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sustainability
Article
A Rough Multi-Criteria Decision-Making Approach
for Sustainable Supplier Selection under
Vague Environment
Huiyun Lu 1 , Shaojun Jiang 2 , Wenyan Song 1,3, * and Xinguo Ming 4
1
2
3
4
*
School of Economics and Management, Beihang University, Beijing 100191, China; [email protected]
School of Information Engineering, Handan University, Handan 056005, China; [email protected]
Beijing Key Laboratory of Emergency Support Simulation Technologies for City Operations,
Beihang University, Beijing 100191, China
School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China;
[email protected]
Correspondence: [email protected]; Tel.: +86-010-8231-3693
Received: 13 June 2018; Accepted: 23 July 2018; Published: 26 July 2018
Abstract: With the growing awareness of environmental and social issues, sustainable supply chain
management (SSCM) has received considerable attention both in academia and industry. Supplier
selection plays an important role in the successful implementation of sustainable supply chain
management, because it can influence the performance of SSCM. Sustainable supplier selection is a
typical multi-criteria decision-making problem involving subjectivity and vagueness. Although some
previous researches of supplier selection use fuzzy approaches to deal with vague information, it has
been criticized for requiring much priori information and inflexibility in manipulating vagueness.
Moreover, the previous methods often omit the environmental and social evaluation criteria in the
supplier selection. To manipulate these problems, a new approach based on the rough set theory
and ELECTRE (ELimination Et Choix Traduisant la REalité) is developed in this paper. The novel
approach integrates the strength of rough set theory in handling vagueness without much priori
information and the merit of ELECTRE in modeling multi-criteria decision-making problem. Finally,
a case study of sustainable supplier selection for solar air-conditioner manufacturer is provided to
demonstrate the application and potential of the approach.
Keywords: sustainability; supplier selection; vague information; rough set theory; ELECTRE
1. Introduction
Manufacturing companies today cannot ignore sustainability concerns in their business
because of increased environmental awareness and ecological pressures from markets and various
stakeholders [1–3]. Sustainable supplier selection is critical to enhance supply chain performance
and competitive advantage [4]. This is because suppliers play an important role in implementing
sustainable supply chain management (SSCM) practices and in achieving social, environmental and
economic goals [5]. In this respect, sustainable supplier selection based on the sustainability criteria
(economic, environmental and social) is a critical strategic decision for SSCM [6,7] and it requires to be
further explored methodically to help achieve sustainability of the whole supply chain.
Although many researchers explore the topic of supplier selection, the study on the sustainable
supplier selection is still in the early stage. Most studies of sustainable supplier selection have only
focused on the economic and environmental aspects of sustainability. The social aspect of sustainability
is often omitted in the decision–making for supplier selection. Besides, the problem of supplier
selection is a typical multi-criteria decision-making (MCDM) problem. The decision makers always
Sustainability 2018, 10, 2622; doi:10.3390/su10082622
www.mdpi.com/journal/sustainability
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need to make trade-offs between conflicting criteria to select the most suitable supplier. It is difficult to
obtain accurate judgments of decision makers in the process of supplier evaluation, because supplier
selection involves large amount of linguistic information and subjective expert knowledge that are
usually imprecise, vague or even inconsistent. To deal with this problem, fuzzy methods are often
used to select suppliers. However, the fuzzy methods need much priori information (e.g., pre-set
fuzzy membership function) which may increase the workload of decision makers [8,9]. The previous
approaches also lack a flexible mechanism to deal with the subjective evaluations of experts [10,11].
Therefore, to manipulate the above problems in sustainable supplier selection, this paper proposes
a novel integrated group decision method based on the ELECTRE (ELimination Et Choix Traduisant
la REalité) approach and rough set theory in vague environments. Different with methods based on
the compensating accumulation principle (e.g., TOPSIS(Technique for Order Preference by Similarity
to an Ideal Solution)), the ELECTRE method is based on a precedence relation and it can meet
different evaluation requirements by defining undifferentiated threshold, strict superior threshold
and rejection threshold and thus, it has stronger flexibility in decision–making of supplier selection.
Furthermore, the rough number originated from the rough set theory can flexibly reflect the uncertainty
in decision–making process of supplier selection and it does not require much priori information.
In this respect, the proposed novel approach integrates the merit of ELECTRE in modeling multi-criteria
decision-making problem and the strength of rough set theory in handling vagueness without much
priori information.
