INTRODUCTION
An ERP is a packaged enterprisewide information system that integrates all
necessary business functions, such as product planning, purchasing, inventory
control, sales, financial and human resources, into a single system with a shared
database (Soffer et al., 2003). A large software
purchase like an ERP system is a project with significant financial impact and
involves making a decision based on both qualitative and quantitative information.
In these types of problems, quantitative tools are extremely valuable; however,
accurately combining fundamentally different data into a single model in such
a way that the answers reflect the correct relative importance is a challenge
(Erol and Ferrell, 2003). As enterprises experience
an increase in data, many look towards implementing enterprisewide data automation
software, in order to organize data and assist in making sensible business decisions.
Software selection is not a technical procedure, but is rather, a subjective
and uncertain decision process (Stamelos et al.,
2000). Selecting a suitable software systems like an ERP system, among many,
depends on the assessment of objective, measurable criteria (e.g., acquisition
costs and training costs), as well as subjective criteria (e.g., compatibility,
vendor selection and technical factors). ERP Software selection decisions including
tangible and intangible factors; so prioritizing these factors can be challenging.
When evaluating and selecting software; suggested that the evaluation should
encompass both performance and technical requirements.
The importance of each criterion may also vary, under different requirements
and situations. It is easier for a decision maker to describe his/her desired
value and the importance of a criterion, by using common language. The purchaser
may state, for example, that vendor support is considered “very important”
in the selection of a software, or that vendor A can provide “good”
technical support during software implementation (HuaY. Lin
et al., 2007). Owing to the imprecise nature of software selection,
there is a need to develop a useful method as Quality Function Deployment (QFD)
to translate the needs of companies to the technical attributes and prioritizing
them based on fuzzy set theory. The fuzzy set theory was originally introduced
by Zadeh (1965) to deal with illdefined problems, characterized
by a certain degree of uncertainty and vagueness.
QFD was originally proposed, through collecting and analyzing the voice of
the customer, to develop products with higher quality to meet or surpass customer’s
needs. Thus, the primary functions of QFD are product development (Moskowitz
and Kim, 1997), quality management and customer needs analysis. Later, QFD’s
functions had been expanded to wider fields such as develop strategy (Kim
et al., 2000), help planning (Milan et al.,
2003), develop software (Herzwurm and Schockert, 2003),
develop services (Enriquez et al., 2004), strategic
management (Bottani and Rizzi, 2006), supplier selection
(Bevilacqua et al., 2006) and the other applications.
The first two reported applications of QFD were in the shipbuilding and electronics
industries. QFD’s early applications focused on such industries as automobiles,
electronics and software. The fast development of QFD has resulted in its applications
to many manufacturing industries. Eventually, QFD has also been introduced to
the service sector such as government, banking and accounting, health care,
education and research. Now it is hardly to find an industry to which QFD has
not yet been applied.
SOFTWARE SELECTION
Software selection review: Decision making in the field of software
selection has become more complex due to a large number of software products
in the market, ongoing improvements information technology and multiple and
sometimes conflicting objectives.
A variety of methodologies and frameworks for software selection and evaluation
have been developed. Lucas and Moore (1976) use of a
scoring method to determine the importance of software, Buss
(1983) proposed a ranking approach to compare computer projects. This method
also has the same limitation with scoring method. Mathematical optimization
such as goal programming, 01 programming and nonlinear programming have been
applied to resource optimization for IS selection. Santhanam
and Kyparisis (1996) developed a nonlinear zeroone programming model which
considered technical interdependencies among Information System (IS) projects.
The model is transformed to a linear mixed integer programming model through
a linearization procedure. Although both of these models improved upon earlier
studies by considering interdependencies inherent in the IS selection process,
the solution procedure is likely to get complicated as the number of IS alternatives
and interactions among them increase Teltumbde (2000)
