
Research Article


Combing Fuzzy Set Theory and Extent Analysis to Construct an Integrated Decision Making Approach in Medical Cosmetology Industry 

ChihYang Chao
and
KaiFu Yang



ABSTRACT

Many researches on how price bundling affects bundle purchasing have been discussed but an analytic evaluation of customer value offered by medical cosmetology industry is still lacking. In this study, an integrated decision making approach that combined fuzzy set theory and extent analysis has developed and applied in evaluating medical cosmetology product bundles. Based on the results, manufacturers and marketers should utilize appropriate strategy in order to deliver functional, symbolic and safety performance to enhance customers’ value image for medical cosmetology product bundles.





Received: March 30, 2011;
Accepted: July 29, 2011;
Published: September 19, 2011


INTRODUCTION
Product bundling is widely practiced in today’s marketplace. For example,
McDonalds offers “value meals” that includes a hamburger, a soft drink
and French fries. Paun (1993) argued that bundling is
a strategic marketing variable. To be able to increase customer’s value,
firms could utilize product bundles as an appropriate strategy to get better
competitive position and then obtain more competitive advantages (Afuah,
2002). Thus, delivering value of bundling will enhance overall evaluation
of a product bundle. From the previous studies, many researchers have mentioned
and described the value concept in terms of a trade off. Sheth
et al. (1991) proposed the dimensions of customer valuefunctional,
social, emotional, epistemic and conditional value; however, the causal relationships
are scanty. Woodruff (1997) pointed out that it is a
good approach to adopt the hierarchy concept to better explain customer value
but at that time the concrete variables are difficult to define. Thus, developing
a systematic structure and a comprehensive methodology for product bundles is
necessary for marketers, so as to understand the preferences of customers and
to create a competitive advantage.
In the marketing field, it is not easy to explore customers’ minds to
understand their needs, wants and demands and therefore, it is difficult to
understand customers’ preferences. This is because when customers are during
decision making process, they would consider multiple criteria for many alternatives.
In this process, there is likely to be interaction among the different criteria.
Thus, the evaluating process is complex (Oliver, 2000).
In addition, the qualitative attributes for human assessment is always imprecise
and subjective. Thus, the descriptions of customer requirements are usually
linguistic and vague. For the question of facing the uncertainty due to customers’
imprecision and vagueness, fuzzy set theory is suggested as a suitable tool
to deal with such a situation (Lin et al., 2009).
Furthermore, the fuzzy multiple criteria decision making approach can deal with
qualitative information in addition to the quantitative data (Tsao,
2006). The benefit and premium capability of adopting fuzzy set theory to
deal with qualitative and quantitative data is to represent unclear and indefinite
data. As we know, perceptive organs of human brain provided and interpreted
incomplete afferent data and vague information. To deal with such imprecise
information and incomplete data, fuzzy set theory could not only perform numerical
computation but also implement systematic calculus by using linguistic terms.
Thus, this study combined fuzzy set theory and extent analysis in order to capture
the subjective and imprecise perceptive data from customers and finally constructed
an effective and integrated decision making approach.
CONCEPTUAL FRAMEWORK Product bundle: Product bundling is the process of securing two or more necessary goods or services from a single vendor. This strategy usually provides several advantages, including saving a great deal of money. Long a practice among businesses, the concept of product bundling has expanded to include residential customers over the last twenty years, especially in regard to various types of communication services. The basic nature of product bundling involves the willingness of a supplier or vendor to provide two or more services to a customer for basic rate. This rate normally provides significant savings over purchasing the products individually.
Product bundling also provides other benefits for the customer. One has to
do with sheer convenience. Rather than having to deal with multiple vendors
in order to have access to multiple products, the customer has one central vendor
for all his or her needs (Yang, 2010; Montinaro
and Sciascia, 2011). This concept of onestop shopping means the client
spends less time keeping up with payments to various vendors and has one central
pipeline when there is a need for customer services. Product bundling is combining
two or more products or services together, creating differentiation, greater
value and therefore enhancing the offering to the customer (Asikhia,
2009; Jiang et al., 2011). Bundling is based
on the idea that consumers value the grouped package more than the individual
items (Sudhahar et al., 2006; Lee
et al., 2009).
Marketers utilize joint pricing for the sale of two or more products and/or
services in a single package (Guiltinan, 1987; Kaicker
et al., 1995; Stremersch and Tellis, 2002).
Most researchers have been interested in issues of how price bundling is a pricing
and promotion tool to decrease price sensitivity and increase purchase likelihood
for customers (Johnson et al., 1999; Soman
and Gourville, 2001; Janiszewski and Cunha Jr., 2004).
Simonin and Ruth (1995) utilized a quasiexperimental
procedure to investigate the effects of bundling influence on consumers’
