Supplier Selection by the Pair of Nondiscretionary Factors-Imprecise Data Envelopment Analysis Models

Abstract

Discretionary models for evaluating the efficiency of suppliers assume that all criteria are discretionary, that is, controlled by the management of each supplier and varied at its discretion. These models do not assume supplier selection in the conditions that some factors are nondiscretionary. The objective of this paper is to propose a new pair of nondiscretionary factors-imprecise data envelopment analysis (NF-IDEA) models for selecting the best suppliers in the presence of nondiscretionary factors and imprecise data. A numerical example demonstrates the application of the proposed method. Yes Yes

Full text

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Supplier Selection by the Pair of Nondiscretionary Factors-Imprecise
Data Envelopment Analysis Models

Reza Farzipoor Saen

Department of Industrial Management, Faculty of Management and Accounting, Islamic Azad University - Karaj Branch,
Karaj, Iran, P. O. Box: 31485-313

Tel: 0098 (261) 4418144-6
Fax: 0098 (261) 4418156
E-mail: farzipour@yahoo.com

Abstract
Discretionary models for evaluating the efficiency of suppliers assume that all criteria are
discretionary, i.e., controlled by the management of each supplier and varied at its discretion. These
models do not assume supplier selection in the conditions that some factors are nondiscretionary. The
objective of this paper is to propose a new pair of Nondiscretionary Factors-Imprecise Data
Envelopment Analysis (NF-IDEA) models for selecting the best suppliers in the presence of
nondiscretionary factors and imprecise data. A numerical example demonstrates the application of the
proposed method.

Keywords: Nondiscretionary factors, Imprecise data envelopment analysis, Supplier selection

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Introduction
Supplier selection models are based on cardinal data with less emphasis on ordinal data. However,
with the widespread use of manufacturing philosophies such as Just-In-Time (JIT), emphasis has
shifted to the simultaneous consideration of cardinal and ordinal data in supplier selection process. On
the other hand, discretionary models for evaluating the efficiency of suppliers assume that all criteria
are discretionary, i.e., controlled by the management of each supplier and varied at its discretion. Thus,
failure of a supplier to produce maximal output levels with minimal input consumption results in a
decreased efficiency score. In any realistic situation, however, there may exist exogenously fixed or
nondiscretionary criteria that are beyond the control of a management. In an analysis of a network of
fast food restaurants, Banker and Morey (1986) illustrate the impact of exogenously determined inputs
that are not controllable. In their study, each of the 60 restaurants in the fast food chain consumes six
inputs to produce three outputs. The three outputs (all controllable) correspond to breakfast, lunch, and
dinner sales. Only two of the six inputs, expenditures for supplies and expenditures for labor, are
discretionary. The other four inputs (age of store, advertising level, urban/rural location, and
presence/absence of drive-in capability) are beyond the control of the individual restaurant manager.
Their analysis clearly demonstrates the value of accounting for the nondiscretionary character of these
inputs explicitly in the Data Envelopment Analysis (DEA) models they employ; the result is
identification of a considerably enhanced opportunity for targeted savings in the controllable inputs and
targeted increases in the outputs. In the case of supplier selection, distance and supply variety are
generally considered nondiscretionary criterion.
Weber (1996) applied DEA in supplier evaluation for an individual product and demonstrated the
advantages of applying DEA to such a system. In Weber’s study, six vendors supplying an item to a
baby food manufacturer were evaluated. Significant reductions in costs, late deliveries, and rejected
materials can be achieved if inefficient vendors can become DEA efficient.

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When suppliers are compared for their overall performances, an aggregate evaluation relevant to
the considerations of a purchasing firm needs to be conducted. Such an overall performance evaluation
of suppliers should be based on performance measures for all part types supplied to the purchasing
company. A potential use of an overall performance evaluation of suppliers is to provide benchmarking
data for reducing the number of suppliers, which in turn results in benefits including reduction in costs
of parts and order processing, and better partnership with suppliers.
This paper depicts the supplier selection process through an Imprecise Data Envelopment Analysis
(IDEA) model, while allowing for the incorporation of nondiscretionary factors. The objective of this
paper is to propose a new pair of Nondiscretionary Factors-Imprecise Data Envelopment Analysis (NF-
IDEA) models for selecting the best suppliers in the presence of nondiscretionary factors and imprecise
data.

