What is the dea projection principle

We illustrate the data envelopment analysis of scale reward invariant input dominated with a simple example including five DMUs (firms). Each DMU is two inputs and one output, and the data are as follows:

Table 1 Data for the DEA example with constant returns to scale

The input-output ratios for this example are plotted in Data 6, which also remits the frontier corresponding to the equivalent equation 12 of the DEA. We can take this to heart, however, this DEA frontier is the result of calculating a linear program once for each of the 5 DMUs. For example, for DMU3 we can rewrite Equation 12 like this.

minθ,λθ,

st -y3+(y1λ1+y2λ2+y3λ3+y4λ4+y5λ5)≥0,

θx13-(x11λ1+x12λ2+x13λ3+x14λ4+x15λ5)≥0,

θx23-(x21λ1+x22λ2+x23λ3+x24λ4+x25λ5)≥0,

λ≥0, (14)

whereλ=(λ1,λ2,λ3,λ4,λ5)′.

The values of θ and λ are provided in the third row of Table 2 for the smallest θ value. We note that the technical efficiency value of DMU3 is 0.833. DMU3 can reduce inputs by 16.7% without reducing outputs. This implies that it should be produced at point 3' of data 6. This estimated point 3' is on the line connecting DMU2 and DMU5 and it is considered as the counterpart of point 3. They define where the relevant part of the frontier is located (e.g. related to DMU3) also defines the efficient production point of DMU3. Point 3' is a linear combination of points 2 and 5, and the weight of the linear combination is the value of λ in the third row of Table 2.

Data 6

Example of Scale-Reward Invariant Input-Dominant DEA

Table 2

Results of Scale-Reward Invariant Input-Dominant DEA

Many studies discuss the target and the corresponding point. the target of DMU is also the corresponding efficiency projection point 3'. This is equal to 0.833 x (2,2) = (1.666,1.666). Thus, for DMU3 to get 3 units of output it would have to use 3 x (1.666,1.666) = (5,5) units of both inputs

One can have a similar discussion for each of the other inefficient DMUs. DMU4 has an efficiency index of 0.714, and has a similar point of correspondence, as does DMU3. DMU1 has an efficiency index of 0.5, and DMU2 is the point of correspondence for his efficiency. the corresponding point for his efficiency. We can note that the estimated point of DMU1 is at the upper end of the efficiency part which is parallel to the axis of x2 . Therefore, it does not represent the efficiency point (according to Koopman's definition.) This is because we can reduce the input x2 by 0.5 units (so the production point is at point 2) and still get the same amount of output. Thus DMU1 can be said to be 50% wasted for input radioactivity and has 0.5 units of x2 of (non-radioactive) input slack variable. This leads to the objective (x1=1,x2=2). That is the corresponding point 2.

3.2 The Variable Returns to Scale Model (VRS) and Scale Efficiencies The Variable Returns to Scale Model (VRS) and Scale Efficiencies

The assumptions of the CRS are only appropriate when all DMUs are operating at the optimal scale appropriate (e.g., a corresponding flat portion of the LRAC curve). Imperfect competition and constraints, finance, etc., may cause DMUs not to operate at the optimal size.Banker, Charnes and Cooper (1984) extend the DEA model with constant returns to scale to the case of changing returns to scale. The use of the CRS specification when DMUs are not operating at the optimal scale may result in the measurement of technical efficiency being confounded by scale efficiency.The VRS model specification will allow for the calculation of technical efficiency excluding the effect of scale efficiency.

The CRS linear programming model can be easily modified into the VRS model by adding convexity constraints. The convexity constraint added to Equation 12 is N1'λ=1 It can be obtained,

minθ,λθ,

st -yi+Yλ≥0,

θxi-Xλ≥0,

N1'λ=1

λ≥0, (15)

N1 is the matrix of all N × 1. This method creates a convex surface that is able to envelope all the data more tightly than the conical surface of the CRS, and therefore the technical efficiency obtained is higher or equal to that obtained using the CRS model.The VRS gauge manual is the most popular manual of the 20th century.

