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Revisiting dual-role factors in data envelopment
analysis: derivation and implications
Wen-Chih Chen a a
Department of Industrial Engineering and Management , National Chiao Tung University, 1001 Ta Hsueh Rd., Hsinchu, Taiwan
Accepted author version posted online: 27 Aug 2012.Published online: 28 Mar 2014.
To cite this article: Wen-Chih Chen (2014) Revisiting dual-role factors in data envelopment analysis: derivation and implications, IIE Transactions, 46:7, 653-663, DOI: 10.1080/0740817X.2012.721943
To link to this article: http://dx.doi.org/10.1080/0740817X.2012.721943
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CopyrightC“IIE”
ISSN: 0740-817X print / 1545-8830 online DOI: 10.1080/0740817X.2012.721943
Revisiting dual-role factors in data envelopment analysis:
derivation and implications
WEN-CHIH CHEN
Department of Industrial Engineering and Management, National Chiao Tung University, 1001 Ta Hsueh Rd., Hsinchu, Taiwan E-mail: [email protected]
Received May 2011 and accepted July 2012
Data Envelopment Analysis (DEA) is a mathematical programming method to evaluate relative performance. Typical DEA studies consider a production process transforming inputs to outputs. In some cases, however, some factors can be both inputs and outputs simultaneously and are termed dual-role factors. For example, research funding can be an input that strengthens a university’s academic performance and the actual funds can be an output. This article investigates the problem of how to incorporate dual-role factors in DEA. Rather than proposing an ad hoc evaluation model directly, this article considers the concept of “joint technology,” two individual production processes acting in common by summarizing the intuitive thinking. The efficiency evaluation models, based on variant assumptions, thus can be axiomatically derived, validated, and extended. How to determine the input/output tendency of a dual-role factor based on the evaluating results is shown and explained from different aspects. It is concluded that the tendency is a property on the projected boundary, not the data point itself.
Keywords: Data envelopment analysis, dual-role factor, efficiency 1. Introduction
Data Envelopment Analysis (DEA), coined and pop-ularized by Charnes et al. (1978), is a mathematical programming method to evaluate relative performance by peer comparison. DEA considers multiple aspects of the performance simultaneously and aggregates different criteria values as a ratio of weighted output to weighted in-put without a priori weight assignments. At this analytical stage, DEA is an aggregating mechanism to aggregate mul-tiple criteria into a single score. On the other hand, DEA is a stream of nonparametric production analysis, originated by Farrell (1957), to measure the technical efficiency of pro-duction units. The efficiency of a unit is measured relative to the production frontier, which is estimated by a set of data. Typical DEA studies consider a production process of transforming multiple inputs to various outputs. In addi-tion to those having a clear input/output role, however, some factors may play roles as both inputs and outputs simultaneously; these factors are referred to as dual-role
factors. Beasley (1990, 1995) is the first to note dual-role
factors in his study evaluating research productivity at a university. He finds that research funding is an important performance criterion (output) and a resource (input) that
Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/uiie.
strengthens the institution’s academic performance. Cook
et al. (2006) looked at graduate students in higher
educa-tion organizaeduca-tions and nurse trainees on staff in hospitals and found that the graduate student and nurse trainee are dual-role factors. In other words, they are the maximizing-oriented performance criteria (outputs) themselves and the resources (inputs) to provide publications and care for more patients.
A literature survey reveals different models for DEA eval-uation studies with dual-role factors. Following the idea by Charnes et al. (1978), the models specify different output-to-input ratio forms while retaining the core spirit of DEA with minimum assumptions about weights determination. In Beasley (1990, 1995), a dual-role factor is in both the de-nominator (as a part of the weighted input) and the numer-ator (as a part of the weighted output). Cook et al. (2006) suggest moving the input role from the denominator to the numerator, the output side, but with the opposite sign in its weight. Specifically, they consider the dual-role factor as an exogenously fixed or non-discretionary variable (Banker and Morey, 1986), which is not controlled but can affect the DEA evaluation. In addition, Bi et al. (2009) attempt to address this issue from the angle of a production process, not multi-criteria performance aggregation. Rather than considering these factors as inputs and outputs simultane-ously, Cook and Zhu (2007) suggest a method to classify them in DEA. To the best of our knowledge, the avail-able literature does not discuss the validity of any proposed
0740-817XC2014 “IIE”
model or the axiomatic arguments. Although Cook et al. (2006) show that Beasley’s treatment of dual-role factors is inappropriate, they do not provide any validation for their modified model.
Therefore, this article investigates how to handle dual-role factors in DEA. Rather than proposing an ad hoc evaluating model directly, we develop a concept of “joint technology,” two individual production processes acting in common, by summarizing the intuitive thinking. Evaluat-ing models based on different assumptions can be derived, validated, and extended axiomatically using the “joint tech-nology.” Thus, the proposed model is theoretically well de-fined and intuitively obvious.
The benefits of the proposed approach are at least three-fold. First, the approach provides a clear framework for analyzing how to incorporate dual-role factors in DEA. In particular, we approach the problem from production process axioms but not the weighting viewpoints. Multiple dual-role factors can also be analyzed. Second, by focus-ing on the basic framework and not a solution to a spe-cific problem, the approach provides a way to validate the proposed models in the extant literature, including Cook
et al. (2006). It clearly shows the flawed logic in Beasley
(1990, 1995) beyond taking a counterexample. Third, the proposed approach is easy to extend and integrate with variant DEA models with detailed specific requirements, such as Free Disposal Hull (FDH; Deprins et al. (1984)), non-discretionary variables, etc.
