Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 1531282, 8 pages

http://dx.doi.org/10.1155/2016/1531282

## Alternative Trade-Offs in Data Envelopment Analysis: An Application to Hydropower Plants

^{1}Department of Applied Mathematics, Islamic Azad University, Lahijan, Iran^{2}Department of Applied Mathematics, Islamic Azad University, Rasht, Iran^{3}Department of Management, Islamic Azad University, Rasht, Iran

Received 27 September 2015; Accepted 6 December 2015

Academic Editor: Fazal M. Mahomed

Copyright © 2016 Mohsen Mirzaei et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In multidimensional input/output space, the behavior of the firms can be analyzed by using efficient frontier or supporting surfaces of production technology. To this end, mathematicians are interested to use marginal rates of substitutions. The piecewise linear frontier of data envelopment analysis (DEA) technology is not differentiable at the extreme points and marginal rates calculation is valid only for small changes in one or more variables. The existing trade-off analysis methods calculate the maximum changes in a specific throughput when another throughput is changed. We will show that binding efficient supporting surfaces of an efficient point may be used to define different marginal rates of substitutions and in this sense, we get different marginal rates to each frontier point.

#### 1. Introduction

Data envelopment analysis (DEA) is a LP-based nonparametric technique for measuring the relative performances of firms that use multiple inputs to produce multiple outputs. The literature on the performance measurement using DEA has grown substantially since Charnes et al. [1] introduced the traditional CCR model based on the maximum radial reduction in inputs. It has been widely used in performance measurement of many business and industry applications. A complete literature and surveys can be found in Cooper et al. [2] and Cook and Seiford [3].

While much DEA-based research has been directed to the application of DEA for performance measurement, fewer works have been made on its properties as a production function model. Production technology in DEA is extrapolated with the observed inputs and outputs and the boundary points of this set construct the efficient frontier or supporting hyperplanes of the technology. Supporting hyperplanes of production technology enable us to analyze the relation between two different throughputs. Mathematicians are very interested to work with efficient frontiers or supporting surfaces of production possibility set to analyze the behavior of firms in the multidimensional input/output space.

The knowledge of trade-offs in a production process is an important subject to the managers. For instance, managers are interested to know the additional amount of a certain input that is required in order to increase a particular output by a small fixed amount.

Huang et al. [4] proposed a general method for calculating the rates of change of outputs to the inputs along efficient surface of DEA production set. Rosen et al. [5] directly studied the problem of marginal rate of substation on efficient frontier and presented a general framework for the calculation of trade-offs between two variables in DEA. Cooper et al. [6] modified the classic additive DEA model and from the optimal slack values of this model they derived the marginal rates and elasticities of substitution.

Krivonozhko et al. [7] have used the supporting hyperplanes and efficient surfaces to calculate marginal rates of substitution in DEA. In the meantime, using supporting hyperplanes of production technology, Førsund and Hjalmarsson [8] have calculated scale elasticity in DEA models. Førsund et al. [9] have also proposed two ways of obtaining numerical values of scale elasticity by direct-indirect approaches and then they compared the two approaches by real data. Balk et al. [10] generalized the concept of scale elasticity to accommodate changes in any given direction in input-output space. Khoshandam et al. [11] proposed a production in which a group of variables are changed in a given direction and the effect of this change on some throughput is calculated. In their second paper, Khoshandam et al. [11] studied the problem of marginal rates of substitution in the presence of nondiscretionary factors.

The abovementioned approaches calculate a single measure of marginal rates of substitution and these trade-off analysis methods are given in optimistic case in the sense that they calculate the maximum changes in a specific throughput when another throughput is changed by a small quantity. In this sense, the new frontier point may be very far from the original frontier point. However we can equivalently look for a piece of the frontier in which the new frontier point is closing as much as possible to the original point.

We show that binding supporting surfaces of production set in an efficient point can be used to define marginal rates of substitutions and hence we may calculate different marginal rates in efficient points. We show this geometrically by a simple example.

The rest of the paper is organized as follows: marginal rates substitutions are introduced in the next section. Section 3 introduces two marginal rates in optimistic and pessimistic cases. Marginal rates in the presence of undesirable outputs and in the absence of explicit inputs are given in Section 4. A real application is used to illustrate the proposed approach. The paper ends with concluding remarks.

#### 2. Marginal Rates of Substitution

In mathematical economics, marginal rate of substitution is employed to calculate the relative marginal utility. Mathematicians are often interested to work with efficient surfaces, supporting hyperplanes and Pareto-efficient surfaces. In multidimensional input-output space, the behavior of a production unit can easily be read using the supporting hyperplanes of the DEA production possibility set without any loss of mathematical rigor. In this section, we briefly introduce the marginal rate of substitution. Consider a general process in which an output vector is produced by consuming the input vector . There are a set of : , each of which is characterized by a throughput vector in which and are nonzero and nonnegative vectors. To simplify, the throughput matrix is shown as follows:Let be the boundary of the production technology and without loss of generality, we assume thatMoreover, we assume that the production technology shows free disposability or in other words is assumed to be continuously differentiable. Let be a frontier point; that is, .

*Definition 1. *The marginal rate of substitution of the th throughput to the th throughput at the frontier point is defined as follows:As we know, in real applications, is not known exactly and hence we instead use the empirical production technologies such as DEA production possibility set. This set is structured axiomatically and the boundary points of this set construct the empirical production function that is a piecewise linear function. Due to the nature of the DEA efficient frontier, marginal rates are not uniquely defined in the extreme efficient units on the frontier and hence, we should calculate the marginal rate of substitution to right and left.

A DEA-based procedure to calculate the marginal rates of substitution of throughput to throughput from right at the frontier point is given by Asmild et al. [12] as follows:in which is a small positive number and is the optimal solution to the following LP problem:Substituting instead of in the foregoing procedure gives the left marginal rates of substitution.

In this procedure, the new frontier point is calculated in optimistic case in the sense that is maximized. There may be another point on the frontier and we can use this point instead of . In the next section, two different marginal rates of substitutions are calculated, one in optimistic case and another one in pessimistic case.

#### 3. Different Marginal Rates

As we stated in the previous section, the classic trade-off analysis is given in the optimistic case and hence they determine the maximum changes in a specific component of the throughput vector when a specific component is increased (or decreased) by a small quantity.

Generally, we solve the following LP problem to calculate the marginal rates of substitution:in which is the new input/output vector after changes. The maximization objective function in the LP problem (6) is given in optimistic case and hence, it finds the maximum value to a specific throughput when another throughput is changed by a small quantity. In model (6), we look for a piece of the frontier in which the specified component is maximized. We can equivalently look for a piece of the frontier in which the component is minimized. Needless to say in both cases the new point is a frontier point and this is what we want.

We show this by the following simple example. We use a simple example with seven DMUs with two inputs and and one output . The data are summarized in Table 1.