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Modelling weak disposability in data envelopment analysis under relaxed convexity assumptions
European Journal of Operational Research 211 (2011) 577–585
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Modelling weak disposability in data envelopment analysis under relaxed convexity assumptions Victor V. Podinovski a,⇑, Timo Kuosmanen b,c a
Warwick Business School, University of Warwick, Coventry, CV4 7AL, UK School of Economics, Aalto University, 00101 Helsinki, Finland c MTT Agrifood Research Finland, 00410 Helsinki, Finland b
a r t i c l e
i n f o
Article history: Received 31 March 2009 Accepted 4 December 2010 Available online 9 December 2010 Keywords: Data envelopment analysis Nonparametric productivity analysis Environmental performance Undesirable outputs Non-convex technologies
a b s t r a c t The treatment of undesirable (bad) outputs in models of efficiency and productivity analysis often requires replacing the assumption of free disposability of outputs by their weak disposability. In a recent publication the authors showed that the Kuosmanen technology is the only correct representation of the fully convex technology exhibiting weak disposability of bad and good outputs. In this paper we relax the assumption of full convexity and consider two further possibilities: the case in which only the output sets are assumed convex and the case in which no convexity is assumed at all. In the first case we show that, although the traditional Shephard technology of nonparametric production analysis satisfies the assumption of convex output sets, it is larger than necessary. Based on the minimum extrapolation principle, we develop a correct model that is based on the assumed axioms. The second case leads to the development of a weakly disposable analogue of the free disposable hull. To complete our study, we give a full axiomatic definition of the Shephard technology. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction In recent years the treatment of undesirable (bad) outputs in models of productivity and efficiency analysis has attracted significant and growing interest in the literature on data envelopment analysis (DEA) (e.g., Chung et al., 1997; Scheel, 2001; Seiford and Zhu, 2002; Kuosmanen, 2005). One particular example of bad outputs is harmful emissions of a production process and, more generally, any undesirable environmental factor that accompanies production. Conventional treatments of undesirable outputs used in DEA, for example, by regarding them as inputs (Dyson et al., 2001; Seiford and Zhu, 2002), involve two problems. Firstly, assuming free disposability of inputs and bad outputs would imply that a finite amount of input can produce an infinite amount of bad output, thus violating the law of mass conservation. Secondly, the free disposability assumption does not recognize the link between the good and bad outputs, specifically, that emissions can be reduced in proportion to the good outputs. This property is known as weak disposability of undesirable and good outputs (Shephard, 1974). The proportion by which the good and bad outputs are reduced is referred to as the abatement factor (Kuosmanen, 2005). In production economics, the standard approach to modelling weak disposability is the model by Shephard (1974). This model
⇑ Corresponding author. Tel.: +44 2476524281. E-mail address: [email protected] (V.V. Podinovski). 0377-2217/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2010.12.003
is similar to the standard variable returns-to-scale (VRS) DEA model (Banker et al., 1984), except that it differentiates between the good and bad output, does not have a provision for free disposal of bad output and allows for the abatement of bad and good output by the use of an abatement factor. Ideally, a (model of) production technology should be constructed based on observed data and the set of explicitly stated assumptions about the shape of the technology, for example, free disposability of inputs and outputs or convexity. The technology should be proved to satisfy the minimum extrapolation principle as in Banker et al. (1984) which guarantees that the model is the smallest possible and therefore does not contain any arbitrary activities (units). Surprisingly, despite the popularity of the Shephard technology and its clear mathematical representation, the axiomatic properties of this technology have not been fully understood. Recently Kuosmanen and Podinovski (2009) proved that a single abatement factor used in the Shephard technology is insufficient to correctly represent a convex technology exhibiting weak disposability of undesirable and good outputs. In practice, this means that efficiency assessment undertaken with the Shephard technology may overestimate technical efficiency of the evaluated firms when the maintained assumptions hold, and this upward bias does not vanish even if the sample size approaches infinity. Moreover, the computed benchmarks for inefficient activities may not be technically efficient. The authors further showed that the Kuosmanen (2005) model based on the use of multiple abatement factors is the only correct technology suitable for this
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purpose. It was also shown that the Kuosmanen technology is the convex hull of the Shephard technology. Although the Shephard technology is not convex, it is easy to verify that its output sets are convex for each given vector of input. In our new development we set out to investigate whether the Shephard technology is the smallest non-convex technology that has this property and incorporates the assumption of weak disposability of bad outputs. We show that this is not the case. In other words, the Shephard technology is unnecessarily large for this purpose. Instead, a smaller technology which we introduce in this paper is the correct approach here. We then develop a weakly disposable technology that does not assume convexity at all – this is a weakly disposable analogue of free disposable hull (FDH) of Tulkens (1993). Together with our previous developments this gives us a range of models to choose from: we can assume full convexity of the technology, convexity of the output sets only or no convexity at all. If needed, an even wider range of models can be obtained based on the assumption that only some inputs and outputs are subject to the convexity axiom. This is not pursued here but the general approach for modelling such technologies is readily available (Podinovski, 2005). Our development contributes to the growing stream of efficiency analysis literature where the full convexity of the production possibility set is relaxed but some weaker form of convexity is maintained (Petersen, 1990; Bogetoft, 1996; Bogetoft et al., 2000; Dekker and Post, 2001; Kuosmanen, 2001, 2003; Briec et al., 2004; Agrell et al., 2005; Podinovski, 2004a, 2005; Ehrgott and Tind, 2009). The practical motivation for these approaches is that the (fully) convex production set is not well suited for modelling economies of scale (Petersen, 1990; Bogetoft, 1996; Bogetoft et al., 2000). Thus, relaxing the full convexity assumption may be appropriate in such sectors as the railroad industry where economies of scale are thought to prevail (Klein, 1947; Hasenkamp, 1976). On the other hand, the convexity of the output sets is closely related with the economies of scope, which are said to prevail if joint production of two or more outputs is less expensive than producing the outputs separately. The joint production of passenger and freight services in the railroad industry is a classic example of this (Panzar and Willig, 1981). Another example of joint production is the combined heat and power (CHP) generation (also known as cogeneration), which is an increasingly important energy production technology (e.g., Lahdelma and Hakonen, 2003; Rong et al., 2006). Yang and Pollitt (2009) is one of the few studies that discusses CHP in the context of DEA. Therefore, the models developed in this study appear to be particularly well suited for sectors such as energy and transportation where both economies of scale and scope are expected to prevail. While the majority of the DEA approaches under relaxed convexity assumptions (cited above) incorporate free disposability of all inputs and outputs, the models developed in this paper are the first to combine the axioms of weak disposability with reduced convexity. The techniques developed in this paper could be used, for example, for modelling accidents (e.g. fatalities) in the already mentioned railroad industry as a bad output that is weakly disposable: if passenger and freight services are scaled down by a certain percent, it should be possible to decrease the accidents by the same percent. Similarly, in the energy production, harmful emissions such as CO2 can be assumed weakly disposable together with the level of heat and electricity produced. Finally, in order to complete our study of the Shephard technology we give its correct and complete axiomatic characterization. This is driven not only by our academic curiosity but the need to understand why this technology is too small to represent convex production sets (in the input–output space) and too large to represent technologies with convex output sets (in the output space only, for any fixed input vector). We prove that the Shephard
technology is characterized by a strange variant of the convexity axiom: it allows for convex combinations of observed activities but does not explicitly assume the same for all activities in the technology. The authors can offer no reason why this assumption might be acceptable in practice. 2. Preliminaries The terminology and notation in our development are consistent with the economic literature on nonparametric productivity analysis. These are different but equivalent to the terminology and notation used in mainstream DEA (Charnes et al., 1978; Cooper et al., 2000; Thanassoulis, 2001). Following our previous development (Kuosmanen, 2005; Kuosmanen and Podinovski, 2009; Kuosmanen and Kazemi Matin, in press) each production activity (decision making unit in the DEA terminology) is characterized by the triplet (v, w, x), where v ¼ ðv 1 ; . . . ; v M Þ 2 RM þ is the vectors of desirable (good) outputs, w ¼ ðw1 ; . . . ; wJ Þ 2 RJþ is the vector of undesirable (bad) outputs and x ¼ ðx1 ; . . . ; xN Þ 2 RNþ is the vector of inputs. We assume we have K observed activities denoted by (vk, wk, xk), k = 1, . . . , K. A production technology is the set of all producible activities Y = {(v, w, x)jx can produce (v, w)}. Equivalently, technology Y can be represented by its output sets P(x) = {(v, w)j(v, w, x) 2 Y}, x 2 RNþ . In the present context, technology Y is usually assumed to satisfy the following axioms: (A1) Strong disposability of inputs and good outputs. If (v, w, x) 2 Y, 0 6 v0 6 v and x0 P x (for each component) then (v0 , w, x0 ) 2 Y. (A2) Weak disposability of undesirable outputs and good outputs. If (v, w, x) 2 Y and h 2 [0, 1] then (hv, hw, x) 2 Y. (A3) Convexity. Technology Y is convex. The multiplier h used in axiom (A2) is referred to as the abatement factor: it enables the reduction of the level of bad outputs, such as harmful emissions, if accompanied by the reduction of good outputs in the same proportion. Unfortunately, the implementation of the abatement factor in DEA models has caused some confusion. The conventional approach to modelling weak disposability b S em(A2) in DEA is due to Shephard (1974), whose technology Y ploys a single abatement factor h but is not convex (Kuosmanen and Podinovski, 2009). The Shephard technology is stated as follows:
( bS ¼ Y
ðv ; w; xÞ : h
K X
zk v km P v m P 0;
8m
ð1Þ
k¼1
h
K X
zk wkj ¼ wj ;
8j
k¼1 K X
zk xkn 6 xn ;
8n
k¼1 K X
zk ¼ 1
k¼1
zk P 0;
8k
0 6 h 6 1g; where variables z = (z1, . . . , zK) are referred to as the intensity weights. In (1) we have explicitly incorporated the requirement that good outputs vm (taken here as model variables) must be nonnegative. The nonnegativity of bad outputs wj and inputs xn follows automatically from the corresponding constraints and need not be specifically stated.
