Supplier Selection Using Fuzzy Logic
CHIRAZ GASMI & SEMA E. ALPTEKIN
Industrial and Manufacturing Engineering Department
California Polytechnic State University
San Luis Obispo, CA 93402
USA
Abstract: Two supplier ranking models that use linguistic variables are presented. The first model was
developed using the traditional methods of rule definition of fuzzy logic, while Combs method was used in the
second. Combs method maps the problem space using a significantly reduced rule base, hence making the size
of the model manageable in terms of computing time and resources. This paper will show how these models
use natural language (i.e. linguistic variables) to rate and rank suppliers in order to facilitate the supplier
selection task. Comparison of both models is presented along with the results of statistical analysis.
Key-Words: Fuzzy Logic, Combs method, Supplier selection
1 Introduction
In the face of global competition, buyers must
maintain a dynamic list of criteria for the selection
of suppliers. These criteria have traditionally been
quantitative values such as price and on-time
delivery. More recently qualitative criteria such as
quality and service have been added to this list [1].
Many leading companies list value-added criteria
among their selection criteria [2-5]. The following is
a list of most commonly shared criteria in the
supplier selection process:
1) value-added to
buyer’s business (i.e. willingness to have a longterm, mutually beneficial working relationship,
dedication to buyer’s success), 2) quality
management and commitment to continuous
improvement, 3) technological competitiveness, 4)
financial viability, 5) knowledge of buyer’s line of
business, 6) customer service and satisfaction, 7)
competitive pricing, and 8) sharing similar values
and ethics as the buyer.
Most of the criteria are in fact vague. For
example how would one measure the supplier’s
ethics and values? Or how would one determine
quantitatively if the supplier were willing to invest
in a mutually beneficial relationship with the buyer?
It is criteria like these that depend on the human
“instinct”, “gut feeling”, or “experience”.
As
mentioned in [4], “Selecting a supplier is a
quantitative and qualitative process. A supplier
should offer more than just ‘parts that meet spec".
It is these types of data that have been very
difficult to incorporate in mathematical models. In
fact, modeling decision making scenarios has
captured engineering interest for over two decades
[6].
Several techniques were developed for
modeling decision-making [7]. Until the inception
of fuzzy logic and the analytic hierarchy process
(AHP), the available models only considered the
tangible data (i.e. data that we can quantify).
However, systems based solely on quantitative data
will not reproduce the desired behavior. This is due
to the fact that systems in the real world usually rely
on quantitative as well as on qualitative data.
The decision makers need models that mimic
human decision making to be able to evaluate
suppliers using both qualitative and quantitative
data. The more comprehensive the decision-making
models are and the closer they are to human
decision-making behavior, the more flexibility the
buyers have at their disposal.
Fuzzy logic is a method that models imprecise,
ambiguous, or vague data [8]. It is based on human
common sense. It can easily handle “vagueness”,
“imprecision”, or what is commonly known as
“partial truths” in the data.
Fuzzy logic has been used in similar decision
making applications. Wang and Lin propose a
system for ranking different configuration items for
software development [9]. They further develop
consensus
measures
to
determine
group
acceptability of the ranking orders. Another
application of ranking using fuzzy logic is that
performed by Beccali et al. where they present
fourteen sets of actions (alternatives) to be ranked
according to twelve criteria [10]. Out of the twelve
criteria, seven are qualitative. They use fuzzy logic
on the seven qualitative criteria in order to come up
with a rank for each alternative.
This paper aims at modeling the behavior of
evaluation and selection of criteria used in the
supplier selection from the buyer’s side. Two
models are developed using fuzzy logic
demonstrating the power of this approach in dealing
with qualitative data.
The structure of the remainder of the paper is as
follows. In Section 2 background literature on
fuzzy logic will be reviewed. The models will be
discussed in Section 3. Research results will be
discussed in Section 4 followed by conclusions and
recommendations for future research in Section 5.
2 Fuzzy Logic Overview
In 1965, Lotfi Zadeh published his theory of fuzzy
set mathematics and by extension, fuzzy logic [11].
Fuzzy logic is based on the concept of fuzzy sets,
whose values are in the range [0 1] with 0
representing absolutely false and 1 representing
absolutely true. All other values in between
represent degrees of truth. The degree of truth of a
statement as it relates to the fuzzy set would be
determined by the associated membership function.
In the supplier evaluation case, the qualitative
expression “the level of trust” can be classified in
the fuzzy set “Trust”.
In comparison with traditional Boolean logic,
Fuzzy logic is similar in that it operates on very
stable rules composed of a series of IF-THEN
statements. The similarity to Boolean logic,
however, stops at the word “logic”. In fact, where
Boolean logic is based on crisp numbers, fuzzy logic
is based on fuzzy numbers that can take on linguistic
values such as “cloudy”, “short”, “empty”, etc. how
cloudy the weather is or how short the person is or
how empty the glass is will take on degrees of truth.
In this way, the fuzzy logic system (FLS) can
accommodate an entire range of values.
