9
A material selection model based on the concept of multiple attribute decision making A. Shanian, O. Savadogo * Laboratoire de nouveaux mate ´riaux pour les syste `mes e ´lectrochimiques, E ´ cole Polytechnique de Montre ´al, P.O. Box 6079, Centre-ville, Montre ´al, Que ´bec, 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. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: ELECTRE; Modeling; Entropy; Rank; Production; Cost; Material selection Re ´sume ´ Le remplacement et le choix des mate ´riaux pour diffe ´rentes applications technologiques est une pratique tre `s populaire de nos jours. On doit noter quÕen choisissant le bon mate ´riau, il nÕy a pas quÕun seul crite `re de choix bien de ´fini. Les concepteurs et les inge ´nieurs doivent tenir compte dÕun grand nombre de crite `res de performance. Les experts utilisent habituellement la me ´thode dÕes- sai et dÕerreur ou se basent sur des expe ´riences pre ´ce ´dentes. Dans cet article, une nouvelle approche a e ´te ´ effectue ´e sur lÕutilization du mode `le dÕELECTRE (ELimination Et Choix Traduisant la REalite ´) pour le choix du mate ´riau. En produisant une matrice de de ´ci- sion des mate ´riaux et une analyse de sensibilite ´ des crite `res, ELECTRE a e ´te ´ applique ´e pour obtenir un choix de mate ´riau plus pre ´cis pour une application particulie `re, y compris le rang logique des mate ´riaux conside ´re ´s. Une liste de tous les mate ´riaux approprie ´s, du meilleur au moins performant, peut e ˆtre obtenue en tenant compte de tous les indices de performance et e ´galement du cou ˆt de pro- duction. Ce travail de ´montre quÕELECTRE peut e ˆtre utilise ´e avec succe `s lors de la se ´lection dÕun mate ´riau approprie ´ pour la fab- rication dÕun conducteur thermique en charge. Les re ´sultats obtenus avec les me ´thodes utilise ´es concordent avec ceux disponibles 0261-3069/$ - see front matter Ó 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]. www.elsevier.com/locate/matdes Materials and Design 27 (2006) 329–337 Materials & Design

A material selection model based on the concept of multiple attribute decision making

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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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