Calculation of Scale Efficiencies

Many studies have decomposed the technical efficiencies obtained from the CRS model into two parts, one due to scale inefficiencies and one due to pure technical inefficiencies. This can be done by implementing both CRS and VRS DEA models on the same data. If the two technical efficiencies are different for a particular DMU, this proves that there is scale inefficiency for that DMU. The scale inefficiency can be calculated by the difference between the technical efficiency of VRS and the technical efficiency of CRS.

Data 7 attempts to elaborate on this issue. Inside this data we have an example of one input and one output, and we draw the efficiency frontiers of VRS and CRS. In the input-dominated technical inefficiency of CRS, the distance to the P-point is PPC, while in the VRS model technical inefficiency is PPV. the difference between the two, PCPV, is scale inefficiency. We can express these in terms of ratio efficiency measures.

TEI,CRS=APC/AP

TEI,VRS=APV/AP

SEI=APC/APV

All of these measures are between 0 and 1. We can also note that

TEI,CRS=TEI,VRS×SEI

because

APC/AP=(APV/AP)×(APC/APV).

This is where the technical efficiency of CRS can be decomposed to be called pure technical efficiency and scale efficiency.

Data 7

Calculation of Economies of Scale in DEA

One of the disadvantages of the scale efficiency approach is that his values do not reflect whether the DMU is running in an area of increasing returns to scale or decreasing returns to scale. This can be determined by running an additional DEA model with non-increasing returns to scale (NIRS). This can be done by changing the DEA model in Equation 15, replacing the restriction of N1'λ = 1 in Equation 15 with N1'λ ≤ 1 can be obtained

minθ,λθ,

st -yi+Yλ≥0,

θxi-Xλ≥0,

N1'λ≤1

λ≥0, (16)

The NIRS DEA frontiers are plotted in Data 7. The type of scale inefficiency (e.g., due to increasing scale or decreasing scale) of a DMU can be determined by looking at whether the NIRS technical efficiency values agree with the VRS technical efficiency values. If they are not equal (which is the example of data point 7, P), then there is scale payoff progression for this DMU. If they are equal (which is the example of data point 7 Q), diminishing returns to scale apply. an example of this approach applied to international airlines is provided in BIE (1994). An example of this approach applied to international airlines is provided in BIE(1994).

Example 2Example 2

This is a simple example consisting of five firms, each producing a single product with a single input. The data are presented in Table 3, and the results of the VRS and CRS input-dominant DEA models are presented in Table 4 and plotted in Data 8. Assuming we use the input-dominant type, efficiency can be measured horizontally through Data 8. When we assume constant returns to scale, we observe that firm 3 is the only efficient firm (on the efficiency frontier of the DEA), but when we assume changes in returns to scale, firms 1, 3, and 5 are all efficient.

The calculation of different efficiency methods can be demonstrated by using firm 2, since firm 2 is technically inefficient under both CRS and VRS models.

Technical efficiency of CRS is equal to 2/4=0.5 and technical efficiency of VRS is 2.5/4=0.625 and scale efficiency is equal to the ratio of technical efficiency of CRS to technical efficiency of VRS and that is 0.5/0.625=0.8.We can also observe that firm 2 has an increasing phase of scale payoffs in the efficiency frontier of VRS.

Table 3

Example Data for DEA Model of Changing Rewards of Scale

Table 4

Results of VRS Input Dominant DEA

Data 8

Example of VRS Input Dominant DEA

3.3 Input and Output Orientations Input and Output Orientations

Inside the Input Orientation model above, which we discussed in sections 3.1 and 3.2, this model attempts to define technological inefficiency as the proportional reduction of wasted inputs. This is equivalent to the technical inefficiency calculated by Farrell's input-based approach. As discussed in section 2.2, we might also calculate technological inefficiency as a proportional increase in output. The values of the two methods are the same when the returns to scale are constant, but are just not the same when the returns to scale change (see Data 3). It is assumed that linear programming models do not suffer from statistical problems such as covariance bias. The choice of appropriate direction will not be as critical as in the econometric estimation example. In many studies, the analysis favors input-dominated models because many DMUs have special orders to meet (e.g., generation). Thus the quantity of inputs appears to be the main determining variable, although this argument may not be strong in all industries. In some industries, DMUs may be given a fixed amount of resources and be asked to produce as much output as possible. In this case, output-dominant is the appropriate one. What is necessary is that our choice of direction is based on those quantities that are within the maximum control of the manager. Further, in many cases you will find that the choice of direction will have a weak effect on the data obtained. (e.g. see Coelli and Perelman 1996).

