- Email: [email protected]
Fuzzy interpretation of efficiency in data envelopment analysis and its application in a non-discretionary model
Knowledge-Based Systems xxx (2013) xxx–xxx
Contents lists available at SciVerse ScienceDirect
Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys
Fuzzy interpretation of efficiency in data envelopment analysis and its application in a non-discretionary model Majid Zerafat Angiz L a,⇑, Adli Mustafa b a b
School of Quantitative Sciences, Universiti Utara Malaysia, Sintok, Kedah, Malaysia School of Mathematical Sciences, Universiti Sains Malaysia, Penang, Malaysia
a r t i c l e
i n f o
Article history: Received 17 October 2011 Received in revised form 29 April 2013 Accepted 7 May 2013 Available online xxxx Keywords: Data envelopment analysis Fuzzy numbers Ranking
a b s t r a c t Data envelopment analysis (DEA) is a nonparametric model which evaluates the relative efficiencies of decision-making units (DMUs). These DMUs produce multiple outputs by using multiple inputs and the relative efficiency is evaluated using a ratio of total weighted output to total weighted input. In this paper an alternative interpretation of efficiency is first given. The interpretation is based on the fuzzy concept even though the inputs and outputs data are crisp numbers. With the interpretation, a new model for ranking DMUs in DEA is proposed and a new perspective of viewing other DEA models is now made possible. The model is then extended to incorporate situations whereby some inputs or outputs, in a fuzzy sense, are almost discretionary variables. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction Data envelopment analysis (DEA), proposed by Charnes et al. [5] (CCR model) is a nonparametric model for identifying the efficient frontier and measuring the relative efficiency of decision making units (DMUs). Based on the CCR model, Banker et al. [2] (BCC model) developed a variable returns to scale variation of the model. Both the CCR and BCC models have undergone various theoretical extensions and also many successful applications [1]. Nowadays, DEA has been applied to a variety of research areas. Most of the performance evaluation models presented in the literature on DEA are divided into two categories. In the first category, the methods focused on the production possibility set (PPS) to provide models for special purposes. The super efficiency models and models based on return to scale are in this category. In the second category, the methods focused on projecting an inefficient DMU onto the efficient frontier using different measures. For instance, the efficient DMU on the efficiency frontier corresponding to an inefficient DMU under evaluation in the CCR model and the additive model are different. Therefore, two different optimal solutions are obtained. This paper falls into both categories as it provides an alternative interpretation of the CCR model using fuzzy concept. It will be demonstrated that the relative efficiency is equivalent to the membership function of a fuzzy number. Based on this new interpretation, a new DEA model equivalent to the CCR model is
⇑ Corresponding author. Tel.: +60 017 3509310; fax: +60 04 6570910. E-mail addresses: [email protected] (M. Zerafat Angiz L), [email protected] (A. Mustafa).
proposed. The new concept is then extended further to describe its application in a non-discretionary model. The CCR model only handles crisp data. When data are fuzzy, fuzzy DEA models are used instead. The existing approaches for solving fuzzy DEA are usually categorised in four categories; (a) the tolerance approaches [10,18], (b) the fuzzy ranking approaches [9], (c) the a-level based approaches [11] and (d) the defuzzification approaches [13]. Zerafat Angiz et al. [27] proposed two different methods basis on multi-objective programming. In the nonradial approach [27], attempt has been made to solve the fuzzy DEA model by retaining the whole information throughout the evaluation process as far as possible. In the discrete approach [25], the a-cut concept is extended. The model that is proposed in this paper however, is not about a fuzzy DEA model. It is about the use of fuzzy concept in handling crisp data. The uses of fuzzy concepts in handling certain crisp mathematical modelling situations have resulted in the formulation of creative and efficient procedures. This can be seen for example in the work of Zerafat Angiz et al. [26] for ranking efficient DMUs. In the literature, a few studies on the relationship between DEA and other optimisation models have been done. Thanassoulis and Allen [20] showed the equivalence between the CCR model and some linear value function model for multiple outputs and multiple inputs. Then, Joro et al. [12] proved the structural correspondences between DEA models and multiple objective linear programming using an achievement scalarising function that was proposed by Wong and Beasley [21]. Furthermore, Yang et al. [23] presented a Multi Objective Linear Programming (MOLP) formulation based on min–max reference point approach in order to solve output-oriented DEA models. Banker et al. [3] studied the
0950-7051/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.knosys.2013.05.001
Please cite this article in press as: M. Zerafat Angiz L, A. Mustafa, Fuzzy interpretation of efficiency in data envelopment analysis and its application in a non-discretionary model, Knowl. Based Syst. (2013), http://dx.doi.org/10.1016/j.knosys.2013.05.001
2
M. Zerafat Angiz L, A. Mustafa / Knowledge-Based Systems xxx (2013) xxx–xxx
relationship between DEA and game theory. Besides that, some authors utilised the fuzzy concept to handle difficulties that arise in some procedures, even though they were not dealing with fuzzy numbers. For example, using fuzzy concept, Yeh and Chang [22] aggregated individual judgements as a group judgements and then applied it in a new approach for evaluating alternatives in a multicriteria decision making (MCDM) environment. Zerafat Angiz et al. [24–26] used a fuzzy structure to select the best alternative in a group decision making environment and Emrouznejad et al. [7] used fuzzy DEA in an assignment problem. None of the previous studies however compares DEA with fuzzy concept. The remainder of this paper is organised as follows: In Section 2, some background information about fuzzy numbers, the DEA model and the non-discretionary DEA model are given. A new interpretation of technical efficiency based on fuzzy approach and a new DEA model equivalent to the CCR model are elaborated in Section 3. A new non-discretionary DEA model is described in Section 4. In Section 5, a numerical example is given. Finally, the conclusion is presented in Section 6. 2. Background 2.1. Fuzzy number
(b) There is no DMU in J whose data domain can be proportionally expressed by that of another DMU.
Definition 2. Given the (empirical) points (Xj, Yj), j = 1, 2, . . . ,n, the Production Possibility Set (PPS) is defined as follows:
T ¼ fðX t ; Y t Þjoutput Y t can be produced by input X t g The production possibility set corresponding to constant return to scale will be as follows:
Tc ¼
8 > < > :
n n X X ðX t ; Y t ÞjX t ¼ kj xij ; Y t ¼ kj yrj ; kj P 0; j¼1
j¼1
9 j ¼ 1; 2; . . . ; n > = B @ i ¼ 1; 2; . . . ; m > ; r ¼ 1; 2; . . . ; s 0
ð2Þ
The set Tc is the production possibility set corresponding to constant returns to scale, and kj is a nonnegative value related to the jth DMU. The vector k = (k1, k2, . . . , kn)t is used to construct a hull that covers all the data points. Constant Return to Scale (CRS) means that an increase in the amount of input consumed leads to a proportional increase in the amount of output produced. If this increase is culminated in larger or smaller than proportional increase in the amount of outputs, return to scale will be Increasing Return to Scale (IRS) or Decreasing Return to Scale (DRS), respectively.
Definition 1. A fuzzy number ~ x is of LR-type if there exists reference functions L (for left), R (for right), and a > 0, b > 0 such that
To evaluate efficiency corresponding to set Tc, consider the following model.
8 < L xxl for xl 6 x 6 xm m xl x l~x ðxÞ ¼ : R xu x for xm 6 x 6 xu xu xm
hp ¼ min hp
m
ð1Þ m
l
x , called the mean value of ~x, is a real number; a = x x and b = xu xm are called the left and right spreads, respectively; and xl and xu are the lower bound and the upper bound of the interval of fuzzy number. Symbolically, ~x is denoted by (xm, a, b)LR or (m, a, b)LR. If ~x is a symmetric triangular fuzzy number, then we have R(x) = L(x) = x.
