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Materials
www.elsevier.com/locate/matdes
Materials and Design 27 (2006) 329–337
&Design
A material selection model based on the conceptof multiple attribute decision making
A. Shanian, O. Savadogo *
Laboratoire de nouveaux materiaux pour les systemes electrochimiques, Ecole Polytechnique de Montreal,
P.O. Box 6079, Centre-ville, Montreal, Quebec, Canada H3C 3A7
Received 30 September 2003; accepted 7 October 2004
Available online 15 December 2004
Abstract
Replacing and selecting materials for different engineering applications is relatively common. It must be noted that in some cases,
there is more than a single definite criterion for selecting the right kind of material. The designers and engineers have to take into
account a large number of material selection criteria. Experts usually apply trial and error methods or build on previous experimen-
tation. In this paper, a new approach has been carried out for the use of the ELECTRE: ELimination Et Choix Traduisant la REa-
lite (ELimination and Choice Expressing the REality) model in material selection. By producing a material selection decision matrix
and criteria sensitivity analysis, ELECTRE has been applied to obtain a more precise material selection for a particular application,
including logical ranking of considered materials. A list of all possible choices from the best to the worst suitable materials can be
obtained taking into account all the material selection criteria, including the cost of production. This work shows that ELECTRE
can be used successfully in selecting a suitable material for the particular application of a loaded thermal conductor. There is good
agreement between the results of the methods being used and available data in Cambridge Engineering Selector (CES) databases. A
computer program, Mathematica has been developed to facilitate the application of the method to other types of material selection
problems.
� 2004 Elsevier Ltd. All rights reserved.
Keywords: ELECTRE; Modeling; Entropy; Rank; Production; Cost; Material selection
Resume
Le remplacement et le choix des materiaux pour differentes applications technologiques est une pratique tres populaire de nos
jours. On doit noter qu�en choisissant le bon materiau, il n�y a pas qu�un seul critere de choix bien defini. Les concepteurs et les
ingenieurs doivent tenir compte d�un grand nombre de criteres de performance. Les experts utilisent habituellement la methode d�es-sai et d�erreur ou se basent sur des experiences precedentes. Dans cet article, une nouvelle approche a ete effectuee sur l�utilization du
modele d�ELECTRE (ELimination Et Choix Traduisant la REalite) pour le choix du materiau. En produisant une matrice de deci-
sion des materiaux et une analyse de sensibilite des criteres, ELECTRE a ete appliquee pour obtenir un choix de materiau plus precis
pour une application particuliere, y compris le rang logique des materiaux consideres. Une liste de tous les materiaux appropries, du
meilleur au moins performant, peut etre obtenue en tenant compte de tous les indices de performance et egalement du cout de pro-
duction. Ce travail demontre qu�ELECTRE peut etre utilisee avec succes lors de la selection d�un materiau approprie pour la fab-
rication d�un conducteur thermique en charge. Les resultats obtenus avec les methodes utilisees concordent avec ceux disponibles
0261-3069/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.matdes.2004.10.027
* Corresponding author. Tel.: +1 514 340 4711x4725; fax: +1 514 340 4468.
E-mail address: [email protected].
330 A. Shanian, O. Savadogo / Materials and Design 27 (2006) 329–337
dans les bases de donnees CES. Un programme Mathematica a aussi ete developpe afin de faciliter l�application de la methode a
d�autres types de problemes de selection de materiaux.
� 2004 Elsevier Ltd. All rights reserved.
Mots-cle: ELECTRE; Modelisation; Entropie; Classifier; Production; Cout; Choix des materiaux
Table 1
Decision matrix in MADM models
Weighted coefficients of criteria P1 P2 – Pj
Criteria candidate materials g1 g2 – gjA1 r11 r12 – r1jA2 r21 r22 – r2j–
Ai ri1 ri2 – rij
1. Introduction
Recently, many materials which have long been
used in industry are being replaced by newer materials
in order to meet the demands of cost reduction andbetter performance. When choosing a new material
or replacing an existing one on a mechanical compo-
nent, experts usually apply trial and error methods
or build on previous experimentation. It must be noted
that in choosing the right material, there is not always
a single definite criterion of selection and the designers
and engineers have to take into account a large num-
ber of material selection criteria. These material selec-tion criteria range from mechanical to electrical
properties and corrosion resistance. With regards to
each material selection criterion, a wide range of mate-
rial properties and performance attributes can be
considered.
