1 Introduction

Soon after Charnes et al. (1978, 1979) developed data envelopment analysis (DEA), the first eco-DEA models were presented (Färe et al. 1986, 1989). These models consider the environmental impact of production operations in addition to applied resources and produced goods. The various environmental impacts in the different dimensions and other undesirable outputs are referred to as bad outputs or simply “bads”.

In DEA, the production processes of different Decision-Making Units (DMUs; the observations examined) are compared to each other and classified as efficient or inefficient ones using the technical and price efficiency measures of Farrell (1957). The DMUs produce several output components with several input factors. In other words, the production processes can be characterized by a multi-input-multi-output structure. The different DMUs must use the same input factors and produce the same output factors to allow comparisons between the DMUs (“homogeneous production”).

In this context, the term “eco-efficiency” refers to the inability to increase the production of one good without increasing the use of at least one resource and/or pollution or reducing the production of another good. Both the possible multidimensionality of outputs and the non-necessity of known prices in DEA are especially advantageous when considering pollutants, which have no valid or stable prices for them in general.

Numerous attempts have been made to integrate undesirable outputs into the DEA concept. Murty and Russell (2020) have provided a comprehensive overview. The by far most used variant is the inclusion of the weak disposability condition defined as an equation (Dakpo et al. 2016:356). The literature about theory is rich, and there are many more applications. Weak disposability means that the bads cannot be reduced without cutting the goods to the same relative extent or increasing the inputs. However, the production of goods can still be decreased arbitrarily. Another condition is zero-joint outputs; the only way to avoid producing any bad is to produce nothing. Färe et al. (1986, 1989) were the first to use this approach in DEA models. However, the assumption of weak disposability has a major disadvantage. As shown in, for example, Chen (2014), the assumption of weak disposability can cause a violation of the monotonicity condition and generate misclassifications of efficiency and incorrect efficiency values. The monotonicity condition indicates that the eco-efficiency may only increase if, ceteris paribus, the use of inputs or the pollution produced decreases, or if the production of goods increases. In other DEA models, the “strong” disposability condition is used for bads (e.g., Cooper et al. 2007:368–371) and thus – in terms of these models – any reduction of bads without changing the other factors is allowed. Hybrid models divide bads into “strong disposable” and “weak disposable” based on their properties (Yang and Pollitt 2010; Cooper et al. 2007; more generally, Mehdiloo and Podinovski 2019).

This article investigates the circumstances under which the monotonicity condition in weak disposability DEA models is violated, as well as the deeper economic reasons for this. In particular, it examines which data structures are associated with this occurrence. The mechanics of the mathematical-economic models are thus centered. The properties of the weak disposability DEA models are of interest in this article. This study does not consider which variant “best” includes bads in DEA models.

Finally, rules are given for identifying some DMUs that lead to violations of the monotonicity condition. Further, it is proven that, under the weak disposability assumption, the efficiency frontier alters with the orientation of the applied DEA model, in contrast to the standard DEA models. Furthermore, this result can also be applied to detect DMUs, which are prone to this phenomenon.

This article provides an important orientation for applied researchers regarding the correct application of modeling bads as weak disposables by providing procedures for finding misclassified and overestimated DMUs. This was done the first time, to my knowledge. The occurrence of this phenomenon depends not only on the data structure, but also on the orientation of the DEA model used. In this way, it contributes to the discussion of the advantages and limitations of modeling bads as weak disposables in DEA models. This helps reduce a gap in the literature.

2 Disposability concepts

A production technology is defined by its production possibility set:

$$P=\left\{\left({\varvec{x}}, \varvec{{y}}\right)\left|\text{outputs}\text{ }\varvec{y}\in {\varvec{R}}_{+\cup \left\{0\right\}}^{R}\text{ }\text{can be produced with inputs}\text{ }\varvec{x}\in {\varvec{R}}_{+\cup \left\{0\right\}}^{I}\right.\right\}$$

A multidimensional output bundle \(\varvec{y}\) can be produced using a multidimensional input bundle \(\varvec{x}\). If wasting of resources and inefficiencies are accepted, one can always produce fewer goods with the same input bundle and/or can produce the same number of goods with a higher level of inputs. \(P\) is non-empty, convex (i.e., convex combinations of feasible production plans are also feasible), closed, (i.e., the boundary is also part of it), and bounded by the abscissa (from below) and the efficiency frontier or envelope (from above). The efficiency frontier contains each efficient production plan. Efficient DMUs cannot increase the production of any output without increasing the usage of at least one input or reducing at least one other output. Different assumptions can be imposed about wasting possibilities, which are modeled by different dependencies between the various production components. It is focused on the output side in the following.

If strong (or free) disposability of outputs is assumed, a single output component can be reduced without affecting the other output or input components:

$$\text{if }\left(\varvec{x},\varvec{y}\right)\in P\wedge {\mathbf{y}}^{\prime}\le \varvec{y}\text{ }\text{then}\text{ }\left(\varvec{x},\varvec{y}^{\prime }\right)\in P$$

If joint weak (or ray) disposability of outputs is assumed, an output component (e.g., \({y}_{1}\)) can only be reduced if and only if either another output component (e.g., \({y}_{2}\)) is also reduced in the same ratio \(\theta\) or an input component is increased:

$$\text{if}\text{ }\left(\varvec{x},\varvec{y}\right)\in P\text{ then}\text{ }\left(\varvec{x},\theta \varvec{y}\right)\in P,\forall 0\le \theta \le 1$$

Output reduction is costly. Kuosmanen (2005) interpreted θ as abatement factor. Another condition of the weak disposability assumption is that if for example, \({y}_{1}\) is weakly disposable and \({y}_{2}\) is not, then if \({y}_{1}=0\), then \({y}_{2}=0\) (zero-joint outputs). In other words, there is no free lunch.

