Computing Efficiency for Decision Making Units with Negative and Interval Data

Data Envelopment Analysis (DEA) is a nonparametric method for identifying sources and estimating the mount of inefficiencies contained in inputs and outputs produced by Decision Making Units (DMUs). DEA requires that the data for all inputs and outputs should be known exactly, but under many qualifications, exact data are inadequate to model real-life situations. So these data may have different structures such as bounded data, interval data, and fuzzy data. Moreover, the main assumption in all DEA is that input and output values are positive, but we confront many cases that discount this condition producing negative data. The purpose of this paper is to compute efficiency for DMUs, which permits the presence of intervals which can take both negative and positive values.


Introduction
Data envelopment analysis (DEA) is used to identify best practices and efficient frontier decision making units (DMUs) in the presence of multiple inputs and outputs (Charnes et al., [1]).DEA provides not only efficiency scores for inefficient DMUs but also frontier projections for such units onto an full-efficient frontier.The old DEA models did not deal with imprecise data assuming that all input and output data are exactly known.In real world situations, this assumption may not always be true.If such imprecise data information is integrated into the standard linear CCR model, the resulting DEA model is a nonlinear and non convex program, called imprecise DEA (IDEA) (Cooper et al. [2,3]).In addition to Lee et al. [4], Zhu [5], Thompson et al. [6] investigated IDEA.Recently, upper and lower bounds for the efficiency scores of the DMUs with imprecise data have been calculated by Despotits and Smirlis [7].In Jahanshahloo et al. [8] the radius of stability for the DMUs with interval data is calculated.Also, in Jahanshahloo et al. [9] Ranking DMUs with interval data using interval super efficiency index is extended.Moreover, the main assumption in all DEA is that input and output values are positive, but we encounter many cases that violate this term ultimately yielding negative inputs and outputs.Among the proposed methods of dealing with negative data, the following models could be provided.

Data Envelopment Analysis and Decision Science
Seiford and Zhu.[10] considered a positive and very small value of negative output.Another method was proposed by Halme et al. [11] and modified slack-based measure model, called MSBM was represented by Sharp et al [12].However, the latest method of behavior with negative data was provided by Emrouznejad et al [13,14], which is based on SORM model where some variables are considered which are both negative and positive for DMUs.Consequently, radial methods of DEA were modified for the evaluation of the efficiency of units by negative data.Data Envelopment Analysis (DEA) with integer and negative inputs and outputs was proposed by Jahnshahloo and Piri [15].The main objective of this paper is to decide how to deal with decision making units that have negative and interval inputs and outputs.This paper is organized as follows.In section 2 we calculate efficiency of decision making units with interval input and output and also with negative and positive input and output.Section 3 discusses how we can calculate efficiency for decision making units using both negative and interval input and output.A numerical example is provided in section 4 and the paper concludes in section 5.

Efficiency of Decision
x , x and y y , y ; x 0 , y 0 The efficiency attained by in model (2.2) serves as a lower bound of its possible efficiency scores.In this case, has the worst conditions and others DMUs are in the best conditions.Models (2.1) and (2.2) provide each DMU with a bounded interval in which possible efficiency scores lie from the worst to the best case.Considering (2.1) and (2.2), it is evident that .On the basis of the above efficiency score intervals, DMUs can be classified in three subsets as follows: Proof.It is evident.
Model (2.7) represents the general case for an input oriented VRS DEA model which has both inputs and outputs which take positive values for some DMUs and negative for others.
Based on this optimal solution, we define a DMU as being SORM-Efficient as follows.

Definition 2.2. (SORM -Efficient). is SORM -Efficient, if
3 Efficiency for Decision Making Unit with Negative and Interval Data.
In this section, we discuss how to calculate the efficiency for decision making units both negative and interval input and output.Now suppose we have n DMUs where input and output levels of each DMU are not known exactly.Let , where lower and upper bounds are exactly known, finite, positive or/ and negative.


At first, we divide the inputs and outputs into two groups, as follows: That is, the set of index of inputs with nonnegative values is represented by I whereas L denotes the set of index of inputs which have negative values in at least one DMU.In the same way, R is the set of index of the outputs with nonnegative values and K is the set of index of outputs with a negative value in at least one observation.Thus, for the following model provides an upper limit of interval efficiency when we encounter negative and interval data.
In this case, has the best conditions and other DMUs are in the worst conditions.Also, the following model provides a lower limit of interval efficiency score for when we encounter negative and interval data.
Proof.Assume are the optimal solutions for models (       the procedure is similar.

A numerical example
In this section, the model used in a numerical example attempts to measure the efficiency of 10 DMUs.Suppose that there are 10 DMUs with two interval inputs and outputs shown in Table (1).The second input and output have an interval value which does not have any sign.It means that there is a positive value for some of DMUs and a negative value for some others.
solution for this model.Because the problem is minimizing, the proof is complete.For proving *L * pp

Making Units with Interval Input and Output.
That is, the set of index of inputs with nonnegative values is indicated by I while L denotes the set of index of inputs which have negative value in at least one DMU.Similarly, R is the set of index of the outputs with nonnegative values and K is the set of index of outputs which have a negative value in at least one observation.Let us take an output variable   which is positive for some DMUs and negative for others.
, respectively.At first, we prove