There are several good definitions about fuzzy
logic systems (FLS). The one used here is Mendel’s.
He defines a FLS as a non-linear mapping of an
input into an output [12]. A rule based FLS contains
four parts: rules, fuzzifier, inference engine, and
output processor, that are all interconnected. Once
the rules have been established, a FLS can be
viewed as a mapping from inputs to outputs, and this
mapping can be expressed quantitatively as y = f(x).
Rules are a collection of IF-THEN statements,
where the “IF” part of the rule is the antecedent and
the “THEN” part of the rule is the consequent.
FLSs interchangeably, referred to as fuzzy logic
controllers, may be used in a variety of decisionmaking applications as discussed in [13-15]. Besides
its application in control areas, FLSs may be used in
manufacturing, science and medicine, medical
diagnostic systems, health monitoring and
automated interpretation of experimental data [16],
business finance [17], credit evaluation, and stock
market analysis [18].
Fuzzy logic is used in a biological resource
management system. The system uses information to
predict stream/watershed quality for fish habitat
[19]. Fuzzy logic has also been used in the banking
sector [18]. Banks review an applicant history and
decide on the level of risk s/he poses with the aid of
a fuzzy logic decision making system. DeTurris et
al. reports on the use of Fuzzy Logic in the control
of a “flying eye” that floats down carrying a camera
sending pictures of its surroundings to the user [20].
3 Supplier Ranking Models
Two supplier ranking models that are developed
using Fuzzy Logic are presented in this section. The
first model was developed using the standard
method, while Combs method was used in the
second.
3.1
Model I – Complete System
This model consists of four inputs C1, C2, C3, and
C4 and one output (score). The inputs are the criteria
used in evaluating the supplier, and the output is the
score obtained for each of the suppliers prior to
ranking them. Examples of criteria could be factors
such as trust, relationship, flexibility, quality,
service, and ethics [21]. Fuzzy-set values for the
input and output are specified as shown in Table 1.
Table 1. Fuzzy set values of inputs and output
Input
Output
C1
C2 C3 C4 Score
P (Poor)
P P P No
A (Average)
A A A M-N (Maybe Not)
G (Good)
G G G M-Y (Maybe Yes)
E (Excellent) E E E Yes
Rules are then developed to describe
relationships between inputs and outputs.
example rule is provide below:
the
An
If “Relationship is Excellent” AND “Flexibility
is Good” AND “Quality is Excellent” AND “Service
is Good” Then “Score is Yes”.
In developing the model the following
methodology was used:
FLS Mapping: In order to cover all the possible
combinations of rules a mapping of the FLS was
developed. There are 256 rules in this model
determined by the complete enumeration of four
membership functions in each of four criteria. Table
2 shows a decision rule matrix example where C1
and C2 vary between "Poor" and "Excellent", C3 in
this case is "Average" and C4 is "Excellent". As
seen, the output values vary from "Maybe not" to
"Yes". 16 tables such as this example were
developed to cover all possible combinations.
Table 2. Sample Decision Rule Matrix
C2
C1
P
A
G
E
P
M-N
M-N
M-Y
M-Y
A
M-N
M-Y
M-Y
M-Y
G
M-Y
M-Y
M-Y
M-Y
E
M-Y
M-Y
M-Y
Y
Computer Model: A computer model was
developed using the Fuzzy Logic Toolbox of
Matlab. The four criteria C1 through C4 were
entered into the system as fuzzy set inputs and score
as an output set, as shown in Figure 1.
is based on "product", the aggregation on
"maximum" and the defuzzification on "centroid".
3.2
Model II with Reduced Rule Set
William E. Combs developed an alternative rule
configuration for mapping problem spaces in control
situations. He presented his findings on a truck
backer-upper comparison of the two methods: the
standard method (or as Combs refers to it the
Intersection Rule Configuration), and the Union
Rule Configuration method [22]. The latter, which
will be referred to as Combs' method, is simpler to
build and faster to process. The idea is based on the
following equivalence:
[(p Λ q) => r]  [(p => r) V (q => r)]
where p and q are the antecedents and r is the
consequent. In the above equation Λ represents the
intersection or the logical AND and V represents the
union or the logical OR (for further detail see [2226].
In Combs method each implication has only one
antecedent and one consequent. An example of an
implication using Combs' method is:
If “Relationship is High” Then “Score is High”
Figure 1. Computer Model I
Membership Functions: Each criterion was
assigned its set of membership functions as
appropriate (see example in Figure 2).
The
membership functions are all triangular functions
with a range from 0 to 1.
Figure 2. Membership Functions for Criteria “i”
Rules: Rules were entered in accordance with the
mapping discussed earlier. The "And" method used
As recommended by Mendel in his comments
[24], this paper aims at going past the mathematical
proof of the logic and verifying whether it is
possible to use Combs' method in areas other than
controls (i.e. in the area of supplier selection). To
do so, the inputs (C1, C2, C3, and C4) and the
output (score) from Model I were maintained.
Similarly, the same fuzzy set values for both inputs
and output were maintained.
