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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

r 2009 The Authors.

Journal compilation r 2009 International Federation of Operational Research Societies

Published by Blackwell Publishing, 9600 Garsington Road, Oxford, OX4 2DQ, UK and 350 Main St, Malden, MA 02148, USA.

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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

Â E jl ; P f ðE j Þn

Â E jn; P f ðE j Þr

Â 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


0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7

Exp(E3)

B13B23

B33



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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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