proposed a methodology framework for evaluating ERP projects based on the Nominal
Group Technique NGT and AHP. Badri et al. (2001)
presented a 01 goal programming model to select an IS project considering multiple
criteria including benefits, hardware, software and other costs, risk factors,
preferences of decision makers and users, completion time and training time
constraints. Wei and Wang (2004) proposed an AHPbased
approach to ERP system selection. This study uses the analytical framework of
AHP to synthesize decision maker’ tangible and intangible measures with
respect to numerous competing objective inherent in ERP system selection and
facilitate the group decisionmaking process. Bernroider
and Stix (2006) combined the utility ranking method and the DEA to overcome
the limitations of DEA in software selection.
Liao et al. (2007) presented a new model, which
is based on the 2tuple linguistic information processing, for dealing with
the problem of ERP system selection. In that study, a similarity degree based
algorithm is proposed to aggregate the objective information about ERP systems
from some external professional organizations, which may be expressed by different
linguistic term sets. The consistency and inconsistency indices are defined
by considering the subject information obtained from internal interviews with
ERP vendors and then a linear programming model is established for selecting
the most suitable ERP system. Karsak and Ozogul (2007)
developed a decision framework for ERP software selection based on Quality Function
Deployment (QFD), fuzzy linear regression and zeroone goal programming. The
proposed framework enables both company demands and ERP system characteristics
to be considered and provides the means for incorporating not only the relationships
between company demands and ERP system characteristics but also the interactions
between ERP system characteristics through adopting the QFD principles. Kutlu
and Akpinar (2009) applied fuzzy logic for Enterprise Resource Planning
software selection. This study has been concentrated on a shipping company as
the case.
Fuzzybased decisionmaking method has been successful employed on a great diversity of applications. The use of fuzzy set theory improves decisionmaking procedure by accommodating the vagueness and ambiguity occurred during human decision makings. The decision makers can use linguistic terms to evaluate criteria and alternatives easily and intuitively. Thus, the objective of this study is to propose a comprehensive vendor ERP software selection procedure, in which the objectives structure is constructed and the appropriate criteria are specified to provide detailed guidance for ERP software evaluation based on fuzzy set theory.
FUZZY QFD
Quality function deployment: QFD belongs to the sphere of quality
management methods, offering us a linear and structured guideline for converting
the customer’s needs into specifications for and characteristics of new
products and services. The method involves developing four matrixes, or ‘houses’,
that we enter by degrees as a project for a given product or production process
is developed on increasingly specific levels (Akao, 1990;
Venkatachalam et al., 2008). In the present article,
our attention focuses on the Planning Matrix, or (HOQ) (Hauser
and Clausing, 1988) (Fig. 1) details of the fuzzy method
is available in appendix.
The HOQ provides the specifications for product design (or engineering characteristics)
in terms of their relative importance and of target values that have to be reached
in design and production. In a sense, the HOQ is the hub of the whole QFD method:
its construction enables us to precede from the customer’s requirements
to the design specifications (Schmidt, 1997; Fariborz
and Rafael, 2002; Bier and Cornesky, 2001).
This study describes the HOQ and its process following the approaches suggested
by Brown (1991). Step 1: Identify the WHATs. The wanted
benefits in a product or service in the customer’s own words are customer
needs and are usually called customer attributes (CA) or “WHATs”,
area (A) in Fig. 1. In assigning priorities to WHATs, it is
necessary to balance efforts in order to accomplish those needs that add value
to the customer. The priorities are usually indicated in the area designated
as (B) in Fig. 1. Step 2: Determination of HOWs. Engineering
characteristics are specified as the “HOWs” of the HOQ and also called
measurable requirements. HOWs are identified by a multidisciplinary team (Hauser,
1993) and positioned on the area marked as (C) on the matrix diagram, Fig.
1. Step 3: Preparation of the relationship matrix (D). A team judges which
WHATs impact which HOWs and to what degree. Step 4: Elaboration of the correlation
matrix. The physical relationships among the technical requirements are specified
on an array known as “the roof matrix” and identified as (E) in Fig.
