Information Sciences 170 (2005) 191–210

www.elsevier.com/locate/ins

Fuzzy multi-attribute selectionamong transportation companies

using axiomatic design and analytichierarchy process

Osman Kulak, Cengiz Kahraman *

Department of Industrial Engineering, Istanbul Technical University, 34367 Macka, Istanbul, Turkey

Received 14 July 2003; received in revised form 18 February 2004; accepted 19 February 2004

Abstract

The basic concept in axiomatic design (AD) is the existence of design axioms. First of

these axioms is the independence axiom and the second one is the information axiom.

Information axiom proposes the selection of the best alternative that has minimum

information. Analytic hierarchy process (AHP) is another multi-attribute method which

is a decision-making method for selecting the best among a set of alternatives, given

some criteria. The method has been extensively applied, especially in large-scale prob-

lems where many criteria must be considered and where the evaluation of alternatives is

mostly subjective. Multi-attribute transportation company selection is a very important

activity for effective supply chain. Selection of the best company under determined

criteria (such as cost, time, damage/loss, flexibility and documentation ability) using

both multi-attribute AD and AHP will be realized in this study. The fuzzy multi-

attribute AD approach is also developed and it is compared by one of fuzzy AHP

methods in the literature. The selection process has been accomplished by aiding a

software that includes crisp AD and fuzzy AD.

� 2004 Elsevier Inc. All rights reserved.

Keywords: Transportation company selection; Axiomatic design; Fuzzy sets; Informa-

tion; AHP

* Corresponding author. Tel.: +90-212-293-13-00x2035; fax: +90-212-240-72-60.

E-mail address: kahramanc@itu.edu.tr (C. Kahraman).

0020-0255/$ - see front matter � 2004 Elsevier Inc. All rights reserved.

doi:10.1016/j.ins.2004.02.021

192 O. Kulak, C. Kahraman / Information Sciences 170 (2005) 191–210

1. Introduction

Today’s high competitive environment forces organizations to find better

ways to create and deliver value to customers. It is a challenging process tosend products to customers in a cost-effective way. While building a trans-

portation budget, decision-makers are faced with the challenge of selecting the

most efficient transportation company. The selection of a few transportation

companies at the same time with limited resources is impossible. Thus, trans-

portation company selection becomes an efficient resources allocation proce-

dure. In this resource allocation problem, the evaluation process of alternatives

brings some difficulties. The major problem is the consideration of multiple

objectives which are generally conflicting with each other and measured indifferent scales. Another problem is about the values of benefit and cost

measures [1]. There can be fewer consensuses about the intangible measures

such as one hour of a passenger or the value of a later dispatch. Besides; it is

difficult to measure the values of qualitative aspects quantitatively. Thus the

evaluation process and methodology must handle, or support users on these

difficult steps.

The approaches to the solution of transportation company selection prob-

lems can be classified in five groups: (1) profile and checklist methods, (2)scoring methods, (3) cost-benefit analysis, (4) mathematical programming

models, (5) fuzzy approaches. Most of these methods are based on compen-

satory models that all the values of criteria are consolidated and then evalu-

ated. These methods enable tradeoffs between the criteria. Such as, a poor

value in a criterion can be offset by another criterion with a high value. On the

other hand, in non-compensatory models no tradeoffs are allowed. The pro-

jects are grouped as pass–fail categories, according to their values compared

with the threshold value of the decision-maker. In consequence, a transpor-tation company with a high overall value can be failed since one of it’s criteria

is below the threshold value.

Having to use crisp values is one of the problematic points in the crisp

evaluation process. As some criteria are difficult to measure by crisp values,

they are usually neglected during the evaluation. Another reason is about

mathematical models that are based on crisp values. These methods cannot

deal with decision makers’ ambiguities, uncertainties and vagueness which

cannot be handled by crisp values.In most of the real-world problems, some of the decision data can be pre-

cisely assessed while others cannot. Humans are unsuccessful in making

quantitative predictions, whereas they are comparatively efficient in qualitative

forecasting. Further, people are more prone to interference from biasing ten-

dencies if they are forced to provide numerical estimates since the elicitation of

numerical estimates forces an individual to operate in a mode which requires

more mental effort than that required for less precise verbal statements. Real

O. Kulak, C. Kahraman / Information Sciences 170 (2005) 191–210 193

numbers are used to represent data which can be precisely measured. For those

data which cannot be precisely assessed, Zadeh’s [29] fuzzy sets can be used to

denote them. The use of fuzzy set theory allows us to incorporate unquanti-

fiable information, incomplete information, non-obtainable information, andpartially ignorant facts into the decision model. When decision data are pre-

cisely known, they should not be faced into a fuzzy format in the decision

analysis. Applications of fuzzy sets within the field of decision making have, for

the most part, consisted of extensions or ‘‘fuzzifications’’ of the classical the-

ories of decision making. While decision-making under conditions of risk and

uncertainty have been modeled by probabilistic decision theories and by game

theories, fuzzy decision theories attempt to deal with the vagueness or fuzziness

inherent in subjective or imprecise determinations of preferences, constraints,and goals.