ð3Þ
s:t: ðhp X p ; Y p Þ 2 T
The variable h is the DEA efficiency score of CCR model. The CCR model (input-oriented) for evaluating the efficiency of DMUp, is written as follows:
hp ¼ min hp n X s:t: kj xij 6 hp xip i ¼ 1; 2; . . . ; m j¼1
2.2. Data envelopment analysis
n X kj yrj P yrp
r ¼ 1; 2; . . . ; s
ð4Þ
j¼1
Data envelopment analysis evaluates the relative efficiency of a set of homogenous DMUs using a ratio of total weighted output to total weighted input. It generalises the usual efficiency measurement from a single-input, single-output ratio to a multiple-input, multiple-output ratio. The DEA approach applies linear programming techniques to observe inputs consumed and outputs produced by decision making units and then constructs an efficient production frontier based on best practices. Each DMU’s efficiency is then measured relative to this frontier. For a clearer description of the DEA models, the following general assumptions are specified: (a) There are n DMUs denoted by j 2 J, each of which produces a nonzero output vector Yj = (y1j, y2j, . . . , ysj)t P 0 using a nonzero input vector Xj = (x1j, x2j, . . . , xmj)t P 0, where the superscript 0 t0 indicates the transpose of a vector. Here, the symbol 0 P 0 indicates that at least one component of Xj or Yj is positive while the remaining X 0j s or Y 0j s are nonnegative.
kj P 0 j ¼ 1; 2; . . . ; n hp free 2.3. Non-discretionary DEA model One of the significant concepts in data envelopment analysis is the non-discretionary variables. An input or output is called a nondiscretionary variable if it cannot be varied at the discretion of management or other users. For example, as stated by Charnes et al. [5] ‘‘In evaluating the performances of different bases for the Fighter Command of U.S. Air Forces, for instance, it was necessary to consider weather as an input since the number of ‘‘sorties’’ (successfully completed missions) and ‘‘aborts’’ (non-completed missions) treated as output, could be affected by the weather (measured in degree days and numbers of ‘‘flyable’’ days) at different bases’’. Banker and Morey [4] were pioneers in this study by including non-discretionary variables in the input-oriented DEA model. The
Please cite this article in press as: M. Zerafat Angiz L, A. Mustafa, Fuzzy interpretation of efficiency in data envelopment analysis and its application in a non-discretionary model, Knowl. Based Syst. (2013), http://dx.doi.org/10.1016/j.knosys.2013.05.001
3
M. Zerafat Angiz L, A. Mustafa / Knowledge-Based Systems xxx (2013) xxx–xxx
Banker and Morey model, considering constant return to scale, is given by the following mathematical programming:
min hp n X s:t: kj xij þ si ¼ hxip i 2 D
/p ¼ i 2 ND
xip xip xip
xip 6 xip
i ¼ 1; 2; . . . ; m
ð6Þ
With the new definition of the variables, Model (3) is converted to the following model:
j¼1 n X kj xij þ si ¼ xip
l~xip ðxip Þ ¼
ð5Þ
maxfminfl~xip ðxip Þgði ¼ 1; 2; . . . ; mÞg
s:t: ðxip ; yrp Þ 2 T
ð7Þ
j¼1 n X kj yrj sþi ¼ yrp
j ¼ 1; 2; . . . ; s
A new linear programming model corresponding to the CCR model is initially proposed as follows:
j¼1
kj P 0 j ¼ 1; 2; . . . ; n þ In Model (5), s i and si are the slack variables for the ith input and rth output, respectively, and the symbols D and ND refer to discretionary and non-discretionary variables, respectively. Muniz et al. [15] reviewed some non-discretionary approaches and compared the relative performance of each by simulation analysis. Ruggiero [16] provided an alternative model by relaxing the convexity constraint with respect to the non-discretionary variables and used simulation analysis to highlight the advantages of his model over the Banker and Morey’s model. A weakness of Ruggiero’s model was the tendency to identify decision-making units (DMUs) as efficient by default as the number of exogenous factors increased. Ruggiero [17] extended this model to allow multiple non-discretionary input. The model requires three stages and uses regression analysis to develop an aggregate index capturing the influence of the non-discretionary factors (see [15]). Muñiz [14] provided an alternative three-stage model to control for the effect that non-controllable inputs have on production. Like the first stage in Ruggiero [17], this model considers only discretionary input and output in the first stage. This model, however, focuses not only on the radial measure of efficiency but also the remaining slacks that remain after equi-proportionate projection to the overall frontier. Moreover, by using only DEA methods, it has the advantage that there is no need to assume any functional form in any of the stages (see [15]). Cordero-Ferrera et al. [6] extend the Simar and Wilson [19] model which is a four-stage model based on the bootstrap procedure, in order to estimate unbiased efficiency scores and to set realistic targets that take into account information concerning non-discretionary variables. Golany and Roll [8] pointed out that in many real-life applications of efficiency studies, a factor is neither fully controllable nor totally uncontrollable. For example, managers can make marginal alterations in personnel scheduling. However they have to comply with general guidelines of their organisation in many other aspects involving the use of their human resources. In other words, the factor is partially controllable. To incorporate this factor into a DEA model, an index taking on the values between 0 and 1 is used to represent the degree of discretion that the DMU has on the factor. In this paper, such factor will be called an almost discretionary variable or fuzzy non-discretionary (FND) variable will also be used. More discussion on this is presented in Section 4.
max min model maxfminfl~x ðxip Þgði ¼ 1; 2; . . . ; mÞg ip
n X kj xij 6 xip i ¼ 1; 2; . . . ; m s:t: j¼1 n X kj yrj P yrp
ð8Þ j ¼ 1; 2; . . . ; s
j¼1
0 6 xip 6 xip
i ¼ 1; 2; . . . ; m
kj P 0 j ¼ 1; 2; . . . ; n In the solution of the model, xip indicates the input levels which makes DMUp efficient. Since the right hand side of the constraints in the Model (8) has remained unchanged, both the CCR and the proposed model have the same production possibility set. Obviously, Model (8) is always feasible and the optimum value of the objective function is non-negative and less than or equal to 1. Since the objective function is the membership function, the optimal value does not exceed the maximum value of the membership values, i.e. a value of 1. The values kp = 1, kj = 0 (j – p) and xip ¼ xip are the feasible solutions in Model (8), thus this is another indication that the model is always feasible. If we assume that a ¼ minfl~xip ð xip Þg, then the following model is obtained:
max
a
ð9Þ
3. A new version of the traditional CCR model Assume that for the new DEA model to be proposed, all postulations to construct the production possibility set corresponding to a constant return to scale model are satisfied. Therefore, the production possibility sets of the CCR model and the proposed model are similar. To present the new model, we introduce a fuzzy number with the following membership function. Assume that xip and yrp are inputs and outputs of the DMU under evaluation, say DMUp. Then, the membership function of the fuzzy number ~ xip is considered as follows:
Fig. 1. A fuzzy view of the CCR model.
Please cite this article in press as: M. Zerafat Angiz L, A. Mustafa, Fuzzy interpretation of efficiency in data envelopment analysis and its application in a non-discretionary model, Knowl. Based Syst. (2013), http://dx.doi.org/10.1016/j.knosys.2013.05.001
4
M. Zerafat Angiz L, A. Mustafa / Knowledge-Based Systems xxx (2013) xxx–xxx
b
a
Fig. 2. (a) Discretionary input. (b) Almost discretionary input.
s:t:
n X kj xij 6 xip
i ¼ 1; 2; . . . ; m
ð9:1Þ
r ¼ 1; 2; . . . ; s
ð9:2Þ
j¼1 n X kj yrj P yrp
PH HC ¼ AB BC Thus:
j¼1
x x a 6 ip ip i ¼ 1; 2; . . . ; m xip xip 6 xip i ¼ 1; 2; . . . ; m
ð9:3Þ ð9:4Þ
kj P 0 j ¼ 1; 2; . . . ; n
aP0
1
HC BC HC BH ¼ ¼ BC BC BC
a
ð11Þ
BH is the technical efficiency score of DMUc. Since AB = 1, the followBC ing expression is obtained from (10) and (11):
Efficiency of DMU C ¼
This model can now be solved for each DMU. For a better understanding of Models (8) and (9), we refer to Fig. 1 which provides a two-dimensional diagram of a simple efficiency case study involving 3 DMUs (D, C and Z) in which only a single input (x) is used to produce a single output (y). Based on the output to input ratio, DMUZ is efficient and the line from the origin passing through point Z is the efficient frontier. Triangles ABC and DEF are the membership functions of the fuzzy numbers which correspond to the inputs for DMUC and DMUD, respectively. These membership functions are super imposed onto the graph. Notice that, the shape of the membership functions reflex the characteristic of the input variables in DEA (i.e., the smaller the level of input the more efficient is the DMU). Based on Thales theorem (basic proportionality theorem), in triangle ABC (Fig. 1), PH is parallel to AB. Therefore, triangle ABC is similar to triangle PHC, so:
ð10Þ
BH ¼ 1 PH BC
ð12Þ
As observed, PH is the membership value of the triangular fuzzy number corresponding to the input of DMUC at point H, which is on the efficient frontier. The situation indicates the relationship between technical efficiency and membership function of a fuzzy number. Similarly, the length of segment LK indicates the efficiency score of DMUD. Since the length of segment PH is less than the length of segment LK, we can conclude that the efficiency score of DMUC is more than efficiency score of DMUD. The main idea that needs to be pointed out here is that there exists a relationship between the efficiency scores that were calculated using the CCR model in Model (4) and the membership values of the fuzzy number ~ xip . The smaller the membership value, the higher is the efficiency score of the DMU under evaluation. The following theorem indicates equivalency between the efficiency scores of Models (4) and (9).
b
Fig. 3. (a) A fuzzy number corresponding to DMUB. (b) Another fuzzy number corresponding to DMUB.