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Kuosmanen (2005) developed an alternative model for a technology whose bad outputs exhibit weak disposability. In contrast to the Shephard technology, the Kuosmanen technology is convex and uses individual abatement factors hk attached to each observed activity k = 1, . . . , K:
( bK ¼ Y
ðv ; w; xÞ :
K X
hk zk v km P v m P 0;
8m
v
U
V
C x
ð2Þ
k¼1 K X
W k k
hz
wkj
¼ wj ;
8j
5
k¼1 K X
zk xkn 6 xn ;
8n
k¼1 K X
B k
z ¼1
E
M
k¼1
zk P 0;
8k
o 0 6 h 6 1; 8k : k
Although both programs (1) and (2) are nonlinear, Kuosmanen (2005) demonstrated that model (2) could be linearized by substitution of variables. To gain further economic insight, Kuosmanen and Kazemi Matin, in press, developed dual formulations of the linearized technology. However, the Shephard technology (1) cannot be linearized by the same approach. We use the suggested substitution again in our development below. It is clear that the Shephard technology is a subset of the Kuosmanen technology, as the former can be obtained from description (2) by the incorporation of the additional condition h1 = h2 = = hK. The difference between the Shephard and Kuosmanen technologies is further illustrated by the example taken from Kuosmanen and Podinovski (2009) which involves just two activities B and C as in Table 1. In the Shephard technology (Fig. 1) the edges BH and CE are obtained by the simultaneous abatement of the good and bad outputs of activities B and C, respectively. The triangle BDH is induced by free disposability of the good output v of the activities represented by the edge BH. The rays BV, CU, DW and HE are induced by free disposability of the input. Since activities B, C, E and H are not in the same plane, the side BCEH is not ‘‘flat’’ and the interior points of the segment CH are not in the technology. This demonstrates that the Shephard technology is generally not convex. The corresponding Kuosmanen technology is shown in Fig. 2. It is larger than the Shephard technology and, in particular, includes the segment CH. This is because formulation (2) allows for convex combinations of the observed activities abated in different proportions hk. In Fig. 2, activity H is obtained by the application of the extreme abatement factor zero to activity B. The segment CH is then obtained by combining C and H. Kuosmanen and Podinovski (2009) proved that the Kuosmanen b K is the smallest technology that incorporates all obtechnology Y served activities and satisfies the axioms (A1)–(A3) stated above. b K is the intersection of all technologies that satisfy Formally, Y these axioms. This approach to the definition of technologies based on a given set of axioms is known as the minimum extrapolation principle, which was introduced to DEA by Banker et al. (1984). Although many other technologies, for example, the standard VRS and constant returns-to-scale (CRS) technologies, also satisfy Table 1 The dataset for illustrations: good output v, undesirable output w, input x.
v w x
Activity B
Activity C
3 4 1
5 1 4
1
1
D
H 0
1
w
4
b S induced by activities B and C. Fig. 1. The Shephard technology Y
v
U
V
C x
W 5
B M E
1
1 D
H 0
1
w
4
b K induced by activities B and C. Fig. 2. The Kuosmanen technology Y
(A1)–(A3), all of these contain arbitrary (in the given context) activities that are not implied by the stated axioms and are, therefore, unnecessarily large. In contrast, no activity in the Kuosmanen b K can be removed without a violation of the stated technology Y axioms. Kuosmanen and Podinovski (2009) further proved that the b K is the convex hull (smallest convex Kuosmanen technology Y
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b S . This fact is clearly illusextension) of the Shephard technology Y trated by Figs. 1 and 2: the Kuosmanen technology is obtained from the Shephard technology by adding to the latter all missing convex combinations of its activities.
v
U
V
C
x
3. Production technologies with convex output sets b S is not convex, its output Although the Shephard technology Y sets P(x) are convex for any input vector x. This formally follows from the Corollary to Theorem 3 and subsequent discussion in Kuosmanen and Podinovski (2009). This can also be seen in Fig. 1. For example, if x = 1,the corresponding set P(1) is the triangle BHD. Färe and Grosskopf (2009) pointed out the noted convexity of output sets as a reason for using the Shephard technology. Given this observation, a question may be asked whether the Shephard technology can indeed be regarded as being based on the disposability axioms (A1), (A2) and the following variant of the convexity axiom:
L
W
5
B M
E
(A4) Convexity of the output sets. The sets P(x) are convex for all x.
1
1 In other words, is the Shephard technology a variant of Bogetoft’s (1996) technology that assumes convex output sets but no other form of convexity, in which free disposability of inputs and outputs is replaced by weak disposability for bad outputs? (Note that in mathematics empty sets are viewed as convex, and so we do not need to limit axiom (A4) to the case of non-empty sets P(x) only.) Below we show that the answer to the above question is negative: the Shephard technology is larger than necessary for the axiom (A4) and is, therefore, an arbitrary technology with respect to the stated specifications. The correct weakly disposable variant of the Bogetoft technology, based on the minimum extrapolation principle, is as follows. b B which contains all observed Theorem 1. The minimum technology Y activities and satisfies the axioms (A1), (A2) and (A4) is obtained from (2) by replacing the inequalities associated with the inputs by the following conditions:
if zk > 0 then xkn 6 xn ;
8n; k:
ð3Þ
The proofs of this and the other theorems presented below are given in Appendix A. Following Podinovski (2004b), we first prove b B satisfies all the stated axioms. We then that the technology Y b B , then ðv ; w; xÞ 2 Y, b where prove that, if some activity ðv ; w; xÞ 2 Y b is any technology that satisfies the same set of axioms. The Y b B is the minimum implication of these two statements is that Y technology that satisfies the stated axioms. b B is as Based on Theorem 1, the full formulation of technology Y follows:
( bB ¼ Y
ðv ; w; xÞ :
K X
hk zk v km P v m P 0;
8m
ð4Þ
k¼1 K X
k k
hz
wkj
¼ wj ;
K X
8j 8n; k
4
b B with convex output sets. Fig. 3. The weakly disposable Bogetoft technology Y
extends this to triangle BDH. The free disposability of the input induces the unbounded polyhedron shaped by this triangle and the rays BV, DW and HE. Similarly, using (A1) and (A2), activity C induces the unbounded polyhedron based on the triangle CEM. Note that we cannot join activities B and C as in the Shephard technology in Fig. 1 because the convexity axiom (A3) is not assumed. Instead, we can only form convex combinations of activities that have the same input level x. For example, activity L on the ray BV has x = 4, the same as activities C and E. Therefore, the convex combinations of L, C and E must be in the technology, and we obtain the triangle CEL. b B in Fig. 3 is the smallest technology According to Theorem 1, Y with convex output sets induced by activities B and C. The output b S are also convex but this technolsets of the Shephard technology Y ogy contains additional activities, for example, the segment BC, which cannot be justified by the assumed axioms (A1), (A2) and (A4). b B in the form (4) has two practical drawRepresenting technology Y backs: this model is not linear and includes conditional inequalities (3). Both drawbacks are easy to overcome. In order to linearize model (4), we can use the substitution from Kuosmanen (2005):
kk ¼ hk zk ;
lk ¼ ð1 hk Þzk ; 8k;
ð5Þ
so that
kk þ lk ¼ zk :
ð6Þ
bB ¼ Y
ðv ; w; xÞ :
K X
kk v km P v m P 0; 8m
k¼1 K X
k¼1
k
1
(
zk ¼ 1
zk P 0;
0
kk wkj ¼ wj ; 8j
k¼1
8k
0 6 h 6 1;
w
Rearranging the terms, we obtain an equivalent statement of techb B: nology Y
k¼1
if zk > 0 then xkn 6 xn ;
D
H
o 8k :
b B induced by activities B and C from Fig. 3 shows technology Y Table 1. The weak disposability of the bad and good output induces the segment BH. The free disposability of good output
if kk þ lk > 0 then xkn 6 xn ; 8n; k K X ðkk þ lk Þ ¼ 1 k¼1
kk ;
o
lk P 0; 8k :
ð7Þ
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Following Kuosmanen et al. (2006, Appendix), the conditional input inequalities in program (7) can be replaced by the linear inequalities
ðkk þ lk Þxkn 6 ðkk þ lk Þxn ;
8n; k:
ð8Þ
If kk + lk > 0, inequalities (8) imply xkn 6 xn as required. The fully linb B is therefore as follows: earized version of technology Y
(
bB ¼ Y
ðv ; w; xÞ :
K X
kk v km P v m P 0;
8m
ð9Þ
k¼1 K X
kk wkj ¼ wj ;
8n; k
K X
ðkk þ lk Þ ¼ 1
kk ; lk P 0;
ð12Þ
For each observed activity k define a binary variable b 2 {0, 1} and introduce the following inequality:
kk þ lk 6 bk ;
8k:
ð13Þ k
The idea behind (13) is that b serves as an indicator that becomes equal to 1 if the sum kk + lk is positive (note that kk + lk cannot exceed 1). If kk + lk = 0 then bk can be either 0 or 1. Condition (12) is now restated in the following linear form:
8n; k:
ð14Þ
Indeed, if kk + lk > 0 then, from (13), bk = 1, and (14) implies xkn 6 x0n /g xn . If kk + lk = 0, then by (13) b can be either 0 or 1 and (14) does not restrict anything as required. With all of the above changes, the value of the general directional distance function is obtained by solving the following mixed integer linear program:
k¼1
k¼1
8n; k: k
bk xkn 6 x0n /g xn ;
8j
ðkk þ lk Þxkn 6 ðkk þ lk Þxn ;
if kk þ lk > 0 then xkn 6 x0n /g xn ;
o
8k :
b B in practical computations is discussed in The use of technology Y the next section.