Literature review
Some mathematical programming approaches have been used for supplier selection in the past.
Zeng et al, (2006) considered a simplified partner selection problem which takes into account only the
bid cost, the bid completion time of subprojects, and the due date of the project. They modeled the
problem as a nonlinear integer programming problem and proved that the decision problem of the
partner selection problem is NP-complete. Then they analysed some properties of the partner selection
problem and construct a branch and bound algorithm.
Hajidimitriou and Georgiou (2002) presented a quantitative model, based on the Goal Programming
(GP) technique, which uses appropriate criteria to evaluate potential candidates and leads to the
selection of the optimal partner (supplier).
ebi
,C
and Bayraktar (2003) proposed an integrated model
for supplier selection. In their model, supplier selection problem has been structured as an integrated
Lexicographic Goal Programming (LGP) and Analytic Hierarchy Process (AHP) model including both

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quantitative and qualitative conflicting factors. Karpak et al, (2001) presented one of the "user-
friendly" multiple criteria decision support systems-Visual Interactive Goal programming (VIG). VIG
facilitates the introduction of a decision support vehicle that helps improve the supplier selection
decisions. To take into account both cardinal and ordinal data in supplier selection, Wang et al, (2004)
developed an integrated AHP and Preemptive Goal Programming (PGP) based methodology.
However, one of the GP problems arises from a specific technical requirement. After the
purchasing managers specify the goals for each selected criterion (e.g., amount of price, quality level,
etc), they must decide on a preemptive priority order of these goals, i.e., determining in which order the
goals will be attained. Frequently such a priori input might not produce an acceptable solution and the
priority structure may be altered to resolve the problem once more. In this fashion, it may be possible to
generate a solution iteratively that finally satisfies the Decision Maker (DM). Unfortunately, the
number of potential priority reorderings may be very large. A supplier selection problem with five
factors has up to 120 priority reorderings. Going through such a laborious process would be costly and
inefficient.
Sha and Che (2006) presented a multi-phased mathematical approach called the Hybrid Multi-
phased-based Genetic Algorithm (HMGA) for network design of supply chain. From the point of view
of network design, the important issues are to find suitable and quality companies, and to decide upon
an appropriate production/distribution strategy. It is based on various methodologies that embrace
Genetic Algorithms (GAs), AHP, and the Multi-Attribute Utility Theory (MAUT) to simultaneously
satisfy the preferences of suppliers and customers at each level of the supply chain network. Bayazit
(2006) provided a good insight into the use of the Analytic Network Process (ANP) that is a multiple
criteria decision making methodology in evaluating supplier selection problems. Dulmin and Mininno
(2003) presented a proposal for applying a decision model to the final vendor-rating phase of a process
of supplier selection. Their model uses a Multiple Criteria Decision Aid (MCDA) technique
(PROMETHEE 1 and 2), with a high-dimensional sensitivity analysis approach. They tried to explain

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how an outranking method and PROMETHEE/GAIA techniques, provides powerful tools to rank
alternatives and analysed the relations between criteria or between DMs. Bhutta and Huq (2002)
illustrated Total Cost of Ownership (TCO) and AHP approaches and provided a comparison. They
concluded that TCO is better suited to those situations where cost is of high priority and detailed cost
data are available to make comparisons. In the case of AHP, it is better suited to solve and decide
between suppliers when several conflicting goals exist and, though cost may be an important factor, it
is not the overriding one.
However, AHP has two main weaknesses. First subjectivity of AHP is a weakness. Second AHP
could not include interrelationship within the criteria in the model.
Chen et al, (2006) presented a fuzzy decision making approach to deal with the supplier selection
problem in supply chain system. They used linguistic values to assess the ratings and weights for the
criteria. These linguistic ratings can be expressed in trapezoidal or triangular fuzzy numbers. Then, a
hierarchy Multiple Criteria Decision Making (MCDM) model based on fuzzy sets theory is proposed to
deal with the supplier selection problems in the supply chain system. According to the concept of the
Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), a closeness coefficient is
defined to determine the ranking order of all suppliers by calculating the distances to the both Fuzzy
Positive Ideal Solution (FPIS) and Fuzzy Negative Ideal Solution (FNIS) simultaneously.
Choy et al, (2002) presented an Intelligent Supplier Management Tool (ISMT) using the Case-
Based Reasoning (CBR) and Neural Network (NN) techniques to select and benchmark suppliers.
Choy and Lee (2003) suggested an intelligent Generic Supplier Management Tool (GSMT) using the
CBR technique for outsourcing to suppliers and automating the decision making process when
selecting them. Choy et al, (2004) discussed an Intelligent Supplier Relationship Management System
(ISRMS) integrating a company’s Customer Relationship Management (CRM) system, Supplier Rating
System (SRS) and Product Coding System (PCS) by the CBR technique to select preferred suppliers
during the New Product Development (NPD) process. In order to develop a flexible data access