Verma et al. [28] proposed a method using a special type of non-linear

(hyperbolic and exponential) membership functions to solve the multi-objective

transportation problem. Teng and Tzeng [27] presented the fuzzy multi-

objective programming using the fuzzy spatial algorithm for the problem of

transportation investment project selection. Chanas and Kuchta [5] proposed

an algorithm as to solve the integer transportation problem with fuzzy supply

and demand values and integrality condition imposed on the solution. Shih [20]applied three fuzzy linear programming models as well as crisp linear pro-

gramming models to solve the cement transportation planning problem in

Taiwan. Li and Lia [15] proposed a new fuzzy compromise programming

approach to multi-objective transportation problems. Avineri et al. [1] pre-

sented a methodology for the selection and ranking of transportation projects

using fuzzy sets theory. Kikuchi [12] proposed a simple adjustment method

finding the most appropriate set of crisp numbers. The fuzzy linear program-

ming was used in this paper. Sakawa et al. [18] presented a fuzzy programmingfor the production and the transportation planning under the assumption

considering multiple production and sales bases. Sakawa et al. [19] dealt with a

transportation problem in a housing material manufacturer and derived a

solution to the problem with respect to the objectives of the housing material

manufacturer. Liu and Kao [16] developed a method to find the membership

function of the fuzzy total transportation cost. The idea was Zadeh’s extension

principle to transform the fuzzy transportation problem to a pair of mathe-

matical programs.One of the multi-attribute mathematical programming models is the analytic

hierarchy process. The purpose of the AHP is to provide vector of weights

expressing the relative importance of the transportation alternatives for each

criterion. AHP requires four steps: (1) structuring the hierarchy of criteria and

alternatives for evaluation; (2) assessing the decision-makers’ evaluations by

pairwise comparisons; (3) using the eigenvector method to yield priorities for

criteria and for alternatives by criteria; and (4) synthesizing the priorities of the

194 O. Kulak, C. Kahraman / Information Sciences 170 (2005) 191–210

alternatives by criteria into composite measures to arrive at set of ratings for

the alternatives. The scale of importance is defined according to Saaty 1–9 scale

for pairwise comparison. Readers are referred to Saaty [17] for detailed dis-

cussion of AHP modeling and solution methodology. 1, 3, 5, 7, 9 are definedrespectively as equal, moderate, strong, very strong and extreme importance.

There are many fuzzy AHP methods proposed by various authors. These

methods are systematic approaches to the alternative selection and justification

problem by using the concepts of fuzzy set theory and hierarchical structure

analysis. The earliest work in fuzzy AHP appeared in van Laarhoven and

Pedrycz [14], which compared fuzzy ratios described by triangular membership

functions. Buckley [4] determined fuzzy priorities of comparison ratios whose

membership functions trapezoidal. Stam et al. [21] explored how recentlydeveloped artificial intelligence techniques can be used to determine or

approximate the preference ratings in AHP. They conclude that the feed-for-

ward neural network formulation appears to be a powerful tool for analyzing

discrete alternative multi-criteria decision problems with imprecise or fuzzy

ratio-scale preference judgments. Chang [6] introduced a new approach for

handling fuzzy AHP, with the use of triangular fuzzy numbers for pairwise

comparison scale of fuzzy AHP, and the use of the extent analysis method for

the synthetic extent values of the pairwise comparisons. Ching-Hsue [9] pro-posed a new algorithm for evaluating naval tactical missile systems by the fuzzy

analytical hierarchy process based on grade value of membership function.

Cheng et al. [8] proposed a new method for evaluating weapon systems by

analytical hierarchy process based on linguistic variable weight. Zhu et al. [30]

make a discussion on extent analysis method and applications of fuzzy AHP.

The information axiom which is the second axiom of axiomatic design (AD)

provides the basis for decision-making when there are many choices [26]. A

new model based on this axiom is generated to support decision-makers intransportation company selection process. In order to avoid the pitfalls of

preceding methods, AD based method enables decision-makers to evaluate

both qualitative and quantitative criteria together.

Many AD applications in designing products, systems, organizations and

software have appeared in the literature in the last 10 years. Suh [22] has

introduced AD theory and principles first time. Kim et al. [13] applied AD

principles on software design. AD principles have also been used in design of

quality systems [23] and general system design [24]. Suh and Cochran [25]provided a manufacturing system design using AD principles. AD principles

have also been applied in designing flexible manufacturing systems [2]. Other

applications of Axiomatic Design include process and product development

[26]. These studies have convincingly shown the applicability and benefits of

AD in solving industrial problems.

In this paper as the first time, a fuzzy multi-attribute axiomatic design ap-

proach for selection of the best transportation company will be introduced and

O. Kulak, C. Kahraman / Information Sciences 170 (2005) 191–210 195

the implementation process will be shown by a real world example. The same

problem will be solved by crisp and fuzzy AHP approaches and the results

obtained by these three approaches will be compared.