Based on the proposed “joint technology” framework, this article makes the following contributions to the lit-erature. We develop a simple three-dimensional case for visualization, and we also generalize to multi-dimensional cases for considering the multiple factors. Furthermore, we determine the input/output behavior (tendency) of a dual-role factor based on the analysis results and explain it from multi-criteria performance aggregation, geometry, and economics perspectives. Importantly, we conclude that the tendency is a property on the projected boundary, not the data point itself. This conclusion sheds light for centralized/decentralized dual-role factor allocation problems.
The remainder of this article is organized as follows. Section 2 introduces DEA as a performance evaluation method. Section 3 proposes the axiomatic framework and derives the efficiency evaluation considering dual-role fac-tors, including Cook et al. (2006). Section 4 provides a visualization example and motivates the geometric inter-pretations. Section 5 generalizes the findings to multiple dimensions, gives economic interpretations, and discusses some possible variant models, and Section 6 concludes.
2. Technology and radial efficiency
Consider a production process transforming various inputs into different outputs. Denote an input set I and an output set O, and let x∈ |I|+ and y∈ |O|+ be the value vectors for
inputs and outputs, respectively. A production technology, represented by a set T, describes the possible input–output transformation as
T≡ {(x, y) : x can produce y}.
Given a set of records R with data (xr, yr)∈ |I|+|O|+ , r∈ R,
we can approximate the underlying but unknown T based on the following assumptions:
1. Strong disposability: (x, y) ∈ T, x≥ x and y≤ y imply (x, y)∈ T.
2. Convexity: (x, y) ∈ T and (x, y)∈ T imply λ(x, y) + (1− λ)(x, y)∈ T for λ ∈ [0, 1].
3. Constant Returns to Scale (CRS): (x, y) ∈ T implies
α(x, y) ∈ T for α ∈ [0, ∞). One approximation is ˆ T≡ (x, y) : r∈R xrλr ≤ x; r∈R yrλr ≥ y; λr ≥ 0, r ∈ R . ˆ
T imposes assumptions 1 to 3 and is referred to as the CRS
technology. The CRS assumption can be relaxed by adding the convexity constraintr∈Rλr = 1 and thus is termed the
Variable Returns to Scale (VRS) technology. An important use of ˆT is the Farrell input efficiency measure (Farrell,
1957). To measure input efficiency of (xk, yk), k∈ R, we
solve the following problem:
min θ, (1) s.t. r∈R xrλr ≤ θxk, r∈R yrλr ≥ yk, λr ≥ 0, r ∈ R.
The optimal valueθ∗of Model (1) is the input efficiency of
k associated with the CRS technology ˆT. Assuming CRS
technology, it suggests that k can reduce its inputs xk to
(100× θ∗)% while maintaining the same level of outputs yk.
There are variant models according to different technology assumptions and objectives, such as the orientations. Com-prehensive explanations and variant models can be found in F¨are et al. (1994a) and Cooper et al. (2007).
Model (1) can be transformed to the equivalent problem as follows (Charnes et al., 1978). For notational simplicity, the vector multiplication in this paper represents the dot product of two vectors.
maxykv xku , (2) s.t. yrv xru ≤ 1, r ∈ R, u≥ 0, v ≥ 0.
We can see Model (2) as a weight aggregation scheme for record k∈ R. u ∈ |I|+ and v∈ |O|+ are weights assigned for inputs and outputs, respectively, and all records’ overall
performance scores are calculated as ratios of weighted out-put to weighted inout-put. Model (2) allows k to select weights favoring its performance score ykv/xku, as long as all
per-formance scores are normalized within one.
3. Modeling dual-role factors
Dual-role factors can play roles as both inputs and out-puts simultaneously. When dual-role factors play roles as outputs, we term them dual-role outputs in contrast with the regular outputs y, and when dual-role factors play roles as inputs, we term them dual-role inputs in contrast to the regular inputs x. Consider the cases with dual-role factors in addition to inputs and outputs. Denote the dual-role factor set D and the value vector w∈ |D|+ . Collect the data (xr, wr, yr)∈ |I|+|D|+|O|+ , r ∈ R.
3.1.A joint technology
The following two statements are apparently true and gen-erally intuitive for production processes having dual-role factors in addition to regular inputs and outputs:
Regular inputs generate regular outputs and
dual-role outputs, (3)
and
Regular inputs and dual-role inputs generate
regular outputs. (4)
Dual-role factors can be considered as part of the outputs, which count for performance to be maximized. However, there is a cost, since dual-role outputs and regular outputs consume the same resources. On the other hand, when dual-role factors are also considered as part of inputs, together with regular inputs, they can contribute to provide more regular outputs. In the study by Cook et al. (2006, pp. 105) on the role of graduate students in evaluating researchers’ performance, the authors make two important arguments and observations:
while published research (articles in referred journals, etc.) is likely the predominant output for evaluating the re-searcher, the extent to which the research contributes to the training of highly qualified personnel is also an impor-tant component in the evaluation.
and
at least two inputs contribute to the generation of research publications: (i) research dollars available to support pub-lication; and (ii) the number of graduate assistants.. . . Note that their first statement coincides with Observation (3) and that (4) generalizes the second argument.
