Duality and profit efficiency for the hyperbolic measure model

European Journal of Operational Research 278 (2019) 410–421

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Duality and profit efficiency for the hyperbolic measure model Margaréta Halická, Mária Trnovská∗ Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Mlynská dolina, Bratislava 842 48, Slovakia

a r t i c l e

i n f o

Article history: Received 14 July 2017 Accepted 3 December 2018 Available online 13 December 2018 Keywords: Data envelopment analysis Semidefinite programming Hyperbolic distance function Return to dollar Profit efficiency

a b s t r a c t The hyperbolic measure (HM) model is a radial, non-oriented model that is often used in Data Envelopment Analysis (DEA). It is formulated as a non-linear programming problem and hence the conventional linear programming methods, customarily used in DEA, cannot be applied to it in general. In this paper, we reformulate the hyperbolic measure model in a semidefinite programming framework which opens the way to solving the HM model by reliable and efficient interior point algorithms and allows us to benefit from simple primal-dual correspondence in semidefinite programming. We derive the dual of the HM model and so, for the first time, establish its multiplier form. We also offer an economic interpretation of the dual HM model via a comparison with the multiplier form of the directional distance model and relate the HM score to the so-called Nerlovian profit efficiency. © 2018 Elsevier B.V. All rights reserved.

1. Introduction The hyperbolic distance function, introduced by Färe, Grosskopf, and Lovell (1985) as a measure of technical efficiency, combines Farrell’s input- and output-oriented radial measures into one measure function that allows simultaneous radial contraction of inputs and expansion of outputs using a single parameter. This feature of the hyperbolic distance measure (HM) has proved to be very useful especially for environmental and other applications with undesirable inputs or outputs. (See Färe, Margaritis, Rouse, & Roshdi, 2016 for further references in this direction.) On the other hand, HM leads to a non-linear convex programming problem and hence requires application of convex programming techniques. Nevertheless, Färe, Grosskopf, and Zaim (2002) show that under constant returns to scale (CRS) the HM score equals the square root of the Farrell output efficiency score or the reciprocal of the square root of the Farrell input score (Farrell, 1957). Consequently, under CRS, the HM score can be computed by means of linear CCR input- or output-oriented models. Moreover, a connection with the return-to-the-dollar notion was established by Färe et al. (2002) in the CRS case and, by Zofí o and Prieto (2006) in the variable returns to scale (VRS) case within a more general framework. In more general settings, however, merely approximation techniques for estimating the HM score are available. Färe, Grosskopf, Lovell, and Pasurka (1989) suggest a linear approximation of HM,

∗

Corresponding author. E-mail addresses: [email protected] (M. Halická), [email protected] (M. Trnovská). https://doi.org/10.1016/j.ejor.2018.12.001 0377-2217/© 2018 Elsevier B.V. All rights reserved.

which is tight enough only near the boundary of the technology set. Färe et al. (2016) present a connection to the direction distance measure (DDM) and propose a linear programming based algorithm for estimating the exact value of the HM score, projections and dual prices. The linear DEA models are usually formulated both in the envelopment and the multiplier forms that are connected together as a pair of primal-dual linear programming problems. Although the envelopment form of any model has a direct economic interpretation, also its dual, multiplier form is interpretable through shadow input and output prices.1 Note that the multiplier forms are useful in the return-to-scale measurement or when some additional information (tradeoffs) on weights must be incorporated in the model. Moreover, the multiplier form of specific models is helpful in establishing links between measures of technical efficiency and various notions of profit inefficiency: the so-called loss function (Aparicio, Borras, Pastor, & Zofío, 2016; Pastor, Lovell, & Aparicio, 2012) the lost profit on outlay (Aparicio, Pastor, & Ray, 2013); and the Nerlovian profit inefficiency (Aparicio, Borras, & Pastor, 2015; Aparicio, Ortiz, & Pastor, 2017; Chambers, Chung, & Färe, 1998; Cooper, Pastor, Aparicio, & Borras, 2011; Färe & Grosskopf, 20 0 0; Halická & Trnovská, 2018a). Nevertheless, to the best of our knowledge, no dual of the hyperbolic measure model has been derived and no relations with the loss function and the profit inefficiency have been established for the HM model.

1 The duality relationship between the general input distance function and the cost function can be established even without the data envelopment framework as is seen from the results of Aparicio and Pastor (2011).

M. Halická and M. Trnovská / European Journal of Operational Research 278 (2019) 410–421

To make headway in the analysis of the HM model, one may profitably use the results of modern (conic) convex optimization (see e.g. Ben-Tal & Nemirovski, 2001; Boyd & Vandenberghe, 2004). These results provide •

•

efficient and reliable interior point methods for solving the problems, which are implemented in a number of available solvers (see e.g. Alizadeh, Haeberly, & Overton, 2006; Helmberg, Rendl, Vanderbei, & Wolkowitz, 2006); a straightforward technique for development of the dual counterparts of convex conic problems and the possibility of applying the results of duality theory (see e.g. Todd, 2001; Trnovská, 2005).

The advantages of convex optimization are accessible as easily in convex conic problems as they are in linear programming. It only suffices to recognize a conical structure of the problem and formulate it in a suitable form, whether as a second-order cone or a semidefinite cone. Sueyoshi and Sekitani (2007) were the first authors in DEA to exploit results from convex conic optimization. They formulated the Russell graph measure (RM) model as a second-order cone programming problem, solved it by means of the corresponding solvers and derived the dual model. Halická and Trnovská (2018a) obtained similar results for the RM model by means of its semidefinite reformulation. However, the multiplier form of the RM model derived by Halická and Trnovská is much more transparent than that obtained by Sueyoshi and Sekitani and has allowed to establish a relation to profit inefficiency and the loss function. In this paper we apply the idea from Halická and Trnovská (2018a) on the case of the hyperbolic measure (HM) model: we formulate the HM model as a semidefinite programming problem and benefit from the duality theory in semidefinite programming. Note that the process of the application to the HM model is not quite straightforward and some specific features of the model must be taken into account. As a result we obtain several new and useful results about the HM model. The paper is organized as follows. In Section 2 we recall the HM model and present some of its properties. In Section 3 we introduce a semidefinite formulation of the HM model and derive a dual to the HM model in a semidefinite programming setting. We also prove strong duality and establish useful implications of complementarity conditions. In Section 4 we reformulate the HM dual into a non-linear programming setting and describe some of its properties. In Section 5 we discuss an interpretation of the HM dual in comparison with the DDM dual. In Section 6 we use the HM dual to reveal relationships of the HM inefficiency with both the loss function concept and the Nerlovian (normalized) profit inefficiency. In Section 7 we derive a relation of the HM (VRS) efficiency to return-to-dollar efficiency. Finally, in Section 8 we present two illustrative examples demonstrating the theoretical results, methodology of decomposing the normalized profit inefficiency and return-to-dollar efficiency, as well as computational tractability and applicability to the HM modification for undesirable outputs. 2. Preliminarities 2.1. Formulation of the model Consider a set of n decision making units DMU j ( j = 1, . . . , n ), each consuming given amounts of m inputs xij (i = 1, . . . , m ) to produce s outputs yrj (r = 1, . . . , s ). Assume that all input and output amounts are positive. Let xj and yj denote the m- and s-dimensional column vectors of inputs and outputs of DMUj ( j = 1, . . . , n ), respectively. Moreover, let X = (x1 , . . . , xn ) and Y =

411

(y1 , . . . , yn ) denote the m × n and s × n matrices of the input and output data, respectively. By o ∈ {1, . . . , n} we denote the index of DMU to be currently evaluated. In order to evaluate DMUo = (xo, yo ), (o ∈ {1, . . . , n} ), the HM model formulates the following nonlinear program:

θ

min θ ,λ

(1)

X λ ≤ θ xo ,

s.t.