Other, not so popular disposability concepts are semi-disposability (Chen et al. 2017), generalized disposability (Roshdi et al. 2018), exponential disposability (ibid.), g-disposability (Chung 1996) and weak g-disposability (Rødseth 2017; Hampf and Rødseth 2015), natural disposability and managerial disposability (Suyeoshi and Goto 2011, 2012), B-disposability (Abad and Briec 2019) and value disposability (Dyckhoff and Souren 2020:34–36).

3 Application to DEA

In this section, the weak disposability concept is introduced into the good-oriented DEA-model with constant returns to scale (CRS). \(J\) DMUs (\(j=1,\dots ,J\)) produce the same \(R\) different good components \({y}_{r}\) (\(r=1,\dots ,R\)) using the same \(I\) different resource components \({x}_{i}\) (\(i=1,\dots ,I\)) causing \(P\) different bad components \({b}_{p}\) (\(p=1,\dots ,P\)). If weak disposability is assumed, the abatement factor \(\theta\) must be introduced and the inequalities associated with the bads must be mutated into equalities (e.g., Färe and Grosskopf 2003: 1071, Färe et al. 2009:536). The envelopment problem of the good-oriented eco-DEA model is represented by (1):

$$\begin{array}{*{20}l} {\mathop {{\text{max}}}\limits_{{\varphi _{0} ,\theta ,\lambda _{j} }} \varphi _{0} \cdot \theta } \hfill \\ {\varphi _{0} y_{{rj_{0} }} - \theta \cdot \sum\limits_{{j = 1}}^{J} {\lambda _{j} y_{{rj}} \le 0;\quad r = 1, \ldots ,R} } \hfill \\ {\quad \quad \quad \theta \cdot \sum\limits_{{j = 1}}^{J} {\lambda _{j} b_{{pj}} = b_{{pj_{0} }} ;\quad p = 1, \ldots ,P} } \hfill \\ {\quad \quad \quad \quad \sum\limits_{{j = 1}}^{J} {\lambda _{j} x_{{ij}} \le x_{{ij_{0} }} ;\quad i = 1, \ldots ,I} } \hfill \\ {\quad \quad \quad \quad \quad \quad 0 \le \theta \le 1;\quad \lambda _{j} \ge 0;\quad\varphi _{0} {\text{ free}}.} \hfill \\ \end{array}$$
(1)

It is said “inputs and goods are strongly disposable, but bads are weakly disposable jointly with goods.” Positive slacks are not allowed because of the equality at the constraints for the bads. Program (1) is non-linear. Linear approximations were presented in, for example, Färe and Grosskopf (2003: 1071), Sahoo et al. (2011:752), and Zhou et al. (2008:7,12). Färe and Grosskopf (2005:50) demonstrated that program (2) satisfies weak disposability for the bads and Färe and Grosskopf (2009:536) indicated that in the models with constant or non-increasing returns to scale the production possibility sets of the linearized model (2) and of model (1) are identical (as a consequence, the sets of efficient DMUs are also identical). The efficiency values resulting from (1) and (2) can therefore be different:

$$\begin{array}{*{20}l} {\mathop {{\text{max}}}\limits_{{\varphi _{0} ,\lambda _{j} }} \varphi _{0} } \hfill \\ {\varphi _{0} y_{{rj_{0} }} - \sum\limits_{{j = 1}}^{J} {\lambda _{j} y_{{rj}} \le 0;\quad r = 1, \ldots ,R} } \hfill \\ {\quad \quad \quad \sum\limits_{{j = 1}}^{J} {\lambda _{j} b_{{pj}} = b_{{pj_{0} }} ;\quad p = 1, \ldots ,P} } \hfill \\ {\quad \quad \quad \sum\limits_{{j = 1}}^{J} {\lambda _{j} x_{{ij}} \le x_{{ij_{0} }} ; \quad i = 1, \ldots ,I} } \hfill \\ {\quad \quad \quad \quad \quad \lambda _{j} \ge 0; \quad \varphi _{0} {\text{ free}}.} \hfill \\ \end{array}$$
(2)

The abatement factor \(\theta\) is dropped in the linear program (2), but the equation sign remains for the constraints associated with the bad components. The corresponding multiplier program reveals the problem:

$$\begin{array}{*{20}l} {\mathop {{\text{min}}}\limits_{{\mu _{r} ,\nu _{i} ,\omega _{p} }} \sum\limits_{{i = 1}}^{I} {\nu _{i} x_{{ij_{0} }} } + \sum\limits_{{p = 1}}^{P} {\omega _{p} b_{{pj_{0} }} } } \hfill \\ { - \sum\limits_{{r = 1}}^{R} {\mu _{r} y_{{rj}} } + \sum\limits_{{p = 1}}^{P} {\omega _{p} b_{{pj}} } + \sum\limits_{{i = 1}}^{I} {\nu _{i} x_{{ij}} } \ge 0; \quad j = 1, \ldots ,J} \hfill \\ {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \sum\limits_{{r = 1}}^{R} {\mu _{r} y_{{rj_{0} }} } = 1} \hfill \\ {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \omega _{p} {\text{ free}}; \quad p = 1, \ldots ,P} \hfill \\ {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \mu _{r} ,\nu _{i} \ge 0; \quad r = 1, \ldots ,R; \quad i = 1, \ldots ,I.} \hfill \\ \end{array}$$
(3)

The shadow prices \({\omega }_{p}\) for the bad components are unbounded on both sides; thus, they can be negative (Hailu and Veeman 2001). If a negative shadow price for a bad occurs, then pollution production reduces the costs so that, ceteris paribus, it is possible to increase the efficiency value by increasing the bads. (A negative shadow price for a bad causes in an input-oriented model extra revenue.) Consequently, the monotonicity property is no longer valid because of this.

