Zhu J., Cook W.D. (Eds.) Modeling Data Irregularities and Structural Complexities in Data Envelopment Analysis

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Chapter 5

NON-DISCRETIONARY INPUTS

John Ruggiero

Department of Economics and Finance, University of Dayton, Dayton, OH 45469-2251,

ruggiero@notes.udayton.edu

A motivating factor for Data Envelopment Analysis (DEA) was the

measurement of technical efficiency in public sector applications with

unknown input prices. Many applications, including public sector production,

are also characterized by heterogeneous producers who face different

technologies. This chapter discusses existing approaches to measuring

performance when non-discretionary inputs affect the transformation of

discretionary inputs into outputs. The suitability of the approaches depends on

underlying assumptions about the relationship between non-discretionary

inputs and outputs. One model treats non-discretionary inputs like

discretionary inputs but uses a non-radial approach to project inefficient

decision making units (DMUs) to the frontier holding non-discretionary inputs

fixed. A potential drawback is the assumption that convexity holds with

respect to non-discretionary inputs and outputs. Alternatively, the assumption

of convexity can be relaxed by comparing DMUs only to other DMUs with

similar environments. Other models use multiple stage models with regression

to control for the effect that non-discretionary inputs have on production.

Data Envelopment Analysis (DEA), Performance, Efficiency, Non-

discretionary inputs

1. INTRODUCTION

Data Envelopment Analysis is programming modeling approach for

estimating technical efficiency assuming constant returns to scale introduced

by Charnes, Cooper and Rhodes (1978) (CCR) and extended by Banker,

Charnes and Cooper (1984) (BCC) to allow variable returns to scale. An

underlying assumption necessary for the comparison of decision making

Abstract:

Key words:

units (DMUs) in both models is homogeneity – production possibilities of

each and every DMU can be represented by the same piecewise linear

technology. In many applications this assumption is not valid. For example,

in education production depends not only on the levels of discretionary

inputs like capital and labor but also on the socio-economic environment

under which the DMU operates. Schools facing a harsher environment,

perhaps represented by high poverty, will not be able to achieve outcome

levels as high as other schools with low poverty levels even if they use the

same amount of discretionary inputs.

Application of the standard DEA models to evaluate efficiency leads to

biased estimates if the non-discretionary inputs are not properly controlled.

Banker and Morey (1986) (BM) extended DEA to allow fixed,

uncontrollable inputs by projecting inefficient DMUs to the frontier non-

radially so that the resulting efficient DMU had the same levels of the non-

discretionary inputs. The BM approach is consistent with short-run

economic analyses with fixed inputs along the appropriate expansion path.

As such, the model could be used for short-run analyses characterized by

fixed inputs. However, the BM model assumes convexity with respect to

outputs and non-discretionary inputs. In situations where such an

assumption is not valid, the BM model will overestimate technical efficiency

by allowing production impossibilities into the referent set.

An alternative approach, introduced by Ruggiero (1996) restricts the

weights to zero for production possibilities with higher levels of the non-

discretionary inputs. As a result, production impossibilities are appropriately

excluded from the referent set. A limitation of this model is the increased

information set necessary to estimate efficiency. In particular, excluding

DMUs from the analysis eliminates potentially useful information and could

lead to classification of efficient by default. However, simulation analyses

in Ruggiero (1996), Syrjanen (2004) and Muñiz, Paradi, Ruggiero and Yang

(2006) show that the model developed by Ruggiero (1996) performs

relatively well in evaluating efficiency.

The problem of limited information in the Ruggiero model is magnified

when there are multiple non-discretionary inputs. As the number of non-

discretionary inputs increases, efficiency scores are biased upward and the

number of DMUs classified as efficient by default increases. In such cases,

multiple stage models have been incorporated. Ray (1991) introduced one

such model where standard DEA models are applied in the first stage using

only discretionary inputs and outputs. The resulting index, which captures

not only inefficiency but also non-discretionary effects, is regressed in a

second stage on the non-discretionary inputs to disentangle efficiency from

environmental effects. The residual from the regression represents

efficiency. Ruggiero (1998) showed that the Ray model works very well in

Chapter 5

controlling for non-discretionary inputs using rank correlation between true

and estimated efficiency as the criterion. However, the approach

overestimates efficiency and is unable to properly distinguish between

efficient and inefficient DMUs.