^dB
v0 ; w0 ; x0 ; gv ; gw ; gx
Subject to
K X
¼ max /
kk v km P v 0m þ /g vm ;
ð15Þ
8m;
k¼1 K X
4. Computational approaches In practice, measuring the efficiency of a given activity (v0, w0, x0) is often conducted using a directional distance function (Chambers et al., 1996, 1998; Chung et al., 1997) that allows for a simultaneous increase in good output and reduction in input and bad output. The conventional radial Farrell input and output efficiency measures can be obtained as special cases of the directional distance functions (Chambers et al., 1998). When using directional b B we need to distinguish bedistance functions with technology Y tween two cases: the output directional distance function
do ðv 0 ; w0 ; x0 ; gv ; gw Þ ¼ sup /ðv 0 þ /gv ; w0 /gw ; x0 Þ 2 Y
ð10Þ
and the general directional distance function
dðv 0 ; w0 ; x0 ; gv ; gw ; gx Þ ¼ sup /ðv 0 þ /gv ; w0 /gw ; x0 /gx Þ 2 Y : ð11Þ In (10) and (11), gv, gw and gx are nonnegative vectors referred to as the direction vectors: these specify the direction in which the evaluated activity (v0, w0, x0) is projected on the frontier. The output directional distance function (10) keeps the input vector x0 at a constant level and seeks to increase the good output v0 in the direction gv and decrease the bad output w0 in the direction gw. The general directional distance function (11) differs from (10) in that it additionally seeks to decrease the input use x0 in the direction gx. Note that (10) is a special case of (11) where gx = 0; any zero elements in the direction vectors imply that the corresponding input or output is held at a constant level in the efficiency evaluation. The use of output directional distance function (10) with techb B leads to a straightforward linear program: we simply nology Y 0 substitute v m ¼ v 0m þ /g vm , wj ¼ w0j /g w m and xn ¼ xn onto the right-hand side of model (9) and set the objective function to maximize /. We can also omit the requirements v m ¼ v 0m þ /g vm P 0 for all m in (9) as obviously redundant. The use of the general directional distance function (11) requires a somewhat more complex treatment. Note that by simply introducing input reductions in the linear model (9) we would render the third group of constraints nonlinear, specifically, kk xkn 6 kk ðx0n /g xn Þ. This problem can be overcome by resorting to mixed integer linear programming as in Podinovski (2005). For b B as a starting this purpose formulation (7) of the technology Y point is more useful. Its conditional input constraints would take on the form:
kk wkj ¼ w0j /g w j ;
8j;
k¼1
bk xkn 6 x0n /g xn ;
8k; n;
K X ðkk þ lk Þ ¼ 1; k¼1
kk þ lk 6 bk ;
8k;
kk ; lk P 0;
8k;
bk 2 f0; 1g;
8k:
Compared to the case of output directional distance function leading to a simple linear program, the general formulation (15) is more complex, involving binary variables bk. We will next show that in practical computations the number of variables in the previous programs can be significantly reduced (under some realistic assumptions discussed below). More specifically, we only need variables kk and can dispose of the variables lk. This approach has obvious computational advantages. e obtained from the formuConsider the following technology Y lation (7) by removing all variables lk:
( e¼ Y
ðv ; w; xÞ :
K X k¼1 K X
kk v km P v m P 0; kk wkj ¼ wj ;
8m
ð16Þ
8j;
k¼1
if kk > 0 then xkn 6 xn ; K X kk 6 1; k¼1 o kk P 0; 8k :
8n; k;
e is not identical to Y b B . For example, the former contains Technology Y the origin (0, 0, 0), which satisfies (16) with k = 0 and is missing from b B (assuming none of the observed activities has a zero input vector). Y The following theorem formalizes this difference and shows that e is slightly larger than Y b B. technology Y e¼Y b B [ fð0; 0; xÞ : x P 0g. Theorem 2. The following holds: Y e is obtained from the As an illustration, in Fig. 3 technology Y b B by adding to it the additional segment depicted technology Y OH.
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e and Y b B stated by TheThe difference between technologies Y orem 2 is unimportant for most practical purposes, and we can e instead of Y b B whenever the asuse the simpler technology Y sessed activity (v0, w0, x0) has at least one strictly positive good output (v0 – 0). Because the good output v0 cannot decrease by a movement in the direction gv P 0, the presence of additional e becomes irrelevant for the assesspoints (0, 0, x) in technology Y ment. Note that the condition v0 – 0 is sufficient but not necese and Y b B when used sary for the equivalence of technologies Y with the directional distance function (11). Other sufficient conditions can be stated in terms of the direction vectors (gv, gw, gx): if gv – 0 or gx = 0, then the directional distance function (11) yields e and Y b B . In practice, situations the same value with respect to Y where activities in the form (0, 0, x) with x P 0 and located outb B are identified as efficient targets are extremely side the set Y rare. Based on the above observation, we can further simplify the two programs developed in the previous section. In the case of the outb B by put directional distance function (10), replacing technology Y e Y leads to the following linear program:
^do ðv 0 ; w0 ; x0 ; gv ; gw Þ ¼ max / B Subject to
K X
ð17Þ
kk v km P v 0m þ /g vm ;
5. Weak disposability without convexity
8m; Following the same approach as above, it is not difficult to construct a technology which is based only on axioms (A1) and (A2) without assuming any variant of the convexity axiom. This can be considered as the analogue of the FDH technology of Tulkens (1993) in which free disposability of inputs and outputs is replaced by weak disposability for bad outputs. We denote such a technolbT. ogy Y
k¼1 K X
k
k
wkj
¼
w0j
/g w j ;
8j;
k¼1
kk xkn 6 kk x0n ; K X
8k; n;
kk 6 1;
k¼1
kk P 0;
models (17) and (18).1 In model (17), we use about a half of the varib B . In model (18), we use ables required in the proper technology Y about one third fewer variables than in program (15) based on techb B. nology Y Model (18) is a mixed integer linear program in which the number of binary variables bk is equal to the number of activities K. This model can in principle be solved by most commercial linear optimizers, which may be a preferred approach if K is not large and, consequently, the computational time is not problematic. However, because in practice we would normally need to solve K programs (18), one for each activity under the assessment, for large K this may present a certain computational challenge. The development of methods of mixed integer programming2 suitable for large scale computations based on model (18) lies beyond the scope of this paper. We only note that computational algorithms for some other mixed integer linear DEA models have been explored in the literature, but these are specific and generally not transferable between different DEA models. Examples include the elimination algorithm for solving the FDH model (Tulkens, 1993) and a column generation technique by Ehrgott and Tind (2009) for the free replicability model of Tulkens (1993).
8k:
In the case of general directional distance function (11) we need to repeat the logic behind program (15), this time starting with ~ In particular, we need to introduce binary variables technology Y. bk that serve as indicators taking on the value of 1 if kk > 0. The resulting mixed integer linear program takes on the following form:
b T which contains all observed Theorem 3. The minimum technology Y activities and satisfies the axioms (A1) and (A2) is obtained from (2) by additionally requiring that the intensity weights zk take on only 0 or 1 values. b T is described as follows: Based on this result the technology Y
(
bT ¼ Y
ðv ; w; xÞ :
K X
hk zk v km P v m P 0;
8m
ð19Þ
k¼1
^dð v 0 ; w0 ; x0 ; gv ; gw ; gx Þ ¼ max / B Subject to
K X
k
k
v
k m
Pv
0 m
ð18Þ
þ /g v ; m
8m;
k¼1 K X
k
k
wkj
¼
w0j
/g xn ;
/g w j ;
8j;
k¼1
bk xkn K X
6
x0n
8k; n;
k
kk 6 1;
k P 0;
8k; 8k:
In program (18), if kk > 0 then bk = 1 and the corresponding input constraints xkn 6 x0n /g xn are satisfied for all n. It is worth emphasizing that the only correct technology based ^ B as described by on the axioms (A1), (A2) and (A4) is technology Y (4) or equivalent formulations (7) and (9). The assessment of effib B by using these forciency can be performed directly in terms of Y mulations. In practice, the same results can be obtained by e using performing computations in the slightly larger technology Y,
8j;
k¼1 K X
zk xkn 6 xn ;
8n;
k¼1 K X
zk ¼ 1;
k¼1
0 6 hk 6 1;
8k;
bk 2 f0; 1g;
hk zk wkj ¼ wj ;
zk 2 f0; 1g; 8k;
k¼1
kk 6 bk ;
K X
o
8k :
b T can be linearized using the For practical purposes technology Y substitution (5) as discussed above. A minor obstacle with this approach is that the sum of variables kk + lk = zk needs to be declared binary. This can be achieved by including equalities (6) and keeping the declaration of variables zk as binary variables in the model. The use of this technology in DEA with the directional distance functions (11) would lead to solving mixed integer programs, one for each activity under the consideration. 1 e and Y b B were discussed The formal conditions of substitutability of technologies Y after Theorem 2. 2 For an overview of methods of integer programming, see Wolsey (1998) and Lübbecke and Desrosiers (2005).
V.V. Podinovski, T. Kuosmanen / European Journal of Operational Research 211 (2011) 577–585
Alternatively, as with the classical implementations of FDH, b T can be regarded as the union of all technologies intechnology Y duced by individual observed activities:
bT ¼ Y
K [ ðv ; w; xÞ : hv km P v m P 0;
8m
k¼1
hwkj ¼ wj ; xkn
6 xn ;
8j; 8n;
0 6 h 6 1g: The value of the directional distance function for any activity can now be computed by solving a required linear program in each of the K separate technologies and taking the largest of the values so obtained. Some of these programs (but not all) may be infeasible and should be ignored. b T induced by activities B and C from Fig. 4 shows technology Y Table 1. It is the smallest of all of the discussed technologies and does not assume any convexity. It is the simple union of two separate technologies induced by activities B and C. The technology induced by B is based on the triangle BHD which, due to free disposability of input, extends along the rays BV, DW and HE to infinity along the input axis. The technology induced by activity C is obtained in a similar way based on the triangle CEM.