Please cite this article in press as: M. Zerafat Angiz L, A. Mustafa, Fuzzy interpretation of efficiency in data envelopment analysis and its application in a non-discretionary model, Knowl. Based Syst. (2013), http://dx.doi.org/10.1016/j.knosys.2013.05.001
5
M. Zerafat Angiz L, A. Mustafa / Knowledge-Based Systems xxx (2013) xxx–xxx
xip 6 xip
Table 1 A list of DMUs with two inputs and two outputs. DMUs
I1
I2
O1
O2
1 2 3 4 5 6
1.50 4.00 3.20 5.20 3.50 3.20
1.50 0.70 1.20 2.00 1.20 0.70
1.40 1.40 4.20 2.80 1.90 1.40
0.35 2.10 1.05 4.20 2.50 1.50
Theorem 1. Assume that h⁄ = min h and a⁄ = max a are the optimal values of Models (4) and (9), respectively. Then, h⁄ = 1 a⁄. In other words, the treatment of Model (9) and the CCR model are similar. Proof. In Model (9), if we consider a 6 so
xip xip , xip
then axip 6 xip xip ,
xip 6 xip axip ¼ ð1 aÞxip
ð13Þ
There is a one-to-one correspondence between the values xip and (1 a)xip, and therefore we can replace Eq. (9.1) by taking into consideration Eq. (13). This resulted in the following linear programming model:
a
max s:t:
ð14Þ
n X kj xij 6 xip 6 ð1 aÞxip
i ¼ 1; 2; . . . ; m
ð14:1Þ
j¼1 n X kj yrj P yrp
r ¼ 1; 2; . . . ; s
ð14:2Þ
i ¼ 1; 2; . . . ; m
ð14:3Þ
j¼1
a6 xip 6 xip
xip xip xip
i ¼ 1; 2; . . . ; m
ð14:4Þ
kj P 0 j ¼ 1; 2; . . . ; n
aP0 Assume that h = 1 a. Then, Model (14) is converted to the following model:
max s:t:
1h
ð15Þ
n X kj xij 6 xip 6 hxip
i ¼ 1; 2; . . . ; m
ð15:1Þ
j¼1 n X kj yrj P yrp
r ¼ 1; 2; . . . ; s
ð15:2Þ
j¼1
1h6
xip xip xip
i ¼ 1; 2; . . . ; m
ð15:3Þ
1 2 3 4 5 6
ð15:4Þ
hP0 x x
In Eq. (15.3), 1 h 6 ipxip ip implies that xip 6 hxip , thus, Eq. (15.3) is x x implied by Eq. (15.1). Since 0 6 ipx ip 6 1, so 0 6 h 6 1 is obtained ip from Eq. (15.3). On the other hand, by considering max (1 h) = 1 min h, Model (5) is resulted. This means that h⁄ = 1 a⁄. By performing some substitutions, Model (4) implies Model (9), analogously. In other words, Model (9) represents a new version of the CCR model. The efficiency score in this model is the membership value of the point located at the intersection of the fuzzy interval (segment BC in Fig. 1) and the efficiency frontier. The example given in the Section 5 demonstrates the correspondence between the proposed model and the CCR model. h
4. A new non-discretionary DEA model In this section, an application of the fuzzy concept that has been discussed in the previous section is presented. Fig. 2a and b illustrate two separate DEA studies with a single input and a single output, in which the input variable corresponding to A is discretionary and the input variable corresponding to B is almost discretionary. The appropriate membership functions are superimposed onto the graphs. Notice that, for the membership function of the discretionary variable, more weight is given for the smaller values, reflecting a characteristic of the input variable in a traditional DEA model, which is, the smaller the level of input the more efficient is the DMU. However, for the membership function of the almost discretionary variable, more weight is given for the measured/current value to reflect the reluctance on the part of the DMU to reduce the value. Notice that, the membership value corresponding to the nondiscretionary variable is a modified form of the membership value in Fig. 3. It is easy to show the relationship between h⁄ and a⁄. For a better understanding of the relationship between h⁄ and a⁄, we refer to Fig. 3a and b. Triangles CAE and CAF are the membership functions of the fuzzy numbers which correspond to the input of DMUA. In referring to Fig. 3, BD is parallel to AE, therefore the two triangles CEA and CDB are similar. So, we have
CB BD ¼ CA AE
ð11Þ
CB It is well known that the value CA is equivalent to the technical efficiency of DMUA. Thus, with AE ¼ 1; BD ¼ h is the technical efficiency AE of DMUA which is also the membership value corresponding to point B, the reference point of DMUA on the efficient frontier. Furthermore we consider Fig. 3b in which triangle CFA is similar to triangle BHA. Therefore, based on Thales theorem, we have
BA BH ¼ CA CF
Table 2 Efficiency scores for the CCR and proposed models. DMUs
i ¼ 1; 2; . . . ; m
kj P 0 j ¼ 1; 2; . . . ; n
ðCF ¼ 1Þ
ð12Þ
And thus,
CCR model
The proposed model
h⁄
a⁄
1 a⁄
0.711111 1 1 1 0.988488 0.893962
0.288889 0 0 0 0.011512 0.106038
0.711111 1 1 1 0.988488 0.893962
Ranking
1 4 1 1 1 2 3
BA CA BA CB ¼ ¼ CA CA CA
ð13Þ
Obviously, the following is obtained from (11), (12) and (13):
BD ¼ 1 BH
or
h ¼ 1 a
ð14Þ
The membership functions associated with the discretionary input variable of DMUA and the almost discretionary input variable of DMUB are defined as follows:
Please cite this article in press as: M. Zerafat Angiz L, A. Mustafa, Fuzzy interpretation of efficiency in data envelopment analysis and its application in a non-discretionary model, Knowl. Based Syst. (2013), http://dx.doi.org/10.1016/j.knosys.2013.05.001
6
M. Zerafat Angiz L, A. Mustafa / Knowledge-Based Systems xxx (2013) xxx–xxx
Table 3 Result of CCR model in the presence of almost discretionary variable.
s:t:
n X kj xij 6 xip
i 2 ID
ð18:1Þ
i 2 IND
ð18:2Þ
i 2 IFND
ð18:3Þ
j¼1
DMU
Almost non-discretionary
1 2 3 4 5 6
n X kj xij 6 xip
x11
x12
a⁄
1 a⁄
1.066667 – – – 3.460165 2.839548
1.163793 – – – 1.186342 0.632889
0.2241378 0 0 0 0.01138147 0.0958722
0.7758622 1 1 1 0.9886185 0.9041278
j¼1 n X kj xij 6 xip j¼1 n X kj yrj P yrp
ð18:4Þ
j¼1
q6a6
l~x1 ðx1 Þ ¼
x1 x1 x1
x1 6 x1 and
6 x2
l~x2 ðx2 Þ ¼
x2 x2
x2 x2
x2 6 x2
q 6 bi 6 ð16Þ
The value x2 is the input level at point C, which is the projection of point B onto the efficient frontier. In fact, x2 ¼ h x2 where h⁄ is the optimal solution of the CCR model related to DMUB (i.e., Model (4) where all the inputs are treated as discretionary variables). The inclusion of the almost discretionary variables or the fuzzy non-discretionary (FND) variables into Model (5) resulted in the following multi-objective linear programming problem:
xip xip xip xip xlip xuip xlip
i 2 ID &IFND
ð18:5Þ
i 2 IFND
ð18:6Þ
xlip 6 xip 6 xuip
ð18:7Þ
kj P 0 j ¼ 1; 2; . . . ; n aP0
qP0 bi P 0 i 2 IFND 5. A numerical example
max
i 2 IFND
bi
max a n X s:t: kj xij 6 xip
ð17Þ
i 2 ID
ð17:1Þ
i 2 IND
ð17:2Þ
i 2 IFND
ð17:3Þ
j¼1 n X kj xij 6 xip j¼1 n X kj xij 6 xip j¼1 n X kj yrj P yrp
ð17:4Þ
j¼1
a6 bi 6
xip xip xip xip xlip xuip xlip
i 2 ID &IFND
ð17:5Þ
i 2 IFND
ð17:6Þ
xlip 6 xip 6 xuip
ð17:7Þ
kj P 0 j ¼ 1; 2; . . . ; n
aP0 bi P 0 i 2 IFND
max q
The above model can be solved using a two-stage method. In the first stage, all variables are treated as discretionary variables and the traditional CCR model (Model (4)) or the newly proposed fuzzy equivalent model (Model (9)) is used to find the efficiency score of each DMU (i.e. hp Þ. These efficiency scores are then used to determine variables xlip of Eq. (17.6). Variables xlip are the projection of the almost discretionary variables xip onto the efficient frontier. They are determined using xlip ¼ hp xip ¼ xip . An example of such variable is the value x2 in Fig. 2b. In the second stage, the multi-objective linear programming model (17) is transformed into a linear programming model as follows and solved.