2. Principles of axiomatic design

The most important concept in axiomatic design is the existence of the de-

sign axioms. The first design axiom is known as the Independence Axiom and

the second axiom is known as the Information axiom. They are stated as

follows [22].

Axiom 1. The Independence Axiom:Maintain the independence of functionalrequirements.

Axiom 2. The Information Axiom: Minimize the information content.

The Independence Axiom states that the independence of functional

requirements (FRs) must always be maintained, where FRs are defined as the

minimum set of independent requirements that characterizes the design goals

[22]. In the real world, engineers tend to tackle a complex problem by

decomposing it into sub-problems and attempting to maintain independentsolutions for these smaller problems. This calls for an effective method that

provides guidelines for the decomposition of complex problems and indepen-

dent mappings between problems and solutions.

The Information Axiom states that among those designs that satisfy the

Independence Axiom, the design that has the smallest information content is

the best design [26]. Information is defined in terms of the information content,

Ii, that is related in its simplest form to the probability of satisfying the given

FRs. Ii determines that the design with the highest probability of success is thebest design. Information content Ii for a given FRi is defined as follows:

Ii ¼ log21

pi

� �ð1Þ

where pi is the probability of achieving the functional requirement FRi and logis either the logarithm in base 2 (with the unit of bits). This definition of

information follows the definition of Shannon, although there are operational

differences. Because there are n FRs, the total information content is the sum of

all these probabilities. If I approaches infinity, the system will never work.

When all probabilities are one, the information content is zero, and conversely,

the information required is infinite when one or more probabilities are equal to

zero [23].

In any design situation, the probability of success is given by what designerwishes to achieve in terms of tolerance (i.e. design range) and what the system

0 5 10 15 20 25 30 35 40 450

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

FR

Pro

babi

lity

dens

ity Design Range

Common Area

System pdf

Common Range

Fig. 1. Design range, system range, common range and probability density function of a FR.

196 O. Kulak, C. Kahraman / Information Sciences 170 (2005) 191–210

is capable of delivering (i.e. system range). As shown in Fig. 1, the overlap

between the designer-specified ‘‘design range’’ and the system capability range‘‘system range’’ is the region where the acceptable solution exists. Therefore, in

the case of uniform probability distribution function pi may be written as

pi ¼Common range

System range

� �ð2Þ

Therefore, the information content is equal to

Ii ¼ log2System range

Common range

� �ð3Þ

As illustrated in Fig. 1, the system range and the common range that is the

overlap between the design range and the system range are calculated as fol-

lows: Common Range¼ 30) 20¼ 10 and System Range¼ 40) 20¼ 20. Infor-

mation content, I , is calculated as I ¼ log2ð2010Þ ¼ 1.

The probability of achieving FRi in the design range may be expressed, if

FRi is a continuous random variable, as

pi ¼Z dru

dr1psðFRiÞdFRi ð4Þ

where ps (FRi) is the system pdf (probability density function) for FRi. Eq. (4)

gives the probability of success by integrating the system pdf over the entire

design range. (i.e. the lower bound of design range, dr1, to the upper bound ofthe design range, dru). In Fig. 2, the area of the common range (Acr) is equal to

the probability of success P [22].

-3 -2 -1 0 1 2 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Design Range

Area within common range (Acr)

System pdf

Common Range System Range FR

Pro

babi

lity

Den

sity

Fig. 2. Design range, system range, common range and probability density function of a FR.

O. Kulak, C. Kahraman / Information Sciences 170 (2005) 191–210 197

Therefore, the information content is equal to

I ¼ log21

Acr

� �ð5Þ

3. Fuzzy axiomatic design approach

The fuzzy data can be linguistic terms, fuzzy sets, or fuzzy numbers. If the

fuzzy data are linguistic terms, they are transformed into fuzzy numbers first.

Then all the fuzzy numbers (or fuzzy sets) are assigned crisp scores. The fol-

lowing numerical approximation systems are proposed to systematically con-vert linguistic terms to their corresponding fuzzy numbers. The system contains

five conversion scales (Figs. 3 and 4).

(x)µ

x

1

Poor Fair Good Very Good

Excellent

α β δ κ γ λ φ ρ ν

Fig. 3. The numerical approximation system for intangible factors.

µ (x)

x

1

Very Low

Low Medium

High Very High

1 1 1 1 1 1 1 1 1α β δ κ γ λ φ ρ ν

Fig. 4. The numerical approximation system for tangible factors.

198 O. Kulak, C. Kahraman / Information Sciences 170 (2005) 191–210

In the fuzzy case, we have incomplete information about the system and

design ranges. The system and design range for a certain criterion will be ex-

pressed by using ‘over a number’, ‘around a number’ or ‘between two num-bers’. Triangular or trapezoidal fuzzy numbers can represent these kinds of

expressions. We now have a membership function of triangular or trapezoidal

fuzzy number whereas we have a probability density function in the crisp case.

So, the common area is the intersection area of triangular or trapezoidal fuzzy

numbers. The common area between design range and system range is shown

in Fig. 5.

