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Product selection in Internet business: a fuzzy approach
B. K. Mohantya and B. Aounib
aIndian Institute of Management, Off Sitapur Road, Lucknow 226 013, India,bDecision Aid Research Group, School of Commerce and Administration, Faculty of Management, Laurentian University,
Sudbury, ON, Canada P3E 2C6
Received 5 June 2007; received in revised form 11 August 2008; accepted 9 March 2009
Abstract
In this paper, we propose a methodology which helps customers buy products through the Internet. This
procedure takes into account the customer’s level of desire in the product attributes, which are normally
fuzzy, or in linguistically defined terms. The concept of fuzzy number will be used to measure the degree of
similarities of the available products to that of the customer’s requirements. The degrees of similarities so
obtained over all the attributes give rise to the fuzzy probabilities and hence the fuzzy expected values of
availing a product on the Internet as per the customer’s requirement. Attribute-wise the fuzzy expected
values are compared with those of the available products on the Internet and the product that is closest to
the customer’s preference is selected as the best product. The multi-attribute weighted average method is
used here to evaluate and hence to select the best product.
Keywords: Internet business; multiple product attribute; customer’s preference; fuzzy probability; fuzzy numbers
1. Introduction
Over the last decade many companies have focussed on using the Internet to commercialise their
products and services. Some examples are available in Alter (2004). The development of advanced
information technology, online business strategies and Internet tools have established more
convenient ways to commercialise their business transactions efficiently. Many organisations
consider this a major paradigm shift from the traditional concept of across-the-table marketing to
the world of easy and effortless business processing. Internet business is considered to be a
technological driving force in the 21st century. For example, in online car purchasing, a customerin India can view cars which are available in the United States. If a car is to his/her liking, he/she
can order the car over the Internet while sitting in India. In a similar way, a company can
broadcast design or model changes to its customers across the globe by advertising on the
Internet. Almost all multinational companies have incorporated IT-enabled systems including
Internet business strategies in order to establish direct channels for organisational business
Intl. Trans. in Op. Res. 17 (2010) 317–331DOI: 10.1111/j.1475-3995.2009.00712.x
INTERNATIONAL
TRANSACTIONS
IN OPERATIONAL
RESEARCH
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transactions, remote information retrieval, processing and management for agile marketing, and
electronic trading. Online business has given a boost to companies in terms of responding rapidly
and accommodating themselves to market changes.
In any business environment (online or traditional), a company presents its products and
features to their customers. A customer uses his product knowledge and the product information
provided to compare different products and product features, and select the best product. Thisdecision is made based upon information from marketing sources, sales personnel, brand
information, product features, and the customer’s own financial budget. Normally a customer
gathers all this information from a variety of sources and combines it implicitly before making
their decision. However, obtaining pertinent, consistent and up-to-date information is a complex
and time-consuming process. With the help of the Internet, companies can provide various aspects
of their business information to their customers. Internet tools aim to improve the way that this
information is gathered, managed, distributed and presented to customers. However, for the
successful implementation of the business process, we need to address two main difficulties.
Firstly, the customer’s product requirements and preferences are imprecise or fuzzy in nature, and
secondly, product assessments by customers are based on multiple attributes.
In practice, a customer’s product requirements are usually represented with multiple attributesthat can be conflicting, non-commensurable and fuzzy. In the traditional method of shopping, a
sales assistant collects the customer’s fuzzy information about the desired product attributes and
suggests an appropriate product. However, this is next to impossible to achieve in the Internet
market as there is no sales assistant available for the job. Moreover, attribute-wise fuzzy inputs to
e-business systems will show an input error. Although some of the e-commerce systems are based
on the concepts of fuzzy logic, the majority of present day online business systems require precise
data, non-ambiguous knowledge and exact information. Additionally, many of them do not
follow any objectively defined procedure of obtaining the similarities between the customer’s
desired attribute levels and the available product’s realisation level (Cao and Li, 2007; Lee and
Widmeyer, 1986; Mohanty and Bhasker, 2005; Ryu, 1999). Non-fuzzy data assumptions in
Internet business will be difficult to implement because imprecision is inherent in the customer’smind and also in almost all business problem domains. Ignoring or approximating the customer’s
fuzzy requirements and using existing precise models will not only show an unpredictable result
but also lead to a business failure because of the misrepresentation of the customer’s real opinion
in terms of their product requirements. For example in buying a shirt, a customer does not
normally consider the precise percentage of cotton, the style and softness of the fabric and the
durability of the shirt. However, a customer fuzzily expresses his requirements on these attributes.