max
q
In Table 1 a list of six DMUs with measurements of two inputs and two outputs is given. A ranking of these DMUs based on their efficiency scores need to be done. The efficiency scores and ranking of the DMUs based on the CCR and the proposed model are given in Table 2. Note that the results are the same. Even though the advantages of the proposed fuzzy-based model over the traditional CCR model are not that apparent, the correspondence between fuzzy concept and technical efficiency could be explored further. In the following example, such exploration is done while dealing with a non-discretionary model. For an illustration of the proposed non-discretionary model, the same example is considered again. Assume that the second input is an almost discretionary variable. In the first stage of the proposed method where all variables are treated as discretionary variables and the traditional CCR model or the newly proposed fuzzy equivalent model is used to find the efficiency score of each DMU, the result is shown in Table 2. From the result, the values for variables xlip of Eq. (17.6) are determined. As an example, for DMU 5 ; xl2 ¼ hx2 ¼ 0:988488 1:2 ¼ 1:186185. Thus, the linear programming model (Model 18) for finding the efficiency of DMU5 in stage 2 is as follows:
ð18Þ
ð19Þ
s:t: 1:50k1 þ 4:00k2 þ 3:20k3 þ 5:20k4 þ 3:50k5 þ 3:20k6 6 x51 1:50k1 þ 0:70k2 þ 1:20k3 þ 2:00k4 þ 1:20k5 þ 0:70k6 6 x52 1:40k1 þ 1:40k2 þ 4:20k3 þ 2:80k4 þ 1:90k5 þ 1:40k6 6 1:90 0:35k1 þ 2:10k2 þ 1:05k3 þ 4:20k4 þ 2:50k5 þ 1:50k6 6 2:50 3:50 x51 q6 3:50 1:20 x52 q6 1:20 x52 1:186185 q6 1:20 1:186185 0 6 x51 6 3:50 0 6 x52 6 1:20
Please cite this article in press as: M. Zerafat Angiz L, A. Mustafa, Fuzzy interpretation of efficiency in data envelopment analysis and its application in a non-discretionary model, Knowl. Based Syst. (2013), http://dx.doi.org/10.1016/j.knosys.2013.05.001
M. Zerafat Angiz L, A. Mustafa / Knowledge-Based Systems xxx (2013) xxx–xxx
The overall result of the efficiency analysis when input 2 is almost discretionary is shown in Table 3. Note that there are some suggestions for improvement of both the input variables. As an example, for DMU5, reducing the input variable 1 from 3.50 to x11 ¼ 3:460165 and the input variable 2 from 1.20 to x12 ¼ 1:186342 is suggested. The efficiency score of the proposed non-discretionary model is 0.9886185 and it is larger than the efficiency score of the CCR model at 0.9884877. 6. Conclusion The uses of fuzzy concepts in handling certain crisp mathematical modelling situations have resulted in the formulation of creative and efficient procedures. This paper describes another use of the fuzzy concept in a crisp situation. A new DEA model equivalent to the traditional CCR model and its application in handling nondiscretionary data is presented. Membership functions are used to represent certain characteristics of the input data. For the newly proposed non-discretionary model, the usage of membership function replaces the need to determine the discretionary index of a non-discretionary variable. The discretionary index concepts are used in some of the existing non-discretionary models. In real life applications, discretionary indexes are usually not known and are arbitrarily determined by decision makers. This paper offers opportunities for studying DEA models involving input and output data with various characteristics. With the usage of the appropriate membership functions to represent these characteristics, suitable mathematical models could be derived for them. References [1] N. Adler, L. Friedman, Z. Sinuany-Stern, Review of ranking methods in the data envelopment analysis context, European Journal of Operational Research 140 (2002) 249–265. [2] R.D. Banker, A. Charnes, W.W. Cooper, Some methods for estimating technical and scale inefficiencies in data envelopment analysis, Management Science 30 (1984) 1078–1092. [3] R.D. Banker, A. Charnes, W.W. Cooper, R. Clark, Constrained game formulations and interpretations for data envelopment analysis, European Journal of Operational Research 40 (1989) 299–308. [4] R.D. Banker, R. Morey, Efficiency analysis for exogenously fixed inputs and outputs, Operations Research 34 (1986) 513–521. [5] A. Charnes, W.W. Cooper, E. Rhodes, Measuring the efficiency of decision making units, European Journal of Operational Research 2 (1978) 429–444. [6] J.M. Cordero-Ferrera, F. Pedraja-Chaparro, D. Santın-Gonz’alez, Enhancing the inclusion of non-discretionary inputs in DEA, Journal of the Operational Research Society 61 (2010) 574–584.
7
[7] A. Emrouznejad, L.M. Zerafat Angiz, W. Ho, Assignment problem with fuzzy data, Journal of Operations Research Society 63 (2011) 59–63. [8] B. Golany, Y. Roll, Some extensions of techniques to handle non-discretionary factors in data envelopment analysis, Journal of Productivity Analysis 4 (1993) 419–432. [9] P. Guo, H. Tanaka, Fuzzy DEA: a perceptual evaluation method, Fuzzy Sets and Systems 19 (2001) 149–160. [10] C. Kahraman, E. Tolga, Data envelopment analysis using fuzzy concept, in: Proceedings of The International Symposium on Multiple-Valued Logic, Fukuoka, Japan, IEEE Comp. Soc., Los Alamitos, CA, USA, 1998, pp. 338–342. [11] C. Kao, S.T. Liu, Fuzzy efficiency measures in data envelopment analysis, Fuzzy Sets and Systems 113 (2000) 427–437. [12] R. Joro, P. Korhonen, J. Wallenius, Structural comparison of data envelopment analysis and multiple objective linear programming, Management Science 44 (1998) 962–970. [13] S. Lertworasirikul, F. Shu-Cherng, J.A. Joines, H.L.W. Nuttle, Fuzzy data envelopment analysis (DEA): a possibility approach, Fuzzy Sets and Systems 139 (2003) 379–394. [14] M.A. Muñiz, Separating managerial inefficiency and external conditions in data envelopment analysis, European Journal of Operational Research 143 (2002) 625–643. [15] M.A. Muñiz, J. Paradi, J. Ruggiero, Z. Yang, Evaluating alternative DEA models used to control for non-discretionary inputs, Computers & Operations Research 33 (2006) 1173–1183. [16] J. Ruggiero, On the measurement of technical efficiency in the public sector, European Journal of Operational Research 90 (1996) 553–565. [17] J. Ruggiero, Non-discretionary inputs in data envelopment analysis, European Journal of Operational Research) 111 (1998) 461–469. [18] J.K. Sengupta, Measuring efficiency by a fuzzy statistical approach, Fuzzy Sets and Systems 46 (1992) 73–80. [19] L. Simar, P.W. Wilson, Estimation and inference in two-stage, semi-parametric models of production processes, Journal of Econometrics 136 (2007) 31–64. [20] E. Thanassoulis, R. Allen, Simulating weights restrictions in data envelopment analysis by means of unobserved DMUs, Management Science 44 (1998) 586– 594. [21] H. Wong, J. Beasley, Restricting weight flexibility in data envelopment analysis, Journal of Operational Research Society 41 (1990) 829–835. [22] C.H. Yeh, Y.H. Chang, Modeling subjective evaluation for fuzzy group multicriteria decision making, European Journal of Operational Research 194 (2009) 464–473. [23] J.B. Yang, B.Y.H. Wong, D.L. Xu, T.J. Stewart, Integrating DEA-oriented performance assessment and target setting using interactive MOLP methods, European Journal of Operations Research 195 (2009) 205–222. [24] M.L. Zerafat Angiz, A. Emrouznejad, A. Mustafa, A.S. Al-Eraqi, Aggregating preference ranking with fuzzy data envelopment analysis, Knowledge-Based Systems 23 (2010) 512–519. [25] M.L. Zerafat Angiz, A. Emrouznejad, A. Mustafa, Fuzzy assessment of performance of a decision making units using DEA: a non-radial approach, Expert Systems with Applications 37 (2010) 5153–5157. [26] M.L. Zerafat Angiz, A. Emrouznejad, A. Mustafa, Ranking efficient decisionmaking units in data envelopment analysis using fuzzy concept, Computers and Industrial Engineering 59 (2010) 712–719. [27] M.L. Zerafat Angiz, A. Emrouznejad, A. Mustafa, Fuzzy data envelopment analysis: a discrete approach, Expert Systems with Applications 39 (2012) 2263–2269.