The paper is organized as follows: Section 2 presents a literature review of supplier selection,
ELECTRE method and rough set. Section 3 develops an integrated rough ELECTRE method for
sustainable supplier selection. In the Section 4, a case study of sustainable supplier selection for solar
air-conditioner manufacturer is used to validate the feasibility and effectiveness of the method and a
comparative analysis is also conducted in this section. In Section 5, conclusions and future research
directions are presented.
2. Literature Review
2.1. Evaluation Criteria for Sustainable Supplier Selection
Supplier selection decisions are important for most of manufacturing firms, because a right
supplier can effectively improve the economic benefit of the manufacturing firm [12,13]. In the past,
economic criteria are usually used for supplier selection. The environment and social criteria are
often overlooked. However, with the development of sustainable supply chain management (SSCM),
both the researchers and practitioners are paying more attention to environment criteria and social
criteria in supplier selection [14]. They find it is important to incorporating the social and environment
criteria into the supplier selection process [15,16]. This paper summarizes the sustainable supplier
selection criteria from the economic, environment and social aspects. The details of the recognized
sustainable supplier selection criteria with their sources and descriptions are summarized in Table 1.
Table 1. Sustainable supplier selection criteria.
Sustainable Supplier
Selection Criteria
Descriptions
Economic criteria
Quality [17,18]
Product quality and reliability level guaranteed by supplier.
Response [5]
The ability for timely response, completing orders on time and reliable delivery.
Cost [19]
Purchasing cost, holding cost, ordering cost and supplier’s bidding price of
the product.
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Table 1. Cont.
Sustainable Supplier
Selection Criteria
Descriptions
Environmental criteria
Environmental
management system
(EMS) [20,21]
A set of systematic processes and practices reducing environmental impacts.
Carbon emission &
resource
consumption [22,23]
Greenhouse gas emissions in producing, transporting, using and recycling the
product and the resource (e.g., energy, power and water) consumption of
the company.
Design for the
environment [14,24]
Design reducing the overall impact of a product, process or service on human
health and environment.
Green image [17]
The image of company in the green aspect, which can be improved by adopting
environmental friendly products or implementing ‘green’ program. It can affect
the purchasing trend of customers, market share and the relationship
with stakeholders.
Social criteria
Product liability [25]
Being responsible for customer health and safety, providing products and
services with high quality and advertising based on real information.
Employee right and
welfare [26,27]
Treating employee with dignity and respect and maintaining a culture of security,
nondiscrimination and equality. Paying to employee shall comply with all
applicable wage laws.
Social commitment [27]
Involving in local community, education, job creation, healthcare and
social investment.
2.2. The Methods of Sustainable Supplier Selection
Selecting the right suppliers to set up optimal supplier networks can help to reduce purchasing
costs and increase the efficiency of the procurement logistics process [28]. Supplier selection is a
multi-criteria decision-making problem. There are some papers concerning sustainable (or green)
suppliers. Dai and Blackhurst (2012) integrate Analytical Hierarchy Process (AHP) with Quality
Function Deployment (QFD) for sustainable supplier selection [18]. The approach consists of four
stages, that is, linking customer requirements with the firm’s sustainability strategy, determining the
sustainable purchasing competitive priority, determining evaluation criteria of sustainable supplier
and evaluating the sustainable suppliers. Hsu and Hu (2009) develop a method for selecting suppliers
with emphasis on issues of hazardous substance management based on Analytic Network Process
(ANP) [29]. Liu and Hai (2005) provide a method called voting analytic hierarchy process for supplier
selection [30]. Although AHP/ANP methods are more popular in the field of the supplier selection,
they are always used to determine the relative importance weightings of criteria and sub-factors merely.
They need to be integrated with other decision–making techniques. Besides, due to the number of
pairwise comparisons that need to be made, the number of supplier selections is practically limited in
the AHP/ANP-based supplier selection methods. Moreover, the conventional AHP/ANP methods do
not consider the vagueness of decision–making information.
To manipulate the increasing number of the suppliers, data envelopment analysis (DEA) is a
prevalent approach used in supplier selection. This is because DEA can easily handle huge number of
suppliers with little managerial input and output required. Kuo et al. (2012) present a green supplier
selection method using an analysis network process as well as data envelopment analysis (DEA) [31].
ANP which is able to consider the interdependency between criteria releases the constraint of DEA that
the users cannot set up criteria weight preferences. Wu and Blackhurst (2009) propose an augmented
DEA approach for supplier evaluation and selection [32]. Sevkli et al. (2007) develop a new supplier
selection method by embedding the DEA approach into AHP methodology [33]. They conclude that
Sustainability 2018, 10, 2622
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the integrated method outperforms the conventional AHP method for supplier selection. However,
DEA-based supplier selection methods have some drawbacks. The practitioners may be confused with
input and output criteria. Besides, DEA is a linear programming to measure the relative efficiencies of
homogenous decision–making units (DMUs). An efficient supplier generating more outputs while
requiring less input may be not an effective supplier. Furthermore, the conventional DEA also does
not consider the subjectivity and vagueness in the decision–making process.
