keshvari2008

November 4, 2008 9:20 WSPC/APJOR
00195.tex
Asia-Pacific Journal of Operational Research
Vol. 25, No. 5 (2008) 689–696
c World Scientific Publishing Co. & Operational Research Society of Singapore
AN EXTENDED NUMERATION METHOD FOR SOLVING
FREE DISPOSAL HULL MODELS IN DEA
Asia Pac. J. Oper. Res. 2008.25:689-696. Downloaded from www.worldscientific.com
by MONASH UNIVERSITY on 12/15/14. For personal use only.
ABOLFAZL KESHVARI∗ and NASIM DEHGHAN HARDOROUDI
Faculty of Mathematics, Iran University of Science
and Technology, Tehran, Iran
∗[email protected]
Received 7 August 2007
Accepted 25 January 2008
Production Possibility Set (PPS) based on Free Disposal Hull assumption describes the
minimum PPS for evaluating efficiency of DMUs and presents one reference for each
unit. Tulkens (Journal of Productivity Analysis, 4(1), 183–210) proposed a mathematical program and a procedure for solving FDH model that can be used for only VRS
technology. In this paper, we extend the method for solving all four standard technologies
(VRS, CRS, NDRS and NIRS) by a numeration algorithm without using LP or MILP
regular solving methods.
Keywords: Free Disposal Hull (FDH); dominant units; efficiency; data envelopment
analysis.
1. Introduction
Free Disposal Hull assumption was initially presented for variable returns to scale
model by Deprins et al. (1984). VRS model (Banker et al., 1984) is a linear programming with continuous variables while FDH model uses the zero-one variables
and presents one reference for each unit. VRS was the unique technology defined
on FDH approach by a mixed integer linear programming modeling for about two
decades. Agrell and Tind (2001) have developed a linear programming form to FDH
model and Leleu (2006) introduced CRS, NDRS and NIRS technologies by a linear
program.
The FDH models are more acceptable in comparison with convex models because
it presents a single reference for each DMU instead of a combination of some units
and it could be more applicable in the real world problems. A problem about evaluating efficiency of bank branches is a regular form of DEA problems, for example.
In this case constructing a convex combination of efficient units as a reference for
inefficient unit may be rejected by the manager of the branches. In this case we
must use a single reference method such as FDH models.
∗Corresponding
author.
689
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A. Keshvari & N. Dehghan Hardoroudi
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00195.tex
The Free Disposal Hull (FDH) model was suggested to form a non-convex hull
imposing strong disposability assumptions by Deprins et al. (1984) and has no
assumptions regarding returns to scale. Tulkens (1993) proposed a mathematical
programming formulation and a procedure for solving FDH model. Agrell and Tind
(2001) introduced an LP model for FDH and Leleu (2006) relied on it and introduced
RTS in FDH models with a LP framework. The first FDH model by Tulkens (1993)
was a VRS FDH model in MILP format and Agrell and Tind (2001) created an
LP for VRS FDH model and Leleu (2006) extended their model to introduce CRS,
NDRS and NIRS FDH models in LP format. The procedure of Tulkens (1993)
computes efficiency of units by a numeration algorithm without solving its MILP
model by regular methods in Operations Research. In this paper we extend the
method of Tulkens (1993) for CRS, NDRS and NIRS technologies on FDH models.
In Sec. 2, the development of FDH models is summarized. Section 3 presents
the extended numeration method. In Sec. 4, a numerical example is used to test the
extended model and finally some concluding remarks are presented in Sec. 5.
2. Free Disposal Hull Models
We start by definition of FDH assumption and notations. The FDH assumption is
a technique for estimating a free disposal hull covering a set of observations called
Decision Making Units (DMUs), and computing radial distance between DMUs
and efficiency frontier. Each DMU is a point in the space of m+s and traditionally
specify each of them via a vector x ∈ m as input vector and a vector y ∈ s
as output vector. We specify jth DMU via uj = (yj , −xj ) for simplifying in other
notations. J is set of all indices of DMUs and R and I are sets of indices of outputs
and inputs, respectively.
