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280

Table 15-4 (continued)

Convex DEA Integer DEA

Target Target

DMU

NG NS NB

Ireland 4.11 4.11 0.94 2.34 4.00 4 2 3

Israel 6.07 6.07 2.80 5.00 6.00 6 4 6

Italy 1.92 20.06 19.36 22.72 1.90 20 19 22

Jamaica 1.05 2.11 1.23 1.86 1.00 2 1 2

Japan 1.61 25.69 25.56 35.11 1.56 25 25 35

Kazakhstan 3.57 7.94 6.24 9.33 3.33 7 6 10

Kenya 2.01 4.30 3.65 3.62 1.83 4 3 4

Korea 1.90 19.27 18.50 20.99 1.87 19 18 20

Latvia 1.55 2.46 1.28 2.82 1.25 2 1 2

Lithuania 2.16 3.30 2.03 3.83 2.00 3 2 3

Mexico 23.56 23.67 23.34 30.67 23.00 23 23 30

Mongolia 1.00 0.00 0.00 1.00 1.00 0 0 1

Morocco 4.74 9.47 7.53 11.71 4.50 9 7 11

Netherlands 2.21 14.00 13.00 13.18 2.16 14 12 13

New Zealand 1.31 3.92 2.62 3.87 1.00 3 3 4

Nigeria 33.00 9.65 7.72 11.97 30.50 9 7 11

Norway 1.00 5.00 0.00 1.00 1.00 5 0 1

Paraguay 7.68 2.62 2.44 1.00 6.00 2 2 1

Poland 4.83 14.49 13.12 17.36 4.67 14 13 17

Portugal 11.63 8.63 7.18 9.28 11.00 8 7 9

Romania 1.26 10.08 8.20 12.61 1.25 10 8 12

Russia 1.00 27.00 27.00 38.00 1.00 27 27 38

Serbia/Montenegro 7.16 5.06 4.19 4.81 7.00 5 4 4

Slovakia 2.13 4.73 3.32 5.54 2.00 4 4 4

Slovenia 2.77 2.27 1.19 2.39 2.40 2 1 2

South Africa 7.96 13.75 12.28 18.01 7.60 13 12 18

Spain 3.25 18.72 17.89 19.78 3.12 18 17 19

Sweden 2.10 8.41 4.59 5.38 2.00 8 7 7

Switzerland 4.98 6.74 5.74 6.39 4.67 6 6 6

Syrian Arab Rep. 42.02 6.29 5.06 6.74 41.00 6 5 7

Thailand 4.84 14.51 13.13 19.27 4.67 14 13 19

Trinidad/Tobago 10.15 1.74 0.66 1.89 7.00 1 1 1

Turkey 5.65 17.31 16.23 23.24 5.56 17 16 23

U. Arab Emirates 3.90 3.90 1.83 3.61 3.00 3 3 4

Ukraine 1.09 9.84 7.94 12.27 1.00 9 7 12

United States 1.00 35.00 39.00 29.00 1.00 35 39 29

Uzbekistan 1.91 3.83 3.30 2.87 1.50 3 3 4

Venezuela 39.30 11.33 9.59 14.10 38.00 11 9 14

Zimbabwe 2.34 2.62 2.44 0.98 1.86 2 2 1

Note that the targets of the convex DEA approach are generally fractional

and that the optimal integer output targets given by the proposed model are

not always a rounding up or down of the fractional output targets. Thus, for

Chapter 15

281

example, the integer DEA target for Belgium is (9, 8, 8) which is not the

result of just rounding up or down its convex DEA target (9.44, 6.04, 6.90).

Analogously, the integer target for Israel (6, 4, 6) differs significantly from

the convex target (6.07, 2.80, 5.00).

Even when the integer values correspond to rounding up or down the

fractional DEA targets there is no rule to ascertain which fractional values

should be rounded up and which ones rounded down. The convex target for

Colombia (11.61, 9.90, 14.91) is optimally rounded to (11, 9, 14) while for

Dominican Republic (5.73, 4.67, 5.86) is optimally rounded to (5, 5, 6).