What is needed here is an e-business system which incorporates fuzzy representations of both the
customer’s requirements and the products available and leads to a matching product, as in
traditional showrooms. Our paper aims to develop such a model to address the issue of
association of the customer’s fuzzy requirements and fuzzily defined product realisations on the
Internet market.Another problem normally encountered in an online business process is due to the customer’s
interest for non-digital product features. For example, a car’s digital attributes include price,
maintenance cost, and mileage, and can be easily expressed over the Internet. However, non-
digital attributes are more difficult. In the same car problem, the non-digital attributes include seat
comfort, surface polishing, and ease of driving. These attributes cannot be assessed over the
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Internet without experiencing them physically. Digitisation of non-digital attributes is beyond the
scope of this paper. The proposed procedure only works on the product attribute information at
hand and does not distinguish between digital and non-digital attributes.
An additional difficulty in handling Internet business is due to the fact that the customer’s
product assessment is of multi-dimensional nature. That is, very often a customer would like to
examine and analyse multiple aspects of the product before making their decision. For example, inthe car-purchasing problem, the multiple attributes may be cost, design, colour, fuel consumption,
resale value, engine type, etc. In this case, the customer has to select a car that offers the best
match to his/her requirements. In the traditional market, a customer can discuss with a sales
person and finally select a product. In this situation, a customer generally makes trade-offs or
compromises when convinced by the sales personnel that this is the only possible way of doing
business in the given market profile. However, in the Internet market, there is no discussion and
also it is difficult for a customer to express his views about the product attributes. This becomes
more so, when product attributes are fuzzy, conflicting and non-commensurable in nature. By
considering the concepts of multiple-attribute decision-making, we have highlighted this case in
the Internet market. The multi-attribute approach is one of the many models that have been
developed to deal with the decision-making situations where several factors such as objectives,constraints, or attributes have to be considered simultaneously (Aouni and Kettani, 2001;
Carlsson and Fuller, 1996; Martel and Aouni, 1998; Mohanty, 1998; Mohanty and
Vijayaraghvan, 1995; Triantaphyllou, 2000). The proposed procedure could be qualified as a
decision aid tool since it enables the customer to get more involved in the decision-making
process.
Personal shopping agents like Jango and DealTime (by http://www.dealtime.com) are early
efforts to provide agent based shopping support to customers. These agents collect the price and
desirable product attributes from the Internet for a specified product. Once the customer selects a
product, these agents assist in identifying the best available deal. However, the buyer’s problem is
in identifying a desirable product in the vast Internet market. Another system like decision guide
by http://www.ActivityBuyersGuide.com assists the consumer by giving the best availableproduct based on the product attribute requirements provided by the customer. This agent
provides a list of matching products after obtaining the attribute-wise product requirements from
the customer. This has its own limitations as a customer cannot define his/her desired level of
satisfaction in each product attribute. Further, in decision guide a customer is unable to state the
importance of the product attributes or specify a range for the product attribute levels from
minimum to maximum. These inputs enable the system to list appropriate products and present
them to the customers. However, there might be a set of products having slight deviations from
the above range but still offering an overall desirability to the customer, yet they are not included
in the list. Thus, the satisfaction level that the customer is likely to derive from the deviated
products becomes hidden and thereby makes it difficult to observe the products in totality and to
assess the superiority of the listed products to that of the deviated ones. This comparison isnecessary because at times a customer may like to give concession and compromise in a certain
attribute and choose a product from the deviated list. The ‘‘decision guide’’ does not account for
the customer’s compromising attitude.