1

Yλ ≥

θ

(2)

yo ,

(3)

λ ≥ 0,

(4)

eT λ = 1,

(5)

where λ, an n-dimensional column vector, and θ , a positive scalar, are the decision variables. Here e denotes the column vector of ones and the superscript T signifies transposition. This model, as described by (1)–(5), associates the hyperbolic graph measure with the production possibility set



T = (x, y ) ∈ Rm × Rs

 | X λ ≤ x, Y λ ≥ y, λ ≥ 0, eT λ = 1 ,

(6)

corresponding to VRS. To obtain the CRS version of the production set and the model, the condition (5) is removed from (6) and from the constraints of the model. To distinguish between the two versions of the HM model we will use the notation HM-V and HM-C, respectively. 2.2. Interpretation and basic properties The HM model has a pleasing geometric interpretation from which it derives its name. Minimization over θ takes us along the hyperbolic curve (θ xo, θ1 yo ) from the evaluated point (xo , yo ) at θ = 1 to a point at the boundary of T . The value of θ , at which the curve reaches the boundary ∂ T of T , is the optimal value of the HM model and is denoted by θ ∗ . The point (θ ∗ xo, θ1∗ yo ) is called the hyperbolic projection of (xo , yo ) onto ∂ T . Clearly, θ ∗ ∈ (0, 1] is the score of technical efficiency and it is unit invariant. If no movement is possible, the point (xo , yo ) belongs to ∂ T and its score is equal to one. Let us note that similarly as in the other radial measures, the hyperbolic model may project (xo , yo ) on the weakly efficient component of ∂ T and hence some part of its inefficiency is not captured. This can be identified by the standard second phase procedure, in which the slacks in inequalities (2) and (3) are maximized under the fixed optimal value of θ ∗ . 2.3. Connection to CCR. It is well known that the HM-C model can be transformed into both the input or the output oriented CCR models (Färe et al., 2002). In fact, if we multiply the inequalities in (2)–(4) by θ > 0 and then use notation λI := λθ , θ I := θ 2 , we obtain the input oriented CCR model (CCR-I). Hence, the following relations hold between the optimal solution (θH∗ , λ∗H ) to the HM-C model given by (1)–(4) and the solution (θI∗ , λ∗I ) to the CCR-I model :

θH∗ =



θI∗ and λ∗H =

λ∗I . θI∗

Let us note that a similar transformation does not work when applied to the HM-V, or to those environmental modifications of

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the HM model where some of input/output components are not being contracted/extended. In Introduction we claim that no dual for the HM model has been derived so far. However, based on the correspondence between the solutions of HM-C and CCR-I described above, one could argue that our claim is not true for the special case of HM-C, since the dual of CCR-I is available. Nevertheless, existence of this correspondence does not mean that the dual of CCR-I could be considered to be a dual of HM-C in the usual mathematical programming sense. Later, after we establish the dual of HM in a general form, we also describe a correspondence between the duals of CCR-I and HM-C in Section 4. 2.4. Linearization and connection to directional distance measure Since θ ∈ (0, 1], we have δ := 1 − θ ∈ [0, 1 ). Substituting θ = 1 − δ into the inequalities in (2) and (3) and using linearization around δ = 0, i.e.

1 ≈ ( 1 + δ ), 1−δ we obtain the linear model:

min{1 − δ | X λ ≤ (1 − δ )xo, Y λ ≥ (1 + δ )yo, δ,λ

λ ≥ 0, eT λ = 1}. (7)

Note that (7) is a version of the directional distance measure (DDM) model (introduced by Chambers et al., 1998) with a special choice of the directional vector (gx , gy ) = (xo, yo ). Hence, the DDM model in (7) represents a linearization of the HM model. This fact was observed already in Färe et al. (2016). However, we can prove even more. Since

1 ≥ 1 + δ, 1−δ we have that Y λ ≥ ≥ (1 + δ )yo implying that the feasible set of the HM model is a subset of the feasible set for the DDM model (7). This implies that 1 y 1−δ o

θ ∗ ≥ 1 − δ∗,

θ Y λ ≥ yo , which can be viewed as s hyperbolic constraints of the type u(aT v ) ≥ c with u > 0 and c > 0. Such constraints can be equivalently reformulated as second-order cone constraints,

 √   2 c  T   u − aT v ≤ u + a v, 2

or as semidefinite constraints,



3. Semidefinite programming approach The HM model formulates a convex nonlinear program on a reasonable domain (where θ > 0), and therefore it can be solved This type of approximation is used also in a recent application by Widiarto, Emrouznejad, and Anastasakis (2017).

√  c  0. aT v

√u c

However, we opt for a different approach based on the idea of semidefinite relaxation, which leads to a simple dual that can be interpreted in a standard way. 3.1. Semidefinite relaxation In this approach, the first step is to introduce a new variable

φ satisfying φ = θ1 , and then relax this (non-convex) equality with a hyperbolic inequality φθ ≥ 1. This leads to an SDP program with just one semidefinite constraint:

θ

(9)

s.t. X λ ≤ θ xo,

(10)

Y λ ≥ φ yo ,

(11)

min

θ ,φ ,λ

(8)

which means that the DDM model (7) provides a lower bound for the HM model. The same result can be obtained by the approximation θ1 ≈ (2 − θ ) used by Färe et al. (1989)2 . It leads to a linear model differing from the HM model only in the inequality (3) which after the approximation takes the form Y λ ≥ (2 − θ )yo. Then, the substitution θ = 1 − δ leads to the problem (7). Let us note that the optimal solution of (7) represents a tight enough lower bound for the exact values of the hyperbolic measure only if θ ∗ is close to one, i.e. for those DMUs that are close to the boundary of T . This is a simple consequence of the fact that the linearization was carried out at θ = 1. Nevertheless, version (7) of the DDM model is a well defined alternative to the HM model and it becomes popular in applications. It is usually formulated as a measure of inefficiency δ , where minimization of 1 − δ is replaced by maximization of δ in (7). Moreover, the model (7) is used as a first step of an algorithm for solving the HM model by Färe et al. (2016).

2

with standard convex programming solvers. However, as mentioned in Section 1, an explicit recognition of the conic structure of the problem is the first step to accessing the benefits of modern convex optimization theory. There are multiple ways to reformulate the problem (1)–(5) as a conic one. Since the data are positive, the first constraint implies θ > 0. The second constraint is therefore equivalent to

P :=

 θ 1

1

φ

  0,

(12)

λ ≥ 0,

(13)

eT λ = 1.

(14)

Here we use the standard notation P  0 for positive semidefinite matrix P. Note that for a 2 × 2 matrix an equivalent condition for P  0 is that the diagonal entries and det P are non-negative, which explains the SDP reformulation. The formulation (9)–(13) corresponds to the CRS case and the formulation (9)–(14) corresponds to the VRS case. This kind of relaxation was applied to the HM-V model by Johnson and McGinnis (2009), who proved that the original and the relaxed problems give the same optimal value. In the next theorem we relate not only the optimal values but also the optimal solutions of both models separately for CRS and VRS. We show that the semidefinite constraint is tight in the case of CRS, but need not be tight in any optimal solution of (9)–(14) (the VRS case), though the tight solution always exists3 . These observations will be important later in the proof of Theorem 4. 3 Note that this phenomenon does not occur in the case of the semidefinite reformulation of the RM model used by Halická and Trnovská where semidefinite constraints are always tight.