Consider FDH model in LP framework by Agrell and Tind (2001) and Leleu
(2006):
min
θpj ,λj ,ωj
J
θjp
p=1
s.t.
yrj (λj + ωj ) ≥ yrp λj ,
xij (λj + ωj ) ≤ xip θjp ,
n
λj = 1,
j=1
λj ≥ 0,
ω j ∈ Γj ,
j∈J
r ∈ R, j ∈ J
i ∈ I, j ∈ J
where
Γj ∈ {NIRS, NDRS, CRS, VRS}
with
NIRS = {ωj : ωj ≤ 0},
(FDH)
NDRS = {ωj : ωj ≥ 0},
CRS = {ωj : ωj unconstrained}
VRS = {ωj : ωj = 0}.
j∈J
ωj is the factor of scaling for DMUs for creating various returns to scale based on
VRS model, which is described in Leleu (2006).
November 4, 2008 9:20 WSPC/APJOR
00195.tex
An Extended Numeration Method for Solving Free Disposal Hull Models in DEA
Asia Pac. J. Oper. Res. 2008.25:689-696. Downloaded from www.worldscientific.com
by MONASH UNIVERSITY on 12/15/14. For personal use only.
CRS
VRS
NDRS
691
NIRS
Fig. 1. Efficiency frontiers in CRS, VRS, NDRS and NIRS FDH models.
For an illustration of various FDH models, consider Fig. 1 that shows the frontier
of various technologies on FDH model.
As Fig. 1 shows VRS has the smallest production possibility set (PPS) and CRS
has the biggest, and no one is convex in the case of multi inputs and outputs.
As mentioned above, Tulkens (1993) FDH model is a variable returns to scale
model and so its procedure for solving FDH models can be used only for VRS FDH.
Although LP framework is a suitable format for FDH models in both theoretical
and applications but we want to extend Tulkens (1993) numeration procedure to
all FDH technologies such that in the case of VRS the extended model is the same
as Tulkens (1993) procedure. The Tulkens (1993) procedure has two capabilities in
comparison with LP format of FDH models. The first is to find all dominant units
of under assessed unit which is important when we want to propose more than one
unit as reference for units under FDH assumption. In this case DMUs can accept
or reject one or more references and finally the most suitable reference for each unit
can be determined. The second capability of procedure is to find efficiency of DMUs
without using linear programs and so it does not need to use LP solvers. Therefore
a code for solving FDH models can be easily prepared.
In the next section we introduce the extended method for solving various FDH
models by a numeration algorithm.
3. Extended Numeration Algorithm
In this section we propose a numeration method for solving VRS, CRS, NIRS
and NDRS FDH technologies that is an extension to Tulkens (1993) procedure. The numeration method is based on specifying all dominant units of under
assessed unit. By this method we can solve FDH problems by a trade off between
units. Consider to MILP FDH model which will be used for describing the
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A. Keshvari & N. Dehghan Hardoroudi
numeration method:
min θ
θj ,λj ,ωj
s.t.
yrj (λj + ωj ) ≥ yrp ,
r ∈ R,
xij (λj + ωj ) ≤ xip θ,
i ∈ I,
where
Γj ∈ {NIRS, NDRS, CRS, VRS}
j
j
n
with
NIRS = {ωj : ωj ≤ 0},
λj = 1,
VRS = {ωj : ωj = 0}.
Asia Pac. J. Oper. Res. 2008.25:689-696. Downloaded from www.worldscientific.com
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j=1
λj ∈ {0, 1},
(MILP FDH)
NDRS = {ωj : ωj ≥ 0},
CRS = {ωj : ωj unconstrained}
j∈J
ω j ∈ Γj ,
ωj is the factor of scaling for DMUs for creating various returns to scale.
The numeration method consists of two steps, filtering DMUs for finding dominant units and specifying reference unit. The numeration method for VRS, CRS,
NDRS and NIRS technologies is as follows:
Suppose we want to evaluate efficiency of pth DMU.
Step 1. Filtering DMUs
Create the set of all units that dominant pth under specified RTS of the Γ and
specify them by DpΓ .
DpVRS = {j ∈ J|uj ≥ up },
DpCRS = {j ∈ J|∃δj , (1 + δj )uj ≥ up },
DpNDRS = {j ∈ J|∃δj ≥ 0, (1 + δj )uj ≥ up }
DpNIRS
and
= {j ∈ J|∃δj ≤ 0, (1 + δj )uj ≥ up }.
For an illustration of dominant units consider Fig. 2 and dominant units for 3rd
DMU.