Summarizing, for this application, the consideration of the integer nature

of the outputs leads to more valid targets, i.e. guaranteed to lie within the

integer PPS and to be non-dominated by any other feasible operation point,

as well as more reliable efficiency scores (from which a more effective

ranking of DMU can be derived).

4. OTHER INTEGER DEA MODELS

The above section has illustrated how integrality constraints can be added

to a radial DEA model in order to handle integer inputs and outputs. It has

been also seen that this is compatible with the existence of weight

restrictions as well as with the non-discretional character of the inputs or

outputs.

Radial models are not the only DEA models that can be thus adapted.

Actually, since the efficiency scores in radial DEA models do not capture all

efficiency gains and since in the case of integer inputs and outputs the

remaining (i.e. after the radial phase) slacks may be relatively important,

other DEA formulations may be more appropriate. In this section we will

expand the integer DEA approach presenting other DEA models that can

also be adapted to handle integer inputs and outputs.

Perhaps the easiest integer DEA model that can be formulated is the

integer additive model, which is mathematically related to Phase II of the

integer radial VRS model, i.e.

Lozano & Villa, Integer DEA Models

282

Integer additive DEA model

'Okegerinty'Iiegerintx

k0y,si0x,sj0

ksyy

'Okyy

isxx

'Iixx

.t.s

ssMax

iij

kJk

kkjj

iiJi

iijj

∈∀∈∀

∀≥∀≥∀≥λ

=λ

∀+=

∈∀≥λ

∉∀=λ

∀−=

∈∀≤λ

∉∀=λ

+−

−

+−

∑

∑∑

(9)

One problem with additive DEA models is that the interpretation of the

objective function is meaningful only when the inputs and output slacks are

in the same units, which only occurs occasionally (e.g. Lozano and Salmerón

2005).

A preferable alternative may be the input-oriented non-radial Russell

DEA model, which differs from the BCC-I model in that instead of using a

single contraction factor for all the inputs it uses a different one for each

input dimension. The Russell input-oriented measure of technical efficiency

is the average of these factors along the different input dimensions. Integer

constraints can be easily added to this type of models as follows.

Let

reduction factor for input i

Chapter 15

283

Input-oriented integer non-radial Russell VRS DEA model

'Okegerinty'Iiegerintx

freek0y,si0xj0

ksyy

'Okyy

ixx

'Iixx

.t.s

Min

kJk

kkjj

iJii

iijj

∈∀∈∀

θ∀≥∀≥∀≥λ

=λ

∀+=

∈∀≥λ

∉∀=λ

∀≤θ

∀θ=

∈∀≤λ

∉∀=λ

⋅ε−θ

∑

∑∑

(10)

The output-oriented non-radial Russell DEA model can analogously be

adapted for integer data. In this model, instead of a single radial expansion

scores applied to all the output dimensions, a different expansion factor for

each output dimension is defined and an efficiency score is computed based

on the average of these expansion factors.

Let

expansion factor for output k

Lozano & Villa, Integer DEA Models

284

Output-oriented integer non-radial Russell VRS DEA model

'Okegerinty'Iiegerintx

freek0yi0x,sj0

kyy

'Okyy

isxx

'Iixx

.t.s

Max

kkiij

kJkk

kkjj

iiJi

iijj

∈∀∈∀

γ∀≥∀≥∀≥λ

=λ

∀≥γ

∀⋅γ=

∈∀≥λ

∉∀=λ

∀−=

∈∀≤λ

∉∀=λ

⋅ε+γ

−

∑

∑∑

(11)

Unfortunately, the oriented non-radial Russell DEA models formulated

above also suffer from the problem of the incommensurability of slacks,

although less than radial models and much less than the additive model. To

completely avoid the problem the following non-oriented integer non-radial

DEA model can be used.

