A MODEL USING DATA ENVELOPMENT ANALYSIS FOR THE
CROSS EVALUATION OF SUPPLIERS UNDER UNCERTAINTY
Nicola Costantino
1
, Mariagrazia Dotoli
2
, Marco Falagario
1
and Fabio Sciancalepore
1
1
Dipartimento di Ingegneria Meccanica e Gestionale, Politecnico di Bari, Via Re David 200, 70126, Bari, Italy
2
Dipartimento di Elettrotecnica ed Elettronica, Politecnico di Bari, Via Re David 200, 70125, Bari, Italy
Keywords: Business Intelligence, Supplier Evaluation, Data Envelopment Analysis, Uncertainty, Monte Carlo Method.
Abstract: The paper addresses one of the key objectives of the purchasing function of a supply chain, i.e., the optimal
selection of suppliers. We present a novel methodology that integrates the well-known cross-efficiency
evaluation called Data Envelopment Analysis (DEA) and the Monte Carlo approach, to manage supplier
selection considering uncertainty in the supply process, e.g. evaluating potential suppliers. The model
allows to distinguish among several suppliers, overcoming the limitation of the traditional DEA method of
not distinguishing among efficient suppliers. Moreover, the technique is able to classify suppliers with
uncertain performance. The method is applied to the selection of suppliers of a Southern Italy SME.
1 INTRODUCTION
Within the purchasing management area, the process
of supplier selection is currently widely investigated
in the scientific literature, particularly as regards the
private sector, due to its strategic role in the success
of a Supply Chain (SC) (Costantino et al., 2011).
Supplier evaluation techniques periodically identify
and verify the best potential candidates able to
provide the expected performance level to the SC.
Typically, supplier selection is a multi-objective
decision problem with conflicting objectives, such
as, besides the obvious goal of (low) price, also
quality, quantity, delivery, performance, capacity,
communication, service, geographical location, etc.
The wide literature of the area is a synonym of the
importance of supplier choice and the interested
reader may find a detailed discussion of the
appeared contributions in Ho et al. (2010).
Roughly, the multi-objective approaches
proposed in the related literature for the solution of
the supplier selection problem may be classified into
individual model and integrated techniques. One of
the best-known individual methods is the so-called
Data Envelopment Analysis (DEA) technique, due
to its robustness and simplicity of application (Ho et
al., 2010). The DEA technique is based on linear
programming to determine the efficiency of several
units subject to the decision (Charnes et al., 1978).
However, a limitation of the classical DEA is that it
distinguishes only between inefficient and efficient
suppliers, without enabling a discrimination among
the elements of the latter set. Such a characteristic
makes it difficult to apply DEA for supplier
selection, particularly in the case of a single sourcing
purchasing, i.e., with one supplier only. With the
aim of improving the method discriminating power,
Sexton et al. (1986) proposed the so-called cross-
efficiency DEA method that evaluates the decision
units efficiency in a crossed way. However, the
resulting technique is quite complex, since it
requires the set-up and solution of a two-level
optimization problem, and deterministic, so that it
cannot manage uncertain data on suppliers.
Nevertheless, uncertainty is one of the most relevant
issues in SC management and this is particularly
apparent in the supplier selection process. Indeed,
such a problem is often concerned with the
evaluation of potential candidates, with which the
buyer has not had previous commercial
relationships, so that the corresponding key
performance indicators are inevitably vague.
This paper proposes an integrated model for
supplier selection based on the cross-efficiency DEA
and Monte Carlo simulation. The technique main
advantages are two, as follows: on the one hand it
enables the purchasing manager to distinguish
among multiple suppliers that according to the
classical DEA method are considered as equally
efficient, on the other hand it is able to classify
152
Costantino N., Dotoli M., Falagario M. and Sciancalepore F..
A MODEL USING DATA ENVELOPMENT ANALYSIS FOR THE CROSS EVALUATION OF SUPPLIERS UNDER UNCERTAINTY.
DOI: 10.5220/0003638901520157
In Proceedings of the International Conference on Knowledge Management and Information Sharing (KMIS-2011), pages 152-157
ISBN: 978-989-8425-81-2
Copyright
c
2011 SCITEPRESS (Science and Technology Publications, Lda.)
suppliers with uncertain performance against some
defined criteria for evaluation of the supply.