The output-dominated model is very similar to the input-dominated model. Consider the following example of an output-dominated VRS model.

maxφ,λφ,

st -φyi+Yλ≥0,

xi-Xλ≥0,

N1'λ=1

λ≥0, (17)

Where 1 ≤ φ<∞, φ-1 is the proportion of output increase the ith DMU can achieve by keeping the quantity of inputs constant. the proportion by which output can be increased while keeping the quantity of inputs constant. Note that φ-1 defines an index of technical efficiency that varies between 0 and 1 (this is the output-dominant technical efficiency derived by the DEAP software) An example of two outputs of an output-dominant DEA can be represented by a segmented linear output possibilities curve, as illustrated in Data 8.

Note that the observation point is below the curve, and a portion of the curve is at right angles to the axes, and when the production point is projected onto this portion of the curve by a radial line of inflated output, the slack variable for output is computed.

For example, point P is projected onto point P', which is on the frontier, but not on the efficiency frontier, because without any increase in output, product Y1 can be increased by AP'. AP' is the output slack of output y1 variable.

One point that needs to be emphasized is that the output-dominant and input-dominant models will identify the same DMU as the most efficient by defining exactly the same estimated frontier,. The efficiency measures for inefficient DMUs may be different in the two approaches. Both methods we will show in the second part of Data 3, where we can observe that both provide the same efficiency value only when the returns to scale are constant.

Data 8

Output-Dominated DEA

3.4 Price Information and Allocative Efficiency

If we have information about prices, we consider behavioral objectives such as minimizing cost or maximizing output so that we can measure both technical and allocative efficiency. For the cost minimization example of VRS, we will use the input dominated DEA model included in Equation 15 to calculate technical efficiency. We then run the following cost-minimizing DEA model.

minλ,xi* wi'xi*,

st -yi+Yλ≥0,

xi*-Xλ≥0,

N1'λ=1

λ≥0, (23)

Where wi is the input of the ith DMU price matrix, and xi* (computed from the linear programming model) is the number of inputs that minimize the cost of the ith DMU for a given input price wi and output level yi. The cost efficiency or economic efficiency of the total ith DMU can then be calculated in this way

CE=wi'xi*/wi'xi.

That is, observe the cost in the proportion that minimizes the cost. We can then apply Equation 4 to calculate the allocative efficiency residual as

AE=CE/TE.

Note that this calculation does not create any slack variables in the measure of allocative efficiency. This is always reasonable in the context of slack variables reflecting inappropriate input ratios.

We also note that we can consider revenue maximization and allocative inefficiency for output mixing choices in the same way. See Lovell (1993, p33) for a detailed discussion of this. Note that this revenue efficiency model cannot be implemented inside DEAP

Example 3 Example 3

In this example we use the data from Example 1 with the added information that all firms have prices of 1 and 3 for Input 1 and Input 2, respectively.Thus we draw a cost curve with a slope of -1/3 on Data 6 who is tangent to the Equivalent yield curves are tangent. From the table we see that firm 5 is the only cost efficient firm and that all the other firms are allocatively inefficient to some degree. A variety of different cost efficiencies and allocative efficiencies are listed in Table 5. The calculation of these efficiencies can be demonstrated using firm 3. We have long found that the technical efficiency of firm 3 is calculated by a ray from the origin (O) to 3. His ratio is the ratio of the distance from 0 to the point 3' over the distance from 0 to the point 3, which is equal to 0.833. The configurational efficiency is the ratio of the distance from 0 to 3'' over the distance from 0 to 3 's distance is 0.9. cost efficiency is the ratio of the distance from 0 to 3'' over the distance from 0 to 3, which equals 0.75. We can also calculate 0.833 x 0.9 = 0.750 in this way.