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framework, and to support the partner selection activity, the combination of OnLine Analytical
Processing (OLAP), and CBR was proposed by Lau et al, (2005).
Lee et al, (2003) proposed a High-Quality-Supplier Selection (HQSS) model to deal with supplier
selection problems in supply chain management. In selecting a supplier, quality management factors
are considered first, and then price, delivery, etc. Linn et al, (2006) proposed a new approach to
supplier selection using a Capability index and Price Comparison (CPC) chart. The CPC chart
integrates the process capability and price information of multiple suppliers and presents them in a
single chart for the management to make supplier selection decisions.
All the aforementioned literature relied on some form of procedures that assigns weights to various
performance measures. The primary problem associated with arbitrary weights is that they are
subjective, and it is often a difficult task for the DM to accurately assign numbers to preferences. It is a
daunting task for the DM to assess weighting information as the number of performance criteria
increased. Therefore, a more robust mathematical technique that does not demand too much and too
precise information, i.e., ordinal preferences instead of cardinal weights, from the DM can strengthen
the supplier evaluation process. To this end, Weber (1996) demonstrated how DEA can be used to
evaluate vendors on multiple criteria and identified benchmark values which can then be used for this
purpose. Weber et al, (2000) presented an approach for evaluating the number of vendors to employ in
a procurement situation using Multi-Objective Programming (MOP) and DEA. The approach advocates
developing vendor-order quantity solutions (referred to as supervendors) using MOP and then
evaluating the efficiency of these supervendors on multiple criteria using DEA. Recently, to select the
best suppliers in the presence of both cardinal and ordinal data, Farzipoor Saen (2007) proposed an
innovative method, which is based on IDEA. However, he did not consider the weights restrictions.
However, all the aforementioned references are based on complete discretion of decision making
criteria (factors) and do not consider supplier selection in the presence of both imprecise data and
nondiscretionary factors.

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To the best of author’s knowledge, there is not any reference that deals with supplier selection in the
conditions that nondiscretionary factor and imprecise data are present.

Proposed model for supplier selection
DEA formulations proposed by Charnes et al, (1978) (CCR model) and developed by Banker et al,
(1984) (BCC model) is an approach for evaluating the efficiencies of Decision Making Units (DMUs).
This evaluation is generally assumed to be based on a set of cardinal (quantitative) output and input
factors. In many real world applications (especially supplier selection problems), however, it is
essential to take into account the existence of ordinal (qualitative) factors when rendering a decision on
the performance of a DMU. Very often, it is the case that for a factor such as supplier reputation, one
can, at most, provide a ranking of the DMUs from best to worst relative to this attribute. The capability
of providing a more precise, quantitative measure reflecting such a factor is generally beyond the realm
of reality. In some situations, such factors can be legitimately quantified, but very often such
quantification may be superficially forced as a modeling convenience. In situations such as that
described, the data for certain influence factors (inputs and outputs) might better be represented as rank
positions in an ordinal, rather than numerical sense. Refer again to the supplier reputation example. In
certain circumstances, the information available may permit one to provide a complete rank ordering of
the DMUs on such a factor. Therefore, the data may be imprecise.
Recently, Wang et al, (2005) developed a new pair of interval DEA models for dealing with
imprecise data such as interval data, ordinal preference information, fuzzy data and their mixture.
Compared with the IDEA model developed by Cooper et al, (1999), Cooper et al, (2001a), and Cooper
et al, (2001b), their interval DEA models are much easier to understand and more convenient to use.
Also, compared with the interval DEA models developed by Despotis and Smirlis (2002), their interval
DEA models utilise a fixed and unified production frontier as a benchmark to measure the efficiencies
of all DMUs, which makes their models more rational and more reliable. Moreover, the means by