In Internet business, we have not come across any paper which represents product attributes in
fuzzy sets and at the same time compares them to the customer’s fuzzily defined opinions to find a
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matching product. A shopping program given in Lee and Widmeyer (1986) aims at the selection of
a desirable product through the Internet. In the situation of non-availability of the product the
program suggests a product that is closest to the requested product in the taxonomy hierarchy as
an alternative. The real shortcoming in this approach is that the search is conducted in a single
generic product hierarchy. However, in real problems, the product selections are based on
multiple attributes. For example, if a customer wants to select a car based on its price only, thecars can be arranged in an ascending order based on their prices. Similarly we can have another
ordering of the cars based on its mileage attribute. However, if a customer prefers to order cars
based on its price as well as mileage (with different degrees of importance) the ordering will be
different. The improvement from a single hierarchy to a multiple one and the preference hierarchy
therein is given in Ryu (1999).
Electronic shopping support given in Ryu (1999) classifies products based on product
attributes, which are in multiple numbers. Depending on the attribute specifications by the
customer this procedure searches the Internet for the desired product. If a product is available
with the prescribed attribute specifications the customer will be satisfied. In case of non-
availability the procedure chooses the next available product which is closest to the target
product. Closeness of the product is measured and compared through product attribute values.The author has introduced attribute flexibility in order to classify different products with different
attribute values into the same preference class. The main drawback in the methodology in Ryu
(1999) is that flexibility values are chosen subjectively. This idea was extended in Mohanty and
Bhasker (2005) by deriving the customer’s flexibility values objectively. However, this work also
concentrates on the hierarchical satisfaction of the customer’s requirements and lacks
simultaneous satisfaction of multiple attributes of the product. Although the procedure in
Mohanty and Bhasker (2005) hierarchically classifies the products according to the customer’s
preferences, no similarity between the customer’s choice and the products’ availability is measured
while making the classifications. By representing the product attributes in fuzzy numbers, our
paper also classifies the products in a preference hierarchy based on multiple product attributes.
Some other papers are available in the literature. In Cao and Li (2007), the consumer specifiesthe product preferences in each attribute and the system provides the products according to his/
her personal choices. However, the procedure of recommendation does not give the similarities
between the buyers’ desired attribute levels and the products’ attainability in the market forum. In
Miao et al. (2007), personalised recommendations are made in an interactive way. Although the
process of interaction somewhat represents an assessment of similarity between the customer’s
choices and the products’ availability, it does not indicate a concrete procedure for obtaining
similarity measures. In Cheng et al. (2006), automated intelligent agents of the trading partners
negotiate on several issues with the aim to arrive at a consensus business pact. Although the
work (Cheng et al., 2006) maintains flexibility in the agents’ strategies during the process of
negotiation, it does not account for similarities among the agents’ views before arriving at a
business deal.This paper is organised as follows: Section 2 is a summary of the procedure introduced in the
paper. In Section 3, we give the fuzzy representation of product attributes as perceived by
customers. Also in this section, we give the company’s fuzzy viewpoint about product attributes.
Section 4 gives the methodology of obtaining similarities between the attributes that customers
require and those of the available products. Section 5 is devoted to problem formulation and the
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solution procedure. In Section 6, we illustrate a numerical example to highlight the procedure.
Finally, in Section 7 we present our conclusions.
2. Brief procedure
Our paper has examined the above shortcomings and presents the following to overcome them.
Product attribute values are incorporated in the model by taking the fuzzy requirements of the
customer as inputs.
(a) Consideration and implementation of multiple product attributes is addressed in the paper.
(b) Each attribute of the available products is compared with the customer’s choices to obtain the
degree of similarity between the available products and customer’s requirements.
Let a customer require k fuzzily defined attributes of a product. Denote the customer’s desired
preference levels of these attributes E j ( j 51,2, . . ., k) as L–R fuzzy numbers
E ¼ ðE 1; E 2; . . . ; E kÞ:
Assume that there are m products available on the Internet. Let their attribute levels be
specified as
Bi ¼ ðBi 1; Bi 2 . . . BikÞ ðfor i ¼ 1; 2; . . . ; mÞ:
Bij represents the j th attribute level of the i th product. Bij s (8 i and j ) are assumed here to be
fuzzy numbers. Details about fuzzy numbers can be found in Dubois and Prade (1978).
Initially take an attribute level E j (say the j th attribute) as prescribed by the customer and match
this attribute to the corresponding j th attribute Bij of all the products. The degree of similarity
T Ej |Bij between E j and Bij (8i ) can be obtained following the procedure given in Yager (1984). As
per this procedure T Ej |Bij is also a fuzzy number. The information on these degrees of similarities,
along with the fuzzy number operations given in Yager (1984), helps the customer predict the
probability and hence the expectation of availing a particular attribute level of E j in the given
market profile. Let the fuzzy number representation of the above expected values be
ðExpðE j ÞÞ ¼ ðExpðE j Þl ; ExpðE j Þn; ExpðE j ÞrÞ ðfor j ¼ 1; 2; . . . ; kÞ: ð1Þ
If I i ( j ) represents the matching or similarity degree between Bij and (Exp (E j )), we can have the
satisfaction level for the i th product as
Pði Þ ¼X
w j I i ð j Þ; ðfor i ¼ 1; 2; . . . ; mÞ; ð2Þ
where w j are the weights attached to the attributes E j . The weights used here can be estimated by
using the procedure given in Mohanty (1998). After calculating all P(i ) values, we have the best
product corresponding to the maximum P(i ) value.
3. Fuzzy representation of the product attributes
Most of the decision making in the real world takes place in an environment in which goals,
constraints, and possible consequent actions are not known precisely. Fuzzy logic deals
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quantitatively with the qualitatively defined items in the decision-making process. In order to
make business decisions more customer focussed, a company needs to analyse a customer’s fuzzily
defined requirements to find matching products for the customer. To do this, the company needs
to maintain flexibility in its products to satisfy diverse groups of customers.
For example, in a car purchasing problem customers’ requirements for the attribute cost will
differ from one customer to another, and might be in fuzzy terms such as ‘‘around US$20,000’’,‘‘close to US$25,000’’, ‘‘approximately US$22,000’’, or ‘‘more or less US$23,000’’, etc. In the
scenario of diverse customer price requirements, a salesperson may like to promote the company’s
products in various ways to different groups of customers. That is, if the price of a car as per the
company’s judgement is US$21,000 (say), the company may claim that it is the most suitable car
for a particular group of customers whose desired price level is US$25,000, of course with a
satisfaction level or a matching degree. Similarly the company can derive different matching
degrees for different groups of customers with respect to the cost attribute. That is, as per the
company’s appraisal, each product can be claimed as a suitable product for each customer group
with a matching degree or a satisfaction level. In other words, the company, on its own, may look
into the desires of the various customer groups in terms of the product attributes and then define
the ranges for attribute deviations. These deviations from the actual attribute values helpcompanies make their products more flexible and customer focussed in the Internet market. Note
that the customer’s desires and the company’s assessments in terms of defining attribute
flexibilities can be represented through fuzzy numbers.
Example 3.1. In any product purchase, a customer normally assesses a product through multiple
numbers of attributes and specifies their requirements in fuzzy terms. In the car purchasing
problem, a customer may opt for attributes such as cost, re-sale value, mileage, comfort,
maintenance cost, etc. By choosing the attribute cost, the customer’s fuzzily defined views might
be defined in the following ways.
(A, mA): Cost should be around US$20,000, (B, mB): Cost should be approximately US$22,000,
(C, mC): Cost should be closer to US$23,000,
(D, mD): Cost should be more or less US$25,000.
In the above the italic words are fuzzy terms. They are denoted by the fuzzy sets A, B, C, and D.
Without loss of generality these fuzzy terms can be represented as fuzzy numbers as shown below.
ðA; mAÞ ¼ ð15; 20; 25Þ; ðB; mBÞ ¼ ð18; 22; 24Þ; ðC; mCÞ ¼ ð17; 23; 25Þ and ðD; mDÞ
¼ ð22; 25; 27Þ;
where mA, mB, mC and mD represent the membership functions of the fuzzy numbers A, B, C and D,respectively. These are graphically shown in Fig. 1.
In Fig. 1, the horizontal axis represents car prices and the vertical axis represents the customer’s
satisfaction level (membership function of the fuzzy numbers) corresponding to a price. Taking
the fuzzy number ‘‘around US$20,000’’, we can say that a buyer is fully satisfied (with membership
value one) if the car price is US$20,000 and his/her satisfaction level gradually decreases when the
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car price deviates from US$20,000 and becomes zero when the price is either below US$15,000 or
above US$25,000.
If a company has a car with price US$23,000, the company’s analyst may present the cost of the
car for all the customer groups in a single fuzzy term (fuzzy number) as ( Price, mPrice
)5 (15, 23, 25)
with membership function mPRICE. Graphically this is shown in Fig. 2.
As before the horizontal axis represents the company’s own assessed flexible price for different
customer groups and the vertical axis represents the expected satisfaction level of the company
(membership values).
Note that in Fig. 2 the company has fuzzi-fied the attribute cost after scanning the market
demand and the customer’s requirements on the cost attribute. The company’s fuzzily defined
attribute is meant for all the customer groups with variant price aspirations. In fact this is assessed
subjectively by the company’s experts without following an analytical or logical procedure.
4. Similarity measure
In this section, we measure the degrees of similarity between the attribute levels of the available
products on the Internet and the desired attribute levels of the customer. As the customer’s
interest is based on multiple product attributes, which are generally conflicting, non-
commensurable and fuzzy in nature, it is very difficult to find similarities amongst the attributes
0
0.2
0.4
0.6
0.8
1
1.2
10 15 20 25 30
Price
S
a t i s f a c t i o n
L e v e l
A
B
C
D
Fig. 1. Buyers requirements of prices in fuzzy terms.
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20 25 30
Price
S a
t i s f a c t i o n
L e v e l
Fig. 2. Company’s price assessment in fuzzy terms.
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when choosing a product. However, in our paper we address this problem and obtain similarities
by representing product attributes in fuzzy numbers and following the technique given in Yager
(1984). The procedure is given below.
Let B1, B2, . . ., Bm be m products available on the Internet. Let these products be assessed in
terms of the satisfaction levels of k attributes. We denote the j th attribute of the i th product as Bij
so the i th product in terms of its attributes is
Bi ¼ ðBi 1; Bi 2 . . . BikÞ:
Similarly we can have the customer’s desired levels ( preference levels) of the product attributes
as
E ¼ ðE 1; E 2; . . . ; E kÞ:
E j ( j 5 1,2, . . ., k) represents the customer’s desired level in the j th attribute.
We have assumed here that the product attributes are in the form of L–R fuzzy numbers
(Dubois and Prade, 1978; Mohanty and Bhasker, 2005). Thus, we have
E j ¼ ðE jl ; E jn; E jrÞ ðfor j ¼ 1; 2; . . . ; kÞ and Bij ¼ ðBijl ; Bijn; BijrÞ
ðfor i ¼ 1; 2; . . . ; m and j ¼ 1; 2; . . . ; kÞ:
In order to determine the satisfaction levels of the customers regarding the available products,
we need to calculate the similarities between the attribute levels E j and the product attributes Bij
(for i 5 1, 2, . . ., m and j 51, 2, . . ., k). Mathematically the attribute-wise comparisons for the i th
product to that of the customer’s requirements are given as:
Bi 1 $ E 1;
Bi 2 $ E 2;
Bik $ E k:
ð3Þ
The symbol $ is used here to represent the similarity. By generalising equation (3) we have
Bij $ E j ðfor i ¼ 1; 2; . . . ; m and j ¼ 1; 2; . . . ; kÞ: ð4Þ
Following the method given in Yager (1984), we derive the degree of similarity between E j and
Bij :
T Ej jBij ði Þ ¼ Max: ½Bij ðxÞ;
E j ðxÞ ¼ i :ð5Þ
Here, T Ej |Bij represents the extent to which the customer’s satisfaction level in the j th attribute
matches Bij . Note that T Ej |Bij is also a fuzzy number (Yager, 1984).
By varying over all products, for the attribute j and the customer’s preference on the j thattribute, we can have the following set of similarity measures, which are in the form of fuzzy
numbers.
ðT Ej jB1 j ; T Ej jB2 j ; . . . ; T Ej jBmj Þ: ð6Þ
Equation (6) represents the similarity degrees between the j th attributes of Bij and E j .
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Given this scenario, the customer may like to know the likelihood that he/she will get a product
with the desired attribute levels. This can be predicted by calculating the probability of availing
the attribute level E j , P f (E j ). Here, the probability needs to be calculated under fuzzy observations
Bij (8 i , j ) and with the fuzzily defined customer’s requirements E j . Following Yager (1984), we
have calculated the probabilities as given below
P f ðE j Þ ¼ 1=mX
T Ej jBij : ð7Þ
The term P f (E j ) reprents the probability of obtaining the customer’s preference level in the j th
attribute E j .
Since each T Ej |Bij are L–R fuzzy numbers their addition and hence the derived probabilities are
again L–R fuzzy numbers. Thus we have
P f ðE j Þ ¼ ðP f ðE j Þl ; P f ðE j Þn; P f ðE j ÞrÞ: ð8Þ
The expected value of the j th attribute under the given market profile can be calculated as
Exp f
ðE j Þ ¼ ½P f ðE j Þl
 E jl ; P f ðE j Þn
 E jn; P f ðE j Þr
 E jr; ð8 j Þ: ð9Þ
From equation (9) it is clear that the expected value is a fuzzy set of fuzzy numbers. The
membership function of the fuzzy set represents the flexibility behaviour or compromising attitude
of the customer while purchasing a product.
5. Solution procedure
This section is devoted to the procedure of articulating the customer’s preference level of the
product attributes and selecting the best available product in the Internet market. For each
product attribute the customer expects at least the level prescribed by the expected value given inequation (9). This needs a comparison of the available products’ attribute levels to the desired
expectation. For a minimisation (maximisation) type attribute (of say Bij ), the comparison can be
expressed as: ‘‘To what extent is the attribute Bij ( jth attribute of the ith product) less (greater)
than the expected value given in (9)?’’
Mathematically this can be written as
Max: Min: ½Bij ðxÞ; ExpðE j Þf yg ¼ I i ð j Þ;x) y: ð10Þ
Similarly for an attribute which is of maximisation type we can have the mathematical equation
as:
Max: Min: ½Bij ðxÞ; ExpðE j Þf yg ¼ I i ð j Þ;x* y: ð11Þ
In equations (10) and (11) we have made fuzzy number comparisons (Dubois and Prade, 1978).
The quantity I i ( j ) represents the degree of satisfaction of the j th attribute of the i th product.
Continuing this procedure over all the attributes and over all the products we have m sets of
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satisfaction levels. They are as given below
½ðI i ð1Þ; I i ð2Þ; . . . ; I i ðkÞ ðfor i ¼ 1; 2; . . . ; mÞ: ð12Þ
If all the I i ( j )s (for every j ) are unity, all the attributes of the i th product are very much up to
expectation and the i th product can be selected as the best product. Unfortunately, in normal
cases this does not occur as the attributes conflict with each other. Generally, for a particularproduct one attribute may achieve to the fullest extent, another to some extent, and another may
be to a negligible extent. In this situation, a customer needs to compromise, and the degree of
compromise depends on his/her attribute preferences.
If W j represents the weights attached to the attribute j , following the weighted average method
given in Mohanty, (1998), we can have the satisfaction level for the i th product as
Pði Þ ¼X
W j I i ð j Þ ði ¼ 1; 2; . . . ; mÞ: ð13Þ
Similarly we can obtain the satisfaction level for other products. The product corresponding to
the highest satisfaction level is selected as the best product. This can be obtained through the
following equation:
P ¼ Max: ½Pð1Þ; Pð2Þ;. . .
; PðmÞ: ð14ÞHere, P represents the best product as per the customer’s preference. The next best product and so
on can be obtained by maximising again over P(i ) values for the remaining products.
