CHAPTER 3 CONCEPT OF DATA ENVELOPMENT ANALYSIS: METHODS AND MATERIALS

Transcription

1 CHAPTER 3 CONCEPT OF DATA ENVELOPMENT ANALYSIS: METHODS AND MATERIALS Data envelopment analysis (DEA) is a linear programming based technique for measuring the relative performance of organizational units where the presence of multiple inputs and outputs makes comparisons difficult. This chapter reviews the literature on DEA mathematical models and their extensions that are germane to this research and the applications of this approach to various sectors including mining sector. This chapter provides the discussion on various DEA models available and their methodologies and applications. The DEA models [111 include Constant Return to Scale (CCR) model, The Variable Return to Scale (VSR) model, Stochastic Data Envelopment Analysis (SDEA) and Non parametric Stochastic Frontier Estimation. DEA is a multi-factor productivity analysis model for measuring the relative efficiency of a homogenous set of coal mines (DMU s). For every inefficient coal mine, DEA identifies a set of corresponding efficient coal mines that can be utilized as benchmarks for improvement of performance and productivity. DEA developed based on two scale of assumptions viz., Constant Return to Scale (CRS) model (17] and Variable Return to Scale (VRS) model. 3.1 Introduction There is an increasing concern with measuring and comparing the efficiency of organizational units such as local authority departments, schools, hospitals, shops, bank branches and similar instances where there is a relatively homogeneous set of units. 52

2 The usual measure of efficiency, i.e.:. output ethciency = input is often inadequate due to the existence of multiple inputs and outputs related to different resources, activities and environmental factors. A formula for relative efficiency incorporating multiple inputs and outputs is introduced now, and the DEA model which allows relative efficiency measures to be determined is developed. This is followed by a discussion of the information made available by solving the model and some issues of practical concern in applying the technique. 3.2 Efficiency Measurement Concepts Relative Efficiency Measurement: The measurement of relative efficiency where there are multiple possibly incommensurate inputs and outputs was addressed by Farrell and developed by Farrell and Fieldhouse focusing on the construction of a hypothetical efficient unit, as a weighted average of efficient units, to act as a comparator for an inefficient unit. A common measure for relative efficiency [23] is, Weighted sum of outputs Efficiency = ; : Weight sum of inputs The primary purpose of this section is to outline a number of commonly used efficiency measures and to discuss how they may be calculated relative to an efficient technology, which is generally represented by some form of frontier 53

3 function. Frontiers have been estimated using many different methods over the past 40 years. The two principal methods are: 1. Data envelopment analysis (DEA) and 2. Stochastic frontiers, Which involve mathematical programming and econometric methods, respectively? 3.3 Data Envelopment Analysis (DEA) Models Data envelopment analysis (DEA) is the non-parametric mathematical programming approach to frontier estimation. The discussion of DEA models presented here is brief, with relatively little technical detail. More detailed reviews of the methodology are presented by Seiford and Thrall (1990), Lovell (1993), Ali and Seiford (1993), Lovell (1994), Chames et al (1995) and Seiford (1996).The piecewise-linear convex hull approach to frontier estimation, proposed by Farrell 1957), was considered by only a handful of authors in the two decades following Farrell s paper.authors such as Boles (1966) and Afriat (1972) suggested mathematical programming methods which could achieve the task, but the method did not receive wide attention until a the paper by Chames, Cooper and Rhodes (1978) which coined the term data envelopment analysis (DEA). There have since been a large number of papers which have extended and applied the DEA methodology. Chames, Cooper and Rhodes (1978) proposed a model which had an input orientation and assumed constant returns to scale (CRS).6 Subsequent papers have considered alternative sets of assumptions, such as Banker, Chames and Cooper (1984) who proposed a variable returns to scale (VRS) model [23]. The following discussion of DEA begins with a description of the inputorientated CRS model was the first to be widely applied. 3.4 The Constant Returns to Scale Model (CRS) A number of different versions of the basic DEA model have been developed to address a series of potential shortcomings of the original DEA model. 54

4 We will examine three variations of the DEA methodology and study the differences in assumptions of the three as well as compare the different outcomes of each. The purpose of this analysis is to understand whether the choice of methodology may have an impact on the outcome of an analysis CRS Input- oriented Model In all variations of the DEA models, [17] the DMU(s) with the best inherent efficiency in converting inputs Xi, X2,...,x into outputs Yi, Y2,...,Ym is identified, and then all other DMUs are ranked relative to that most efficient DMU. For DMU 0, the basic CRS Input Oriented model (so-called CCR after Chames, Cooper, and Rhodes) is calculated as follows: max /?o = r i 2>-iv <1 subject to y r,xij Mr,V, >0 for each unit j 3.1 The interpretation of ur and w is that they are weights [101 applied to outputs yn and inputs xij and the are chosen to maximize the efficiency score h0 for DMUo. The constraint forces the efficiency score to be no greater than 1 for any DMU. An efficiency frontier is calculated, enveloping all data points in a convex hull. The DMU(s) located on the frontier represent an efficiency level of 1.0, and those located inside the frontier are operating at a less than full efficiency level, i.e. less than 1.0. The above fractional program is executed once for each participating DMU, resulting in the optimal weights 55