2 THE DEA METHOD FOR THE
EVALUATION OF SUPPLIERS
2.1 The Traditional DEA Technique
The DEA method (Charnes et al., 1978) is a
technique for classifying multiple Decision Making
Units (DMU) in a compared way while measuring
the maximum unit efficiency with respect to the
performance of all the analyzed DMU. In particular,
the units are characterized by the sharing of a set of
resources used to produce goods or services. In the
DEA method the efficient DMU are the vertices of a
Pareto face: based on these, the other DMU
efficiencies are evaluated. As regards the supplier
evaluation and selection area, DEA allows
determining, among a set of partners, a subset
composed by the suppliers using the given (input)
resources to produce the required (output)
goods/services in the most efficient way. To this
aim, several inputs express the contribution required
to the supplier (e.g., the purchasing price, the lead
time, etc.) and several outputs identify its
performance in the procurement process (e.g.,
delivery timeliness, finite product quality).
In general, a supplier selection problem may be
defined by a set of suppliers offering the requested
product
{}
12
, ,.....,
F
Sss s= and by a set of
conflicting objectives
{}
12
, ,.....,
n
Ccc c= , against
which the suppliers are to be classified. We assume
that the set of criteria is partitioned as
I
O
CC C=∪,
with
{}
12
, ,.....,
IH
Ccc c= and
{}
12
, ,.....,
OHH HK
Ccc c
++ +
= , representing the set of
input and output criteria respectively, with H+K=n
being the overall number of criteria.
The generic supplier
f
s
S
has efficiency:
1
1
K
kkf
k
f
H
hhf
h
uy
E
vx
=
=
=
,
(1)
that is the ratio between the weighted sum of the
outputs and the weighted sum of the inputs, where:
y
kf
is the k-th type output (k=1, 2,…K) referred to
supplier
f
s
S ; x
hf
is the h-th type input (h=1,
2,…,H) referred to supplier
f
s
S ; u
k
is the weight
assigned to the
k-th type output; v
h
is the weight
associated with the h-th type input.
The aim of the DEA method is associating to the
outputs and inputs of supplier
f
s
S , given their
values,
a set of weights leading to maximize
efficiency, while taking into account that this cannot
by definition be higher than 1. In such a way the
efficiency of the
f-th supplier (f=1,2,…,F, F being
the total number of analyzed suppliers) is evaluated
solving the following mathematical programming
problem, determining the sets of weights
u
k
and v
h
that maximize
E
f
:
1
1
max
K
kkf
k
f
H
hhf
h
uy
E
vx
=
=
=
with f=1,2,…,F,
subject to
1
1
1
K
kkf
k
H
hhf
h
uy
vx
=
=
with f=1,2,…,F,
,0
kh
uv for k=1,2,…,K; h=1,2,…,H.
(2)
The first constraints of (2) represent an upper
bound (equal to 1) in terms of absolute efficiency for
all suppliers using the optimal weights for the
f-th
vendor, while the second constraints of (2) impose
that all weights are non negative. The programming
problem (2) is non-linear with its unknowns: hence,
determining the solution with numerous potential
suppliers and evaluation criteria is computationally
intensive. However, this problem may be simplified
by linearizing it, using the so-called
output-oriented
procedure
, as follows (Charnes et al., 1978):
1
max max
K
f
kkf
k
Euy
=
=
with f=1,2,…,F
subject to
11
0
KH
kkf hhf
kh
uy vx
==
−⋅
∑∑
with f=1,2,…,F,
1
1
H
hhf
h
vx
=
=
,
,0
kh
uv for k=1,2,…,K; h=1,2,…,H.
(3)
Solving (3) we compute the maximum efficiency of
each vendor: a supplier
f
s
S is efficient if it
exhibits a unitary efficiency value
E
f
. Suppliers may
accordingly be classified in a descending order of
efficiency, thus leading to a ranking.