4 Illustrating example

A violation of the monotonicity property is exemplified in this section by an example. Four DMUs (A to D) produce a single good \(y\) and a single bad \(b\) with a single input \(x\), which is kept constant. A good-oriented eco-DEA with constant returns to scale (CRS) is applied. Table 1 contains the data of this example, including four versions of DMU D (D0 to D3).

Table 1 Data from the given example
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Multiplier space: The constraints of the multiplier model are formed by four halfplanes defined by the production schemes of each DMU, which are the same for each analyzed DMU. Equivalence transformations provide the bounding ray to each halfplane:

$$\begin{aligned} &{P}_{A}:10\ge 2\frac{\mu }{\nu }-2\frac{\omega }{\nu }\\ &{P}_{B}:10\ge 3\frac{\mu }{\nu }-4\frac{\omega }{\nu } \\ &{P}_{C}:10\ge 2.5\frac{\mu }{\nu }-6\frac{\omega }{\nu }\\ &{P}_{D}:10\ge 2\frac{\mu }{\nu }-5\frac{\omega }{\nu }\end{aligned}$$

Figure 1 depicts the multiplier space. The intersection of the halfplanes forms the production possibility set, which is represented by the dashed lines. The shadow prices for the bads can be negative (dashed lines left of the origin).

Fig. 1
figure 1

Multiplier space of the given example

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Two-outputs space: As the input \(x\) is kept constant, the good-bad space respective input-isoquant (Luptáčik 2010:171) can be drawn. As the bads and goods are outputs, the good-bad space is interpreted in Fig. 2 as a two-outputs space. Under the strong disposability assumption, DMU A can reduce the bads solely through a horizontal movement to the left; however, under the weak disposability assumption, DMU A is limited to reducing the bads along with the goods along a ray towards the origin.

Fig. 2
figure 2

Output space

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The constraints of the envelopment model for DMU C are presented by (the left-hand side is identical for each DMU, only the right-hand side changes for different DMUs \({j}_{0}\)):

$$\begin{aligned} &{D}_{O}:2.5\varphi -\left(2{\lambda }_{A}+3{\lambda }_{B}+2.{5\lambda }_{C}+2{\lambda }_{D}\right)\leq 0\\ &{D}_{B}:2{\lambda }_{A}+4{\lambda }_{B}+{6\lambda }_{C}+5{\lambda }_{D}=6 \\ &{{D}_{I}:10\lambda }_{A}+10{\lambda }_{B}+10{\lambda }_{C}+10{\lambda }_{D}\leq 10 \\ &{\lambda }_{A},{\lambda }_{B},{\lambda }_{C},{\lambda }_{D}\geq 0 \end{aligned}$$

From constraint \({D}_{I}\), \({\lambda }_{A}+{\lambda }_{B}+{\lambda }_{C}+{\lambda }_{D}\leq 1\), which equals the non-increasing return to scale (NIRS) condition.

Input-output space: If the bads are interpreted as inputs, the same space can be interpreted as input-output space in Fig. 3. If strong disposability is assumed, then the DMUs C and D would be projected onto the horizontal dashed line and then to DMU B via slacks. If weak disposability is assumed, DMU C is evaluated as efficient, even though it produces fewer goods but pollutes more than DMU B; DMU C is misinterpreted as efficient.

Fig. 3
figure 3

Input-output space

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With constraint \({D}_{I}\) together with constraint \({D}_{B}\) produce \({\lambda }_{A},{\lambda }_{B},{\lambda }_{D}=0;{\lambda }_{C}=1\). There is no linear combination without including DMU C as a possibility. Putting these \(\lambda\)s into constraint \({D}_{O}\) results in \(0.25\varphi -0.25\leq 0\), from which follows \(\varphi \leq 1\). In a good-oriented model, the efficiency value must be \(\varphi \ge 1\), thus \(\varphi =1\).

Constraint \(\sum _{j=1}^{J}{\lambda }_{j}{b}_{pj}={b}_{p{j}_{0}}\) forces the construction of a peer DMU via a linear combination with the same pollution level as DMU C. A combination of the other DMUs fulfilling \({\lambda }_{A}+{\lambda }_{B}+{\lambda }_{C}+{\lambda }_{D}\leq 1\) does not exist, which produces the same amount of bads, as DMU C pollutes most. DMU C can only be projected on itself because of the equation in constraint \({D}_{B}\).

The inefficient DMU D is projected on the line BC under the weak disposability condition, resulting in a wrong peer for improving. If DMU D would ceteris paribus increase its pollution (e.g., from point D0 to D1), the distance to the line BC would decrease, meaning the efficiency value would improve. If pollution increased further, DMU D would be considered efficient (e.g., point D2 to D3). Consequently, the efficiency value for DMU D is incorrect. Table 2 gives the model results.

