#### 4.2. Additive-Based DEA Models

In order to solve the negative data problem, many new DEA models were proposed, most of which were characterized as additive-based models, because of the translation invariance property of the additive model. The translation invariance, as is pointed out by Lovell and Pastor, is critical when the data contain zero or negative values and must be translated prior to analysis with available software packages [

20]. The first additive-based DEA model, named the constant weighted additive model (CWA-DEA), was proposed by Pastor in 1994. It shared the translation invariance property with the original additive model, while neither of them was unit invariant [

20]. In order to obtain a model that shared both the translation invariance and unit invariance properties, Lovell and Pastor in 1995 proposed the normalized weighted additive model (NWA-DEA), which was a great step forward in the history of additive-based DEA models [

20]. They used the sample standard deviations of the output variables and the input variables, respectively, to replace the constant weight in the CWA-DEA model, and they pointed out that any first order dispersion measures can also be used to normalize the input excess and output slack variables. Apart from the translation invariance and unit invariance properties, three other important properties were proposed by Cooper

et al. to testify to the quality of the additive-based DEA models [

21]. Furthermore, based on the five properties, Cooper

et al., 1999, and Cooper

et al., 2011, extended the NWA-DEA model and proposed the famous RAM and BAM models [

21,

22]. The five properties were:

- (P1)
The optima is between 0 and 1;

- (P2)
The optima is 0 when DMU_{o} is fully inefficient, while the optima is 1 when DMU_{o} is fully efficient;

- (P3)
The optima is well defined and unit invariant;

- (P4)
The optima is strongly monotonic;

- (P5)
The optima is translation invariant.

According to Cooper

et al., 1999, the “strong monotonicity” property is described as follows: holding all other inputs and outputs constant, an increase in any of its inputs will increase the inefficiency score for an inefficient DMUo. The same is true for a decrease in any of its outputs [

21].

Table 2 describes whether current DEA models satisfy these five properties. According to Lovell and Pastor, 1994, the CCR model and the normalized weighted CCR model were not translation invariant, while the BCC model and the normalized BCC model are translation invariant in a limited sense, being invariant with respect to the translation of inputs or outputs, but not both. The radial component of the efficiency measure obtained from the BCC model and the CCR model is unit invariant, but the slack component is not [

20]. The additive model can only measure the inefficiencies of the DMUs, and the optima are not between 0 and 1. With respect to P4, it is obvious that only the RAM model satisfied the strongly monotonic property, while the others are all monotonic.

**Table 2.**
Comparisons of traditional DEA models on the five properties.

**Table 2.**
Comparisons of traditional DEA models on the five properties.
Model | P1 | P2 | P3 | P4 | P5 |
---|

CCR | Yes | Yes | Partially Units Invariance | Monotonic | NO |

Normalized weighted CCR | Yes | Yes | Units Invariance | Monotonic | NO |

BCC | Yes | Yes | Partially Units Invariance | Monotonic | Partially Translation Invariance |

Normalized weighted BCC | Yes | Yes | Units Invariance | Monotonic | Partially Translation Invariance |

Additive model | No | No | No | Monotonic | Yes |

Normalized weighted Additive Model | No | No | Units Invariance | Monotonic | Yes |

SBM | Yes | Yes | Yes | Monotonic | Yes |

RAM | Yes | Yes | Yes | Strongly Monotonic | Yes |

BAM | Yes | Yes | Yes | Monotonic | Yes |

#### 4.3. Generalized Additive-Based DEA Model-BMA Model

Based on the previous research, we proposed a generalized additive-based DEA model, which was called the big M additive-based DEA model. We here not only show that the previous additive-based DEA models were the particular form of the big M additive-based DEA model, but also show that other different forms of additive-based models can be derived from the big M additive-based DEA model.

Consider the following model:

here, Ω denotes the sample space, Ψ(Ω) and Φ(Ω) are non-zero mappings from the sample space to ℝ with homogeneous properties,

i.e., θΦ(Ω) = Φ(θΩ), θΨ(Ω) = Ψ(θΩ), θ ϵ ℝ, and big M is a large real number. The following theorem motivates the proposition of the homogeneity property of the mappings Ψ(Ω) and Φ(Ω).

Theorem 1: (Lovell and Pastor [

20]) In an additive DEA model, scaling an input (output) by multiplying it by a constant α > 0 is equivalent to leaving the input (output) unscaled and multiplying the corresponding input excess (output slack) variable in the objective function by the same constant.