A MODEL USING DATA ENVELOPMENT ANALYSIS FOR THE CROSS EVALUATION OF SUPPLIERS UNDER
UNCERTAINTY
153
2.2 The Cross-efficiency DEA Method
The described DEA method can only distinguish
between efficient and inefficient suppliers, typically
providing a set of maximally efficient vendors,
without discriminating among these. Such a
characteristic makes it difficult its application to the
supplier selection problem, particularly in the case
of single sourcing purchasing. The so-called cross-
efficiency DEA method (Sexton et al. 1986)
improves the discriminating power of the traditional
DEA technique, evaluating the efficiency of each
supplier not only with respect to the set of weights
that is optimal supplier itself, but also with respect to
the sets of weights that are optimal for the other
vendors, i.e., that maximize their efficiencies. In
such a way, the
f-th supplier efficiency is evaluated
as the cross-efficiency CE
f
given by the average of
all the efficiency values that the supplier obtains by
varying the considered weights: hence the evaluation
becomes a cross-evaluation rather than a self-
evaluation. The resulting cross-efficiency matrix
CE
={CE
fi
}
F
xF
+
\ is determined by evaluating,
with respect to the i-th supplier, considered each
time as a pivot, the related efficiency
CE
fi
of all the
other suppliers with index f. Hence, the generic
diagonal element of CE, indicated by
ii
CE for
i=1,…,F, is determined as the solution of problem
(3) solved with a pivot
i, i.e., with f=i. Hence we set
ii i
CE E= , where E
i
is the optimal value of the
objective function of the linear programming
problem (3) for the
i-th supplier. As summarized
earlier, for each supplier
f
s
S the values of the
optimal input and output weights of its efficiency
with respect to the
i-th pivot supplier
i
k
u and
i
h
v are
successively employed to determine
CE
fi
as follows:
1
1
K
i
kkf
k
fi
H
i
hhf
h
uy
CE
vx
=
=
=
with f,i=1,2,…,F.
(4)
However, (4) is not univocally applicable, since
there exist multiple weight combinations that
maximize the
i-th supplier efficiency. With them,
also the efficiencies of the other suppliers
CE
fi
with
respect to s
i
would vary. Hence, it is necessary to
univocally choose, for each supplier, a set of weights
among the combinations that maximize the
efficiency. To solve the issue, Green and Doyle
(1995) propose a second-level optimization
procedure that is to be executed for each
i-th
supplier after the solution of the described linear
programming problem, as follows:
11,
min
KF
i
kkf
kffi
uy
==
⎛⎞
⎜⎟
⎝⎠
∑∑
subject to:
11,
1
HF
i
hhf
hffi
vx
==
⎛⎞
=
⎜⎟
⎝⎠
∑∑
,
11
0
KH
ii
kki i hhi
kh
uy E vx
==
=
∑∑
,
11
0
KH
ii
kkf hhf
kh
uy vx
==
−⋅
∑∑
with f=1,2,…,F,
,0
ii
kh
uv per k=1,2,…,K; h=1,2,…,H.
(5)
In (5), it is imposed that each pivot supplier
i
s
S
chooses, among the sets of weights
maximizing its efficiency
E
i
, the set that minimizes
the overall efficiency of other vendors. The set of
weights
(
)
,
ii
kh
uv with k=1,2,...,K and h=1,2,…,H
solving this problem is employed to determine all
elements
CE
fi
with f=1,2,…,i-1,i+1,…,F according
to (4). Therefore, all suppliers
f
s
S with f=1,…,F
can be classified according to their overall cross-
efficiency value that can be determined by averaging
the elements of the
f-th row of the cross-efficiency
matrix CE as follows:
1
1
F
f
fi
i
CE CE
F
=
=
with f=1,…,F.
(6)
3 ADDRESSING UNCERTAINTY
IN SUPPLIER SELECTION BY
THE DEA TECHNIQUE
The DEA technique, both in its traditional version
and in its cross-efficiency extension, is
deterministic. In other words, the inputs and outputs
of each supplier are assumed as certain and the
model does not consider any uncertainty element.
On the contrary, such aspects usually characterize
the supply process and should, even more
importantly, be taken into account when the
evaluation is referred to potential commercial
partners, with whom no previous relations exist. The
issue of uncertainty on DEA data was already
addressed in the related literature. Dyson and Shale
(2010), in particular, distinguish among four
different approaches: Imprecise DEA,
Bootstrapping, Chance-Constrained DEA, and
Monte Carlo simulation. The Imprecise DEA model
allows employing performance values that are
KMIS 2011 - International Conference on Knowledge Management and Information Sharing
154
imprecise, i.e., expressed either as values in a range
or as ordinal ranked values (that are defined by a
ranking of the alternatives for the single attribute)
rather than cardinal values. Bootstrapping, instead, is
a technique consisting in re-sampling a sample of
real observations of the uncertain variables: for each
new extraction, the values of the corresponding
objective function are computed, so as to obtain a
sample distribution of the variable. The Chance-
Constrained DEA allows employing stochastic
performance values both in input and output: the
probability constraints assure that the probability
that the observed outputs (inputs) are higher (lower)
than the best possible values overcomes a given
threshold.