Decision making in the presence of multiple, gener-
ally conflicting criteria is known as multiple criteria deci-
sion making (MCDM). Depending on the domain ofalternatives, MCDM problems are usually divided into
continuous or discrete types. MCDM problems have
two classifications: multiple objective decision making
and multiple attribute decision making [1–3]. MODM
have decision variable values which are determined in
a continuous or integer domain with either an infinitive
or a large number of choices, the best of which should
satisfy the decision maker�s constraints and preferencepriorities. MADM on the other hand are generally dis-
crete, have a limited number of prespecified alternatives.
They require both intra and inter attributes comparisons
and involve explicit tradeoffs which are appropriate for
the problem explained [4]. The MADM problem under
consideration is depicted by a decision matrix which is
given in Table 1.
Each decision matrix in MADM models has threemain parts, namely: (a) alternatives (b) criteria and (c)
relative importance of each criterion (weight). In the
decision matrix all the elements must be normalized to
the same units so that we can consider all the possible
criteria in our decision problem. Of the many MADM
methods, we present here the ELECTRE methods,
which have a high potential to solve material selection
problems.The original method ELECTRE I (Elimination and
Et Choice Translating REality) was developed by Ber-
nard Roy [5] in 1968. This method was published in
RIRIO, ‘‘Revue Francaise d� Informatique et de Recher-
che Operationnelle’’. This paper presents a compressive
description of the ELECTRE and the principals of out-
ranking approach; it is also suggested that the reader
consult the book in graph theory by Bernard Roy [6].
Several related techniques have been developed:ELECTRE II [7–10], a method for dealing with
the problem of ranking alternatives from the best to the
worst, ELECTRE III [11,12], a method which uses the
pseudo-criteria [13] and fuzzy binary outranking rela-
tions, and ELECTRE IV [14–18], a method in which it
is possible to rank alternatives without using the relative
criteria importance coefficients. This method is equipped
with an embedded outranking relations frame work.ELECTRE IS [19] is a method used for modeling situa-
tions in which the data are imperfect. ELECTRE TRI
[20,21] is designed to classify the alternatives in various
categories. These categories are separated by ‘‘reference
alternatives’’. As the comparison is done between alter-
natives and reference alternatives, this method permits
the decision maker to deal with many alternatives.
ELECTRE uses the concept of an outranking rela-tion, S, for modeling the preference. The outranking
relation of Mk ! Ml says that MK outranks Ml, if MK
is at least as good as Ml on a majority of criteria and
is not significantly based on any of the other criteria
[22,23]. On the other hand, when considering two alter-
natives, four situations can occur:
� MkSMl and not MlSMk, i.e., MkPMl (MK is strictlypreferred to Ml).
� MlPMk and not MkSMl, i.e., MlPMk (Ml is strictly
preferred to MK).
� MkSMl and not MlSMk, i.e., MkIMl (MK is indiffer-
ent to Ml).
� Neither MkSMl nor MlSMk, i.e., MkRMl (MK is
incomparable to Ml).
ELECTRE methods operate with one or several (crispy,
fuzzy or embedded) outranking relations. Modeling of
preferences by outranking relation presents an incompa-
Nomenclature
gj jth attribute in decision matrix
Ai ith alternative in decision matrixPj weight of jth attribute
M bending required moment
t thickness of the sheet
rh the stress in h direction
Y yield stress
r0 the radius of curvature of the sheet before
bending
r 0 the new radius of curvature of the sheet after
unloadingm the poison ratio
E elastic modulus
rij performance of ith alternative (candidate
material) with respect to jth criterion
A. Shanian, O. Savadogo / Materials and Design 27 (2006) 329–337 331
rability relation. This relation is helpful when the analyst
is not able to compare two alternatives [24].
By successively assessing the outranking relations of
the other alternatives, one can eliminate the dominated
alternatives defined by the outranking relationship.
However, the construction of this partial order is not
an unambiguous task for the decision maker. ELEC-
TRE sets the criteria for the mechanical assessment ofthe outranking relationships [1,2,25].
This method consists of a pairwise comparison of
alternatives, based on the degree to which evaluations
of the alternatives and the preference weights confirm
or contradict the pairwise dominance relationship be-
tween alternatives. It examines both the degree to which
the preference weights are in agreement with pairwise
dominance relationships, and the degree to whichweighted evaluations differ from each other. These stages
are based on a ‘‘concordance and discordance’’ set; hence
this method is also called concordance analysis [1,2,25].