Therefore, information content is equal to

I ¼ log2TFN of System Design

Common Area

� �ð6Þ

µ (x)

x

1

α δ κ λ

Common Area

TFN of Design Range

TFN of System Design

Fig. 5. The common area of system and design ranges.

O. Kulak, C. Kahraman / Information Sciences 170 (2005) 191–210 199

4. Fuzzy AHP

In the following, first the outlines of Chang’s [7] extent analysis method on

fuzzy AHP are given and then the method is applied to a transportationcompany selection problem. Some papers recently published which used the

extent analysis method are as follows: Kahraman et al. [10] provided an ana-

lytical tool to select the best catering firm providing the most customer satis-

faction. Fuzzy AHP is used to compare three catering firms. Kahraman et al.

[11] proposed a tool based on fuzzy AHP for facility location selection. Bozda�get al. [3] used the extent analysis method for selection among computer inte-

grated manufacturing systems.

Let X ¼ fx1; x2; . . . ; xng be an object set, and U ¼ fu1; u2; . . . ; umg be a goalset. According to the method of Chang’s [7] extent analysis, each object is

taken and extent analysis for each goal, gi, is performed respectively. There-

fore, m extent analysis values for each object can be obtained, with the fol-

lowing signs:

M1gi;M2

gi; . . . ;Mm

gi; i ¼ 1; 2; . . . ; n ð7Þ

where all the Mjgi(j ¼ 1; 2; . . . ;m) are triangular fuzzy numbers (TFNs) whose

parameters are l, m, and u. They are the least possible value, the most possible

value, and the largest possible value respectively. A TFN is represented as

ðl;m; uÞ.The steps of Chang’s extent analysis can be given as in the following:

Step 1. The value of fuzzy synthetic extent with respect to the ith object is

defined as

Si ¼Xmj¼1

Mjgi�Xni¼1

Xmj¼1

Mjgi

" #�1

ð8Þ

To obtainPm

j¼i Mjgi, perform the fuzzy addition operation of m extent

analysis values for a particular matrix such that

Xmj¼1

Mjgi¼

Xmj¼1

lj;Xmj¼1

mj;Xmj¼1

uj

!; i ¼ 1; 2; . . . ; n ð9Þ

and to obtain ½Pn

i¼1

Pmj¼1 M

jgi�1

, perform the fuzzy addition operation

of Mjgi(j ¼ 1; 2; . . . ;m) values such that

Xni¼1

Xmj¼1

Mjgi¼

Xni¼1

Xmj¼1

lij;Xni¼1

Xmj¼1

mij;Xni¼1

Xmj¼1

uij

!ð10Þ

200 O. Kulak, C. Kahraman / Information Sciences 170 (2005) 191–210

and then compute the inverse of the vector in Eq. (11) such that

Xni¼1

Xmj¼1

Mjgi

" #�1

¼ 1Pni¼1 ui

;1Pni¼1 mi

;1Pni¼1 li

� �ð11Þ

Step 2. The degree of possibility of M2 ¼ ðl2;m2; u2ÞPM1 ¼ ðl1;m1; u1Þ is de-fined as

V ðM2 PM1Þ ¼ supyx

½minðlM1ðxÞ; lM2

ðyÞ ð12Þ

and can be equivalently expressed as follows:

V ðM2 PM1Þ ¼ hgtðM1 \M2Þ ¼ lM2ðdÞ

¼1; if m2 Pm1

0; if l1 P u2l1�u2

ðm2�u2Þ�ðm1�l1Þ; otherwise

8<: ð13Þ

where d is the ordinate of the highest intersection point D between lM1

and lM2(see Fig. 6).

To compare M1 and M2, we need both the values of V ðM1 PM2Þ andV ðM2 PM1Þ.

Step 3. The degree possibility for a convex fuzzy number M to be greater thank convex fuzzy numbers Mi (i ¼ 1; 2; . . . ; k) can be defined by

V ðM PM1;M2; . . . ;MkÞ ¼ V ½ðM PM1Þ and M PM2Þ and; . . . ; andðM PMkÞ ¼ MinV

iðM PMiÞ; i ¼ 1; 2; 3; . . . ; k: ð14Þ

Assume that

d 0ðAiÞ ¼ MinVi

ðSi P SkÞ ð15Þ

M 2 M 1

l 2m 1 u1u2dl1m 2

1

V(M 2 M1)

Fig. 6. The intersection between M1 and M2.

O. Kulak, C. Kahraman / Information Sciences 170 (2005) 191–210 201

For k ¼ 1; 2; . . . ; n; k 6¼ i. Then the weight vector is given by

W 0 ¼ ðd 0ðA1Þ; d 0ðA02Þ; . . . ; d 0ðAnÞÞT ð16Þ

where Ai (i ¼ 1; 2; . . . ; n) are n elements.Step 4. Via normalization, the normalized weight vectors are

W ¼ ðdðA1Þ; dðA2Þ; . . . ; dðAnÞTÞ ð17Þwhere W is a non-fuzzy number.