Beside the multi-criteria decision–making method, some researchers use heuristic optimization
approaches to select proper suppliers. Basnet and Leung (2005) develop an incapacitated mixed
linear integer programming which minimizes the aggregate purchasing, ordering and holding costs
subject to demand satisfaction [34]. They solve the problem with an enumerative search algorithm
and a heuristic procedure. Veres et al. (2017) propose a heuristic method for optimizing supply
chain including intelligent transportation systems (ITS) based vehicles for transportation operations
problems [35]. To solve the multi-product multi-period inventory lot sizing with supplier selection
problem, Cárdenas-Barrón et al. (2015) propose a heuristic algorithm based on reduce and optimize
approach (ROA) and a new valid inequality [36]. Unfortunately, the heuristic optimization approaches
omit the vagueness and subjectivity in the decision–making, which may lead to inaccurate results of
supplier selection.
In order to deal with the imprecise or vague nature of linguistic assessment in evaluation and
selection of suppliers, fuzzy set theory is introduced into the conventional approaches. Considering
time pressure and lack of expertise in sustainable supplier selection, Büyüközkan and Çifçi (2011)
developed a method based on fuzzy analytic network process within group decision-making schema
under incomplete preference relations [37]. To manipulate the subjectivity of decision makers’
evaluations, Amindoust et al. (2012) develop a new ranking method on the basis of fuzzy inference
system (FIS) for sustainable supplier selection problem [6]. Azadnia et al. (2015) developed an
integrated method based on rule-based weighted fuzzy approach [38], fuzzy analytical hierarchy
process and multi-objective mathematical programming for sustainable supplier selection and order
allocation. Grisi et al. (2010) propose a fuzzy AHP method for green supplier selection using a
seven-step approach [39]. Fuzzy logic is used to overcome uncertainty caused by human qualitative
judgments. ELECTRE (ELimination Et Choix Traduisant la REalité) methods are able to make a
successful assessment of each alternative based on knowledge of the concordance and discordance
sets for all pairs of alternatives. They are often used to select right suppliers [40]. Thus, Sevkli (2010)
proposes a fuzzy ELECTRE for supplier selection [41]. Although the fuzzy methods can deal with
the imprecise or vague nature of linguistic assessment, it requires priori information (e.g., pre-set
membership function). Moreover, the fuzzy methods always convert linguistic variables into fuzzy
numbers with fixed intervals. Therefore, computation results usually do not exactly match initial
linguistic terms, which easily cause loss of information and lack of precision in the final results.
Although these methods have brought great insights to supplier selection literature, most of them
lack flexible mechanisms to handle the subjectivity and the vagueness of decision makers’ assessments.
Although some fuzzy methods of supplier selection (e.g., fuzzy ELECTRE) consider the vagueness in
decision–making information, they require much priori information (e.g., pre-set fuzzy membership
function) which consumes much time and effort of managers. Moreover, the previous fuzzy approaches
use fuzzy number with fixed interval to indicate the uncertainty, which cannot identify the changes in
decision makers’ judgments. For those reasons, there is a clear need for a new formal decision support
methodology for the sustainable supplier selection under vague environment.
3. The Proposed Method
The main objective of this paper is to propose an integrated method for sustainable supplier
selection based on rough set theory and ELECTRE. Besides, vagueness manipulation is also considered
in the proposed approach. A flowchart of the proposed approach is shown in Figure 1.
Sustainability 2018, 10, 2622
Sustainability 2018, 10, x FOR PEER REVIEW
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Figure 1.
1. The
Theframework
framework of
of rough
rough ELimination
ELimination Et
Et Choix
Choix Traduisant
Traduisant la REalité (ELECTRE).
Figure
3.1. Determine
Determine the
the Supplier
Supplier Evaluation
Evaluation Criteria and Their Weights
Step 1: determine the evaluation criteria of sustainable suppliers
Step 1: determine the evaluation criteria of sustainable suppliers
First of all, a panel of expert who are knowledgeable about supplier selection is established. The
First of all, a panel of expert who are knowledgeable about supplier selection is established.