Please cite this article in press as: M. Zerafat Angiz L, A. Mustafa, Fuzzy interpretation of efficiency in data envelopment analysis and its application in a non-discretionary model, Knowl. Based Syst. (2013), http://dx.doi.org/10.1016/j.knosys.2013.05.001
Contents lists available at SciVerse ScienceDirect
Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys
Fuzzy interpretation of efficiency in data envelopment analysis and its application in a non-discretionary model Majid Zerafat Angiz L a,⇑, Adli Mustafa b a b
School of Quantitative Sciences, Universiti Utara Malaysia, Sintok, Kedah, Malaysia School of Mathematical Sciences, Universiti Sains Malaysia, Penang, Malaysia
a r t i c l e
i n f o
Article history: Received 17 October 2011 Received in revised form 29 April 2013 Accepted 7 May 2013 Available online xxxx Keywords: Data envelopment analysis Fuzzy numbers Ranking
a b s t r a c t Data envelopment analysis (DEA) is a nonparametric model which evaluates the relative efficiencies of decision-making units (DMUs). These DMUs produce multiple outputs by using multiple inputs and the relative efficiency is evaluated using a ratio of total weighted output to total weighted input. In this paper an alternative interpretation of efficiency is first given. The interpretation is based on the fuzzy concept even though the inputs and outputs data are crisp numbers. With the interpretation, a new model for ranking DMUs in DEA is proposed and a new perspective of viewing other DEA models is now made possible. The model is then extended to incorporate situations whereby some inputs or outputs, in a fuzzy sense, are almost discretionary variables. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction Data envelopment analysis (DEA), proposed by Charnes et al. [5] (CCR model) is a nonparametric model for identifying the efficient frontier and measuring the relative efficiency of decision making units (DMUs). Based on the CCR model, Banker et al. [2] (BCC model) developed a variable returns to scale variation of the model. Both the CCR and BCC models have undergone various theoretical extensions and also many successful applications [1]. Nowadays, DEA has been applied to a variety of research areas. Most of the performance evaluation models presented in the literature on DEA are divided into two categories. In the first category, the methods focused on the production possibility set (PPS) to provide models for special purposes. The super efficiency models and models based on return to scale are in this category. In the second category, the methods focused on projecting an inefficient DMU onto the efficient frontier using different measures. For instance, the efficient DMU on the efficiency frontier corresponding to an inefficient DMU under evaluation in the CCR model and the additive model are different. Therefore, two different optimal solutions are obtained. This paper falls into both categories as it provides an alternative interpretation of the CCR model using fuzzy concept. It will be demonstrated that the relative efficiency is equivalent to the membership function of a fuzzy number. Based on this new interpretation, a new DEA model equivalent to the CCR model is
⇑ Corresponding author. Tel.: +60 017 3509310; fax: +60 04 6570910. E-mail addresses: [email protected] (M. Zerafat Angiz L), [email protected] (A. Mustafa).
proposed. The new concept is then extended further to describe its application in a non-discretionary model. The CCR model only handles crisp data. When data are fuzzy, fuzzy DEA models are used instead. The existing approaches for solving fuzzy DEA are usually categorised in four categories; (a) the tolerance approaches [10,18], (b) the fuzzy ranking approaches [9], (c) the a-level based approaches [11] and (d) the defuzzification approaches [13]. Zerafat Angiz et al. [27] proposed two different methods basis on multi-objective programming. In the nonradial approach [27], attempt has been made to solve the fuzzy DEA model by retaining the whole information throughout the evaluation process as far as possible. In the discrete approach [25], the a-cut concept is extended. The model that is proposed in this paper however, is not about a fuzzy DEA model. It is about the use of fuzzy concept in handling crisp data. The uses of fuzzy concepts in handling certain crisp mathematical modelling situations have resulted in the formulation of creative and efficient procedures. This can be seen for example in the work of Zerafat Angiz et al. [26] for ranking efficient DMUs. In the literature, a few studies on the relationship between DEA and other optimisation models have been done. Thanassoulis and Allen [20] showed the equivalence between the CCR model and some linear value function model for multiple outputs and multiple inputs. Then, Joro et al. [12] proved the structural correspondences between DEA models and multiple objective linear programming using an achievement scalarising function that was proposed by Wong and Beasley [21]. Furthermore, Yang et al. [23] presented a Multi Objective Linear Programming (MOLP) formulation based on min–max reference point approach in order to solve output-oriented DEA models. Banker et al. [3] studied the
0950-7051/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.knosys.2013.05.001
Please cite this article in press as: M. Zerafat Angiz L, A. Mustafa, Fuzzy interpretation of efficiency in data envelopment analysis and its application in a non-discretionary model, Knowl. Based Syst. (2013), http://dx.doi.org/10.1016/j.knosys.2013.05.001
2
M. Zerafat Angiz L, A. Mustafa / Knowledge-Based Systems xxx (2013) xxx–xxx
relationship between DEA and game theory. Besides that, some authors utilised the fuzzy concept to handle difficulties that arise in some procedures, even though they were not dealing with fuzzy numbers. For example, using fuzzy concept, Yeh and Chang [22] aggregated individual judgements as a group judgements and then applied it in a new approach for evaluating alternatives in a multicriteria decision making (MCDM) environment. Zerafat Angiz et al. [24–26] used a fuzzy structure to select the best alternative in a group decision making environment and Emrouznejad et al. [7] used fuzzy DEA in an assignment problem. None of the previous studies however compares DEA with fuzzy concept. The remainder of this paper is organised as follows: In Section 2, some background information about fuzzy numbers, the DEA model and the non-discretionary DEA model are given. A new interpretation of technical efficiency based on fuzzy approach and a new DEA model equivalent to the CCR model are elaborated in Section 3. A new non-discretionary DEA model is described in Section 4. In Section 5, a numerical example is given. Finally, the conclusion is presented in Section 6. 2. Background 2.1. Fuzzy number
(b) There is no DMU in J whose data domain can be proportionally expressed by that of another DMU.
Definition 2. Given the (empirical) points (Xj, Yj), j = 1, 2, . . . ,n, the Production Possibility Set (PPS) is defined as follows:
T ¼ fðX t ; Y t Þjoutput Y t can be produced by input X t g The production possibility set corresponding to constant return to scale will be as follows:
Tc ¼
8 > < > :
n n X X ðX t ; Y t ÞjX t ¼ kj xij ; Y t ¼ kj yrj ; kj P 0; j¼1
j¼1
9 j ¼ 1; 2; . . . ; n > = B @ i ¼ 1; 2; . . . ; m > ; r ¼ 1; 2; . . . ; s 0
ð2Þ
The set Tc is the production possibility set corresponding to constant returns to scale, and kj is a nonnegative value related to the jth DMU. The vector k = (k1, k2, . . . , kn)t is used to construct a hull that covers all the data points. Constant Return to Scale (CRS) means that an increase in the amount of input consumed leads to a proportional increase in the amount of output produced. If this increase is culminated in larger or smaller than proportional increase in the amount of outputs, return to scale will be Increasing Return to Scale (IRS) or Decreasing Return to Scale (DRS), respectively.
Definition 1. A fuzzy number ~ x is of LR-type if there exists reference functions L (for left), R (for right), and a > 0, b > 0 such that
To evaluate efficiency corresponding to set Tc, consider the following model.
8 < L xxl for xl 6 x 6 xm m xl x l~x ðxÞ ¼ : R xu x for xm 6 x 6 xu xu xm
hp ¼ min hp
m
ð1Þ m
l
x , called the mean value of ~x, is a real number; a = x x and b = xu xm are called the left and right spreads, respectively; and xl and xu are the lower bound and the upper bound of the interval of fuzzy number. Symbolically, ~x is denoted by (xm, a, b)LR or (m, a, b)LR. If ~x is a symmetric triangular fuzzy number, then we have R(x) = L(x) = x.
