 American Journal of Computational Mathematics, 2011, 1, 256-263 doi:10.4236/ajcm.2011.14031 Published Online December 2011 (http://www.SciRP.org/journal/ajcm) Copyright © 2011 SciRes. AJCM Context-Dependent Data Envelopment Analysis with Interval Data Mohammad Izadikhah Department of Mat hematics, Islamic Azad University, Arak, Iran E-mail: m_izadikhah@yahoo.com, m-izadikhah@iau-arak.ac.ir Received August 11, 2011; revised Septembe r 20, 2011; accept ed September 28, 2011 Abstract Data envelopment analysis (DEA) is a non-parametric method for evaluating the relative efficiency of deci-sion making units (DMUs) on the basis of multiple inputs and outputs. The context-dependent DEA is intro-duced 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 be-comes a nonlinear programming problem and is called imprecise DEA (IDEA). In this paper the con-text-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. Keywords: DEA, Context-Dependent, Interval Data, Interval Attractiveness, Interval Progress 1. Introduction Data envelopment analysis (DEA), developed by Char-nes et al. , usually evaluates decision making units (DMUs) from the angle of the best possible relative effi-ciency. If a DMU is evaluated to have the best possible relative efficiency of unity, then it is said to be DEA ef-ficient; otherwise it is said to be DEA inefficient. Per-formance of inefficient DMUs depends on the efficient DMUs, that is, the inefficiency sco res 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 [2-4] is introduced to measure the relative attrac-tiveness 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 con-text-dependent DEA provides several benchmark targets by setting evaluation context . 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 con-structed 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 as-sumption may not be true. In the real world, some out-puts and inputs may be only known as in forms of inter-val data, ordinal data and ratio interval data. If we incor-porate such imprecise data information in to the standard linear CCR model, the resulting DEA model is a nonlin-ear and non-convex program, and is called imprecise DEA (IDEA), (Cooper et al.  and Kim et al. ). Th e 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  and then extended their model  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 dis-tinct ranks of it. The value for such an auxiliary variable corresponding to DMUj and a rank position is 1, if M. IZADIKHAH257 DMUj 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  calculated upper and lower bounds for the efficiency scores of the DMUs with imprecise data. They developed an alterna-tion approach for dealing with imprecise data. They transformed the non-linear DEA model to a linear pro-gramming equivalent by using a straightforward formu-lation, 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 rea-son we consider each DMUj, with interval data, as two DMUs that have exact data. Then by a procedure similar to that one in , 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 at-tractiveness and progress of these DMUs and also we can determine the interval attractiveness and interval pro-gress for each original DMU with interval data. Further by combination of these measures we can also character-ize the performance of DMUs. The rest of the paper is organized as follows: next sec-tion introduces the basic definitions of interval data and notations of the original context-dependent DEA. In Sec-tion 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. 2. Preliminaries Now suppose we have DMUs which utilize in-puts nm,1,,ijximto produce s outputs rj,1,,,1,,yrs1jn,,. Also, assume that input and output levels of each DMU are not known exactly. We define jjmjXxx and 1,,jjsjYy y, . 