On Some Relations between Several Generalized. Convex DEA Models. Louisa ANDRIAMASY Walter BRIEC Stéphane MUSSARD

Transcription

1 Groupe de Recherche en Économie et Déveoppement Internationa Cahier de Recherche / Woring Paper On Some Reations between Severa Generaized Convex DEA Modes Louisa ANDRIAMASY Water BRIEC Stéphane MUSSARD

2 On Some Reations between Severa Generaized Convex DEA Modes Louisa Andriamasy Caepem Université Evry Va D Essonne Stéphane Mussard Chrome Université de Nîmes Water Briec Caepem Université de Perpignan Abstract The purpose of this paper is to estabish a topoogica reation between severa casses of nown generaized convex modes extending the approach proposed by Baner, Charnes and Cooper [6]. Using some basic agebraic convex structures proposed by Avrie [4] and Ben- Ta [9] we anayze the Painevé-Kuratowsi imit of the CES-CET and Apha-returns to scae modes. It is shown that their topoogica imits yied the B-convex and Cobb-Dougas production modes. Aong this ine some semi-attice production modes satisfying an α-returns to scae assumption are proposed. Keywords: Non-parametric production modes, Kuratowsi-Painevé imit, attice, CES-CET mode, generaized convexity, α-returns to scae. 1 Introduction Traditionay there exist two basic approaches to estimate a production technoogy over a sector of the economy. The first is based on the econometric estimation of the production frontier, which invoves a parametric specification of some functiona form to describe the frontier of the technoogy. Caepem, Université Evry Va D Essonne, Bouevard Franois Mitterrand, Evry; Emai: ar.ouisa@gmai.com Caepem, Université de Perpignan Via Domitia, 52 Avenue Pau Aduy, Perpignan Cedex ; Emai: briec@univ-perp.fr Chrome Université de Nîmes, Rue du Dr Georges Saan, Nîmes, - e-mai: mussard@ameta.univ-montp1.fr, Associate researcher at Lameta Universit Montpeier In Grédi Université de Sherbrooe and Liser Luxembourg. 1

3 The second approach is based upon operation research methods and nonparametric modes that do not specify a functiona form of the production technoogy. In their papers, Charnes, Cooper and Rhodes [15] and Baner, Charnes, Cooper [6] show how to determine the efficient observed production units in a sampe of firms operating on the same sector of the economy. In their approach the production set is derived from the convex hu of a production vectors representing each firm. Using a inear programming the measure of technica efficiency can be computed to compare the decision maing units and to determine the efficient ones. Impicity this yieds an estimation of the production frontier. From Charnes et a. [15] and Baner et a. [6], severa extensions of the non-parametric production mode have been proposed. A piecewise Cobb- Dougas enveopment was introduced by Charnes et a. [16] and Baner and Maindiratta [7]. In Färe, Grossopf and Njineu [19] a CES-CET (Constant- Easticity-Substitution-Transformation) was investigated. The point-wise imit of the CES-CET mode were anayzed in a production context by Post [23] from the transformations proposed in Aczé [1], Avrie [4] and Ben-Ta [9]. A reaxation of the CES-CET mode was proposed in Boussemart et a. [10]. This mode invoves a structure of α-returns to scae where the returns to scae of the technoogy can be either increasing or decreasing according to the choice of some parameters. More recenty, some casses of path-connected semi-attice production modes were introduced by Briec and Horvath [13] and extended by Briec and Liang [14]. These modes are caed B-convex and are issued from the upper (or ower) imit of the convex hu of a finite number of points. It is worth mentioning that the aforementioned data enveopment techniques rey on the transformation of input/output vectors. As advocated by Post [23], data transformation is crucia because it imits the number of observations to be non attainabe, that is, to avoid input/output combinations to be far-off the enveopment of the data this is particuary chaenging when the sampes are of imited size. Aso, data transformation aows a more important quantity of data to be expoitabe, and as a consequence, accurate indicators of technica efficiency may be derived. This paper shows that, given an observed set of decision maing units, the Painevé-Kuratowsi imit of a sequence of CES-CET modes yieds either a B-convex or a Cobb-Dougas non-parametric estimation of the technoogy. The same hods true for a sequence of non-parametric technoogy satisfying an apha-returns to scae assumption. The suitabe sequence of generaized means (power means) is derived from the transformation scheme suggested by Ben-Ta [9]. To achieve these aims, some genera convergence resuts are investigated. To estabish the fina resuts, the convergence of the sequence of the generaized convex hu of a finite number of points pays a crucia roe. 2

4 The paper is organized as foows. Section 2 presents the standard DEA mode. The CES-CET and Cobb-Dougas DEA modes are aso presented. Section 3 focuses on the notion of generaized convexity and power means. Section 4 estabishes some ey resuts concerning the convergence of a generaized convex hu. A notion of imit set is aso derived with a typoogy of those imits. Section 5 deas with B-convex production technoogies and Section 6 exhibits the resuts for the imit of α-returns to scae modes. Section 7 coses the paper. 2 The Non-Parametric Production Mode The mathematica toos presented in the Introduction can now be appied to production modes. Subsections 1, 2 and 3 are devoted to the exposition of the basic concepts: the production technoogy, the methods used to estimate the production frontier, and by the way, the technoogy set. 2.1 The Bacground of the Production Mode We first define the notations used in this section. Let R d + be the non negative d-dimensiona Eucidean space. For z, w R d +, we denote z w z i w i i [d] where [d] = 1,..., d. For a m, n N, such that d = m + n, a production technoogy transforms inputs x = (x 1,..., x m ) into outputs y = (y 1,..., y n ). The set T R m+n + of a input-output vectors that are feasibe is caed the production set. It is defined as foows: T = (x, y) R m+n + : x can produce y. T can aso be characterized by an input correspondence L : R n + 2 Rm + an output correspondence P : R m + 2 n respectivey defined by and and Finay, et us denote L(y) = x R m : (x, y) T, P (x) = y R n : (x, y) T. K = R m + ( R n +). There are some assumptions that can be made on the production technoogy (see Shephard [24]): T1: T is a cosed set. T2: T is a bounded set, i.e for any z T, (z K) T is bounded. T3: T is strongy disposabe, i.e. T = (T + K) R m+n +. 3