5. Numerical applications

In the following, numerical applications of fuzzy AD and weighted fuzzy

AD approaches and crisp and fuzzy AHP approaches to select the besttransportation company are given.

5.1. A numerical application of fuzzy AD approach

An international company needs a freight transportation company to carry

its goods. The company determined four possible transportation companies.

The criteria considered in the selection process are transportation costs,

defective rate, tardiness rate, flexibility and documentation ability. Transpor-

tation cost is the cost to carry one ton along one kilometer. Tardiness rate is

computed as ‘‘the number of days delayed/the number of days expected for

delivery. Transportation costs, defective rate and tardiness rate whose proba-

bility distribution are uniform are accepted to be crisp variables. The othercriteria ‘‘flexibility’’ and ‘‘documentation ability’’ are linguistic variables. The

company’s design ranges are as follows:

FRTC ¼Transportation cost (TC) must be in the range of cent 5.5 to cent 6,

FRDR ¼Defective rate (DR) must not be over 1%,

FRTR ¼Tardiness rate (TR) must not be over 5%,

FRF ¼Flexibility (F) must be over 9:(9, 20, 20),

FRDA ¼Documentation ability (DA) must be over 13: (13, 20, 20).

Alternative companies’ transportation cost and performance scores evalu-

ated by the company’s managers with respect to criteria are given in Table 1.

The data given in the Table 1 are arranged to include the minimum and

maximum performance values supplied by the transportation company. The

managers produce the system range data and use the linguistic expressions

about flexibility and documentation ability as in Table 1. Fig. 7 shows the

membership functions of the linguistic expressions about flexibility and doc-umentation ability. The decision-maker subjectively evaluates the alternatives

Table 1

The system range data for transportation companies

Alternative com-

panies

Criteria

Transporta-

tion cost

(cent)

Defective

rate (%)

Tardiness

rate (%)

Flexibility Documenta-

tion ability

Trans.-Comp. 1 5.5–6.5 0–1 3–8 Very good Good

Trans.-Comp. 2 5.5–6.0 2–4 0–5 Fair Excellent

Trans.-Comp. 3 5.8–6.5 0–2 1–4 Excellent Very good

Trans.-Comp. 4 5.5–5.8 0–3 2–10 Good Very good

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

x

f(x)

Poor Fair Good Very Good Excellent

Fig. 7. TFNs for intangible factors.

202 O. Kulak, C. Kahraman / Information Sciences 170 (2005) 191–210

with the linguistic term ‘‘poor’’ if these two criteria are assigned a score of (0, 0,

6) over 20; ‘‘fair’’ with a score of (4, 7, 10) over 20; ‘‘good’’ with a score of (8,

11, 14) over 20; ‘‘very good’’ with a score of (12, 15, 18) over 20; ‘‘excellent’’

with a score of (16, 20, 20) over 20.

Using these design and system ranges, the information content for each FR

in each Transportation company may be computed using Eqs. (3) and (6).

The information contents for the criteria with respect to the alternatives are

given in Table 2. As the transportation company with minimum informationcontent is the best one, Transportation company-3 is selected.

5.2. Weighted fuzzy AD approach

Crisp AHP was used to obtain the weight of each criterion. The scale of

importance in Saaty’s [17] crisp AHP is defined as follows: 1 for equal

importance; 3 for moderate importance; 5 for strong importance; 7 for verystrong or demonstrated importance; 9 for extreme importance; and 2, 4, 6, 8

Table 2

The results of Suh’s information content for transportation companies

Alternative

companies

ITC IDR ITR IF IDA

PI

Trans.-Comp. 1 1.000 0.000 1.322 0.365 5.907 8.594

Trans.-Comp. 2 0.000 Infinite 0.000 6.381 0.000 Infinite

Trans.-Comp. 3 1.807 1.000 0.000 0.000 2.466 5.273�

Trans.-Comp. 4 0.000 1.585 1.415 1.748 2.466 7.214

Table 3

Comparison matrix of the criteria

Criteria TC DR TR F DA Priority vector

TC 1 5 3 5 9 0.494

DR 1/5 1 1/2 1/2 7 0.114

TR 1/3 2 1 1/2 7 0.164

F 1/5 2 2 1 8 0.199

DA 1/9 1/7 1/7 1/8 1 0.029

O. Kulak, C. Kahraman / Information Sciences 170 (2005) 191–210 203

for compromise. Readers are referred to Saaty [17] for obtaining the priority

vector in Table 3.

An expert in the end user company determined the preference numbers in

the choice for each pairwise comparison. In Table 3, the comparison matrix of

the criteria and the priority weights are shown.

Saaty’s [17] consistency ratio, consistency index/random index, was used to

test the consistency of the eigenvector of the comparison matrix of the criteria.

The consistency ratio of the matrix in Table 3 was obtained as 0,078. Since thisratio is less than 0.10, the comparison matrix is reasonable consistent.