5
2
4
3
1
Fig. 2. Dominant units for 3rd DMU in CRS, VRS, NDRS and NIRS FDH models.
November 4, 2008 9:20 WSPC/APJOR
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An Extended Numeration Method for Solving Free Disposal Hull Models in DEA
693
We have D3VRS = {2, 3}, D3CRS = {1, 2, 3, 4, 5}, D3NDRS = {1, 2, 3} and D3NIRS =
{2, 3, 4, 5}.
Step 2. Finding the Reference
Efficiency could be computed by the following formula:
θp∗ = min {max(xij /xip ) × max(yrp /yrj )}
j∈DpΓ
r
i
(for CRS, NDRS and NIRS),
θp∗ = min {max(xij /xip )} (for VRS).
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j∈DpVRS
i
As could be seen in the algorithm, first it finds all dominant units for pth and then
calculates the optimum θ. In the case of VRS FDH model, the proposed model is
the same as Tulkens (1993) procedure. Now we prove the validity of the algorithm in
the next lemma and theorem. In the next lemma we prove only members of DpΓ are
eligible to be select as the reference unit and in the theorem we prove the efficiency
value computed in the algorithm.
Lemma. If j ∈
/ DpΓ then jth unit is not reference of pth DMU under Γ-RTS, where
Γ ∈ {VRS, CRS, NDRS, NIRS}.
Proof. Suppose j ∈
/ DpΓ is reference of pth unit. Therefore there is ∀δj
(= 0, f ree, ≥ 0, ≤ 0) : (1 + δj )uj ≥ up . But it is the reference of pth unit so by the
model FDH we have (1 + δj )yrj ≥ yrp , r ∈ R, therefore ∀ δj (= 0, f ree, ≥ 0, ≤ 0),
∃i ∈ I, (1 + δj )xij > xip . We know that if unit j is reference of pth unit in model
MILP FDH then also λj = 1, so the second group of constraints in this model
impose that θj ≥ (xij /xip )(1 + δj ), i ∈ I and θk ≥ 0, k = j, so θp∗ > 1 and this is
impossible.
Theorem. θp∗ which is defined in the algorithm is optimal solution of FDH model.
Proof. There is j∈J λj = 1, λj ∈ {0, 1}, j ∈ J by MILP FDH model, so ∃j ∈
J, λj = 1, λk = 0, k = j, so (λj + ωj ) ≥ 0 and (λk + ωk ) = 0, k = j because
(λj + ωj ) = λj (1 + δj ). There is 1 + δj ≥ (yrp /yrj ), r ∈ R from the first group of
constraints and from the second group of constraints we have the following:
θp∗ ≥ (xij /xip )(1 + δj ) ≥ (xij /xip )(yrp /yrj ).
Based on the Lemma, we must find the minimum value θ over the members of
DpΓ . So the final expression is as θp∗ = minj∈DpΓ {maxi (xij /xip ) × maxr (yrp /yrj )} for
CRS, NDRS and NIRS technologies. In the case of VRS technology, δj = 0 and
then θp∗ ≥ (xij /xip ) and θp∗ = minj∈DpVRS {maxi (xij /xip )}.
4. Numerical Example
Consider the sample data set in Table 1 consists of 28 units with three inputs and
three outputs (Charnes et al., 1989).
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Table 1. Sample data for example.
DMU
Input 1
Input 2
Input 3
Output 1
Output 2
Output 3
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
483.01
371.95
268.23
202.02
197.93
178.96
146.04
189.93
23.33
116.91
129.62
106.26
89.7
109.26
85.5
72.17
76.18
73.21
86.72
89.09
77.69
97.42
54.96
67.03
46.3
65.12
20.09
69.81
397736
855509
685584
452713
471650
423124
3670112
408311
245542
305316
295812
198703
210891
282209
184992
223327
161159
144163
190043
158439
135046
206926
79563
144092
100431
96873
50717
117790
616961
385453
341941
117429
112634
189743
97004
111904
91861
91710
92409
53499
95642
84202
49357
73907
47977
43312
55326
66640
46198
66120
43192
43350
31428
28112
54650
30976
6785798
2505984
2292025
1158016
1244124
1187130
658910
993238
854188
606743
736545
454684
494196
842854
776285
490998
482448
515237
625514
382880
867467
830142
521684
869973
604715
601299
145792
319218
1594957
545140
406947
135939
204909
190178
86514
1411954
135327
78357
114365
67154
78992
149186
116974
117854
67857
114883
173099
74126
65229
128279
37245
86859
55989
37088
11816
31726
1088699
835745
473600
336165
317709
605037
239760
353896
239360
208188
298112
233733
1188553
243361
234875
118924
158250
101231
130423
123968
262876
242773
184055
194416
127586
224855
24442
169051
Table 2. Basic results of FDH efficiency evaluation.