5 T1-T3 define a convex technoogy with freey disposabe inputs and outputs. The foowing subsection presents a cassica way to estimate the production technoogy. 2.2 Non-Parametric Convex and Non-Convex Technoogy Foowing the wors initiated by Farre [20], Charnes et a. [15] and Baner et a. [6], the production set is traditionay defined by the convex hu that contains a the observations under a free disposa assumption. Suppose that A = (x 1, y 1 ),..., (x, y ) R m+n + is a finite set of production vectors. Let Co(A) denotes the convex hu of A. From Baner et a. [6], the production set under an assumption of variabe returns to scae is defined by T DEA = (Co(A) + K) R m+n + (2.1) or equivaenty, for any given vector t of size, by T DEA = (x, y) R m+n + : x t x, y t y, t 0, =1 =1 t = 1. This approach is the so-caed DEA method (Data Enveopment Anaysis) that eads to an operationa definition of the production set. This subset represents some ind of convex hu of the observed production vectors. In ine with Charnes et a. [15], under an assumption of constant returns to scae, the production set can aso be represented by the smaest convex cone containing a the observed firms. In such a case the constraint =1 t = 1 is dropped from the above mode. Technica efficiency can be measured by introducing the usua concept of input distance function and finding the cosest point to any observed firms on the boundary of the production set. Aong this ine, the probem of efficiency measurement can be readiy soved by inear programming. Among the most usua measures of technica efficiency, the Farre efficiency measure (see Farre [20] and Debreu [17]) is essentiay the inverse of the Shephard s [24]. The input Farre efficiency measure is the map E i : R m+n + R + defined as foows: E i (x, y) = inf λ 0 : ( λx, y ) T =1. (2.2) It measures the greatest contraction of an input vector unti to reach the isoquant of the input correspondence, and can be computed by inear programming. In the output case, the output Farre efficiency measure is the map E o : R m+n + R + defined as: E o (x, y) = sup θ 0 : ( x, θy ) T. (2.3) 4

6 It is aso possibe to exogenousy fixed input and outputs to measure efficiency [8]. it is possibe to provide a non-parametric estimation that does not postuate the convexity of the technoogy. It is the FDH approach deveoped by Deprins, Simar and Tuens [18] FDH stands for Free Disposa Hu. The FDH hu of a data set yieds the foowing non-parametric production set: T F DH = (A + K) R n+m +, or in a perhaps more expicit form, T F DH = x R m+n + : x t x, y =1 t y, t 0, 1, t = 1. =1 =1 The main difference with the convex non-parametric technoogy is that t 0, 1. The FDH technoogy is non-convex but it ony postuates the free disposa assumption. The Shephard distance function can aso be computed over the FDH production set by enumeration, see Tuens and Vanden Eecaut [25]. One can aso consider mixed approaches combining both DEA and FDH approaches (see Podinovsi [22]). The next section presents the parametric viewpoint to estimate the production set. 2.3 The CES-CET and Cobb-Dougas Modes This subsection focuses on a modification of the Constant Easticity of Substitution (CES)-Constant Easticity of Transformation (CET) mode introduced by Färe et a. [19] and extended by Boussemart et a. [10]. It consists in two parts: the output part is characterized by a Constant Easticity of Transformation formua and the input part is characterized by a Constant Easticity of Substitution formua. This CES-CET mode can be seen as a generaization of the traditiona inear modes proposed by Charnes et a. [15] and Baner et a. [6]. Moreover, it admits as a imiting case the mutipicative mode proposed by Charnes et a. [16], which is aso discussed in the next subsection. To do that, et us fix d = m + n. Suppose that r > 0 and et us consider the map Φ r : R d + R d + defined as: Φ r (z) = (z r 1,, z r ). (2.4) This function is an isomorphism from R d + to itsef and its reciproca is defined on R d + as: ( ) Φ 1 r (z) = z 1/r 1,, z 1/r. (2.5) If r < 0 the map Φ r is an isomorphism from R d ++ to itsef. For the sae of simpicity suppose that A = (x, y ) : [] R m+n Moreover, et us denote (r) the Φ r simpex defined by: ++. 5

7 (r) = (t 1,..., t ) R + : t r = 1. (2.6) Now, et us consider the foowing set: T (r) CES = ( (x, y) : x Φ 1 r t Φ r (x ) ) (, y Φ 1 r t Φ r (y ) ), t (r). (2.7) Accordingy, the input Farre efficiency measure may be computed as foows: E i (x, y) = inf λ s.t. λx i ( t x ) r 1 r,i i = 1,..., m (2.8) y j ( t y ) r 1 r,j t r = 1, t 0. j = 1,..., n It is then straightforward to convert the above program to a inear program. Based on Charnes et a. [16], we now consider the piecewise Cobb- Dougass (CD) mode. as: Let us define the map Φ 0 : R d ++ R d ++ defined Φ 0 (z) = (n(z 1 ),..., n(z d )). (2.9) This function is an isomorphism from R d ++ to itsef and its reciproca is defined on R d ++ by: Φ 1 0 (z) = (exp(z 1 ),..., exp(z d )). (2.10) The map Φ 1 0 is an isomorphism from R d ++ to itsef. Again, in order to simpify the notations, et us denote (0) the Φ 0 simpex defined by: (0) = (λ 1,..., λ ) R ++ : λ = 1. (2.11) Therefore, the Cobb-Dougas technoogy is defined by: T CD = (x, y) : x x λ, y y λ, λ (0). (2.12) 6

8 The program soving for the technica efficiency in the Cobb-Dougas case is: D i (x, y) = inf λ s.t. λ 1 x x λ y y λ (2.13) λ = 1, λ 0. Appying a og-inear transformation to this program yieds a inear program. 3 Isomorphism of Vector Space Structures This section introduces a notion of generaized convexity based on some particuar agebraic operators. These preiminary properties were estabished and anayzed in detais by Ben-Ta [9]. 3.1 Isomorphism of a Vector Space Structure Let d be a positive natura number and et Φ : X R d be a bijective map, where X is an arbitrary set. From Ben-Ta [9] we consider on X the agebraic operators Φ + and Φ. defined x, y X and for a α R by: x Φ + y = Φ 1( Φ(x) + Φ(y) ) (3.1) α Φ x = Φ 1( α Φ(x) ). (3.2) The subset X endowed with these agebraic operators has some properties very simiar to those of a vector space. Indeed, et K be an arbitrary nonempty set and et φ : K R be an isomorphism. One can define over K the operations defined λ, µ K by λ φ + µ = φ 1 (φ(λ) + φ(µ)) (3.3) λ φ. µ = φ 1 (φ(λ).φ(µ)). (3.4) From Ben-Ta [9] the set φ(r) endowed with the agebraic operators φ + and φ. is a scaar fied. A vector space can then be constructed as the cartesian product of an isomorphic transformation of the scaar fied R, that is K d, in the case where the bijective map Φ is defined for a u R d and a x X = K d by: Φ(x) = (φ(x 1 ),..., φ(x d )) and Φ 1 (u) = ( φ 1 (u 1 ),..., φ 1 (u d ) ). (3.5) 7

