*3.2. Multi-Plant Firm Environmental Production Technology*

Assume that each plant consumes not only M inputs and produces N desirable outputs but also P undesirable outputs. Let *x<sup>j</sup> ik* be the *i*-the input (1 ≤ *i* ≤ *M*), *y j rk* be the *r*-the desirable output (1 ≤ *r* ≤ *N*), and *z j hk* be the *h*-the undesirable output (1 ≤ *h* ≤ *P*) of plant *k* at firm *j*. We considered the general production technology that considers the undesirable outputs for the *j*-th plant by the output correspondence *P<sup>j</sup>* : R*<sup>M</sup>* <sup>+</sup> → *<sup>P</sup><sup>j</sup>* (*x*) <sup>⊆</sup> <sup>R</sup>*N*+*<sup>P</sup>* <sup>+</sup> . For dealing with both desirable and undesirable outputs, we considered the strong disposability for desirable output and the weak disposability for undesirable outputs proposed by [21] as follows. Strong disposability says if *<sup>y</sup>* ∈ *<sup>P</sup><sup>j</sup>* (*x*) then *<sup>y</sup>* ∈ *<sup>P</sup><sup>j</sup>* (*x*) for *y* ≤ *y* while weak disposability implies that *<sup>y</sup>* ∈ *<sup>P</sup><sup>j</sup>* (*x*) then *θy* ∈ *P*(*x*) for 0 ≤ *θ* ≤ 1. Considering *y* and *z* as desirable and undesirable outputs, we assumed that desirable outputs are strong disposal, and undesirable outputs are weak disposal in the context that if (*y*, *<sup>z</sup>*) ∈ *<sup>P</sup><sup>j</sup>* (*x*), then (*y* , *<sup>z</sup>*) ∈ *<sup>P</sup><sup>j</sup>* (*x*) for *y* ≤ *y*.

Considering this and the constant returns to scale, we found the following output set for plant *j*:

$$T\_{EL}^j = \{(x, y) \, \middle| \, x \ge \sum\_{k=1}^{K\_j} \lambda\_k^j x\_{k'}^j y \le \sum\_{k=1}^{K\_j} \lambda\_k^j y\_{k'}^j z = \sum\_{k=1}^{K\_j} \lambda\_k^j z\_{k'}^j \lambda \ge 0\}.$$

Considering this environmental production technology for the multi-plant firm, we may use the following linear programming model for assessing the performance of plant "*o*" in the *j*-the firm.

$$\boldsymbol{\varrho}\_{ELo}^{\dot{j}} = \underset{\mathbf{s},t}{\text{Max}}\boldsymbol{\varrho}$$

$$\sum\_{k=1}^{K\_{\dot{j}}} \lambda\_k^{\dot{j}} \mathbf{x}\_{\text{rk}}^{\dot{j}} \le \mathbf{x}\_{\text{io}}^{\dot{j}} \quad i = 1, 2, \dots, m \text{(3)}$$

$$\sum\_{k=1}^{K\_{\dot{j}}} \lambda\_k^{\dot{j}} \mathbf{y}\_{\text{rk}}^{\dot{j}} \ge \boldsymbol{\varrho} \mathbf{y}\_{\text{ro}}^{\dot{j}} \quad r = 1, 2, \dots, \text{s}$$

$$\sum\_{k=1}^{K\_{\dot{j}}} \lambda\_k^{\dot{j}} z\_{\text{rk}}^{\dot{j}} = z\_{\text{lo}}^{\dot{j}} \quad h = 1, 2, \dots, P$$

$$\lambda\_k^{\dot{j}} \ge 0, \ k = 1, 2, \dots, K\_{\dot{j}}$$

In contrast with the classical efficiency analysis and associated efficiency measure that was dealt in the previous section, in the environmental efficiency analysis of multi-plant firms, we considered any undesirable output that may be produced as a byproduct of the desired output.

**Theorem 1.** *The environmental efficiency of an arbitrary plan in a firm is not greater than its classical efficiency measure*.

The above theorem says if a production unit is efficient, then it is not necessarily environmentally efficient. In order to have acceptable environmental performance, production units need to take care of associated environmental issues that may not be considered in the classical efficiency analysis (Proof of Theorem 1 is available in Appendix A).