Next, consider the example of nurse trainees in a hospital. Part of the responsibilities and performance of certified
nursing staff is to guide and educate more trainees, which, of course, consumes more of the hospital’s resources, such as the certified nurses’ time and labor. However, the trainees help the nurses with several routine tasks requiring the least skill and experience.
Observations (3) and (4) describe two individual produc-tion processes. Observaproduc-tion (3) is formalized by T1as
T1 ≡ {(x, w, y) : x can produce (w, y)}. Observation (4) is represented by T2as
T2 ≡ {(x, w, y) : (x, w) can produce y}.
Observations (3) and (4) are observed simultaneously. In other words, we can formalize the process as a joint tech-nology based on both Observations (3) and (4). The overall underlying technology with dual-role factors is the inter-section of technologies T1and T2as
T1∩ T2.
It is important to bear in mind that dual-role factors play two roles, inputs and outputs, simultaneously. They are neutral in terms of their input/output roles. We do not treat a dual-role factor as either an input or an output ex
ante. We only ex post conclude that a particular record
prefers the dual-role factor to be an input or output by comparing what it gains from both roles as discussed later. Without loss of generality, denote ψ(y) = {(x, w) : (x, w, y) ∈ T1∩ T2} as the feasible set of (x, w) for a given y. The following proposition shows that ψ(y) is nested in y due to the strong disposability of y.
Proposition 1. ψ(y) ⊆ ψ(y) if y ≤ y.
Proof. y exhibits strong disposability for T1and T2, and we have If y≤ y, then (x, w, y) ∈ T1 ⇒ x, w, y∈ T1. If y≤ y, then (x, w, y) ∈ T2 ⇒ x, w, y∈ T2. Therefore, for y≤ y, (x, w, y) ∈ T1∩ T2implies (x, w, y)∈
T1∩ T2. According to the definition of ψ(y), we can also rewrite the statement as (x, w) ∈ ψ(y) ⇒ (x, w) ∈ ψ(y);
i.e.,ψ(y) ⊆ ψ(y), for y≤ y.
Following the idea of approximating technology based on collected data set R, a possible approximation of T1∩ T2 is T1∩ T2 ≡ (x, w, y) : r∈R xrλr ≤ x; r∈R yrλr ≥ y; r∈R wrλr ≥ w; r∈R xrλr ≤ x; r∈R yrλr ≥ y; r∈R wrλr ≤ w; λr ≥ 0, r ∈ R .
The approximation imposes assumptions 1 to 3 as does ˆT. The first three constraints associate with T1, and the next three associate with T2. Any (x, w, y) (as seen on the right-hand side of the inequalities) that satisfies all constraints are inT1∩ T2. Note thatT1∩ T2 approximates T1∩ T2 directly and are joint by applying the sameλrvalues corresponding
to two technologies.
3.2.Radial efficiency
The classic Farrell input efficiency is a radial efficiency measure relative to the boundaries of the technology by proportionately reducing all “inputs,” including any fac-tor that could provide function as inputs. We thus reduce role factors together with regular inputs since dual-role factors are part of inputs. As an analog to Model (1), solving the following problem evaluates (xk, wk, yk),
k∈ R: θradial k = min θ, (P1) s.t. r∈R xrλr ≤ θxk, r∈R yrλr ≥ yk, r∈R wrλr ≥ θwk, r∈R xrλr ≤ θxk, r∈R yrλr ≥ yk, r∈R wrλr ≤ θwk, λr ≥ 0, r ∈ R.
Constraints represent a joint technology ofT1∩ T2, where the top three constraints associate with T1 and the next three constraints associate with T2. The optimal value of (P1),θkradial, is the efficiency of k in a process with dual-role factors; i.e., (P1) proportionately reduces all “inputs” to θ times of its current level, including xk and wk.
Re-ducing wk together with xk is clear with respect to T2, in which dual-role factors play roles of inputs, and this reduc-tion result is denoted as (θxk, θwk, yk). A feasibleθ means
that (θxk, θwk, yk)∈T1∩ T2, and (θxk, θwk, yk) should
sat-isfy the constraints associated with both T1 and T2. Thus, we haveθwk, not wk, in the right-hand side of the third
constraint. Rather than interpreting it as simply minimiz-ing (dual-role) outputs, this settminimiz-ing truthfully and passively reflects the results of reducing both regular and dual-role inputs xk and wk, which define the radial efficiency
mea-sure here. Note that reducing wkimplies that the dual-role
factors are adjustable (other efficiency measures with addi-tional assumptions are discussed in Sections 3.3 and 5.3).
(P1) is simplified by removing identical constraints as
θradial k = min θ, s.t. r∈R xrλr ≤ θxk, r∈R yrλr ≥ yk, r∈R wrλr ≥ θwk, r∈R wrλr ≤ θwk, λr ≥ 0, r ∈ R.
The dual problem follows as
max ykv, (5)
s.t. − xru+ yrv+ wrγo− wrγi ≤ 0, r ∈ R,
xku− wkγo+ wkγi = 1,
u≥ 0, v ≥ 0,γo ≥ 0,γi ≥ 0, where u∈ |I|+, v∈ |O|+ ,γo ∈ |D|
+ , andγi ∈ |D|+ are dual variables corresponding to the constraints from the top to the bottom. The equivalent linear fractional program-ming problem (Charnes and Cooper, 1962; Charnes et al., 1978) is max ykv xku− wkγo+ wkγi, s.t. − xru+ yrv+ wrγo− wrγi ≤ 0, r ∈ R, u≥ 0, v ≥ 0,γo ≥ 0,γi ≥ 0. It can be rearranged as max ykv xku+ wk(γi −γo) (P1R) s.t. yrv xru+ wr(γi−γo) ≤ 1, r ∈ R, u≥ 0, v ≥ 0,γo≥ 0,γi ≥ 0.