Table 2 Results of the example
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With assumed strong disposability only A and B are classified as efficient; C and D are projected onto B. With assumed weak disposability C is erroneously classified as efficient; and the bad shadow price is negative (\(-0.1\)). D0 is projected on the line BC, and it also has a negative bad shadow price (\(-0.125\)). With increasing bad (from D0 to D3), the efficiency value improves; finally, D3 is erroneously classified as efficient, with a further decreasing negative shadow price for the bad (\(-0.5\)).

This example illustrates that DMUs can be misclassified with the weak disposability assumption. The subsequent sections analyze the circumstances under which violations of the monotonicity condition occur. Due to some similarities, the more-for-less paradox is the starting point.

5 The more-for-less paradox

The more-for-less paradox can arise in transportation, distribution, or production problems. For example, \(J\) different goods (\(j=1,\dots ,J\)) may be manufactured using \(I\) different technologies (\({z}_{i}\), \(i=1,\dots ,I\)). The single use of technology \({z}_{i}\) costs \({c}_{i}\) with \({a}_{ji}\) pieces of good \(j\) produced. Exact \({b}_{j}\) pieces of good \(j\) must be manufactured (no waste of goods!). The objective is to minimize production costs. The primal and dual models of this production problem are given by (4) and (5):

$$\begin{array}{l}\underset{{z}_{i}}{\text{min}}\sum\limits_{i=1}^{I}{c}_{i}{z}_{i}\\ \sum\limits_{i=1}^{I}{a}_{ji}{z}_{i}={b}_{j};\quad j=1,\dots ,J\\{z}_{i}\ge 0;i=1,\dots ,I\end{array}$$
(4)
$$\begin{array}{l}\underset{{u}_{j}}{\text{max}} \sum\limits_{j=1}^{J}{b}_{j}{u}_{j}\\ \sum\limits_{j=1}^{J}{a}_{ij}{u}_{j}\leq {c}_{i};\quad i=1,\dots ,I\\ {u}_{j}\,\text{free};\quad j=1,\dots ,J\end{array}$$
(5)

The more-for-less paradox occurs if \({b}_{j}\leq \stackrel{-}{{b}_{j}}, \forall j=1,\dots ,J\) and \({\text{min}}_{{z}_{i}}\sum_{i=1}^{I}{c}_{i}{z}_{i}>{\text{min}}_{{\stackrel{-}{{z}_{i}}}}\sum_{i=1}^{I}{c}_{i}\stackrel{-}{{z}_{i}}\) at the same time. That is, it is possible to produce more with fewer total costs (e.g., Plevný 2003:1). The shadow price \({u}_{j}\) (\(j=1,\dots ,J\)) denotes how much the production of an additional piece of good \(j\) costs. The equations in the primal model (4) lead to unbounded shadow prices.

Chobot and Turnovec (1974)Footnote 1 revealed that this paradox occurs if and only if at least one shadow price is negative in each optimal solution (also in Charnes et al. 1987:195). Charnes et al. (1980) proved that the problem degenerates into unconnected subproblems if the paradox arises. Luptáčik (2001) indicated that if a change of the structure of the output requirements \(\left({b}_{1},\dots , {b}_{J}\right)\) forces a change of the used technologies, then the paradox can ascend. Ultimately, those technologies that most closely resemble the required output structure are applied. In this paradox, solely the right side (\({b}_{j}\),\(j=1,\dots ,J\)) of the primal model is changed.

Good-oriented eco-DEA: To apply the similarities between the more-for-less paradox and the weak disposability assumption, the envelopment model of the good-oriented eco-DEA will be brought into the same form as (4), which can be done by

$$\begin{aligned}\varvec{z}& =\left(\begin{array}{c}{\varphi }_{0}\\ {\varvec{\lambda }}_{J}\end{array}\right)\\ \varvec{c} & =\left(\begin{array}{c}1\\ {\varvec{0}}_{J}\end{array}\right); \\ \varvec{b}&=\left(\begin{array}{c}{\varvec{0}}_{R}\\ {\varvec{b}}_{0P}\\ {\varvec{x}}_{0I}\end{array}\right) \\ \varvec{A}&=\left[\begin{array}{cc}{\varvec{y}}_{0P}& -\varvec{Y}\\ {\varvec{0}}_{P}& \varvec{B}\\ {\varvec{0}}_{I}& \varvec{X}\end{array}\right]\end{aligned}$$

The “technology matrix” \(\varvec{A}\) contains a few negative elements. The more-for-less paradox occurs when the inefficiency value of the DMU in consideration \({j}_{0}\) increases as \(\stackrel{-}{{\varphi }^{\text{*}}}>{\varphi }^{\text{*}}\) and the corresponding bads \(\stackrel{-}{{b}_{p{j}_{0}}}{<b}_{p{j}_{0}}\) decrease. The problem looks as follows:

$$\begin{array}{l}\underset{{\varphi }_{0},{\lambda }_{j}}{\text{max}}{\varphi }_{0}\\ {{\varphi }_{0}y}_{r{j}_{0}}-\sum\limits_{j=1}^{J}{\lambda }_{j}{y}_{rj}\leq 0;\quad r=1,\ldots ,R\\ \sum\limits _{j\ne {j}_{0}}{\lambda }_{j}{b}_{pj}+{\lambda }_{{j}_{0}}\stackrel{-}{b}_{p{j}_{0}}=\stackrel{-}{b}_{p{j}_{0}}{={b}_{p{j}_{0}}-{s}_{p}\leq b}_{p{j}_{0}};\quad p=1,\dots ,P\\ \sum\limits_{j=1}^{J}{\lambda }_{j}{x}_{ij}\leq {x}_{i{j}_{0}};\quad i=1,\dots ,I\\ {\lambda }_{j},{s}_{p}\geq 0;\quad \sum\limits_{p=1}^{P}{s}_{p}>0,{\varphi }_{0}\text{ free}. \end{array}$$
(6)

