
Research Article


Developing a Nondiscretionary Slacksbased Measure Model for Supplier Selection in the Presence of Stochastic Data 

Majid Azadi
and
Reza Farzipoor Saen



ABSTRACT

Supplier selection has a strategic importance for every company. Nondiscretionary
Slacksbased Measure (SBM) model is one of the models in Data Envelopment Analysis
(DEA). In many real world applications, data are often stochastic. A successful
approach to the address uncertainty in data is to replace deterministic data
via random variables, leading to Chanceconstrained DEA (CCDEA). In this study,
the concept of chanceconstrained programming approach is used to develop nondiscretionary
SBM model in the presence of stochastic data and also its deterministic equivalent
which is a nonlinear program is derived. Furthermore, it is shown that the deterministic
equivalent of the stochastic nondiscretionary SBM model can be converted into
a quadratic program. Finally, a numerical example demonstrates the application
of the proposed model.





Received: April 27, 2012;
Accepted: June 30, 2012;
Published: December 05, 2012


INTRODUCTION
According to Sonmez (2006), supplier selection is the
process of finding the suppliers that are able to present products and/or services
to the customer with appropriate quality, at the appropriate cost, quantities
and time. As Amid et al. (2011) address, within
new strategies for purchasing and manufacturing, suppliers play a key role in
achieving corporate competitiveness. Consequently, correct selection of suppliers
is a critical element. Main cost of the most industries in manufacturing belongs
to cost of raw materials and component parts which in most cases constitutes
up to 70% of the total costs. Therefore, purchasing department plays an important
role in efficiency and effectiveness of a firm, due to the contribution of supplier
performance on expenditure, quality, delivery and service in accomplishing the
objectives of a supply chain.
As Chamodrakas et al. (2010) describe, advanced
industries should conform to market environment in which accessibility to global
competition is an important factor. Consequently, in order to reduce production
expenditures, it is important that expenses of companies be logical and reasonable.
To this end, reducing the purchasing prices through selection of right supplier
can be beneficial. In addition, some advanced production systems like just in
time and mass customization manufacturing pay attention to quick provision of
raw materials and outsourced components within expected quality and quantity.
The fact that many businesses are turning to outside suppliers and manufacturers
to obtain universal resources more effectively, emphasizes to importance of
requirements of these issues.
Some mathematical programming approaches have been used for supplier selection
in the past. Table 1 categorizes the reviewed papers based
on applied techniques. Because of the intricacy of the decision making process
involved in supplier, all the aforementioned references in Table
1, except for Data Envelopment Analysis (DEA), rely on some form of procedures
that assign weights to various performance measures. The primary problem associated
with arbitrary weights is that they are subjective and it is often a complex
task for the decision maker to precisely assign numbers to preferences. As well,
it is a daunting task for the decision maker to assess weighting information
as the number of performance criteria is increased. In the meantime, they do
not consider stochastic data.
Standard DEA models suppose that Decision Making Units (DMUs) carry out same
obligations with same goals, employ similar inputs and create similar outputs.
In real world, some factors are out of the control of decision makers and are
called nondiscretionary factors (Syrjanen, 2004).
Instances from the DEA literature include snowfall or weather in evaluating
the efficiency of maintenance units, soil characteristics and topography in
different farms, number of competitors in the branches of a restaurant chain
and age of facilities in different universities (Saen, 2005).
It is suitable for solving optimization problems with random variables included
in constraints and sometimes in the objective function as well (Charnes
and Cooper, 1959). As Olson and Swenseth (1987)
discuss, CCP was developed as a means of describing constraints in the form
of probability levels of attainment. Consideration of chance constraints allows
the decisionmaker to consider objectives in terms of their attainment probability.
If α is a predetermined confidence level desired by the decisionmaker,
the implication is that a constraint will have a probability of satisfaction
of α. The probabilistic nature of this approach lends itself to multiobjective
analysis.
Table 1: 
A summary of methods for suppliers selection 