Therefore, Model (3) is unit invariant if and only if Ψ(Ω) and Φ(Ω) satisfy the homogeneity property. In this sense, the additive-based DEA models mentioned above are only the particular form of the big M additive-based DEA model. Obviously, Model (3) can be transformed into the additive model, if we set Ψ(Ω) = 1, Φ(Ω) = 1 and M = 1. It can be transformed into the normalized weighted additive model when we set Ψ(Ω)_{r} = σ_{r}, Φ(Ω)_{i} = σ_{i} and M = 1. Here, σ_{r}, σ_{i} denote the sample standard deviation of the r-th output variable and the i-th input variable. It can be transformed into the SBM model when we set Ψ(Ω)_{r} = s·y_{ro}, Φ(Ω)_{i} = m**·**x_{io} and M = 1. It can be transformed into the RAM model when we set Ψ(Ω)_{r} = ${\text{R}}_{r}^{+}$, Φ(Ω)_{i} = ${\text{R}}_{i}^{-}$ and M = (m + s), where ${R}_{i}^{-}={\overline{x}}_{i}-{\underset{\xaf}{x}}_{i}$ with ${\overline{x}}_{i}=\text{max}\{{x}_{ij},j=1,\dots ,n\}$, ${\underset{\xaf}{x}}_{i}=\text{min}\{{x}_{ij},j=1,\dots ,n\}$ and ${R}_{r}^{+}={\overline{y}}_{r}-{\underset{\xaf}{y}}_{r}$ with ${\overline{y}}_{r}=\text{max}\{{y}_{rj},j=1,\dots ,n\}$, ${\underset{\xaf}{y}}_{r}=\text{min}\{{y}_{rj},j=1,\dots ,n\}$. It can be transformed into the BAM model when we set Ψ(Ω)_{r} = ${\text{U}}_{ro}^{+}$, Φ(Ω) = ${\text{U}}_{io}^{-}$ and M = (m + s), where ${L}_{io}^{-}={x}_{io}-{\underset{\xaf}{x}}_{i}$ and ${U}_{ro}^{+}={\overline{y}}_{r}-{y}_{ro}$. Moreover, it is obvious that the mappings Ψ(Ω) and Φ(Ω) control the unit invariance property, i.e., property (P2) and M control the properties of (P1), (P2) and (P4).

Furthermore, we can get various additive-based DEA models when we set different types of Ψ(Ω), Φ(Ω) and M. We listed several types of Ψ(Ω), Φ(Ω) and M as follows:

Case 1: p, q order geometric moment. $\text{\Psi}(\text{\Omega})\hspace{0.17em}=\hspace{0.17em}{({\displaystyle \sum _{j=1}^{n}{y}_{rj}^{p}})}^{\frac{1}{p}},p\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1,\dots ,\text{\u221e}$, and $\text{\Phi}(\text{\Omega})\hspace{0.17em}=\hspace{0.17em}{({\displaystyle \sum _{j=1}^{n}{x}_{ij}^{q}})}^{\frac{1}{q}},q\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1,\dots ,\text{\u221e}$. It is noted that Ψ(Ω) = max{y_{rj}, j = 1, …, n} and Φ(Ω) = max{x_{ij}, j = 1, …, n} when p,q = ∞.

Case 2: p, q order central moment. $\text{\Psi}(\text{\Omega})\hspace{0.17em}=\hspace{0.17em}{({\displaystyle \sum _{j=1}^{n}{({y}_{rj}-{\overline{y}}_{r})}^{p}})}^{\frac{1}{p}},p\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1,\dots ,\text{\u221e}$, and $\text{\Phi}(\text{\Omega})\hspace{0.17em}=\hspace{0.17em}{({\displaystyle \sum _{j=1}^{n}{({x}_{ij}-{\overline{x}}_{i})}^{q}})}^{\frac{1}{q}},q\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1,\dots ,\text{\u221e}$, where ${\overline{x}}_{i},{\overline{y}}_{r}$ denote the mean value of the i-th input and the r-th output, respectively.

Then, we researched whether the generalized additive-based DEA model satisfies the five properties mentioned above.

Theorem 2: (P1) The optimum of Model (3) is between 0 and 1 only when M is big enough.