(P1R) is a standard ratio form of DEA (Charnes et al., 1978) and has properties similar to Model (2). u and v are the weight vectors for regular inputs and outputs, re-spectively. γo is the weight vector for dual-role outputs,
and γi is for the w
r serving as inputs. γi−γo can be
either positive or negative, although γo and γi are both non-negative.
We give the implication on the sign ofγdi∗− γdo∗ d∈ D
in a more intuitive and straightforward manner. Similar to Model (2), Problem (P1R) allows k to select weights to favor its performance (efficiency). In (P1R),γi∗
d < γdo∗
d∈ D implies that k prefers factor d as an output rather
than being an input and weights factor d more on its output role. As a result, γi∗
d − γdo∗< 0 provides a smaller
denominator value and results in a better performance.
γi∗
d < γdo∗suggests that factor d benefits k from an output
role and has an overall negative impact on d as an input. In
contrast,γdi∗> γdo∗means that k prefers factor d as an
in-put.γdi∗= γdo∗ suggests indifference to d being an output or an input.
3.3.Being exogenously fixed
Instead of minimizing both xkand wkproportionately as
shown in (P1), we may want to treat dual-role factors as ex-ogenously fixed or non-discretionary (Banker and Morey, 1986). The concept is adopted by Cook et al. (2006). Here, we adopt this assumption and derive the formulation using joint technology. Solving the following problem evaluates
k, but we assume that the dual-role factors are exogenously
fixed: θfix k = min θ, (P2) s.t. r∈R xrλr ≤ θxk, r∈R yrλr ≥ yk, r∈R wrλr ≥ wk, r∈R xrλr ≤ θxk, r∈R yrλr ≥ yk, r∈R wrλr ≤ wk, λr ≥ 0, r ∈ R.
The interpretation of (P2) is similar to (P1). However, unlike (P1), (P2) minimizes only the regular inputs.r∈Rwrλr ≥
wkspecifies exogenously fixed dual-role outputs associated
with T1, and
r∈Rwrλr ≤ wkassociates non-discretionary
dual-role inputs associated with T2.
Following the same procedure deriving (P1R) from (P1), the dual of the equivalence of (P2) is
max ykv+ wkγo− wkγi, (6)
s.t. − xru+ yrv+ wrγo− wrγi ≤ 0, r ∈ R,
xku= 1,
u≥ 0, v ≥ 0,γo ≥ 0,γi ≥ 0. The equivalent DEA ratio form is
max ykv+ wk(γ o−γi) xku , (P2R) yrv+ wr(γo−γi) xru ≤ 1, r ∈ R, u≥ 0, v ≥ 0,γo ≥ 0,γi ≥ 0.
The interpretations of decision variables are identical to (P1R), but the dual-role factors are now in the numerator with weight vectorγo−γi, not in the denominator with
γi−γo as in (P1R).
Indeed, (P2R) derived from (P2) is identical to the for-mulation in Cook et al. (2006); more precisely, (P2R) is a general model considering multiple dual-role factors. Ar-guing thatγo∗
d γdi∗= 0, Cook et al. (1986) propose the
im-plication of sign by comparing (P2R) to the VRS ratio form (Banker et al., 1984) without dual-role factors. The differ-ence between Model (2) and (P2R) is the second term of the numerator, as is the difference between Model (2) and the VRS ratio form. Their reasoning is based on the Returns To Scale (RTS) characteristics drawn from the VRS ratio form.γdo∗− γdi∗< 0 is similar to the decreasing RTS cases, and thus the dual-role factor d acts like an input. Simi-larly,γdo∗− γdi∗> 0 indicates that d behaves like an output, andγo∗
d − γdi∗= 0 suggests that d is at an equilibrium or
optimal level.
Cook et al. (2006) are correct in judging a dual-role fac-tor’s input/output behavior based on the sign ofγdo∗− γdi∗, but it is not necessary to haveγo∗
d γdi∗= 0. Without loss of
generality, ifγdo∗= 0 and γdi∗> 0, γdo∗+ δ and γdi∗+ δ for
δ ≥ 0 give the same optimal value, then γo∗
d + δ and γdi∗+ δ
forδ ≥ 0 are also optimal solutions to (P2R). Although the mathematical formulation structure of Cook et al. (2006) gives a nice comparison to judge the importance, or ten-dency, of the role of dual factors, the managerial interpre-tations are not yet clear. Thus, we suggest using the same arguments for (P1R): k prefers d as an output if and only ifγdi∗< γdo∗gives a positive weighted value of d in the nu-merator and d contributes positively as an output.
Comparing (P1R) and (P2R) shows that (P1R) has dual-role factors in the denominators with weight vectorγi −γo while (P2R) has them in the numerators withγo−γi. Our proposed judging rules and interpretations yield a con-sistent result in both cases. In contrast, relying on RTS characteristics may limit to (P2R). Moreover, to derive the problem in a ratio form, note that both wkγi and
wkγo should always be in either the unity constraint as
in Model (5) or the objective function as in Model (6). Otherwise, since the objective function and unity con-straint correspond to the decision whether to reduce dual-role factors in T1∩ T2, wk will be minimized by θ for
its input role asr∈Rwrλr ≤ θwk and also being
exoge-nously fixed for its output role asr∈Rwrλr ≥ wk(or,
simi-larly,r∈Rwrλr ≥ θwkand
r∈Rwrλr ≤ wk); thusθ∗= 1.
Therefore, we can conclude that the model proposed by Beasley (1990, 1995) is problematic and does not align with the general intuitive understanding of the dual-role factors.
4. Visualization and geometric implications
Here, we visualize the joint technology using a simple, three-dimensional case with one regular input, one regu-lar output, and one dual-role factor, based on a portion of the real-world data collected by Beasley (1990) in his study evaluating research productivity at a university. The
purpose of the visualization is to understand the gener-alized DEA models with dual-role factors, the difference between (P1) and (P2), and the possible extensions.
4.1.Data set
First, we recall that (P1) and (P1R) are equivalent, as are (P2) and (P2R). For the purpose of visualization, from Beasley (1990), we arbitrarily use equipment expenditure as the only input (x), PG research as the only output (y), and research income as the only dual-role factor (w). We exclude Universities 7 and 49 due to zero or extremely little equipment expenditure; however, this exclusion does not affect our visualization. We approximate and visualize
T1∩ T2, in whichw is dual-role, and also ˆT1,w as an output, and ˆT2, w as an input, for comparison. All technology approximations employ the CRS.
Table 1 shows the data and the analysis results for our simple visualization. Columns 5 and 6 are the input-oriented efficiency measures associated with ˆT1 and ˆT2. Forθk2,w is minimized together with x regarding ˆT2, but
w is not minimized together with x regarding ˆT1 to ob-tain θ1
k. Universities 24 and 36 have θk1= 1 and are
effi-cient with respect to ˆT1. Universities 9, 19, and 36 have
θ2
k = 1 and are efficient with respect to ˆT2. Column 7 is the radial efficiency measures θradial
k for T1∩ T2, which is ob-tained by (P1), and followed by the associated weight forw,
γi∗
d − γdo∗computed by (P1R).θkfixis the efficiency measure
for k consideringw as non-discretionary computed by (P2). Column 10 is the weight ofw under this assumption. As expected, the efficiency measures, weights of the dual-role factors, and signs of the weights are different for the two models.
Fig. 1. A partial three-dimensional visualization of ˆT1.
Fig. 2. A partial three-dimensional visualization of ˆT2.
Figures 1 to 3 visualize ˆT1, ˆT2, and T1∩ T2. One block represents 20 in x, 500 in w, and 10 in y. All figures are bounded at x= 120 and Fig. 3 is also bounded at w = 3000. Doing so produces a more focused illustration of the technology boundaries. Figure 1 represents ˆT1, in which research incomew is an output. Universities 24 and 36 form the boundaries of ˆT1(also see Table 1). Figure 2 represents
ˆ
T2, where research income is an input. Universities 9, 19, and 36 form the boundaries of ˆT2(also see Table 1). Figure 3 shows the joint technologyT1∩ T2. Comparing three figures show the differences of treating a dual-role factor in three different settings.
Fig. 3. A partial three-dimensional visualization ofT1∩ T2.
Table 1. Data set and analysis results (P1) & (P1R) (P2) & (P2R) Univ. x w y θ1 k θk2 θkr adi al γi∗− γo∗ θ f i x k γi∗− γo∗ 1 64 254 26 0.59 0.68 0.68 0.000 654 0.61 0.000 784 2 301 1485 54 0.31 0.29 0.29 0.000 134 0.31 −0.000 063 3 23 45 3 0.19 0.24 0.24 0.001 986 0.19 −0.000 826 4 485 940 48 0.15 0.18 0.18 0.000 094 0.15 −0.000 039 5 90 106 22 0.35 0.48 0.48 0.009 434 0.43 0.000 557 6 767 2967 166 0.33 0.36 0.36 0.000 055 0.33 −0.000 025 8 126 776 32 0.42 0.39 0.39 0.000 304 0.42 −0.000 151 9 32 39 17 0.77 1.00 1.00 0.001 477 1.00 0.001 568 10 87 353 27 0.45 0.52 0.52 0.000 479 0.45 0.000 071 11 161 293 20 0.18 0.23 0.23 0.000 286 0.18 −0.000 118 12 91 781 37 0.65 0.59 0.59 0.000 065 0.65 −0.000 209 13 109 215 19 0.25 0.32 0.32 0.000 419 0.25 0.000 057 14 77 269 24 0.45 0.53 0.53 0.000 554 0.45 0.000 081 15 121 392 31 0.37 0.44 0.44 0.000 357 0.37 0.000 051 16 128 546 31 0.37 0.40 0.40 0.000 323 0.37 −0.000 148 17 116 925 24 0.40 0.30 0.30 0.000 051 0.40 −0.000 164 18 571 764 27 0.08 0.09 0.09 0.000 082 0.08 −0.000 033 19 83 615 57 0.99 1.00 1.00 0.000 441 1.00 0.000 075 20 267 3182 153 0.91 0.83 0.88 −0.000 092 0.91 −0.000 071 21 226 791 53 0.35 0.40 0.40 0.000 189 0.35 −0.000 084 22 81 741 29 0.60 0.52 0.52 −0.000 284 0.60 −0.000 234 23 450 347 32 0.10 0.21 0.21 0.002 882 0.10 0.000 014 24 112 2945 47 1.00 0.61 1.00 −0.000 339 1.00 −0.000 170 25 74 453 9 0.26 0.19 0.19 0.000 519 0.26 −0.000 257 26 841 2331 65 0.14 0.14 0.14 0.000 052 0.14 −0.000 023 27 81 695 37 0.71 0.66 0.66 0.000 073 0.71 −0.000 234 28 50 98 23 0.66 0.84 0.84 0.000 914 0.82 0.001 003 