D1,DD12, D
,…,
DkD)k )who
The group
has
k decision-makers
whoare
areresponsible
responsiblefor
for determining
determining and
and the
group
has k
decision-makers
(i.e.,(i.e.,
2 , …,
ranking each criterion (i.e., C1 , C2 , …, Ck ). For the sustainable supplier selection, three aspects we
C2,…,are
Ckeconomic
ranking
eachinto
criterion
(i.e., C1,They
). For thecriteria,
sustainable
supplier selection,
three
aspects
we
should take
consideration.
environmental
criteria and
social
criteria.
should take into consideration. They are economic criteria, environmental criteria and social criteria.
Step 2: determine the weights for the evaluation criteria of sustainable suppliers
Step 2: determine the weights for the evaluation criteria of sustainable suppliers
Experts have their own individual experience and knowledge. Therefore, they may have different
Experts have their own individual experience and knowledge. Therefore, they may have
cognitive vagueness for alternatives and criteria. Let us assume a judgment set P = { p1 , p2 , · · · , ph }
different cognitive vagueness for alternatives and criteria. Let us assume a judgment set
with h ordered judgments, in the manner of p1 ≤ p2 ≤ · · · ≤ ph . Let pi be a random judgment in the
p1 ≤ approximation
p2 ≤≤ ph . Let
P =P and
p1, pd2is,
, ph as
with
orderedof
judgments,
of lower
a
set
defined
thehdistance
P, where din=the
phmanner
− p1 . The
Apr (ppii ) be
and
the upper approximation Apr ( pi ) of the judgment pi can be identified as follows.
random judgment in the set P and d is defined as the distance of P , where d = ph − p1 . The
Lower approximation set:
{
}
lower approximation
Apr( pi ) and the upper approximation Apr
( pi ) of the judgment pi can
Apr ( pi ) = ∪ p j ∈ P p j ≤ pi , pi − p j ≤ d
be identified as follows
Lower
Upper approximation
approximation set:
set:
{
≤d
Apr
( (ppi ))==∪∪pp j ∈
j ≤p p
j)d
Apr
∈P
P |pp ≥
, i ,p( p−i −p p ≤
i
Upper approximation set:
j
j
i
i
h
RN ( pi ) = piL , pU
i
{
j
i
Apr ( pi ) = ∪ p j ∈ P | p j ≥ pi , ( p j − pi ) ≤ d
(1)
}
}
(1)
(2)
(3)
(2)
Sustainability 2018, 10, 2622
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q
Where piL = m ∏ xij
q
n
pU
=
∏ yij
i
(4)
(5)
where xij and yij are the elements of the lower approximation set Apr ( pi ) and the upper approximation
set Apr ( pi ) of pi respectively and m and n are the number of elements in the two sets respectively.
For different criteria, experts might give different weights. Use wkj indicate the weight of jth
criterion with kth expert.
With the Formulas (1)–(5)
n
o
n
d j = MAX wm
(6)
j − wj
o
n
m
n
≤ dj
(7)
= ∪ wnj ∈ P wnj ≤ wm
Apr wm
j , wj − wj
j
o
n
n
m
≤ dj
(8)
Apr wm
= ∪ wnj ∈ P wnj ≥ wm
j , wj − wj
j
q
Lim wkj = m ∏ x j
(9)
q
Lim wkj = n ∏ y j
(10)
where x j and y j are the elements of the lower approximation set Apr (wkj ) and the upper approximation
set Apr (wkj ) of wkj respectively and m and n are the number of elements in the two sets respectively.
h
i h
i
kU
RN wkj = Lim wkj , Lim wkj
= wkL
,
w
j
j
s
w jL =
s
(11)
s
∏ wkL
j
(12)
k =1
s
s
s
wU
∏ wkU
j =
j
(13)
k =1
h
i
We could get the weight of each criterion w j = w jL , wU
j .
3.2. Evaluate the Sustainable Suppliers with the Proposed Rough ELECTRE
Step 1: Construct the rough decision matrix
Apart from the decision for the weight of criteria, the experts should give the assessment of the
alternatives with consideration of all the criteria. Let’s use rijk to represent the kth expert scores on jth
criterion in ith alternative. The following is the scoring matrix. Aggregate all the scoring matrix.
k
r11
k
r21
Rk =
..
.
k
rm1
k
r12
k
r22
..
.
k
rm2
···
···
..
.
···
k
r1n
k
r2n
..
.
k
rmn
(14)
rf
rf
· · · rf
11
12
1n
f f
r21 r22 · · · rf
2n
e
R= .
..
..
..
.
.
.