evaluations and reservation price judgments. They found product bundling also
could be a strategy for new product introduction through bundling with an existing
product. Beside price information about bundling, Mulhern
and Leone (1991) and Harlam et al. (1995)
observed complementary effects in their study. Bundles composed of complements
will have higher purchase intentions than the bundles of unrelated products.
Sarin et al. (2003) applied product bundling
as a strategy to reduce the perceived risk with new hightech products. From
the above literature, the value of a product bundle is not only in the momentary
savings but it also involves other attributes for customers. Besides price factor,
this study want to find the other influential factors and understand the preferences
of customers which will help manufacturers to focus on the product attributes
that concern customers.
Integrated decision making approach: The integrated decision making approach, the focus of this research, can be used in selection problems where decisions involve a finite number of alternatives and a set of performance attributes. The decision variables can be either quantitative or qualitative. The key difference in integrated decision making approach, as compared to multiple attributes decision making models and multiple objectives decision making ones, is that they include discreet variables with a number of prespecified alternatives and more importantly, they do not require an explicit relation between input and output variables. Concerning multiple attributes decision making problems and multiple objectives decision making ones, the following methods could be utilized when the information of attribute preference is available from the decision maker: (1) The Simple Additive Weight (SAW) method; (2) The Weighted Product Method (EPM); (3) The Elimination et Choice Translating Reality method (ELECTRE); (4) The technique for order preference by similarity to ideal solution (TOPSIS) and (5) The Analytic Hierarchy Process (AHP).
Fuzzy set theory: Fuzzy set theory has been studied extensively over
the past 30 years. Most of the early interest in fuzzy set theory pertained
to representing uncertainty in human cognitive processes. Fuzzy set theory is
now applied to problems in engineering, business, medical and related health
sciences and the natural sciences (Lin et al., 2009).
In an effort to gain a better understanding of the use of fuzzy set theory in
product bundles research and to provide a basis for future research, a literature
review of fuzzy set theory in product bundles has been conducted. While similar
survey efforts have been undertaken for other topical areas, there is a need
in product bundles for the same. Over the years there have been successful applications
and implementations of fuzzy set theory in product bundles. Fuzzy set theory
is being recognized as an important problem modeling and solution technique.
Kaufmann and Gupta (1988) report that over 7,000 research
papers, reports, monographs and books on fuzzy set theory and applications have
been published since 1965. Mahmoud (2003) reviews the
literature on fuzzy industrial controllers and provide an index of applications
of fuzzy set theory to 12 subject areas including decision making, economics,
engineering and operations research.
As evidenced by the large number of citations found from the previous researches,
fuzzy set theory is an established and growing research discipline. The use
of fuzzy set theory as a methodology for modeling and analyzing decision systems
is of particular interest to researchers in product bundles due to fuzzy set
theory’s ability to quantitatively and qualitatively model problems which
involve vagueness and imprecision. Karwowski and Evans (1986)
identify the potential applications of fuzzy set theory to the following areas
of product bundles: new product development, facilities location and layout,
production scheduling and control, inventory management, quality and cost benefit
analysis. Karwowski and Evans identify three key reasons why fuzzy set theory
is relevant to product bundles research. First, imprecision and vagueness are
inherent to the decision maker’s mental model of the problem under study.
Thus, the decision maker’s experience and judgment may be used to complement
established theories to foster a better understanding of the problem. Second,
in the product bundles environment, the information required to formulate a
model’s objective, decision variables, constraints and parameters may be
vague or not precisely measurable. Third, imprecision and vagueness as a result
of personal bias and subjective opinion may further dampen the quality and quantity
of available information. Hence, fuzzy set theory can be used to bridge modeling
gaps in descriptive and prescriptive decision models in product bundles research.
CONSTRUCTION OF FUZZY JUDGMENT MATRIX
After a hierarchical model is constructed, experts are asked to compare a series
of pairwise comparisons and establish the relative importance of customer requirements
in achieving the upper level criterion. The linguistic variables that people
used to express their feelings or judgment are vague. For example, the question
asked to experts is: “what is the relative impact on the value by performance
A when compared to performance B in using various medical cosmetology products?”
In such comparisons, linguistic terms are used to assess any two elements, namely
equally, moderately, strongly, very strongly, or extremely preferred. In this
study, the widely adopted triangular fuzzy number technique (Chan
et al., 1999) is used to represent a pairwise comparison of customer
requirements. Experts express their preferences between options based on the
definition of linguistic variables in Table 1.
Table 1: 
Linguistic variables and triangular fuzzy number 