1, ,jn2.1. Interval Data Let input and output values of any DMU be located in a certain interval, where Lijx and Uijxare the lower and upper bounds of the i-th input of the DMUj, respecti- vely, and Lrjy, are the lower and upper bounds of the r-th output of the UrjyDMUj, respectively, that is to say, LUijij ijxxx and LUryj rjyyrj . Such data are called interval data, because they are located in intervals. Note that always LUij ijxx and LUrjyrjy. If LUijijxx, then the i-th input of the DMUj 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 ,LUijij ijxxx and rj rj,LUyrjyy, where lower and upper bounds are known exactly, positive and finite. 2.2. Context-Dependent Data Envelopment Analysis with Exact Data The original context-dependent DEA model is developed by using the following radial efficiency measure. Let 1I be the set of all DMUs, and kI and interactively defined as kE1kkkIIE, where consists of all the radially efficient DMUs by following linear program-ming: kE*00.. oojjjj okkkstX XYkjFE()()max 0, kkjFIjFIjY (1) where kjFI means DMU kjIE. When k = 1, model (1) 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. 1EkAssume that, , 0. We can calculate relative attractiveness measures for with respect to k-th level efficient frontier. Based upon the evaluation context , relative attractiveness meas-ure of o can obtained by the following con-text-dependent DEA: 0kkEDMUoEkE01, ,knDMUoDMU 00()()ax0, kkkkooEE0*000m 1,...,-.. jjjFjj ojFkkjHkHXkkLst XYHkYjFEk (2) Then **o1oAkHkDMUo0, ,kk is called the (output oriented) attractiveness of from a specific level . kEModel (2) is same as  with slightly change. In model (2) we set 0L in order to consider-ing as evaluation context for . 0kE0DMU koENote: In this paper, only, attractiveness measure is used. By the same mann er one can use progress m easure Copyright © 2011 SciRes. AJCM M. IZADIKHAH 258 for DMUs with interval data. 3. Interval Context-Dependent DEA Now, assume that there are DMUs which produce ns outputs by using inputs rjymijx such that ,Urjrj rjyyyL and ,LUijij ijxxx. By considering each DMUjas two DMUs with exact data, namely, U,LjjXY and ,ULjjXY we will have DMUs. 2nNow, to identify efficiency levels, we apply the pro-cedure introduced in  for these DMUs. L2nLet 1,, 1,,LUjjJXY jn,, 1,,ULjj and 1IXY jn be the sets of all these DMUs. Now, we consider two following models: 2n   *max .. , , 0, 0, kkkkooLULjjjj ojFJjFIULjojjj ojFJjFIkjjkkstXXXYYkjFJ kjFIUY (3) and   *max() .. , , 0, 0, kkkkooLUUjjjj ojFJjFIULjLjjjojFJjFIkjjkkstXXXYYkjFJ kjFIoY (4) 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 kjFJ means ,LUjj kXYJ and means kjFI,UL kjjXYI. Then we define 11kkkJJE and 12kkkIIE, where*1,;kLUkojjEXYJk 1 and *2,;kULkjj oEXYIk1k1k1, then to identify k th- level efficient DMUs, set . 12kkEEE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 fron-tiers, where 12 consist the th-level efficient frontier. By following steps we can identify these effi-cient frontiers using models (3) and (4): 1EE11EEkEkStep 1. Set k = 1, evaluate the DMUs belong to 1JI using models (3) and (4) to obtain the first-level efficient DMUs, . 12kkEEStep 2. Set 11kkkJJE , 12kkkIIE . (If 1kJ and 1kI then stop). Step 3. Evaluate the new subset “inefficient” DMUs, 1kJ, using models(3) and (4) to obtain new sets of effi-cient DMUs 11kE and . Set k = k + 1 and go to step 2. 12kEIn the first section, we said that, the DMUs in the ref-erence set can be used as benchmark targets for ineffi-cient DMU. The context-dependent DEA provides sev-eral 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 out-put (see Table 1). Assume th at L levels of efficient fron-tiers are identified by the above algorithm. It can be seen that in this example L = 5. Now based upon these evaluation contexts , 1,,,kEk L the context dependent DEA meas-ures the relative attractiveness of each DMU (DMUs with interval data) as follows: Suppose for , by the above algorithm, we ob- DMUotain 2,ULoo lXYE and 1,lLUooXYE, clearly 12ll. Output DMU2 DMU1 DMU3DMU5 DMU4 E4E5E3E2 E1Input Figure 1. Levels of efficient frontiers. Copyright © 2011 SciRes. AJCM M. IZADIKHAH259 Table 1. Data of numerical example. DMUj