*3.1. Directional Distance Functions with Undesirable Outputs*


Consider a technology T with inputs *x R N* +, desirable outputs *y R M* + , and undesirable outputs (such as bad loans) *b RJ* +. The directional distance function introduced by Chung et al. (1997), which seeks to directionally increase desirable outputs, while decreasing inputs and undesirable outputs, can be defined as follows:

$$D\_T\left(\mathbf{x},\,y,\,b;\,\mathbf{g}\right) = \text{Sup}\left\{\boldsymbol{\beta} \,:\, (\mathbf{x} - \beta \mathbf{g}\_{x}, y + \beta \mathbf{g}\_{y}, b - \beta \mathbf{g}\_{b}) \in T\right\},\tag{1}$$

where the nonzero vector g = ( <sup>−</sup>*gx*, *gy*, −*gb*) determines the directions in which the inputs, desirable outputs, and undesirable outputs are scaled. The reference technology set *T* = {(*y*, *b*): *x* can produce (*y*, *b*)} and is assumed to satisfy the assumptions of constant returns to scale, strong disposability of desirable outputs and inputs, and weak disposability of undesirable outputs.

Supposed there were *k* = 1, 2, ... *K* decision-making units (DMUs). Then, according to Chung et al. (1997), the directional distance function can be obtained by solving the following DEA problem:

$$\mathbf{Max}\boldsymbol{\beta} = \stackrel{\rightharpoonup}{D} (\mathbf{x}\_{\prime} \mathbf{y}\_{\prime} \mathbf{b}\_{\prime} \mathbf{g}\_{x\_{\prime}} \mathbf{g}\_{y\_{\prime}} \mathbf{g}\_{b})\_{\prime}$$

subject to

$$\sum\_{k=1}^{K} z\_k y\_{km} \ge y\_{km} + \beta y\_{km}, \quad m = 1, 2, \dots, M; \tag{2}$$

$$\sum\_{k=1}^{K} z\_k y\_{kj} = y\_{kj} - \beta y\_{kj}, \quad j = 1, 2, \dots, l;\tag{3}$$

$$\sum\_{k=1}^{K} z\_k \mathbf{x}\_{kn} \le \mathbf{x}\_{kn} - \beta \mathbf{x}\_{kn}, \quad n = 1, 2, \dots, N; \tag{4}$$

$$\sum\_{k=1}^{K} z\_k = 1,\tag{5}$$

$$z\_k \ge 0, \qquad k = 1, 2, \dots, \mathbb{K};$$

where ∑*Kk*=<sup>1</sup> *zkykm* is the efficient frontier formed as a linear combination of outputs of the other firms. Equation (2) states that the actual output (*ykm*) produced by the firm, plus the possible expansion (*βykm*), should be at most as large as the output represented by the efficient frontier. Likewise, Equation (3) requires that the input use (*xkn*), minus the possible input contraction (*βxkn*), should be at least as large as ∑*Kk*=<sup>1</sup> *zkxkn*, the linear combination of inputs used by the other firms. The undesirable output, i.e., the bad loans are represented by *ykj*, while *βykj* is the possible reduction in the undesirable output. The weights *zk* are the intensity variables for expanding or shrinking the individual observed activities of DMUs to construct convex combinations of the observed inputs and outputs.

Given the network DEA, the efficiency scores were obtained separately for the deposit mobilization stage and the loan financing stages. The overall efficiency score for the bank was obtained by multiplying the two sores.