This work focuses on the use of Monte Carlo
simulation for the application of DEA to the supplier
selection problem based on stochastic data. We
propose a novel method relying on the idea that the
uncertain input and output values may be modelled
by suitable probability distributions, based on real
observations or on a simple estimate of such data.
The chosen distributions (and their opportunely
estimated characteristic parameters) may be
employed for a series of casual extractions useful to
determine the efficiency of each DMU. Such a
procedure was already adopted in the SC supplier
choice with the traditional DEA method by Wong et
al. (2008). The work by Kao and Liu (2009) is also
significant, evaluating the efficiency of Taiwan
banks with the DEA technique using stochastic input
and output methods that are evaluated by a Beta
distribution: they demonstrate that 2000 iterations
are sufficient to obtain convergent results.
This paper integrates the DEA stochastic
methodology proposed in Kao and Liu (2009) with
the cross-efficiency DEA for application to the
supplier evaluation and selection problem. In
particular, let
X
hf
and Y
kf
be the h-th stochastic input
value (with
h=1,…,H) and the k-th stochastic output
value (with
k=1,…,K), respectively, for supplier
f
s
S . These variables are modelled by a specific
Beta probability distribution, called Beta-PERT
(Vose, 2008). Such a distribution suits very well the
cases in which no information is available on the
values assumed in the past by the stochastic
variables and it is thus impossible to define a
frequency-based probability density function (Law
and Kelton, 2000). Indeed, the characteristic
parameters of such a distribution may be determined
using a three-estimates approach that is inspired
from the stochastic PERT technique for project
management, while obtaining a lower standard
deviation (and thus a more faithful reproduction of
the expert estimate) than the well-known triangular
distribution (Vose, 2008). In particular, calling
min
hf
x
,
mod
hf
x
, and
max
hf
x
respectively the lowest
possible, most probable, and highest possible
estimated values, respectively, that the generic input
variable
X
hf
may assume, the corresponding Beta (or
Beta-PERT) distribution is defined by the following
characteristic parameters (Vose, 2008):
min mod min max
1,
mod max min
()(2 )
()()
hi
hf
hf
Xhf hf hf hf
X
hf X hf hf
xxxx
xxx
μ
α
μ
−−
=
−−
,
max
1,
2,
min
()
()
hf hf
hf
hf
Xhf X
X
Xhf
x
x
αμ
α
μ
=
,
(7)
with
min mod max
4
6
hf
hf hf hf
X
x
xx
μ
++
=
.
(8)
The resulting Beta-PERT distribution is defined
as follows (Vose, 2008):
min mod max
max min min
12
(, , )
( , )( )
hf hf hf
hf hf hf
BetaPERT x x x
Beta x x x
αα
=
=−+
.
(9)
Analogously, considering the stochastic variable
Y
kf
, given
min
kf
y
,
mod
kf
y
, and
max
kf
y
respectively the
lowest possible, most probable, and highest possible
estimated values of the variable we set:
min mod min max
1,
mod max min
()(2 )
()( )
kf
kf
kf
Ykf kf kf kf
Y
kf Y kf kf
yyyy
yyy
μ
α
μ
−−
=
−−
,
max
1,
2,
min
()
()
kf kf
kf
kf
Ykf Y
Y
Ykf
y
y
αμ
α
μ
=
,
(10)
with
min mod max
4
6
ki
kf kf kf
Y
yyy
μ
++
=
,
(11)
so that
min mod max
max min min
12
(, , )
( , )( )
kf kf kf
kf kf kf
BetaPERT y y y
B
eta y y y
αα
=
=−+
.
(12)
As a consequence, all the input and output
stochastic indicators
x
kf
and y
hf
may be modelled as
stochastic variables that are distributed according to
(9) and (12), respectively. Applying the Monte Carlo
method allows assigning at each iteration a casual
value to all the stochastic variables according to the
cited probability density functions. Such values are
employed to solve at each iteration the cross-
efficiency problem (3) and (5). The final cross-
A MODEL USING DATA ENVELOPMENT ANALYSIS FOR THE CROSS EVALUATION OF SUPPLIERS UNDER
UNCERTAINTY
155
efficiency values associated to the f-th supplier for
f=1,2,…,F are evaluated by averaging the overall
cross-efficiency index
CE
f
computed for all the
iterations, so that a ranking of the suppliers is
established by the descending order of such values.