583
b S is the minimum set which Theorem 4. The Shephard technology Y contains all observed activities and satisfies the axioms (A1), (A2) and (A5). Note that Theorem 4 does not state that the Shephard technology is convex: axiom (A5) explicitly allows only for observed activities to be combined. As an illustration to this comment, the Shephard technology in Fig. 1 includes all convex combinations of observed activities B and C that form the edge BC. However, activities C and H cannot be combined because activity H is not observed. This does not mean, however, that all convex combinations involving unobserved activities are missing from this technology: some of such combinations are present due to other axioms. For example, convex combinations of activities B and H, of which H is not observed, form the segment BH which is in the technology as required by the weak disposability axiom (A2). It is difficult to see reasons why in practice one should assume convexity only for observed activities but not allow convexity for unobserved activities deduced from the maintained axioms. This makes it difficult to justify the use of this technology for efficiency assessment as there is no guarantee that the computed benchmarks and targets would be substantiated. 7. Conclusion
6. Axioms of the Shephard technology In this section we offer a simple axiomatic definition of the b S as defined in (1). Our previous developShephard technology Y ment has proved that, although this technology satisfies both disposability axioms (A1) and (A2), it is too small to represent a fully convex technology and too large to represent a technology in which only the output sets are assumed convex. Consider the following ‘‘unusual’’ variant of the convexity axiom: (A5) Convexity for observed activities. Technology Y contains all convex combinations of observed activities.
v
U C
V x
W 5
B E
1
M
1 D
H 0
1
w
4
b T without any convexity Fig. 4. The weakly disposable Tulkens technology Y assumptions.
In this paper we concluded our investigation of production technologies that incorporate weak disposability of bad and good outputs and satisfy various definitions of convexity. In addition b S and the Kuosmanen techto the classical Shephard technology Y b nology Y K introduced earlier, in this paper we developed two furb B and Y b T , where the subscripts refer to the fact ther technologies: Y that these technologies can be interpreted as weakly disposable variants of the Bogetoft (1996) and Tulkens (1993) technologies. If we construct the four technologies based on the same data set, the following embeddings will be observed:
bB # Y bS # Y bK: bT # Y Y
ð20Þ
b T does not assume any convexity for its activities and Technology Y can be regarded as the weakly disposable analogue of free disposable hull. It is the smallest of the four technologies. Technology b B assumes that all output sets P(x) are convex. This technology Y b T but smaller than technology Y b K which is obviously larger than Y is fully convex. b S , whose analysis initially motivated The Shephard technology Y our investigation, is not convex but allows for convex combinations of the observed activities. It is smaller than the Kuosmanen b K , and the latter is its convex hull (see Theorem 2 in technology Y Kuosmanen and Podinovski, 2009). At the same time, as follows from the Corollary to Theorem 3 in Kuosmanen and Podinovski (2009), all output sets of the Shephard technology are convex. This b B is a subset of technology Y b S. implies that technology Y The above embeddings (20) are illustrated by Figs. 1–4 in which the four technologies in question are constructed based on the same data set shown in Table 1. The smallest of these is the techb T in Fig. 4 which is not explicitly based on any variant of nology Y the convexity assumption. The larger and next in the order of b B shown in Fig. 3 is obtained from Y b T by embedding technology Y the assumption that the output sets should be convex for any input vector. As a result, this technology additionally includes the triangle ECL and the activities obtained from it by the strong disposabilb S enlarges the ity of the input. The Shephard technology Y b B by additionally including all convex combinations technology Y of the observed activities. In Fig. 1 this is represented by the segb K shown in Fig. 2 ment BC. Finally, the Kuosmanen technology Y is the convex hull of any of the other three technologies. In partic-
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b K inular, in contrast with the Shephard technology in Fig. 1, Y cludes the segment HC which is formed by all convex combinations of the observed activity C and hypothetical activity H. Appendix A
b T . It must satisfy (19) with Consider any activity (v, w, x) from Y b Without b ^ some vectors h and z. We need to prove that ðv ; w; xÞ 2 Y. loss of generality, let z1 = 1 and zk = 0 for k – 1. Then conditions (19) are reduced to
h1 v 1m P v m ; 1
h b B satisfies (A1), Proof of Theorem 1. First we need to prove that Y (A2) and (A4). The first part of this is identical to the corresponding part of the proof of Lemma 1 in Kuosmanen and Podinovski (2009), and is not given here. The proof of (A4) requires a minor modification. Consider any ~ ~ b B with the same input ~ x ^Þ and ðv ~ x ^Þ from Y ~ ; w; ~ ; w; two activities ðv ~ ~ ^ z, ~ h vector x. These must satisfy (4) with some vectors ~ z, ~ h and ~ ^ ¼ aðv ~ ^ ; wÞ ~ ; wÞþ respectively. For any a 2 [0; 1] define ðv ~ ~ ~ ^ x ^Þ ^ ; w; ~ ; wÞ. To prove (A4), we need to show that ðv ð1 aÞðv satisfies (4) with some vectors ^ z and ^ h. These are defined by formulae (A.1) and (A.2) in Kuosmanen and Podinovski (2009). The proof that the inequalities for good and bad outputs are satisfied is unchanged. To prove the conditional inequalities (3) note that if ^zk > 0 then, according to (A.1) in Kuosmanen and Podinovski (2009), either ~zk > 0, or ~ ~zk > 0, or both. In each of these cases, according to (3), we have xkn 6 ^ xn for all n = 1, . . . , N. Therefore the input inequalities (3) b B satisfies (A4). are satisfied, and Y b b B is a subset of any technology Y We now need to prove that Y that includes all observed activities and satisfies the axioms (A1), b B . It must satisfy (A2) and (A4). Consider any activity (v, w, x) from Y b (4) with some vectors ^ h and ^z. We need to prove that ðv ; w; xÞ 2 Y. ^ Let K0 be the set of all observed activities k such that ^zk > 0. Let x be the input vector whose components are defined as ^ xn ¼ maxk2K 0 fxkn g. For each observed activity (vk, wk, xk), k 2 K0 , ^Þ by changing its input introduce the modified activity ðv k ; wk ; x b satisfies (A1), all so modified ^. Since technology Y vector xk to x b activities are in Y. Then the abated activities k k k ^k k ^ b ^ a ¼ ðh v ; h w ; xÞ 2 Y for all k 2 K0 . Define activity a^ as the convex combination of activities ak taken with weights ^zk , k 2 K0 . Since the input vector ^ x is the same for all ak, by (A4) we have b a^ 2 Y. ^ do not ^n of activity a According to the definition, the inputs x exceed the inputs xn of activity (v, w, x). The summation terms on the left-hand side of (4) describe the good and bad outputs of ^ . Taking into account the corresponding inequalities and activity a b h equalities of (4), axiom (A1) implies that ðv ; w; xÞ 2 Y. b B satisfies (7) with some Proof of Theorem 2. Any ðv ; w; xÞ 2 Y ~ . Then it satisfies (16) with the same ~ vectors e k and l k. In particular, ~ k > 0 in (7) and inequalities xkn 6 xn folif ~ kk > 0 in (16), then ~ kk þ l low. Also, (0, 0, x) satisfies (16) with k = 0 for any x P 0. Therefore b B [ fð0; 0; xÞ : x P 0g # Y. e Y ~ satisfies (16) with some ~ Conversely, any ðv ; w; xÞ 2 Y k. Suppose first that ~ k – 0. Without loss of generality, let ~ k1 > 0. Define P l~ 1 ¼ 1 Kk¼1 ~k1 and l~ k ¼ 0 for k – 1. Then (v, w, x) satisfies (7) b B . If ~ ~ , and ðv ; w; xÞ 2 Y with ~ k and l k ¼ 0, then from (16) we must e#Y b B [ fð0; 0; xÞ : x P 0g. h have v = w = 0. Therefore, Y b T satisfies (A1) and (A2) Proof of Theorem 3. The proof that that Y is identical to the proof referred to in Theorem 1. We now need to b T is a subset of any technology Y b that prove that technology Y includes all observed activities and satisfies the axioms (A1) and (A2).
w1j
¼ wj ;
x1n 6 xn ;
8m; 8j;
8n:
ð21Þ ð22Þ ð23Þ
The activity (h1v1, h1w1, x1) is obtained by applying the abatement factor h1 to the observed activity (v1, w1, x1) and is, by axiom (A2), b From (21)–(23) and axiom (A1), ðv ; w; xÞ 2 Y. b h in technology Y. b S satisfies (A1) and (A2) is Proof of Theorem 4. The proof that Y identical to the proof referred to in Theorem 1. To prove that (A5) is satisfied, note that any convex combination of observed activities with weights zk satisfies (1) where h = 1. b S . It must satisfy (1) with Consider any activity (v, w, x) from Y b The some scalar ^ h and vector ^ z. We need to prove that ðv ; w; xÞ 2 Y. activity on the left-hand side of (1) is the convex combination of observed activities taken with the weights zk and further abated by b S . By (A1), the the factor h. According to (A2) and (A5), this is in Y b S. h activity (v, w, x) on the right-hand side of (1) must be in Y
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V.V. Podinovski, T. Kuosmanen / European Journal of Operational Research 211 (2011) 577–585 Kuosmanen, T., Kazemi Matin, R. Duality of weakly disposable technology, Omega, in press, doi:10.1016/j.omega.2010.10.008. Kuosmanen, T., Podinovski, V.V., 2009. Weak disposability in nonparametric production analysis: Reply to Färe and Grosskopf. American Journal of Agricultural Economics 91, 539–545. Lahdelma, R., Hakonen, H., 2003. An efficient linear programming algorithm for combined heat and power production. European Journal of Operational Research 148, 141–151. Lübbecke, M.E., Desrosiers, J., 2005. Selected topics in column generation. Operations Research 53, 1007–1023. Panzar, J.C., Willig, R.D., 1981. Economies of scope. American Economic Review 71, 268–272. Petersen, N.C., 1990. Data envelopment analysis on a relaxed set of assumptions. Management Science 36, 305–314. Podinovski, V.V., 2004a. On the linearisation of reference technologies for testing returns to scale in FDH models. European Journal of Operational Research 152, 800–802. Podinovski, V.V., 2004b. Bridging the gap between the constant and variable returns-to-scale models: Selective proportionality in data envelopment analysis. Journal of the Operational Research Society 55, 265–276.