1. Step 5: Action plan. The weights of the HOWs, identified as area (F),
are placed at the base of the quality matrix. These weights are one of the main
outputs of the HOQ and are determined by:
Weight(how)_{i }= V(how)_{i1}ximp(WHAT_{i
}) + … + V(HOW)_{in}ximp(WHAT_{n }), 
where, V(HOW)_{in }is the correlation value of HOW_{i} with WHAT_{n} and imp(WHAT_{n}) represents the importance or priority of WHAT_{n}.
Fuzzy logic: In dealing with a decision process, the decisionmaker
is often faced with doubts, problems and uncertainties. To cope with and “handle”
such uncertainties and inaccuracies, he generally relies on tools provided by
probability theory, accepting the principle that an inaccuracy, whatever its
nature, is governed by random law. In a real decisionmaking process, however,
we have to deal with different types of uncertainty and inaccuracy, each of
which needs to be treated with the aid of a specific tool. Probability theory
is fine for representing the stochastic nature of decisional analysis, but is
unable to measure the inaccuracies or uncertainty that stem from human behavior,
which is neither stochastic nor random. The fundamental role of the decisionmaker
or other parties involved in the decisional process poses a number of problems
that cannot be handled appropriately by probability theory.
Referring specifically to a multicriterion analysis, this means that the values
of a certain alternative concerning a given attribute often cannot be precisely
defined, the decisionmaker is unable (or unwilling) to express his preferences
precisely, the evaluations or opinions are expressed in linguistic terms and
so on. To deal with this type of uncertainty correctly we can resort to fuzzy
logic (Zadeh, 1965). The logical tools that people can
rely on are generally considered the outcome of a bivalent logic (yes/no, true/false),
but the problems posed by reallife situations and human thought processes and
approaches to problemsolving are by no means bivalent (Tong
and Bonissone, 1980). Just as conventional, bivalent logic is based on classic
sets, fuzzy logic is based on fuzzy sets. A fuzzy set is a set of objects in
which there is no clearcut or predefined boundary between the objects that
are or are not members of the set. The key concept behind this definition is
that of “membership”: each element in a set is associated with a value
indicating to what degree the element is a member of the set. This value comes
within the range [0, 1], where 0 and 1, respectively, indicate the minimum and
maximum degree of membership, while all the intermediate values indicate degrees
of “partial” membership.
Nature of decisional analysis, but is unable to measure the inaccuracies or uncertainty that stem from human behavior, which is neither stochastic nor random. The fundamental role of the decisionmaker or other parties involved in the decisional process poses a number of problems that cannot be handled appropriately by probability theory.
There are various types of fuzzy number, each of which may be more suitable
than others for analyzing a given ambiguous structure; the present analysis
uses triangular fuzzy numbers. These numbers are represented by triplets of
the type A = (x^{L}, x^{a}, x^{R}) where x^{L}
and x^{R} are respectively the lower and upper limits of the fuzzy number
considered, while x^{a} is the element that denotes the closest fit.
Triangular fuzzy numbers are often used to quantify linguistic data. The use
of triangular fitness functions is fairly common in the literature (Karsak,
2004; Chan and Wu, 2005), because triangular fuzzy
numbers are among the few fuzzy number forms that are easy to manage from the
computational point of view.
For instance, let U = (VL, L, M, H, VH) a linguistic set used to express opinions on a group of attributes (VL = very low, L = low, M = medium, H = high, VH = very high). The linguistic variables of U can be quantified using triangular fuzzy numbers as Table 1 (Fig. 2).
The linguistic variable M for example means that the decisionmaker’s assessment contains elements of grades x^{L} = 4 up to a grade x^{R} = 6 with a maximum degree of membership in x^{a} =5.
FuzzyQFD: Research on fuzzyQFD has received a certain amount of attention
(Temponi et al., 1999; Harding
et al., 2001) and made substantial progress. Khoo
and Ho (1996) proposed an approach centered on the application of possibility
theory and fuzzy arithmetic to address the ambiguity in QFD operations. Fung
et al. (1998) developed a hybrid system to incorporate the principles
of QFD, AHP and fuzzy set theory to determine design targets. Wang
(1999) proposed a fuzzy outranking approach to prioritize HOWs. Shen
et al. (2001) proposed a fuzzy procedure to examine the sensitivity
of the ranking of HOWs to the defuzzification strategy and degree of fuzziness
of fuzzy numbers.