Eq. (18) is proposed for the weighted crisp and fuzzy AD approaches:

Iij ¼log2

1pij

�h i 1wj; 06 Iij < 1

log21pij

�h iwj

; Iij > 1

wj; Iij ¼ 1

8>><>>:

9>>=>>; ð18Þ

Using the weighted fuzzy AD approach, Table 4 is obtained. Transportation

company-3 is again the selected alternative.

5.3. Numerical application of crisp AHP

In the crisp AHP, the degree of preference of the decision maker in the

choice of each pairwise comparison is quantified on a scale of 1–9, and this

quantities are placed in a matrix of comparisons. Even numbers (2, 4, 6, and 8)

Table 4

The results of weighted information content for transportation companies

Alternative

companies

ITC IDR ITR IF IDA

PI

Trans.-Comp. 1 0.494 0.000 1.047 0.006 1.053 2.600

Trans.-Comp. 2 0.000 Infinite 0.000 1.446 0.000 Infinite

Trans.-Comp. 3 1.339 0.114 0.000 0.000 1.027 2.480�

Trans.-Comp. 4 0.000 1.054 1.059 1.118 1.027 4.257

Selection of the best transportation company

Cost Defectiverate

Tardinessrate Flexibility Documentation

ability

Transportationcompany-1

Transportationcompany-2

Transportationcompany-3

Transportationcompany-4

Fig. 8. The hierarchy of the problem.

204 O. Kulak, C. Kahraman / Information Sciences 170 (2005) 191–210

can be used to represent to compromises among the preferences below. Fig. 8

shows the hierarchy of the problem.

The weight vector from Table 5 is calculated as WTC ¼ ð0:099; 0:284;0:099; 0:518Þ. The other matrices of pairwise comparisons and the weight

vector of each matrix are given in the following.

The weight vector from Table 6 is calculated as WDR ¼ ð0:540; 0:047;0:275; 0:138Þ.

The weight vector from Table 7 is calculated as WTR ¼ ð0:093; 0:319;0:534; 0:055Þ.

Table 5

Evaluation of the transportation companies with respect to transportation costs

Alternatives Trans.-Comp. 1 Trans.-Comp. 2 Trans.-Comp. 3 Trans.-Comp. 4

Trans.-Comp. 1 1 1/3 1 1/5

Trans.-Comp. 2 3 1 3 1/2

Trans.-Comp. 3 1 1/3 1 1/5

Trans.-Comp. 4 5 2 5 1

O. Kulak, C. Kahraman / Information Sciences 170 (2005) 191–210 205

The weight vector from Table 8 is calculated as WF ¼ ð0:251; 0:056;0:584; 0:109Þ.

The weight vector from Table 9 is calculated as WDA ¼ ð0:079; 0:519;0:201; 0:201Þ.

Table 10 shows the combination of the weight vectors obtained by pairwise

comparisons. Alternative 4 is the selected transportation company. The result

is different from the ones of crisp and fuzzy AD approaches. One reason may

Table 8

Evaluation of the transportation companies with respect to flexibility

Alternatives Trans.-Comp. 1 Trans.-Comp. 2 Trans.-Comp. 3 Trans.-Comp. 4

Trans.-Comp. 1 1 5 1/3 3

Trans.-Comp. 2 1/5 1 1/7 1/3

Trans.-Comp. 3 3 7 1 7

Trans.-Comp. 4 1/3 3 1/7 1

Table 9

Evaluation of the transportation companies with respect to documentation ability

Alternatives Trans.-Comp. 1 Trans.-Comp. 2 Trans.-Comp. 3 Trans.-Comp. 4

Trans.-Comp. 1 1 1/5 1/3 1/3

Trans.-Comp. 2 5 1 3.00 3

Trans.-Comp. 3 3 1/3 1 1

Trans.-Comp. 4 3 1/3 1 1

Table 6

Evaluation of the transportation companies with respect to defective rate

Alternatives Trans.-Comp. 1 Trans.-Comp. 2 Trans.-Comp. 3 Trans.-Comp. 4

Trans.-Comp. 1 1 7 3 5

Trans.-Comp. 2 1/7 1 1/5 1/3

Trans.-Comp. 3 1/3 5 1 3

Trans.-Comp. 4 1/5 3 1/3 1

Table 7

Evaluation of the transportation companies with respect to tardiness rate

Alternatives Trans.-Comp. 1 Trans.-Comp. 2 Trans.-Comp. 3 Trans.-Comp. 4

Trans.-Comp. 1 1 1/5 1/5 2

Trans.-Comp. 2 5 1 1/3 7

Trans.-Comp. 3 5 3 1 7

Trans.-Comp. 4 1/2 1/7 1/7 1

Table 10

The results of crisp AHP for transportation companies

Cost Defective

rate

Tardiness

rate

Flexibil-

ity

Documenta-

tion ability

Priority

weight

Weight 0.494 0.114 0.164 0.199 0.029

Alternative

Trans.-Comp. 1 0.099 0.540 0.093 0.251 0.079 0.178

Trans.-Comp. 2 0.284 0.047 0.319 0.056 0.519 0.224

Trans.-Comp. 3 0.099 0.275 0.534 0.584 0.201 0.290

Trans.-Comp. 4 0.518 0.138 0.055 0.109 0.201 0.308�

206 O. Kulak, C. Kahraman / Information Sciences 170 (2005) 191–210

be that AD approach’s information content becomes infinitive (this means the

elimination of that alternative) if a certain range is above or below its design

range, while AHP never eliminates an alternative in this way.