ð3Þ
s:t: ðhp X p ; Y p Þ 2 T
The variable h is the DEA efficiency score of CCR model. The CCR model (input-oriented) for evaluating the efficiency of DMUp, is written as follows:
hp ¼ min hp n X s:t: kj xij 6 hp xip i ¼ 1; 2; . . . ; m j¼1
2.2. Data envelopment analysis
n X kj yrj P yrp
r ¼ 1; 2; . . . ; s
ð4Þ
j¼1
Data envelopment analysis evaluates the relative efficiency of a set of homogenous DMUs using a ratio of total weighted output to total weighted input. It generalises the usual efficiency measurement from a single-input, single-output ratio to a multiple-input, multiple-output ratio. The DEA approach applies linear programming techniques to observe inputs consumed and outputs produced by decision making units and then constructs an efficient production frontier based on best practices. Each DMU’s efficiency is then measured relative to this frontier. For a clearer description of the DEA models, the following general assumptions are specified: (a) There are n DMUs denoted by j 2 J, each of which produces a nonzero output vector Yj = (y1j, y2j, . . . , ysj)t P 0 using a nonzero input vector Xj = (x1j, x2j, . . . , xmj)t P 0, where the superscript 0 t0 indicates the transpose of a vector. Here, the symbol 0 P 0 indicates that at least one component of Xj or Yj is positive while the remaining X 0j s or Y 0j s are nonnegative.
kj P 0 j ¼ 1; 2; . . . ; n hp free 2.3. Non-discretionary DEA model One of the significant concepts in data envelopment analysis is the non-discretionary variables. An input or output is called a nondiscretionary variable if it cannot be varied at the discretion of management or other users. For example, as stated by Charnes et al. [5] ‘‘In evaluating the performances of different bases for the Fighter Command of U.S. Air Forces, for instance, it was necessary to consider weather as an input since the number of ‘‘sorties’’ (successfully completed missions) and ‘‘aborts’’ (non-completed missions) treated as output, could be affected by the weather (measured in degree days and numbers of ‘‘flyable’’ days) at different bases’’. Banker and Morey [4] were pioneers in this study by including non-discretionary variables in the input-oriented DEA model. The
Please cite this article in press as: M. Zerafat Angiz L, A. Mustafa, Fuzzy interpretation of efficiency in data envelopment analysis and its application in a non-discretionary model, Knowl. Based Syst. (2013), http://dx.doi.org/10.1016/j.knosys.2013.05.001
3
M. Zerafat Angiz L, A. Mustafa / Knowledge-Based Systems xxx (2013) xxx–xxx
Banker and Morey model, considering constant return to scale, is given by the following mathematical programming:
min hp n X s:t: kj xij þ si ¼ hxip i 2 D
/p ¼ i 2 ND
xip xip xip
xip 6 xip
i ¼ 1; 2; . . . ; m
ð6Þ
With the new definition of the variables, Model (3) is converted to the following model:
j¼1 n X kj xij þ si ¼ xip
l~xip ðxip Þ ¼
ð5Þ
maxfminfl~xip ðxip Þgði ¼ 1; 2; . . . ; mÞg
s:t: ðxip ; yrp Þ 2 T
ð7Þ
j¼1 n X kj yrj sþi ¼ yrp
j ¼ 1; 2; . . . ; s
A new linear programming model corresponding to the CCR model is initially proposed as follows:
j¼1
kj P 0 j ¼ 1; 2; . . . ; n þ In Model (5), s i and si are the slack variables for the ith input and rth output, respectively, and the symbols D and ND refer to discretionary and non-discretionary variables, respectively. Muniz et al. [15] reviewed some non-discretionary approaches and compared the relative performance of each by simulation analysis. Ruggiero [16] provided an alternative model by relaxing the convexity constraint with respect to the non-discretionary variables and used simulation analysis to highlight the advantages of his model over the Banker and Morey’s model. A weakness of Ruggiero’s model was the tendency to identify decision-making units (DMUs) as efficient by default as the number of exogenous factors increased. Ruggiero [17] extended this model to allow multiple non-discretionary input. The model requires three stages and uses regression analysis to develop an aggregate index capturing the influence of the non-discretionary factors (see [15]). Muñiz [14] provided an alternative three-stage model to control for the effect that non-controllable inputs have on production. Like the first stage in Ruggiero [17], this model considers only discretionary input and output in the first stage. This model, however, focuses not only on the radial measure of efficiency but also the remaining slacks that remain after equi-proportionate projection to the overall frontier. Moreover, by using only DEA methods, it has the advantage that there is no need to assume any functional form in any of the stages (see [15]). Cordero-Ferrera et al. [6] extend the Simar and Wilson [19] model which is a four-stage model based on the bootstrap procedure, in order to estimate unbiased efficiency scores and to set realistic targets that take into account information concerning non-discretionary variables. Golany and Roll [8] pointed out that in many real-life applications of efficiency studies, a factor is neither fully controllable nor totally uncontrollable. For example, managers can make marginal alterations in personnel scheduling. However they have to comply with general guidelines of their organisation in many other aspects involving the use of their human resources. In other words, the factor is partially controllable. To incorporate this factor into a DEA model, an index taking on the values between 0 and 1 is used to represent the degree of discretion that the DMU has on the factor. In this paper, such factor will be called an almost discretionary variable or fuzzy non-discretionary (FND) variable will also be used. More discussion on this is presented in Section 4.
max min model maxfminfl~x ðxip Þgði ¼ 1; 2; . . . ; mÞg ip
n X kj xij 6 xip i ¼ 1; 2; . . . ; m s:t: j¼1 n X kj yrj P yrp
ð8Þ j ¼ 1; 2; . . . ; s
j¼1
0 6 xip 6 xip
i ¼ 1; 2; . . . ; m
kj P 0 j ¼ 1; 2; . . . ; n In the solution of the model, xip indicates the input levels which makes DMUp efficient. Since the right hand side of the constraints in the Model (8) has remained unchanged, both the CCR and the proposed model have the same production possibility set. Obviously, Model (8) is always feasible and the optimum value of the objective function is non-negative and less than or equal to 1. Since the objective function is the membership function, the optimal value does not exceed the maximum value of the membership values, i.e. a value of 1. The values kp = 1, kj = 0 (j – p) and xip ¼ xip are the feasible solutions in Model (8), thus this is another indication that the model is always feasible. If we assume that a ¼ minfl~xip ð xip Þg, then the following model is obtained:
max
a
ð9Þ
3. A new version of the traditional CCR model Assume that for the new DEA model to be proposed, all postulations to construct the production possibility set corresponding to a constant return to scale model are satisfied. Therefore, the production possibility sets of the CCR model and the proposed model are similar. To present the new model, we introduce a fuzzy number with the following membership function. Assume that xip and yrp are inputs and outputs of the DMU under evaluation, say DMUp. Then, the membership function of the fuzzy number ~ xip is considered as follows:
Fig. 1. A fuzzy view of the CCR model.
Please cite this article in press as: M. Zerafat Angiz L, A. Mustafa, Fuzzy interpretation of efficiency in data envelopment analysis and its application in a non-discretionary model, Knowl. Based Syst. (2013), http://dx.doi.org/10.1016/j.knosys.2013.05.001
4
M. Zerafat Angiz L, A. Mustafa / Knowledge-Based Systems xxx (2013) xxx–xxx
b
a
Fig. 2. (a) Discretionary input. (b) Almost discretionary input.
s:t:
n X kj xij 6 xip
i ¼ 1; 2; . . . ; m
ð9:1Þ
r ¼ 1; 2; . . . ; s
ð9:2Þ
j¼1 n X kj yrj P yrp
PH HC ¼ AB BC Thus:
j¼1
x x a 6 ip ip i ¼ 1; 2; . . . ; m xip xip 6 xip i ¼ 1; 2; . . . ; m
ð9:3Þ ð9:4Þ
kj P 0 j ¼ 1; 2; . . . ; n
aP0
1
HC BC HC BH ¼ ¼ BC BC BC
a
ð11Þ
BH is the technical efficiency score of DMUc. Since AB = 1, the followBC ing expression is obtained from (10) and (11):
Efficiency of DMU C ¼
This model can now be solved for each DMU. For a better understanding of Models (8) and (9), we refer to Fig. 1 which provides a two-dimensional diagram of a simple efficiency case study involving 3 DMUs (D, C and Z) in which only a single input (x) is used to produce a single output (y). Based on the output to input ratio, DMUZ is efficient and the line from the origin passing through point Z is the efficient frontier. Triangles ABC and DEF are the membership functions of the fuzzy numbers which correspond to the inputs for DMUC and DMUD, respectively. These membership functions are super imposed onto the graph. Notice that, the shape of the membership functions reflex the characteristic of the input variables in DEA (i.e., the smaller the level of input the more efficient is the DMU). Based on Thales theorem (basic proportionality theorem), in triangle ABC (Fig. 1), PH is parallel to AB. Therefore, triangle ABC is similar to triangle PHC, so:
ð10Þ
BH ¼ 1 PH BC
ð12Þ
As observed, PH is the membership value of the triangular fuzzy number corresponding to the input of DMUC at point H, which is on the efficient frontier. The situation indicates the relationship between technical efficiency and membership function of a fuzzy number. Similarly, the length of segment LK indicates the efficiency score of DMUD. Since the length of segment PH is less than the length of segment LK, we can conclude that the efficiency score of DMUC is more than efficiency score of DMUD. The main idea that needs to be pointed out here is that there exists a relationship between the efficiency scores that were calculated using the CCR model in Model (4) and the membership values of the fuzzy number ~ xip . The smaller the membership value, the higher is the efficiency score of the DMU under evaluation. The following theorem indicates equivalency between the efficiency scores of Models (4) and (9).
b
Fig. 3. (a) A fuzzy number corresponding to DMUB. (b) Another fuzzy number corresponding to DMUB.
