*3.3. Joint Scaling of Desirable and Undesirable Outputs*

Proposed models in the previous subsection only consider the desirable output and look for possible expansion of this type of output. However, we may take care of both desirable and undesirable outputs simultaneously. In previous models, we sought a possible expansion of desirable output while keeping undesirable output. But, there might be a possibility of areduction of undesirable outputs that are not considered in the previous models. Therefore, we proposed the following mixed model taking both desirable and undesirable factors into consideration.

*o*.

$$\phi\_{ILLo}^{j} = \text{Max1} + \text{q}$$

$$\text{s.t}$$

$$\sum\_{k=1}^{K\_j} \lambda\_k^j x\_{ik}^j \le x\_{\text{no}}^j \quad i = 1, 2, \dots, m(5)$$

$$\sum\_{k=1}^{K\_j} \lambda\_k^j y\_{rk}^j \ge (1 + \varrho) y\_{r\alpha}^j \quad r = 1, 2, \dots, s$$

$$\sum\_{k=1}^{K\_j} \lambda\_k^j z\_{hk}^j = (1 - \varrho) z\_{h\alpha'}^j \quad h = 1, 2, \dots, P$$

$$\lambda\_k^j \ge 0, \ k = 1, 2, \dots, K\_j$$

The optimal value of the above model is less than or equal to zero. If it is zero, then the under-evaluation unit is efficient, otherwise it is inefficient.

**Theorem 5**. *If a plant is locally mixed efficient then its local efficiency score is greater than or equal to the mixed local efficiency score*.

An associated global model that simultaneous changes desirable and undesirable outputs can also be proposed by the following:

$$\begin{aligned} \boldsymbol{\varrho}\_{1EGo}^{j} &= \text{Max1} + \boldsymbol{\varrho} \\ \text{s.t} \\ \sum\_{j=1}^{I} \sum\_{k=1}^{K\_{j}} \lambda\_{k}^{j} \boldsymbol{x}\_{ik}^{j} &\leq \boldsymbol{x}\_{io}^{j}, \ \boldsymbol{i} = 1, 2, \dots, m \\ \sum\_{j=1}^{I} \sum\_{k=1}^{K\_{j}} \lambda\_{k}^{j} \boldsymbol{y}\_{rk}^{j} &\geq (1 + \boldsymbol{\varrho}) \boldsymbol{y}\_{ro}^{j}, \ \boldsymbol{r} = 1, 2, \dots, \boldsymbol{s}, \\ \sum\_{j=1}^{I} \sum\_{k=1}^{K\_{j}} \lambda\_{k}^{j} \boldsymbol{z}\_{hk}^{j} &= (1 - \boldsymbol{\varrho}) \boldsymbol{z}\_{ho}^{j} \boldsymbol{h} = 1, 2, \dots, \boldsymbol{P} \\ \lambda\_{k}^{j} \geq 0, \ \boldsymbol{k} &= 1, 2, \dots, K\_{j}, \boldsymbol{j} = 1, 2, \dots, \boldsymbol{J}. \end{aligned}$$

**Theorem 6**. *If a plant is a locally mixed efficient plant, then its local efficiency score is greater than or equal to the mixed local efficiency score*.

**Proof of Theorem 6**. It is similar to the proof of Theorem 5.

The previous subsection's proposed indices can be updated by the new mixed uniform measure of the model (5) and model (6). However, we cannot compare the efficiency measures using mixed models and peer models in the previous subsection, since the structure production technologies are different. Therefore, associated indices may be meaningless. However, we can still compare the local and global environmental performance of production units by the new mixed index of

$$\text{Uniform Enviornmental Load-Global Index} = \frac{\rho\_{\text{LIEGo}}^j}{\rho\_{\text{LIELo}}^j}$$

**Theorem 7**. *The global mixed efficiency measure of a production unit is greater than or equal to its local mixed efficiency measure*.

**Proof of Theorem 7.** It is similar to the proof of Theorem 4.

Using Theorem 7, we then have

$$\text{Uniform Enviornmental Load-Global Index} = \frac{\varrho\_{ILGeo}^{\circ}}{\varrho\_{ILLo}^{\circ}} \ge 1.1$$

The percentage of the desirable output expansion and the undesirable output reduction may not necessarily be equal; thus, we proposed the following model for the local and global environmental efficiency measurement of *P<sup>j</sup> o*.

$$q^{j}\_{\text{NILLo}} = \text{Max1} + q + \gamma$$

$$\sum\_{k=1}^{K\_i} \lambda\_k^i x\_k^j \le x\_{\text{lo}}^j \; i = 1, 2, \dots, m(7)$$

$$\sum\_{k=1}^{K\_i} \lambda\_k^i y\_{rk}^j \ge (1 + q) y\_{\text{ro}}^j \; r = 1, 2, \dots, s$$

$$\sum\_{k=1}^{K\_i} \lambda\_k^i z\_{\text{hi}}^j = (1 - \gamma) z\_{\text{lo}}^j \; h = 1, 2, \dots, P$$

$$\lambda\_k^j \ge 0, \; k = 1, 2, \dots, K\_j \; q^j\_{\text{NILGe}} = \text{Max1} + q + \gamma$$

$$\text{s.t}$$

$$\sum\_{j=1}^{I} \sum\_{k=1}^{K\_i} \lambda\_k^j x\_{ik}^j \le x\_{\text{lo}}^j \; i = 1, 2, \dots, m(8)$$

$$\sum\_{j=1}^{I} \sum\_{k=1}^{K\_j} \lambda\_k^j y\_{rk}^j \ge (1 + q) y\_{\text{ro}}^j \; r = 1, 2, \dots, s$$

$$\sum\_{j=1}^{I} \sum\_{k=1}^{K\_j} \lambda\_k^j z\_{kk}^j = (1 - \gamma) z\_{\text{lo}}^j \; h = 1, 2, \dots, P$$

$$\lambda\_j^j \ge 0, \; k = 1, 2, \dots, K\_j, j = 1, 2, \dots, J$$

**Theorem 8.** *The global mixed efficiency measure of a production unit is greater than or equal to its local mixed efficiency measure*.

**Proof of Theorem 8.** It is similar to the proof of Theorem 7.

Using the model (7) and (8), we proposed the non-uniform environmental localglobal index as follows:Non-uniform Environmetal Local-Global Index <sup>=</sup> *<sup>ϕ</sup><sup>j</sup> NUEGo ϕj NUELo* and using Theorem 8, we could see that Non-uniform Environmetal Local-Global Index = *ϕj NUEGo ϕj* ≥ 1.

*NUELo* The local model of (7) and global model (8) consider the non-uniform scaling factor for desirable and undesirable output. In contrast, this factor is considered to be uniform in the local model (5) and global model (6). If we consider *γ* = *ϕ* in the model (7) and model (8), then we get models (5) and (6), respectively. Therefore, we could consider model (5) and model (6) as a particular case of the model (7) and model (8), respectively.

Table 1 lists and summarizes the variables and parameters used in the current paper.


**Table 1.** Decision variables and parameters.
