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Two-stage Data Envelopment Analysis Foundation and recent developments Dimitris K. Despotis, University of Piraeus, Greece ICOCBA 2012, Kolkata, India

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A Data Envelopment Analysis (DEA) primerOpening the black-boxTwo-stage processes: The two fundamental approachesA novel additive efficiency-decomposition approachConclusions 2

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Data Envelopment Analysis (DEA) (based on the seminal work of Farrell, 1957) William W. Cooper Abraham Charnes Edwardo Rhodes Charnes, Cooper and Rhodes, Measuring the efficiency of decision making units, European Journal of Operational Research 2 (1978), pp Banker, Charnes and Cooper, Some models for estimating technical and scale inefficiencies in data envelopment analysis, Management Science, 30 (1984), pp Charnes, Cooper and Rhodes, Measuring the efficiency of decision making units, European Journal of Operational Research 2 (1978), pp Banker, Charnes and Cooper, Some models for estimating technical and scale inefficiencies in data envelopment analysis, Management Science, 30 (1984), pp Rajiv Banker 3

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What is DEA DEA is a linear programming technique for evaluating the relative efficiency of a set of peer entities, called Decision Making Units (DMUs), which use multiple inputs to produce multiple outputs. DEA identifies an efficient mix of DMUs that achieve specified levels of the outputs with the minimal deployment of resources (inputs). The resources deployed by the efficient mix are then compared with the actual resources deployed by a DMU to produce its observed outputs. This comparison highlights whether the DMU under evaluation is efficient or not. 4

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Decision making units homogeneous Independent “black box” ▫internal structure unknown ▫transformation mechanism (production function) unknown 5 DMU InputsOutputs

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Efficiency The efficiency of a DMU is defined as the ratio of a weighted sum of the outputs yielded by the DMU over a weighted sum of its inputs s outputs: y 1, y 2, …,y s m inputs: x 1, x 2, …, x m 6 Virtual output Virtual input

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Returns to scale Constant returns-to-scale (CRS–CCR model) Variable returns-to scale (VRS – BCC model) Input Output A C B CRS VRS O Production possibility set 7

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Orientation Input oriented model ▫The objective is to minimize inputs while producing at least the given output levels Output oriented model ▫The objective is to maximize outputs while using no more than the observed amount of any input 8 Input Output A C B O Input oriented projection Output oriented projection D P Q R Efficiency of unit D: CRS: PQ/PD VRS: PR/PD VRS ≥ CRS

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The fractional form (CRS-input oriented) 9 n DMUs s outputs m inputs j 0 the evaluated unit

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Input oriented model - CRS The multiplier formThe envelopment form At optimality: 0<θ≤1 10 Dual

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Input oriented model - VRS The multiplier formThe envelopment form 11 Dual

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Projections on the frontier 12

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A Data Envelopment Analysis (DEA) primerOpening the black-boxTwo-stage processes: The two fundamental approachesA novel additive efficiency-decomposition approachConclusions 13

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Opening the black box DMU X Y - DM subunits (DMSU) - (Sub)processes - Components In some contexts, the knowledge of the internal structure of the DMUs can give further insights for the DMU performance evaluation 14 L. Castelli, R. Pesenti, W. Ukovich, A classification of DEA models when the internal structure of the Decision Making Units is considered, Ann Oper Res (2010)

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A Data Envelopment Analysis (DEA) primerOpening the black-boxTwo-stage processes: The two fundamental approachesA novel additive efficiency-decomposition approachConclusions 15

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The fundamental two-stage production process xjxj Stage 1 Stage 2 zjzj yjyj DMU j The external inputs entering the first stage of the process are transformed to a number of intermediate measures that are then used as inputs to the second stage to produce the final outputs. DMUs are homogeneous. 16

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Profitability and marketability of the top 55 U.S. Commercial Banks (Seiford and Zhu, 1999) 17 Profits Profitability Marketability Revenues Employees Assets Equity Market value Total returns to investors Earnings per share Stage 1Stage 2

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The multiplicative approach 1/5 Kao and Hwang (2008) Stage 1 X Z Stage 2 Y Stage -1 efficiency Stage-2 efficiency Overall DMU efficiency = stage 1. stage 2 A series relationship is assumed between the stages. The value of the intermediate measures Z is assumed the same, no matter they are considered as outputs of the first stage or inputs to the second stage. 18

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The multiplicative approach 2/5 Kao and Hwang (2008) 19

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The multiplicative approach 3/5 Kao and Hwang (2008) 20

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The multiplicative approach 4/5 Kao and Hwang (2008) 21

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The multiplicative approach 5/5 Kao and Hwang (2008) The multiplicative model is not extendable to VRS situations Chen, Cook and Zhu (2010) provide a modeling framework to derive the efficient frontier 22

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The additive approach 1/4 Chen, Cook, Li and Zhu (2009) Stage 1 X Z Stage 2 Y Stage -1 efficiency Stage-2 efficiency Overall DMU efficiency = t 1. stage 1 + t 2. stage 2 (t 1 +t 2 =1) A series relationship is assumed between the stages. The value of the intermediate measures Z is assumed the same, no matter they are considered as outputs of the first stage or inputs to the second stage. 23

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The additive approach 2/4 Chen, Cook, Li and Zhu (2009) 24

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The additive approach 3/4 Chen, Cook, Li and Zhu (2009) 25

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The additive approach 4/4 Chen, Cook, Li and Zhu (2009) The additive decomposition approach is extendable to VRS situations Does not comply with the rule that VRS efficiency scores >= CRS scores Does not provide sufficient information to derive the efficient frontier 26

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A Data Envelopment Analysis (DEA) primerOpening the black-boxTwo-stage processes: The three fundamental approachesA novel additive efficiency-decomposition approachConclusions 27

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An alternative additive model Stage 1 X Z Stage 2 Y Stage -1 efficiency Stage-2 efficiency Overall DMU efficiency = ½ stage 1 + ½ stage 2 A series relationship is assumed between the stages. The value of the intermediate measures Z is assumed the same, no matter they are considered as outputs of the first stage or inputs to the second stage. 28

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An alternative additive model 29 Stage 1 X Z Stage 2 Y Output orientedInput oriented

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An alternative additive model 30 Stage 1 X Z Stage 2 Y Output orientedInput oriented Common constraints, bi-objective LP

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An alternative additive model 31 Simple average … Stage-1Stage-2

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An alternative additive model 32 … or a weighted average a 1, a 2 user defined weights, or weights reflecting the “size” of the stages with respect to the portion of total resources used in each stage (in raw quantities)

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An alternative additive model 33 The dual modelThe primal model

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An alternative additive model The model is extendable to VRS situations The new model suffers from the same irregularities with other additive-decomposition models 34

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Deriving the efficient frontiers 35 DualPrimal

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Deriving the efficient frontiers The assumption that the weights of the intermediate measures are equal is sufficient to drive the efficiency assessments in two-stage DEA processes in compliance with the DEA standards 36

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Extensions - Conclusions Two-stage DEA: A fundamental approach Extensions to multi-stage processes Other two-stage schemes 37 ….. X YZ1Zk X Y Z E H

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38 Thank you for your attention!

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