A DMU in consideration can be evaluated in three ways: (1) inefficient; (2) efficient but it lies not at a corner of the efficiency frontier or (3) efficient but it lies at a corner:

  1. (1)

    If DMU \({j}_{0}\) is deemed inefficient, the peer production plan for DMU \({j}_{0}\) can be constructed as a linear combination from at least one other production plan \(j\ne {j}_{0}\), resulting in \({\lambda }_{{j}_{0}}=0\). Hence, only the right side of (6) changes, similar to the more-for-less paradox. If DMU \({j}_{0}\) is inefficient, then negative shadow prices in each optimal solution are necessary and sufficient for the identification of the undesired behavior under the weak disposability assumption.

  2. (2)

    If DMU \({j}_{0}\) is considered efficient (i.e., efficiency value \({\varphi }_{0}=1\)) and the DMU is not located at a corner, then it lies on a facet or an edge of the efficiency frontier with a given slope. The production plan of DMU \({j}_{0}\) can then be constructed as a linear combination from other production plans \(j\ne {j}_{0}\). Thus, \({\lambda }_{{j}_{0}}=0\) at at least one optimal solution, and only the right side of (6) changes as in the more-for-less paradox. Consequently, negative shadow prices in each optimal solution are necessary and sufficient for a violation of the monotonicity condition.

  3. (3)

    If the efficiency value of DMU \({j}_{0}\) equals one and the DMU lies on a corner of the efficiency frontier, then \({\lambda }_{{j}_{0}}>0\) and \({\lambda }_{j}=0,\forall j\ne {j}_{0}\). Thus, the right and left sides of (6) change simultaneously. The similarities to the more-for-less paradox end here. It is not possible to conclude from (6) that negative shadow prices in all optimal solutions are necessary and sufficient for a violation of the monotonicity condition. However, negative shadow prices of the bads are generally economically implausible (Hailu and Veeman 2001:608)Footnote 2 and therefore represent a paradox in themselves.

To clarify, if a DMU \({j}_{0}\) “is assigned (considered, etc.) as efficient,” then in this article it does not mean that this DMU is efficient, it just means that this DMU is classified as efficient. However, this could be a misclassification.

In summary, DMUs for which this phenomenon occurs are projected on themselves, like efficient ones, or on peers which are formed by a linear combination of DMUs, including another such DMU.

6 Sufficient conditions for the violation of monotonicity

This section considers the mathematical and economic circumstances under which a violation of the monotonicity condition occurs, as well as the associated data structures. Constant return to scale (CRS) models are used and have three dimensions – one input component, one output component and one bad component. The results are no longer interpretable for higher dimensions because the formulas have become very complicated. The corresponding computations can be requested by the author. The conditions presented are sufficient for the occurrence of violations of the monotonicity condition. If a DMU is found to be affected, it must be taken out of the dataset, and a recalculation must be done based on this new dataset to identify other DMUs that may have been affected. The outcomes presented can be extended to other eco-DEA models.

If at least one DMU \(j\ne {j}_{0}\) meets the following three conditions simultaneously, a violation of the monotonicity condition arises:

  1. 1.

    \(\frac{{y}_{{j}_{0}}}{{x}_{{j}_{0}}}<\frac{{y}_{j}}{{x}_{j}}\): the technical efficiency

  2. 2.

    \(\frac{{y}_{{j}_{0}}}{{b}_{{j}_{0}}}<\frac{{y}_{j}}{{b}_{j}}\): the environmental efficiency

  3. 3.

    \(\frac{{x}_{{j}_{0}}}{{b}_{{j}_{0}}}<\frac{{x}_{j}}{{b}_{j}}\): the input-pollution indices

are higher than the corresponding values of the DMU in consideration \({j}_{0}\) and the efficiency value equals one in a good-oriented eco-DEA model. Similar to the more-for-less paradox, the relationships between inputs, outputs, and bads to each other and between the different DMUs are decisive. The indices for the constructed example in Sect. 4 are shown in Table 3:

Table 3 Identification indices in the example
Full size table

Each of these indices is lower for DMUs C, D0 to D3 than the corresponding indices for DMU B. Thus, for these DMUs the monotonicity condition is violated. The gray-shaded cells demonstrate that with increasing bads for D0 to D3, the environmental efficiency, the input-pollution indices, and negative shadow prices decrease, but the efficiency values increase (Table 2).

The first condition states that DMU \(j\) is relatively more technically efficient (i.e., produces more output per input), the second condition states that it produces more output per bad (i.e., the production is less polluting), and the third condition states that it extracts less bad per input (i.e., the resource usage is less polluting) than the DMU in consideration \({j}_{0}\). We therefore have the following relation ranking: \(\frac{{b}_{j}}{{b}_{{j}_{0}}}<\frac{{x}_{j}}{{x}_{{j}_{0}}}<\frac{{y}_{j}}{{y}_{{j}_{0}}}\); the relative gap between the DMUs \(j\) and \({j}_{0}\) increases from the bads to the inputs to the outputs. If a DMU \({j}_{0}\) has several shadow price bundles as optimal solutions and a DMU \(j\) exists with \(\frac{{b}_{j}}{{b}_{{j}_{0}}}<\frac{{x}_{j}}{{x}_{{j}_{0}}}<\frac{{y}_{j}}{{y}_{{j}_{0}}}\), each of these bundles will have a negative shadow price for the bad in each optimal solution.