M. Halická and M. Trnovská / European Journal of Operational Research 278 (2019) 410–421

Theorem 1. (a) CRS case: If (θ ∗ , λ∗ ) is an optimal solution to (1)– ˆ ) is (4), then (θ ∗ , θ1∗ , λ∗ ) is an optimal solution to (9)–(13). If (θˆ , φˆ , λ ˆ ) is an optimal an optimal solution to (9)–(13), then θˆ = 1 and (θˆ , λ φˆ

solution to (1)–(4). (b) VRS case: If (θ ∗ , λ∗ ) is an optimal solution to (1)–(5), then ˆ ) is an op(θ ∗ , θ1∗ , λ∗ ) is an optimal solution to (9)–(14). If (θˆ , φˆ , λ ˆ ) is an optimal solution to (1)– timal solution to (9)–(14), then (θˆ , λ ˆ ) to (9)–(14) (5). Moreover, if there exists an optimal solution (θˆ , φˆ , λ such that θˆ φˆ > 1, then each component of the s-dimensional vector ˆ ) in inequality of slacks corresponding to the optimal solution (θˆ , φˆ , λ y ˆ ˆ (11) is positive, i.e. sˆ := Y λ − φ yo > 0. Proof. (a) Assume that (θ ∗ , λ∗ ) is an optimal solution of (1)–(4). Then clearly (θ ∗ , θ1∗ , λ∗ ) is feasible for (9)–(13) and hence θ ∗ ≥θˆ . ˆ ) is optimal for (9)–(13), then the constraints Conversely, if (θˆ , φˆ , λ ˆ ≥ φˆ yo ≥ 1 yo. Therefore (θˆ , λ ˆ ) is feasible for (11), (12) imply Y λ θˆ

θ ∗ ≤θˆ .

θ∗

(1)–(4) and On combining the inequalities for and θˆ we ˆ ) be an opget θ ∗ = θˆ . It remains to show that θˆ = 1ˆ . Let (θˆ , φˆ , λ φ

timal solution of (9)–(13) such that θˆ > equalities (10), (11) by 1

θˆ φˆ

1

φˆ

. Then multiplying in-



we obtain that also



is a feasible solution to (9)–(13). Moreover,

θˆ , φˆ



φˆ λˆ , θˆ θˆ φˆ



θ

3.2. The dual HM formulation and duality results In this section we introduce the dual of the HM model and some duality results. The derivation of the dual is based on the standard Lagrange function approach and is stated in the following theorem: Theorem 2. The dual to the semidefinite form (9)–(14) of the HM-V model is of the form

− σ − 2z,

max

(15)

s.t. Y T u − X T v ≤ σ e,



1 − vT xo z

z uT yo

(16)

  0,

(17)

Proof. To derive the dual of the problem (9)–(14), we introduce s dual variables v ∈ Rm + , u ∈ R+ , σ ∈ R corresponding to the constraints (10), (11) and (14), respectively. The dual variable corresponding to the semidefinite constraint P  0 in (12) is a 2 × 2 symmetric, positive semidefinite matrix Z  0 so that the trace of the product of matrices P and Z is nonnegative4 , i.e.

P=

1

L(θ , φ , λ; v, u, σ , Z ) = θ − vT (θ xo − X λ ) − uT (Y λ − φ yo ) − σ (1 − eT λ ) − tr (P Z ) = −σ − 2z + θ (1 − vT xo − z1 ) + φ (uT yo − z2 ) + λT (X T v − Y T u − σ e ), where λ ≥ 0, u ≥ 0, v ≥ 0 and Z  0. The Lagrangian is linear in the primal variables θ , φ , λ and hence the corresponding dual is

max

u,v,Z,σ

− σ − 2z

(20)

s.t. Y T u − X T v ≤ σ e,

(21)

z1 = 1 − vT xo ,

(22)

z2 = uT yo ,

(23)

u, v ≥ 0, Z  0.

(24)

If we substitute (22), (23) into Z, we obtain the formulation (15)– (18), as stated in the theorem.  We proceed to derive several duality results. We will only focus on the VRS case, since results for the CRS case can be obtained analogously. First, we prove the weak and strong duality property for the primal dual pair (9)–(14) and (15)–(18). In particular, we specify the relation between the optimal values and the existence of the optimal solutions of both programs. These properties follow from the general theory if the Slater conditions are satisfied, i.e. if there exist interior points in the primal and dual feasible set (see e.g. Boyd & Vandenberghe, 2004; Todd, 2001; Trnovská, 2005). Theorem 3. (a) Weak duality. Between the feasible solutions (θ , φ , λ) of (9)–(14) and (u, v, z, σ ) of (15)–(18) the relation θ ≥ −σ − 2z holds. (b) Strong duality. Both the primal and the dual attain optimal solutions and the optimal value of the primal is equal to the optimal value of the dual, i.e. θ ∗ = −σ ∗ − 2z∗ . (c) Complementarity conditions. Assume that (θ ∗ , φ ∗ , λ∗ ) is the optimal solution of (9)–(14) and (u∗ , v∗ , z∗ , σ ∗ ) is the optimal solution of (15)–(18). Then the following complementarity conditions hold:

(Y λ∗ − φ ∗ yo )T u∗ = 0,

(25)

(18)

The version for CRS can be obtained by setting σ = 0.

θ

In this case, the Lagrangian is

(X λ∗ − θ ∗ xo )T v∗ = 0,

u, v ≥ 0.



(19)

θˆ < θˆ which contraφˆ

dicts optimality of θˆ . (b) The first part of the statement can be shown analogously as in a). We now show that the slack sˆy is positive, provided θˆ φˆ > ˆ ) is an optimal solution of (9)–(13). This follows 1, where (θˆ , φˆ , λ ˆ ) is also a solution and hence Y λ ˆ ≥ from the fact that (θˆ , 1/θˆ , λ 1 ˆ yo .  y > φ o ˆ

u,v,z,σ

tr (P Z ) := θ z1 + φ z2 + 2z ≥ 0.

413

1

φ





 0,

Z :=

z1 z

z z2



 0,

4 The trace tr(PZ) represents an inner product in the vector space of symmetric 2 × 2 matrices.

(Y T u∗ − X T v∗ − σ ∗ e )T λ∗ = 0,

(1 − eT λ∗ )σ ∗ = 0,

tr (P ∗ Z ∗ ) := θ ∗ z1∗ + φ ∗ z2∗ + 2z∗ = 0.

(26)

(27)

Proof. (a) The weak duality is a standard property of any primaldual pair and the result follows from the construction of the dual. (b) To show the strong duality result we prove that both the primal and the dual satisfy the Slater condition, i.e. we show that a point in the relative interior of the primal and dual feasible set exists. More specifically, the Slater condition for the primal problem ¯ , θ¯ , φ¯ ≥ 0 such that can be formulated as follows: there exist λ

¯ ≤ θ¯ xo, Xλ

¯ ≥ φ¯ yo, Yλ

λT e = 1, θ¯ φ¯ > 1.

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¯ := 1 e = 1 (1, 1, . . . , 1 )T , x¯ := X λ ¯ = 1 Xe and y¯ := Y λ ¯ = We set λ n n n y¯ r 1 ¯ Ye. Then φ = minr r > 0 and it suffices to choose n

yo



θ¯ = max max j

¯j

x

xoj

1 φ¯

,



+ ε,

for some value ε > 0. Hence the primal is strictly feasible. The Slater condition for the dual reads: there exist u¯ , v¯ ≥ 0, z¯, σ such that

Y T u¯ − X T v¯ ≤ σ e,

1 − v¯ T x0 > 0, u¯ T y0 > 0, (1 − v¯ T x0 )(u¯ T y0 ) > z¯2 .

If we set z¯ = 0, σ = 0, the Slater condition becomes the following: there exist u¯ , v¯ ≥ 0 such that T

T

T

Y u¯ − X v¯ ≤ 0,

v¯ xo < 1, u¯ yo > 0.

Now we need to guarantee that

For this purpose it is sufficient to choose ε as ε = μ mini

xTi e . y¯ i

Bellow, we establish several interesting duality results that follow from the complementarity conditions stated in Theorem 3. Theorem 4. Complementarity implications for the VRS case. Assume that (θ ∗ , φ ∗ , λ∗ ) is the optimal solution of (9)–(14) and that (u∗ , v∗ , z∗ , σ ∗ ) is the optimal solution of (15)–(18). Then the following relation holds: (i) If there is no solution of (9)–(14) such that θ ∗ > φ1∗ , then

σ ∗ < −2z∗ ,

θ ∗2 =

) − ( z ) = 0, ∗ 2

θ∗ − σ∗ θ∗ + σ∗ ∗ , yTo u∗ = θ . ∗ 2θ 2

φ∗ =

1 θ ∗ into the complementarity

z∗ z1

θ∗ = − ∗ .

(33)

From the former we obtain θ ∗ = 1/φ ∗ = −z2∗ /z. So, we have two alternative expressions for z∗ :

−z∗ =

z2∗

θ∗

and − z∗ = θ ∗ z1∗

(34)

yielding (30). On the other hand, by substituting the two expressions for z∗ from (34) into the strong duality property θ ∗ = −σ ∗ − 2z∗ we obtain (after some simple algebraic manipulation)

θ∗ + σ∗ 2

θ ∗ and z1∗ =

θ∗ + σ∗ , 2θ ∗

(35)

which yields (31). (ii) The assumption of θ ∗ > φ1∗ implies regularity of P. Then, by Lemma 1 from Appendix and (27) we obtain that P vanishes and hence z1∗ = z2∗ = z∗ = 0 implying the statements in (ii).  Remark 1. The CRS case: To obtain results for the CRS case analogous to the statements of Theorems 3 and 4(i) it suffices to set σ = 0 or σ ∗ = 0. It is interesting that then (31) implies

θ ∗2 = 2yTo u∗ =

yTo u∗ yTo u∗ = . xTo v∗ 1 − xTo v∗

Moreover, if θ ∗ = 1, then φ ∗ = 1 and xTo v∗ = yTo u∗ = −z∗ =

1 2.

Remark 2. If, in addition to the assumption of Theorem 4, the optimal value satisfies θ ∗ = 1, then we obtain

xTo v∗ =

1 − σ∗ , 2

yTo u∗ = −z∗ =

1 + σ∗ 2

in case (i) and

σ ∗ = −1, xTo v∗ = 1, yTo u∗ = 0 = z∗ (28)

yTo u∗ , 1 − xTo v∗

xTo v∗ =

z∗ and z2∗

xTo v∗ = 1/2 and

3.3. Duality implications

v )(

φ∗ = −

z2∗ =

We have shown that both the primal and the dual SDP formulation is strictly feasible. From the SDP duality theory it follows that both of the problems have optimal solutions and the optimal values are equal. (c) The complementarity conditions directly follow from the duality theory in semidefinite programming. 

(1 −

1 φ ∗ and then alternatively

condition (27) and solve the resulting quadratic equations in φ ∗ and θ ∗ , respectively. Here we use the fact that due to (32), the discriminant in both equations vanishes. We obtain the solutions

o

ε y¯ i ≤ μxTi e, for all i = 1, . . . , n.

y∗o u∗

θ∗ =

T

We can now just set u¯ = ε e, v¯ = μe for some sufficiently small positive ε , μ. Since μeT xo < 1, we choose μ ∈ (0, eT1x ). Denote Y T e = y¯ .

xTo ∗

which proves (29). Using Lemma 1 and (27) again, we get θ ∗ z1 = 1 θ ∗ z2 from which (30) follows. To prove (31) we first substitute

(29)

(30)

in case (ii). Theorem 4 has the following interesting corollary providing a geometric insight into the HM model: Corollary 1. Let (θ ∗ , λ∗ ) be the optimal solution of (1)–(5) and let (u∗ , v∗ , σ ∗ ) be the optimal solution of (20)–(24). Denote by (xˆ, yˆ ) = (θ ∗ xo, 1/θ ∗ yo ) the projection point of the DMUo . Then (a) yˆT u∗ + xˆT v∗ = θ ∗ ; (b) yˆT u∗ − xˆT v∗ = σ ∗ ; and (c) H∗ = {(x, y ) | yT u∗ − xT v∗ = σ ∗ } is a supporting hyperplane of the set T at the point (xˆ, yˆ ).

Proof. In this proof we recycle the definition of z1 and z2 from the proof of Theorem 1, i.e.

Proof. The statements (a) and (b) directly follow from Theorem 4, more specifically, from the statement (31) of (i), or from the statements of (ii). We now prove (c). By (b) we have that (xˆ, yˆ ) ∈ H∗ . It remains to show that for any (x, y ) ∈ T it holds yT u∗ − xT v∗ ≤ σ ∗ . ¯ ≥0 Let (x¯, y¯ ) be an arbitrarily chosen point in T . Then for some λ ¯ = 1 we have X λ ¯ ≤ x¯ and Y λ ¯ ≥ y¯ . Since u∗ , v∗ ≥ 0 it such that eT λ holds

z1∗ := 1 − xTo v∗ and z2∗ := yTo u∗ .

(X λ¯ )∗ v∗ ≤ x¯T v∗ , (Y λ¯ )T u∗ ≥ y¯ T u∗ .

(i) The property (28) follows from the strong duality result in Theorem 3, since −σ ∗ − 2z∗ = θ ∗ > 0. To prove the other statements of (i) we use θ ∗ = φ1∗ , which implies that P in (12) is

From these relations and the complementarity conditions in (26) we obtain

(31)

(ii) If there is a solution of (9)–(14) such that θ ∗ > φ1∗ , then xTo v∗ = 1, yTo u∗ = z∗ = 0, and −σ ∗ = θ ∗ > 0.

nonzero but singular. Using Lemma 1 from Appendix and the complementarity condition (27), we obtain that also Z is a nonzero singular matrix and hence

z1∗ z2∗ = z∗2 ,

(32)

¯ = σ ∗, ¯ T (Y T u∗ − X T v∗ ) = σ ∗ eT λ y¯ T u∗ − x¯T v∗ ≤ λ which proves (c).



Remark 3. Similar results can be obtained analogously for the CRS case, where the supporting hyperplane of the CRS technology set at

M. Halická and M. Trnovská / European Journal of Operational Research 278 (2019) 410–421

the point (xˆ, yˆ ) passes through the origin, i.e. H∗ = {(x, y ) | yT u∗ − xT v∗ = 0}. Theorem 4 and Corollary 1 lead to the following result that will be useful in Section 4.1. Corollary 2. Let (u∗ , v∗ , z∗ , σ ∗ ) be the optimal solution of (15)–(18). Then



σ ∗ = max yTj u∗ − xTj v

 ∗

j

= max



(x,y )∈T

yT u∗ − xT v

 ∗

σ∗

and hence can be interpreted as a maximum shadow profit (i.e. the maximal profit in shadow prices u∗ , v∗ ) over T . Proof. Since

( u ∗ , v∗ , z ∗ , σ ∗ )

is feasible, it clearly holds

yTj u∗

−

xTj v∗ ≤ σ ∗ for all j = 1, . . . , n. For the first equality, it is suffi-

415

u, v ≥ 0.

(43)

Note that the second constraint in (42) is redundant since we assume that the data is positive. If we rewrite the objective (40) as follows:

2



(1 − vT xo )(uT yo ) = 1 + uT yo − vT xo   − 1 − vT xo + uT yo − 2 (1 − vT xo )(uT yo )  2  = 1 + uT yo − vT xo − 1 − vT xo − uT yo ,

we get the formulation as stated in (36)–(39).

