Sensitivity analysis of efficient units in the presence of non-discretionary inputs

doi:10.1016/j.amc.2007.02.002

Applied Mathematics and Computation

Volume 190, Issue 2, 15 July 2007, Pages 1185-1197

https://doi.org/10.1016/j.amc.2007.02.002 Get rights and content

Abstract

Discretionary models of data envelopment analysis (DEA) assume that all inputs and outputs can be varied at the discretion of management or other users. In any realistic situation, however, there may exist “exogenously fixed” or non-discretionary factors that are beyond the control of a DMU’s management, which also need to be considered. This paper discusses and reviews the use of super-efficiency approach in data envelopment analysis (DEA) sensitivity analyses when some inputs are exogenously fixed. Super-efficiency data envelopment analysis (DEA) model is obtained when a decision making unit (DMU) under evaluation is excluded from the reference set. In this paper by means of modified Banker and Morey’s (BM hereafter) model [R.D. Banker, R. Morey, Efficiency analysis for exogenously fixed inputs and outputs, Operations Research 34 (1986) 513–521], in which the test DMU is excluded from the reference set, we are able to determine what perturbations of discretionary data can be tolerated before frontier DMUs become nonfrontier.

Introduction

Data envelopment analysis (DEA) initially proposed by Charnes et al. [5] is a non-parametric technique for measuring and evaluating the relative efficiencies of a set of entities, called decision making units (DMUs), with the common inputs and outputs. Examples include school, university, hospital, library, bank, business firm, court and, more recently, whole economic and society systems, in which outputs and inputs are always multiple in character [4]. Standard data envelopment analysis implicitly assumes that all inputs and outputs are discretionary, i.e., can be controlled by the management of each DMU and varied at its discretion. However, there may exist exogenously fixed (or non-discretionary) inputs or outputs that are beyond the control of a DMU’s management, which also need to be considered [14], [16], [19]. Instances from the DEA literature include snowfall or weather in evaluating the efficiency of maintenance units, soil characteristics and topography in different farms, the number of competitors in the branches of a restaurant chain, local unemployment rates which affect the ability to attract recruits by different US Army recruitment stations, age of facilities in different universities, and the number of transactions (for a purely gratis service) in library performance. DEA identifies empirical efficient frontier of a set of DMUs. In data envelopment analysis efficient decision making units are of primary importance as they define the efficient frontier. It is well known that adding or deleting an inefficient DMU or a set of inefficient DMUs does not alter the efficiencies of the existing DMUs and the efficient frontier. The inefficiency scores change only if the efficient frontier is altered, i.e. the performance of DMUs depends only on the identified efficient frontier.The efficient frontier is characterized by the DMUs with an unity efficiency score. One important issue in DEA which has been studied by many DEA researchers is sensitivity analysis of a specific DMU_o, unit under evaluation [6], [7], [20]. One type of DEA sensitivity analysis is based on super-efficiency DEA approach, which is obtained when a decision making unit (DMU) under evaluation is excluded from the reference set [1], [8], [13], [17], [18], [21], [22]. By updating the inverse of the basis matrix associated with a specific efficient DMU in a DEA linear programming problem, Charnes and Neralic have been studied the sensitivity of DEA models by a series of sensitivity analysis articles in which sufficient conditions preserving efficiency are determined [9], [10], [11], [12], [15]. This paper discusses and reviews the use of super-efficiency approach in data envelopment analysis (DEA) sensitivity analyses, when some inputs are non-discretionary. For this task we first introduce the BM model [2], then by means of modified the BM model, in which the test DMU is excluded from the reference set, we determine what perturbations of data can be tolerated before frontier DMUs become nonfrontier. The sensitivity analysis approach developed in this paper can be applied to all DMUs on the entire frontier. Also this paper attempts to a situation where proportional percentage data changes are assumed for a DMU under evaluation and for the remaining DMUs. Necessary and sufficient conditions for preserving a DMU’s BM-efficiency classification are developed when proportional data changes are applied to all DMUs. Note that in this paper we assume that the factors are either fully discretionary or fully non-discretionary. Also we assume that all models have no non-discretionary outputs. The current article proceeds as follows. Section 2 provides preliminary information that will be used in the succeeding sections. In Section 3 we will discuss super-efficiency and DEA sensitivity analysis in the BM model. Section 4 is the main part of this study where we will discuss simultaneous changes in all the discretionary data. In Section 5 we will apply the proposed method to an empirical data set. Conclusions are provided in the last section.