5 being determined for each DMU. Before solving the problem, the denominator in the objective function is removed and instead an additional constraint is added. Also, the original constraint is manipulated in order to convert the fractional program to a linear program. These two steps result in the following: max *o= X"r. V, 2X>v r -X v'v i - subject to =1 i II,-V, >0 3.2 In simpler notation, this is written as: max(v.u) = liy0 subject to itv0 =1 -\'X + ny< v > 0,// > 0 Finally, before solving, the linear program is converted to its dual for computational efficiency reasons: 56

6 mm( 6./.) = 0 subject to IA > V0 0kq - XX > A >0 With the addition of slack variables, the dual problem becomes: min( $./,) = 0 0\ o - XX = s subject to YX = V0 + S A > 0. s+ > 0..s~ >0 The slack variables can be interpreted as the output shortfall and the input overconsumption compared to the efficient frontier. The following is linear program in which the slacks are taken to their maximal values. m ma x][>~ + 2>r i-i r=l subject to s z -VjT + s~ = / = 1.2,...,///; FI 21) rf^- j ~~ ~ y ro Ars;,s; >o ytj, r ii Where we note the choices of si and -do not affect the optimal which is determined from model (3.5 ) CRS Output-oriented Model Alternately, one could have started with the output side and considered instead the ratio of virtual input to output. This would reorient the objective from max to min, as in (3.1), to obtain: 57

7 Mill /Zrttr.V Subject to ZiY,xv / I> 1 for; = ur. r, > s > 0 for all / and r Again, the Chames-Cooper (1962) transformation for linear fractional programming yields model (1.8) (multiplier model) below, with associated dual problem, (1.9) (envelopment model), as in the following pair, mine/ = 2>vt10 i l subject to m s Zn*tf >o 7=1 r=l 3.8 r=l >. Vr,7 m max # + (X s7 + Z K) 1=1 7 = 1 subject to s zv,+*;=.- M n X M *5. IV v, I *7 4 1! i 7 = nr. r = 1*2*. 0;=ix Here we are using a model with an output oriented objective as contrasted with the input orientation in (3.6). However, as before, model (3.9) is calculated in a two-stage process. First, we calculate First, we calculate <p* by ignoring the slacks. Then we optimize the slacks by fixing 9* in the following linear programming problem, 58

8 max^s. r=l +ZK r=l subject to Zx^J+s:=xi j-i 2-3 tj^ j <P.v, J-l Aj > 0 / = * = s: j = 1.2,..., Table 3.1 presents the CRS model in input- and output-oriented versions, each in the form of a pair of dual linear programs. Table 3.1 CRS DEA Model Envelopment model m s + ZK) M /-=! subject to Z.x *,+s; i = 1.2,..., w; j-l n Z.Vn^j ~K = yn r = 1,2,...s; M kj'z 0 j ~ 1,2... n. Envelopment model max + (!> ; m + sv.s;) /-l r-1 subject to ixva.+sj =xi0 i w; j-i 1 Zy^Aj -5* = r = 1,2... s; 7»t Xj>0 j = 1,2,...,». Input-oriented Output-oriented Multiplier model maxz = ZMrXro subject to 2>,.v s - m 0 r-1 M m TVX ~l l ftr.v, > t: > 0 Multiplier model ming m = 7-1 subject to m s Zv,x9 - ZVr>\: * 0 M r-l Z/yyro = 1 r»l /lr>v, > >0 3.5 VRS (Variable to Returns Scale) or BCC Model If the constraint is f=i 7.j = 1 is adjoined, they are known as BCC (Banker, Cooper, 1984) models. This added constraint introduces an additional variable, fi0 into the (dual) multiplier problems. As will be seen in 59

9 the next chapter, this extra variable makes it possible to effect retums-to-scale evaluations (increasing, constant and decreasing). So the BCC model is also referred to as the VRS (Variable Returns to scale) model and distinguished form the CCR model which is referred to as the CRS (Constant Returns to Scale) model. The CRS model is designed with the assumption of constant returns to scale. This means that there is no assumption that any positive or negative economies of scale exist. It is assumed is that a small unit should be able to operate as efficiently as a large one - that is, constant returns to scale. In order to address this, Banker, Chames, and Cooper developed the BCC model. It is also referred as VRS model. The VRS model is closely related to the standard CRS model as is evident in the dual of the BCC model: min(6u)= 6 subject to - XA = s YA = vq + s* ea = l A>0.s+ >0..$~ > The difference compared to the CRS model is the introduction of the convexity condition = 1. This additional constraint gives the frontiers piecewise linear and concave characteristics. 3.6 Increasing Returns Scale (IRS) and Decreasing Returns Scale (DRS) Models Returns to scale refers to a technical property of production that examines changes in output subsequent to a proportional change in all inputs (where all inputs increase by a constant). The output increases by that same proportional 60