(44)



cient to show that there exists 1 ≤ l ≤ n such that σ ∗ = yTl u∗ − xTl v∗ . Assume by contradiction that yTj u∗ − xTj v∗ < σ ∗ for all j = 1, . . . , n. Then by complementarity condition (26) we have λ∗ = 0, which is infeasible due to (14). The second equality easily follows from Corollary 1 (c) and the definition of the supporting hyperplane. The maximum is attained at the projection point (xˆ, yˆ ) = (θ ∗ xo, 1/θ ∗ yo ) ∈ T . 

Let us note that (36)–(39) is equivalent to (20)–(24). Hence the duality results formulated for the primal-dual pair given by (1)–(5) and (20)–(24) in Theorems 3, 4 and Corollaries 1,2 apply also to the pair given by (1)–(5) and (36)–(39). Hence if θ ∗ is optimal for (1)–(5), then

4. Dual HM model in non-linear programming setting

s.t. Y T u − X T v ≤ σ e,

(46)

The dual formulation (15)–(18) is not suitable for an economic interpretation. Therefore we reformulate it as a problem with a nonlinear objective function and linear constraints.

1 − vT xo ≥ 0, uT yo ≥ 0,

(47)

Theorem 5. The dual hyperbolic measure model (in the VRS case) can be formulated as follows:

u, v ≥ 0,

(48)

− σ + 1 + uT yo − vT xo − F (u, v ),

1 − θ ∗ = min u,v,σ

− θ∗

+ σ − (uT yo − vT xo ) + F (u, v ),

(45)

(36)

where 1 measure.

s.t. Y T u − X T v ≤ σ e,

(37)

4.1. Interpretation and comparison with the dual DDM

1 − vT xo ≥ 0,

(38)

Model (45)–(48) for the HM inefficiency is very similar to the dual of the DDM model. In fact, if δ ∗ is the optimal value of DDM model, then the DDM dual satisfies:

max u,v,σ

uT yo ≥ 0,

u, v ≥ 0,

(39)

can be interpreted as the hyperbolic inefficiency

δ ∗ = min σ − (uT yo − vT xo ),

(49)

s.t. Y T u − X T v ≤ σ e,

(50)

vT xo + uT yo = 1,

(51)

u, v ≥ 0.

(52)

u,v,σ

where F is a nonlinear term defined as

F (u, v ) =



1 − vT xo −



2

uT yo

.

The formulation for CRS can be obtained by setting σ = 0. Proof. Recall that the semidefinite constraint (11) is equivalent to

1 − xTo v ≥ 0, yTo u ≥ 0, (1 − xTo v )(yTo u ) ≥ z2 . Moreover, the complementarity results (28) and (29) imply that we always have



z∗ = −

(1 − xTo v∗ )(yTo u∗ )

at the optimum. Therefore, the SDP dual (15)–(18) can be reformulated5 as

max

−σ +2



(1 − vT xo )(uT yo ),

(40)

s.t. Y T u − X T v ≤ σ e,

(41)

1 − vT xo ≥ 0,

(42)

uT yo ≥ 0,

5 It can be seen that the problem (40)–(43) is actually the Lagrange dual of the original HM model (1)–(4), see Appendix for the details.

In both of these models the decision variables u and v represent shadow input and shadow output prices, respectively. The expressions yTj u − xTj v represent virtual profits of DMUj for j = 1, . . . , n, in the prices u and v. Hence DMUo = (xo, yo ) tries to choose its prices and σ under the condition that the virtual profits for each DMU is less than or equal to σ . We can also see that the difference of σ and the virtual profit of (xo , yo ) appears in the objective function of both models. However, the dual HM model does not contain the condition 1 − vT xo = uT yo from the dual DDM, only its relaxed version 1 − vT xo ≥ 0 is required. Hence it may happen that 1 − vT xo = uT yo. In this case, the deviation of 1 − vT xo ≥ 0 from uT yo is penalized by the nonlinear term F (u, v ) in the objective function of the HM model. Let us note that the relaxation of the condition 1 − vT xo = uT yo allows to find a lower value of the difference of σ and the virtual profit in the HM model than in the DDM model. This is in accord with the relation (8) by which the

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HM inefficiency score 1 − θ ∗ is less or equal to the DDM inefficiency score δ ∗ . Remark 4. To see better how the penalization in the objective √ √ function works, consider the function F˜ (z1 , z2 ) := ( z1 − z2 )2 , i.e. F˜ (1 − vT xo, uT yo ) = F (u, v ). It can be seen that

x,y

Hence, the nonlinear term in the objective function serves as a barrier against 1 − vT xo or uT yo vanishing from one side, and a penalty for violation of the equality 1 − vT xo = uT y from the other side. Remark 5. Let us note that in the CRS case, the hyperbolic model maximizes the geometric average (mean) of the values z1 , z2 (defined by (22), (23)) on the corresponding feasible set, while the directional distance model maximizes the arithmetic average of the values z1 , z2 with the feasible set restricted by an additional condition z1 = z2 (under which both the averages give the same value). The penalty term F(z1 , z2 ) measures the difference between the two averages. 4.2. Correspondence between the HM-C and CCR-I duals In Section 2.3 we have described the correspondence between optimal solutions of CCR-I and HM-C models. From the results of Remark 1, we now establish a similar correspondence between optimal solutions of their duals. We recall that the dual to CCR-I reads:

uT yo ,

u,v

(53)

s.t. Y T u − X T v ≤ 0,

vT xo = 1, u, v ≥ 0.

(54)

θH∗

is the optimal value of the HM-C model Proposition 1. (a) If and, u∗H and v∗H are the optimal weights of the HM-C dual, then θI∗ := (θH∗ )2 is the optimal value of the CCR-I model and, u∗I := 2u∗H and v∗I := 2v∗H are the optimal weights of the CCR-I dual. (b) If θI∗ is the optimal value of the CCR-I model and, u∗I and v∗I are the optimal weights of the CCR-I dual, then θH∗ := HM-C model and, of the HM-C dual.

u∗H

:=

1 ∗ 2 uI

and

v∗H



θI∗ is the optimal value of the := 12 v∗I are the optimal weights

Proof. The first part of (a), i.e. θI∗ := (θH∗ )2 , follows from the correspondence between the optimal solutions of CCR-I and HM-C described in Section 2.3. From the strong duality result, optimality of u∗H and v∗H , and from Remark 1 we have that

θH∗ = 1 + yTo u∗H − xTo v∗H − Y T u∗H − X T v∗H ≤ 0,



xTo v∗H

1 − xTo v∗H −

1 = , 2

u∗H ,



yTo u∗H

2

,

(55)

v∗H ≥ 0.

(56)

By multiplying the inequalities in (56) by 2, we obtain that the weights u∗I := 2u∗H and v∗I := 2v∗H satisfy Y T u∗I − X T v∗I ≤ 0, xTo v∗I = 1 and u∗I , v∗I ≥ 0 and hence they are feasible for the CCR-I dual. To prove optimality of u∗I and v∗I it suffices to show that θI∗ = yTo u∗I , which follows from

⎛

θ

θ

∗ ∗2 I = H =

⎝1 +

yTo u∗H

1 − − 2



1 − 2



We now consider the VRS technology T defined by (6). The profit function π corresponding to T associates positive vectors s p ∈ Rm + and q ∈ R+ with the maximal profit over all (x, y ) ∈ T . i.e.

π ( p, q ) = max{ pT y − qT x | (x, y ) ∈ T }.

∂ F˜ (z1 , z2 ) ∂ F˜ (z1 , z2 ) = lim = −∞, ∂ z1 ∂ z2 z1 ↓0 z2 ↓0 ∂ F˜ (z2 , z2 ) ∂ F˜ (z1 , z1 ) while = = 0. ∂ z1 ∂ z2 lim

max

5. Relation to the profit function: the VRS case

2 ⎞2

yTo u∗H

The proof of the statement (b) is analogous.

⎠ 

= 2yTo u∗H

=

yTo u∗I .