9 It foows that K = φ 1 (R) is endowed with a tota order defined by: λ φ µ φ(λ) φ(µ). (3.6) ( Obviousy K d, +, φ φ. ) is a vector space where the agebraic operators + φ and φ are those defined above. It is then cear that if B = v 1,..., v d is a basis of R d ( then B φ = Φ 1 (B) = Φ 1 (v 1 ),..., Φ 1 (v d ) is a basis of the vector space K d, +, φ φ. ). One can then define some convexity notion, from the agebraic operators φ + and φ. defined over K. Notice that the scaar fied the agebraic structure is based upon may not be R. However, it is shown beow that such a formuation aso yieds a number of geometrica properties, in particuar when φ is a bijective endomorphism defined on R. Definition Let φ be a bijective map defined from a nonempty set K to R. A subset C of K d is φ-convex if for a x, y C and a t 1, t 2 φ 1 ([0, 1]), with t 1 φ + t2 = φ 1 (1) we have t 1 φ x φ + t2 φ y C. Now, we can define a notion of convex hu given a finite number of points in K d. Definition Let φ be a bijective map defined from a nonempty set K to R. Let A = x 1,..., x K d. The subset Φ φ Co φ (A) = t Φ x : t φ φ 1 (0), t = φ 1 (1) is the φ-convex hu of A. The convex hu defined in can be written in a mixed form. Such a particuarity wi be of importance in the remainder of the paper. The φ- convex hu of A Co φ (A) is said to be expressed in mixed form if, for s = Φ(t), we have: ( Co φ (A) = Φ 1 s = 1, s 0. (3.7) s Φ( x )) : 3.2 The Exampe of Power functions In this subsection, the concepts deveoped above are appied to a specia transformation of a rea scaar fied, which is the usua power function. For a r ]0, + [, et φ r : R R be the map defined by: φ r (λ) = λ r if λ 0 λ r if λ 0. (3.8) 8

10 For a r 0, the reciproca map is φ 1 r = φ 1. Let Z be the set of integers. r If r 2Z + 1, then φ r (x) = x r, x R. In the foowing subsection, we successivey distinguish severa cases. It is first quite straightforward to state that: (i) φ r is defined over R; (ii) φ r is continuous over R; (iii) φ r is bijective. Throughout the section, for any vector x = (x 1,..., x d ) R d we use the foowing notations: If x R d +, then Φ r (x) = ( φ r (x 1 ),..., φ r (x d ) ). (3.9) Φ r (x) = ( x 1 r,..., x d r ) = x r. (3.10) It is then natura to introduce the foowing agebraic operation over R n : x + r ( y = Φ 1 r Φr (x) + Φ r (y) ) (3.11) λ r x = Φ 1 r (φ r (λ)φ r (x)). (3.12) Let us consider A = x 1,..., x R d. The φ r -convex hu of the set A is: Co φ r (A) = φr t r x : φ r If A R d +, then: Co φ r ( (A) = t r x ) r 1 r : ( t ) r 1 r t = 1, t 0. = 1, t 0. Let us focus on the case r ], 0[. The map x x r is not defined at point x = 0. Thus, it is not possibe to construct a bijective endomorphism on R. Set K = R \ 0. For a r ], 0[ we consider the function φ r defined by: λ r if λ > 0 φ r (λ) = ( λ ) r if λ < 0 0 if λ = +. Ceary, the function φ r is an isomorphism from K to R. Moreover, et us construct the isomorphism Φ r : K d R d, defined by Φ r (x 1,..., x d ) = ( φ r (x 1 ),..., φ r (x d )). For a r < 0, et us consider the agebraic operators + r and r defined by: x + r 1 y = Φ r ( Φr (x) + Φ r (y) ) λ r ( 1 x = Φ r φr (λ). Φ r (x) ). ( Then K d, +, r r ) is a Φ r -vector space. One can remar that if r < 0 then Φ r = Φ 1 (Φ r ). Hence, if A R d ++, then we aso have: Co φ r (A) = Φ 1( Co φ r (Φ 1 (A)) ). 9

11 Now, to depict the geometrica form of the convex hu induced by the power function, it is usefu to distinguish the cases r > 1 and r < 1. If r = 1, one retrieves the standard convex hu. The curvature of the facets changes with respect to r. This is is depicted in Figure and 3.2.2, when r > 1 and r < 1 respectivey. The shaded ines represents the usua convex hu, i.e. when r = 1. x x 2 Co φr (A) x x 2 x 1 Co φr (A) x 4 x 1 0 x 3 x 0 x 3 x Figure Convex hu for r > 1. Figure Convex hu for r < 1. 4 Power Functions and Limit Sets This section introduces the notion of a imit set when r r 0 R. In particuar the geometric deformation of the φ r -convex hu with respect to r is studied. To simpify the notations, et us denote Co r (A) = Co φr (A) for a finite subsets A of R d. The Kuratowsi-Painevé upper imit of the sequence of sets ( Co r (A) ) r N, where A is finite, wi be denoted by Co (A). By definition, a B-poytope is a set of the form Co (A) for some finite subset of R d. We wi see that in R d + the upper-imit is in fact a imit and that the eements of Co (A) have a simpe anaytic description. The Kuratowsi- Painevé ower [upper] imit of the sequence of sets A n n N is denoted Li n A n [Ls n A n ]. For a set of points p for which there exists a sequence p n of points such that p n A n for a n and p = im n p n, a sequence A n n N of subsets of R m is said to converge, in the Kuratowsi- Painevé sense, to a set A if Ls n A n = A = Li n A n, in which case we write A = Lim n A n. Our first statement, Lemma 4.1.1, gives a simpe agebraic description of Co (A); it has been extended to arbitrary sets by Briec and Horvath [12]. 4.1 Typoogy of Limit Sets We denote by =1 x the east upper bound of x 1,..., x R d, that is: x = (maxx 1,1,..., x,1,..., maxx 1,d,..., x,d ). =1 10