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Contents lists available at ScienceDirect
European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Decision Support
Modelling weak disposability in data envelopment analysis under relaxed convexity assumptions Victor V. Podinovski a,⇑, Timo Kuosmanen b,c a
Warwick Business School, University of Warwick, Coventry, CV4 7AL, UK School of Economics, Aalto University, 00101 Helsinki, Finland c MTT Agrifood Research Finland, 00410 Helsinki, Finland b
a r t i c l e
i n f o
Article history: Received 31 March 2009 Accepted 4 December 2010 Available online 9 December 2010 Keywords: Data envelopment analysis Nonparametric productivity analysis Environmental performance Undesirable outputs Non-convex technologies
a b s t r a c t The treatment of undesirable (bad) outputs in models of efficiency and productivity analysis often requires replacing the assumption of free disposability of outputs by their weak disposability. In a recent publication the authors showed that the Kuosmanen technology is the only correct representation of the fully convex technology exhibiting weak disposability of bad and good outputs. In this paper we relax the assumption of full convexity and consider two further possibilities: the case in which only the output sets are assumed convex and the case in which no convexity is assumed at all. In the first case we show that, although the traditional Shephard technology of nonparametric production analysis satisfies the assumption of convex output sets, it is larger than necessary. Based on the minimum extrapolation principle, we develop a correct model that is based on the assumed axioms. The second case leads to the development of a weakly disposable analogue of the free disposable hull. To complete our study, we give a full axiomatic definition of the Shephard technology. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction In recent years the treatment of undesirable (bad) outputs in models of productivity and efficiency analysis has attracted significant and growing interest in the literature on data envelopment analysis (DEA) (e.g., Chung et al., 1997; Scheel, 2001; Seiford and Zhu, 2002; Kuosmanen, 2005). One particular example of bad outputs is harmful emissions of a production process and, more generally, any undesirable environmental factor that accompanies production. Conventional treatments of undesirable outputs used in DEA, for example, by regarding them as inputs (Dyson et al., 2001; Seiford and Zhu, 2002), involve two problems. Firstly, assuming free disposability of inputs and bad outputs would imply that a finite amount of input can produce an infinite amount of bad output, thus violating the law of mass conservation. Secondly, the free disposability assumption does not recognize the link between the good and bad outputs, specifically, that emissions can be reduced in proportion to the good outputs. This property is known as weak disposability of undesirable and good outputs (Shephard, 1974). The proportion by which the good and bad outputs are reduced is referred to as the abatement factor (Kuosmanen, 2005). In production economics, the standard approach to modelling weak disposability is the model by Shephard (1974). This model
⇑ Corresponding author. Tel.: +44 2476524281. E-mail address: [email protected] (V.V. Podinovski). 0377-2217/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2010.12.003
is similar to the standard variable returns-to-scale (VRS) DEA model (Banker et al., 1984), except that it differentiates between the good and bad output, does not have a provision for free disposal of bad output and allows for the abatement of bad and good output by the use of an abatement factor. Ideally, a (model of) production technology should be constructed based on observed data and the set of explicitly stated assumptions about the shape of the technology, for example, free disposability of inputs and outputs or convexity. The technology should be proved to satisfy the minimum extrapolation principle as in Banker et al. (1984) which guarantees that the model is the smallest possible and therefore does not contain any arbitrary activities (units). Surprisingly, despite the popularity of the Shephard technology and its clear mathematical representation, the axiomatic properties of this technology have not been fully understood. Recently Kuosmanen and Podinovski (2009) proved that a single abatement factor used in the Shephard technology is insufficient to correctly represent a convex technology exhibiting weak disposability of undesirable and good outputs. In practice, this means that efficiency assessment undertaken with the Shephard technology may overestimate technical efficiency of the evaluated firms when the maintained assumptions hold, and this upward bias does not vanish even if the sample size approaches infinity. Moreover, the computed benchmarks for inefficient activities may not be technically efficient. The authors further showed that the Kuosmanen (2005) model based on the use of multiple abatement factors is the only correct technology suitable for this
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purpose. It was also shown that the Kuosmanen technology is the convex hull of the Shephard technology. Although the Shephard technology is not convex, it is easy to verify that its output sets are convex for each given vector of input. In our new development we set out to investigate whether the Shephard technology is the smallest non-convex technology that has this property and incorporates the assumption of weak disposability of bad outputs. We show that this is not the case. In other words, the Shephard technology is unnecessarily large for this purpose. Instead, a smaller technology which we introduce in this paper is the correct approach here. We then develop a weakly disposable technology that does not assume convexity at all – this is a weakly disposable analogue of free disposable hull (FDH) of Tulkens (1993). Together with our previous developments this gives us a range of models to choose from: we can assume full convexity of the technology, convexity of the output sets only or no convexity at all. If needed, an even wider range of models can be obtained based on the assumption that only some inputs and outputs are subject to the convexity axiom. This is not pursued here but the general approach for modelling such technologies is readily available (Podinovski, 2005). Our development contributes to the growing stream of efficiency analysis literature where the full convexity of the production possibility set is relaxed but some weaker form of convexity is maintained (Petersen, 1990; Bogetoft, 1996; Bogetoft et al., 2000; Dekker and Post, 2001; Kuosmanen, 2001, 2003; Briec et al., 2004; Agrell et al., 2005; Podinovski, 2004a, 2005; Ehrgott and Tind, 2009). The practical motivation for these approaches is that the (fully) convex production set is not well suited for modelling economies of scale (Petersen, 1990; Bogetoft, 1996; Bogetoft et al., 2000). Thus, relaxing the full convexity assumption may be appropriate in such sectors as the railroad industry where economies of scale are thought to prevail (Klein, 1947; Hasenkamp, 1976). On the other hand, the convexity of the output sets is closely related with the economies of scope, which are said to prevail if joint production of two or more outputs is less expensive than producing the outputs separately. The joint production of passenger and freight services in the railroad industry is a classic example of this (Panzar and Willig, 1981). Another example of joint production is the combined heat and power (CHP) generation (also known as cogeneration), which is an increasingly important energy production technology (e.g., Lahdelma and Hakonen, 2003; Rong et al., 2006). Yang and Pollitt (2009) is one of the few studies that discusses CHP in the context of DEA. Therefore, the models developed in this study appear to be particularly well suited for sectors such as energy and transportation where both economies of scale and scope are expected to prevail. While the majority of the DEA approaches under relaxed convexity assumptions (cited above) incorporate free disposability of all inputs and outputs, the models developed in this paper are the first to combine the axioms of weak disposability with reduced convexity. The techniques developed in this paper could be used, for example, for modelling accidents (e.g. fatalities) in the already mentioned railroad industry as a bad output that is weakly disposable: if passenger and freight services are scaled down by a certain percent, it should be possible to decrease the accidents by the same percent. Similarly, in the energy production, harmful emissions such as CO2 can be assumed weakly disposable together with the level of heat and electricity produced. Finally, in order to complete our study of the Shephard technology we give its correct and complete axiomatic characterization. This is driven not only by our academic curiosity but the need to understand why this technology is too small to represent convex production sets (in the input–output space) and too large to represent technologies with convex output sets (in the output space only, for any fixed input vector). We prove that the Shephard
technology is characterized by a strange variant of the convexity axiom: it allows for convex combinations of observed activities but does not explicitly assume the same for all activities in the technology. The authors can offer no reason why this assumption might be acceptable in practice. 2. Preliminaries The terminology and notation in our development are consistent with the economic literature on nonparametric productivity analysis. These are different but equivalent to the terminology and notation used in mainstream DEA (Charnes et al., 1978; Cooper et al., 2000; Thanassoulis, 2001). Following our previous development (Kuosmanen, 2005; Kuosmanen and Podinovski, 2009; Kuosmanen and Kazemi Matin, in press) each production activity (decision making unit in the DEA terminology) is characterized by the triplet (v, w, x), where v ¼ ðv 1 ; . . . ; v M Þ 2 RM þ is the vectors of desirable (good) outputs, w ¼ ðw1 ; . . . ; wJ Þ 2 RJþ is the vector of undesirable (bad) outputs and x ¼ ðx1 ; . . . ; xN Þ 2 RNþ is the vector of inputs. We assume we have K observed activities denoted by (vk, wk, xk), k = 1, . . . , K. A production technology is the set of all producible activities Y = {(v, w, x)jx can produce (v, w)}. Equivalently, technology Y can be represented by its output sets P(x) = {(v, w)j(v, w, x) 2 Y}, x 2 RNþ . In the present context, technology Y is usually assumed to satisfy the following axioms: (A1) Strong disposability of inputs and good outputs. If (v, w, x) 2 Y, 0 6 v0 6 v and x0 P x (for each component) then (v0 , w, x0 ) 2 Y. (A2) Weak disposability of undesirable outputs and good outputs. If (v, w, x) 2 Y and h 2 [0, 1] then (hv, hw, x) 2 Y. (A3) Convexity. Technology Y is convex. The multiplier h used in axiom (A2) is referred to as the abatement factor: it enables the reduction of the level of bad outputs, such as harmful emissions, if accompanied by the reduction of good outputs in the same proportion. Unfortunately, the implementation of the abatement factor in DEA models has caused some confusion. The conventional approach to modelling weak disposability b S em(A2) in DEA is due to Shephard (1974), whose technology Y ploys a single abatement factor h but is not convex (Kuosmanen and Podinovski, 2009). The Shephard technology is stated as follows:
( bS ¼ Y
ðv ; w; xÞ : h
K X
zk v km P v m P 0;
8m
ð1Þ
k¼1
h
K X
zk wkj ¼ wj ;
8j
k¼1 K X
zk xkn 6 xn ;
8n
k¼1 K X
zk ¼ 1
k¼1
zk P 0;
8k
0 6 h 6 1g; where variables z = (z1, . . . , zK) are referred to as the intensity weights. In (1) we have explicitly incorporated the requirement that good outputs vm (taken here as model variables) must be nonnegative. The nonnegativity of bad outputs wj and inputs xn follows automatically from the corresponding constraints and need not be specifically stated.