..
rf
nm
m2 · · · rg
m1 rf
n
o
reij = rij1 , rij2 , · · · , rijh
(15)
(16)
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Determine the rough matrix with expert ratings.
d = max rijm − rijn
(17)
o
n
Apr rijm = ∪ rijn ∈ P rijn ≤ rijm , rijm − rijn ≤ d
o
n
Apr rijm = ∪ rijn ∈ P rijn ≥ rijm , rijn − rijm ≤ d
(18)
q
Lim rijk = m
∏ xij
(19)
(20)
q
Lim rijk = n ∏ yij
(21)
where xij and yij are the elements of the lower approximation set Apr (rijk ) and the upper approximation
set Apr (rijk ) of rijk respectively and m and n are the number of elements in the two sets respectively.
i
h
RN rijk = Lim, Lim = rijkL , rijkU
(22)
i h
i
h
io
nh
RN reij = rij1L , rij1U , rij2L , rij2U , · · · , rijsL , rijsU
i
h
RN reij = rijL , rijU
(23)
s
rijL =
s
s
s
∏
L , rU
r11
11
L , rU
r21
21
R=
..
.
L U
rm1 , rm1
s
∏ rijkU
rijkL , rijU =
s
L U
r12 , r12
L U
r22 , r22
···
k =1
..
.
L U
rm2 , rm2
(24)
(25)
k =1
···
..
.
···
L U
r1n , r1n
L U
r2n , r2n
..
.
L U
rmn , rmn
(26)
Then, we normalize the rough decision matrix with the weight of criteria.
h
i h
i
L U
sij = rij · w j = rijL w jL , rijU wU
ij = sij , sij
(27)
” L U#
h
i
sij sij
tij =
,
= tijL , tU
ij
Cj Cj
n o
Where Cj = MAX sU
ij
L , tU
t11
11
L , tU
t21
21
T=
..
.
L U
tm1 , tm1
L U
t12 , t12
L U
t22 , t22
···
..
.
L U
tm2 , tm2
..
···
.
···
L U
t1n , t1n
L U
t2n , t2n
..
.
L U
tmn , tmn
(28)
(29)
(30)
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Step 2: Construct the rough concordance matrix and discordance matrix
In this step, we construct some field for the comparison among all the alternatives. We compare
different alternatives in two aspects. One is the concordance and the other is the discordance. Construct
the concordance and discordance matrices.
CS pq = Fj t pj ≥ tqj
(31)
DS pq = Fj t pj < tqj
(32)
CS pq represents the areas that alternative p is better than alternative q and DS pq represents the
areas that alternative p is worse than alternative q.
c pq =
∑
wj
(33)
Fj ∈CS pq
max Fj ∈ DS pq d t pj , tqj
d pq =
max Fj ∈ J d t pj , tqj
L U
L U
−
c12 , c12
· · · c1m
, c1m
L , cU
L , cU
−
·
·
·
c
c21
2m 2m
21
C=
..
..
..
..
.
.
.
.
L U L U
···
−
cm1 , cm1
cm2 , cm2
− d12 · · · d1m
− · · · d2m
d21
D= .
..
..
..
.
.
.
.
.
dm1 dm2 · · · −
(34)
(35)
(36)
By means of the calculation, we could get the rough concordance matrix C and discordance
matrix D.
Step 3: Determine the general Boolean matrix
After we get the concordance matrix and discordance matrix, we should determine the threshold
value. Using it to transform the matrix into Boolean matrix. First, we calculate the mean of the all
factors in matrix C and matrix D.
m
m
∑
∑ c pq
c=
p=1,p6=q q=1,q6= p
m
∑
d=
(37)
m ( m − 1)
m
∑
p=1,p6=q q=1,q6= p
d pq
m ( m − 1)
(38)
Compare the factors in matrix C with c and the factors in matrix D with d. According the result of
the comparison, we get the concordance Boolean matrix F and discordance Boolean matrix G.
(
f pq =
(
g pq =
1
i f : c pq ≥ c
0
i f : c pq < c
1
i f : d pq ≤ d
0
i f : d pq > d
(39)
(40)
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F = f pq m×m , G = g pq m×m
(41)
Then we could construct the general Boolean matrix H.
h pq = f pq · g pq
(42)
H = h pq m×m
(43)
According to the above calculations, we could get the general Boolean matrix. It is a basis for the
ranking of the alternatives. If h pq = 1, that means alternative p is better than alternative q.
Step 4: Calculate the pure concordance index and discordance index
By the Boolean general matrix, we could get part relations between all alternatives. Since if
h pq = 1, we know that alternative p is better than alternative q. But if h pq = 0 and we could not infer
the relationship of alternative p and alternative q from other alternatives, then we do not know which
is better. In order to get a rank of all the alternatives, we bring into pure concordance index cