Two-stage Data Envelopment Analysis Foundation and recent developments Dimitris K. Despotis, University of Piraeus, Greece I would like to thank Professor Mukherjee and the organizing committee of the Conference for inviting me to deliver this talk. My presentation will be on two-stage data envelopment analysis, which in fact is a framework for performance evaluation, when there is some knowledge of the internal structure of the evaluated units. Next slide ICOCBA 2012, Kolkata, India

A Data Envelopment Analysis (DEA) primer Opening the black-box Two-stage processes: The two fundamental approaches A novel additive efficiency-decomposition approach Conclusions I will start with a quick reference to the basic background of DEA, which will help the eventually non-familiar with DEA part of the audience comprehend the main definitions and techniques, that will then be transferred to the two-stage DEA framework. Next slide

Data Envelopment Analysis (DEA) (based on the seminal work of Farrell, 1957) Based on the seminal work of Farrell in ’50s, Charnes, Cooper and Rhodes developed the Data Envelopment Analysis technique, as a methodological framework for performance evaluation, based on linear programming. A few years later, Banker, Charnes and Cooper extended the original models to capture different assumptions on returns-to-scale. These models are in the core of DEA literature, which since then evolved rapidly, counting these days over 4500 journal publications with both theoretical developments and applications. Next slide William W. Cooper 1914-2012 Abraham Charnes 1917-1992 Edwardo Rhodes Rajiv Banker Charnes, Cooper and Rhodes, Measuring the efficiency of decision making units, European Journal of Operational Research 2 (1978), pp. 429-444. Banker, Charnes and Cooper, Some models for estimating technical and scale inefficiencies in data envelopment analysis, Management Science, 30 (1984), pp. 1078-1092.

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. DEA is a linear programming technique for evaluating the relative efficiency of a set of peer entities, called Decision Making Units (DMUs), that use multiple incommensurable inputs to produce multiple incommensurable outputs. In the core of DEA is the identification of 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. A first outcome of DEA is a categorical classification of the DMUs in efficient and inefficient DMUs Next slide

Decision making units homogeneous Independent “black box” DMU Inputs Outputs homogeneous Independent “black box” internal structure unknown transformation mechanism (production function) unknown DEA allows for a great flexibility in defining DMUs University departments, bank branches, hospitals, …. Homogeneous: This means that they all have the same type of inputs and outputs. The are assumed independent: This means that there is no constraint binding the input and output levels of a DMU with the inputs and outputs of other DMUs. The DMUs are viewed as “black-boxes”: The internal structure of the DMUs as well as the transformation mechanism of inputs to outputs are unknown. Next slide

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 The efficiency of a DMU is defined as the ratio of a weighted sum of its outputs over a weighted sum of its inputs The numerator of the fraction is called “Virtual output”. The denominator is called “Virtual input”. The weights of the inputs and the outputs are not predefined, but they are problem variables that are estimated in favor of the evaluated unit, so as to maximize its efficiency score In general, Different DMUs use different weights to compute their efficiency. Supplement … s outputs: y1, y2, …,ys m inputs: x1, x2, …, xm Virtual output Virtual input

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 Next, we must think about returns to scale. The distinction between constant (CRS) and variable (VRS) returns to scale is trivial mathematically but can make a significant difference in the results. CRS assumes that multiplying the input by a constant implies multiplication of the output by the same constant. For example, doubling the machine capacity in a production line, it is reasonable to assume that you will get twice as much products from the production process. Describe the figure …. Next slide

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 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 Finally, we must think about orientation. The orientation prescribes the way the inefficient units are projected on the efficient frontier. What kind of advice are we seeking to give to the inefficient DMUs? Do we want them to reduce inputs? Then use an input orientation. Do we want them to increase outputs? Then use an output orientation One might also consider mixed orientations in which we want the DMUs to both reduce inputs and increase outputs.  … In my opinion, these are the key steps in creating a DEA model that will produce trustworthy results that managers can actually use to improve efficiency. Next slide

The fractional form (CRS-input oriented) n DMUs s outputs m inputs j0 the evaluated unit Explain …. The model is solved for one DMU at a time. Only the objective function changes. The weights (decision variables) are estimated in favor of the evaluated unit so as to maximize its efficiency score. Fractional program Linear equivalent by the C-C transformation Next slide

Input oriented model - CRS The multiplier form The envelopment form Dual The linear form … explain : Objective function, normalization constraint, variable alteration Multiple optimal solutions. At optimality, If Ej=1 and there is at least one optimal solution with non-zero weights then the evaluated unit is efficient. Otherwise it is inefficient. The multiplier form provides a managerial aspect of efficiency. The envelopment form provides a technical aspect of efficiency. At optimality, If θ=1 and all slacks are zero (they are not indicated explicitly in the model) then the evaluated unit is efficient. Otherwise it is inefficient. If θ<1 (the case of an inefficient unit), the non-zero λ’ς (lambda) indicate the efficient peers for the evaluated unit. Next slide At optimality: 0<θ≤1