10 change with input then there are constant returns to scale (CRTS), sometimes referred to simply as returns to scale. If output increases by less than that proportional change, there are decreasing returns to scale (DRS). If output increases by more than that proportion, there are increasing returns to scale (IRS). Example: Where all inputs increase by a factor of 2, new values for output should be: Twice the previous output given a constant return to scale (CRTS) less than twice the previous output given a decreased return to scale (DRS) more than twice previous output given an increased return to scale (IRS). 3.7 Cross Efficiency (CE) Model Cross efficiency in DEA allows for effective discrimination between niche performers and good overall performers. Cross efficiency [48] score of a DMU represents how well the unit is performing with respect to the optimal weights of another DMU. A DMU that achieves high cross efficiency scores is considered to be a good overall perform to improve the discrimination power of DEA, Sexton et al (1986) first introduces the concept of a cross-efficiency measure [62l in DEA. The basic idea is to use DEA in a peer-appraisal instead of a self-appraisal, which is calculated by the CRS (constant returns to scale) model. Peer evaluation is done by constituting a cross efficiency matrix of efficiency value given to each DMUs. This technique can also identify overall efficient and false positive DMUs, and it selects appropriate targets for poorly performing DMUs to learn as a benchmark. 61

11 Cross Efficiency Models: Aggressive and Benevolent Approaches Aggressive Model s, f 1 min Z v*z^a k=\ V i*p m ( \ sj Z uj'lxji =1 j=1 v i*p / m 4=1 y=i ZV*JV -0PlLujxjP =0 4=1 y=l V*,Uj> 0 V*, j Where 0p\$ the relative efficiency score of DMU p obtained from the CCR model Benevolent Model s max v*z yu 4=1 V >*P m f ) S-f Z ujtxj> =1 y=i y i*p J m Zv*->^ Z yx;> *» Vi*P k=\ j=1 s Z v^kp - p1lujx,p =0 4=1 j=1 m vk,uj> 0 Vk,j Where 6^ is the relative efficiency score of DMU p obtained from the CCR model 62

12 3.8 Scale Efficiency (SE) Model In DEA, Scale Efficiency [15] is routinely calculated. This measure may, however, tell us very little about whether a production unit is over- or undersized. An empirical case is used to illustrate that, under some circumstances, scale inefficiency may simply reflect that a production unit is producing too little, given its use of factors of production, and not that is over- or undersized. A fictitious sample based on a production function with variable returns to scale is used for demonstrating that in small samples with large deviations from the efficiency frontier and limited variability between units in terms of factor proportions scale efficiency may not reflect very well how far the production units are from being of an optimal size. 3.9 Super Efficiency (SE) Model Data Envelopment Analysis (DEA) evaluates the relative efficiency of decision-making units (DMUs) but does not allow for a ranking of the efficient units themselves. A modified version of DEA based upon comparison of efficient DMUs relative to a reference technology spanned by all other units is developed (called super-efficiency model). The procedure provides a framework for ranking efficient units and facilitates comparison with rankings based on parametric methods. Super efficiency model allows for effective ranking of efficient DMUs max S m st 2>a=1 m vk,uj>0 Vk,j 63

13 Super-efficiency data envelopment analysis (DEA) model is obtained when a decision making unit (DMU) under evaluation is excluded from the reference set. Because of the possible infeasibility of super-efficiency DEA model, the use of super-efficiency DEA model has been restricted to the situations where constant returns to scale (CRS) are assumed. It is shown that one of the inputoriented and output-oriented super-efficiency DEA models must be feasible for a any efficient DMU under evaluation if the variable returns to scale (VRS) frontier consists of increasing, constant, and decreasing returns to scale DMUs. We use both input- and output-oriented super-efficiency models to fully characterize the super-efficiency. When super-efficiency is used as an efficiency stability measure, infeasibility means the highest super-efficiency (stability). If super-efficiency is interpreted as input saving or output surplus achieved by a specific efficient DMU, infeasibility does not necessary mean the highest super-efficiency Slacks-Based Measure of Efficiency (SBM) Model Finally, the second adjustment to the basic CRS model is the Slacks- Based Measure of efficiency (SBM), proposed by Tone. The motivation for the development of this model is the observation that while both the CRS and the BCC models calculate efficiency scores, neither is able to take into accounts the resulting amount of slack for inputs and outputs. Consequently, the purpose of this model is to minimize the input and output slacks, resulting in this fractional program, which is converted to a linear program before solving: min(/!.s~ j) p = x0 XA =.s" subject to YA = V0 + s~

