Developing a nondiscretionary model of slacks-based measure in data envelopment analysis

Abstract

Discretionary models of data envelopment analysis (DEA) assume that all inputs and outputs are discretionary, i.e., controlled by the management of each decision making unit (DMU) and varied at its discretion. In any realistic situation, however, there may exist exogenously fixed or nondiscretionary inputs or outputs that are beyond the control of a DMU’s management. The objective of this paper is to present a methodology for treating nondiscretionary in slacks-based measure (SBM) formulation. A numerical example is presented. At last, concluding remarks have been mentioned.

Introduction

Discretionary models of data envelopment analysis (DEA) assume that all inputs and outputs are discretionary, i.e., controlled by the management of each decision making unit (DMU) and varied at its discretion. Thus, failure of a DMU to produce maximal output levels with minimal input consumption results in a decreased efficiency score. In any realistic situation, however, there may exist exogenously fixed or nondiscretionary inputs or outputs that are beyond the control of a DMU’s management.¹ Instances from the DEA literature include snowfall or weather in evaluating the efficiency of maintenance units, soil characteristics and topography in different farms, number of competitors in the branches of a restaurant chain, age of facilities in different universities, and number of transactions (for a purely gratis service) in library performance. For example, Banker and Morey [2] illustrate the impact of exogeneously determined inputs that are not controllable in an analysis of a network of fast food restaurants. In their study, each of the 60 restaurants in the fast food chain consumes six inputs to produce three outputs. The three outputs (all controllable) correspond to breakfast, lunch, and dinner sales. Only two of the six inputs, expenditures for supplies and expenditures for labor, are discretionary. The other four inputs (age of store, advertising level, urban/rural location, and presence/absence of drive-in capability) are beyond the control of the individual restaurant manager. Their analysis clearly demonstrates the value of accounting for the nondiscretionary character of these inputs explicitly in the DEA models they employ; the result is identification of a considerably enhanced opportunity for targeted savings in the controllable inputs and targeted increases in the outputs.

Here, a literature review is presented briefly. The initial formulation is in terms of additive model. Suppose that the input and output variables may each be partitioned into subsets of discretionary (D) and nondiscretionary (N) variables. Thus,

$$I=\{1\text{,}2\text{,}\dots \text{,}m\}=I_{\mathrm{D}}\cup I_{\mathrm{N}}\text{,}{I}_{\mathrm{D}}\cap I_{\mathrm{N}}=\Phi$$

and

$$O=\{1\text{,}2\text{,}\dots \text{,}s\}=O_{\mathrm{D}}\cup O_{\mathrm{N}}\text{,}{O}_{\mathrm{D}}\cap O_{\mathrm{N}}=\Phi$$

The basic model formulation for the additive model (with nondiscretionary variables) is given by

$$\begin{array}{cc}\hfill & \underset{{\lambda }_j\text{,}{s}_r^{+}\text{,}{s}_r^{-}}{\min }-\left(\sum _{r\in O_D}{s}_r^{+}+\sum _{i\in I_D}{s}_i^{-}\right)\hfill \\ \hfill & \mathrm{s}\text{.}\mathrm{t}\text{.}\sum _{j=1}^n{y}_{\mathit{rj}}{\lambda }_j-s_r^{+}=y_{r0}\text{,}r=1\text{,}\dots \text{,}s\text{,}\hfill \\ \hfill & -\sum _{i=1}^n{x}_{\mathit{ij}}{\lambda }_j-s_i^{-}=-x_{i0}\text{,}i=1\text{,}\dots \text{,}m\text{,}\hfill \\ \hfill & \sum _{j=1}^n{\lambda }_j=1\text{,}\hfill \\ \hfill & {\lambda }_j⩾0\text{,}j=1\text{,}\dots \text{,}n\text{,}\hfill \\ \hfill & s_r^{+}⩾0\text{,}r=1\text{,}\dots \text{,}s\text{,}\hfill \\ \hfill & s_i^{-}⩾0\text{,}i=1\text{,}\dots \text{,}m\text{,}\hfill \end{array}$$

where the index sets r ∈ O_D and i ∈ I_D are confined to the discretionary outputs and inputs. It is assumed that there are n DMUs to be evaluated. Each DMU consumes varying amounts of m different inputs to produce s different outputs. Specifically, DMU_j consumes amounts X_j = {x_ij} of inputs (i = 1, … ,m) and produces amounts Y_j = {y_rj} of outputs (r = 1, … ,s). λ = {λ_j} is vector of DMU loadings, determining “best practice” for the DMU being evaluated. The variable s⁺ is the amount of slack and s⁻ is excess amount of input i.