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Kuosmanen (2005) developed an alternative model for a technology whose bad outputs exhibit weak disposability. In contrast to the Shephard technology, the Kuosmanen technology is convex and uses individual abatement factors hk attached to each observed activity k = 1, . . . , K:
( bK ¼ Y
ðv ; w; xÞ :
K X
hk zk v km P v m P 0;
8m
v
U
V
C x
ð2Þ
k¼1 K X
W k k
hz
wkj
¼ wj ;
8j
5
k¼1 K X
zk xkn 6 xn ;
8n
k¼1 K X
B k
z ¼1
E
M
k¼1
zk P 0;
8k
o 0 6 h 6 1; 8k : k
Although both programs (1) and (2) are nonlinear, Kuosmanen (2005) demonstrated that model (2) could be linearized by substitution of variables. To gain further economic insight, Kuosmanen and Kazemi Matin, in press, developed dual formulations of the linearized technology. However, the Shephard technology (1) cannot be linearized by the same approach. We use the suggested substitution again in our development below. It is clear that the Shephard technology is a subset of the Kuosmanen technology, as the former can be obtained from description (2) by the incorporation of the additional condition h1 = h2 = = hK. The difference between the Shephard and Kuosmanen technologies is further illustrated by the example taken from Kuosmanen and Podinovski (2009) which involves just two activities B and C as in Table 1. In the Shephard technology (Fig. 1) the edges BH and CE are obtained by the simultaneous abatement of the good and bad outputs of activities B and C, respectively. The triangle BDH is induced by free disposability of the good output v of the activities represented by the edge BH. The rays BV, CU, DW and HE are induced by free disposability of the input. Since activities B, C, E and H are not in the same plane, the side BCEH is not ‘‘flat’’ and the interior points of the segment CH are not in the technology. This demonstrates that the Shephard technology is generally not convex. The corresponding Kuosmanen technology is shown in Fig. 2. It is larger than the Shephard technology and, in particular, includes the segment CH. This is because formulation (2) allows for convex combinations of the observed activities abated in different proportions hk. In Fig. 2, activity H is obtained by the application of the extreme abatement factor zero to activity B. The segment CH is then obtained by combining C and H. Kuosmanen and Podinovski (2009) proved that the Kuosmanen b K is the smallest technology that incorporates all obtechnology Y served activities and satisfies the axioms (A1)–(A3) stated above. b K is the intersection of all technologies that satisfy Formally, Y these axioms. This approach to the definition of technologies based on a given set of axioms is known as the minimum extrapolation principle, which was introduced to DEA by Banker et al. (1984). Although many other technologies, for example, the standard VRS and constant returns-to-scale (CRS) technologies, also satisfy Table 1 The dataset for illustrations: good output v, undesirable output w, input x.
v w x
Activity B
Activity C
3 4 1
5 1 4
1
1
D
H 0
1
w
4
b S induced by activities B and C. Fig. 1. The Shephard technology Y
v
U
V
C x
W 5
B M E
1
1 D
H 0
1
w
4
b K induced by activities B and C. Fig. 2. The Kuosmanen technology Y
(A1)–(A3), all of these contain arbitrary (in the given context) activities that are not implied by the stated axioms and are, therefore, unnecessarily large. In contrast, no activity in the Kuosmanen b K can be removed without a violation of the stated technology Y axioms. Kuosmanen and Podinovski (2009) further proved that the b K is the convex hull (smallest convex Kuosmanen technology Y
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b S . This fact is clearly illusextension) of the Shephard technology Y trated by Figs. 1 and 2: the Kuosmanen technology is obtained from the Shephard technology by adding to the latter all missing convex combinations of its activities.
v
U
V
C
x
3. Production technologies with convex output sets b S is not convex, its output Although the Shephard technology Y sets P(x) are convex for any input vector x. This formally follows from the Corollary to Theorem 3 and subsequent discussion in Kuosmanen and Podinovski (2009). This can also be seen in Fig. 1. For example, if x = 1,the corresponding set P(1) is the triangle BHD. Färe and Grosskopf (2009) pointed out the noted convexity of output sets as a reason for using the Shephard technology. Given this observation, a question may be asked whether the Shephard technology can indeed be regarded as being based on the disposability axioms (A1), (A2) and the following variant of the convexity axiom:
L
W
5
B M
E
(A4) Convexity of the output sets. The sets P(x) are convex for all x.
1
1 In other words, is the Shephard technology a variant of Bogetoft’s (1996) technology that assumes convex output sets but no other form of convexity, in which free disposability of inputs and outputs is replaced by weak disposability for bad outputs? (Note that in mathematics empty sets are viewed as convex, and so we do not need to limit axiom (A4) to the case of non-empty sets P(x) only.) Below we show that the answer to the above question is negative: the Shephard technology is larger than necessary for the axiom (A4) and is, therefore, an arbitrary technology with respect to the stated specifications. The correct weakly disposable variant of the Bogetoft technology, based on the minimum extrapolation principle, is as follows. b B which contains all observed Theorem 1. The minimum technology Y activities and satisfies the axioms (A1), (A2) and (A4) is obtained from (2) by replacing the inequalities associated with the inputs by the following conditions:
if zk > 0 then xkn 6 xn ;
8n; k:
ð3Þ
The proofs of this and the other theorems presented below are given in Appendix A. Following Podinovski (2004b), we first prove b B satisfies all the stated axioms. We then that the technology Y b B , then ðv ; w; xÞ 2 Y, b where prove that, if some activity ðv ; w; xÞ 2 Y b is any technology that satisfies the same set of axioms. The Y b B is the minimum implication of these two statements is that Y technology that satisfies the stated axioms. b B is as Based on Theorem 1, the full formulation of technology Y follows:
( bB ¼ Y
ðv ; w; xÞ :
K X
hk zk v km P v m P 0;
8m
ð4Þ
k¼1 K X
k k
hz
wkj
¼ wj ;
K X
8j 8n; k
4
b B with convex output sets. Fig. 3. The weakly disposable Bogetoft technology Y
extends this to triangle BDH. The free disposability of the input induces the unbounded polyhedron shaped by this triangle and the rays BV, DW and HE. Similarly, using (A1) and (A2), activity C induces the unbounded polyhedron based on the triangle CEM. Note that we cannot join activities B and C as in the Shephard technology in Fig. 1 because the convexity axiom (A3) is not assumed. Instead, we can only form convex combinations of activities that have the same input level x. For example, activity L on the ray BV has x = 4, the same as activities C and E. Therefore, the convex combinations of L, C and E must be in the technology, and we obtain the triangle CEL. b B in Fig. 3 is the smallest technology According to Theorem 1, Y with convex output sets induced by activities B and C. The output b S are also convex but this technolsets of the Shephard technology Y ogy contains additional activities, for example, the segment BC, which cannot be justified by the assumed axioms (A1), (A2) and (A4). b B in the form (4) has two practical drawRepresenting technology Y backs: this model is not linear and includes conditional inequalities (3). Both drawbacks are easy to overcome. In order to linearize model (4), we can use the substitution from Kuosmanen (2005):
kk ¼ hk zk ;
lk ¼ ð1 hk Þzk ; 8k;
ð5Þ
so that
kk þ lk ¼ zk :
ð6Þ
bB ¼ Y
ðv ; w; xÞ :
K X
kk v km P v m P 0; 8m
k¼1 K X
k¼1
k
1
(
zk ¼ 1
zk P 0;
0
kk wkj ¼ wj ; 8j
k¼1
8k
0 6 h 6 1;
w
Rearranging the terms, we obtain an equivalent statement of techb B: nology Y
k¼1
if zk > 0 then xkn 6 xn ;
D
H
o 8k :
b B induced by activities B and C from Fig. 3 shows technology Y Table 1. The weak disposability of the bad and good output induces the segment BH. The free disposability of good output
if kk þ lk > 0 then xkn 6 xn ; 8n; k K X ðkk þ lk Þ ¼ 1 k¼1
kk ;
o
lk P 0; 8k :
ð7Þ
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Following Kuosmanen et al. (2006, Appendix), the conditional input inequalities in program (7) can be replaced by the linear inequalities
ðkk þ lk Þxkn 6 ðkk þ lk Þxn ;
8n; k:
ð8Þ
If kk + lk > 0, inequalities (8) imply xkn 6 xn as required. The fully linb B is therefore as follows: earized version of technology Y
(
bB ¼ Y
ðv ; w; xÞ :
K X
kk v km P v m P 0;
8m
ð9Þ
k¼1 K X
kk wkj ¼ wj ;
8n; k
K X
ðkk þ lk Þ ¼ 1
kk ; lk P 0;
ð12Þ
For each observed activity k define a binary variable b 2 {0, 1} and introduce the following inequality:
kk þ lk 6 bk ;
8k:
ð13Þ k
The idea behind (13) is that b serves as an indicator that becomes equal to 1 if the sum kk + lk is positive (note that kk + lk cannot exceed 1). If kk + lk = 0 then bk can be either 0 or 1. Condition (12) is now restated in the following linear form:
8n; k:
ð14Þ
Indeed, if kk + lk > 0 then, from (13), bk = 1, and (14) implies xkn 6 x0n /g xn . If kk + lk = 0, then by (13) b can be either 0 or 1 and (14) does not restrict anything as required. With all of the above changes, the value of the general directional distance function is obtained by solving the following mixed integer linear program:
k¼1
k¼1
8n; k: k
bk xkn 6 x0n /g xn ;
8j
ðkk þ lk Þxkn 6 ðkk þ lk Þxn ;
if kk þ lk > 0 then xkn 6 x0n /g xn ;
o
8k :
b B in practical computations is discussed in The use of technology Y the next section.