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which they treat ordinal preference information also seems more reasonable than the way Zhu (2003)
did. However, Wang et al, (2005) did not consider nondiscretionary factors.
In this section, a new pair of NF-IDEA models is proposed that can overcome the shortcoming
mentioned above, to consider nondiscretionary factors of the suppliers (DMUs) while ordinal and
cardinal data are present. The final efficiency score for each DMU will be characterised by an interval
bounded by the best lower bound efficiency and the best upper bound efficiency of each DMU.
Suppose that there are n DMUs to be evaluated. Each DMU consumes m inputs to produce s
outputs. In particular, DMUj consumes amounts Xj ={ }
ij x
of inputs (i=1, …, m) and produces amounts
Yj={ }
rj
y
of outputs (r=1, …, s). Without loss of generality, it is assumed that all the input and output
data xij and yrj (i=1, …, m; r=1, …, s; j=1, …, n) cannot be exactly obtained due to the existence of
uncertainty. They are only known to lie within the upper and lower bounds represented by the intervals
[
ij
x ,
rj
y ,
, where
0
>
and
0
>
rj
y
.
]
UL
ijx
and []
UL
rjy
L
ij x
L
In order to deal with such an uncertain situation, the following pair of linear programming models
have been developed to generate the upper and lower bounds of interval efficiency for each DMU
(Wang et al, 2005):

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. ,
r
,
) 2 (,,, 1, 0
, 1
. .
t s
. ,
r
,
) 1 (,,, 1, 0
, 1
. .
t s
11
1
1
11
1
1
ivu
njxvyu
xv
yuMax
ivu
njxvyu
xv
yu Max
ir
m
i
L
ij i
s
r
U
rjr
m
i
U
ij i
s
r
L
rjr
L
jo
ir
m
i
L
iji
s
r
U
rjr
m
i
L
iji
s
r
U
rjr
U
jo
o
o
o
o
∀≥
=≤−
=
=
∀≥
=≤−
=
=
∑
=
∑
=
∑
=
∑
=
∑
=
∑
=
∑
=
∑
=
ε
θ
ε
θ




where jo is the DMU under evaluation (usually denoted by DMUo); ur and vi are the weights assigned to
the outputs and inputs;
U
jo
θ stands for the best possible relative efficiency achieved by DMUo when all
the DMUs are in the state of best production activity, while
L
jo
θ stands for the lower bound of the best
possible relative efficiency of DMUo. They constitute a possible best relative efficiency interval
[
jo
θ ,
. ε is the non-Archimedean infinitesimal.
]
UL
joθ
In order to judge whether a DMU is DEA efficient or not, the following definition is given.

Definition 1. A DMU, DMUo, is said to be DEA efficient if its best possible upper bound efficiency
; 1
*=
jo
θ
U
otherwise, it is said to be DEA inefficient if
. 1
*<
jo
θ
U

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Now, to demonstrate how to consider nondiscretionary factors in the model, the new pair of NF-
IDEA models is proposed. The envelopment formulation (dual problem) of Models (1) and (2)
becomes
.,, 10
,,, 1
=
0
,,, 10
free,
j
=
,,, 10
) 4 (,, , 1
. .
t s
,
.,, 10
,,, 1
=
0
,, , 10
free,
j
=
,, , 10
) 3 (,, , 1
. .
t s
,
1
1
11
1
1
11
srs
mis
n
misxx
srysy
ssMinc
srs
mis
n
misxx
srysy
ssMinc
r
i
+
j
o
ij
n
j
L
ijo
U
ijo
L
rjorj
n
j
U
rj
s
r
r
m
i
io
L
jo
r
i
+
j
o
ij
n
j
L
ijo
L
ijo
U
rjorj
n
j
U
rj
s
r
r
m
i
io
U
jo