Fig. 2: 
Linguistic scale for relative 
Table 1: 
Quantified linguistic variables by triangular fuzzy numbers 

All these works aim to determine a rating of the HOWs. In this study we propose
for the first time the fuzzyQFD methodology for ERP software vendor selection.
The conceptual and procedural approach of the HOQ remains, though the roles
have been inverted: in traditional QFD applications, the company has to identify
its customers’ expectations and their relative importance (external variables)
in order to identify which design characteristics (internal variables) should
be allocated the most resources; when the HOQ is used in ERP software vendor
selection. on the other hand, the company starts with the features that the
ERP software systems must have in order to meet certain requirements that the
company has establishedand consequently knows very well (so the customer’s
expectations become internal variables, since the company itself is the customer)and
then tries to identify which of the alternative attributes (external variables)
have the greatest impact on the achievement of its established objectives. Finally,
we have used fuzzy suitability index FSI to express the degree to which each
supplier satisfies a given requirement (Teng and Tzeng,
1996).
ERP SOFTWARE SELECTION PROCEDURE WITH A CASE STUDY
To test the efficacy of the proposed method, it was applied to an ERP software selection process for an automotive parts assembling company at the end of Oct. 2009 to Feb. 2010. . The data used as input to implement the proposed software selection method were collected by means of interviews with a team that consist of three experts that involve General Manager and senior Management Information System (MIS) and Purchasing manager.
The whole ERP software selection procedure is characterized by the following steps:
• 
Identifying the characteristics or criteria that the ERP software
being purchased must have (internal variables or “WHAT”) in order
to meet the Company’s Needs (CNs) 
• 
Identifying the technical attributes of the ERP software (TAs)
relevant to vendor assessment (external variables or “HOW”) 
• 
Determining the relative importance of the “WHATs” 
• 
Determining the “WHAT”“HOW” correlation
scores and constructing the HOQ 
• 
Determining the weight of the “HOWs” 
• 
Preparing the matrix for correlating the “HOWs” 
• 
Determining each potential vendor’s impact on the attributes
considered (“HOWs”) 
• 
Drawing up the final ranking on the FSI (fuzzy suitability
index) 
Identifying the (“WHATs”): There are some fundamental criteria
required of ERP software purchased from outside vendors, that recommended by
some experts and studies (Teltumbde, 2000; Erol
and Ferrell, 2003; Wei et al., 2005; Karsak
and Ozogul (2007)), total cost of ownership, functional fit of the system,
user friendliness, flexibility, vendor’s reputation and service and support
quality are identified as the company needs (CNs or WHATs), in the ERP system
selection study. The definition of each company needs is as follow:
• 
Total cost of ownership (TCW) consists of cost components
including software, hardware, consulting, training, implementation team,
etc. 
• 
Functional fit of the ERP system (FF) shows the amount of
customization and additional development needed for a close fit to intended
processes or customer requirements. Functional fit is preferred over functionality
since superfluous functionalities bring nothing but unnecessary complexity
to user 
• 
User Friendliness (UF) is essential since intuitive and selfexplained
screens and menus would shorten the adaptation process for endusers and
reduce the required basic trainings 
• 
Flexibility Implies (FI) not only the ease of adapting the
system to optimal business processes but also the ease of addon development,
system administration and platform independence. A flexible ERP system should
offer multiple languages support and conformity to country specific accounting
and costing schemes 
• 
Vendor’s Reputation (VR), internationality, sales references
and in particular, completed successful projects in the same industry is
a key factor to be considered for customers 
• 
Service and Support Quality (SSQ), Customers also assign high
importance to reliable and rapid responsive support. Partner networks are
crucial to constituting vendor’s service and support infrastructure 
Identifying the (“HOW”): In this section we should determine
the technical attributes (TAs) of the ERP software relevant to vendor assessment.
By a careful review of the ERP software selection literature and the opinions
of experts and on the base of Karsak and Ozogul (2007),
this study identified seven technical attributes as follow:
• 
Percentage of Supported Needs (PSN) 
• 
Percentage of Supported Needs via customization(PSNC) 
• 
Number Of Customers(NOC) 
• 
Vendor’s Total Revenues(VTR) 
• 
Number of Solution Partners(NSP) 
• 
Average duration of User Training (AVUT) 
• 
Vendor’s Operating Countries (VOC) 
Table 2: 
The judge of decision makers about each ERP software criteria 