^dB
v0 ; w0 ; x0 ; gv ; gw ; gx
Subject to
K X
¼ max /
kk v km P v 0m þ /g vm ;
ð15Þ
8m;
k¼1 K X
4. Computational approaches In practice, measuring the efficiency of a given activity (v0, w0, x0) is often conducted using a directional distance function (Chambers et al., 1996, 1998; Chung et al., 1997) that allows for a simultaneous increase in good output and reduction in input and bad output. The conventional radial Farrell input and output efficiency measures can be obtained as special cases of the directional distance functions (Chambers et al., 1998). When using directional b B we need to distinguish bedistance functions with technology Y tween two cases: the output directional distance function
do ðv 0 ; w0 ; x0 ; gv ; gw Þ ¼ sup /ðv 0 þ /gv ; w0 /gw ; x0 Þ 2 Y
ð10Þ
and the general directional distance function
dðv 0 ; w0 ; x0 ; gv ; gw ; gx Þ ¼ sup /ðv 0 þ /gv ; w0 /gw ; x0 /gx Þ 2 Y : ð11Þ In (10) and (11), gv, gw and gx are nonnegative vectors referred to as the direction vectors: these specify the direction in which the evaluated activity (v0, w0, x0) is projected on the frontier. The output directional distance function (10) keeps the input vector x0 at a constant level and seeks to increase the good output v0 in the direction gv and decrease the bad output w0 in the direction gw. The general directional distance function (11) differs from (10) in that it additionally seeks to decrease the input use x0 in the direction gx. Note that (10) is a special case of (11) where gx = 0; any zero elements in the direction vectors imply that the corresponding input or output is held at a constant level in the efficiency evaluation. The use of output directional distance function (10) with techb B leads to a straightforward linear program: we simply nology Y 0 substitute v m ¼ v 0m þ /g vm , wj ¼ w0j /g w m and xn ¼ xn onto the right-hand side of model (9) and set the objective function to maximize /. We can also omit the requirements v m ¼ v 0m þ /g vm P 0 for all m in (9) as obviously redundant. The use of the general directional distance function (11) requires a somewhat more complex treatment. Note that by simply introducing input reductions in the linear model (9) we would render the third group of constraints nonlinear, specifically, kk xkn 6 kk ðx0n /g xn Þ. This problem can be overcome by resorting to mixed integer linear programming as in Podinovski (2005). For b B as a starting this purpose formulation (7) of the technology Y point is more useful. Its conditional input constraints would take on the form:
kk wkj ¼ w0j /g w j ;
8j;
k¼1
bk xkn 6 x0n /g xn ;
8k; n;
K X ðkk þ lk Þ ¼ 1; k¼1
kk þ lk 6 bk ;
8k;
kk ; lk P 0;
8k;
bk 2 f0; 1g;
8k:
Compared to the case of output directional distance function leading to a simple linear program, the general formulation (15) is more complex, involving binary variables bk. We will next show that in practical computations the number of variables in the previous programs can be significantly reduced (under some realistic assumptions discussed below). More specifically, we only need variables kk and can dispose of the variables lk. This approach has obvious computational advantages. e obtained from the formuConsider the following technology Y lation (7) by removing all variables lk:
( e¼ Y
ðv ; w; xÞ :
K X k¼1 K X
kk v km P v m P 0; kk wkj ¼ wj ;
8m
ð16Þ
8j;
k¼1
if kk > 0 then xkn 6 xn ; K X kk 6 1; k¼1 o kk P 0; 8k :
8n; k;
e is not identical to Y b B . For example, the former contains Technology Y the origin (0, 0, 0), which satisfies (16) with k = 0 and is missing from b B (assuming none of the observed activities has a zero input vector). Y The following theorem formalizes this difference and shows that e is slightly larger than Y b B. technology Y e¼Y b B [ fð0; 0; xÞ : x P 0g. Theorem 2. The following holds: Y e is obtained from the As an illustration, in Fig. 3 technology Y b B by adding to it the additional segment depicted technology Y OH.
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e and Y b B stated by TheThe difference between technologies Y orem 2 is unimportant for most practical purposes, and we can e instead of Y b B whenever the asuse the simpler technology Y sessed activity (v0, w0, x0) has at least one strictly positive good output (v0 – 0). Because the good output v0 cannot decrease by a movement in the direction gv P 0, the presence of additional e becomes irrelevant for the assesspoints (0, 0, x) in technology Y ment. Note that the condition v0 – 0 is sufficient but not necese and Y b B when used sary for the equivalence of technologies Y with the directional distance function (11). Other sufficient conditions can be stated in terms of the direction vectors (gv, gw, gx): if gv – 0 or gx = 0, then the directional distance function (11) yields e and Y b B . In practice, situations the same value with respect to Y where activities in the form (0, 0, x) with x P 0 and located outb B are identified as efficient targets are extremely side the set Y rare. Based on the above observation, we can further simplify the two programs developed in the previous section. In the case of the outb B by put directional distance function (10), replacing technology Y e Y leads to the following linear program:
^do ðv 0 ; w0 ; x0 ; gv ; gw Þ ¼ max / B Subject to
K X
ð17Þ
kk v km P v 0m þ /g vm ;
5. Weak disposability without convexity
8m; Following the same approach as above, it is not difficult to construct a technology which is based only on axioms (A1) and (A2) without assuming any variant of the convexity axiom. This can be considered as the analogue of the FDH technology of Tulkens (1993) in which free disposability of inputs and outputs is replaced by weak disposability for bad outputs. We denote such a technolbT. ogy Y
k¼1 K X
k
k
wkj
¼
w0j
/g w j ;
8j;
k¼1
kk xkn 6 kk x0n ; K X
8k; n;
kk 6 1;
k¼1
kk P 0;
models (17) and (18).1 In model (17), we use about a half of the varib B . In model (18), we use ables required in the proper technology Y about one third fewer variables than in program (15) based on techb B. nology Y Model (18) is a mixed integer linear program in which the number of binary variables bk is equal to the number of activities K. This model can in principle be solved by most commercial linear optimizers, which may be a preferred approach if K is not large and, consequently, the computational time is not problematic. However, because in practice we would normally need to solve K programs (18), one for each activity under the assessment, for large K this may present a certain computational challenge. The development of methods of mixed integer programming2 suitable for large scale computations based on model (18) lies beyond the scope of this paper. We only note that computational algorithms for some other mixed integer linear DEA models have been explored in the literature, but these are specific and generally not transferable between different DEA models. Examples include the elimination algorithm for solving the FDH model (Tulkens, 1993) and a column generation technique by Ehrgott and Tind (2009) for the free replicability model of Tulkens (1993).
8k:
In the case of general directional distance function (11) we need to repeat the logic behind program (15), this time starting with ~ In particular, we need to introduce binary variables technology Y. bk that serve as indicators taking on the value of 1 if kk > 0. The resulting mixed integer linear program takes on the following form:
b T which contains all observed Theorem 3. The minimum technology Y activities and satisfies the axioms (A1) and (A2) is obtained from (2) by additionally requiring that the intensity weights zk take on only 0 or 1 values. b T is described as follows: Based on this result the technology Y
(
bT ¼ Y
ðv ; w; xÞ :
K X
hk zk v km P v m P 0;
8m
ð19Þ
k¼1
^dð v 0 ; w0 ; x0 ; gv ; gw ; gx Þ ¼ max / B Subject to
K X
k
k
v
k m
Pv
0 m
ð18Þ
þ /g v ; m
8m;
k¼1 K X
k
k
wkj
¼
w0j
/g xn ;
/g w j ;
8j;
k¼1
bk xkn K X
6
x0n
8k; n;
k
kk 6 1;
k P 0;
8k; 8k:
In program (18), if kk > 0 then bk = 1 and the corresponding input constraints xkn 6 x0n /g xn are satisfied for all n. It is worth emphasizing that the only correct technology based ^ B as described by on the axioms (A1), (A2) and (A4) is technology Y (4) or equivalent formulations (7) and (9). The assessment of effib B by using these forciency can be performed directly in terms of Y mulations. In practice, the same results can be obtained by e using performing computations in the slightly larger technology Y,
8j;
k¼1 K X
zk xkn 6 xn ;
8n;
k¼1 K X
zk ¼ 1;
k¼1
0 6 hk 6 1;
8k;
bk 2 f0; 1g;
hk zk wkj ¼ wj ;
zk 2 f0; 1g; 8k;
k¼1
kk 6 bk ;
K X
o
8k :
b T can be linearized using the For practical purposes technology Y substitution (5) as discussed above. A minor obstacle with this approach is that the sum of variables kk + lk = zk needs to be declared binary. This can be achieved by including equalities (6) and keeping the declaration of variables zk as binary variables in the model. The use of this technology in DEA with the directional distance functions (11) would lead to solving mixed integer programs, one for each activity under the consideration. 1 e and Y b B were discussed The formal conditions of substitutability of technologies Y after Theorem 2. 2 For an overview of methods of integer programming, see Wolsey (1998) and Lübbecke and Desrosiers (2005).