12 The foowing resut is an immediate adaptation of the resut estabished by Briec [11]. Lemma Let A = x 1,..., x be a finite subset of R d +. For a positive rea number r et A (r) = x (r) 1,..., x (r) be a finite coection of vectors in R d +. (a) If there exists an increasing sequence r s s N of positive rea numbers such that im s r s = and im s x (r s) = x with = 1,...,, then: im s φ r s x (rs) = (b) If for = 1,..., im s x (r s) = x, then Co (A) = Lim s Co rs (A (rs) ) = t x : max t = 1, t 0. Our first resut, Lemma 4.1.1, gives a simpe agebraic description of Co (A). A these properties are ined to the notion of B-convexity, defined in Briec and Horvath [12]. A subset C of R d + is B-convex, if for a subset A of C, Co (A) C. A simiar resut can be obtained in the case where im s r s =. In such a case one shoud, however, assume that A is a subset of R d ++. We denote by =1 x the east upper bound of x 1,..., x R d, that is: x. x = (minx 1,1,..., x,1,..., minx 1,d,..., x,d ). =1 The resut is a straightforward consequence of that obtained by Adiov and Yesice [3], where a suitabe notion of B 1 -convexity was introduced. Lemma Let A = x 1,..., x be a finite subset of R d ++. For a rea number r et A (r) = x (r) 1,..., x (r) be a finite coection of vectors in R d ++. (a) If there exists a decreasing sequence r s s N of rea numbers such that im s r s = and im s x (r s) = x for = 1,...,, then: im s φ rs x (rs) = (b) If for = 1,..., im s x (r s) = x, then Lim s Co r s (A (rs) ) = t x : min t = 1, t 1 = Co (A). x. 11

13 In the foowing ines, we consider the case where r s 0. For this purpose, et use denote Co 0 (A) = λ = 1, λ 0, for a subset A of R d ++. x λ : Lemma Let A = x 1,..., x be a finite subset of R d ++. For a rea number r et A (r) = x (r) 1,..., x (r) be a finite coection of vectors in R d ++ and et λ (r) R + be an eement of (1). (a) If there exists some λ R + and a decreasing sequence r s s N of positive rea numbers such that im s r s = 0 +, im s λ (rs) = λ and im s x (rs) = x then: ( im Φ 1 r s s (b) If for = 1,..., im s x (r s) Lim s Co r s (A (rs) ) = Proof: (a) For a j [d], we have [ ( 1 Φ rs λ (rs) ( (r Φ rs x s) ) ) = x λ. = x, then x λ : ( (r λ Φ rs x s )) )] ( = j λ = 1, λ 0 = Co 0 (A). ( (r λ x s )) ) 1 rs rs. Since (1) is a cosed set, λ (1). Hence, λ = 1. Taing the ogarithm and appying the Lhôspita rue for r s 0 + yieds the desired resut. (b) We first remar that setting φ r (t ) = λ for a [] yieds ( Co r s ( 1 (r (A) = Φ rs λ Φ r x s) ) ) : λ = 1, λ 0. We first estabish that Co 0 (A) = x λ : λ = 1, λ 0 Li s Co r s (A). Let y = x λ with λ1,, λ [0, 1] and λ = 1. Define y (rs) Co rs (A) by ( ) y (rs) 1 = Φ rs λ 1 Φ rs (x (r s) 1 )+ +λ Φ rs (x (r s) ). Since x (r s) 1,, x (r s) R d ++ we deduce from (a) that im s y(rs) = y. 12

14 This competes the first part of the proof. Next, we estabish that Ls s Co r s (A) Co 0 (A). Tae z Ls s Co rs (A); there is an increasing sequence s N and a sequence of points z N such that z Co rs (A (rs ) ) and im z = z. Each z being in Co r s (A (r s ) ), we can write ( ) 1 z = Φ rs λ,1 Φ rs (x (rs ) 1 )+ +λ, Φ rs (x (rs ) ). Since λ = (λ,1,..., λ, ) [0, 1] one can extract a subsequence (λ q ) q N that converges to a point λ = (λ 1,..., λ ) [0, 1]. From (a) we deduce that x = =1 xλ with λ = 1. The first and the second part of the proof show that Ls s Co r s (A (rs) ) Co 0 (A) Li s Co r s (A (rs) ), and this competes the proof since we aways have the incusion Li s Co r s (A (rs) ) Ls s Co r s (A (rs) ). The foowing tabe provides a synthesis of the imit sets with respect to the form of the vector space structure. Tabe 4. 1: Typoogy of the imit sets Convex Hu Limit r s 0 Co 0 (A) = =1 x λ : λ 0, =1 λ = 1 r s + Co (A) = =1 t x : t 0, =1 t = 1 r s Co (A) = =1 t x : t 0, =1 t = 1 For a finite and nonempty set A contained in R d, Co r (A) beongs to K(R d ), the space of nonempty compact subsets of R d, which is metrizabe by the Hausdorff metric D H (C 1, C 2 ) = inf ε > 0 : C 1 B(x, ε), and C 2 B(x, ε), x C 2 x C 1 where B(x, ε) is the ba of center x and radius ε. In the remainder, we assume that A R d ++. Notice, however, that the case where s hods true when A R d + (see Briec and Horvath [12]). Lemma Let r, 0,. Let r s s N be a sequence of rea numbers which converges to r. For a finite nonempty subsets A of R d ++, the sequence Co rs (A) s N converges to Co r (A) in K(R d ), with respect to the Hausdorff metric. 13

15 Proof: First, remar that since A R d ++, Co r (A) is we defined for a r [, + ]. Choose δ > 0 such that A [0, δ] d ; we have Φ rs (A) [0, δ 2rs+1 ] d, and therefore aso Co(Φ rs (A)) [0, δ 2rs+1 ] d. Taing the inverse image by Φ rs yieds Co rs (A) [0, δ] d ; a the terms of the sequence Co r s (A) s N are contained in the compact set [0, δ] d. To concude, reca that on compact metric spaces, Kuratowsi-Painevé convergence of a sequence of compact sets impies convergence in the Hausdorff metric. It is noteworthy that in the three cases the imit set reduces to a singeton. The foowing figures depict the geometric form of the string joining two points with respect to the parameter r of the power function. Figure 4.1 depicts the case where x 1 and x 2 are not ordered. x x 1 1 < r < r = B 0 < r < 1 A r = r = 1 Figure 4.1 String joining x 1 and x 2 when x 1 and x 2 are not ordered. The maximum-semi-attice hu Co (x 1, x 2 ) is the broen ine joining the points x 1, B and x 2. The minimum-semi-attice hu Co (x 1, x 2 ) is the broen ine joining the points x 1, A and x 2. The intermediary strings corresponding to < r < 1, r = 1, 1 < r < are incuded in the rectange x 1 Ax 2 B. In particuar, the imit set in mixed form Co 0 (x 1, x 2 ) is between the string r = 1 and r =. Figures 4.2 and 4.3 depict two cases where x 1 and x 2 are ordered. x 2 x 14