(57)

Given the output and input prices (p, q) the profit inefficiency PI(p, q, xo , yo ) of some (xo, yo ) ∈ T can now be viewed as the difference between the maximal profit π (p, q) and the observed profit pT yo − qT xo, i.e.

PI( p, q, xo, yo ) := π ( p, q ) − ( pT yo − qT xo ). A drawback is, however, that the profit function is positively homogeneous, i.e.,

απ ( p, q ) = π (α p, α q ) for any α > 0

(58)

and hence PI(p, q, xo , yo ) depends on the units in which the prices are expressed. This is not usually the case if the prices are provided by the multiplier version of some DEA model, since a certain normalization is naturally incorporated in the model (see for example the NC(u, v ) conditions in the loss function concept by Pastor et al., 2012). The prices provided by the multiplier models are called shadow prices and the corresponding profit inefficiency is called shadow profit inefficiency. However, if the prices (p, q) represent the market prices, different types of normalization of PI(p, q, xo , yo ) are considered in the literature leading to the socalled Nerlovian profit inefficiency

NPIα ( p, q, xo, yo ) := α (π ( p, q ) − ( pT yo − qT xo )), where α > 0 is a normalization factor. The aim is to find α > 0 allowing a following relationship between technical efficiency TE(xo , yo ) (or the technical inefficiency defined as TI(xo, yo ) := 1 − TE(xo, yo )) and normalized profit efficiency:

TI(xo, yo ) = 1 − TE(xo, yo ) ≤ NPIα ( p, q, xo, yo ).

(59)

This type of correspondence was established in Chambers et al. (1998) and Färe and Grosskopf (20 0 0) for the directional distance measure, in Cooper et al. (2011) for the weighted additive models, in Aparicio et al. (2015) for the input or output Russell measures, in Aparicio et al. (2017) for the SBM model, and in Halická and Trnovská (2018a) for the Russell measure model. The relation (59) allows to decompose the normalized profit inefficiency into technical inefficiency TI(xo , yo ) plus a residual that can be interpreted (Farrell, 1957) as a allocative inefficiency for DMU = (xo, yo ), i.e.

AIα ( p, q, xo, yo ) := NPIα ( p, q, xo, yo ) − TI(xo, yo ).

(60)

5.1. Relation to the shadow profit inefficiency We first establish the relation of the HM model to the shadow profit inefficiency. Assume u∗ , v∗ , σ ∗ are attained at the optimum of (36)–(39), F ∗ := F (u∗ , v∗ ) and θ ∗ is the optimal value of the HM model. Then Corollary 2 yields σ ∗ = π (u∗ , v∗ ) and hence the difference σ ∗ − (yTo u∗ − xTo v∗ ), i.e. the difference between the maximal profit in shadow prices σ ∗ = π (u∗ , v∗ ) and the shadow observed profit (yTo u∗ − xTo v∗ ), can be viewed as a “shadow profit inefficiency”. Then, the duality gives us the following relation:

σ ∗ − (yTo u∗ − xTo v∗ ) + F ∗ = (1 − θ ∗ ). F∗

(61)

Therefore the term expresses the difference between the hyperbolic inefficiency (1 − θ ∗ ) and the shadow profit inefficiency where the only restriction on the shadow prices, besides the usual (39), is xTo v∗ ≤ 1.

M. Halická and M. Trnovská / European Journal of Operational Research 278 (2019) 410–421

Using the concept of production function (57) one can show even more. The hyperbolic measure of inefficiency actually minimizes a sum of shadow profit inefficiency and penalty under the condition that the shadow cost is not greater than one. This can be written as follows:

π (u, v ) − (uT yo − vT xo ) + F (u, v ),

min u,v,σ

s.t.

(62)

1 − vT xo ≥ 0, u ≥ 0, v ≥ 0,

(63)

which relates our model to the loss function introduced by Pastor et al. (2012) (see also Aparicio et al., 2016). In our case, the corresponding set of normalization conditions NC(u, v ) is given by (63) and, the difference between the shadow optimal profit π (u, v ) and the shadow observed profit (uT yo − vT xo ) in the definition of the loss function is affected by the penalty term F (u, v ). Let us note that these results are similar to the results obtained for another nonlinear model, the Russell graph measure, by Halická and Trnovská (2018a). 5.2. Relation to the profit inefficiency at market prices s Now, let p ∈ Rm + and q ∈ R+ be the given market input and output prices, respectively. We aim to establish the decomposition (60) for the technical inefficiency provided by the hyperbolic model, i.e. for TI(xo, yo ) := 1 − θo∗ . In order to find a suitable normalization α , one option is to use the result obtained for the DDM model by Chambers et al. (1998) and Färe and Grosskopf (20 0 0) and apply the relation (8) between the hyperbolic and the directional distance models derived in Section 2.4. Putting the results together we get

HI∗o := 1 − θo∗ ≤ δo∗ ≤ NPIαˆ (xo, yo; p, q ), where

αˆ =

1 pT yo + qT xo

and δ ∗ is the DDM technical inefficiency. Then the allocative inefficiency residual can be defined by (60), with α = αˆ . We now propose a different kind of normalization, under which the normalized profit inefficiency attains lower values than the one discussed above and hence it leads to a smaller allocative inefficiency residual. Moreover, under this normalization it is true that if (xo , yo ) is projected onto a point with maximal profit, then the allocative inefficiency is zero. Proposition 2. Let (p, q) be market prices and θo∗ be the hyperbolic technical efficiency score of (xo , yo ). Denote

α∗ =

417

(67) Finally, (67) proves (64). If π ( p, q ) = pT (1/θo∗ yo ) − qT (θo∗ xo ), then the inequalities (65)–(67) hold as equalities and hence

1 − θo∗

π ( p, q ) − pT yo − qT xo =

α∗

.

 From Proposition 2 it follows that it is reasonable to define the allocative inefficiency as

AIα ∗ (xo, yo; p, q ) := NPIα ∗ (xo, yo; p, q ) − HI∗o . 1/θo∗

(68)

α∗

Since ≥ 1 we have that ≤ αˆ and hence the newly proposed normalization yields a smaller allocative inefficiency residual. Note that in the case of nonlinear models such as the RM model or the HM model, the technical inefficiency 1 − θo∗ is not equal the shadow profit inefficiency PI∗o := σ ∗ − yTo u∗ + xTo v∗ in general. See (61) for the case of HM model and Halická and Trnovská (2018a) for the RM model. In the case of the HM model the equality holds if the optimal weights satisfy yTo u∗ + xTo v∗ = 1 (which is satisfied for the efficient units). Using (61) and (68) we get the following decomposition:

NPIα ∗ (xo, yo; p, q ) = AIα ∗ (xo, yo; p, q ) + HI∗o = AIα ∗ (xo, yo; p, q ) + PI∗o + Fo∗ . 6. Relation to return-to-dollar: the VRS case An alternative viewpoint on the profit efficiency can be obtained via the return-to-dollar notion6 . In this section we establish the relation between the HM-V technical efficiency and return-todollar efficiency measure. Similar results were obtained by Zofí o and Prieto (2006) for the generalized graph distance function introduced by Chavas and Cox (1999). The return-to-dollar function (see Gregorescu-Roegen, 1951) associated with the VRS technology T in (6) is defined as



ρ ( p, q ) = max

pT y qT x

| (x, y ) ∈ T ,

s where p ∈ Rm + and q ∈ R+ . Following Zofí o and Prieto (2006) we define the efficiency measure as

RDE(xo, yo; p, q ) =

pT yo qT xo

ρ ( p, q )

.