The structure of an outranking relation is constructed
based on this analysis. The concordance concept states
that for an outranking to be validated, a sufficient
majority of criteria should be in favor of this affirma-
tion. The discordance concept states that when the con-cordance condition is satisfied, none of the criteria in the
majority can oppose the affirmation too strongly [24].
Van Delft and Nijkamp [25] introduced the net con-
cordance and discordance values for the complementary
analysis of the ELECTRE method. Obviously,MK has a
higher chance of being accepted with higher net concor-
dance values and lower net discordance values.
The detailed operations of the ELECTRE methodscan be found in references [23,26–30]. Some other works
deserve mention because they include information con-
cerning the ELECTRE methods: these are references
[22,31–42]. For more information on the operation
and details of the ELECTRE methods, the authors
encourage the reader to review the sources provided.
In the context of material selection, compensation of
the loss on a given material selection criterion by a gainin another one cannot be accepted by the material de-
signer. A physical, metallurgical and mechanical crite-
rion must stand on its own. Therefore, such a
situation requires the use of noncompensatory aggrega-tion procedures which exist in ELECTRE methods (see
[43]). This ability makes ELECTRE methods unique
amongst all decision aiding models. The ELECTRE
methods are quick, operate with simple logic, and have
the strength of being able to detect the presence of
incomparability. It uses a systematic computational pro-
cedure, an advantage of which is an absence of strong
axiomatic assumptions. Compared to other decision aid-ing techniques such as Maximax (min) and Lexico-
graphic [1–3] which utilize a part of the decision
matrix, the ELECTRE makes full use of material selec-
tion decision matrices and is thus able to provide us with
more reliable results. Another advantageous point is
that when a large number of alternatives are considered,
the high number of comparisons required by methods
such as the analytical hierarchy process (AHP) [44,45]is avoided. In this case, the ELECTRE method is com-
pletely suitable for linking with computer databases
dealing with material selection.
In this paper, among of ELECTRE methods, we have
chosen the ELECTRE I–II methods to solve material
selection problems. The ELECTRE I allows one to
choose the best candidate material with respect to a
set of material selection criteria, while the ELECTREII method has also been proposed for the ranking of
candidate materials.
2. Methods for assessing the relative importance of
material selection criteria
The importance coefficients in the ELECTRE meth-ods refer to intrinsic ‘‘weight’’. Some work has been
done on the topic of the relative importance of the crite-
ria, for example [38,46,26,47–49,12,50]. The entropy
method [1,51–54] is the method used for assessing the
weight in a given problem because with this method,
the decision matrix for a set of candidate materials con-
tains a certain amount of information. The entropy
method works based on a predefined decision matrix.
332 A. Shanian, O. Savadogo / Materials and Design 27 (2006) 329–337
Since in material selection problems, there is straight
access to the values of the decision matrix, the Entropy
method is the appropriate method to use. Entropy in
information theory is a criterion for the amount of
uncertainty represented by a discrete probability distri-
bution, in which there is agreement that a broad distri-
bution represents more uncertainty than does a
sharply packed one [3]. The entropy idea is particularlyuseful for investigating contrasts between sets of data
and a good tool in criteria evaluation.
Using the entropy method, it is possible to combine
the material designer�s priorities with that of the sensitiv-
ity analysis. Final weights defined are a combination of
two sets of weights. The first is the set of objective
weights that are derived directly from the nature of the
design problem and materials properties using the entro-py method, with no concern to the design performer�swill. The second is the set of subjective weights that
are defined by the material designer�s preferences to
modify the previous weights and find the total weights.
When the designer finds no reason to give preference
to one criterion over another, the principle of insufficient
reason [54] suggests that each one should be equally pre-
ferred. This rule is applied in the given problem. If thedesigner has a prior, subjective weight, then this can
be adopted with the help of weight information [3].
It is believed that that the combined weighting scheme
introduced is important for material selection problems.
It can take into account both the nature of conflicts
among criteria and the practicality of the decisions. This
opportunity reflects the advantage of more controllable
design selections. This possibility makes the entropymethod very flexible and efficient formaterial design. This
work shows how the entropy results can help the designer
select the proper criteria in the design of the component.
Fig. 2. Schematic of thermal loaded conductor.
3. Case study
This paper focuses on the material selection of mass-produced non-heat-treatable cylindrical cover material.
z0L
0r
wt
rr /
Fig. 1. Geometry definition and stress
These materials usually operate under a static load
and carry out heat efficiency very close to room temper-
ature. The design is known as the late conceptual kind.