16 x r = x A 2 < r < 1 x 2 r = 1 B r = 1 < r < x 0 Figure 4.2 x 2 under the ray spanned from x 1. x 2 x r = 1 1 < r < r = A B r = x 1 < r < 1 x 0 Figure 4.3 x 2 upper the ray generated by x 1. In Figure 4.2 we consider the situation where x 1 x 2 and x 2 is under the ray spanned from x 1 that is λx 1 : λ 1. In the the case r =, the imit set Co (x 1, x 2 ) is the broen ine joining the points x 1, B and x 2. If r = then the imit set Co (x 1, x 2 ) is the broen ine joining the points x 1, A and x 2. If r = 1, then one retrieve the usua convex hu between the points x 1 and x 2. The intermediary cases < r < 1 and 1 < r < are respectivey represented between the string r = 1 and r = on the one hand and between the string r = 1 and r = on the other hand. The string Co 0 (x 1, x 2 ) in the mixed case r 0 can aso be represented between the string r = 1 and r =. Figure 4.3 depicts the case where x 1 x 2 and x 2 is above the ray spanned from x 1. It foows that the geometric form of Co (x 1, x 2 ) and Co (x 1, x 2 ) are significanty modified. The string Co (x 1, x 2 ) is then the broen ine joining x 1, B and x 2. Moreover Co (x 1, x 2 ) is the broen ine joining x 1, A and x 2. Geometricay, comparing to Figure 4.3, the respective positions of the maximum and minimum enveopments are reversed. The same hods considering the intermediary cases < r < 1 and 1 < r <. Of course the imit set Co 0 (x 1, x 2 ) coud be represented between the string r = 1 and r =. Figure 4.2 and 4.3 show that the reative position of x 1 and x 2 has a strong impication on the curvature of the string joining them. 4.2 Some Genera Properties Proposition Let C n n N, be a sequence of compact sets of points of R d which converges, in the Kuratowsi-Painevé sense, to a set C of R d, that is Lim n C n = C. Then for a cosed subset K of R d, we have: Lim n (C n + K) = C + K. Proof: Suppose that z Ls n (C n + K). We first prove that z C + K. By hypothesis there is a sequence z n N such that z n C n + K for a 15

17 natura number and im z n = z. By definition, for a there exists (u n, v n ) C n K with z n = u n + v n. Since C n n N is a sequence of compact subsets of R d its Kuratowsi-Painevé imit C is cosed, bounded and therefore compact. However, on compact metric spaces, Kuratowsi- Painevé convergence of a sequence of compact sets impies convergence in the Hausdorff metric. Consequenty, the sequence u n n N is bounded. Moreover, since z n N is a convergent sequence it is aso bounded and it foows that the sequence v n N is bounded. Therefore one can extract from the sequence u n, v n n N a subsequence u n, v n N which converges to some (u, v ) R d R d. By hypothesis u C and since K is cosed v K. Moreover, im u n + v n = u + v = z. Hence, z C + K which proves the first incusion. Conversey, if z C + K, there exists (u, v) C K such that z = u + v. Moreover, there exists a sequence u n n N C such that im n u n = u. Hence z = im n (u n + v) and it foows that z Li n (C n + K). Hence, we deduce that Ls n (C n + K) C + K Li n (C n + K). Consequenty, since Li n (C n + K) Ls n (C n + K), we deduce that: Ls n (C n + K) = Li n (C n + K) = Lim n (C n + K) = C + K. Given a subset C of R d, span(c) denotes the set homogeneousy spanned from C, equivaenty span(c) = λv : v C, λ R. Proposition Let C n n N, be a sequence of compact sets of points of R d which converges, in the Kuratowsi-Painevé sense, to a set C of R d, that is Lim n C n = C. Then, we have: Lim n span(c n ) = span(c). Proof: Suppose that z Ls n span(c n ). We first prove that z span(c). By hypothesis there is a sequence z n N such that z n span(c n ) for a natura numbers and im z n = z. By definition for a there exists (u n, λ n ) C n R with z n = λ n u n. Since C n n N is a sequence of compact subsets of R d its Kuratowsi-Painevé imit C is cosed, bounded and thereby compact. However, on compact metric spaces, Kuratowsi-Painevé convergence of a sequence of compact sets impies convergence in the Hausdorff metric. Consequenty, the sequence u n n N is bounded. Moreover, since z n N is a convergent sequence it is aso bounded and it foows that the rea sequence λ n N is bounded. Therefore one can extract from the sequence u n, λ n n N a subsequence u n, λ n N which converges to some (u, λ ) R d R. By hypothesis u C and λ R. Moreover, im λ n u n = λ u = z. Hence, z span(c) which proves 16

18 the first incusion. Conversey, if z span(c), there exists (u, λ) C R such that z = λu. Moreover, there exists a sequence u n n N C such that im n u n = u. Hence z = im n λu n and it foows that z Li n span(c n ). Hence, we deduce that Ls n span(c n ) span(c) Li n span(c n ). Consequenty, since Li n span(c n ) Ls n span(c n ), we deduce that: Ls n span(c n ) = Li n span(c n ) = Lim n span(c n ) = span(c). Proposition Let C n n N be a sequence of compact sets of points of R d + which converges, in the Kuratowsi-Painevé sense, to a set C of R d +, that is Lim n C n = C. Let K = R d 1 + ( R d 2 + ) with d 1 + d 2 = d. Then we have: Lim n [ (Cn + K) R d +] = (C + K) R d +. Proof: For the sae of simpicity, et us denote D n = C n + K for a n and D = C + K. From Proposition 4.2.1, Lim n D n = D. Suppose that z Ls n (D n R d +). We first prove that z D R d +. By hypothesis there is a sequence z n N such that z n D n R d + for a natura numbers and im z n = z. Since z n D n R d +, z n D n. It foows that z = im z n Ls n D n. Moreover, since R d + is cosed then z R d +. Hence, z D R d +, which proves the first incusion. Suppose now that z D R d +. By definition, for a n there exists z n D n such that im n z n = z. For a n, D n R d +. Since R d + is cosed and C n is compact, it foows that D n R d + is a nonempty cosed subset of R d. Consequenty, for a natura numbers n, there exists some z n D n R d + such that: z n z n = min y D n R d + z n y, where is the Eucidean norm. It is easy to show that, since z n D n = C n + K, we have for a n and a i [d] : z z n,i = n,i if 1 i d 1 max0, z n,i if d i d 1 + d 2 Since z R d +, we have z z n z z n. It foows that im n z n = z. Hence, z Li n (D n R d +). Therefore, we deduce that: Ls n (D n R d +) D R d + Li n (D n R d +). Consequenty, since Li n (D n R d +) Ls n (D n R d +), we deduce that: Ls n (D n R d +) = Li n (D n R d +) = Lim n (D n R d +) = D R d +. 17