≥
=≥
≥
==−−
==−
−−=
≥
=≥
≥
==−−
==−
−−=
−
−
=
+
=
=
+
=
−
−
−
=
+
=
=
+
=
−
∑
∑
∑∑
∑
∑
∑∑
λ
θ
λθ
λ
εεθ
λ
θ
λθ
λ
εεθ


where
o θ ,
+
r
−
ij
ss
and,,
λ
are the dual variables.
o θ is the radial input shrinkage factor (eventually
to become efficiency measure) and
{ }
λ
j
λ =
is the vector of DMU loadings, determining "best
practice" for the DMU being evaluated.
U
jo
c stands for the best possible relative efficiency achieved by

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DMUo when all the DMUs are in the state of best production activity, while
L
jo
c stands for the lower
bound of the best possible relative efficiency of DMUo. They constitute a possible best relative
]
jo
c ,
. The variable
efficiency interval [
UL
joc
+
rs is shortfall amount of output r and
−
is is excess amount of
input i. From the duality theory in linear programming, for an inefficient DMUo,
0
*>
j λ
in the optimal
dual solution implies that DMUj is a unit of the peer group. A peer group of an inefficient DMUo is
defined as the set of DMUs that reach the efficiency score of 1 using the same set of weights that result
in the efficiency score of DMUo. It is the existence of this collection of DMUs that forces the DMUo to
be inefficient.
Now, the model that considers both imprecise data and nondiscretionary factors is introduced.
Suppose that the input variables may be partitioned into subsets of discretionary (D) and
nondiscretionary (N) variables. Thus,



The pair of NF-IDEA models is then finally given by
{
2 , 1
}
Φ=∩∪==
NDND
IIIImI
,,...,

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., , 10
,0
,0
,, , 10
free,
j
=
,0
,0
) 6 (,,, 1
. .
t s
,
., , 10
,0
,0
,, , 10
free,
j
=
,0
,0
) 5 (,,, 1
. .
t s
,
1
1
1
1
1
1
1
1
srs
Nis
Dis
n
Nisxx
Disxx
srysy
ssMinc
srs
Nis
Dis
n
Nisxx
Disxx
srysy
ssMinc
r
i
+
i
−
j
o
ij
n
j
L
ij
U
ijo
ij
n
j
L
ijo
U
ijo
L
rjorj
n
j
U
rj
s
r
r
Di
io
L
jo
r
i
+
i
−
j
o
ij
n
j
L
ij
L
ijo
ij
n
j
L
ij o
L
ijo
U
rjorj
n
j
U
rj
s
r
r
Di
io
U
jo






=≥
∈=
∈≥
≥
∈=−−
∈=−−
==−
−−=
=≥
∈=
∈≥
≥
∈=−−
∈=−−
==−
−−=
−
−
=
−
=
+
=
=
+
∈
−
−
−
=
−
=
+
=
=
+
∈
−
∑
∑
∑
∑∑
∑
∑
∑
∑∑
λ
θ
λ
λθ
λ
εεθ
λ
θ
λ
λθ
λ
εεθ


It is to be noted that the
o θ to be minimised appears only in the constraints for which
Di∈
,
whereas the constraints for which
Ni∈
operate only indirectly (as they should) because the input

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levels
ijo
x
are not subject to managerial control. Therefore this is recognised by entering all
Nixijo
∈
,

at their fixed (observed) value. Note that the slacks
Nisi
∈
−,
are omitted from the objective function.
Hence these nondiscretionary inputs do not enter directly into the efficiency measures being optimised
in (5) and (6). They can, nevertheless, affect the efficiency evaluations by virtue of their presence in the
constraints. For models (5) and (6), it is not relevant to minimise the proportional decrease in the entire
input vector. Such minimisation should be determined only with respect to the subvector that is
composed of discretionary inputs.
In order to judge whether a DMU is DEA efficient or not, the following definition is given.