This topic can also be treated in different ways. For instance, Charnes et al. [6] extended the additive model in order to accommodate nondiscretionary variables in the following form:

$$\begin{array}{cc}\hfill & \max \sum _{i=1}^m{s}_i^{-}+\sum _{r=1}^s{s}_r^{+}\text{,}\hfill \\ \hfill & \mathrm{s}\text{.}\mathrm{t}\text{.}\sum _{j=1}^n{x}_{\mathit{ij}}{\lambda }_j+s_i^{-}=x_{\mathit{io}}\text{,}i=1\text{,}\dots \text{,}m\text{,}\hfill \\ \hfill & \sum _{j=1}^n{y}_{\mathit{rj}}{\lambda }_j-s_r^{+}=y_{\mathit{ro}}\text{,}r=1\text{,}\dots \text{,}s\text{,}\hfill \\ \hfill & s_i^{-}⩽{\beta }_i{x}_{\mathit{io}}\text{,}i=1\text{,}\dots \text{,}m\text{,}\hfill \\ \hfill & s_r^{+}⩽{\gamma }_r{y}_{\mathit{ro}}\text{,}r=1\text{,}\dots \text{,}s\text{,}\hfill \end{array}$$

where the β_i,γ_r represent parameters (to be prescribed) and all variables are constrained to be nonnegative.

Assigning values from 0 to 1 accord different degrees of discretion to input i with β_i = 0 characterizing this input as completely nondiscretionary and β_i = 1 changing the characterization to completely discretionary. Similarly, setting γ_r = 0 consigns output r to a fixed (nondiscretionary) value while allowing γ_r → ∞, or, equivalently, removing this constraint on $s_r^{+}$ allows its value to vary in a freely discretionary manner [7].

In the case of the input- or output-oriented CCR [5] and BCC [1] models², the treatment of nondiscretionary inputs and outputs is similar. In particular, for an input orientation, it is not relevant to maximize the proportional decrease in the entire input vector. Such maximization (actually, the minimum value of θ) should be determined only with respect to the subvector that is composed of discretionary inputs. Thus, the formulation for the input-oriented BCC model with nondiscretionary variables is given by

$$\begin{array}{cc}\hfill & \underset{\theta \text{,}{\lambda }_j\text{,}{s}_r^{+}\text{,}{s}_i^{-}}{\min }\theta -\epsilon \left(\sum _{r\in O_D}{s}_r^{+}+\sum _{i\in I_D}{s}_i^{-}\right)\hfill \\ \hfill & \mathrm{s}\text{.}\mathrm{t}\text{.}\sum _{j=1}^n{y}_{\mathit{rj}}{\lambda }_j-s_r^{+}=y_{\mathit{ro}}r=1\text{,}\dots \text{,}s\text{,}\hfill \\ \hfill & \theta x_{\mathit{io}}-\sum _{j=1}^n{x}_{\mathit{ij}}{\lambda }_j-s_i^{-}=0i\in I_D\text{,}\hfill \\ \hfill & -\sum _{j=1}^n{x}_{\mathit{ij}}{\lambda }_j-s_i^{-}=-x_{\mathit{io}}i\notin I_D\text{,}\hfill \\ \hfill & \sum _{j=1}^n{\lambda }_j=1\text{,}\hfill \\ \hfill & {\lambda }_j⩾0\text{,}j=1\text{,}\dots \text{,}n\text{,}\hfill \\ \hfill & s_r^{+}⩾0\text{,}r=1\text{,}\dots \text{,}s\text{,}\hfill \\ \hfill & s_i^{-}⩾0\text{,}i=1\text{,}\dots \text{,}m\text{,}\hfill \end{array}$$

where θ is radial input contraction factor (eventually to become efficiency measure).

It is to be noted that the θ to be minimized appears only in the constraints for which i ∈ I_D, whereas the constraints for which i ∉ I_D operate only indirectly (as they should) because the input levels x_io are not subject to managerial control [4].

With respect to many practical applications of SBM model, the objective of this paper is to present a methodology for treating nondiscretionary in slacks-based measure (SBM) formulation. To the author’s knowledge, there is not any reference that discusses nondiscretionary in SBM.

This paper proceeds as follows. In Section 2 proposed model is introduced. Numerical example and concluding remarks are discussed in Section 3 and Section 4, respectively.