Inputs Ljx Ujx Outputs Ljy Ujy 1 3.5 6 6 7 2 2.1 5 3 4.2 3 1.5 3 0.6 1.5 4 7.5 8 1.6 4.5 5 5 6 0.5 1 Now, we introduce two following context dependent DEA models to obtain the relative attractiveness meas-ure.  221222122*21max, 0,, .. , , 0, 0, kl klkl klooLULjjjj ojFEjFEULojjjj ojFEjFEkljjHkHkk LlstXXXYYHjFE22 kljFEUkY (5) and  221222122*21max,0,, .. , , 0, 0, kl klkl klooLUUjjjj ojFEjFEULjojjj ojFEjFEkljjHkHkk LlstXXXYYHjFE22 kljFELkY (6) Clearly, *1oHk and *1oHk for 20,, kLl, and also,  *1o*oHkH k and  **1ooHkH k. Theorem 1. For we have DMUo **,ooHkHk 2Proof. First assume that 0, ,.kLl**,,oHk * be optimal solution for model (5). Thus we have 2212klklULULoojjjjoojFEjFEYYHkYHkY therefore **,,oHk * is a feasible solution for model (6). Since model (6) is maximization and has an optimal value as *oHk, then  **ooHkHk. Corollary 1. If DMU ,XY has exact data, where LUooXXX and LUoYYYo, then we must have   ***2,,0,,ooHkHkHkk Ll, where *Hk is the relative attractiveness measure for DMU . Assume that has interval data, that is, DMUoUo,LooXXX and ,LUoooYYY. By Corollary 1,   ***2,,0,,oo.HkHkHkk Ll Definition 1. If we call k-degree attractiveness of (which is lie on the specific level DMUoll221EEE22l) by *oAk, then we h ave   ***11,oooAk HkHk. That is, attractiveness score of with respect to efficiency level is DMUokE*oAk. Definition 2. We define  **11,ooHkHk as k-de- gree attractiveness of . DMUoSince th e parameter v alues are located in intervals and their exact values being unable to be identified, hence value of *oAk is unknown. Also, one can rank the DMUs in each level based upon their attractiveness scores. 4. Application In order to illustrate the use of the methodology for de-termining 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. 4.1. 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 DMUj as two DMUs with exact data, namely, Ujj,LXY and ,ULjjXY we will have 10 DMUs. Now, to identify the efficient levels of these DMUs, we apply models (3) and (4). Results are shown below:  111 22,,,LU LUEXYXY  21133,,,UL LUEXYXY  322 44,,,UL LUEXYXY   433 44 55,, ,,,UL ULLUEXYXYXY  555,ULEXY Copyright © 2011 SciRes. AJCM M. IZADIKHAH 260 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]. L1DMU3EThat 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 . 1DMUDMU(3)3E*1A*11.67(3)3.33ADiscussion In the above mentioned example, suppose that, we fix the data of these DMUs in their intervals, in th e other words, assume that we have 5 DMUs, namely, DMU,, 1,,5,jjjXY j which have exact data such that ,LUjjjXXX, ,LUjjjYYY. These data are as Table 3. We will show that, the attractiveness score of these DMUs lies in the interval attractiveness presented in Ta- Table 2. Interval attractiveness. DMUj k-degree Efficiency Levels k-degree inte rval attractiveness 1 Zero-degree 2E [1,2] First-degree 3E [1.67,3.33] Second-degree 4E [5,10] Third-degree 5E [12,24] 2 Zero-degree 3E [1,3.33] First-degree 4E [3,10] Second-degree 5E [7.2,24] 3 Zero-degree 4E [1,5] First-degree 5E [2.4,12] 4 Zero-degree 4E [1,3] First-degree 5E [2.4,7.2] 5 Zero-degree 5E [1,2.4] Table 3. DMUs with exact data. DMUj Input Output 1 5 6 2 3 4 3 2 1 4 7.5 4.5 5 6 0.5 ble 2. Hence, we employ models (3) and (4) for these DMUs. Let kE be the th-level of efficient frontier. See Figure 2 and compare kkE and . kEWe define kSE as follows: 11,,,0,,knn kjjjj jjjjjSExyxX yYXYE For example, Figure 3 illustrates region of kSE . It can be seen that, 5454 332121SE SESESESESESE SE SESE Assume that, we want to calculate the attractiveness degree of 1DMU with respect to efficiency level 4E, that is, we want to calculate . Since *14A21DMU E and 434SE SESE, hence the attractiveness OutputInput DMU1 DMU2 DMU3 DMU5 DMU4 E1E3E4E55 3214 1E 2E E2 3E5E Figure 2. Levels of efficient froutiers for DMUs with exact data and interval data. OutputInput 3E3SE Figure 3. Region of 3SE. Copyright © 2011 SciRes. AJCM M. IZADIKHAH Copyright © 2011 SciRes. AJCM 261with respect to efficiency level 4E, and *114H score of 1DMU lies in the following interval: and *113H are taken from Table 2.   *1**1111443AHH Therefore, without direct calculation of attractiveness scores of these DMUs (which have exact data) we must have where is the attractiveness degree of *14A1DMU *221,3.33A *13 1.67,3.33A *231,3.33A *141.67,10A *241,10A *441,3A *1512,24A *257.2,24A *352.4,12A *452.4,7.2A However, if we calculate attractiveness of this set of DMUs we have the Table 4 which shows that the above statement is true. 4.2. Example 2: Empirical Data Consider a performance measurement problem of manu-facturing industry, in which there are eight manufactur-ing industries from different cities (DMUs) participating in the evalua tion, each consuming two inputs ( Labor and working funds) and producing three outputs (Gross in-dustrial 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:   11122 77 88,,,,,,,LU LU LU LUE XYXYXYXY  244 55,,,LULUEXYXY  31133 66 88,,,,,,,LU LU LU LUE XYXYXYXY   422 33 55 77,, ,, ,, ,UL ULULULE XYXYXYXY  544,ULEXY  666,ULEXYTherefore, we see that for these DMUs, 6 efficient lev-els are identified, that is . 6L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 Table 6, one can see that interval attractiveness of 1 with respect to efficiency level is [1.18, 1.5]. Nevertheless, if one can find the exact value of and again calculate the attractiveness score of 1 with respect to , it must lie in interval [1.18, 1.5]. For more details see discussion presented in exam-ple 1. DMU4DMU1DMUDMU4EENote: All the computations in this example are carried out by a computer program using GAMS software. 5. 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 in-terval 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 attrac-tiveness we merge the results of these models. Also we show that, if we choose n arbitrary DMUs with exact data, then the attractiveness of th ese DMUs are belong to that intervals. After this manager decided that, what combination of each interval is appropriate. 2nAlthough the proposed method presented in this paper is illustrated by a personal selectio n data, however, it can also be applied to many problems of decision manage- Table 4. Exact attractiveness for exact data of Table 3. DMUj Levels Attractiveness 1 3E 2 4E 2.4 5E 15 2 2E 1.11 3E 2.22 4E 2.67 5E 16.67 3 3E 6.25 4 4E 1.2 5E 7.5 M. IZADIKHAH Copyright © 2011 SciRes. AJCM 262 Table 5. Data for eight DMUs. DMU Input Output Labor Working Funds GIO Profit and Taxes Retail Sales 1 [66, 73] [1354, 1540] [3200, 3800] [1100, 1200] [1000,1150] 2 [54, 70] [1205, 1425] [3000, 3350] [1000, 1150] [800,930] 3 [65, 80] [950, 985] [2800, 3000] [800, 900] [650, 700] 4 [55, 63] [850, 1000] [2500, 2750] [800, 850] [600, 850] 5 [72, 85] [1105, 1200] [3050, 3700] [950, 1150] [1000, 1050] 6 [63, 80] [1250, 1380] [2700, 2900] [800, 950] [700, 750] 7 [57, 72] [950, 1150] [2950, 3250] [950, 1200] [800, 900] 8 [60, 71] [800, 970] [2700, 2800] [800, 950] [900, 1000] Table 6. Interval attractiveness of Example 2. DMUj k-degree Efficiency Levels k-degree interval attractiveness 1 Zero-degree 3E [1, 1.31] First-degree 4E [1.18, 1.5] Second-degree 5E [1.43, 1.83] Third-degree 6E [1.56, 2] 2 Zero-degree 4E [1, 1.51] First-degree 5E [1.2, 1.81] Second-degree 6E [1.42, 2.13] 3 Zero-degree 4E [1, 1.2] First-degree 5E [1.14, 1.26] Second-degree 6E [1.45, 1.63] 4 Zero-degree 5E [1, 1.67] First-degree 6E [1.39, 1.96] 5 Zero-degree 4E [1, 1.3] First-degree 5E [1.39, 1.58] Second-degree 6E [1.64, 1.86] 6 Zero-degree 6E [1, 1.51] 7 Zero-degree 4E [1, 1.58] First-degree 5E [1.17, 1.66] Second-degree 6E [1.42, 2.18] 8 Zero-degree 3E [1, 1.4] First-degree 4E [1.11, 1.5] Second-degree 5E [1.55, 2.08] Third-degree 6E [1.83, 2.5] M. IZADIKHAH Copyright © 2011 SciRes. AJCM 263 ment. By the same manner one can use the proposed procedure to calculate interval progress for each DMU. 6. References  A. Charnes, W. W. Cooper and E. 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