14 3.11 Window Analysis In the examples of the previous sections, each DMU was observed only once, i.e., each example was a cross-sectional analysis of data. In actual studies, observations for DMUs are frequently available over multiple time periods (time series data), and it is often important to perform an analysis where interest focuses on changes in efficiency over time. In such a setting, it is possible to perform DEA over time by using a moving average analogue, where a DMU in each different period is treated as if it were a "different" DMU. Specifically, a DMU's performance in a particular period is contrasted with its performance in other periods in addition to the performance of the other DMUs. The window analysis technique represents one area for further research extending DEA. For example, the problem of choosing the width for a window (and the sensitivity of DEA solutions to window width) is currently determined by trial and error. Similarly, the theoretical implications of representing each DMU as if it were a different DMU for each period in the window remain to be worked out in full detail Allocative and Overall Efficiency To this point we have confined attention to technical efficiency [31] which, does not require a use of prices or other weights. Now we extend the analysis to situations in which unit prices and unit costs are available. This allows us to introduce the concepts of allocative and overall efficiency and relate them to technical efficiency in a manner first introduced by M.J. Farrell (1957). For this introduction we utilize Figure 3.1 in which the solid line segments connecting points ABCD constitute an isoquant or level line that represents the different amounts of two inputs (xt, xz) which can be used to produce the same amount (usually one unit) of a given output. This line represents the efficiency frontier of the production possibility set because 65

15 it is not possible to reduce the value of one of the inputs without increasing the other input if one is to stay on this isoquant. The dashed line represents an isocost (=budget) line for which (xi, X2) pairs on this line yield the same total cost, when the unit costs are cl and c2, respectively. When positioned on C the total cost is k. However, shifting this budget line upward in parallel fashion until it reaches a point of intersection with R would increase the cost to k > k fact, as this Figure shows, k is the minimum total cost needed to produce the specified output since any parallel shift downward below C would yield a line that fails to intersect the production possibility set. Thus, the intersection at C gives an input pair (xi, X2) that minimizes the total cost of producing the specified output amount and the point C is therefore said to be allocatively as well as technically efficient. Now let R represent an observation that produced this same output amount. The ratio 0 < OQ / OR < 1 is said to provide a radial measure of technical efficiency with 0 < 1 - (OQ / OR) < 1 yielding a measure of technical inefficiency. Now consider the point P which is at the intersection of this cost line through C with the ray from the origin to R. We can also obtain a radial measure of overall efficiency from the ratio 0 < OP / OR < 1 In addition, we can form the ratio 0 < OP / OR < 1 to obtain a measure of what Farrell (1957) referred to as price efficiency but is now more commonly called allocative efficiency. Finally, we can relate these three measures to each other by noticing that OP OO OP Which we can verbalize by saying that the product of allocative and technical efficiency equals to overall efficiency in these radial measures. 66

16 C4 To implement these ideas we use the following model, as taken from Cooper, Seiford and Tone (2000, p. 236), min Sc " to x. i i* 1 subject to n V r X < Y i = 1 j=l L < ZJL, < V j=l r = l...m s 3.14 where the objective is to choose the X; and Xj values to minimize the total cost of satisfying the output constraints. The Cj0 in the objective represent unit costs. This formulation differs from standard models, as in Fare, Grosskopf and Lovell (1985, 1994), in that these unit costs are allowed to vary from one DMUco to another in (3.14). Finally, using the standard approach, we can obtain a measure of relative cost (= overall efficiency) by utilizing the ratio: 67

17 0 m M TC. X i=l f m * * T c. W x. 10 M Where the** are the optimal values obtained from (3.14) and the Xj0 are the observed values for DMUo Profit Efficiency We now introduce another type of model called the additive model to evaluate technical inefficiency. First introduced in Chames et al. (1985) this model has the form max ts"+t.s' r~l r i*l * subject to v = fr./l-r. r = s ro *-> t} J r j=l U j-i o <Ars+ $~:'xflj.r. This model uses a metric that differs from the one used in the radial measure model.3 It also dispenses with the need for distinguishing between an output and an input orientation as was done in the discussion with above mentioned models. Because the objective in (3.16) simultaneously maximizes outputs and minimizes inputs-in the sense of vector optimizations. This can be seen by utilizing the solution to (3.16) to introduce new variables yro, xlo defined as follows, V = V 4- S' > V. ro ro r. ro \ X S~ < X io io i io' r = 1... s\ f = ,/;/ Now note that the slacks are all independent of each other. Hence an optimum is not reached until it is not possible to increase an output%0 or reduce an 68