Please cite this article in press as: M. Zerafat Angiz L, A. Mustafa, Fuzzy interpretation of efficiency in data envelopment analysis and its application in a non-discretionary model, Knowl. Based Syst. (2013), http://dx.doi.org/10.1016/j.knosys.2013.05.001
5
M. Zerafat Angiz L, A. Mustafa / Knowledge-Based Systems xxx (2013) xxx–xxx
xip 6 xip
Table 1 A list of DMUs with two inputs and two outputs. DMUs
I1
I2
O1
O2
1 2 3 4 5 6
1.50 4.00 3.20 5.20 3.50 3.20
1.50 0.70 1.20 2.00 1.20 0.70
1.40 1.40 4.20 2.80 1.90 1.40
0.35 2.10 1.05 4.20 2.50 1.50
Theorem 1. Assume that h⁄ = min h and a⁄ = max a are the optimal values of Models (4) and (9), respectively. Then, h⁄ = 1 a⁄. In other words, the treatment of Model (9) and the CCR model are similar. Proof. In Model (9), if we consider a 6 so
xip xip , xip
then axip 6 xip xip ,
xip 6 xip axip ¼ ð1 aÞxip
ð13Þ
There is a one-to-one correspondence between the values xip and (1 a)xip, and therefore we can replace Eq. (9.1) by taking into consideration Eq. (13). This resulted in the following linear programming model:
a
max s:t:
ð14Þ
n X kj xij 6 xip 6 ð1 aÞxip
i ¼ 1; 2; . . . ; m
ð14:1Þ
j¼1 n X kj yrj P yrp
r ¼ 1; 2; . . . ; s
ð14:2Þ
i ¼ 1; 2; . . . ; m
ð14:3Þ
j¼1
a6 xip 6 xip
xip xip xip
i ¼ 1; 2; . . . ; m
ð14:4Þ
kj P 0 j ¼ 1; 2; . . . ; n
aP0 Assume that h = 1 a. Then, Model (14) is converted to the following model:
max s:t:
1h
ð15Þ
n X kj xij 6 xip 6 hxip
i ¼ 1; 2; . . . ; m
ð15:1Þ
j¼1 n X kj yrj P yrp
r ¼ 1; 2; . . . ; s
ð15:2Þ
j¼1
1h6
xip xip xip
i ¼ 1; 2; . . . ; m
ð15:3Þ
1 2 3 4 5 6
ð15:4Þ
hP0 x x
In Eq. (15.3), 1 h 6 ipxip ip implies that xip 6 hxip , thus, Eq. (15.3) is x x implied by Eq. (15.1). Since 0 6 ipx ip 6 1, so 0 6 h 6 1 is obtained ip from Eq. (15.3). On the other hand, by considering max (1 h) = 1 min h, Model (5) is resulted. This means that h⁄ = 1 a⁄. By performing some substitutions, Model (4) implies Model (9), analogously. In other words, Model (9) represents a new version of the CCR model. The efficiency score in this model is the membership value of the point located at the intersection of the fuzzy interval (segment BC in Fig. 1) and the efficiency frontier. The example given in the Section 5 demonstrates the correspondence between the proposed model and the CCR model. h
4. A new non-discretionary DEA model In this section, an application of the fuzzy concept that has been discussed in the previous section is presented. Fig. 2a and b illustrate two separate DEA studies with a single input and a single output, in which the input variable corresponding to A is discretionary and the input variable corresponding to B is almost discretionary. The appropriate membership functions are superimposed onto the graphs. Notice that, for the membership function of the discretionary variable, more weight is given for the smaller values, reflecting a characteristic of the input variable in a traditional DEA model, which is, the smaller the level of input the more efficient is the DMU. However, for the membership function of the almost discretionary variable, more weight is given for the measured/current value to reflect the reluctance on the part of the DMU to reduce the value. Notice that, the membership value corresponding to the nondiscretionary variable is a modified form of the membership value in Fig. 3. It is easy to show the relationship between h⁄ and a⁄. For a better understanding of the relationship between h⁄ and a⁄, we refer to Fig. 3a and b. Triangles CAE and CAF are the membership functions of the fuzzy numbers which correspond to the input of DMUA. In referring to Fig. 3, BD is parallel to AE, therefore the two triangles CEA and CDB are similar. So, we have
CB BD ¼ CA AE
ð11Þ
CB It is well known that the value CA is equivalent to the technical efficiency of DMUA. Thus, with AE ¼ 1; BD ¼ h is the technical efficiency AE of DMUA which is also the membership value corresponding to point B, the reference point of DMUA on the efficient frontier. Furthermore we consider Fig. 3b in which triangle CFA is similar to triangle BHA. Therefore, based on Thales theorem, we have
BA BH ¼ CA CF
Table 2 Efficiency scores for the CCR and proposed models. DMUs
i ¼ 1; 2; . . . ; m
kj P 0 j ¼ 1; 2; . . . ; n
ðCF ¼ 1Þ
ð12Þ
And thus,
CCR model
The proposed model
h⁄
a⁄
1 a⁄
0.711111 1 1 1 0.988488 0.893962
0.288889 0 0 0 0.011512 0.106038
0.711111 1 1 1 0.988488 0.893962
Ranking
1 4 1 1 1 2 3
BA CA BA CB ¼ ¼ CA CA CA
ð13Þ
Obviously, the following is obtained from (11), (12) and (13):
BD ¼ 1 BH
or
h ¼ 1 a
ð14Þ
The membership functions associated with the discretionary input variable of DMUA and the almost discretionary input variable of DMUB are defined as follows:
Please cite this article in press as: M. Zerafat Angiz L, A. Mustafa, Fuzzy interpretation of efficiency in data envelopment analysis and its application in a non-discretionary model, Knowl. Based Syst. (2013), http://dx.doi.org/10.1016/j.knosys.2013.05.001
6
M. Zerafat Angiz L, A. Mustafa / Knowledge-Based Systems xxx (2013) xxx–xxx
Table 3 Result of CCR model in the presence of almost discretionary variable.
s:t:
n X kj xij 6 xip
i 2 ID
ð18:1Þ
i 2 IND
ð18:2Þ
i 2 IFND
ð18:3Þ
j¼1
DMU
Almost non-discretionary
1 2 3 4 5 6
n X kj xij 6 xip
x11
x12
a⁄
1 a⁄
1.066667 – – – 3.460165 2.839548
1.163793 – – – 1.186342 0.632889
0.2241378 0 0 0 0.01138147 0.0958722
0.7758622 1 1 1 0.9886185 0.9041278
j¼1 n X kj xij 6 xip j¼1 n X kj yrj P yrp
ð18:4Þ
j¼1
q6a6
l~x1 ðx1 Þ ¼
x1 x1 x1
x1 6 x1 and
6 x2
l~x2 ðx2 Þ ¼
x2 x2
x2 x2
x2 6 x2
q 6 bi 6 ð16Þ
The value x2 is the input level at point C, which is the projection of point B onto the efficient frontier. In fact, x2 ¼ h x2 where h⁄ is the optimal solution of the CCR model related to DMUB (i.e., Model (4) where all the inputs are treated as discretionary variables). The inclusion of the almost discretionary variables or the fuzzy non-discretionary (FND) variables into Model (5) resulted in the following multi-objective linear programming problem:
xip xip xip xip xlip xuip xlip
i 2 ID &IFND
ð18:5Þ
i 2 IFND
ð18:6Þ
xlip 6 xip 6 xuip
ð18:7Þ
kj P 0 j ¼ 1; 2; . . . ; n aP0
qP0 bi P 0 i 2 IFND 5. A numerical example
max
i 2 IFND
bi
max a n X s:t: kj xij 6 xip
ð17Þ
i 2 ID
ð17:1Þ
i 2 IND
ð17:2Þ
i 2 IFND
ð17:3Þ
j¼1 n X kj xij 6 xip j¼1 n X kj xij 6 xip j¼1 n X kj yrj P yrp
ð17:4Þ
j¼1
a6 bi 6
xip xip xip xip xlip xuip xlip
i 2 ID &IFND
ð17:5Þ
i 2 IFND
ð17:6Þ
xlip 6 xip 6 xuip
ð17:7Þ
kj P 0 j ¼ 1; 2; . . . ; n
aP0 bi P 0 i 2 IFND
max q
The above model can be solved using a two-stage method. In the first stage, all variables are treated as discretionary variables and the traditional CCR model (Model (4)) or the newly proposed fuzzy equivalent model (Model (9)) is used to find the efficiency score of each DMU (i.e. hp Þ. These efficiency scores are then used to determine variables xlip of Eq. (17.6). Variables xlip are the projection of the almost discretionary variables xip onto the efficient frontier. They are determined using xlip ¼ hp xip ¼ xip . An example of such variable is the value x2 in Fig. 2b. In the second stage, the multi-objective linear programming model (17) is transformed into a linear programming model as follows and solved.