Technology
CRS
VRS
NDRS
NIRS
Reference unit
Efficiency
24
15
24
15
0.61
0.73
0.61
0.65
We want to find FDH-efficiency for 10th unit in CRS, NDRS, NIRS and VRS
technologies. First we solve LP FDH models as Leleu (2006) and present results in
Table 2.
This is a regular result table for a FDH problem and we can suggest to unit
10 to increase efficiency by reducing its inputs. For example targets of inputs are
∗
= [71.778, 187452.32, 56306] for CRS.
X10
Now we use the numeration method and create its results in Table 3.
As can be seen in Table 3, we can compute efficiency by this method that is
faster than regular LP models.
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An Extended Numeration Method for Solving Free Disposal Hull Models in DEA
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Table 3. Numeration method results of 10th unit.
Technology
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CRS
VRS
NDRS
NIRS
Dominant units
Reference unit
Efficiency
8, 9, 10, 11, 14, 15, 17, 21, 22, 23, 24, 25
10, 14, 15, 22
10, 14, 15, 17, 21, 22, 23, 24, 25
8, 9, 10, 11, 14, 15, 22
24
15
24
15
0.61
0.73
0.61
0.65
Also, using numeration method we can find all dominant units. Finding all
dominant units can be important if the under assessed unit doesn’t accept the
results of evaluation. So, we can suggest another reference and compute efficiency
by θ∗ formulation as step 2. This case occurs when suggested reference doesn’t has
similar situations to under assessed unit, for example if suggested reference is a
branch of a bank in a commercial section in a big city and under assessed unit is a
branch in a medium or little city.
5. Conclusion
There is a numeration method that could solve FDH problems proposed by Tulkens
(1993). We extend the numeration method for CRS, NDRS and NIRS FDH technologies and prove the algorithm. Using numeration method can result a set of
potentially references that can be used for step by step improvement.
References
Agrell, PJ and J Tind (2001). A dual approach to nonconvex frontier models. Journal of
Productivity Analysis, 16(2), 129–147.
Banker, RD, A Charnes and WW Cooper (1984). Some method for estimating technical
and scale inefficiencies in data envelopment analysis. Management Science, 30(9),
1078–1092.
Charnes, A, WW Cooper and S Li (1989). Using DEA to evaluate relative efficiencies in
the economic performance of Chinese-Key cities. Socio-Economic Planning Sciences,
23, 325–344.
Deprins, D, L Simar and H Tulkens (1984). Measuring labor efficiency in post offices. In
M. Marchand, P Pestieu and H Tulkens (eds.), The Performance of Public Enterprises: Concepts and Measurements. Amsterdam: North Holland, pp. 247–263.
Leleu, H (2006). A linear programming framework for free disposal hull technologies and
cost functions: Primal and dual models. European Journal of Operational Research,
168, 340–344.
Tulkens, H (1993). On FDH efficiency: Some methodological issues and application to
retail banking, courts, and urban transit. Journal of Productivity Analysis, 4(1),
183–210.
Abolfazl Keshvari graduated in 2005 with a Master’s degree in Applied Mathematics at Iran University of Science and Technology. Since 2006 he has been a
November 4, 2008 9:20 WSPC/APJOR
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A. Keshvari & N. Dehghan Hardoroudi
PhD student at the same university. His research activities mainly concern data
envelopment analysis, mathematical programming and optimization.
Asia Pac. J. Oper. Res. 2008.25:689-696. Downloaded from www.worldscientific.com
by MONASH UNIVERSITY on 12/15/14. For personal use only.
Nasim Dehghan Hardoroudi graduated in 2005 in Applied Mathematics at Iran
University of Science and Technology. Her main research interests are focused on
optimization, mathematical programming and computer science.
00195.tex