6. Numerical example
In this section, we have illustrated a numerical example to highlight the procedure. Here, we have
taken three products B1, B2, and B3, which are available on the Internet. A customer assesses these
products based on three attributes. We can represent these product attributes as
Bi ¼ ðBi 1; Bi 2; Bi 3Þ ðfor i ¼ 1; 2; 3Þ:
Bij represents the j th attribute of the i th product. Let Bi 1, Bi 3 be of minimisation type and Bi 2 of
maximisation type attributes. The attribute values are given as fuzzy numbers
For the product B1:
B115 (1, 2.5, 4),
B125 (0.6, 0.7, 0.8),
B135 (4, 5, 6).
For the product B2:
B215 (2, 4, 6),
B225 (0.6, 0.7, 0.9),
B235 (3, 5, 6).
For the product B3:
B315 (2, 3, 6),
B325 (0.5, 0.6, 0.9),
B335 (2, 3, 5).
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Let the customer’s preferences on the product attributes 1, 2 and 3 be as follows:
E 15 (1, 3, 5),
E 25 (0.2, 0.8, 0.9),
E 35 (2, 4, 6).
From equation (5), we have derived the following degrees of similarity of the product attribute 1,with the customer’s desire level E 1:
T E 1jB11 ¼ f0=0; 0:53=0:4; 1=0:75; 0:94=0:8; 0:65=1g ¼ ð0; 0:75; 1Þ;
T E 1jB21 ¼ f0:5=0; 0:84=0:4; 1=0:5; 0:7=0:8; 0:5=1g ¼ ðÀ0:12; 0:5; 1:5Þ;
T E 1jB31 ¼ f0:3=0; 0:6=4; 0:9=0:8; 1:0=1:0g ¼ ðÀ0:5; 1; 0Þ:
ð15Þ
Degrees of similarity of the product attribute 2, with the customer’s desire level E 2:
T E 2jB12 ¼ f0=0; 0=0:4; 1=0:75; 0:8=0:8; 1=0:85; 0:7=0:9; 0=1g ¼ ð0:6; 0:85; 1Þ;
T E 2jB22 ¼ f0=0; 0:2=0:4; 1=0:75; 0:3=0:6; 0:8=0:8; 0:5=1g ¼ ð0:37; 0:83; 1:8Þ;
T E 2jB32 ¼ f0=0; 0=0:4; 0:6=0:6; 1=0:67; 0:75=0:8; 0:35=1g ¼ ð0:5; 0:67; 1:16Þ:
ð16Þ
Degrees of similarity of the product attribute 3, with the customer’s desire level E 3:
T E 3jB13 ¼ f0=0; 0:72=0:4; 1=0:5; 0:50:8; 0=1g ¼ ð0:13; 0:5; 1Þ;
T E 3jB23 ¼ f0:2=0:1; 0:6=0:3; 1=0:5; 0:68=0:8; 0:5=1g ¼ 0; 0:5; 1:5Þ;
T E 3jB33 ¼ f0=0; 0:1=0:1; 1=0:5; 0:78=0:7; 0:5=1g ¼ ð0:05; 0:5; 1:4Þ:
ð17Þ
Using equation (7), we can have the probabilities of attaining the attribute levels E 1, E 2 and E 3as follows (after limiting the values between 0 and 1):
P f ðE 1Þ ¼ ð0; 0:75; 1:0Þ;
P f ðE 2Þ ¼ ð0:49; 0:78; 1:0Þ;
P f ðE 3Þ ¼ ð0:08; 0:5; 1:0Þ:
ð18Þ
Note that the above probabilities are in the form of fuzzy numbers.
Now using equation (9) we can find the customer’s expected values in different attributes under
the given market profile as:
Exp ðE 1Þ ¼ ð0; 2:25; 5Þ;
Exp ðE 2Þ ¼ ð0:1; 0:62; 0:9Þ;
Exp ðE 3Þ ¼ ð0:16; 2; 6Þ:
ð19Þ
In order to see whether the available products live up to the expectation, the customer needs to
compare each product attribute to the fuzzily defined expected values given in equation (19). This
comparison is done by considering that Bi 1, Bi 3 are of minimisation type and Bi 2 is of
maximisation type. Following equations (3)–(5) we have:
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For the product B1:
Deg [B114Exp (E 1)]5 I 1(1)51.0,
Deg [B12XExp (E 2)]5 I 1(2)5 1.0,
Deg [B134Exp (E 3)]5 I 1(3)50.4.