max
q
In Table 1 a list of six DMUs with measurements of two inputs and two outputs is given. A ranking of these DMUs based on their efficiency scores need to be done. The efficiency scores and ranking of the DMUs based on the CCR and the proposed model are given in Table 2. Note that the results are the same. Even though the advantages of the proposed fuzzy-based model over the traditional CCR model are not that apparent, the correspondence between fuzzy concept and technical efficiency could be explored further. In the following example, such exploration is done while dealing with a non-discretionary model. For an illustration of the proposed non-discretionary model, the same example is considered again. Assume that the second input is an almost discretionary variable. In the first stage of the proposed method where all variables are treated as discretionary variables and the traditional CCR model or the newly proposed fuzzy equivalent model is used to find the efficiency score of each DMU, the result is shown in Table 2. From the result, the values for variables xlip of Eq. (17.6) are determined. As an example, for DMU 5 ; xl2 ¼ hx2 ¼ 0:988488 1:2 ¼ 1:186185. Thus, the linear programming model (Model 18) for finding the efficiency of DMU5 in stage 2 is as follows:
ð18Þ
ð19Þ
s:t: 1:50k1 þ 4:00k2 þ 3:20k3 þ 5:20k4 þ 3:50k5 þ 3:20k6 6 x51 1:50k1 þ 0:70k2 þ 1:20k3 þ 2:00k4 þ 1:20k5 þ 0:70k6 6 x52 1:40k1 þ 1:40k2 þ 4:20k3 þ 2:80k4 þ 1:90k5 þ 1:40k6 6 1:90 0:35k1 þ 2:10k2 þ 1:05k3 þ 4:20k4 þ 2:50k5 þ 1:50k6 6 2:50 3:50 x51 q6 3:50 1:20 x52 q6 1:20 x52 1:186185 q6 1:20 1:186185 0 6 x51 6 3:50 0 6 x52 6 1:20
Please cite this article in press as: M. Zerafat Angiz L, A. Mustafa, Fuzzy interpretation of efficiency in data envelopment analysis and its application in a non-discretionary model, Knowl. Based Syst. (2013), http://dx.doi.org/10.1016/j.knosys.2013.05.001
M. Zerafat Angiz L, A. Mustafa / Knowledge-Based Systems xxx (2013) xxx–xxx
The overall result of the efficiency analysis when input 2 is almost discretionary is shown in Table 3. Note that there are some suggestions for improvement of both the input variables. As an example, for DMU5, reducing the input variable 1 from 3.50 to x11 ¼ 3:460165 and the input variable 2 from 1.20 to x12 ¼ 1:186342 is suggested. The efficiency score of the proposed non-discretionary model is 0.9886185 and it is larger than the efficiency score of the CCR model at 0.9884877. 6. Conclusion The uses of fuzzy concepts in handling certain crisp mathematical modelling situations have resulted in the formulation of creative and efficient procedures. This paper describes another use of the fuzzy concept in a crisp situation. A new DEA model equivalent to the traditional CCR model and its application in handling nondiscretionary data is presented. Membership functions are used to represent certain characteristics of the input data. For the newly proposed non-discretionary model, the usage of membership function replaces the need to determine the discretionary index of a non-discretionary variable. The discretionary index concepts are used in some of the existing non-discretionary models. In real life applications, discretionary indexes are usually not known and are arbitrarily determined by decision makers. This paper offers opportunities for studying DEA models involving input and output data with various characteristics. With the usage of the appropriate membership functions to represent these characteristics, suitable mathematical models could be derived for them. References [1] N. Adler, L. Friedman, Z. Sinuany-Stern, Review of ranking methods in the data envelopment analysis context, European Journal of Operational Research 140 (2002) 249–265. [2] R.D. Banker, A. Charnes, W.W. Cooper, Some methods for estimating technical and scale inefficiencies in data envelopment analysis, Management Science 30 (1984) 1078–1092. [3] R.D. Banker, A. Charnes, W.W. Cooper, R. Clark, Constrained game formulations and interpretations for data envelopment analysis, European Journal of Operational Research 40 (1989) 299–308. [4] R.D. Banker, R. Morey, Efficiency analysis for exogenously fixed inputs and outputs, Operations Research 34 (1986) 513–521. [5] A. Charnes, W.W. Cooper, E. Rhodes, Measuring the efficiency of decision making units, European Journal of Operational Research 2 (1978) 429–444. [6] J.M. Cordero-Ferrera, F. Pedraja-Chaparro, D. Santın-Gonz’alez, Enhancing the inclusion of non-discretionary inputs in DEA, Journal of the Operational Research Society 61 (2010) 574–584.
7
[7] A. Emrouznejad, L.M. Zerafat Angiz, W. Ho, Assignment problem with fuzzy data, Journal of Operations Research Society 63 (2011) 59–63. [8] B. Golany, Y. Roll, Some extensions of techniques to handle non-discretionary factors in data envelopment analysis, Journal of Productivity Analysis 4 (1993) 419–432. [9] P. Guo, H. Tanaka, Fuzzy DEA: a perceptual evaluation method, Fuzzy Sets and Systems 19 (2001) 149–160. [10] C. Kahraman, E. Tolga, Data envelopment analysis using fuzzy concept, in: Proceedings of The International Symposium on Multiple-Valued Logic, Fukuoka, Japan, IEEE Comp. Soc., Los Alamitos, CA, USA, 1998, pp. 338–342. [11] C. Kao, S.T. Liu, Fuzzy efficiency measures in data envelopment analysis, Fuzzy Sets and Systems 113 (2000) 427–437. [12] R. Joro, P. Korhonen, J. Wallenius, Structural comparison of data envelopment analysis and multiple objective linear programming, Management Science 44 (1998) 962–970. [13] S. Lertworasirikul, F. Shu-Cherng, J.A. Joines, H.L.W. Nuttle, Fuzzy data envelopment analysis (DEA): a possibility approach, Fuzzy Sets and Systems 139 (2003) 379–394. [14] M.A. Muñiz, Separating managerial inefficiency and external conditions in data envelopment analysis, European Journal of Operational Research 143 (2002) 625–643. [15] M.A. Muñiz, J. Paradi, J. Ruggiero, Z. Yang, Evaluating alternative DEA models used to control for non-discretionary inputs, Computers & Operations Research 33 (2006) 1173–1183. [16] J. Ruggiero, On the measurement of technical efficiency in the public sector, European Journal of Operational Research 90 (1996) 553–565. [17] J. Ruggiero, Non-discretionary inputs in data envelopment analysis, European Journal of Operational Research) 111 (1998) 461–469. [18] J.K. Sengupta, Measuring efficiency by a fuzzy statistical approach, Fuzzy Sets and Systems 46 (1992) 73–80. [19] L. Simar, P.W. Wilson, Estimation and inference in two-stage, semi-parametric models of production processes, Journal of Econometrics 136 (2007) 31–64. [20] E. Thanassoulis, R. Allen, Simulating weights restrictions in data envelopment analysis by means of unobserved DMUs, Management Science 44 (1998) 586– 594. [21] H. Wong, J. Beasley, Restricting weight flexibility in data envelopment analysis, Journal of Operational Research Society 41 (1990) 829–835. [22] C.H. Yeh, Y.H. Chang, Modeling subjective evaluation for fuzzy group multicriteria decision making, European Journal of Operational Research 194 (2009) 464–473. [23] J.B. Yang, B.Y.H. Wong, D.L. Xu, T.J. Stewart, Integrating DEA-oriented performance assessment and target setting using interactive MOLP methods, European Journal of Operations Research 195 (2009) 205–222. [24] M.L. Zerafat Angiz, A. Emrouznejad, A. Mustafa, A.S. Al-Eraqi, Aggregating preference ranking with fuzzy data envelopment analysis, Knowledge-Based Systems 23 (2010) 512–519. [25] M.L. Zerafat Angiz, A. Emrouznejad, A. Mustafa, Fuzzy assessment of performance of a decision making units using DEA: a non-radial approach, Expert Systems with Applications 37 (2010) 5153–5157. [26] M.L. Zerafat Angiz, A. Emrouznejad, A. Mustafa, Ranking efficient decisionmaking units in data envelopment analysis using fuzzy concept, Computers and Industrial Engineering 59 (2010) 712–719. [27] M.L. Zerafat Angiz, A. Emrouznejad, A. Mustafa, Fuzzy data envelopment analysis: a discrete approach, Expert Systems with Applications 39 (2012) 2263–2269.
Please cite this article in press as: M. Zerafat Angiz L, A. Mustafa, Fuzzy interpretation of efficiency in data envelopment analysis and its application in a non-discretionary model, Knowl. Based Syst. (2013), http://dx.doi.org/10.1016/j.knosys.2013.05.001