The heat conduction through a thermally loaded con-
ductor requires using a sheet of metal, which is bent
around a heat transfer medium. The sheet thickness de-
pends on the required heat transfer circumstances. This
sheet has to be able to support an immobile compressiveweight and be able to hold against any denting during
the hardening process.
3.1. Modeling and simulation
An analytical model in [55] is considered for analyz-
ing and evaluating the criteria and their related attri-
butes in the studied case. The schematic of theproposed model is shown in Figs. 1 and 2. This model
allows the sheet on the outside of the bend to be at
some uniform value in tension. At the same time, the
inner part of the bend is also at the same uniform va-
lue in compression. The central part of the sheet is
elastic. The required bending moment is computed as
follows:
0t
drrr +/
dr
t
analyses of the sheet for bending.
Table 2
List of materials
Material
number
Material name Material standard
number
1 Copper-2-beryllium (cast) UNS C82400
2 Copper–cobalt–beryllium (cast) UNS C82000
3 Electrolytic tough-pitch, h.c.
copper, soft (wrought)
UNS C11000
4 Electrolytic tough-pitch, h.c.
copper, hard (wrought)
UNS C11000
5 Wrought aluminum alloy 5052 H34
6 Wrought austenitic stainless steel AISI 304,HT grade D
7 Commercial bronze, CuZn10,
soft (wrought)
UNS C22000
8 Carbon steel (annealed) AISI 1020
A. Shanian, O. Savadogo / Materials and Design 27 (2006) 329–337 333
M ¼Z t=2
�t=2wrhzdz ¼
wt2Y
2ffiffiffi3
p : ð1Þ
Since the applied bending moment approaches zero
upon unloading, and as a result the internal moment
also approaches zero [55], the spring back can be com-
puted. By considering this point, the spring back value
can be computed as follows:
1
r0¼ 1
r0� 6ð1� m2ÞYffiffiffi
3p
Et: ð2Þ
Eqs. (1) and (2) are useful in predicting the spring back
and bending moment respectively.
Based on the defined analytical model and the ex-
pected thermal properties, the criteria are subdivided
into three main sections: (a) thermal, (b) cost (tooling
and base material cost), and (c) mechanical.
Since high thermal conductivity is desirable, one canmake the amount of transferred heat and the time of
steady state conduction (thermal diffusivity, higher is
desirable) the thermal criteria. Cost criteria are divided
into two main parts: (1) Tooling costs, which are speci-
fied by the bend force index and the spring back (a lower
value is desirable for both). The first one is proportional
to Yt2 and the second one is proportional to Y/tE,
according to Eqs. (1) and (2), respectively. Small num-bers for both indices denote desirable manufacturability.
(2) The price of the base material, which is specified by
the density and price of the material (lower values are
desirable for both). Mechanical component design fac-
tors are the elastic modulus, yield stress, compressive
and the ultimate tensile strength, for which higher values
are desirable. Other criteria are the ability to resist dent-
ing during hardening, or hardness, for which a highervalue is desirable, as well as the ability to carry a static.
For the ability to carry the static load, one defines the
Table 3
Material propertiesa
Materials properties 1 2 3
Density (Mg/m3) 8.25 8.65 8.94
Compressive stress (Mpa) 560 460 50
Ultimate tensile stress (MPa) 940 600 210
Spring back indexb 0.78 0.71 0.08
Bend force indexb 15,183 12,472 1355
Static load indexb 2916 2395 260
Hardness (vickers) 380 220 45
Yield stress (Mpa) 560 460 50
Elastic modulus (GPa) 138 125 122
Thermal diffusivity (cm2/h) 465 465 460
Thermal conductivity (W/m K) 105 205 398
Thickness (mm) 5.207 5.207 5.207
Cost of base material (CAN$/kg) 18.64 13.99 3.00
a These data are taken from the CES selector database.b Static-load-carrying capability is proportional to YS(t) and large num
spring back is proportional to YS/(tE). Small numbers for both indices deno
static load index, which is proportional to the Yt based
on the presented analytical model.