Section snippets

Background

Assume that there are n DMUs, where each ${DMU}_{j} (j = 1, 2, \dots, n$ ), uses m different discretionary inputs, $x_{ij} (i = 1, 2, \dots, m)$ , and $p$ different non-discretionary inputs, $z_{ij} (i = 1, 2, \dots, p)$ , to produce s different outputs, $y_{rj} (r = 1, 2, \dots, s)$ . We assume that the data set are positive.

Assuming constant returns to scale, the BM model to evaluate the efficiency of any DMU – in the input-oriented case – is given by the following modification of the CCR model: $[BM_{CCR}] θ_{ND}^{CCR *} = \min θ$ $s.t. \sum_{j = 1}^{n} x_{ij} λ_{j} + s_{i}^{-} = θ x_{io}, i \in D,$ $\sum_{j = 1}^{n} z_{ij} λ_{j} + s_{i}^{-} = z_{io} i$

Super-efficiency and DEA sensitivity analysis in the BM model

In evaluating DMU_o by the BM model, if $θ_{ND}^{CCR *} = 1$ then we call this unit is a frontier point. Therefore, based on Definition 1, a set of DMUs can be partitioned into two groups: frontier DMUs and non-frontier DMUs. Furthermore, the frontier DMUs consist of DMUs in set E (extreme Full-BM_CCR-efficient), set E′ (Full-BM_CCR-efficient but not an extreme point), set E″ (BM_CCR-efficient but with non-zero non-discretionary slacks) and set F (weakly BM_CCR-efficient or frontier point but with non-zero

Simultaneous changes in all the discretionary data

The frontier points in DEA are of primary importance as they define the DEA frontier. In this section we will discuss the stability of efficiency classification for such units. First we answer the following question: if the outputs of DMU_o are fixed and we desire the super-BM_CCR-efficieny index $θ_{ND}^{\sup *}$ to remain greater than one (that is, DMU_o remains extreme BM_CCR-efficient), how much could the discretionary inputs of DMU_o increased?

Theorem 2

Let DMU_o be an extreme BM_CCR-efficient point, and let $θ_{ND}^{\sup *}$

An application

In this section to complete our paper, we apply our approach to the data set of 23 public libraries in Tokyo: L1 to L23 (Which are taken from Cooper et al. [4]). In total, 4 inputs and 2 outputs were employed. The inputs were floor area (unit = 1000 m²) [Area], the number of books (unit = 1000) [Book], staffs (unit = 1000) [Staff], and the population of the area (unit = 1000) [Population]. The outputs were the number of registered residents (unit = 1000) [Register] and the number of borrowed books (unit =

Conclusion

In this paper, we have expanded the super-efficiency and DEA sensitivity analysis concept in the CCR model to the BM model. This expansion is important since in any realistic situation there may exist “exogenously fixed” or non-discretionary factors that are beyond the control of a DMU’s management, which also need to be considered. The new sensitivity analysis approach simultaneously considers the data perturbations in all DMUs, For instance, the proportional data changes of the DMU under

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View full text

Sensitivity analysis of efficient units in the presence of non-discretionary inputs

Abstract

Introduction

Section snippets

Background

Super-efficiency and DEA sensitivity analysis in the BM model

Simultaneous changes in all the discretionary data

An application

Conclusion

European Journal of Operational Research

Glasnik Matematicki

Glasnik Matematicki

Annals of Operations Research

European Journal of Operational Research

A procedure for ranking efficient units in data envelopment analysis

Management Science

Efficiency analysis for exogenously fixed inputs and outputs

Operations Research

Linear Programming and Network Flows

second ed.

Data Envelopment Analysis: A Comprehensive Text with Models, Applications, References and DEA Solver Software

Sensitivity and stability analysis in DEA: some recent developments

Journal of Productivity Analysis

A structure for classifying and characterizing efficiencies and inefficiencies in DEA

Journal of Productivity Analysis