Context-Dependent Data Envelopment Analysis with Interval Data

Data envelopment analysis (DEA) is a non-parametric method for evaluating the relative efficiency of decision making units (DMUs) on the basis of multiple inputs and outputs. The context-dependent DEA is introduced to measure the relative attractiveness of a particular DMU when compared to others. In real-world situation, because of incomplete or non-obtainable information, the data (Input and Output) are often not so deterministic, therefore they usually are imprecise data such as interval data, hence the DEA models becomes a nonlinear programming problem and is called imprecise DEA (IDEA). In this paper the context-dependent DEA models for DMUs with interval data is extended. First, we consider each DMU (which has interval data) as two DMUs (which have exact data) and then, by solving some DEA models, we can find intervals for attractiveness degree of those DMUs. Finally, some numerical experiment is used to illustrate the proposed approach at the end of paper.


Introduction
Data envelopment analysis (DEA), developed by Charnes et al. [1], usually evaluates decision making units (DMUs) from the angle of the best possible relative efficiency.If a DMU is evaluated to have the best possible relative efficiency of unity, then it is said to be DEA efficient; otherwise it is said to be DEA inefficient.Performance of inefficient DMUs depends on the efficient DMUs, that is, the inefficiency scores change only if the efficiency frontier is altered.
Although the performance of efficient DMUs is not influenced by the presence of inefficient DMUs, it is often influenced by the context.The context-dependent DEA [2][3][4] is introduced to measure the relative attractiveness of a particular DMU when compared to others.We know that the DMUs in the reference set can be used as benchmark targets for inefficient DMUs.The context-dependent DEA provides several benchmark targets by setting evaluation context [3].The context-dependent DEA is introduced to measure the relative attractiveness of a particular DMU when compared to others.Relative attractiveness depends on the evaluation context constructed from alternative DMUs.The original DEA method evaluates each DMU against a set of efficient DMUs and cannot identify which efficient DMU is a better option with respect to the inefficient DMU.This is because all efficient DMUs have an efficiency score of one.The standard DEA models assume that all data are known exactly without any variation.However, this assumption may not be true.In the real world, some outputs and inputs may be only known as in forms of interval data, ordinal data and ratio interval data.If we incorporate such imprecise data information in to the standard linear CCR model, the resulting DEA model is a nonlinear and non-convex program, and is called imprecise DEA (IDEA), (Cooper et al. [5] and Kim et al. [6]).The approach in IDEA is to transform the non-linear and non-convex model to a linear programming equivalent, by imposing scale transformation on the data variable alterations (products of variables are replaced by new variables).Prior to IDEA, pertinent work was that of Cook et al. [7,8], which, is confined only to mixture of exact and ordinal data.They started by dealing with only one ordinal input [8] and then extended their model [7] to handle multiple cardinal and ordinal criteria.The basic idea in these models is to assign new auxiliary variables, one for every combination of ordinal variables and distinct ranks of it.The value for such an auxiliary variable corresponding to DMU j and a rank position is 1, if DMU j is rated in the k-th place of ordinal variable and 0 otherwise.Extensions of these basic ideas are reported in [9,10].Recently, Despotits and Smirlis [11] calculated upper and lower bounds for the efficiency scores of the DMUs with imprecise data.They developed an alternation approach for dealing with imprecise data.They transformed the non-linear DEA model to a linear programming equivalent by using a straightforward formulation, completely different than that in IDEA.Contrarily to IDEA their transformation on the variables are made on the basis of the original data set, without applying any scale transformations on data.Also, in Jahanshahloo et al. [12,13] the radius of stability for the DMUs with interval data is calculated.In this paper we concentrate on the context-dependent DEA with interval data.For this reason we consider each DMU j , with interval data, as two DMUs that have exact data.Then by a procedure similar to that one in [3], the evaluation contexts are obtained by partitioning these DMUs into several levels of efficient frontier.Then by introducing some models based upon these efficient frontiers we can measure the relative attractiveness and progress of these DMUs and also we can determine the interval attractiveness and interval progress for each original DMU with interval data.Further by combination of these measures we can also characterize the performance of DMUs.
The rest of the paper is organized as follows: next section introduces the basic definitions of interval data and notations of the original context-dependent DEA.In Section 3, interval context-dependent DEA is presented.In Section 4 we illustrate our proposed DEA method with two numerical examples and some discussion.Finally, some conclusions are pointed out in the end of this paper.

Now suppose we have
DMUs which utilize inputs . Also, assume that input and output levels of each DMU are not known exactly.We define , then the i-th input of the DMU j has a definite value.Interval problems are those whose parameter values are located in intervals, their exact values being unable to be identified.
Therefore we consider problems with data such as ,  , where lower and upper bounds are known exactly, positive and finite.

Context-Dependent Data Envelopment Analysis with Exact Data
The original context-dependent DEA model is developed by using the following radial efficiency measure.Let 1 I be the set of all DMUs, and k I and interactively defined as consists of all the radially efficient DMUs by following linear programming: where becomes the original output oriented CCR model and define the first-level efficient frontier.When k = 2, model (1) gives the second-level efficient frontier after the exclusion of the first-level efficient DMUs.And so on.In this manner we identify several levels of efficient frontiers.We call the k-th level efficient frontier.2) is same as [3] with slightly change.In model ( 2  Now, to identify efficiency levels, we apply the procedure introduced in [3] for these DMUs.