Conditions 1 to 3 apply to the DMUs assigned as efficient \(\left({\varphi }_{0}=1\right)\). For any efficiency value \(\left({\varphi }_{0}\ge 1\right)\), the first, and second conditions change to \(\frac{{\varphi }_{0}{y}_{{j}_{0}}}{{x}_{{j}_{0}}}<\frac{{y}_{j}}{{x}_{j}}\) respective \(\frac{{{\varphi }_{0}y}_{{j}_{0}}}{{b}_{{j}_{0}}}<\frac{{y}_{j}}{{b}_{j}}\), whereas the third condition remains the same. The comparison is conducted with DMU’s \({j}_{0}\) projection point \({j}_{0}\), not with DMU \({j}_{0}\) itself.

For an input-oriented eco-DEA model, the first condition changes to \(\frac{{x}_{j}}{{{\sigma }_{0}x}_{{j}_{0}}}<\frac{{y}_{j}}{{y}_{{j}_{0}}}\), the second condition remains unchanged, the third condition is not necessary. As a third condition is necessary in the good-oriented model, a violation of the monotonicity condition may in principle occur more often in the input-oriented model than in the good-oriented model.

If the efficiency value equals zero, the same conditions as for the good-oriented eco-DEA model apply for an additive eco-DEA model. In contrast to the other models, the interpretation of the conditions for inefficient DMUs is more complicated in the additive model. The conditions generalize in two different variants and require the efficiency value \(\left(-{s}^{+}-{s}^{-}\right)\) as \({L}_{1}\)-distance from the efficiency frontier.

The most important condition of the three conditions 1 to 3 to be fulfilled simultaneously for a violated monotonicity condition in an additive, good- and input-oriented eco-DEA-model is that if the DMU in consideration \({j}_{0}\) is assigned as efficient and a DMU \(j\ne {j}_{0}\) with \(\frac{{b}_{j}}{{b}_{{j}_{0}}}<\frac{{y}_{j}}{{y}_{{j}_{0}}}\) exists (i.e., the gap between the DMUs \(j\) and \({j}_{0}\) is higher for the goods than for the bads). In other words, if a DMU \(j\ne {j}_{0}\) with better environmental efficiency \(\frac{{y}_{{j}_{0}}}{{b}_{{j}_{0}}}<\frac{{y}_{j}}{{b}_{j}}\) exists, except for the bad-oriented eco-DEA model.

The bad-oriented eco-DEA model was presented in Färe et al. (2004:348) and in Korhonen and Luptáčik (2004:440–441, Model C). The shadow price for the bad cannot be negative, because the second constraint of the multiplier model follows \({\omega }_{0}=\frac{1}{{b}_{{j}_{0}}}\). A higher bad is compensated for with a smaller shadow price. In the envelopment program, the constraint concerning the bads is \({\tau }_{0}{b}_{{j}_{0}}={\sum }_{j=1}^{J}{\lambda }_{j}{b}_{j}\). In contrast to the other models, it is not necessary to find a composite unit which pollutes at exactly the same level as DMU \({j}_{0}\), rather, the unit should pollute at the reduced level of \({\tau }_{0}{b}_{{j}_{0}}\) because the efficiency value \(0\leq {\tau }_{0}\leq 1\), which is always possible. The pollution level can be adjusted. Because the three-dimensional bad-oriented weak disposability eco-DEA model seems to be an exception concerning the possibility of a violation of the monotonicity condition, a short examination including more bad dimensions is presented in Sect. 6.1.

The monotonicity condition is also violated if the virtual input \({\nu }_{0}{x}_{{j}_{0}}\) exceeds the efficiency value \({\varphi }_{0}\) in a three-dimensional good-oriented eco-DEA model. It is violated in an input-oriented eco-DEA model if the virtual output \({\mu }_{0}y\) is lower than the efficiency value \({\sigma }_{0}\), in an additive eco-DEA model if holds: \({\mu }_{0}{y}_{{j}_{0}}+{s}^{+}<{\nu }_{0}{x}_{{j}_{0}}-{s}^{-}\).

Table 4 lists the cited sufficient conditions for violating the monotonicity condition. Two sets of conditions are presented. Comparisons between the production plans of the DMUs are required for the first set. The shadow prices received from the evaluation of the DMU in consideration \({j}_{0}\) are considered in the conditions of the second set. Please note that it is sufficient if at least one of these conditions is fulfilled.

Table 4 Identification indices in three-dimensional models
Full size table

6.1 Multiple bads

In the three-dimensional bad-oriented weak disposability eco-DEA model, the shadow price for the bad cannot be negative because of the constraint \(\omega {b}_{{j}_{0}}=1\) in the multiplier program. The aforementioned statement holds true for any weak disposability bad-oriented eco DEA-model with only one bad component, as well as for models which are oriented only in one bad and no other direction.

This is not the case with multiple bads because of the constraint \(\sum _{p=1}^{P}{\omega }_{p}{b}_{p{j}_{0}}=1\) in the multiplier program. A negative shadow price at one bad can be outweighed by positive shadow prices of other bads. For example, with two bads: \({\omega }_{1}=\frac{1-{\omega }_{2}{b}_{2{j}_{0}}}{{b}_{1{j}_{0}}}\) and vice versa. Thus, if \({\omega }_{2}{b}_{2{j}_{0}}>1\), then \({\omega }_{1}\) is negative.