An alternative multi-stage model was introduced by Ruggiero (1998) that

extends Ray (1991) and Ruggiero (1996). Like the Ray model, this model

uses DEA on discretionary inputs and outputs in the first stage and

regression in the second stage. Unlike Ray, the residual from the regression

is not used as the measure of efficiency. Instead, the regression is used to

construct an overall index of environmental harshness, which is used in a

third stage model using Ruggiero (1996). Effectively, this approach reduces

the information required by using only one index representing overall non-

discretionary effects. The simulation in Ruggiero (1998) shows that the

multiple stage model is preferred to the original model of Ruggiero when

there are multiple non-discretionary inputs.

Other models exist to control for environmental effects. Muñiz (2002)

introduced a multiple stage model that focuses on excess slack and uses

DEA models in all stages to control for the non-discretionary inputs. Yang

and Paradi (2003) presented an alternative model that uses a handicapping

function to adjust inputs and outputs to compensate for the non-discretionary

factors. Muñiz, Paradi, Ruggiero and Yang (2006) compare the various

approaches using simulated data and conclude that the multi-stage Ruggiero

model performed well relative to all other approaches. Each approach has its

own advantages. For purposes of this chapter, the focus will be on the BM

model, Ray’s model and the Ruggiero models.

The rest of the chapter is organized as follows. The next section presents

the BM model to control for non-discretionary inputs. Using simulated data

sets, the advantages and disadvantages are discussed. The models due to

Ruggiero are then examined in the same way. The last section concludes.

2. PRODUCTION WITH NON-DISCRETIONARY

INPUTS

We assume that DMU

(j = 1, …, n) uses m discretionary inputs x

(i =

1,…,m) to produce s outputs y

(k = 1,...,s) given r non-discretionary inputs

(l = 1,…,r). For convenience, we assume that in an increase in any non-

discretionary input leads to an increase in at least one output, ceteris paribus.

As a result, higher levels of the non-discretionary inputs lead to a more

favorable production environment. Naïve application of the either the CCR

or the BCC models, without properly considering the non-discretionary

variables, leads to biased efficiency estimates. In this section, we consider

Ruggiero, Non-discretionary Inputs

such applications using the BCC model. The same arguments apply to the

CCR model as well.

Consider first an application of the BCC model that only considers

outputs and discretionary inputs. This DEA model can be written as

min

subject to

1,..., ;

0 1,2,..., .

jkj ko

jij oio

yk s

xi m

θθ

λθ

≥=

≤=

≥=

∑

(1)

This formulation ignores the important role of the non-discretionary

inputs. For illustrative purposes, we show how model (1) leads to biased

efficiency estimates. In Figure 5-1, it is assumed that six A – F produce one

output using one discretionary input and one non-discretionary input. It is

further assumed that all DMUs are operating efficiently. DMUs A – C [D –

F] all have the same non-discretionary input z

(with z

= z

)

] where z

< z

. Because DMUs D – F have a higher level of the non-

discretionary input, each is able to produce a higher level of output y

with

input usage x

0 5 10 15 20

= f(x

)

= f(x

)

Figure 5-1. Production with Non-discretionary Inputs

Chapter 5

DMUs A – F are all on the frontier producing output with the minimum

discretionary input level possible. Effectively, multiple frontiers arise

depending on the level of the non-discretionary input. This set-up is

consistent with microeconomic principles of short-run production with fixed

inputs. Application of the BCC model results in incorrect classification of

inefficiency for DMUs A – C with projections to the frontier consistent with

non-discretionary input level z

. In the case of DMU A, “inefficiency”

results from output slack and not radial contraction of the discretionary

input. Efficient DMU C achieves an efficiency rating of 0.44 with a

weighting structure of ⅔ (⅓) on DMU D (E). The resulting index is not

capturing production inefficiency but rather an index of inefficiency and the

effect that non-discretionary inputs have on production.

An alternative model would apply the BCC model with non-discretionary

inputs treated the same as the discretionary inputs. This approach was

adopted by Bessent, Bessent, Kennington and Reagan (1982) in an analysis

of elementary schools in the Houston Independent District. The BCC

version of this model is given by:

min

subject to

1,..., ;

0 1,2,..., .

jkj ko

jij oio

jij olo

yk s

xi m

zzl r

θθ

λθ

≥=

≤=

≥=

∑

(2)

The problem with this model is the potential contraction of the non-

discretionary inputs, which violates the assumption that these variables are

exogenously fixed. The resulting projection for inefficient DMUs will be to

a production possibility inconsistent with the DMUs’ fixed level on the non-

discretionary input. This is shown in Figure 5-2, where four DMUs A – D

are observed producing the same output with one discretionary input x

and

one nondiscretionary input x

. Application of (2) for DMU D leads to a peer

comparison of DMU B. This is problematic because it implies that DMU D

Ruggiero, Non-discretionary Inputs

is able to reduce the amount of exogenously fixed input x

from 20 units to

10.