19 Proposition Let r s s N be an increasing sequence on rea numbers. Let T (rs) s N be a sequence of production sets of R m+n +, cosed and free disposabe for a natura number. If T = Lim s T (rs), then T is cosed and satisfies a free disposa assumption. Proof: The Painevé-Kuratowsi imit of a sequence of sets is cosed. Therefore T is aso cosed. Let K = R m + ( R n +) be the free disposa cone. K is cosed, moreover R m+n + is cosed and convex. From Propositions and we have: [ ] Lim s (T (rs) + K) R m+n + = (T + K) R m+n +. Since by hypothesis, T (rs) is free disposabe for a s, (T (rs) + K) R m+n + = T (rs). It foows that [ ] T = Lim s T (rs) = Lim s (T (rs) + K) R m+n Thus T is free disposabe which ends the proof. + = (T + K) R m+n +. Notice that if the condition T2, hods for a sequence of production sets, it may not be true for the imit set. Indeed, the Painevé-Kuratowsi imit of a sequence of bounded sets may not be bounded. 5 B-convex Production Technoogies and Painevé- Kuratowsi Limit of CES-CET Modes 5.1 B-convex Modes We come now to the introduction of B-convexity which was defined by Briec and Horvath [12]. A subset L R d + is said to be a B-convex set, if u, z L, and a t [0, 1] u tz L. The basic properties of B-convex sets are anayzed in Briec and Horvath [12]. From this definition a set C such that u, z C for a s, t 0, su tz C is caed a B-convex cone. Aong this ine, a notion of B-convex hu can be provided. Let A = z 1,..., z R d + then the set B(A) = t z, t 0, max t = 1, (5.1) =1... is caed the B-convex hu of A. Paraeing this definition of B-convexity, inverse B-convexity (denoted by B 1 -convexity) is obtained from usua convexity maing the forma substitution + min. It is shown in the remainder of this section that B 1 -convex sets can be derived from B-convex sets via a suitabe isomorphism. This mean that these notions are identica maing a exica change based on the forma substitution max min. Hence, a the 18

20 resuts satisfy by B-convex sets can be transposed to B 1 -convex sets via a suitabe isomorphism, see e.g. Adiov and Yesice [3] and Adiov and Rubinov [2] for B 1 maps and B maps, respectivey. Let M (R ++ + ) d. M is B 1 -convex, if u, z M, and t [1, + ] we have u tz M. Inverse B-convex sets are isomorphicay ined to B-convex sets. To see that et φ : R + R ++ + be the inverse map defined by φ(α) = 1 α. A subset M R d ++ is a B 1 -convex set if, and ony if, L = ϕ 1 (M) is a B-convex set, where ϕ(z 1,..., z d ) = (φ(z 1 ),..., φ(z d )). (5.2) In other words, a subset M (R ++ + ) d is B 1 -convex if and ony if its inverse is B-convex. Though the respective geometric representation of B- convex sets and B 1 -convex sets are different, they are both ined through an isomorphism over (R ++ + ) d. We then provide the foowing definition. For a A = z 1,..., z (R ++ + ) d, the set B 1 (A) = s z, s 0, min s = 1 is caed the inverse B-convex hu of A. Accordingy, one can expose the B-convex non-parametric mode introduced by Briec and Horvath [13]. We consider a coection A = (x, y ) : = 1,..., of observed firms. The subset of R m+n + defined by T max = ( B(A) + K ) R m+n + (5.3) is caed a B-convex non-parametric estimation of the production technoogy. One can equivaenty write: T max = (x, y) R m+n + : x t x, y t y, max t = 1, t 0. (5.4) Simiary, one can define a B 1 -convex production mode defined by Briec and Liang [14]. It is derived by anaogy to the DEA mode and the B- convex structure proposed in the previous section. Let A = (x, y ) : = 1,..., R m+n + a coection of observed production vectors. The subset T min = (x, y) R m+n + : x s x, y s y, min s = 1 is caed the B 1 -convex non-parametric estimation of the production technoogy. Note that if A R m+n ++ then its B 1 -convex hu B 1 (A) is we defined and one can equivaenty write: T min = ( B 1 (A) + K ) R m+n +. (5.5) 19

21 y y z 2 z 2 z 1 z 1 T max z 3 z 4 T min z 3 z 4 0 Figure 5.1 B-convex estimation x 0 Figure 5.2 Inverse B-convex estimation x 5.2 Painevé-Kuratowsi Limit From Briec and Horvath [12] and Proposition 4.1.1, if Lim s r s = +, then: B(A) = Lim s Co r s (A). (5.6) Moreover, from Proposition and from Adiov and Yesice [3], if A R d ++, and if im s r s =, then: Notice aso that for a r R\0 B 1 (A) = Lim s Co r s (A). (5.7) T (r) CES = (Cor (A) + K) R m+n +. (5.8) From Figures 4.1 and 4.2, it is possibe to depict the geometric form of the production set wit respect to r. An eyeba shows that when the parameter r respectivey tends toward + and, then the geometric deformations yieds figures 5.1 and 5.2. y y T (r) CES (r > 1) z 1 z 2 T (r) CES (r < 1) z 1 z 2 z 4 z 4 z 3 z 3 0 x 0 x Figure 5.3 CES enveoppement in the case r > 1 Figure 5.4 CES enveoppement in the case r < 1 20