For the product B2:
Deg [B214Exp (E 1)]5 I 2(1)50.66,
Deg [B22XExp (E 2)]5 I 2(2)5 1.0,
Deg [B234Exp (E 3)]5 I 2(3)50.6.
For the product B3:
Deg [B314Exp (E 1)]5 I 3(1)50.85,
Deg [B32XExp (E 2)]5 I 3(2)5 0.97,
Deg [B334Exp (E 3)]5 I 3(3)50.81.
The above expressions are shown in Figs 3, 4 and 5.
In general a customer’s preferences are better reflected by attaching weights to the product
attributes. By using equation (13) and taking the weights as W 15 0.33, W 250.34 and W 35 0.33
we have the product satisfaction levels according to equation (6) as follows:
For the product B1:
1(0.33)11(0.34)10.6 (0.33)50.76.
For the product B2:
0.66(0.33)11(0.34)10.81(0.33)50.82.
For the product B3:
0.85(0.33)10.97(0.34)10.81(0.33)5 0.88.
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6 7
Exp(E1)
B11
B21
B31
Fig. 3. Comparison of product attribute 1 with the expectation Exp (E 1).
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Using equation (14) we will have the best product corresponding to the maximum product
satisfaction level. Thus we have
Max: ½0:76; 0:82; 0:88 ¼ 0:88:
The value 0.88 corresponds to the product B3. Hence as per the customer’s preference the best
product in the Internet market is B3. The second best is B2 corresponding to the level 0.82 and
third is B1 with satisfaction level 0.76.
7. Conclusion
This paper has introduced a model which takes into account the customer’s opinion in linguistic
terms while purchasing a product on the Internet. Very often customers express their assessmentsor opinions in fuzzy terms. Customers’ fuzzy behaviours and their expectations are modelled here
by using the concepts of fuzzy logic and fuzzy probability respectively. Additionally, products’
attributes as perceived by customers are also incorporated into the model. The appropriation of
these additional concepts makes the online business more practical and corresponds to real world
business decision-making situations.
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1
Exp(E2)
B12
B22
B32
Fig. 4. Comparison of product attribute 2 with the expectation Exp (E 2).
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6 7
Exp(E3)
B13B23
B33
Fig. 5. Comparison of product attribute 3 with the expectation Exp (E 3).
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The methodology in our paper helps customers in the following ways.
(1) In general customers assess products by their attributes, which are usually conflicting, non-
commensurable and fuzzy in nature. Our model considers all these aspects and suggests a
viable product(s) to the customer.
(2) Based on customers’ fuzzy preference of product attributes, a company’s available products
are also defined fuzzily. Using this method the company’s products are compared with the
customer’s requirements and products with high matching degrees are presented to the
customer. Thus, the customer can select a product of his/her own choice with a maximum
possible satisfaction level.
(3) The earlier electronic shopping agents like Jango, DealTime (http://www.dealtime.com) and
decision guide (http://www.activityBuyersGuide.com) assist customers in finding the best
available product based on the product attributes required by the customers. However, these
shopping agents do not have any feature to entertain the fuzziness of the customer’s views
about the product attributes. Simultaneous consideration of multiple numbers of product
attributes is an added difficulty. Our proposed procedure handles these problems by using the
concepts of fuzzy sets, fuzzy probability and multi-attribute decision analysis.
This paper has the following limitations and these topics can be taken as a scope for future
research:
(1) Our methodology does not consider the non-digital attributes inherent in the products. The
explanation of non-digital attributes on the Internet is a challenging problem in Internet
business.
(2) It is explained in Section 2 that in order to make business decisions more customer focussed, a
company needs to analyse fuzzily defined customer requirements about product features and
also assess its own available products in a flexible way. In the company’s point of view we
have not used any methodology to derive product attribute flexibility values. This is merely
done by looking into the market scenario, customers’ attribute-wise requirements and the
company’s own judgement.
Acknowledgement
The authors would like to thank the anonymous referees for their valuable comments which have
improved the paper significantly.
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