For example, someone may consider that element i is strongly important as compared
with the element j under certain criteria; he/she may set e_{ij} = (3,
5, 7). If element j is thought to be less important than element i, the pairwise
comparison between j and i can be presented by using the fuzzy number, e_{ij}
= (1/7, 1/5, 1/3).
APPLICATION OF EXTENT ANALYSIS METHOD
In previous research, many scholars have engaged in the fuzzy extension of
Saaty’s priority theory; for example, Van Laarhoven
and Pedrycz (1983) in the Netherlands, proposed a method, where the fuzzy
comparison judgment is represented by triangular fuzzy numbers. They used fuzzy
numbers with triangular membership function and simple operation laws. According
to the method of Logarithmic least squares (LLSM), the priority vectors were
obtained. However, there is one defect in the calculation process. Researchers
used fuzzy numbers to describe the “linguistic vagueness” and the
crisp values could then be obtained by using a “clear” defuzzification
formula for calculating the priority vector. However, this does not seem to
match the original concept of ambiguity. In a better alternative, the extent
analysis method and the principles for comparing fuzzy numbers are employed
to obtain weight vectors of individual levels for customer requirements (Chang,
1996). The extent analysis method is used to consider the extent to which
an object can satisfy the goal, that is, satisfaction extent. In this method,
the “extent” is quantified by using a fuzzy number. On the basis of
the fuzzy values for the extent analysis of each object, a fuzzy synthetic degree
value can be obtained, which is defined as follows:
Let X = {x_{1}, x_{2},…, x_{n}} be an object set
and G = {g_{1}, g_{2}, …, g_{m}} be a goal set.
According to the method of Chang (1996) extent analysis,
each object is taken and the extent analysis for each goal, g_{i},
is performed, respectively. Therefore, m extent analysis values for each object
can be obtained, with the following signs:
where, all the
are triangular fuzzy numbers.
The steps of extent analysis can be given as follow: Step 1: The value of fuzzy synthetic extent with respect to the ith object is defined as:
where,
is defined as the fuzzy synthetic extent value and ⊗ is defined as the fuzzy
operation multiplication.
To obtain
perform the fuzzy addition operation of m extent analysis values for a particular
matrix such that:
and to obtain ,
perform the fuzzy addition operation of
values such that:
and then compute the inverse of the vector in Eq. 8 such that: Step 2: The degree of possibility of
is defined as:
and can be equivalently expressed as follows:
where, d is the ordinate of the highest intersection point D between μ_{M1} and μ_{M2} (Fig. 1).