Input oriented model - VRS The multiplier form The envelopment form Dual Explain the differences from the CRS model Next slide

Projections on the frontier Explain Next slide

A Data Envelopment Analysis (DEA) primer Opening the black-box Two-stage processes: The two fundamental approaches A novel additive efficiency-decomposition approach Conclusions One of the characteristics of DEA is that the DMUs are considered as black-boxes. No internal structure is considered. As a consequence, there is no clear evidence of the transformations to which the inputs are subject to, within the considered DMUs. In the last two decades, several investigators explored the possibility of measuring efficiency relative to the sub-processes of DMUs within the broad DEA framework. Next slide

Opening the black box In some contexts, the knowledge of the internal structure of the DMUs can give further insights for the DMU performance evaluation DMU X Y - DM subunits (DMSU) - (Sub)processes - Components It is assumed now that, in some particular contexts, the knowledge of the internal structure of the DMUs can give further insights for the DMU performance evaluation. As an example, such knowledge allows to determine whether better performances can be theoretically obtained by merging the technologies of some components of the observed DMUs. In addition, assessing the efficiency of each of the sub-units might prevent the inefficiency of some of them to be compensated by the efficiency of other sub-units. Next slide 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)

This structure is part of a network a architecture. A Data Envelopment Analysis (DEA) primer Opening the black-box Two-stage processes: The two fundamental approaches A novel additive efficiency-decomposition approach Conclusions Now, we will focus on the special structure, known as “two-stage process”. This structure is part of a network a architecture. However, because of its popularity it has attracted special attention in the literature. Next slide

The fundamental two-stage production process xj Stage 1 Stage 2 zj yj DMU j In a two-stage production process, 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. Nothing but the external inputs to the first stage enters the system and nothing but the outputs of the second stage leaves the system. The DMUs are homogeneous, that is, the same type of inputs, intermediate measures and outputs are considered for all the units. Notice here the conflicting role of the intermediate measures, which are considered outputs in terms of the first stage and inputs in terms of the second stage. This conflicting role of the intermediate measures makes the state-of the –art efficiency assessment models suffer from a number of irregularities. Next slide 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.

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

The multiplicative approach 1/5 Kao and Hwang (2008) 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. Stage 1 X Z Stage 2 Y Stage -1 efficiency Stage-2 efficiency Overall DMU efficiency = stage 1 . stage 2 The multiplicative approach: A series relationship is assumed between the stages. The value of the intermediate measures Z (total virtual measure) is assumed the same, no matter they are considered as outputs of the first stage or inputs to the second stage. This reasonable assumption, is the key to join the efficiency assessments in the two-stages with the overall efficiency. Next slide

The multiplicative approach 2/5 Kao and Hwang (2008) Describe ….. Next slide

The multiplicative approach 3/5 Kao and Hwang (2008) Describe …. In the objective function only the final outputs and the external inputs are taken into account and in this manner it provides a direct measure of overall efficiency. The two stages are linked by means of the intermediate measures in the constraints. Next slide

The multiplicative approach 4/5 Kao and Hwang (2008) The linear equivalent derives by the C-C transformation Once we obtain an optimal solution of this linear program the calculation of the individual and the overall efficiencies is straightforward Next slide

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 The multiplicative model is not extendable to VRS situations Chen, Cook and Zhu (2010) provided a modeling framework to derive the efficient frontier Next slide

The additive approach 1/4 Chen, Cook, Li and Zhu (2009) 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. Stage 1 X Z Stage 2 Y Stage -1 efficiency Stage-2 efficiency Overall DMU efficiency = t1 . stage 1 + t2 . stage 2 (t1+t2=1) The additive approach: It is based on the same assumption to conceptualize the series relationship between the two stages. It differs, however, in the aggregation-disaggregation scheme. The overall efficiency is defined as a weighted sum of the individual efficiencies of the two-stages. Next slide

The additive approach 2/4 Chen, Cook, Li and Zhu (2009) Describe …. Considering that t1 and t2 are user defined parameters, the model turns to be non-linear and cannot be linearized by the C-C transformation. For the sake of linearization, the weights t1 and t2 are defined as functions of the weights. They reflect the portion of the total resources used (the denominators) in each stage. As the optimal weights differ among the DMUs, the values of the parameters differ as well. So each DMU, apart from selecting its optimal weights through the optimization process, it selects its own parameter values through the same process. This is a “DMU-centric” approach to derive the overall efficiency from the efficiencies of the two stages. Next slide

The additive approach 3/4 Chen, Cook, Li and Zhu (2009) This particular way to define the parameters t1 and t2 allows for a tractable model that can be transformed to a linear equivalent by the C-C transformation. Once an optimal solution to the linear program is obtained, one can calculate the individual and the overall efficiency of the evaluated DMU. Next slide

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 The additive decomposition approach is extendable to VRS situations However, it does not comply with the rule that VRS efficiency scores >= CRS scores Another deficiency is that the additive approach does not provide sufficient information to derive the efficient frontier. That is the dual variables are not sufficient, as in the standard DEA models, to derive correctly the projections of the inefficient units on the efficient frontier. Next slide

A Data Envelopment Analysis (DEA) primer Opening the black-box Two-stage processes: The three fundamental approaches A novel additive efficiency-decomposition approach Conclusions The last part of my presentation will be devoted to some recent developments that treat the deficiencies reported in the literature, as for deriving the efficient frontier in the additive decomposition approach as well as the compliance of the VRS and CRS efficiency scores with the standard DEA principle.