Proof: Since whatever Ψ(Ω) and Φ(Ω) are, we could always find an M big enough, such that $\sum _{r=1}^{s}\frac{{s}_{r}^{+}}{\text{\Psi}\left(\text{\Omega}\right)}}+{\displaystyle \sum _{i=1}^{m}\frac{{s}_{i}^{-}}{\text{\Phi}\left(\text{\Omega}\right)}\text{\hspace{0.17em}}}=\hspace{0.17em}{\displaystyle \sum _{r=1}^{s}\frac{{\displaystyle \sum _{j=1}^{n}{\lambda}_{j}{y}_{rj}}-{y}_{ro}}{\text{\Psi}\left(\text{\Omega}\right)}}+{\displaystyle \sum _{i=1}^{m}\frac{{x}_{io}-{\displaystyle \sum _{j=1}^{n}{\lambda}_{j}{x}_{ij}}}{\text{\Phi}\left(\text{\Omega}\right)}}\hspace{0.17em}\le \hspace{0.17em}M$; thus $1-\frac{{\displaystyle \sum _{r=1}^{s}\frac{{s}_{r}^{+}}{\text{\Psi}\left(\text{\Omega}\right)}}+{\displaystyle \sum _{i=1}^{m}\frac{{s}_{i}^{-}}{\text{\Phi}\left(\text{\Omega}\right)}}}{M}\ge 0$. Furthermore, since $\sum _{r=1}^{s}\frac{{s}_{r}^{+}}{\text{\Psi}\left(\text{\Omega}\right)}}+{\displaystyle \sum _{i=1}^{m}\frac{{s}_{i}^{-}}{\text{\Phi}\left(\text{\Omega}\right)}}\ge 0$, $1-\frac{{\displaystyle \sum _{r=1}^{s}\frac{{s}_{r}^{+}}{\text{\Psi}\left(\text{\Omega}\right)}}+{\displaystyle \sum _{i=1}^{m}\frac{{s}_{i}^{-}}{\text{\Phi}\left(\text{\Omega}\right)}}}{M}\le 1$.

Theorem 3: (P2) The optimum of Model (3) is 1 when DMU_{o} is fully efficient, while the optimum of Model (3) is not 0 when DMU_{o} is fully inefficient.

Proof: It is obvious that ${s}_{i}^{-}$, ${\text{s}}_{r}^{+}$ are all equal to zero when DMU_{o} is fully efficient; thus, the optimum is one. However, when DMU_{o} is fully inefficient, the optimum varies when different M are chosen.

Theorem 4: (P3) The optimum of Model (3) is well defined and unit invariant.

Proof: Based on Theorem 1, it is obvious that Model (3) is unit invariant if and only if Ψ(Ω) and Φ(Ω) satisfy the homogeneity property. Moreover, since Ψ(Ω) and Φ(Ω) are non-zero mappings, Model (3) is well defined.

With respect to (P4) and (P5), it is obvious that the optima of Model (3) is not necessarily monotonic (the proof is similar to that of (P4’) in Cooper et al., 2011, and the optimum is not monotonic when we simply choose Ψ(Ω) = x_{i}.), and it satisfies the translation invariance property.

#### 4.4. BMAS Model

Since classical DEA models can only recognize the efficiency of each unit, when there are many efficient DMUs, they cannot be ranked. Classic super-efficiency DEA models can effectively solve the problem of ranking effective decision making units, but they cannot deal with the negative and nil data problem. Based on this, this paper presents the BMAS model to solve these problems.

Consider the model below:

The difference between the BMA model and BMAS model is only in the construction of the production possibility set. The BMAS model excludes the DMU evaluated from the production possibility set and only considers the new production possibility set constructed by the remaining DMUs. Then, we could evaluate this DMU based on the new production possibility set.

Thus, when the DMU being evaluated is efficient in the original DEA model, it would be outside of the new production possibility set which is constructed by the remaining DMUs. As a result,

${s}_{i}^{-}$,

${\text{s}}_{r}^{+}$ would be less than zero, so that the value of the objective function would be greater than one. In addition, since the distance between DMUs and the frontier varies from different DMUs, the efficiency would be measured by these distances. If the distance is short, the efficiency is relatively low, while if the distance is long, the efficiency is relatively high.

Figure 1 shows that the production possibility set of the original DEA models (BCC) is the district constructed by EBACF, while the new production possibility set of the super-efficiency DEA models is the district constructed by EBCF. With respect to A,

Figure 3 shows that

${s}_{i}^{-}$,

${\text{s}}_{r}^{+}$ are less than zero.

**Figure 3.**
The improvement of inefficient DMU on the production possibility set.

**Figure 3.**
The improvement of inefficient DMU on the production possibility set.