If we consider all firms and owned plants, we face a more competitive environment, and then we can use the following model for environmental efficiency of plants "*o*". In fact, in the global environmental analysis, we considered all plants' environmental performance belonging to all firms.

$$\boldsymbol{q}\_{EGo}^{j} = \text{Max}\boldsymbol{q}$$

$$\text{s.t}$$

$$\sum\_{j=1}^{I} \sum\_{k=1}^{K\_{j}} \lambda\_{k}^{j} \mathbf{x}\_{ik}^{j} \le \mathbf{x}\_{io'}^{j} \quad i = 1, 2, \dots, m(4)$$

$$\sum\_{j=1}^{I} \sum\_{k=1}^{K\_{j}} \lambda\_{k}^{j} \mathbf{y}\_{rk}^{j} \ge q \mathbf{y}\_{ro}^{j} \quad r = 1, 2, \dots, \text{s}$$

$$\sum\_{j=1}^{I} \sum\_{k=1}^{K\_{j}} \lambda\_{k}^{j} \mathbf{z}\_{hk}^{j} = \mathbf{z}\_{ho'}^{j} \quad h = 1, 2, \dots, P$$

$$\lambda\_{k}^{j} \ge 0, \quad k = 1, 2, \dots, K\_{j}, j = 1, 2, \dots, I$$

**Theorem 2**. *The classical efficiency of an arbitrary plant within its firm is not greater than its efficiency measure when considering all firms*.

**Corollary 1.** *The global classical environmental efficiency is greater than the local environmental efficiency, and we have the following relationship for the classical efficiency of P<sup>j</sup> <sup>o</sup> and its classical and environmental efficiency ϕ<sup>j</sup> Go* <sup>≥</sup> *<sup>ϕ</sup><sup>j</sup> Lo* <sup>≥</sup> *<sup>ϕ</sup><sup>j</sup> ELo*.

**Proof of Corollary 1.** This can be concluded while considering Theorems 1 and 2.

In order to measure the local-global efficiency measure of *P<sup>j</sup> <sup>o</sup>* we proposed Local-Globalindex <sup>=</sup> *<sup>ϕ</sup><sup>j</sup> Go ϕj Lo* and for estimating the classical-environmental efficiency measure of *Pj <sup>o</sup>* we proposed Local-Environmental Index <sup>=</sup> *<sup>ϕ</sup><sup>j</sup> Lo ϕj ELo* . Regarding Corollary 1, we saw that Local-Global Index <sup>=</sup> *<sup>ϕ</sup><sup>j</sup> Go ϕj Lo* ≥ 1. If this index is equal to unity, then it means that the evaluated plant could survive in the competitive global environment, since the local efficiency measure and the global efficiency measure are identical. Corollary 1 also concludes that Local-Environmental Efficiency <sup>=</sup> *<sup>ϕ</sup><sup>j</sup> Lo ϕj ELo* ≥ 1. This index determines whether a plant is environmentally friendly or not. Unity value shows that the efficiency measure does not depend on the technology's environmental structure and is environmentally friendly if we assess it in a classical or environmental production technology context. Suppose this index is greater than unity, then the environmental issue matters and can affect its performance *P<sup>j</sup> o*.

**Theorem 3.** *The global environmental efficiency of a plant is not greater than its local environmental efficiency*.

**Theorem 4.** *The global efficiency of a plant is not greater than its global environmental efficiency*.

**Corollary 2.** *The global classical environmental efficiency is not greater than the local environmental efficiency, and we have the following relationship for the classical efficiency of P<sup>j</sup> <sup>o</sup> and its classical and environmental efficiency: ϕ<sup>j</sup> Go* <sup>≥</sup> *<sup>ϕ</sup><sup>j</sup> EGo* <sup>≥</sup> *<sup>ϕ</sup><sup>j</sup> ELo*.

If we are interested in analyzing the global environmental performance, we can use the newly introduced index Global-Environmental Index <sup>=</sup> *<sup>ϕ</sup><sup>j</sup> Go ϕj EGo* . Regarding Corollary 2, this index is also greater than or equal to unity *<sup>ϕ</sup><sup>j</sup> Go ϕj EGo* ≥ 1. If it is unity, then the environmental and classical efficiency of *P<sup>j</sup> <sup>o</sup>* are equal globally. Otherwise, the environmental issue matters. We could see that a production unit's technically well performance does not necessarily imply an acceptable environmental performance. In other words, if a production unit is economically efficient, then we may not necessarily conclude that it is environmentally efficient too. For analyzing the environmental performance of *P<sup>j</sup> <sup>o</sup>* the local and global production space, we introduced Environmetal Local-Global Index <sup>=</sup> *<sup>ϕ</sup><sup>j</sup> EGo ϕj ELo* ≥ 1. This index is also less than or equal to unity *<sup>ϕ</sup><sup>j</sup> EGo ϕj ELo* ≥ 1 and it indicates the situation of the environmental performance of *P<sup>j</sup> <sup>o</sup>* in the local and global production. If it is equal to unity, then the environmental performance of *P<sup>j</sup> <sup>o</sup>* is resistance globally. This means *<sup>P</sup><sup>j</sup> <sup>o</sup>* has a similar environmental performance both locally and globally. However, if it is greater than unity, then we face a sub-optimal local performance for *P<sup>j</sup>*
