiming EECS 290A Sequential Logic Synthesis and Verification.
Approximation Algorithms based on linear programming.
Section 3.5 Linear Programing In Two Variables. Optimization Example Soup Cans (Packaging) Maximize: Volume Minimize: Material Sales Profit Cost When.
Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 1 S ystems Analysis Laboratory Helsinki University of Technology Decision support.
preference statements
Chapter 2 An Introduction to Linear Programming
Mikko Harju*, Juuso Liesiö**, Kai Virtanen*
3 THE CCR MODEL AND PRODUCTION CORRESPONDENCE
Linear Programming.
Incomplete ordinal information in value tree analysis and comparison of DMU’s efficiency ratios with incomplete information Antti Punkka supervisor Prof.
Decision support by interval SMART/SWING Methods to incorporate uncertainty into multiattribute analysis Ahti Salo Jyri Mustajoki Raimo P. Hämäläinen.
Linear Programming Problem
Juuso Liesiö, Pekka Mild and Ahti Salo Systems Analysis Laboratory
Graphical solution A Graphical Solution Procedure (LPs with 2 decision variables can be solved/viewed this way.) 1. Plot each constraint as an equation.
1.6 Linear Programming Pg. 30.
Presentation transcript:

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

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 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

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 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 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

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 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 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

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 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 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