5.4. Numerical application of fuzzy AHP

Table 11 shows triangular fuzzy conversion scale. This scale is not optimal

and robust. Slight changes in the parameters of the scale will not create an

important influence on the results of the decision unless the attribute prefer-

ences are reordered. Tables 12–17 are the pairwise comparison matrices of the

criteria and alternatives.

The weight vector from Table 12 is calculated as WG ¼ ð0:38; 0:17;0:21; 0:24; 0:00Þ using Chang’s [7] extent analysis method presented previ-

ously. Readers can find the details of the similar calculations in Kahramanet al. [10].

The weight vector from Table 13 is calculated as WTC ¼ ð0:15; 0:31;0:18; 0:36Þ. The other matrices of pairwise comparisons and the weight vector

of each matrix are given in the following.

The weight vector from Table 14 is calculated as WDR ¼ ð0:48; 0:00; 0:32;0:20Þ.

Table 11

Triangular fuzzy conversion scale

Linguistic scale Triangular fuzzy scale Triangular fuzzy reciprocal scale

Just equal (1, 1, 1) (1, 1, 1)

Equally important (1/2, 1, 3/2) (2/3, 1, 2)

Weakly more important (1, 3/2, 2) (1/2, 2/3, 1)

Strongly more important (3/2, 2, 5/2) (2/5, 1/2, 2/3)

Very strongly more important (2, 5/2, 3) (1/3, 2/5, 1/2)

Absolutely more important (5/2, 3, 7/2) (2/7, 1/3, 2/5)

Table 14

Evaluation of the transportation companies with respect to defective rate

Alternatives Trans.-Comp. 1 Trans.-Comp. 2 Trans.-Comp. 3 Trans.-Comp. 4

Trans.-Comp. 1 (1, 1, 1) (2, 5/2, 3) (1, 3/2, 2) (3/2, 2, 5/2)

Trans.-Comp. 2 (1/3, 2/5, 1/2) (1, 1, 1) (2/5, 1/2, 2/3) (2/5, 1/2, 2/3)

Trans.-Comp. 3 (1/2, 2/3, 1) (3/2, 2, 5/2) (1, 1, 1) (1, 3/2, 2)

Trans.-Comp. 4 (2/5, 1/2, 2/3) (3/2, 2, 5/2) (1/2, 2/3, 1) (1, 1, 1)

Table 15

Evaluation of the transportation companies with respect to tardiness rate

Alternatives Trans.-Comp. 1 Trans.-Comp. 2 Trans.-Comp. 3 Trans.-Comp. 4

Trans.-Comp. 1 (1, 1, 1) (2/5, 1/2, 2/3) (2/5, 1/2, 2/3) (1, 3/2, 2)

Trans.-Comp. 2 (3/2, 2, 5/2) (1, 1, 1) (2/3, 1, 2) (2, 5/2, 3)

Trans.-Comp. 3 (3/2, 2, 5/2) (1/2, 1, 3/2) (1, 1, 1) (2, 5/2, 3)

Trans.-Comp. 4 (1/2, 2/3, 1) (1/3, 2/5, 1/2) (1/3, 2/5, 1/2) (1, 1, 1)

Table 16

Evaluation of the transportation companies with respect to flexibility

Alternatives Trans.-Comp. 1 Trans.-Comp. 2 Trans.-Comp. 3 Trans.-Comp. 4

Trans.-Comp. 1 (1, 1, 1) (3/2, 2, 5/2) (1/2, 2/3, 1) (1, 3/2, 2)

Trans.-Comp. 2 (2/5, 1/2, 2/3) (1, 1, 1) (1/3, 2/5, 1/2) (1/2, 2/3, 1)

Trans.-Comp. 3 (1, 3/2, 2) (2, 5/2, 3) (1, 1, 1) (3/2, 2, 5/2)

Trans.-Comp. 4 (1/2, 2/3, 1) (1, 3/2, 2) (2/5, 1/2, 2/3) (1, 1, 1)

Table 12

The fuzzy evaluation matrix with respect to the goal

Criteria Transporta-

tion cost

Defective

rate

Tardiness

rate

Flexibility Documenta-

tion ability

Cost (1, 1, 1) (3/2, 2, 5/2) (1, 3/2, 2) (3/2, 2, 5/2) (5/2, 3, 7/2)

Defective

rate

(2/5, 1/2, 2/3) (1, 1, 1) (4/7, 4/5, 4/3) (4/7, 4/5, 4/3) (2, 5/2, 3)

Tardiness

rate

(1/2, 2/3, 1) (3/4, 5/4, 7/4) (1, 1, 1) (4/7, 4/5, 4/3) (2, 5/2, 3)

Flexibility (2/5, 1/2, 2/3) (3/4, 5/4, 7/4) (3/4, 5/4, 7/4) (1, 1, 1) (9/4, 11/4, 13/

4)

Documen-

tation A.