Summing up, the proposed method integrates
both the advantages of traditional DEA (evaluating
the supply efficiency and avoiding data
normalization) and cross-efficiency DEA
(optimizing the weight set so as to shift from self-
evaluation to peer-evaluation) with respect to
alternative supplier selection techniques. In addition,
the technique is able to deal with uncertain data in
the supply, typical of the real context.
4 USING THE STOCHASTIC DEA
MODEL FOR THE SUPPLIER
SELECTION OF AN SME
We apply the described model for the cross-
efficiency evaluation of suppliers to the supplier
selection process of a small enterprise of southern
Italy, operating in the areas of sale, set-up, and
maintenance of hydraulic plants. Before the
subsequently reported investigation, the case study
SME used the classical lowest price approach to
choose suppliers. On the contrary, thanks to the
proposed supplier selection tool, it was able to
compare the performance of suppliers that it
employed in the past with that of other potential
partners, based not only on price but also
considering additional uncertain data. The
considered supply refers to a specific component, a
cast iron mainspring check valve. While for the
current suppliers the available data are deterministic,
the uncertainty referring to some data of the
performance of potential partners was modelled
using Monte Carlo simulation and the previously
introduced Beta-PERT distribution. In particular, we
consider two currently existing suppliers (denoted
by E1 and E2) for the product supply. Their
performance is compared to that of six potential
additional suppliers (P1, P2, P3, P4, P5, and P6).
The considered efficiency indicators are five. In
particular, three are input indices, i.e., refer to
resources that are required to the SME in the supply
process: 1) component purchasing price (in
€); 2) lead time for the order execution, i.e., time
elapsing between order emission and product
delivery (in days, d); and 3) geographical distance
between supplier and SME (in Km). In addition, we
consider two output indices: 1) quality of the
provided component, defined as a percentage ratio
between working components and overall number of
delivered components (%); and 2) delivery
reliability, expressed as a percentage ratio between
orders that are dealt with on time and overall number
of orders (%).
Table 1: Input and output data for current and potential
suppliers.
Suppl.
Input criteria Output criteria
Price
[€]
Lead Time
[d]
Distance
[Km]
Quality [%]
Reliability
[%]
E1
110 40 1000 99.3 50
E2
91 50 900 99 37.5
P1
95 (30,35,45) 970 (90,95,99) (30,50,75)
P2
78 (42,45,50) 20 (90,92,95) (30,50,75)
P3
132 (32,40,50) 833 (96,99,100) (40,60,85)
P4
130.67 (32,40,50) 897 (90,95,99) (40,60,75)
P5
114.50 (35,45,50) 813 (90,95,99) (35,55,77)
P6
133.10 (32,40,50) 898 (90,95,99) (50,60,80)
Table 2: Cross-efficiency matrix and average cross-
efficiency index of suppliers.
Suppl.
CE
fi
- Efficiency with respect to
CE
f
Cross
Eff.
E1 E2 P1 P2 P3 P4 P5 P6
E1
0.95 0.87 0.84 0.02 0.83 0.87 0.89 0.85 0.77
E2
0.82 0.94 0.70 0.02 0.62 0.66 0.74 0.63 0.64
P1
0.99 0.95 1.00 0.02 0.90 0.93 0.96 0.92 0.83
P2
0.97 1.00 0.82 1.00 0.85 0.90 0.99 0.87 0.92
P3
0.94 0.75 0.84 0.03 0.97 0.92 0.92 0.91 0.79
P4
0.90 0.73 0.82 0.02 0.88 0.94 0.89 0.89 0.76
P5
0.87 0.79 0.78 0.02 0.80 0.84 0.91 0.82 0.73
P6
0.92 0.74 0.84 0.02 0.91 0.93 0.92 0.96 0.78
With reference to the data, we remark that
purchasing prices and geographical distances are
deterministic values. Indeed, prices may be
determined using the price list of the suppliers,
neglecting price variations during the year. On the
contrary, the remaining indices, i.e., lead time,
quality, and reliability, are deterministically defined
for the current suppliers E1 and E2, using historical
data of the SME, while they can be only
stochastically defined for potential suppliers using
the previously described the Beta-PERT approach.
Table 1 summarizes the problem data. For each of
the eight considered suppliers, we evaluate its cross-
efficiency value according to the presented method.