There have been a few different approaches to measuring efficiency

relative while controlling for the fixed levels of the nondiscretionary inputs.

In the next section, we consider the Banker and Morey (1986) model that

introduced the problem of inappropriate projections via equi-proportional

reduction in all inputs. In particular, nondiscretionary variables are treated

as exogenous and projections to the frontier are based on holding the

nondiscretionary inputs fixed.

3. THE BANKER AND MOREY MODEL

3.1 Input-Oriented Model

Banker and Morey (1986) introduced the first DEA model that allowed

for nondiscretionary inputs by modifying the input constraints to disallow

input reduction on the fixed factor. The input-oriented model seeks the equi-

proportional reduction in discretionary inputs while holding non-

discretionary inputs and outputs constant.

The differences between this model and the standard BCC model are

(i) the separation of input constraints for discretionary and non-discretionary

inputs and (ii) the elimination of the efficiency index from the non-

discretionary inputs. Importantly, this programming model provides the

proper comparison for a given DMU by fixing the nondiscretionary inputs at

the current level. Returning to Figure 5-2, the solution of the BM model for

DMU D leads to a benchmark of DMU A which is observed producing the

same level of output with the same level of the non-discretionary input.

On page 516 of the Banker and Morey (1986) paper, the CCR equivalent model is presented.

Equation 32 multiplies the observed nondiscretionary input by the sum of the weights.

However, decreasing returns to scale will lead to frontier projections to levels of the

nondiscretionary inputs above the observed level. This appears to be at typo; in the

paragraph above this model, the authors state that the CCR version of their model is

obtained via the standard method of removing the convexity constraint. The simulation

analysis in this section confirms this point.

Chapter 5

0 5 10 15 20 25

Figure 5-2. Radial Contraction

Assuming variable returns to scale, the BM model for evaluating the

efficiency of DMU “o” is given by:

min

subject to

1,..., ;

0 1,2,..., .

jkj ko

jij oio

jlj lo

yk s

xi m

zz l r

θθ

λθ

≥=

≤=

≥=

∑

(3)

Ruggiero, Non-discretionary Inputs

3.2 Illustrative Example Using Simulated Data

In this sub-section, the BM model is applied to simulated data for

illustrative purposes. We assume that efficient production is characterized

by constant returns to scale where two inputs are used to produce one output

according to the following production function:

0.4 0.6

112

yxx=

Both inputs are generated from a normal distribution with mean 100 and

standard deviation 25. It is assumed that input x

is fixed. Ln θ is generated

from a |N(0,0.2)| distribution and observed production data is calculated in

two ways:

Scenario 1:

0.4 0.6

112

;

jj j

yxx

and

Scenario 2: observed x

= efficient x

/θ

The first scenario leads to inefficiency in the output, but since the production

function is characterized by constant returns to scale, the input-oriented

constant returns to scale model should be appropriate. The second scenario

assumes inefficient usage of only the discretionary input. Tables 3-1 and 3-2

present the data and efficiency results using the BM model for scenarios 1

and 2, respectively.

The results in Table 5-1 show that the BM has a tendency to over-

estimate inefficiency given the data generating process in scenario 1.

Notably, the mean absolute difference (MAD) between true and estimated

efficiency is 0.10. However, the rank correlation is a high 0.95. The

overestimation arises because efficiency is measured relative to the

discretionary inputs only, ignoring the contribution to output from the

nondiscretionary inputs. This problem was identified in Ruggiero (2005)

which provided an alternative BM model consistent with this data generating

process. To evaluate the performance of DMU “o” under variable returns to

scale, the alternative model is:

min

subject to

1,..., ;

0 1, 2,..., .

jkj ko

jij oio

jlj lo

mr mr

yy k s

xi m

zz l r

θθ

λθ

≥=

≤=

≥=

∑

(4)