Obviously, the ideal material cannot be found due to
the conflicting tradeoffs between selection criteria. The
limiting factor for copper and aluminum would be their
high tooling cost due to their thickness. Steel also has
limiting factors due to its low thermal conductivity coef-
ficient, which is difficult to improve.For modeling a given problem, at the initial stage,
one should select all the material properties related to
the given functional requirements. Also, minimum con-
straints on the materials under question should be ap-
plied to screen a number of candidate materials from
all the materials available in a database. One can use
the Cambridge Engineering Selector (CES) software
and database for finding the proper candidate materi-als and related properties, which are developed by Ash-
by and Cambridge University. We can generate the
decision matrix by having the as well the chosen materi-
als. This information has been presented in Tables 2 and
3, which indicate the decision matrix and the direction
of the performance of the criteria. The elements of the
4 5 6 7 8
8.95 2.67 8.06 8.63 7.08
340 190 690 95 267
380 295 1030 270 355
0.48 0.25 1.55 0.17 0.48
9218 20,317 5909 2711 1957
1770 1966 2174 520 720
115 87 350 63 110
340 191 800 100 265
135 73.59 190 116 205
460 741 189 174 329
390 152 17 185 50
5.207 10.238 2.7178 5.207 2.7178
3.46 2.81 5.99 3.32 1.04
bers are desirable. Bending force is proportional to YS(t)2. Extent of
te desirable manufacturability.
Table 5
334 A. Shanian, O. Savadogo / Materials and Design 27 (2006) 329–337
decision matrix for each criterion are taken as the inputs
for the above-mentioned methods. The entropy
weighted coefficients and the output of the ELECTRE
method are obtained using a Mathematica program
developed for this purpose.
3.2. Results and discussion
Table 4 summarizes the weighted coefficients of dif-
ferent attributes obtained using the entropy method
without considering cost, while Table 5 presents the
same coefficients with the cost criteria considered. For
the first case (Table 4), the thermal diffusivity and elastic
modules have a very low value compared to other attri-
butes. For the second case (Table 4), the criterion of
Table 4
Weighted coefficients without the criterion of cost
Attribute Weighted coefficients
Step I
Weighted coefficients
Step II
Compressive strength 0.152260 0.136167
Static load index 0.13166 0.118237
Hardness 0.167986 0.165818
Yield stress 0.166146 0.152534
Elastic modulus 0.0277938 0.0327805
Thermal diffusivity 0.0614223 0.0480244
Thermal conductivity 0.184494 0.233305
Ultimate tensile strength 0.10823 0.113134
Table 6
Final score of candidates for thermal loaded conductor without the criterion
Material number 1 2 3 4
ELECTRE I
Rank 3 2 5 1
ELECTRE II
Step I
Net concordance 3.32113 2.80244 �2.91667 1
Rank 1 2 7 4
Net discordance �2.68347 �1 �0.8646 �1
Rank 3 1–2 5 1–
Step II
Net concordance 2.99928 2.1349 �2.83924 0
Rank 1 2 7 4
Net concordance �1.33468 �0.9751 �1.21466 �4
Rank 2 4 3 1
Table 7
Final score of candidates for thermal loaded conductor with the criterion of
Material number 1 2 3 4
ELECTRE I
Rank 8 6 1 2
ELECTRE II
Net concordance 0.173369 0.405841 �0.2798 0.45
Rank 5 4 6 3
Net discordance 2.79747 2.86245 �3.59575 �2.81
Rank 7 8 1 2
density has a low value similar to thermal diffusivity
and elastic modules. One sees that for the attributes with
a low range, which possess no critical points due to a
uniform rate of increase, the weighted coefficients are
negligible. It can be concluded that those attributes
whose weighted coefficients are of low value have no
major effect on the final decision.
The results of ELECTRE I and II are given in Tables6 and 7 and Figs. 3 and 4, respectively, for the two cases
considered above.
When one considers the decision matrix without the
criterion of cost and its related attributes (Table 6 and
Weighted coefficients with the criterion of cost
Attribute Weighted coefficients
Bend force index 0.12144
Spring back index 0.117335
Ultimate tensile strength 0.0644855
Density 0.0172585
Compressive strength 0.0907241
Static load index 0.0784452
Hardness 0.100089
Yield stress 0.098922
Elastic modulus 0.01656
Thermal diffusivity 0.0365964
Thermal conductivity 0.109924
Cost of base materials 0.148151
of cost
5 6 7 8
6 4 7 8
.27619 �1.1902 1.80253 �3.33068 �1.76475
5 3 8 6
1.78908 �1.70198 +1 5.40134
2 7 4 8 6
.648722 �2.1422 1.71696 �2.51842
5 3 6
.24636 2.57202 0.121993 5.07679
6 5 7
cost
5 6 7 8
5 7 3 4
1625 �0.5906 0.552961 �1.31084 0.597434
7 2 8 1
959 1.53392 1.04964 �1.15423 �0.67390
6 5 3 4
1
2
3
4
5
6
7
8
1 2 3 4 5 6 7 8
Material number
Ran
k
ELECTRE I
Net-concordance(StepI)
Net-discordance(StepI)
Net-concordance(StepII)
Net-discordance(StepII)
Fig. 3. Ranks of candidate materials without the criterion of cost.
