*3.1. Classical Multi-Plant Firm Production Technology*

Consider J multi-plant firms, numbered *j* = 1, 2, ... , *J* and each firm has *Kj* plants numbered *k* = 1, 2, ... , *Kj*. Assume each plant consumes M inputs and produces N desirable 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*) of plant *k* at firm *j*. The general production technology can be represented by the output correspondence of *P<sup>j</sup>* : R*<sup>M</sup>* <sup>+</sup> → *<sup>P</sup><sup>j</sup>* (*x*) <sup>⊆</sup> <sup>R</sup>*<sup>N</sup>* <sup>+</sup> . Considering this setting and constant returns to scale for the production technology that consists of only the desirable output, we can *Kj Kj*

consider the production set of *T<sup>j</sup> <sup>C</sup>* = {(*x*, *y*) *x* ≥ ∑ *k*=1 *λj kxj <sup>k</sup>*, *y* ≤ ∑ *k*=1 *λj ky j <sup>k</sup>*, *λ* ≥ 0} for *j*-th firm. The following linear programming model can be found based on this production set for

assessing the performance of the plant "*o*" of *j*-th firm.

$$\boldsymbol{\wp}\_{\boldsymbol{\text{Lo}}}^{j} = \underset{\mathbf{s}, \mathbf{t}}{\text{Max}} \boldsymbol{\wp}$$

$$\sum\_{k=1}^{K\_{\boldsymbol{\text{t}}}} \lambda\_{k}^{j} \mathbf{x}\_{ik}^{j} \le \mathbf{x}\_{i\boldsymbol{\text{o}}\prime}^{j} \ \boldsymbol{i} = 1, 2, \dots, m(1)$$

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

$$\lambda\_{k}^{j} \ge 0, \ \boldsymbol{k} = 1, 2, \dots, K\_{\boldsymbol{\text{t}}}$$

If we consider all plants operating in all production spaces, we have a broader production system, and plants may face more competitive environments. This setting is seen in the global industry, and thus, for assessing the performance plant "*o*" considering all firms and associated plants, we may use the following linear programming:

$$\boldsymbol{q}\_{Co}^{\boldsymbol{l}} = \boldsymbol{M} \boldsymbol{x} \boldsymbol{q}$$

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

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

$$\lambda\_{k}^{\boldsymbol{j}} \ge 0, \; k = 1, 2, \dots, K\_{\boldsymbol{j}}, \; j = 1, 2, \dots, \boldsymbol{l}.$$

Please note that in the classical efficiency analysis, we consider only inputs and desirable outputs; thus, other measures like emission, etc., that can be considered as undesirable outputs are not considered in the above models and associated measures. The following subsection deals with undesirable outputs in the production process of multi-plant firms.