29 170 879 38 0.36 0.35 0.35 0.000 234 0.36 −0.000 112 30 628 4838 217 0.56 0.50 0.50 0.000 009 0.56 −0.000 030 31 77 490 26 0.52 0.51 0.51 0.000 494 0.52 −0.000 247 32 61 291 25 0.59 0.66 0.66 0.000 664 0.59 0.000 102 33 39 327 18 0.71 0.67 0.67 0.000 151 0.71 −0.000 487 34 131 956 50 0.59 0.56 0.56 0.000 280 0.59 −0.000 145 35 119 512 48 0.58 0.66 0.66 0.000 347 0.59 0.000 422 36 62 563 43 1.00 1.00 1.00 0.000 095 1.00 0.000 100 37 235 714 36 0.24 0.27 0.27 0.000 185 0.24 −0.000 081 38 94 297 23 0.35 0.42 0.42 0.000 461 0.35 0.000 066 39 46 277 19 0.61 0.63 0.63 0.000 838 0.61 −0.000 413 40 28 154 7 0.40 0.39 0.39 0.001 404 0.40 −0.000 678 41 40 531 23 0.94 0.83 0.92 −0.000 635 0.94 −0.000 475 42 68 305 23 0.49 0.55 0.55 0.000 602 0.49 −0.000 279 43 82 85 9 0.16 0.24 0.24 0.011 765 0.17 0.000 612 44 26 130 11 0.61 0.68 0.68 0.001 543 0.61 0.000 238 45 123 1043 39 0.54 0.46 0.46 0.000 048 0.54 −0.000 154 46 149 1523 51 0.60 0.49 0.51 −0.000 158 0.60 −0.000 127 47 89 743 30 0.56 0.49 0.49 0.000 066 0.56 −0.000 213 48 82 513 47 0.83 0.87 0.87 0.000 466 0.83 0.000 076 50 95 485 32 0.50 0.54 0.54 0.000 420 0.50 −0.000 200 4.2.ψ(23)
Figure 4, the x-w cutting plane of T1∩ T2 at y= 23, can represent the feasible collection of (x, w) associated with
y= 23—i.e., ψ(23)—because ψ(y) is nested as shown in
Proposition 1. The x-axis is the value of x and the y-axis is the value of w. The east region, bounded by aABCDd, represents part of the feasible region of (x, w) when
y= 23. In particular, (xA, wA)= (43.29, 52.8), (xB, wB)=
(33.49, 248.1), (xC, wC)= (33.16, 301.2), and (xD, wD)=
Fig. 4. A partial x–w cutting plane at y = 23.
(54.81, 1441.2). The lower boundary Aa is parallel to the x-axis, not the extension of OA (the dashed line in Fig. 4); the lower boundary indicates the strong disposability of regu-lar input x. Segments AB and BC are downward sloping; a reduction of one dimension requires the other dimension to increase for any points on AB and BC. Segments CD and Dd are upward sloping, meaning that for points on these boundaries, increasingw while keeping x the same will result in a smaller y, since the point is outsideψ(23) andψ(y) is nested. To increase w for points on the upward sloping boundaries also requires an increase in x, but the opposite is not true.
Furthermore, in Fig. 4, the upward sloping boundary segments occur only whenw is larger than 301.2 at C. We also note that the normal vector values for AB and BC have the same sign for x andw. In contrast, segments CD and
Dd have opposite signs on the normal vector values for x
andw.
4.3.Projection
In Fig. 4, U28, U38, U41, and U42are x-w values for Univer-sities 28, 38, 41, and 42, for which y= 23; Table 1 shows the detailed values. We can compute the efficiency measure based on (P2), wherew is non-discretionary and is fixed at its current level, by reducing only x horizontally. Therefore, (P2) suggests U28 and U38 project onto segments AB and
BC, respectively, and both U41 and U42 project onto CD, which is upward sloping.
On the other hand, we apply (P1) if adjustment onw is permitted. (P1) considers and measures the proportionate reduction on both x andw. As a result, all records evaluated try to move toward the origin, and we can observe different
projected points on the boundaries by comparing with (P2). For example, U42 projects onto AB, which is downward sloping, instead of CD.
Interestingly, regardless of the model, a projection on up-ward sloping boundaries has a negative weightγi∗− γo∗, such as U41and U42in (P2) and U41in (P1) (Table 1). Pro-jecting onto downward-sloping boundaries coincides with positive γi∗− γo∗, such as U
28, U38, and U42 in (P1) and
U28and U38in (P2). U42has opposite signs in the two mod-els; one projects onto downward-sloping AB, leading to
γi∗− γo∗> 0, and the other projects onto upward-sloping
CD, leading to γi∗− γo∗< 0. In summary, the sign of
γi∗− γo∗ only depends on the projected point, not the data point itself. A model determines the improvement (projection) path. Different improvement paths, due to dif-ferent objectives and assumptions, lead to difdif-ferent pro-jected points on the boundary. Two different points and two different models may project onto the same point on the boundary, leading to the same sign ofγi∗− γo∗. This explains why, in Table 1, (P2R) gives more universities with negativeγi∗− γo∗ than does (P1R). It is simply because (P1R) proportionately reduces x andw, for which the pro-jection moves toward the southwest and, thus, the project-ing points will more likely be on the downward-slopproject-ing boundaries.