(2/7, 1/3, 2/5) (1/3, 2/5, 1/2) (1/3, 2/5, 1/2) (4/13, 4/11, 4/9) (1, 1, 1)

Table 13

Evaluation of the transportation companies with respect to transportation costs

Alternatives Trans.-Comp. 1 Trans.-Comp. 2 Trans.-Comp. 3 Trans.-Comp. 4

Trans.-Comp. 1 (1, 1, 1) (1/2, 2/3, 1) (1/2, 1, 3/2) (2/5, 1/2, 2/3)

Trans.-Comp. 2 (1, 3/2, 2) (1, 1, 1) (1, 3/2, 2) (2/3, 1, 2)

Trans.-Comp. 3 (2/3, 1, 2) (1/2, 2/3, 1) (1, 1, 1) (2/5, 1/2, 2/3)

Trans.-Comp. 4 (3/2, 2, 5/2) (1/2, 1, 3/2) (3/2, 2, 5/2) (1, 1, 1)

O. Kulak, C. Kahraman / Information Sciences 170 (2005) 191–210 207

Table 18

Combination of the weight vectors

Cost Defective

rate

Tardiness

rate

Flexibility Documen-

tation

ability

Priority

weight

Weight 0.38 0.17 0.21 0.24 0.00

Alternative

Trans.-Comp. 1 0.15 0.48 0.13 0.34 0.12 0.248

Trans.-Comp. 2 0.31 0.00 0.43 0.00 0.37 0.273

Trans.-Comp. 3 0.18 0.32 0.44 0.51 0.25 0.319�

Trans.-Comp. 4 0.36 0.20 0.00 0.15 0.26 0.199

Table 17

Evaluation of the transportation companies with respect to documentation ability

Alternatives Trans.-Comp. 1 Trans.-Comp. 2 Trans.-Comp. 3 Trans.-Comp. 4

Trans.-Comp. 1 (1, 1, 1) (2/5, 1/2, 2/3) (1/2, 2/3, 1) (1/2, 2/3, 1)

Trans.-Comp. 2 (3/2, 2, 5/2) (1, 1, 1) (1, 3/2, 2) (1, 3/2, 2)

Trans.-Comp. 3 (1, 3/2, 2) (1/2, 2/3, 1) (1, 1, 1) (1/2, 1, 3/2)

Trans.-Comp. 4 (1, 3/2, 2) (1/2, 2/3, 1) (2/3, 1, 2) (1, 1, 1)

208 O. Kulak, C. Kahraman / Information Sciences 170 (2005) 191–210

The weight vector from Table 15 is calculated as WTR ¼ ð0:13; 0:43; 0:44;0:00Þ.

The weight vector from Table 16 is calculated as WF ¼ ð0:34; 0:00; 0:51;0:15Þ.

The weight vector from Table 17 is calculated as WDA ¼ ð0:12; 0:37; 0:25;0:26Þ. Table 18 shows the combination of the weight vectors obtained by

pairwise comparisons.

Alternative 3 is the selected transportation company. The result is the sameas crisp and fuzzy AD approaches. The reason for selecting different alterna-

tives when using crisp and fuzzy AHP approaches is that fuzzy AHP uses fuzzy

numbers which mean intervals instead of crisp numbers.

6. Conclusions

In most of the real-world problems, some of the decision data can be pre-

cisely assessed while others cannot. Real numbers are used to represent data

which can be precisely measured. For those data which cannot be precisely

assessed, fuzzy sets can be used to denote them. The use of fuzzy set theory

allows us to incorporate unquantifiable information, incomplete information,

non-obtainable information and partially ignorant facts into the decisionmodel. When decision data are precisely known, they should not be forced into

O. Kulak, C. Kahraman / Information Sciences 170 (2005) 191–210 209

a fuzzy format in the decision analysis. Crisp MADM methods solve problems

in which all decision data are assumed to be known and must be represented by

crisp numbers. The methods are to effectively aggregate performance scores.

Fuzzy MADM methods have difficulty in judging the preferred alternativesbecause all aggregated scores are fuzzy data. We propose crisp multi-attribute

AD approach when all decision data are known whereas we propose fuzzy

multi-attribute AD approach when unquantifiable or incomplete information

exists.

The proposed crisp and fuzzy AD approaches use the design ranges deter-

mined by the decision-makers to select best alternative. However, these ap-

proaches that depend on the minimum information axiom do not let an

alternative to be selected even if that alternative meets the design ranges of allother criteria successfully, but not any of these ranges. However, the decision-

maker can assign a numerical value instead of ‘infinitive’ in order to make

possible the selection of an alternative which meets all other criteria success-

fully, except the criterion having an ‘infinitive’ value.

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