Now, we denote θo∗ the optimal value of (9)–(14) for (xo , yo ). Then clearly the point (θo∗ xo, θ1∗ yo ) ∈ T and hence





o

pT 1∗ yo 1 pT yo ρ ( p, q ) ≥ T θo∗ = , q ( θo x o ) (θo∗ )2 qT xo

1 . 1/θo∗ pT yo + qT xo

Then

which yields to

1 − θo∗ ≤ NP Iα ∗ (xo, yo; p, q ) = α ∗ (π ( p, q ) − ( pT yo − qT xo )).

(64)

Moreover, if (xo , yo ) is projected onto a point with maximal profit, i.e. π ( p, q ) = pT (1/θo∗ yo ) − qT (θo∗ xo ), then (64) holds with equality.

(θo∗ )2 ≥ RDE(xo, yo; p, q ). The corresponding residual defined as

1

pT yo qT xo ∗ 2 p, q o

Proof. Obviously (1/θo∗ yo, θo∗ xo ) ∈ T and hence from the definition of profit function (57) it follows that:

AE(xo, yo; p, q ) := RDE(xo, yo; p, q )

π ( p, q ) ≥ pT (1/θo∗ yo ) − qT (θo∗ xo ).

By simple manipulation in (65) we obtain the equivalence of (65) with

can now be naturally interpreted as the allocative efficiency. It is a fraction of the observed return-to-dollar in the scale of HM-V projections with respect to the maximum return to dollar. This leads to the following decomposition:

π ( p, q ) − pT yo + qT xo ≥ (1/θo∗ − 1 ) pT yo − (θo∗ − 1 )qT xo

RDE(xo, yo; p, q ) = AE(xo, yo; p, q )(θo∗ )2 .

(65)

(66)

(θo∗ )2

=

(θ ) ρ (

)

and also the equivalence of (66) with

π ( p, q ) − pT yo − qT xo ≥ (1 − θo∗ )(1/θo∗ pT yo + qT xo ) =

1 − θo∗

α

∗

.

6 As one of referees pointed out the return-to-dollar efficiency is also known in the literature as profitability efficiency.

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M. Halická and M. Trnovská / European Journal of Operational Research 278 (2019) 410–421

Fig. 1. Projections of inefficient units D, E, F, G to different parts of the efficient frontier by the HM-V model (the full line) and by the DDM-V model (dashed line) in Example 1.

Note that in the CRS case the value of ρ (p, q) equals one and in this sense our results can be considered as a direct extension of the result of Färe et al. (2002) to the case of the variable returns to scale.

In this section we present two numerical examples. The first illustrates the duality aspects as well as the application of the methodology of Section 5 to the profit inefficiency decomposition. In the second example we demonstrate the computational tractability of the SDP reformulation of the HM-V model on a real dataset in an environmental application. The results are computed using the CVX modeling system (Grant & Boyd, 2013; 2008) and Sedumi solver (Sturm, 1999) for semidefinite programs.

We now additionally assume that the observed output and input prices are p = q = 1. It is easy to see that the value of the profit function is π ( p, q ) = 2 and the value of the return-to-dollar function is ρ ( p, q ) = 5/3. Table 2 lists the results obtained by the approach proposed in Sections 5 and 6 for the HM-V model. For each DMU we report technical inefficiency score HI, and the values of normalized profit inefficiencies NPIαˆ and NPIα ∗ , as well as the corresponding allocative inefficiencies AIαˆ and AIα ∗ , every time with respect to the two alternative normalizations αˆ and α ∗ , respectively. The last three columns express the return-to-dollar efficiency RDE decomposition into the hyperbolic efficiency squared (θo∗ )2 = HE2 and the corresponding allocative efficiency AE. We can see that B is the only unit that is profit efficient and hence also technically and allocatively efficient, while A and C are technically efficient but profit inefficient and this inefficiency is purely allocative.

7.1. Duality aspects, profit inefficiency and return-to-dollar efficiency

7.2. Application to an environmental dataset

Consider 7 DMUs with one input and one output. The corresponding production possibility set T for VRS and the results for the HM-V model are illustrated in Fig. 1. The units A, B, C are efficient. In this case we always have ∗ z1∗ = z2∗ = −z∗ = 1+2σ , which corresponds to the theory derived in Section 3 (see Theorem 4). The unit D is inefficient, and moreover, the optimum is attained in the relative interior of the feasible set, which is expressed by the value θ ∗ φ ∗ = 1.55 > 1. In this case z1∗ = z2∗ = z∗ = 0 and D is projected on the point DH = (2, 2.32 ), which lies on the weakly efficient boundary. In this case we also have sy = 3 − 2.32 = 0.67 > 0 (see Theorem 1 (b)). On the other hand, for the unit E, θ ∗ φ ∗ = 1, yet it is projected on the point EH = (6.67, 6 ) lying on the weakly efficient boundary, with the corresponding slack sx = 6.67 − 5 = 1.67 > 0. Note that the value of θ ∗ φ ∗ > 1 is a sufficient condition for the unit being projected onto the part of weakly efficient boundary with fixed inputs. The numerical results are shown in Table 1.

In this section we use the semidefinite programming modification of the HM-V model to deal with undesirable outputs. In this modification the inputs of the modified problem are not controlled by the variable θ , the undesirable outputs of the modified problem are contracted in the same way as the inputs of the original problem (see (2)), and the desirable outputs of the modified problem are expanded as the outputs of the original problem (see (3)). We apply the SDP formulation7 of this model to an environmental dataset from 92 U.S. powerplants (1995) with 3 inputs, 1 desirable and 2 undesirable outputs. The same data set was used also by Färe, Grosskopf, and Pasurka (2007) and Färe et al. (2016). According to this model, 20 plants were identified as efficient. In Table 3 we list the results for the first 20 DMUs. The second column reports the HM technical efficiency, the third column re-

7. Illustrative examples

7 The dual model for the environmental modifications of the HM model is derived and some of its properties are analyzed in Halická and Trnovská (2018b).

M. Halická and M. Trnovská / European Journal of Operational Research 278 (2019) 410–421

419

Table 1 Data and optimal solutions by the HM-V model. DMU

θ∗

θ ∗φ∗

σ∗

θ ∗ xo

φ ∗ yo

z1∗

z2∗

z∗

u∗

v∗

F∗

A(2, 3) B(3, 5) C(5, 6) D(3, 1) E(8, 5) F(6, 4) G(6, 2)

1.00 1.00 1.00 0.67 0.83 0.71 0.45

1.00 1.00 1.00 1.55 1.00 1.00 1.00

−0.50 0.11 0.69 −0.67 0.83 0.32 −0.05

2.00 3.00 5.00 2.00 6.67 4.26 2.71

3.00 5.00 6.00 2.32 6.00 5.63 4.42

0.25 0.55 0.85 0.00 1.00 0.73 0.45

0.25 0.55 0.85 0.00 0.69 0.37 0.09

−0.25 −0.55 −0.85 −0.00 −0.83 −0.52 −0.20

0.08 0.11 0.14 0.00 0.13 0.09 0.05

0.37 0.15 0.03 0.33 0.00 0.05 0.09

0.00 0.00 0.00 0.00 0.03 0.06 0.14

Table 2 Profit inefficiency and return to dollar efficiency decomposition. DMU

TI

NPIαˆ

AIαˆ

NPIα ∗

AIα ∗

RDE

HE2

AE

A(2, 3) B(3, 5) C(5, 6) D(3, 1) E(8, 5) F(6, 4) G(6, 2)