γi∗− γo∗ is a boundary property, not the property of
the point before a projection, and it associates the ideal benchmark without inefficiency. We thus suggest interpret-ing the sign of γi∗− γo∗ as preferring dual-role factor w for its input (output) role, without further interpretation for any improvement direction. As addressed in Section 3,
γi∗− γo∗> 0 indicates that dual-role factor w benefits k from being an input; however, this does not lead to the conclusion of reducingw for improvement. Since the pro-jected point is on the downward-sloping boundary, any reduction onw results in infeasible or less y. In contrast, when γi∗− γo∗< 0, the projecting boundary is upward
sloping, and any increase onw results in infeasible or less
y. Now, factor w does not contribute as do the regular
inputs.
The reallocation ofw, such that the efficiency of (P2) is maximized, or other objectives can give multiple solutions. Taking U41 as an example, both increasing and reducing
w will yield identity efficiency in (P2), although the
corre-sponding weightγi∗− γo∗indicates that U41 prefersw as an output.
5. Generalization and extensions
Now, we generalize the observations in Section 4 and in-vestigate γi∗−γo∗ from the viewpoints of geometry and
economics in addition to the weights in the performance aggregation scheme (see Section 3). We also discuss some possible extensions.
5.1.Supporting hyperplanes
Note that the first type of constraints in Models (5) and (6) are identical. This formulation takes the joint technology composed by the two technologies as one technology. In DEA, these constraints associate with a hyperplane, −xu∗+ yv∗− w(γi∗−γo∗)= 0, of the joint technology
T1∩ T2 (also termed the supporting hyperplane; see e.g., Banker et al. (1984)). All (xr, wr, yr) r ∈ Rare either on this
hyperplane (if= holds) or beneath it (<). (xk, wk, yk)
satis-fying the equation is on the supporting hyperplane (bound-ary, frontier), and k is efficient. “<” indicates (xk, wk, yk) is
beneath the hyperplane and withinT1∩ T2, and there must exist (xr, wr, yr) r ∈ R\{k} on the plane. In this case, this
hyperplane is the one on which k should project.
(−u∗, v∗, −(γi∗−γo∗)) is the normal vector for the
hy-perplane −xu∗+ yv∗− w(γi∗−γo∗)= 0, and u∗≥ 0 and v∗≥ 0 in standard DEA. Comparing the normal vector val-ues,−(γdi∗− γdo∗)< 0 d ∈ D indicates that the hyperplane’s orientation on d is like the orientation of the inputs, since it has the same sign with−u∗. In contrast,−(γi∗
d − γdo∗)> 0
d ∈ D suggests that the hyperplane’s orientation on d is like
an output. This observation gives geometric implication on the sign ofγi∗
d − γdo∗, and the input/output behavior
judg-ing result is consistent with the results discussed in Sections 3.2 and 3.3.
5.2.Economic implications
u and v are dual variables corresponding to the input and output constraints, and u∗and v∗specify the shadow prices for the inputs and outputs. γo and γi are dual variables associated with constraints of dual-role factors in their out-put and inout-put roles, respectively, andγo∗and γi∗ are the shadow prices for their output and input roles.γo∗−γi∗(or γi∗−γo∗) is thus the (additive) composited shadow prices for dual-role factors.γo∗
d − γdi∗ can be deemed as the “net
value” per unit of d.γdo∗− γdi∗> 0 suggests an additional unit of d gives a net benefit. γo∗
d − γdi∗< 0 indicates the
input tendency of d since it shows a net cost. Regardless of the setting of evaluation models—e.g., (P1) and (P2)—the shadow price gives a simple and consistent interpretation.
Without loss of generality, consider the boundary (hyper-planes, isoquant) ofψ(y) as defined in Section 3.1. Every point on the boundary has a corresponding normal vector (−u∗, −(γi∗−γo∗)). A positive vector entry value, say en-try i, indicates that the boundary at this point is backward bending in the dimension i and a negative marginal pro-ductivity of the corresponding factor i. Namely, increasing factor i at this point causes a decline in level of y.
If−(γi∗
d − γdo∗)> 0 d ∈ D, d has negative marginal
pro-ductivity on y, and an increase in d will not be in ψ(y) but in aψ(y) such that y ≤ y, since ψ(y) is nested in y. It also suggests that the composted shadow priceγi∗
d − γdo∗
is negative. Bear in mind that the discussion here concerns the projected boundary corresponding to (xk, wk, yk), not
(xk, wk, yk) itself; the projecting hyperplane depends on the
projection paths, such as radial or non-discretionary. The change along the boundaries of a dual-role factor
d’s orientation and composted shadow priceγi∗
d − γdo∗
re-flect the tendency and importance of its input/output role. The input-output tendency is for factor d but not for any particular point under evaluation. That is, we consider cases without inefficiency and show the interactions among d and other factors. Starting from a small value for d, the major role of d is to support regular inputs to produce regular out-puts at this stage. It should be noted that a reduction in d leads to less regular outputs. The composted shadow prices are relative prices among d and other inputs and relate to the marginal rates of substitution in consumption. The law of diminishing marginal rates of substitution, which results inψ(y) being convex, can be observed in this region (e.g., Fig. 4).