Interval Context-Dependent DEA
be the sets of all these DMUs.Now, we consider two following models: and Models ( 3) and ( 4) are similar to model (1) because these models simultaneously identify several levels of efficient frontiers as follows: In the above models Then we define 1 1 When k = 1, then first-level efficient frontier is defined by DMUs in , that is, 2 .When k = 2, models (3) and ( 4) give the second-level efficient frontier after the exclusion the first-level efficient DMUs.In this manner we can identify several levels of efficient frontiers, where 1 2 consist the th-level efficient frontier.By following steps we can identify these efficient frontiers using models (3) and ( 4): Step 1. Set k = 1, evaluate the DMUs belong to 1 J I  using models (3) and ( 4) to obtain the first-level efficient DMUs, .
1 2 Step 3. Evaluate the new subset "inefficient" DMUs, 1 k J  , using models(3) and ( 4) to obtain new sets of efficient DMUs  In the first section, we said that, the DMUs in the reference set can be used as benchmark targets for inefficient DMU.The context-dependent DEA provides several benchmark targets by setting evaluation context.DMUs can reach (if possible) to the nearest efficiency level as the first target to improve their efficiency.
Figure 1 plots the five levels of efficient frontiers of 5 DMUs with single interval input and single interval output (see Table 1).Assume that L levels of efficient frontiers are identified by the above algorithm.It can be seen that in this example L = 5.Now based upon these evaluation contexts the context dependent DEA measures the relative attractiveness of each DMU (DMUs with interval data) as follows: Suppose for , by the above algorithm, we ob-  Now, we introduce two following context dependent DEA models to obtain the relative attractiveness measure. and l , and also, solution for model (5).Thus we have is a feasible solution for model (6).Since model ( 6) is maximization and has an optimal value as is the relative attractiveness measure for DMU .
Assume that has interval data, that is, as k-degree attractiveness of .DMU o Since the parameter values are located in intervals and their exact values being unable to be identified, hence value of

 
* o A k is unknown.Also, one can rank the DMUs in each level based upon their attractiveness scores.

Application
In order to illustrate the use of the methodology for determining the interval attractiveness developed here, first, in example 1 we use the data in Table 1 and in example 2 we use an empirical data.

Example 1: Personal Selection Data
Assume that, we have 5 DMUs in one input and one output and these data are interval as shown in Table 1.By considering each DMU j as two DMUs with exact data, namely,   X Y we will have 10 DMUs.Now, to identify the efficient levels of these DMUs, we apply models (3) and ( 4).
Results are shown below: We see that for these DMUs, 5 efficient levels are identified, i.e., = 5.By models ( 5) and ( 6), interval attractiveness of each DMU can be calculated.In Table 2 interval attractiveness of each original DMUs (which have interval data) are shown in column 4. For example, in Table 2, first-degree interval attractiveness of with respect to efficiency level is [1.67, 3.33].
That is, if we choose exact value for input and output of and then calculate the attractiveness degree of 1 with respect to efficiency level , namely , we must have .

Discussion
In the above mentioned example, suppose that, we fix the data of these DMUs in their intervals, in the other words, assume that we have 5 DMUs, namely,   .These data are as Table 3.
We will show that, the attractiveness score of these DMUs lies in the interval attractiveness presented in Ta- For example, Figure 3 illustrates region of   k S E .It can be seen that, Assume that, we want to calculate the attractiveness degree of  However, if we calculate attractiveness of this set of DMUs we have the Table 4 which shows that the above statement is true.

Example 2: Empirical Data
Consider a performance measurement problem of manufacturing industry, in which there are eight manufacturing industries from different cities (DMUs) participating in the evaluation, each consuming two inputs (Labor and working funds) and producing three outputs (Gross industrial output value, profit and taxes, and retail sales).The data are all estimated and are thus imprecise and only known within the prescribed bounds, which are listed in Table 5.
By using the DEA models (3) and ( 4) and by same manner as shown in example 1, we obtain the following levels of efficient frontiers: Therefore, we see that for these DMUs, 6 efficient levels are identified, that is .6 L  If we apply models (5) and ( 6) for these DMUs, we obtain interval attractiveness of each DMU as Table 6.
For instance, consider 1 which has interval data.By

Conclusions
We developed in this paper an approach for dealing with interval data in context dependent DEA.It is done by considering each DMU (which have interval data) as two DMUs (which have exact data) and then we obtain interval attractiveness for each DMU.For this reason, we introduced some DEA models for evaluating these DMUs, and in the next step to obtain the interval attractiveness we merge the results of these models.Also we show that, if we choose n arbitrary DMUs with exact data, then the attractiveness of these DMUs are belong to that intervals.After this manager decided that, what combination of each interval is appropriate.2n Although the proposed method presented in this paper is illustrated by a personal selection data, however, it can also be applied to many problems of decision manage- In this paper, only, attractiveness measure is used.By the same manner one can use progress measure for DMUs with interval data.
. Set k = k + 1 and go to step 2.
All the computations in this example are carried out by a computer program using GAMS software.