22 The foowing resut estabishes that the Painevé-Kuratowsi imit of the CES-CET production technoogy is the B-convex technoogy when the parameter r tends toward +. Proposition Let A = (x 1, y 1 ),..., (x, y ) be a finite number of production vectors of R m+n +. Let T (r) CES be the CES piecewise estimation of the production technoogy with respect to A. Suppose that r s s N is an increasing sequence of rea numbers such that im s r s = +. Then: Proof: From equation (5.8), T max = Lim s T (r s) CES. Ls r T (r) CES = Ls r (Co r (A) + K) R m+n +. From equation (5.6) and Proposition 4.2.1: Hence, from Proposition 4.2.3: Ls r (Co r (A) + K) = B(A) + K. Lim r (Co r (A) + K) R m+n + = (B(A) + K) R m+n + = T max. In the next statement, it is estabished that the Painevé-Kuratowsi imit of the CES-CET production technoogy is the B 1 -convex technoogy when the parameter r tends toward. In addition, when the parameter r tends toward 0, it is the piecewise Cobb-Dougas mode. Notice that one shoud assume that A R m+n ++. Proposition Let A = (x 1, y 1 ),..., (x, y ) be a finite number of production vectors of R m+n ++. Let T (r) CES be the CES piecewise estimation of the production technoogy with respect to A. (a) Suppose that r s s N is a decreasing sequence of rea numbers such that im s r s =. Then: T min = Lim s T (rs) CES. (b) Suppose that r s s N is a sequence of rea numbers such that im s r s = 0. Then: T CD = Lim s T (r s) CES. Proof: (a) From equation (5.7) and Proposition 4.2.1: Hence, from Proposition Ls s (Co rs (A) + K) = B 1 (A) + K. Lim s (Co rs (A) + K) R m+n + = (B 1 (A) + K) R m+n + = T min. (b) From Lemma and Proposition 4.2.1: Hence, from Proposition Ls s (Co r s (A) + K) = Co 0 (A) + K. Lim s (Co r s (A) + K) R m+n + = (Co 0 (A) + K) R m+n + = T CD. 21

23 6 Limit of α-returns to scae Modes We investigate the modification of the Constant Easticity of Substitution (CES)-Constant Easticity of Transformation (CET) mode of Färe et a. [19], extended by Boussemart et a. [10], by introducing the so-caed α-returns to scae mode. It consists in two parts: the output part is characterized by a Constant Easticity of Transformation formua and the input part is characterized by a Constant Easticity of Substitution formua. This mode can be seen as a generaization of the traditiona constant returns to scae inear modes proposed by Charnes et a. [15]. As in the earier section it wi be shown that, it admits as a imiting case a variant of the mutipicative mode proposed by Baner and Maindiratta [7], which is aso discussed in the next subsection. For the sae of simpicity, assume that A = (x, y ) : [] R m+n ++. Now, et us consider the foowing set: T (q,r) apha = ( (x, y) : x Φ 1 q t Φ q (x ) ) (, y Φ 1 r t Φ r (y ) ), t 0, (6.1) where qr > 0. This production mode sighty extends the one proposed by Boussemart et a. [10] because it aows negative power means. However, it is assumed that r and q have the same sign. Notice that, compared with the CES-CET mode, the variabe returns to scae constraint t = 1 is dropped. In the foowing, we consider the notion of α-returns to scae proposed by Boussemart et a. [10]. We say that a technoogy T satisfies α-returns to scae if for a λ > 0, (x, y) T impies (λx, λ α y) T. (6.2) Assuming that r and q may be jointy negative yieds the foowing resut. Proposition Let A = (x, y ) R m+n ++ be a set of observed production vectors. Suppose that qr > 0, the production technoogy T (q,r) apha defined in (6.1) satisfies α-returns to scae with α = q/r. The proof is identica to the one given in Boussemart et a. [10]. Note that the CES-CET mode as defined by Färe et a. [19] does not satisfy q/r-returns to scae because of the constraint =1 t = The Constant Returns to Scae Case For a subset C of R m+n +, et us denote span + (C) = tz : z C, t 0. If r = q, then by construction, we have: 22

24 T (r,r) apha = ( span + (Co r (A)) + K ) R m+n +. (6.3) Briec and Horvath [13] propose a CRS B-convex mode defined as foows: Tmax c = (x, y) R m+n + : x t x, y t y, t 0. (6.4) Simiary, a B 1 -convex production mode may be defined foowing Briec and Liang [14]. It is constructed by anaogy to the DEA mode and the B-convex structure proposed in the previous section. The subset Tmin c = (x, y) R m+n + : x s x, y s y, s 0 is caed the CRS B 1 -convex non-parametric estimation of the production technoogy. It is obtained by dropping the constraint min s = 1 from the initia mode. We have by construction: and T c max = ( span + (Co (A)) + K ) R m+n +, (6.5) T c min = ( span + (Co (A)) + K ) R m+n +. (6.6) Proposition Let A = (x 1, y 1 ),..., (x, y ) be a finite number of production vectors of R m+n +. Let T (r) CES be the CES piecewise estimation of the production technoogy with respect to A. (a) Suppose that r s s N is an increasing sequence of rea numbers such that im s r s =. Then: T c max = Lim s T (r s,r s ) apha. (b) Suppose that A R m+n ++ and r s s N is a decreasing sequence of rea numbers such that im s r s =. Then: Proof: (a) From Proposition 4.2.4: Hence, from Proposition 4.2.1: T c min = Lim s T (rs,rs) apha. Ls s span + (Co r s (A)) = span + (B(A)). Ls s span + ( (Co r s (A)) + K ) = span + (B(A)) + K. Proposition yieds: Lim s span + ( (Co r s (A))+K ) R m+n + = (span + (B(A))+K) R m+n + = T max. (b) The proof is simiar. 23