Table 3: 
Translated linguistic variables in to triangular fuzzy numbers 

Table 4: 
Average fuzzy weights of each ERP criteria 

Determining the relative importance of the “WHATs”: Each of the three decisionmakers established the level of importance (or weight) of each “what” by means of a linguistic variable. Five different levels of importance were used in this study, i.e. very low, low, medium, high and very high, subsequently indicated as VL, L, M, H and VH. The linguistic variables were translated into fuzzy numbers by defining appropriate fitness functions. Triangular fuzzy numbers were used, characterized as in Table 1 showed. The outcome of this stage is shown in Table 2 and Table 3.
In this study the weights assigned by the decisionmakers were aggregated using
the average operator, as described by the following equation:
Weights_{WHAT} = {w_{i},
where i = 1,…, k}, w_{i }= 1/nx(w_{i1} + w_{i2}
+ … + w_{in}) 
where, k is the number of “WHATs” and n is the number of decisionmakers (k = 6 and n = 3 in our case). Each element on the Weights_{WHAT} vector is a triangular fuzzy number defined by the triplet w_{i} = (w_{iá}, w_{iâ}, w_{iã}). The weights obtained by aggregating the opinions expressed by each decision maker are shown in the Table 4.
Determining the “HOWs””WHATs” correlation scores and weighting the “HOWs”: Each decisionmaker was asked to express an opinion, using one of the five linguistic variables, on the impact of each “HOW” on each “WHAT”. The opinions expressed by the five decisionmakers are shown in Table 5.
Table 5: 
Correlation between HOWs and WHATs 

Table 6: 
The matrix of “HOWsWHATs” correlation as triangular
fuzzy numbers 

Here again, triangular fuzzy numbers were used to quantify the linguistic variables
and, as in the previous case, the fuzzy numbers obtained for each decisionmaker
were aggregated by means of the following equation:
Rating = {r_{ij}, where i =
1,…,k and j = 1,…,m}, r_{ij} = 1/nx(r_{ij1}
+ r_{ij2} + … + r_{ijn} ) 
where, k = number of the “WHATs”, m = number of the “HOWs” and n = number of the decisionmakers (in this example, k = 6, m = 7 and n = 3). This time, the RATING is the matrix of the “how”“what” correlation scores, whose rij elements represent an aggregate correlation score between the ith “what” and the jth “how”. Here again, the r_{ij} elements are triangular fuzzy numbers defined by the triplets r_{ij} = (r_{ijá}, r_{ijâ}, r_{ijã}) as shown in Table 6.
We can now complete the HOQ, calculating the weights of the “HOWs”,
averaging the aggregate weighted r_{ij} correlation scores with the
aggregate weights of the “WHATs” w_{i}, according to the equation:
Weight_{HOW} = {W_{j},
where j = 1,…,m}, W_{j} = 1/k x [(r_{j1} x w_{1})
+ … + (r_{jk} x w_{k})] 
where, the usual conventions are assumed for k and m. Each W_{j} on the Weight_{HOW} vector represents the weight of each technical attribute (matrix F of Fig. 3). The W_{j} are, once again, triangular fuzzy numbers defined by means of the triplets W_{j} = (W_{já}, W_{jâ}, W_{jã}). The fuzzy values for the weights of the “HOWs” are shown in Table 7.
Developing the matrix of correlations between the “HOWs”:
The correlations between the technical attributes (“HOWs”) are contained
in the “roof” of the HOQ (matrix E of Fig. 3). This
step in the construction of the HOQ enables the team members to keep track of
pairs of “HOWs” needing parallel improvements and/or comprising “HOWs”
in potentially difficult relationships that consequently imply measures that
are inconsistent with each other.