Let

relative reduction of input i

relative increase of output k

Chapter 15

285

Non-oriented integer non-radial VRS DEA model

()

'Okegerinty'Iiegerintx

k0y,i0x,j0

k1yy

'Okyy

i1xx

'Iixx

.t.s

Max

kkiij

kkJk

kkjj

iiJi

iijj

∈∀∈∀

∀≥β∀≥α∀≥λ

=λ

∀β+⋅=

∈∀≥λ

∉∀=λ

∀α−⋅=

∈∀≤λ

∉∀=λ

⎟

⎠

⎞

⎜

⎝

⎛

β+α

∑

∑∑

(12)

This model is equivalent to the following formulation, which can be

labelled the integer MIP (Measure of Inefficiency Proportions, Charnes et al

1985) model

Lozano & Villa, Integer DEA Models

286

Integer MIP VRS DEA model

'Okegerinty'Iiegerintx

k0y,si0x,sj0

ksyy

'Okyy

isxx

'Iixx

.t.s

Max

iij

kJk

kkjj

iiJi

iijj

∈∀∈∀

∀≥∀≥∀≥λ

=λ

∀+=

∈∀≥λ

∉∀=λ

∀−=

∈∀≤λ

∉∀=λ

+−

−

∑

∑∑

(13)

Models (12) and (13) have the added advantage of being units-invariant.

Units-invariance is a very desirable, from a practical point of view, feature

of DEA models since the dependency of targets and efficiency gains on the

units of measurement of the data does not contribute to increase the

credibility of DEA in the eyes of the final user of the results. Even if the

final user remains unaware of the possible lack of units-invariance in the

results, that does not solve the issue. Note, however, that we are more

interested in the units-invariance in relation to the continuous inputs and

outputs than in relation to the integer ones since for the integer data with

which we are concerned here there are usually no alternative units of

measurement, i.e. they take small values (e.g. number of machines) that are

absolute numbers.

Another units-invariant model that can be considered here is the RAM

(Range Adjusted Measure of Inefficiency, Cooper et al 1999) model.

Let

=max

– min

range of input i values

R’

=max

– min

range of output k values

Chapter 15

287

Integer RAM DEA model

'Okegerinty'Iiegerintx

k0y,si0x,sj0

ksyy

'Okyy

isxx

'Iixx

.t.s

Max

iij

kJk

kkjj

iiJi

iijj

∈∀∈∀

∀≥∀≥∀≥λ

=λ

∀+=

∈∀≥λ

∉∀=λ

∀−=

∈∀≤λ

∉∀=λ

+−

−

∑

∑∑

(14)

As it can be seen, treatment of integer inputs and outputs in DEA consists

basically in the addition of appropriate integrality constraints. This is easy to

do in all LP DEA models transforming them in MILP but without significant

increase in computational burden. Thus, other DEA models such as the

normalized weighted additive model (Lovell and Pastor 1995) and the

preference structure models of Thanassoulis and Dyson (1992) or Zhu

(1996) can be similarly adapted to handle integer inputs and outputs.

5. CONCLUSIONS

In this chapter the necessity and convenience of taking into account the

integer nature of certain inputs and outputs of DEA applications has been

discussed and an effective approach for adapting common DEA models have

been presented and illustrated. The proposed approach consists in

Lozano & Villa, Integer DEA Models

288

introducing variables in the model to explicitly define the target inputs and

outputs and imposing integrality constraints on these as required.

Note however, that not all DEA models can be adapted in this simple

way. Thus, Non Linear Programming (NLP) DEA models such as the

Russell Graph model (Färe et al 1985) or the MEP (Measure of Efficiency

Proportions, Cooper et al 1999, Green et al 1997) model cannot be so easily

adapted since the resulting model would be a difficult Mixed Integer NLP.

Neither can other DEA models such as the enhanced Russell Graph (Pastor

et al 1999) a.k.a. SBM (Slacks-Based Measure of Efficiency, Tone 2001)

that are NLP but can be linearized.

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Lozano & Villa, Integer DEA Models