The selection of α can be a managerial decision. Chance constraints for
stochastic functions based upon sampling information would often be normally
distributed. Sampling information has long been used in business as a means
of determining the expected value of functional coefficients in linearly constrained
systems.
A main contribution on the stochastic DEA might be found in the work of Sengupta
(1982, 1987, 1990, 1997,
1998, 2000) who has widely studied
the research theme, using the CCP proposed by Charnes and
Cooper (1963). An important feature of his studies is that stochastic variables
are incorporated into DEA and afterward the stochastic DEA is reformulated into
a deterministic equivalent. Land et al. (1993)
utilized CCP (Charnes and Cooper, 1961; Cooper
et al., 1996) to develop efficient frontiers which envelop a given of
observation most of time. Olesen and Petersen (1995)
proposed a Chanceconstrained DEA (CCDEA) model that uses piecewise linear envelopment
of confidence regions for observed stochastic multiple inputs and multiple outputs.
Cooper et al. (1996) incorporated Simon
(1957) “satisficing concepts” into DEA model with chance constrained.
Also, stochastic DEA approaches can be found in but not limited to Cooper
et al. (1998), Huang and Li (1996) and Li
(1998). Morita and Seiford (1999), proposed a measure
for reliability of efficient DMUs as the amount of stochastic variations that
remain the efficient DMU being efficient. A minimum efficiency score at a specified
probability level is also used as a robustness measure. Moreover, they discussed
some stochastic measures such as an expected efficiency score, a probability
being efficient, an α percentile of efficiency score. Sueyoshi
(2000) proposed a “DEA future analysis” that incorporates future
information on outputs into its analytical framework. A stochastic DEA model
is used as an initial starting formulation and then it is reformulated by both
CCP and the estimation technique of PERT/CPM. Besides Huang
and Li (2001) generalize two conventional DEA model by incorporating two
conventional DEA model by incorporating random disturbances into input and output
data. Cooper et al. (2002) proposed CCP models
that are directed to determine where efficient and inefficient behavior will
occur with associated probabilities. Their method replaces ordinary DEA formulations
with stochastic counterparts in the form of a series of CCP models. Emphasis
is on technical efficiency and inefficiencies which do not require costs or
prices but which are nevertheless basic in that the achievement of technical
efficiency is necessary for the attainment of “allocative”, “cost”
and other types of efficiencies.
Talluri et al. (2006) utilized the CCP model
proposed by Land et al. (1993) for supplier selection,
since it is a wellestablished methodology and provides an innovative and simple
method to incorporate variability in input and output measures into the decision
making process. The developed model in this paper uses CCP model proposed by
Cooper et al. (2004). Since it has the advantages
of model proposed by Land et al. (1993), it opens
possible new routes for “sensitivity analysis”. Additionally, it can
be solved by a deterministic equivalent. Also, model utilized by Talluri
et al. (2006) does not consider nondiscretionary factors while model
utilized in this paper takes into account the nondiscretionary factors.
In summary, the approach presented in this study has some distinctive contributions
that are as below:
• 
A stochastic nondiscretionary SBM model is developed and also
its deterministic equivalent which is a nonlinear program is derived 
• 
It is shown that the deterministic equivalent of the stochastic nondiscretionary
SBM model can be converted into a quadratic program 
• 
Sensitivity analysis of the stochastic nondiscretionary SBM model is discussed
with respect to changes in parameters 
• 
For the first time, the proposed model is used for the problem of supplier
selection 
• 
The proposed model deals with stochastic data in a direct manner 
The objective of this paper was to propose a new stochastic SBM nondiscretionary
model for supplier selection.
PROPOSED MODEL
DEA is a decision technique that has been widely used for performance analysis
in public and private sectors. DEA developed by Charnes et
al. (1978), is a nonparametric estimation method, in the sense that
no choice of a parametric functional form is needed in the estimation of the
frontier. DEA can be practical to any organization/industry where a rationally
homogeneous set of DMUs use the identical set of inputs to produce an identifiable
range of outputs. Traditional DEA models can merely measure radial efficiency
(weak efficiency). To measure strong efficiency in DEA, Tone
(2001) proposed SBM. This model deals directly with the input excesses and
output shortfalls. SBM uses the Additive model and provides a scalar measure
ranging from 0 to 1 that encompasses all of the inefficiencies that the model
can identify (Cooper et al., 2007). SBM does not
deal with stochastic data and assume that all input and output data are exactly
known.
The formulation for the SBM nondiscretionary model given by Saen
(2005) is as below:
We can replace model (1) with:
Now, the new nondiscretionary SBM model is developed which permits the possible
existence of stochastic variability in the data. As we know, the typical DEA
models do not permit stochastic variations in input and output, hence, DEA efficiency
measurement may be sensitive to such variations. For instance, a DMU which is
measured as efficient relative to other DMUs, might turn inefficient if such
random variations are considered. In what follows, the output oriented nondiscretionary
SBM model is presented which allows for the possibility of stochastic alterations
in input and output data.
We suppose that all inputs and outputs are random variables with a multivariate
normal distribution and known parameters:
Definition 1: (Stochastic efficiency) DMU_{o} is DEA stochastic
efficient if and only if the following two conditions are both satisfied:
• 
γ* = 1 
• 
S¯* = S^{+}* = 0 
Now assume ς_{r} is the "external slack" for the rth output. Via
'external slack' we refer to slack outside the braces. We can select the value
of this external slack which is not stochastic, so it satisfies:
There must then exist a positive number s_{r}^{+}>0 such
that:
This positive value of s_{r}^{+} permits a still further raise
in _{
}for any set of sample observations devoid of worsening any other input
or output. It is easy to see that ς_{r} = 0 if and only if s_{r}^{+}
= 0.
In an analogous manner, presume ξ_{i}>0 represents 'external
slack' for the ith input chanceconstraint. We select its value to satisfy:
There must then exist a positive number s¯_{i}>0 such that:
Such a positive value of s¯_{i} permits a decrease in
for any sample without worsening any other input or output to the indicated
probabilities. It is easy to show that ξ_{i} = 0 if and only if
s¯_{i} = 0.
Consequently for the constraint 3 of Model (3) we have:
and
Using relations (49), can replace Model (3) with following model:
We can replace the first constraint of Model (10) with:
This reorients the inequality in the braces and replaces (1α) with α.
It next follows that:
Where:
We can also write Eq. 13 as:
Where:
This comes from:
with f the density function for the standard normal variable. From Eq.
15, we have:
Therefore:
We can replace the constraint 2 of Model (10) with following relation:
It follows that:
i.e.:
Where:
This comes from:
Now Φ^{1} (α) = Φ^{1} (1α), by virtue
of the symmetry related with the normal distribution, so from relation (20)
we have:
Thus:
Therefore, the deterministic equivalent for model (10) is as below:
To derive equations for σ^{I}_{i}(λ) note that:
Therefore:
Similarly, for the constraints 2 and 3 of model (25) we have:
It is obvious, from the forms of σ^{o}_{r} (λ), σ^{I}_{i}
(t, λ) and σ^{o}_{r} (t, λ) that model (25) is
a nonlinear program. We demonstrate that this nonlinear program can be transformed
to a quadratic program. Assume that, w^{o}_{r}, w^{I}_{i}
are nonnegative variables. Replacing w^{o}_{r}, w^{I}_{i},
respectively, by σ^{o}_{r} (λ), σ^{I}_{i}
(λ) and adding the following quadratic equality constraints:
Hence, model (25) is transformed to a quadratic programming problem:
Where:
NUMERICAL EXAMPLE
The idea for this example is taken from Saen (2009c)
and Maital and Vaninsky (2001). The example contains
specifications on twenty suppliers (DMUs). These DMUs consume two inputs to
produce two outputs. The data set are in Table 2. Distance
and cost were used as inputs for the DEA model. The outputs utilized in the
study are supplier variety and R and D expenditures. Moreover, assume that cost,
supplier variety and distance are nondiscretionary variables, i.e., these factors
are exogenously fixed and cannot be changed by suppliers (at least in the short
term).
In summary, the suppositions are as below:
• 
Distance is not controllable 
• 
Cost is 50% under control 
• 
Supplier variety is not controllable 
• 
R and D expenditure is controllable 
Table 2: 
Related attributes for 20 suppliers 