18 Input xi0 without decreasing some other output or increasing some other input Stochastic Data Envelopment Analysis (SDEA) Stochastic Data Envelopment Analysis (SDEA) is an extension of Data Envelopment Analysis (DEA) technique. This SDEA model allows for the possibility of random errors in model specification or measurement via a symmetric random error component, in addition to the one sided deviation attributable to inefficiency in the use of input resources Additive Model In basic models of DEA, we distinguish between input-oriented and outputoriented models. In this model, we combine both orientations in a single model, called Additive model. max z =es~+es* s x0 = X A + s~ y0 =7 l~s + ea = l A>0, s',s*> Weight Restrictions in DEA The DEA problem allows for unrestricted weight flexibility [63] in determining the efficiency scores of DMUs. This allows units to achieve relatively high efficiency scores by indulging in inappropriate input and output factor weights. Weight restrictions allow for the integration of managerial preferences in terms of relative importance levels of various inputs and outputs. For example, if output 1 is at least twice as important as output 2 then this can be incorporated into the DEA model by using the linear constraint vl > 2v2. Methods for incorporating weight restrictions have been suggested by several researchers. Included in this stream of research are works by Chames 69

19 et al. (1990), Dyson and Thannassoulis (1988), Thompson et al. (1986, 1990, 1995), and Wong and Beasley (1990). Although weight restrictions effectively discriminate between efficient and inefficient units, ranking DMUs can still be an issue. In order to allow for a ranking of units in the presence of weight restrictions, a combination of models proposed by Talluri and Yoon (2000) 3.17 Extensions to DEA models These are the four models commonly available in literature and described a profound basis for an efficiency analysis with different returns to scale, different envelopment surfaces and different ways to project inefficient units to the efficient frontier [Chames A. et al. (1994)]. During the last ten (10) years a lot of extensions to these four models have been developed that allow further fine tuning to the basic models. Most of these extensions of DEA {67j have been a result of the application of the technique to real life problems (Allen et. al. (1997)). In this section, some of these numerous methodological extensions are briefly presented that could be utilized to improve its discriminatory power in performance evaluation. a) The basic DEA models always assume that inputs and outputs can be altered by the DMUs. In realistic situation there are often exogenous variables that cannot be altered. For example the distribution of competitors may influence efficiency scores without being alterable by the DMUs [Banker R. D.; Morey R. C. (1986a)]. These variables are called nondiscretionary variables. b) Thompson et al. (1986) and Tone (1999) offer a framework for consensus-making based on expert opinion. Tone, in particular, offers 70

20 a framework for deriving quantitative estimates using expert opinion. Toshiyuki discussed about extended DEA - discriminate analysis, proposed new methodology for the same and done detailed theoretical and mathematical analysis. Then he used data set from Japanese banking industry and compared the performance of the extended DEA- DA with other discriminate analysis methods. c) Although benchmarking in DEA allows for the identification of targets for improvements, it has certain limitations. A difficulty addressed in the literature regarding this process is that an inefficient DMU and its benchmarks may not be inherently similar in their operating practices. This is primarily due to the fact that the composite DMU that dominates the inefficient DMU does not exist in reality. To overcome these problems researchers have utilized performance-based clustering methods for identifying more appropriate benchmarks (Doyle & Green (1994); Talluri & Sarkis (1997)). These methods cluster inherently similar DMUs into groups, and the best performer in a particular cluster is utilized as a benchmark by other DMUs in the same cluster. d) Traditional DEA models do not allow for ranking DMUs, specifically the efficient ones. Also, it is possible in DEA that some of the inefficient DMUs are in fact better overall performers than certain efficient ones. This is because of the unrestricted weight flexibility problem in DEA. Thus, a DMU can achieve a high relative efficiency score by being involved in an unreasonable weight scheme (Dyson & Thannassoulis(1988);Wong & Beasley, (1990)). Such DMUs heavily 71

21 weigh few favorable measures and completely ignore other inputs and outputs. These DMUs can be considered as niche members and are not good overall performers. Cross-efficiencies in DEA is one method that could be utilized to identify good overall performers and effectively rank DMUs (Sexton et al. (1986)). Cross-efficiency methods evaluate the performance of a DMU with respect to the optimal input and output weights of other DMUs. The resulting evaluations can be aggregated in a crossefficiency matrix (CEM). In the CEM, the element in /th row and/h column represents the efficiency of DMU j when evaluated with respect to the optimal weights of DMU,. A DMU, which is a good overall performer, should have several high cross efficiency scores along its column in the CEM. On the other hand, a poorly performing DMU should have several low values. The column mean can be computed to effectively differentiate between good and poor performers (Boussofiane et. al, 1991). e) A limitation in using the CEM is that the factor weights obtained may not be unique. This undermines the effectiveness of the CEM in discriminating between good and poor performers. Some techniques have been proposed for obtaining robust factor weights for use in the construction of the CEM. Doyle and Green (1994) have developed a set of formulations for this purpose. The one that is most appropriate for this discussion is the aggressive formulation, which identifies optimal weights that not only maximize the efficiency of a unit but also 72