25 6.2 α-returns to Scae Case We first notice that if q = αr, then Φ q = Φ α Φ r. Hence, the constraint ( x Φ 1 q t Φ q (x ) ) (6.7) can be rewritten: ( x α Φ 1 r t Φ r (x α ) ). (6.8) Let Ψ α : R m + R n + R m + R n + be the map defined by Ψ α (x, y) = (x α, y). This map is an homeomorphism (a continuous bijective map whose reciproca is aso continuous) and its reciproca is Ψ α 1 (x, y) = (x 1 α, y). It foows that: Hence: Equivaenty: T (αr,r) apha T (q,r) apha = (x, y) : Ψ α(x, y) T (αr,r) apha. (6.9) T (αr,r) apha = Ψ α 1 ( Let us consider the two foowing modes: and T α max = T α min = T (q,r) apha ). (6.10) ( (Co = Ψ α 1 r (Ψ α (A)) + K ) ) R m+n +. (6.11) (x, y) R m+n + : x (x, y) R m+n + : x t 1 α x, y s 1 α x, y An exercise of cacuus shows that: ( Tmax α = Ψ ( 1 α B(Ψ α (A)) ) ) + K and T α min = t y, t 0 s y, s 0. (6.12) R m+n + (6.13) ( Ψ ( 1 α B 1 (Ψ α (A)) ) ) + K R m+n +. (6.14) Suppose now that r s s N is an increasing sequence of rea number such that im s r s =, then Lim s Co r s (Ψ α (A)) = B(Ψ α (A)). (6.15) If r s s N is a decreasing sequence of rea number such that im s r s =, then Lim s Co rs (Ψ α (A)) = B 1 (Ψ α (A)). (6.16) We can then deduce the foowing resut. 24

26 Proposition Let A = (x 1, y 1 ),..., (x, y ) be a finite number of production vectors of R m+n +. For a r > 0, et T (αr,r) apha be a piecewise estimation of the production technoogy with respect to A satisfying an assumption of α-returns to scae. (a) Suppose that A R m+n ++ and that r s s N is an increasing sequence of rea numbers such that im s r s =. Then: T α max = Lim s T (αrs,rs) apha. (b) Suppose that A R m+n ++ and that r s s N is a decreasing sequence of rea numbers such that im s r s =. Then: T α min = Lim s T (αr s,r s ) apha. Proof: Let C rs s N be a sequence of compact convex sets of R m+n + which converges in the Kuratowsi-Painevé sense ( ) to C R m+n +, then, since Ψ α is an homeomorphism, it foows that Ψ α Crs s N converges in the Kuratowsi- Painevé sense to Ψ α (C). Using Propositions and 4.2.3, the proof of (a) and (b) foow. 7 Concusion In this paper, we have provided a generaization of the traditiona DEA modes thans to the semina wors of Avrie [4] and Ben-Ta [9]. The first generaization is based on the power mean (i.e. generaized mean) initiated by Hardy, Littewood and Poya [21]. Non-parametric technoogies as we as CES-CET Cobb-Dougas technoogies are obtained from the generaized mean. Accordingy, inear programs reated to those nonparametric modes are derived in order to compute technica efficiency. The second generaization is buit on some imiting cases of convex hus of isomorphisms due to Ben-Ta [9], the so-caed B-convex sets introduced by Briec and Horvath [13]. It is shown that α-returns to scae modes (increasing or constant returns to scae) are particuar cases of technoogies inherent to semi-attice structures being either B-convex sets or inverse B-convex sets. References [1] Aczé, J. (1948), On Mean Vaues, Buetin of the American Mathematica Society, 54, [2] Adiov, G. and A.M. Rubinov (2006), B-convex Sets and Functions, Numerica Functiona Anaysis and Optimization, 27, [3] Adiov, G. and I. Yesice (2012), B 1 -convex Sets and B 1 -measurabe Maps, Numerica Functiona Anaysis and Optimization, 33,

27 [4] Avrie, M. (1972), R-convex Functions, Mathematica Programming, 2, [5] Avrie, M. (1976), Noninear Programming: Anaysis and Methods, Prentice Ha, New Jersey. [6] Baner, R.D., Charnes, A. and W.W. Cooper (1984), Some modes for Estimating Technica and Scae Inefficiencies in Data Enveopment Anaysis, Management Science, 30, [7] Baner, R.D. and A. Maindiratta (1986), Piecewise Loginear Estimation of Efficient Production Surfaces, Management Science, 32, [8] Baner, R.D and R.C. Morey (1986), Efficiency Anaysis for Exogenousy Fixed Inputs and Outputs, Operations Research, 34, 4, [9] Ben-Ta, A. (1977), On Generaized Means and Generaized Convex Functions, Journa of Optimization Theory and Appications, 21(1), [10] Boussemart, J.-P., Briec, W., Peypoch, N. and C. Tavéra (2009), α- Returns to Scae and Muti-output Production Technoogies, European Journa of Operationa Research, 197, [11] Briec. W. (2015), Some Remars on an Idempotent and Non-Associative Convex Structure, Journa of Convex Anaysis, 22, [12] Briec, W. and C.D. Horvath (2004), B-convexity, Optimization, 53, [13] Briec, W. and C.D. Horvath (2009), A B-convex Production Mode for Evauating Performance of Firms, Journa of Mathematica Anaysis and Appications, 355, [14] Briec, W. and Q.B. Liang (2011), On some Semiattice Structures for Production Technoogies, European Journa of Operationa Research, 215, [15] Charnes, A., Cooper, W.W. and E. Rhodes (1978), Measuring the Efficiency of Decision Maing Units, European Journa of Operationa Research, 2, [16] Charnes, A., Cooper, W.W., Seiford, L. and J. Stutz (1982), A Mutipicative Mode for Efficiency Anaysis, Socio-Economic Panning Science, 2, [17] Debreu, G. (1951), The Coefficient of Resource Utiization, Econometrica, 19,

28 [18] Deprins, D., L. Simar and H. Tuens (1984), Measuring abour efficiency in post offices. In: Marchand, M., Pestieau, P. and Tuens, H., Editors, The Performance of Pubic Enterprises, North-Hoand, Amsterdam, pp [19] Färe, R., Grossopf, S. and D. Njineu (1988), On Piecewise Reference Technoogies, Management Science, 34, [20] Farre, M.J. (1957), The Measurement of Productive Efficiency, Journa of the Roya Statistica Society, 120, [21] Hardy, G.H., Littewood, J.E. and G. Poya (1934), Inequaities, Cambridge University Press. [22] Podinovsi, V. (2005), Seective convexity in DEA modes, European Journa of Operationa Research, 161, 2, [23] Post, T. (2001), Transconcave Data Enveopment Anaysis, European Journa of Operationa Research, 132(2), [24] Shephard, R.W. (1953), Cost and Production Functions, Princeton University Press, Princeton. [25] Tuens, H. and P. Vanden Eecaut (2001), Non-frontier Measures of Efficiency, Progress and Regress for Time Series Data, Internationa Journa of Production Economics, 39(1-2),