Data 9

Example of CRS Cost Efficiency DEA

Table 5

Results of CRS Cost Efficiency DEA

3.5 Panel Data,DEA and the Malmquist Index Panel Data,DEA and the Malmquist Index

When we have panel data, we can use a DEA-like linear programming and a ( input or output based) Malmquist TFP index to measure changes in productivity and decompose changes in productivity into changes in technical progress and technical efficiency.

Fare et al (1994) define an output-based Malmquist productivity index as follows:

It represents the productivity at production point (xt+1,yt+1) compared to production point (xt,yt). A value larger than 1 represents a positive TFP growth from period t to t+1. In fact, this index is the equalized middle term of two output-based Malmquist TFP indices. One index uses technology in period t and the other in period t+1. In order to compute Equation 24, we have to compute the function with four parts who contains four linear programs (their construction is similar to the method used to compute Farrell's technical efficiency.) of the problem.

We assume that the returns to scale are constant (we'll break this down further later to look at the problem of scale efficiency)

The CRS output-dominated linear program used to compute d0t(xt,yt) is equivalent to Equation 17, with the exception that the restriction on the VRS is removed, yet it contains a time subscript. It is

[d0t(xt,yt)]-1 =maxφ,λφ,

St -φyit+Ytλ≥0,

xit-Xtλ≥0,

λ≥0, (25)

The other three linear programming problems are simple variations of this:

[d0t+1(xt+1 ,yt+1)]-1=maxφ,λφ,

St -φyi,t+1+Yt+1λ≥0,

xi,t+1-Xt+1λ≥0,

λ≥0, (26)

[d0t(xt+1,yt+1)]-1=maxφ,λφ,

St -φyi,t +1+Ytλ≥0,

xi,t+1-Xtλ≥0,

λ≥0, (27)

[d0t+1(xt,yt)]-1=maxφ,λφ,

st -φyit+Yt+1λ≥0,

xit-Xt+1λ≥0,

λ≥0, ( 28)

Note that in linear programming models 27 and 28, the technologies of the production points are compared across time, and the φ parameter is required to be ≥1 as in the calculation of Farrell's efficiency. points may be above the feasible production mix. This situation is most likely to occur in linear programming 27 when the production points in period t+1 are compared with the technology of the production points in period t. If technological progress occurs, a value of φ<1 is possible. Note that if a technological regression occurs, this may also occur inside linear programming28, but it is very unlikely that a technological regression occurs.

We have to keep in mind the points φ and λ because they may have different values inside the four linear programs.

Furthermore, the four linear programs described above must be computed for every firm in the sample. So, if you have 20 companies and two stages, you will have to perform 80 linear programming operations. Notice also that when you add extra time, you have to compute an additional three linear programs for each firm (to construct a ring index). If you have T time, you must compute (3T-2) linear programs for each sample firm.

So, if you have N companies, you have to compute N x (3T-2) linear programs. For example, when N = 10 companies and T = 10 times, this would involve calculating 20 x (3 x 10-2) = 560 linear programs.

The results for each company for each adjacent time can be tabulated. Or they could suggest brief measures across time or space.

Scale Efficiency Scale Efficiency

The above approach can be extended by decomposing the (CRS) technical efficiency into two components, scale efficiency and pure (VRS) technical efficiency. This involves calculating two additional linear programs (when comparing the two production points) This involves repeated calculations of the linear programs that incorporate the convexity restriction (N1'λ=1) 25 and 26.This is where we want to calculate the VRS (rather than the CRS) efficiency function technical efficiency. We can then use the values of CRS and VRS to calculate the scale efficiency residuals, using the methodology in Section 3.2. For the case with N firms and T times, we have to increase the number of linear programs from N × (3T-2) to N × (4T-2). See Fare et al (1994, p75) for more on scale efficiency.

Example 4 Example 4

In this example we use the data from example 2. And add an extra year of data. This data is presented in Table 6 and also depicted in Data 9. The frontiers of the CRS and VRS DEA models for both times are also plotted in Data 9. The different distances (technical efficiencies) require us to compute the Malmquist index, which is also presented in Table 10c in Section 5.4.

Table 6

Example data for Malmquist DEA

Data 8

Example of VRS input-dominated DEA