Definition 2. A DMU, DMUo, is said to be DEA efficient if its best possible upper bound efficiency
; 1
*=
jo
U
c
otherwise, it is said to be DEA inefficient if
. 1
*<
jo
U
c


Therefore, one unified approach that deals with all aspects of the imprecise data and
nondiscretionary factors in a direct manner has been introduced.
Now, the method of transforming ordinal preference information into interval data is discussed, so
that the pair of NF-IDEA models presented in this paper can still work properly even in these
situations.
Suppose some input and/or output data for DMUs are given in the form of ordinal preference
information. Usually, there may exist three types of ordinal preference information: (1) strong ordinal
preference information such as yrj>yrk or xij>xik, which can be further expressed as
rkr rj
yy
χ≥
and
,
ik iij
xx
η≥
where
1
>
r
χ
and
1
>
i η
are the parameters on the degree of preference intensity provided
by decision maker; (2) weak ordinal preference information such as
rqrp
yy
≥
or
;
iqip
xx ≥
(3)
indifference relationship such as yrl = yrt or xil = xit. Since a DEA model has the property of unit-

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invariance, the use of scale transformation to ordinal preference information does not change the
original ordinal relationships and has no effect on the efficiencies of DMUs. Therefore, it is possible to
conduct a scale transformation to every ordinal input and output index so that its best ordinal datum is
less than or equal to unity and then give an interval estimate for each ordinal datum.
Now, consider the transformation of ordinal preference information about the output yrj (j=1,…, n)
for example. The ordinal preference information about input and other output data can be converted in
the same way.
For weak ordinal preference information
,
21
rnrr
yyy
≥≥≥

we have the following ordinal
relationships after scale transformation:
,
ˆ
y
ˆ
y
ˆ
y
1
21
r rnrr
σ≥≥≥≥≥


where
r
σ is a small positive number reflecting the ratio of the possible minimum value {yrj| j=1,…, n}
to its possible maximum value. As well, it can be approximately estimated by the decision maker. It is
referred as the ratio parameter for convenience. The resultant permissible interval for each
rj
y ˆ is given
by
[]
.,, 1 , 1 ,
r
ˆ
ynj
rj

=∈ σ

For strong ordinal preference information
,
21
rnrr
yyy
>>>

there is the following ordinal
relationships after scale transformation:
,
ˆ
y
and ) 1
−
, , 1(
ˆ
y
ˆ
y
,
ˆ
y
1
1,1
r rnjrrrjr
nj
σχ≥=≥≥
+


where
r
χ is a preference intensity parameter satisfying
r
χ >1 provided by the decision maker and
r
σ
is the ratio parameter also provided by the decision maker. The resultant permissible interval for each
rj
y ˆ can be derived as follows:
[
σ
]
.with,, 1,,
ˆ
y
1
r
1
r
n
r
jj
n
rr rj
nj
−−−
≤=∈χσχχ


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15
Finally, for an indifference relationship, the permissible intervals are the same as those obtained for
weak ordinal preference information.
Through the scale transformation above and the estimation of permissible intervals, all the ordinal
preference information is converted into interval data and can thus be incorporated into the pair of NF-
IDEA models.
In the next section, a numerical example is presented.

Numerical example
The data set for this example is partially taken from Farzipoor Saen (2007) and contains
specifications on 18 suppliers (DMUs). In particular, this example is used to show how ordinal and
bounded data, as well as nondiscretionary factors, can be combined into the one unified approach
provided by NF-IDEA. The cardinal inputs considered are Total Cost of shipments (TC) and Distance
(D). D is generally considered as a nondiscretionary input variable. Supplier Reputation (SR) is
included as a qualitative input while Number of Bills received from the supplier without errors (NB)
will serve as the bounded data output. SR is an intangible factor that is not usually explicitly included
in evaluation model for supplier. This qualitative variable is measured on an ordinal scale so that, for
instance, reputation of supplier 18 is given the highest rank, and supplier 17, the lowest. Note that, the
measures selected in this paper are not exhaustive by any means, but are some general measures that
can be utilised to evaluate suppliers. In an application of this methodology, DMs must carefully
identify appropriate inputs and outputs to be used in the decision making process. The second and third
column of Table 1 depicts the inputs and output of suppliers.
Suppose the preference intensity parameter and the ratio parameter about the strong ordinal
preference information are given (or estimated) as
,01. 0and12 . 1
=
33
=ση
respectively. To show the