22 minimize the efficiency of the average unit that is constructed from the other n - 1 units (Doyle and Green (1994)), f) Talluri (2000) proposed a variation to the Doyle and Green model, which compares a pair of DMUs each time. In this model, the target DMU (evaluator) not only maximizes its efficiency score but also minimizes the efficiency score of each competitor in turn. Therefore, the optimal weights of the target DMU may vary depending on the competitor being evaluated. In essence, the target DMU can involve multiple strategies (optimal solutions or the input and output weights), that is, it emphasizes its strengths, which are weaknesses of a specific competitor. These results can be incorporated into a CEM to identify good overall performers. Sarkis and Talluri (1999) extended the above case to include both cardinal and ordinal input and output factors, which is based on the work by Cook et al. (1996). They proposed a combination of models that allowed for effective ranking of DMUs in the presence of both quantitative as well as qualitative factors. These models are also based on cross-evaluations in DEA. g) Other ranking methods that do not specifically include crossefficiencies were proposed by Rousseau and Semple (1995), and Andersen and Petersen (1993). Rousseau and Semple (1995) approached the same problem as a two-person ratio efficiency game. Their formulation provides a unique set of weights in a single phase as opposed to the two-phase approaches presented above. Andersen and Petersen (1993) proposed a ranking model, which is a revised version. 73

23 In this version of DEA based upon comparison of efficient DMU s relative to a reference technology spanned by all other units is developed. In this model, the test DMU is removed from the constraint set allowing the DMU to achieve an efficiency score of greater than 1, which provides a method for ranking efficient and inefficient units. The procedure provides a frame work for ranking efficient units and facilities comparison with rankings based on parametric methods. h) The size of the data set is also an important factor when using some of the traditional DEA models. As a general rule, with five inputs and five outputs, at least 25 or so units will appear efficient and so the set needs to be much greater than 25 for any discrimination. However, some of these sample size problems can be overcome by using cross efficiency models. i) As mentioned in the previous section, DEA allows for unrestricted weight flexibility in determining the efficiency scores of DMUs. This allows units to achieve relatively high efficiency scores by indulging in inappropriate input and output factor weights. Weight restrictions allow for the integration of managerial preferences in terms of relative importance levels of various inputs and outputs. For example, if output 1 is at least twice as important as output 2 then this can be incorporated into the DEA model by using the linear constraint vl > 2v2. Methods for incorporating weight restrictions have been suggested by several researchers. Included in this stream of research are works by Chames et al.(1990), Dyson and Thannassoulis (1988), Golony et. al. (1988), 74

24 Thompson et al. (1986, 1990, 1995), and Wong and Beasley (1990), Roll et.al. (1993). Roll et.al. (1993) suggested a conceptual frame work for the treatment of factor weights in DEA. They proposed general guide lines for setting bounds on factor weights, then, it develops and presents alternative methods to limit the range within which these factor weights are allowed to vary. Allen et.al (1997) classified the approaches for imposing the weight restrictions into broadly three categories. 1. Direct Restrictions on weights (Absolute Weight Restrictions, Assurance Region I, Assurance Region II) 2. Adjusting the observed input-output levels to capture value judgments. 3. Restricting weight flexibility by restricting the weighted inputs and outputs. All of these methods involve additional information, which is entered into the analysis the form of constraints, bounds or different objective functions. j) Although weight restrictions effectively discriminate between efficient and inefficient units, ranking DMUs can still be an issue. In order to allow for a ranking of units in the presence of weight restrictions, a combination of models proposed by Talluri and Yoon (2000)could be utilized. 75

25 k) The introduction of categorical variables extends the application focus of the DEA, The comparison of quite different units was also addressed by Banker, R. D.; Morey, R.C. (1986). For example bank branches that are difficult to compare because of different demographics can be categorized or even binary-coded variables can be computed. The incorporation of categorical variables into DEA evaluations can be found in Banker and Morey (1986) and Kamakura (1988). Some work in the consideration of both cardinal and ordinal factors in DEA can be found in Cook et al. (1993, 1996), Sarkis and Talluri (1999). l) The former described models, all focuses analysis of one period situation only. In order to capture the variations of efficiency over time, Chames et al. (1985) proposed a technique called window analysis in DEA. Window analysis assesses the performance of a DMU over time by treating it as a different entity in each time period. This method allows for tracking the performance of a unit or a process. For example, if there are n units with data on their input and output measures in k periods, then a total of nk units need to be assessed simultaneously to capture the efficiency variations over time. m) In the traditional window analysis described above, when a new period is introduced into the window the earliest period is dropped out. A variation to this method was proposed by Talluri et al. (1997) to 76