Fig. 1: 
The intersection between M_{1} and M_{2} 
To compare M_{1} and M_{2}, the values of V (M_{1}≥M_{2}) and V (M_{2}≥M_{1}) are needed.
Step 3: The degree of possibility for a convex fuzzy number to be greater
than k convex fuzzy numbers
can be defined by:
Assume that: Then the weight vector is given by: where, A_{i} (i = 1, 2,…, n) = are n elements. Step 4: Via normalization, the normalized weight vectors are: where, W is a nonfuzzy number.
Empirical example of pairwise comparison for hierarchy dependency:
Assume i =1~4 and j =1~4, to be able to evaluate the value of medical cosmetology
product bundles, with the use of Step 1, the fuzzy synthetic degree values of
all elements for the performance level can be calculated as shown below:
Hence, the fuzzy synthetic degree values of the clement V1, D_{V1}
can be calculated as follows:
Following a similar calculation, the fuzzy synthetic degree values of all elements for the category level of the hierarchy can be obtained as shown below:
• 
G: D_{V1}= (0.10, 0.29, 0.83) 
• 
D_{V2}= (0.15, 0.41, 1.00) 
• 
D_{V3}= (0.08, 0.24, 0.67) 
• 
D_{V4}= (0.03, 0.06, 0.24) 
Next, the following comparison results are derived based on step 2 in order
lo calculate the weight vectors of the category level of the hierarchy:
Based on Step 3, the weight vector P of the performance level of the hierarchy can be calculated by using the following formula:
After the normalization of Step 4, the normalized weight vectors of the performance level are as shown below:
DISCUSSION
In the previous studies, most researchers has proposed decision making as the
process of making an appropriate choice in order to realize one or more aims
(Zanjani et al., 2009). Thus, multiple criteria
decision making is used to balance the conflicts or tradeoff among different
aims. There have been many studies on decision making algorithms and methods
in recent years (Chen and Lee, 2003; Bharati
and Chaudhury, 2004; Tasa and Whyte, 2005; Forgionnea
and Newman, 2007; Hewahi, 2009; Dong
et al., 2010; Hosseinpourtehrani and Ghahraman,
2011). The above mentioned researches have some general approaches that
not only can support this study but also can be used to solve a multiple criteria
decision making problem. The general approaches that used in these researches
are multiple objectives decision making methods and multiple attributes decision
making. The objectives are sometimes in conflict with one another, meaning an
optimal solution of one objective does not meet the optimal solution of another.
The planner should then make a compromise between the objectives to come up
with the best solution. This gives rise to an infinite number of compromised
solutions, usually called Paretooptimum solutions (Opricovic
and Tzeng, 2004). These types of models employ decision variables that are
determined in a continuous domain with either an infinite or a large number
of choices. The best decision is then made so as to satisfy the planner’s
preference information as well as the problem constraints and objectives (Lee
et al., 2008).
In this study, by combing fuzzy set theory and extent analysis, an integrated decision making approach has developed and applied in evaluating medical cosmetology product bundles. In addition, it is an aid for manufactures and marketers to understand customers in making prudent decisions when the complexities make their decision task quite complicated. Besides, this decision making approach is capable of taking into consideration both qualitative and quantitative information. Furthermore, this decision making approach could also be used for evaluating the most valuable medical cosmetology product bundles to serve as a competitive forecast for marketers. CONCLUSION Concerning medical cosmetology product development, the efforts of manufacturers in bringing product attributes in line with customers’ expectations is worthwhile. Marketers, moreover, should focus on customer preferences to plan the multiple consequences of the product bundle. Both manufacturers and marketers should not only utilize “price” as a main marketing strategy for medical cosmetology product bundles but also to deliver functional, symbolic and safety performance to consumers and enhance the value image in their minds. In practice, medical cosmetology companies utilize direct mail, email and media as advertising tools to attract customers’ attention. Through this communication process, manufacturers and marketers could understand customers’ preferences and concerns and then help them purchase the medical cosmetology products they need to be able to satisfy their needs, wants and demands. Thus, manufacturers and marketers could focus on the product attributes that match customers’ preferences and through advertisements appeal to customers.

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