Chapter 5

Table 5-1. Simulated Data, Scenario 1

DMU

Alternative

92.49 68.06 67.48 0.88 0.73 0.86

106.11 131.91 113.06 0.94 0.88 0.94

129.96 143.33 133.88 0.97 0.96 0.98

45.41 94.15 62.37 0.89 0.86 0.93

127.38 72.83 88.31 0.97 0.98 0.99

82.74 57.74 42.66 0.64 0.36 0.68

53.83 75.56 64.24 0.97 0.97 0.99

80.66 47.05 44.75 0.77 0.53 0.76

85.80 89.90 85.52 0.97 0.95 0.97

103.37 90.86 86.52 0.90 0.80 0.90

91.83 90.74 70.06 0.77 0.55 0.77

133.57 97.87 97.92 0.88 0.74 0.87

95.35 87.17 77.80 0.86 0.71 0.86

149.31 121.64 128.22 0.97 0.95 0.98

159.39 83.63 103.35 0.95 0.95 0.98

141.54 59.69 80.87 0.96 1.00 1.00

113.47 122.55 86.69 0.73 0.51 0.76

147.97 97.89 114.61 0.99 1.00 1.00

86.91 116.88 100.83 0.97 0.97 0.98

90.47 118.94 73.93 0.69 0.48 0.74

63.90 78.82 63.88 0.88 0.76 0.88

61.96 90.93 60.49 0.78 0.61 0.81

99.19 100.70 99.79 1.00 1.00 1.00

91.93 154.86 124.10 0.99 1.00 1.00

56.44 81.59 51.02 0.72 0.53 0.77

35.56 136.19 74.03 0.93 1.00 1.00

68.01 83.66 70.56 0.92 0.84 0.92

118.94 111.67 109.26 0.95 0.90 0.95

121.87 114.89 102.61 0.87 0.74 0.87

65.70 72.11 68.53 0.99 0.99 0.99

117.35 108.07 96.40 0.86 0.72 0.86

76.50 93.98 69.50 0.80 0.63 0.82

103.29 113.94 105.18 0.96 0.93 0.97

103.47 77.23 71.34 0.82 0.63 0.82

147.12 112.18 84.83 0.68 0.41 0.70

All calculations by author.

Ruggiero, Non-discretionary Inputs

Table 5-2. Simulated Data, Scenario 2

DMU

Alternative

105.47 68.06 76.94 0.88 0.89 0.95

113.47 131.91 120.91 0.94 0.96 0.98

133.78 143.33 137.82 0.97 0.98 0.99

51.21 94.15 70.33 0.89 0.94 0.97

131.37 72.83 91.08 0.97 1.00 1.00

129.34 57.74 66.68 0.64 0.65 0.82

55.28 75.56 65.97 0.97 1.00 1.00

105.23 47.05 58.37 0.77 0.79 0.89

88.53 89.90 88.24 0.97 0.98 0.99

114.31 90.86 95.67 0.90 0.92 0.96

119.50 90.74 91.17 0.77 0.77 0.89

151.18 97.87 110.83 0.88 0.90 0.95

110.73 87.17 90.35 0.86 0.87 0.94

153.74 121.64 132.03 0.97 0.99 1.00

166.94 83.63 108.24 0.95 0.99 1.00

147.56 59.69 84.31 0.96 1.00 1.00

155.56 122.55 118.84 0.73 0.74 0.87

149.09 97.89 115.48 0.99 1.00 1.00

89.48 116.88 103.81 0.97 1.00 1.00

130.46 118.94 106.61 0.69 0.71 0.86

72.49 78.82 72.47 0.88 0.90 0.95

79.90 90.93 77.99 0.78 0.80 0.90

99.49 100.70 100.09 1.00 1.00 1.00

93.12 154.86 125.71 0.99 1.00 1.00

77.88 81.59 70.41 0.72 0.74 0.87

38.24 136.19 79.59 0.93 1.00 1.00

74.22 83.66 77.01 0.92 0.94 0.97

124.67 111.67 114.52 0.95 0.97 0.98

139.71 114.89 117.63 0.87 0.88 0.94

66.61 72.11 69.47 0.99 1.00 1.00

135.96 108.07 111.69 0.86 0.88 0.94

95.28 93.98 86.55 0.80 0.82 0.91

107.58 113.94 109.56 0.96 0.97 0.99

125.91 77.23 86.81 0.82 0.84 0.92

216.84 112.18 125.03 0.68 0.69 0.85

All calculations by author.

Chapter 5