26 effectively monitor the performance of a unit over time and assist in process improvement and benchmarking. Essentially, this technique, referred to as the modified window analysis, drops the poorest performing period instead of the earliest period. This allows for a new period to be challenged against the best of the previous periods and, thereby, assisting in process improvement and benchmarking. n. Apart from the above mentioned areas, there is significant work in the areas of stochastic DEA; sensitivity analysis in DEA, target setting in DEA, more effective ways of weight restrictions in DEA is being carried. Some of the interesting extensions in this area can include the improvement of discriminatory power of non-constant returns to scale models, better methods for benchmarking, developing the robustness of cross-efficiency models, etc.. Multi-output forms of stochastic production frontiers have been developed but remain highly complex (Kumbhakar and Lovell (2000)). The development of stochastic DEA models is currently a key area of research (Resti (2000), and Ruggiero (2000). Approaches have been developed to capture some of the random variability in data (e.g. chance constrained DEA). Details on some of the recent developments in this area are given in Cooper, Seiford and Tone (2000). Bootstrapping techniques have also been applied to estimate the effects of random variation on the estimates of efficiency and methods have been developed to compensate for some of these effects (Simar and Wilson, 2000). Each of these new models and methods can be useful in a variety of manufacturing and service areas. 77

27 3.18 Application of DEA Methodology in Various Sectors This section provides a review of various applications of DEA in various sectors with special focus on weight restriction models that have been published in the DEA literature (Amit K, 2001). In each application, determination of bound values and affect of weight restrictions on the results were discussed. DEA is receiving increasing importance as a tool for evaluating and improving the performance of manufacturing and service operations. It is most useful when a comparison is sought against "best practices" where the analyst doesn't want the frequency of poorly run operations to affect the analysis. This approach has been used in a variety of empirical settings. Examples include program evaluation (Chames, Cooper and Rhodes (1981)), evaluation of school district efficiencies (Bessent et. al. (1983)), productivity measurement for manufacturing operations (Banker (1985), Banker and Maindiratta(1986)). Some other settings in which the DEA technique has been employed are steam-electric power generation (Banker (1984)), coal mines (Byrnes, Fare and Grosskopf (1984)), Pharmacy stores (Banket and Morey (1986a)) and fast-food restaurants (Banker and Morey (1986b)). Ramanathan (2001) used DEA method for comparative performance appraisal of schools in Netherlands. He has improved the results over quality cards and also some post optimal analysis on the basis of program of studies and locations of schools. Finally he has done regression analysis of the DEA efficiencies. Beasley (1989) presents a quantitative model for comparing various university departments concerning the same discipline. His model is 78

28 based upon ideas drawn from the DEA. Computational results are given for chemistry and physics departments in the United Kingdom. Smith (1989) applied DEA to financial statements. According to him, ratio - analysis has been tool of analysis for as long as financial statements have been prepared. Yet it is limitations to considering only one numerator and one denominator severely limit its usefulness. This paper extends the traditional ratio analysis to permit the incorporation of any number of dimensions of production measures of corporate efficiency, together with a wealth of supporting information. The strength and weakness of the method applied to financial statements are appraised. The study made by Roy et. all. (1991) combined DEA with regression modeling to estimate relative efficiency in the public school districts of Connecticut. Factors effecting achievements were classified as school inputs and other socio economic factors. DEA was performed with the school inputs only. Efficiency measures obtained from DEA are subsequently related to socio - economic factor in a regression model a one - sided disturbance term. The findings suggest that while productivity of school inputs varies considerably across districts this can be ascribed to a large extent to differences in the socio - economic back ground of the communities served. Variation in the managerial efficiency is much less than what is implied by the DEA results. DEA has also been utilized as a resource allocation tool. A good example of its use in resource allocation can be found in Bessent et al.(1983) Doyle et. al. (1991) tried to compare products which vary excellence along a number of 79

29 dimensions using DEA, this method was illustrated by comparing published bench marks of 37 computer printers. Mahendra Raj et al.(2002) used this approach to provide a benchmark to measure for the operational efficiency of restructured companies that have reduced staff numbers and also companies that have found it necessary to downsize due to declining demand for its product. Srinivas.T. et al. (2000) demonstrated the applicability of Cone-Ratio DEA (CRDEA) to the Advanced Manufacturing Technology (AMT) selection process. Srinivas.T. et al. (2006) used this approach in the area of purchasing, in general, and vendor evaluation, in particular. Cook et al. (1990) used absolute weight restrictions DEA model for measuring the Relative Efficiency of 14 Highway Maintenance Patrols Thompson et al. (1992) a DEA/AR efficiency analysis of 7 years ( ) was made for 45 oil / gas firms called independents. A comparative evaluation of six competing sites was carried out using DEA is carried out by Thompson et.al (1996a) to determine the ideal site for locating a high energy physics lab. The assurance region was applied to the site location problem; it was found that the region of dominance of one of the sites, in the weight space, completely enclosed the assurance region. Therefore, this site was the only one with efficiency score of 1 and was also the preferred site. Thompson et.al (1996b) the efficiency and profit potential of 14 integrated oil companies were measured by using DEA model AR bounds. Thompson et al. (1996c) solved the AR-DEA model for 48 banks for 80