Fig. 3: 
The completed fuzzy HOQ 
Table 7: 
Average weighed of each HOWs as a triangular fuzzy numbers 

This matrix contains positive and negative correlations between pairs of “HOWs”
using the same symbols as Hines et al. (1998).
The completed fuzzyHOQ is illustrated in Fig. 3.
Determining the impact of each ERP systems on the attributes considered: Having completed the weighting of each attribute, all we have to do is assess each ERP systems versus the attribute in question and combine said assessments with the weight of each attribute in order to establish a final ranking. Table 8 shows each decisionmaker’s opinions on the various ERP systems in relation to each attribute.
In the same way as before, the linguistic variables were quantified by means
of triangular fuzzy numbers, then the five decisionmakers’ assessments
were aggregated according to the following equation:
Vendor Rating = {VR_{hj}, where
h =1,…,p, j = 1,…,m}, VR_{hj} = 1/nx(VR_{hj1}
+ … + VR_{hjn} ) 
where, m is the number of attributes (“HOWs”), p is the number of
vendors, n is the number of decisionmakers and VR_{hjn} is the (fuzzy)
evaluation expressed by the nth decision maker for the hth vendor as regards
the jth attribute.
Table 8: 
Impact of each ERP system on the technical attributes 

Table 9: 
FSI index for each ERP system 

The Vendor Rating matrix will contain the aggregate assessments VR_{hj}
of the hth vendor for the jth attribute; the elements in this matrix are also
triangular fuzzy numbers identified by the triplets VR_{hj} = (VR_{hjá},
VR_{hjâ}, VR_{hjã}).
Vendor ranking: The last step in the procedure involves calculating
the FSI for each vendor; this index expresses the degree to which each vendor
satisfies a given requirement. The FSI_{h} index is a triangular fuzzy
number obtained from the previously calculated aggregate scores, multiplied
by the weights for each assessment criterion (Bevilacqua
et al., 2006). The Eq. is as follows:
FSI = {FSI_{h}, h = 1,…,p},
FSI_{h} = 1/m x [(VR_{hj} xW_{1}) + … + (VR_{hj}
x W_{m})] 
where, the previously adopted conventions apply for p and m. The FSI vector
contains the FSI_{h} indexes for each vendor, which is triangular fuzzy
numbers as usual, defined by the triplets FSI_{h} = (FSI_{há},
FSI_{hâ}, FSI_{hã}), the components of which can
be calculated as follows:
FSI_{há} = 1/m Σ VR_{hjá} . W_{já;
}j = 1,…,m
FSI_{hâ} = 1/m Σ VR_{hjâ} . Wjâ;
j = 1,…,m
FSI_{hã} = 1/m Σ VR_{hjã} . Wjã;
j = 1,…,m