In order to maintain consistency and a basis for
comparison, the methodology used in Model I was
used again with the exception of the FLS mapping.
FLS Mapping: In Combs' method, every antecedent
is directly mapped to a consequent. In other words,
every membership function of every criterion
implies an output. These implications are then
aggregated using the logical OR. The problem
space mapping is shown in Table 3.
It should be noted that we are not using Combs’
method with a strict sense of mathematical
completeness. As pointed out in [25], for Combs’
method to be universally equivalent to the traditional
FLS, it must be modified with corrective rules if the
rule matrix is not additively separable. In our
example we violate this condition and do not use the
corrective rules, however we believe we obtain
reasonable results without absolute mathematical
rigor.
Table 3. Combs' Method Decision Rule Matrix
If
Or
C1
P
A
G
E
C2
P
A
G
E
C3
P
A
G
E
C4
P
A
G
E
Or
The t statistic was computed and found to be
equal to 1.918469. For n = 50, and level of
significance α = 0.05, t49 = 2.0096. Therefore, since
t < t49, there was failure to reject the null hypothesis
and it was concluded that there was evidence to
suggest that there was no difference in the
population means.
The three dimensional control surfaces in Figure
3 show the output “Score” that corresponds to select
combinations of sets of inputs.
Or
Then
Output No
M-N M-Y
Yes
4 Results and Discussion
50 quadruplet values were randomly generated for
the four criteria. Due to the lack of "random fuzzy
value generators" a random number generator was
used to provide as much input data as needed to
establish a basis for the comparison of the two
models. Each quadruplet emulates an expert opinion
of a supplier. The data was then entered into both
models.
A statistical analysis of the results from the two
models was performed. In order to establish whether
or not there is evidence of a statistically significant
association between the two sets of outputs, the
coefficient of correlation r was computed and found
as r = 0.8075. This value of r indicates a positive
association between the two sample outputs. Once
the association between the samples was established,
the hypothesis that there was no association between
the population outputs was tested. Let ρ be the
population coefficient.
H0: ρ = 0 (there is no correlation)
H1: ρ ≠ 0 (there is correlation)
The t statistic was computed and found to be
equal to 9.482888. For n = 50, and a level of
significance α = 0.05, t48 = 2.0106. Therefore,
since t > t48, the hypothesis was rejected and it was
concluded that there was evidence of association
between the two methods.
Having established evidence of association
between the two methods, a paired t-test was
conducted. The hypothesis that there is no difference
in the means of the two methods was tested.
H0: μD = 0 (μD = μ1 – μ2)
H1: μD ≠ 0
Figure 3. Surface Plot of Output
When the scores from both models were ranked,
the results showed some differences. For instance, if
the first 5 ranks were to be examined, it would be
noted that Model I provided the following sequence:
S25, S33, S20, S16, and S18.
Model II provided the following sequence of
ranking: S20, S25, S18, S33, S31.
S16 was ranked as # 4 in Model I but ranked as #
8 in Model II. S31 was ranked # 5 in Model II and
equally # 5 in Model I (S18 and S31 were both
ranked # 5). Although the models do not provide
perfect duplication in rankings, the same suppliers
are clustered at the top for each method.
Similarly, when considering the lowest ranking
suppliers, Model I provided the following sequence:
S49, S50, S47, S2, and S5.
Model II ranks the lowest ranking suppliers in the
following sequence: S2, S4, S10, S49, and. S4 and
S10 both ranked equally as # 34 in Model I. S47 and
S50 ranked as # 32 and # 40 respectively in Model
II.
Both the statistical analysis and the ranking of the
results suggest that there is an association between
the two models. Both methods show the same 4 out
of 5 selected suppliers in the top 5 group.
5 Conclusion and Recommendations
Both models developed provides two advantages:
(1) the flexibility of changing weights assigned to
each criteria as company policy changes or as other
factors are incorporated in the decision-making, and
(2) the flexibility for different experts to
independently rate suppliers. Both models
developed provide a flexible ranking tool that takes
into consideration the dynamic nature of decisionmaking. They provide flexibility for policy change
as well as individual choice and preference. The
model is easy and simple enough to implement in
any decision-making process, from supplier ranking,
to choosing a spouse.
Generally, fuzzy logic has the advantage of being
understandable, logical, and closer to human
reasoning that crisp logic. The advantages of
Combs’ method in mapping problem spaces are the
speed of design and computation and the simplicity
in mapping (easier to get expert opinion and
feedback).
We recognize that the supplier selected by these
methods may not the unanimous choice of all
experts. However these methods do provide a
methodology for selection with subjective criteria.
Since the design of these models relies solely on
the knowledge and expertise of the designer, we
recommend that both models be tested and verified
through expert opinion. We also recommend the
incorporation of the corrective rules as outlined in
[25] to verify that exact equivalance between the two
methods would be obtained.
Acknowledgements: We are thankful to Jon Ervin
for introducing us to Combs’ method, and for
providing feedback on the manuscript. We also are
indebted to the students in the graduate level
simulation class in Spring 2003 for their insightful
comments.
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