Due to the presence of stochastic elements among
the problem data, we solve the problem using the
described Monte Carlo simulation, thanks to 1000
iterations that are executed in the well-known
MATLAB software environment. The results of the
procedure are in Table 2. The last column of the
KMIS 2011 - International Conference on Knowledge Management and Information Sharing
156
table show that the most efficient supplier is
potential supplier P2 (that is favoured by the
geographical proximity to the SME, see Table 1),
followed by potential supplier P1. A key factor to
this result is the optimal set of coefficients chosen
for P2: it is presumable that the higher incidence
falls back on the coefficients related to price and
(especially) to geographical distance, two factors
thanks to which supplier P2 is predominant over the
remaining vendors. As a consequence, the cross-
efficiency values associated to the other suppliers
are two orders of magnitude lower than those
characterizing P2 (ranging between 0.02 and 0.03, as
the fifth column of Table 2 shows). Moreover, the
last column of Table 2 remarks that the four most
efficient suppliers in terms of cross-efficiency are all
in the set of potential suppliers, i.e., they are P2, P1,
P3, and P6, in a descending order of efficiency.
The SME purchasing manager evaluated the
obtained results, underlying as major advantages of
the method its ability to take into account multiple
criteria, its capability to distinguish between the
required resources and the overall performance, and
ultimately its skill in assessing both the supply
process effectiveness and efficiency.
5 CONCLUSIONS
We propose a novel method for the optimal selection
of suppliers based on the well-known Data
Envelopment Analysis (DEA) technique. In
particular, we extend a DEA method for the cross-
evaluation of efficiency, previously presented in the
literature to discern among maximally efficient
suppliers, using the Monte Carlo simulation method,
so as to enable the treatment of uncertain data. The
technique application to the supplier selection
process of an Italian SME, shows its
straightforwardness and its ability to discerning
among suppliers, also in case of uncertain data.
Future developments include further validation of
the method and detailed comparison with alternative
approaches presented in the scientific literature.
ACKNOWLEDGEMENTS
This work was supported by the TRASFORMA
“Reti di Laboratori” network funded by the Italian
Apulia Region.
REFERENCES
Charnes, A., Cooper, W. W., Rhodes, E., 1978. Measuring
the efficiency of decision making units. Eur. J. of
Operational Research, Vol. 2, pp. 429–444.
Costantino, N., Dotoli, M., Falagario, M., Fanti, M.P.,
Mangini, A.M., Sciancalepore, F., 2011. Supplier
selection in the public procurement sector via a data
envelopment analysis approach, 6 pp., Proc. 19th
IEEE Mediterranean Conf. on Control and
Automation (MED 2011), Corfu, Greece, June 23-25.
Dyson, R. G., Shale, E.A., 2010. Data envelopment
analysis, operational research and uncertainty, J Oper
Res Soc, Vol. 61, Issue 1, pp. 25 – 34
Green, R., Doyle, J., 1994. Efficiency and cross-efficiency
in DEA: Derivation, Meanings and Uses. J Oper Res
Soc Vol. 45, pp. 567-578
Ho, W., Xu, X., Dey, P. K., 2010. Multi-criteria decision
making approaches for supplier evaluation and
selection: A literature review. Eur. J. of Operational
Research, Vol. 202, pp. 16-24.
Kao, C., Liu, T.-S., 2009. Stochastic data envelopment
analysis in measuring the efficiency of Taiwan
commercial banks. Eur. J. of Operational Research,
Vol. 196, pp. 312–322.
Law, A. M., Kelton, W. D., 2000. Simulation Modeling
and Analysis. Quarta ed., McGraw-Hill, New York.
Sexton, T. R., Silkman, R.H., Hogan, A.J., 1986. Data
envelopment analysis: Critique and extensions, in R.
H. Silkman (Ed.), Measuring efficiency: An
assessment of data envelopment analysis. San
Francisco, CA: Jossey-Bass.
Vose, D., 2008. Risk analysis: a quantitative guide. John
Wiley and Sons, Second Edition.
Wong, W. P., Jaruphongsa, W., Lee, L. H., 2008. Supply
chain performance measurement system: a Monte
Carlo DEA-based approach. Int. J. of Industrial and
Systems Engineering ,Vol. 3, pp. 162 – 188.
A MODEL USING DATA ENVELOPMENT ANALYSIS FOR THE CROSS EVALUATION OF SUPPLIERS UNDER
UNCERTAINTY
157