30 the years computing DEA/AR Efficiency and Profit Ratio Measures with an Illustrative Bank Application. Zhu etal (1996) employed the data envelopment analysis/assurance region (DEA/AR) methods to evaluate the efficiency of the 35 textile factories of the Nanjing Textiles Corporation (NTC), Nanjing China. By specifying input and output cones, a cone-ratio assurance region (CR-AR) was set up. While most existing approaches involving Assurance Regions use "price/cost" data to determine values of the bounds, AR's were developed based on expert opinions on the relative importance between various inputs/outputs. NTC uses the Analytic Hierarchy Process (AHP) to gather and present expert opinion for systematically evaluating the overall industrial performance. The results from the AHP were used in this paper to set bounds on the weights. DEA and linked-cone assurance region models were used by Taylor et.al (1997) to investigate the efficiency and profitability potential of Mexican banks. Schanffnit et.al (1997) has carried out a best practice analysis of the Ontario based branches of a large Canadian Bank using AR/DEA. Thanassoulis et al. (1995) explores the use of DEA to assess units providing perinatal care (District health Authorities, DHAs) in England Chilingerian and Sherman (1997) used the assurance region model to spot inefficiencies in the practice patterns of primary care physicians (PCPs) Applications of DEA in Mining Sector Bynes et. al. (1984) applied a generalized version of the Farrell measure of technical efficiency to a sample of Illinois mines. They disagreed the original 81

31 Farrell measure (which was designed to measure lost output or wasted inputs due to under utilization of inputs) into three mutually exclusive and exhaustive components: (1) a measure of purely technical efficiency, (2) a measure of input congestion and (3) a measure of scale efficiency. Their paper describes the details of non - parametric approach of performance measurements, case study, discussion and conclusion. Ramani et, al. (1988) discussed about problem with traditional mine management and its unique characteristics. They identified the need for improvement of performance in mining systems and suggested some alternative approaches of mine system design and its analysis. Bhattacherjee (1999) carried out the analysis using 21 UG coal mines by taking total salaries and wages, total store expenses, power expenses, interest, administrative expenses and other expenses as inputs and annual revenue as output by using CCR model and compared the efficiencies of the UG mines. Thomson et. al. (1995) considered the case of application of a modified DEA to Illinois coal mine. The conventional DEA measures technical efficiency and it is recognized that bounds must be introduced into the measurement process proceedings from technical efficiency to overall efficiency. Thomson et. al. (1986), (1990) used economic/environmental price data (beyond the input - output data per se) and expert opinion to bind the multipliers. These modified models have a profit maximizing linear objective function. The authors content that DEA technical efficiency does not imply a DEA maximum profit ratio and vice - versa. The analysis of Illinois coal mines is used to substantiate this claim. 82

32 The data used in Bynes et. al. (1984) was reformulated and used for this study by Thomson. The capacity inputs were defines in real capital term and aggregated into a single capital input: K=kl+K2+k3 The four geographical characteristics were consolidated into one mine quality input: T= (T1/D1+T2/D2) The input corresponding to mine quality gives a measure equivalent to inverse of an aggregated stripping ratio i.e., tons of coal per ton overburden. The output was taken as thousand tons of coal mined and inputs were thousand labor days. Thousand dollars of investment is capital (to a base year) and tons of clean coal per thousand tons of overburden. One notable feature in this study is that by aggregation of input factors. The total number of inputs and outputs has been brought within the limit set by widely acknowledged thumb rules with respect to the number of DMUs. Another feature of this study is that the role of profitability in decision making is examined. If both profitability and efficiency scores are available, units can be assessed on an efficiency profitability matrix (boussofiane et.al 1991). Units which both profitable and efficient provide examples for good operating practices; those which are not efficient (irrespective of profitability) can be subjected to an efficiency drive and those which are efficient but not profitable calls for close scrutiny of operating environments. The study shows that the numbers of DMUs which are efficient in the presence of additional assurance 83

33 region constructs are lower than in a normal DEA which shows the need to restrict weight flexibility to achieve more meaningful results. Niraj Kumar et al (2002) used DEA and fuzzy logic techniques for comparing 40 UG mine. They had chosen capacity, Man shifts and cost per tonne as inputs and production, Financial Operating Efficiency Index (FOEI), Environmental Efficiency Index (EEI) and Safety Efficiency Index (SEI) as outputs. Niraj Kumar et.al used a two stage DEA model to rank the mining units. In the first stage, the DEA is used to get efficiency scores of various mining units. In the second stage, the AHP is applied to differentiate among mines, which have the same efficiency score based upon the DEA method. This helps in further ranking of each mining units. This combined approach not only helps to overcome the limitations of both the methods, but also enables the full ranking of all the mines. The application of this combined approach is demonstrated through a case study example of a group of 20 open cast coal mines by taking capacity, Man shifts and cost per tonne as inputs and productivity, Financial Operating Efficiency Index (FOEI), Environmental Efficiency Index (EEI) and Safety Efficiency Index (SEI) as outputs. 84