An alternative additive model 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. Stage 1 X Z Stage 2 Y Stage -1 efficiency Stage-2 efficiency Overall DMU efficiency = ½ stage 1 + ½ stage 2 We still refer to the additive decomposition approach, with the same assumptions regarding the series relationship between the two stages. The modeling approach, however, is different (simpler, I could say) that resorts to a bi-objective programming framework.

An alternative additive model Stage 1 X Z Stage 2 Y Output oriented Input oriented The key is to select different orientations for the two stages. Actually, an output orientation is selected for the first stage and an input orientation for the second stage. This allows the two models to have common normalization constraints, on the basis of the intermediate measures. Cross-appending the last constraints of the one model to the other, we obtain two augmented models with the same constraints, that is with same feasible region. Next slide

An alternative additive model Stage 1 X Z Stage 2 Y Output oriented Input oriented For both stages, an optimal solution of the augmented model is also optimal to the original model. So working with the augmented models is equivalent to working with the original models. Now, as the two models have common constraints, they can be joined in a bi-objective linear model if we take the maximization counterpart of the stage-1 program. Next slide Common constraints, bi-objective LP

An alternative additive model Simple average … Stage-1 Stage-2 Notice, that this model does not provide a measure of overall efficiency, but only the efficiencies of the two stages. However, Once an optimal solution is obtained, we can get the efficiencies of the two-stages. The minimal value that the objective function can take is zero (0) and this happens when both stages are efficient. Having the two individual efficiency scores, the overall efficiency is obtained as a simple average. The individual efficiency scores are the same with those obtained by the weighted additive model of Chen et al (2009). Obviously, however, the overall efficiency is generally different. Next slide

An alternative additive model … or a weighted average a1, a2 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) If it is to assign different importance to the two stages, one can define weights to get a weighted average. The weights now are user defined parameters and, unlike the previous additive model, they are common for all the DMUs. However, if the importance of the two stages should reflect the “size” of the two-stages , the parameters could be defined in a way similar to the previous model, by taking into account the portion of the total resources used in each stage by all the DMUs, in terms of the raw quantities. Actually, the raw quantities are max-normalized to make them units free. Again, The derived parameter values are common for all the DMUs. This a “stage-centric” approach to derive the overall efficiency from the efficiencies of the two stages. Next slide

An alternative additive model The primal model The dual model The dual model is given in the right side. The dual variables λ and μ correspond to the first and the second stage respectively. Notice that the intermediate measures undergo the same proportional change θ. Next slide

An alternative additive model The model is extendable to VRS situations The new model suffers from the same irregularities with other additive-decomposition models The derivation of the VRS variant is straightforward However, the new model still suffers from the same deficiencies. In fact, The rule that the “VRS scores >= CRS scores” does not generally hold. And the dual variables λ and μ cannot sufficiently be used to derive correctly the projections of the inefficient units. Next slide

Deriving the efficient frontiers Dual Primal We modify the dual model in an attempt to overcome these deficiencies. In the frame of our additive decomposition approach, we are concerned for the DEA frontiers of the two individual stages. In the modified model , the proportional change θ for each intermediate measure is followed by a shift αp. This modification grants the flexibility to the model to estimate new λ’s and μ’s as projection parameters. The last constraint imposes that the total volume of the intermediate measures remains unchanged, no matter what the shifts ap are. The model has been extensively tested with different randomly generated datasets. The results showed that both deficiencies are treated. Taking the multiplier form of the model, uncovers a sufficient condition to generate the efficient frontiers in an additive two-stage DEA framework. In addition to the principal assumption that the value of the intermediate measures remain the same in both stages when a unit is evaluated, the weights assigned to these measures should be equal.

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 Assuming that the weights of the intermediate measures are equal is sufficient to drive the efficiency assessments in two-stage DEA processes in a manner that complies with the DEA standards. VRS scores >= CRS scores Correct derivation of the efficient frontiers Next slide

Extensions - Conclusions Two-stage DEA: A fundamental approach Extensions to multi-stage processes Other two-stage schemes ….. X Y Z1 Zk X Y Z E H Two-stage DEA: A fundamental approach If it is to remain in the grounds of DEA, before considering to deal with complex processes, it is advised to delve into the fundamental architectural schemes, such as the two-stage DEA, in order to derive robust models that comply with the DEA principles. Extensions to multi-stage processes Other two-stage schemes END

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