In the envelopment program, the constraints concerning the bads are \({\tau }_{0}{b}_{{pj}_{0}}={\sum }_{j=1}^{J}{\lambda }_{j}{b}_{pj}\), for each bad \(p\) is the same \({\tau }_{0}\) necessary. Since no slacks are allowed, the pollutions \({b}_{p{j}_{0}}\) can only be reduced radially by \(\left(1-{\tau }_{0}\right)\). If it is not possible to find a composite unit that pollutes at exactly the same reduced levels \({\tau }_{0}{b}_{p{j}_{0}} \forall p\), then the monotonicity condition is violated. If the relationships between the bads are unfavorable, a violation of the monotonicity condition occurs.

7 A changing efficiency frontier under the assumption of weak disposability

Korhonen and Luptáčik (2004:441–442) indicated that the efficiency frontier is the same regardless of orientation under the strong disposability assumption. In this section, it is shown that the efficiency frontier is not independent of orientation under the assumption of weak disposability. The proof partially follows Korhonen, Luptáčik (ibid.), but contrary to them, it suffices to provide a counterexample to the hypothesis that the efficiency frontier would not change with the orientation under the assumption of weak disposability. A general model (such as presented in Korhonen and Luptáčik 2004:441) adapted to the weak disposability assumption is given by

$$\begin{aligned} & \underset{\sigma ,{\lambda }_{j}}{\text{max}}\sigma +\epsilon {\varvec{1}}^{T}\left({\varvec{s}}^{y}+{s}^{\varvec{x}}\right) \\ & \varvec{Y}\varvec{\lambda }-\sigma {\varvec{w}}^{y}-{\varvec{s}}^{y}={\varvec{y}}_{0} \\ & \varvec{X}\varvec{\lambda }+\sigma {\varvec{w}}^{x}+{\varvec{s}}^{x}={\varvec{x}}_{0}\\ & \varvec{B}\varvec{\lambda }+\sigma {\varvec{w}}^{b}={\varvec{b}}_{0} \end{aligned}$$

The only difference from the general model under the strong disposability assumption in Korhonen and Luptáčik (2004:441) is that \({\varvec{s}}^{b}=0\) for any given orientation.

Imagine a situation in which a DMU \(j\ne {j}_{0}\) exists with \({b}_{pj}<{b}_{p{j}_{0}}\) for a specific \(p=1,\dots ,P\) and the other factors are identical \({\varvec{y}}_{j}={\varvec{y}}_{{j}_{0}}, {\varvec{x}}_{j}={\varvec{x}}_{{j}_{0}}, {b}_{\stackrel{-}{p}j}={b}_{\stackrel{-}{p}{j}_{0}}\forall \stackrel{-}{p}\ne p\). The other given DMUs are even worse than these two. For instance, the DMUs D0 to D3 in the example illustrated in Sect. 4 meet this condition.

Imagine the bad \(p\) \({w}_{p}^{b}=0\) and \({\varvec{w}}^{x}\ge 0\) or \({\varvec{w}}^{y}\ge 0\). Then \({\sigma }^{*}=0\) and \({\varvec{s}}^{y}={\varvec{s}}^{x}=0\) (\({\varvec{s}}^{b}=0\) by definition). The constraints can be fulfilled only by \({\lambda }_{{j}_{0}}=1,{\lambda }_{{j\ne j}_{0}}=0\). The DMU \({j}_{0}\) is projected on itself as if it were strongly efficient. The DMU \({j}_{0}\) is classified as strongly efficient even though it clearly not. This makes it a part of the efficiency frontier. This misclassification happens to DMU D3 in the example illustrated in Sect. 4 under output orientation.

In contrast, consider \({w}_{p}^{b}>0\) and \({\varvec{w}}^{x},{\varvec{w}}^{y}=0,{w}_{\stackrel{-}{p}}^{b}=0 \forall \stackrel{-}{p}\ne p\). Then \({\sigma }^{*}=\frac{{b}_{p{j}_{0}}-{b}_{pj}}{{w}_{p}^{b}}>0\) as \({b}_{pj}<{b}_{p{j}_{0}}\). This alternative orientation renders DMU \({j}_{0}\) inefficient, and it is no longer part of the efficiency frontier. In the example illustrated in Sect. 4, the DMU D3 would be projected onto the DMU A and deemed as inefficient under bad orientation.

Thus, under the weak disposability assumption, the efficiency frontier can alter depending on orientation.

This outcome of an efficiency frontier changing with orientation can be utilized for the detection of DMUs that exhibit this phenomenon. In addition to the DEA model, which is used for examinations (the main model), further models should be done using the same dataset. For each bad, a DEA model should be applied that has an orientation solely in the direction of this single bad (the auxiliary models). Then, the sets of efficient DMUs in the different models should be compared. If these sets are different, violations of the monotonicity condition occur. DMUs that are considered efficient in the main model but not in each auxiliary model are misclassified and are not efficient. The efficiency values of inefficient DMUs with misclassified DMUs in their peer set are overestimated.