1
2
3
4
5
6
7
8
1 2 3 4 5 6 7 8Material number
Ran
k
ELECTRE I
Net- concordance
Net-discordance
Fig. 4. Ranks of candidate materials with the criterion of cost.
A. Shanian, O. Savadogo / Materials and Design 27 (2006) 329–337 335
Fig. 3), Materials 1, 2 and 4 are considered as the first
three choices using both ELECTRE I and II methods.The first three choices are reasonable, since these materi-
als have the best thermal and mechanical properties com-
pared to other candidates in the decision matrix. As can
be seen, Materials 1, 2 and 4 (when the cost is not a fac-
tor) can be used for selecting a material with high safety
requirements, such as aerospace components.
Note that the values of infinity in Table 7 denote the
candidates that are fully dominated (plus infinity) orthat fully dominate other weak candidate(s) with respect
to all attributes (minus infinity). For instance, recalling
the decision matrix, one notices that Material 2 domi-
nates Material 7 in all attributes. As a result, we may re-
peat the solution eliminating Material 7, which has the
plus infinity value in net discordance, and thus, no
power to compete in the decision route. In addition to
enhancing the decision precision in problems with alarge number of attributes, such action may yield faster
decisions. The new results are shown under Step II in
Table 6. As seen, there is no change in the ranks with re-
spect to the net concordance values in Steps I and II.
However, considering the net discordance values, Step
II is in close agreement with ELECTRE I. It should no-
ticed that in Step II, the number of candidates is one less
than in ELECTRE I. Furthermore, the entropy weightsfor the ELECTRE I and ELECTRE II-step methods re-
main identical, as seen in Table 4.
When considering the cost criterion (Table 7 and Fig.
4), the ranking of candidate materials changes signifi-
cantly, particularly for Materials 1, 2 and 8, in compari-
son to the first case. For mass production of these
336 A. Shanian, O. Savadogo / Materials and Design 27 (2006) 329–337
components, the cost criterion plays an essential role and
as seen, Material 4 is preferred in all cases; it is therefore
the most appropriate. For Material 6 on the other hand,
different cases result in considerably different ranks. This
brings us to rank these materials with respect to only one
method. It has been argued [1] that ELECTRE II may
yield more visible solutions compared to ELECTRE I
since net concordances and discordances are accountedfor separately. In this case, materials with a higher net
concordance and lower discordance are preferred.
3.2.1. Final decisions
Both Materials 4 and 5 have an almost stable rank,
with and without the criterion of cost, in all methods.
This encourages us to select Material 4 as the best choice
and shows that in the unstable market, these materialscan be used for mass production.
The materials whose mechanical and thermal proper-
ties as well as cost are far from these solutions are
ranked as the last choices. In addition, the materials
which are selected as the best choices by the ELECTRE
I–II models are in agreement with the Cambridge Engi-
neering Selector (CES) databases which contain infor-
mation about the applicability of these materials forthe studied case in this paper. This confirms the applica-
bility and validity of the proposed models.
4. Summary and conclusions
In this paper, ELECTRE models for material selec-
tion problems were shown to be a suitable and efficient
tool. A decision matrix is introduced for the selection of
the appropriate materials for the thermally loaded con-
ductor, based on the design criteria and candidate mate-
rials. The weighted coefficients are obtained for everyattribute, using the entropy method. The decision matrix
and weighted coefficients are taken as the input for the
ELECTRE model. This is done both with and without
considering the criterion of cost. These models list can-
didate materials from best to worst, taking into account
all the material selection criteria. The results show good
agreement with available data in CES databases.
Acknowledgments
The authors acknowledge Prof. Sylvain Turenne,
Ecole Polytechnique of Montreal, for assistance in pro-
viding data as well as his valuable advice. The authors
are very thankful to Prof. Jose Figueria and Prof. Luis
Dias, University of Coimbra and Dr. Vincent Mous-seau, LAMSADE, University of Paris Dauphine for
their valuable helps and suggestions for revising this pa-
per. The comments of the referees are gratefully
acknowledged.
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