Table 10: 
Final rating of ERP systems 

For the case in point, the FSI_{h} indexes are given in Table
9.
Applied to a triangular fuzzy number FN = (FN_{á}, FN_{â},
FN_{ã}), the Facchinetti et al. (1998)
approach produces a score identified by the value:
(Fn_{á} + 2FN_{â}
+ FN_{ã}) / 4 
So, the final scores are shown in Table 10.
Using the fuzzy ranking principle, these fuzzy ratings produce the following
ranking order for the ERP systems:
Sys 10>Sys1>Sys 4>Sys 3>Sys
8>Sys 9>Sys 6>Sys 2>Sys 5>Sys 7 
CONCLUSION
Selecting a suitable ERP system is the basis of implementing ERP project successfully.
This study presents a new model for ERP selection, which is based on fuzzy QFD approach. The main characteristics of this model are:
• 
As a tool for the identification of the best criteria to ERP
software selection 
• 
As a decision model for the finalchoice phase in ERP software
selection process 
The conceptual approach proposed in this study is based on the distinction between the company needs about the ERP software and the technical attributes of ERP software. It becomes evident, in fact that the company’s ultimate aim is to have access to ERP systems that ensure certain standards. It is equally clear, however, that achieving these objectives depends largely on the characteristics of the ERP system itself. It becomes impossible, or at least conceptually unwise, to attempt to achieve such objectives by restricting the assessment to only one of these two categories of attributes.
Constructing an HOQ enables these two groups of attributes to be correlated, so that we can identify how well each ERP system succeeds in meeting the requirements established for the company; having done so, we can go on to draw up a ERP systems rating list. The use of fuzzy logic enables the decisionmakers to eliminate, or at least contain the problems stemming from the subjective and ambiguous nature of their information, so that they can formally treat (and thus implement in calculation systems) even those variables that conventional techniques cannot manage without sacrificing the expressive power typical of verbal language, that still cannot be reproduced by artificial intelligence. Whenever it is impossible to establish clearly distinct constraints due to the variables that define the problem, decisionmakers interpret their values on the strength of their experience and understanding of the problem and then draw appropriate decisions.
APPENDIX
Fuzzy methods: Fuzzy set theory was developed for solving problems
in which descriptions of objects are subjective, vague and imprecise, i.e.,
no boundaries for the objects can be well defined. Let X={x} be a traditional
set of objects, called the universe. A fuzzy set
in X is characterized by a membership function μ_{P}(x) that associates
each object in X with a membership value in the interval [0,1], indicating the
degree of the object belonging to .
A fuzzy number is a special fuzzy set when the universe X is the real line
R^{1}: 8<x<+8. A Symmetrical Triangular Fuzzy Number (STFN),
denoted as =
[0,1], is a special fuzzy number with the following symmetrical triangular type
of membership function:
STFN is widely used in practice to represent a fuzzy set or concept =
“approximately b” where b = (a + c) / 2. For example, if an ERP criterion
TCW is rated as having “very high” importance by a decision maker,
then traditionally we may assign TCW a number 9 using crisp scale. To capture
the vagueness of the decision maker’s subjective assessment, we can according
to the same scale assign TCW an STFN [8,10] which means “approximately
9” and is represented by the following membership function:
This means that, for example, the membership value or “possibility”
that TCW is assigned a number 9 is μ_{[8,10]}(9) = 1, the “possibility”
that TCW is assigned a number 8.5 or 9.5 is μ_{[8,10]}(8.5) = 0.5
or μ_{[8,10]}(9.5) = 0.5. So assigning TCW a number 8.5 or 9.5
is acceptable or “possible” to the degree of 50%. The basic arithmetic
rules for STFNs are as follows:
.

For any two STFNs, =
[a, b] and, =
[c, d], if one interval is not strictly contained by another then their ranking
order can be easily and intuitively determined. That is:
• 
If d>b and ca,
or db
and c>b, then >,
where ">" means "is more importance or preferred than" 
• 
If a=c, b=d, then = 
But if one interval is strictly contained by another, i.e., if d<b and c>a,
or d>b and c<a, then the ranking problem becomes complex and many possibilities
may occur. For more details about fuzzy set theory, STFNs and fuzzy ranking
methods, Zimmermann (1987).