DMU: Decisionmaking units, R and D: Research and development 
Table 3 reports the results of efficiency assessments for
the 20 suppliers obtained by model 25 which are calculated with α = 0.05.
The efficient suppliers are 4, 5, 6, 9, 10, 11 and 19. These suppliers are efficient
because the following two conditions are both satisfied:
• 
γ* = 1 
• 
S¯* = S^{+}* = 0 
This example shows the applicability of the proposed model using chanceconstrained
DEA with nondiscretionary factors and stochastic data in SBM model context.
As is seen, in Table 3 suppliers were selected in uncertain
environment with α = 0.05. Supplier selection in such uncertain environment
reduces the material purchasing cost and enhances company competitiveness which
is why many experts suppose that the supplier selection is the most significant
activity of a purchasing department.
Sensitivity analysis is the study of how the variation (uncertainty) in the
output of a mathematical model can be apportioned to different sources of variation
in the input of a model. Table 4 shows the sensitivity of
results in terms of different α values. In fact, sensitivity analysis performed
in Table 4 shows that how the uncertainty in the output of
a model can be apportioned to different source of uncertainty in the model input.
With respect to definition 1, Table 5 implies that the DMUs
3, 6, 9, 11, 12, 13, 17 and 18 are efficient. The ranking results of Tables
3 and 5 depict there are some differences among the ranks.
Table 3: 
The efficiency scores for the 20 suppliers with α =
0.05 