V.V. Podinovski, T. Kuosmanen / European Journal of Operational Research 211 (2011) 577–585
Alternatively, as with the classical implementations of FDH, b T can be regarded as the union of all technologies intechnology Y duced by individual observed activities:
bT ¼ Y
K [ ðv ; w; xÞ : hv km P v m P 0;
8m
k¼1
hwkj ¼ wj ; xkn
6 xn ;
8j; 8n;
0 6 h 6 1g: The value of the directional distance function for any activity can now be computed by solving a required linear program in each of the K separate technologies and taking the largest of the values so obtained. Some of these programs (but not all) may be infeasible and should be ignored. b T induced by activities B and C from Fig. 4 shows technology Y Table 1. It is the smallest of all of the discussed technologies and does not assume any convexity. It is the simple union of two separate technologies induced by activities B and C. The technology induced by B is based on the triangle BHD which, due to free disposability of input, extends along the rays BV, DW and HE to infinity along the input axis. The technology induced by activity C is obtained in a similar way based on the triangle CEM.
583
b S is the minimum set which Theorem 4. The Shephard technology Y contains all observed activities and satisfies the axioms (A1), (A2) and (A5). Note that Theorem 4 does not state that the Shephard technology is convex: axiom (A5) explicitly allows only for observed activities to be combined. As an illustration to this comment, the Shephard technology in Fig. 1 includes all convex combinations of observed activities B and C that form the edge BC. However, activities C and H cannot be combined because activity H is not observed. This does not mean, however, that all convex combinations involving unobserved activities are missing from this technology: some of such combinations are present due to other axioms. For example, convex combinations of activities B and H, of which H is not observed, form the segment BH which is in the technology as required by the weak disposability axiom (A2). It is difficult to see reasons why in practice one should assume convexity only for observed activities but not allow convexity for unobserved activities deduced from the maintained axioms. This makes it difficult to justify the use of this technology for efficiency assessment as there is no guarantee that the computed benchmarks and targets would be substantiated. 7. Conclusion
6. Axioms of the Shephard technology In this section we offer a simple axiomatic definition of the b S as defined in (1). Our previous developShephard technology Y ment has proved that, although this technology satisfies both disposability axioms (A1) and (A2), it is too small to represent a fully convex technology and too large to represent a technology in which only the output sets are assumed convex. Consider the following ‘‘unusual’’ variant of the convexity axiom: (A5) Convexity for observed activities. Technology Y contains all convex combinations of observed activities.
v
U C
V x
W 5
B E
1
M
1 D
H 0
1
w
4
b T without any convexity Fig. 4. The weakly disposable Tulkens technology Y assumptions.
In this paper we concluded our investigation of production technologies that incorporate weak disposability of bad and good outputs and satisfy various definitions of convexity. In addition b S and the Kuosmanen techto the classical Shephard technology Y b nology Y K introduced earlier, in this paper we developed two furb B and Y b T , where the subscripts refer to the fact ther technologies: Y that these technologies can be interpreted as weakly disposable variants of the Bogetoft (1996) and Tulkens (1993) technologies. If we construct the four technologies based on the same data set, the following embeddings will be observed:
bB # Y bS # Y bK: bT # Y Y
ð20Þ
b T does not assume any convexity for its activities and Technology Y can be regarded as the weakly disposable analogue of free disposable hull. It is the smallest of the four technologies. Technology b B assumes that all output sets P(x) are convex. This technology Y b T but smaller than technology Y b K which is obviously larger than Y is fully convex. b S , whose analysis initially motivated The Shephard technology Y our investigation, is not convex but allows for convex combinations of the observed activities. It is smaller than the Kuosmanen b K , and the latter is its convex hull (see Theorem 2 in technology Y Kuosmanen and Podinovski, 2009). At the same time, as follows from the Corollary to Theorem 3 in Kuosmanen and Podinovski (2009), all output sets of the Shephard technology are convex. This b B is a subset of technology Y b S. implies that technology Y The above embeddings (20) are illustrated by Figs. 1–4 in which the four technologies in question are constructed based on the same data set shown in Table 1. The smallest of these is the techb T in Fig. 4 which is not explicitly based on any variant of nology Y the convexity assumption. The larger and next in the order of b B shown in Fig. 3 is obtained from Y b T by embedding technology Y the assumption that the output sets should be convex for any input vector. As a result, this technology additionally includes the triangle ECL and the activities obtained from it by the strong disposabilb S enlarges the ity of the input. The Shephard technology Y b B by additionally including all convex combinations technology Y of the observed activities. In Fig. 1 this is represented by the segb K shown in Fig. 2 ment BC. Finally, the Kuosmanen technology Y is the convex hull of any of the other three technologies. In partic-
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b K inular, in contrast with the Shephard technology in Fig. 1, Y cludes the segment HC which is formed by all convex combinations of the observed activity C and hypothetical activity H. Appendix A
b T . It must satisfy (19) with Consider any activity (v, w, x) from Y b Without b ^ some vectors h and z. We need to prove that ðv ; w; xÞ 2 Y. loss of generality, let z1 = 1 and zk = 0 for k – 1. Then conditions (19) are reduced to
h1 v 1m P v m ; 1
h b B satisfies (A1), Proof of Theorem 1. First we need to prove that Y (A2) and (A4). The first part of this is identical to the corresponding part of the proof of Lemma 1 in Kuosmanen and Podinovski (2009), and is not given here. The proof of (A4) requires a minor modification. Consider any ~ ~ b B with the same input ~ x ^Þ and ðv ~ x ^Þ from Y ~ ; w; ~ ; w; two activities ðv ~ ~ ^ z, ~ h vector x. These must satisfy (4) with some vectors ~ z, ~ h and ~ ^ ¼ aðv ~ ^ ; wÞ ~ ; wÞþ respectively. For any a 2 [0; 1] define ðv ~ ~ ~ ^ x ^Þ ^ ; w; ~ ; wÞ. To prove (A4), we need to show that ðv ð1 aÞðv satisfies (4) with some vectors ^ z and ^ h. These are defined by formulae (A.1) and (A.2) in Kuosmanen and Podinovski (2009). The proof that the inequalities for good and bad outputs are satisfied is unchanged. To prove the conditional inequalities (3) note that if ^zk > 0 then, according to (A.1) in Kuosmanen and Podinovski (2009), either ~zk > 0, or ~ ~zk > 0, or both. In each of these cases, according to (3), we have xkn 6 ^ xn for all n = 1, . . . , N. Therefore the input inequalities (3) b B satisfies (A4). are satisfied, and Y b b B is a subset of any technology Y We now need to prove that Y that includes all observed activities and satisfies the axioms (A1), b B . It must satisfy (A2) and (A4). Consider any activity (v, w, x) from Y b (4) with some vectors ^ h and ^z. We need to prove that ðv ; w; xÞ 2 Y. ^ Let K0 be the set of all observed activities k such that ^zk > 0. Let x be the input vector whose components are defined as ^ xn ¼ maxk2K 0 fxkn g. For each observed activity (vk, wk, xk), k 2 K0 , ^Þ by changing its input introduce the modified activity ðv k ; wk ; x b satisfies (A1), all so modified ^. Since technology Y vector xk to x b activities are in Y. Then the abated activities k k k ^k k ^ b ^ a ¼ ðh v ; h w ; xÞ 2 Y for all k 2 K0 . Define activity a^ as the convex combination of activities ak taken with weights ^zk , k 2 K0 . Since the input vector ^ x is the same for all ak, by (A4) we have b a^ 2 Y. ^ do not ^n of activity a According to the definition, the inputs x exceed the inputs xn of activity (v, w, x). The summation terms on the left-hand side of (4) describe the good and bad outputs of ^ . Taking into account the corresponding inequalities and activity a b h equalities of (4), axiom (A1) implies that ðv ; w; xÞ 2 Y. b B satisfies (7) with some Proof of Theorem 2. Any ðv ; w; xÞ 2 Y ~ . Then it satisfies (16) with the same ~ vectors e k and l k. In particular, ~ k > 0 in (7) and inequalities xkn 6 xn folif ~ kk > 0 in (16), then ~ kk þ l low. Also, (0, 0, x) satisfies (16) with k = 0 for any x P 0. Therefore b B [ fð0; 0; xÞ : x P 0g # Y. e Y ~ satisfies (16) with some ~ Conversely, any ðv ; w; xÞ 2 Y k. Suppose first that ~ k – 0. Without loss of generality, let ~ k1 > 0. Define P l~ 1 ¼ 1 Kk¼1 ~k1 and l~ k ¼ 0 for k – 1. Then (v, w, x) satisfies (7) b B . If ~ ~ , and ðv ; w; xÞ 2 Y with ~ k and l k ¼ 0, then from (16) we must e#Y b B [ fð0; 0; xÞ : x P 0g. h have v = w = 0. Therefore, Y b T satisfies (A1) and (A2) Proof of Theorem 3. The proof that that Y is identical to the proof referred to in Theorem 1. We now need to b T is a subset of any technology Y b that prove that technology Y includes all observed activities and satisfies the axioms (A1) and (A2).
w1j
¼ wj ;
x1n 6 xn ;
8m; 8j;
8n:
ð21Þ ð22Þ ð23Þ
The activity (h1v1, h1w1, x1) is obtained by applying the abatement factor h1 to the observed activity (v1, w1, x1) and is, by axiom (A2), b From (21)–(23) and axiom (A1), ðv ; w; xÞ 2 Y. b h in technology Y. b S satisfies (A1) and (A2) is Proof of Theorem 4. The proof that Y identical to the proof referred to in Theorem 1. To prove that (A5) is satisfied, note that any convex combination of observed activities with weights zk satisfies (1) where h = 1. b S . It must satisfy (1) with Consider any activity (v, w, x) from Y b The some scalar ^ h and vector ^ z. We need to prove that ðv ; w; xÞ 2 Y. activity on the left-hand side of (1) is the convex combination of observed activities taken with the weights zk and further abated by b S . By (A1), the the factor h. According to (A2) and (A5), this is in Y b S. h activity (v, w, x) on the right-hand side of (1) must be in Y
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