8 Conclusions and further research

The most applied variant for integrating undesirable outputs of production processes like pollutant emissions into DEA models is using the weak disposability assumption regarding undesirable outputs. A drawback of this approach is the potential violation of the monotonicity condition, which results in the possibility of misclassifications of efficient and inefficient units and interpretation difficulties associated with the efficiency values. This implausible phenomenon was examined in this article. In particular, the article focused on the circumstances under which the monotonicity condition in weak disposability DEA models is violated, the deeper economic meaning of this, and the data structures associated with this phenomenon. This phenomenon was also revealed to be partly related to the more-for-less-paradox. If a DMU \({j}_{0}\) “is assigned (considered, etc.) as efficient,” it does not mean in this article that this DMU is efficient; rather, it just means that this DMU is classified as efficient. However, this may be a misclassification.

The bad-related equation constraints force us to construct a peer DMU via linear combination with the same pollution level as the just-examined DMU. If such a combination of the other DMUs does not exist, the examined DMU can only be projected on itself. As a result, the DMU is considered efficient; the other circumstances are not relevant. In sum, DMUs where this phenomenon occurs are projected on themselves (like efficient DMUs), or on a linear combination of DMUs, including another such DMU.

Sufficient conditions for this violation were identified in the three dimensional case. These conditions are related to technical efficiency, environmental efficiency, and input-pollution index and are presented in Table 4 in Sect. 6. For any efficiency values, the conditions generalize in such a way that the comparison is done with the projection point of DMU \({j}_{0}\) and not DMU \({j}_{0}\) itself, with the exception of the additive eco-DEA-model in which the \({L}_{1}\)-distance is included in the conditions. The most important condition of the three conditions 1 to 3 to be fulfilled simultaneously for a violated monotonicity condition in an additive, good- and input-oriented eco-DEA-model is that if the DMU in consideration \({j}_{0}\) is assigned as efficient and a DMU \(j\ne {j}_{0}\) with \(\frac{{b}_{j}}{{b}_{{j}_{0}}}<\frac{{y}_{j}}{{y}_{{j}_{0}}}\) exists (i.e., the gap between the DMUs \(j\) and \({j}_{0}\) is higher for the goods than for the bads). In other words, if a DMU \(j\ne {j}_{0}\) with better environmental efficiency \(\frac{{y}_{{j}_{0}}}{{b}_{{j}_{0}}}<\frac{{y}_{j}}{{b}_{j}}\) exists, except for the bad-oriented eco-DEA model. If an affected DMU is identified, it must be excluded from the dataset and a recalculation must be done based on this new dataset to find other potentially affected DMUs. The outcomes presented can also be extended to other eco-DEA models.

The shadow price for the bad cannot be negative in the bad-oriented weak disposability eco-DEA model with three dimensions. This result is also applicable to any bad-oriented eco-DEA model with only one bad component and to models that are oriented toward only in one bad and no other direction. Unfortunately, this result is no longer valid for weak disposability models oriented toward several bads.

Finally, the findings indicate that the efficiency frontier changes with the orientation of the applied DEA model under the weak disposability assumption, in contrast to the strong disposability DEA models. This result of a with the orientation-changing efficiency frontier can be applied for the detection of paradox DMUs, it is presented how. This was done the first time, to my knowledge.

In conclusion, the weak disposability assumption corresponding to the equation at the constraints to the bads in the envelopment program does not necessarily lead to a violation of the monotonicity condition, but this assumption corresponding to the equation can cause implausible results. However, if this formulation is accepted, it is not possible to demand that all shadow prices be non-negative.

The models in this article were kept simple, but the insights gained are not. Higher-dimensional models should be analyzed as a next step. It should be suspected that for a violation of the monotonicity condition, more different bads will be more problematic as the constraints on the bads at the intensities \({\sum }_{j=1}^{J}{\lambda }_{j}{\varvec{b}}_{j}={\varvec{b}}_{{j}_{0}}\) demand a discovery of a composite unit which pollutes at the same level for all bads. Furthermore, in this article only models with constant return to scale were examined. In a next step, variable return to scale models should also be analyzed. As the constraints at the bads on the intensities are central for violations of the monotonicity condition, it should be expected that a further constraint by the variable return to scale \({\sum }_{j=1}^{J}{\lambda }_{j}{b}_{j}=1\) would exacerbate the problem. The conditions presented are sufficient to cause violations of the monotonicity condition but not necessary. Other circumstances not presented in this article can also produce this phenomenon. Specific intensity structures are examples of this.

There are three ways to prevent misclassified or overestimated DMUs in models that incorporate bads as weak disposables. Checking the non-negativity of each shadow price for each solution is a well-known possibility. In practice, however, most solvers return only one solution, which makes this examination difficult. This provides a necessary but not sufficient hint. This article discusses two additional alternatives. Table 4 in Sect. 6 identifies sufficient conditions regarding the relations between either the production plans or the shadow prices for three-dimensional models. However, it also indicates that the occurrence of this phenomenon depends not only on the data structure but also on the orientation of the DEA model used. Furthermore, Sect. 7 describes a procedure for finding misclassified or overestimated DMUs in general, using differently oriented eco-DEA models to check if the efficiency frontier changes.

It is also important to address misclassified or overestimated DMUs. The easiest way to do this is to exclude the misclassified DMUs (which can cause overestimated DMUs) from the dataset and repeat the full procedure from Sect. 7 until no misclassified DMUs are found. This ensures that the results of the non-excluded DMUs are correct. Further research is needed to address what should be done with the excluded DMUs.

Finally, it is important to emphasize the necessity of identifying misclassified DMUs. It is possible that these misclassified DMUs act as peers for other inefficient DMUs, and thus seduce them in the wrong direction for improvement.