DMU: Decisionmaking units, R and D: Research and development 
Table 4: 
Efficiency scores (γ*) for different α 

DMU: Decisionmaking units, R and D: Research and development 
Therefore, stochastic data leads to different results. This shows that if there
are stochastic data, then we must apply stochastic models. Note that the results
rely on the specified probability level α.
Table 5: 
The efficiency scores for 20 suppliers using Model (6) 

The stochastic model applied in this numerical example permits the data errors
and provides probabilistic results. In general, if the data are under uncertainty
and probabilistic situations and a rough estimate is required, the stochastic
models might be favored.
CONCLUSION
Supplier plays an important role in company successes. Though, as the marketplace
becomes more global, supplier is now seen as a significant area where industries
can cut expenditures and improve their patron service quality. In order to raise
their competitive advantages, many firms consider Supply Chain Management (SCM)
outsourcing as very significant. A successful supplier choice plays a significant
role in building the longterm relationships between the outsourcing firm and
a supplier.
In this study, nondiscretionary SBM model was discussed. In addition to developing
stochastic version of the nondiscretionary SBM model, we attained the deterministic
equivalent of the stochastic version which can be changed to a quadratic problem.
As a numerical example, the proposed approach was also applied to data of twenty
suppliers. Sensitivity analysis of the proposed model was illustrated.
The problem considered in this study is at initial stage of investigation and
further researches can be done based on the results of this study. Some of them
are as follow:
• 
Similar research can be repeated for supplier selection in
the presence of both deterministic data and fuzzy data 
• 
Similar research can be repeated for supplier selection in the presence
of both stochastic data and slightly nonhomogeneous DMUs 
• 
This study used the proposed model for supplier selection. It seems that
more fields (e.g., market selection, technology selection, personnel selection,
etc.) can be applied 
NOMENCLATURE
j 
= 
I, … , n collection of suppliers (DMUs) 
r 
= 
I, … , s the set of outputs 
i 
= 
I, … , m the set of inputs 
A∈R^{mxn } 
= 
A matrix with m rows and n columns: 
X = [x_{1}, …, x_{n}]∈R^{mxn} 
Y = [y_{1}, …, y_{n}]∈R^{mxn} 
DMU_{o } 
= 
The DMU under investigation 
y_{rj } 
= 
The rth output of jth DMU 
x_{ij } 
= 
The ith input of jth DMU 
y_{ro } 
= 
The rth output of the DMU_{o} 
x_{io } 
= 
The ith input of the DMU_{o} 
~ 
= 
Used to identify the inputs and outputs as random variables with a known
joint probability distribution 
γ 
= 
The best possible relative efficiency achieved by DMU_{o} 
Φ^{1 } 
= 
Inverse of standard normal distribution function 
S¯∈R^{m } 
= 
Excesses in inputs 
S^{+ } 
= 
Shortage in outputs 

= 
rth output shortfalls 

= 
ith input excesses 

= 
Standard deviation of rth output 

= 
Standard deviation of ith input 
α 
= 
Risk that is between zero and 1 
Vary_{ro } 
= 
rth output variance of the DMU_{o} 
Varx_{1O } 
= 
ith input variance of the DMU_{o} 
ξ, ς and ζ 
= 
The external slacks 

= 
Standard normal random variable 
β_{i}, γ_{γ } 
= 
Represent parameters (to be prescribed) 
t 
= 
A variable which helps a nonlinear model to be converted to a linear model 
∧ 
= 
[λ_{j}] vector of DMU loadings, determining best practice
for the DMU_{o} 

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