0.0 0 0 0.0 0 0 0.0 0 0 0.333 0.167 0.290 0.548

0.200 0.0 0 0 0.091 1.0 0 0 0.385 0.400 0.750

0.200 0.0 0 0 0.091 0.667 0.218 0.110 0.202

0.200 0.0 0 0 0.091 0.889 0.357 0.344 0.576

0.200 0.0 0 0 0.091 0.556 0.190 0.054 0.028

0.900 1.00 0.720 0.200 0.375 0.400 0.200

1.0 0 0 1.0 0 0 1.0 0 0 0.445 0.694 0.504 0.204

0.900 1.0 0 0 0.720 0.449 0.540 0.794 0.980

Table 3 Results for the first 20 powerplants. DMU

eff

F

sB1

sB2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0.92775 0.90771 0.97525 0.96035 0.77458 0.74685 0.90590 0.88020 0.79075 0.75211 0.67236 0.75012 0.75868 0.67045 0.82033 1.0 0 0 0 0 0.87588 0.87484 0.83561 0.79268

0.00445 0.00844 0.0 0 035 0.00118 0.02543 0.03819 0.00881 0.00719 0.04248 0.06122 0.10735 0.03626 0.03462 0.10323 0.01827 0.0 0 0 0 0 0.01541 0.01558 0.02690 0.02518

0.0 0 0 0 0 0.0 0 0 0 0 0.0 0 0 0 0 0.0 0 0 0 0 0.0 0 0 0 0 0.00494 0.0 0 0 0 0 0.0 0 0 0 0 0.02055 0.0 0 0 0 0 0.01597 0.01327 0.04028 0.0 0 0 0 0 0.0 0 0 0 0 0.0 0 0 0 0 0.33068 0.0 0 0 0 0 0.0 0 0 0 0 0.00407

0.0 0 0 0 0 0.04135 0.05842 0.0 0 0 0 0 0.0 0 0 0 0 0.0 0 0 0 0 0.21666 0.0 0 0 0 0 0.0 0 0 0 0 0.15179 0.0 0 093 0.0 0 0 0 0 0.0 0 0 0 0 0.02816 0.0 0 0 0 0 0.0 0 0 0 0 0.37085 0.58412 0.11498 0.0 0 0 0 0

ports the value of the penalty term F appearing in the dual formulation of the model. Note that for the efficient DMU 16 the penalty term vanishes. The last two columns represent the slacks corresponding to the undesirable outputs. 8. Conclusions In this paper we have presented a new approach for dealing with the hyperbolic measure model. Our approach is based on a semidefinite reformulation of the model and has several advantages. Firstly, SDP is a subclass of convex conic problems for which efficient and reliable optimization methods are known and readily available. Secondly, duality theory for SDP is well developed and allows for an easy dual model derivation. We have taken advantage of the SDP reformulation and demonstrated its computational tractability on an environmental dataset with undesirable outputs. We have also derived the dual of the HM model in the SDP framework and proved several new results, including existence of the optimal solution of the primal and dual

HM models, strong duality property and interesting complementarity implications. Let us note that the SDP formulation is important mainly for the computational purposes and once the dual is derived and the duality relations are established, they can be converted into the usual non-linear programming setting. Hence, we have transformed the SDP dual into a program with a non-linear objective function and linear constraints. This reformulation allowed a direct comparison with the multiplier form of the DDM model. Significant similarities have been observed and, for an explanation of the nonlinearity in the objective, a concept of a penalty function has been introduced. Subsequently, the hyperbolic inefficiency has been interpreted as the sum of penalty and shadow profit inefficiency, and finally, has been related to the concept of the loss function. Moreover, for the case when the market prices are known, we have established a connection of the HM inefficiency score with Nerlovian profit inefficiency. Our approach offers the method for decomposing the normalized profit inefficiency into hyperbolic technical inefficiency and allocative inefficiency. We have used two types of normalization, one taken from the DDM model, and a new normalization yielding a lower value of allocative inefficiency and vanishing when DMU is projected onto the point of maximal profit. Since the profit efficiency can be alternatively viewed via the return-to-dollar notion, we have completed the results by decomposing the return-to-dollar efficiency using the HM-V efficiency. Analogous approach can be applied to various modifications of the HM, for example to the environmental output model by Färe et al. (1989) as shown in Halická and Trnovská (2018b). It is an open question whether similar results can be achieved for the case of the generalized distance function introduced by Chavas and Cox (1999).

Acknowledgments ˇ The authors thank Pavel Brunovský, Aleš Cerný, and three anonymous reviewers for their constructive and helpful comments on earlier versions of this paper. The authors also thank Israfil Roshdi for sharing the data set of 92 U.S. powerplants. The research of the authors was supported by Slovak VEGA grant 01/0062/18.

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Appendix A

Then also (a1 b1 − a2 b2 )2 ≤ 0, which implies

A1. Lagrange dual

a1 b1 = a2 b2 .

We formally derive the Lagrange dual of (1)–(4). The corresponding Lagrangian is

To show the statement a) assume A is singular and nonzero. Then a1 a2 = a23 . By substituting this into the inequality relation in (72), we get

L(θ , λ; v, u, σ ) =θ − vT (θ xo − X λ ) − uT (Y λ −

= − σ + θ ( 1 − vT xo ) +

1

θ

1

θ

yo ) − σ ( 1 − eT λ )

uT yo + λT (X T v − Y T u − σ e ),

(69) (70)

where λ ≥ 0, u ≥ 0, v ≥ 0. In order to derive the Lagrange dual we evaluate infθ ,λ≥0 L(θ , λ; v, u, σ ). Note that the Lagrangian can be viewed as a sum of a convex non-linear function of θ > 0 and a linear function of λ. The nonlinear part  attains its minimum if 1 − vT xo > 0, and the optimal value is 2 (1 − vT xo )(uT yo ). The linear part attains its minimum value (which is σ ) if X T v − Y T u − σ e ≥ 0. Therefore the dual problem can be formulated as maximizing the sum of σ and the geometric mean of 1 − vT xo and uT yo with linear constraints:



−σ + 2 (1 − vT xo )(uT yo ) Y T u − X T v ≤ σ e, u, v ≥ 0,

max(σ ,u,v ) s.t.

(71)

where the domain of the objective is {(u, v ) | 1 > vT xo}. It can be shown that the RM model satisfies the Slater condition and hence the strong duality results apply to the primal-dual pair of problems (1)–(5) and (71) (as well as to the CRS variant). A2. Some properties of positive-semidefinite matrices A (real) n × n symmetric positive-semidefinite matrix A (in standard notation A  0) is a symmetric matrix satisfying the property xT Ax ≥ 0 for any x ∈ Rn . It is known that the set of all n × n positive-semidefinite matrices, viewed as a subset of the vector space of all n × n symmetric matrices forms a closed, convex, solid and pointed cone, which is in addition self-dual - nice properties that have useful implications in duality theory and algorithms (see e.g. Boyd & Vandenberghe, 2004; Todd, 2001). Many equivalent criteria for positive-semidefiniteness are reported in literature. We recall one of them (in terms of 2 × 2 symmetric positive-semidefinite matrices) to be useful later:



A=

a1 a3

a3 a2



0

a1 ≥ 0, a2 ≥ 0, det A = a1 a2 − a23 ≥ 0.

if f

Below we state and prove an auxiliary lemma that is used in the proof of Theorem 4. This result is known for general positivesemidefinite matrices, however we prove this special case for reader’s convenience. Lemma 1. Let



A=

a1 a3

a3 a2



  0,

B=

b1 b3

b3 b2

 0

and assume that A = 0 and tr (AB ) = a1 b1 + a2 b2 + 2a3 b3 = 0. Then, a1 b1 = a2 b2 . Moreover, (a) if A is singular, then B is singular; (b) if A is non-singular, then B = 0. Proof. First note that from the positive-semidefiniteness assumption we have a1 a2 ≥ a23 and b1 b2 ≥ b23 . Since tr (AB ) = 0, we have (a1 b1 + a2 b2 ) = −2a3 b3 and hence

(a1 b1 + a2 b2 )2 = 4a23 b23 ≤ 4a1 a2 b1 b2 .

(72)

a1 a2 b23 ≤ a1 a2 b1 b2 .

(73)

(74)

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