Unlike regular inputs, increasing d comes with the cost of consuming regular inputs. As mentioned, nurse trainees who need guidance from certified staff consume the hospi-tal’s resources of staff time and labor. This phenomenon is significant when the value of dual-role factor d is large; a further increasing d has a negative impact on regular out-puts. Thus, we will observe a backward bending hyperplane and the negative composted shadow priceγdi∗− γdo∗.
Increasing d with negative impact on y does not neces-sarily imply that an increasing d is “bad.” In contrast, d contributes to overall output directly, because d itself is an important component of outputs, yet it does not contribute indirectly as an input to generate regular outputs. There-fore, the net benefit derives more from the output side than the input side. The corresponding hyperplane orientation and shadow price behave like an output. Increasing d at this stage yields a reduction of y, which is the tradeoff between
d and y or, in the real world, the possible transformation in
production between d and regular output.
5.3.Extensions
It is easy and straightforward to extend our proposed mod-els and to interpret the results. Assumptions of variant DEA models can be classified (Cherchye and Post, 2003) as (i) production technology, the underlying characteris-tics of the input/output transformation processes, such as assumptions 1 to 3 listed in Section 2; (ii) the data gen-erating process, the sample data used to estimate T; and (iii) the objective function, the uses of T or its estimation; e.g., (P1) and (P2). Any DEA model is a combination of these three components. Discussions of the variant tech-nologies can be found in Kuosmanen (2003) and Briec et al. (2004).
Based on the three classifications and given a sample data set, it is simple to model different combinations of pro-duction technologies and objective functions. For example, both (P1) and (P2) are based on the technologies assuming CRS and convexity, which may be strong in some cases.
When we adopt VRS together with assumptions 1 and 2, the joint technology is
(x, w, y) : r∈R xrλr ≤ x; r∈R yrλr ≥ y; r∈R wrλr ≥ w; r∈R xrλr ≤ x; r∈R yrλr ≥ y; r∈R wrλr ≤ w; r∈R λr = 1; λr ≥ 0, r ∈ R . (7)
Furthermore, if we relax the convexity, the FDH technol-ogy is applied as (x, w, y) : r∈R xrλr ≤ x; r∈R yrλr ≥ y; r∈R wrλr ≥ w; r∈R xrλr ≤ x; r∈R yrλr ≥ y; r∈R wrλr ≤ w; r∈R λr = 1; λr = {0, 1} , r ∈ R . (8)
Given any technology approximations such as Models (7) and (8), we can apply different objectives with differ-ent interpretations. For example, if the dual-role factors are fixed, we can apply the objective similar to (P2) to Model (7) or (8). Note that deriving the performance eval-uation model is more flexible and easier to interpret than using the ratio form proposed by (P1R) and (P2R). For ex-ample, there is no ratio form if FDH is applied, and the ratio will not be similar to standard VRS or CRS DEA models if we use other objective functions, such as additive models, cost minimization, or profit maximization. Moreover, it is simple to measure the distance function (Shephard, 1970) for any given (x, w, y) using the joint technology proposed. Thus, we can measure the Malmquist productivity index and its decomposition (F¨are et al., 1994b) for cases with dual-role factors.
6. Conclusions
Typical DEA studies consider processes transforming in-puts to outin-puts. In some cases, however, some factors can be both inputs and outputs simultaneously. Such factors are termed dual-role factors and their ambiguous role def-initions make performance evaluation challenging. Rather than proposing an ad hoc model directly, this article pro-posed an axiomatic framework using joint technology, which is developed based on intuitive thinking. Under the proposed joint technology, evaluation models can be mathematically derived and/or axiomatically validated. We noted that variant models based on different assumptions and needs can be easily but rigorously extended. We showed that the input/output behavior of dual factors can be ex-plained from different perspectives such as multi-criteria
performance aggregation, geometry, and economics. We developed a simple three-dimensional case that we also generalized to multi-dimensional cases for considering the multiple dual-role factors. We found that the input/output tendency of a dual-role factor is a property on the projected boundary, not the data point itself. Different projecting paths associated with different objectives produced differ-ent weights and differdiffer-ent implications. We concluded that the weights relate to the ideal performance improvement target. In other words, the benchmark on the boundary is the status after the improvement and does not imply the future improvement.
The finding of negative weights for dual-role factors in
ψ(y) suggests that increasing dual-role factors comes with
a cost. We can simplify two types of constraints associated with dual-role factors as equality constraints when com-puting efficiency scores, but the corresponding dual vari-ables should be interpreted with care. The literature imposes weak disposability to model the cases where increasing a factor comes with a cost (e.g., F¨are et al. (1994a)). Noting that the real connection to weak disposability is beyond the scope of this article, we propose it as a topic for fu-ture research. In addition, more research is needed on RTS and the rate of technical substitutability for the production process with dual-role factors.
Funding
This research is partially supported by grants from the Na-tional Science Council, Taiwan (NSC 99-2628-E-009-087 and NSC 100-2628-E-009-005).
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Biography
Wen-Chih Chen is an Associate Professor at the Department of Industrial Engineering and Management, National Chiao Tung University, Taiwan. He received his B.S. in Industrial Engineering from National Tsing Hua University, Taiwan, and his M.S. and Ph.D. from the H. Milton Stewart School of Industrial and Systems Engineering at Georgia Institute of Technology. His research interests include pro-ductivity and efficiency analysis, material handling, and semiconductor manufacturing.
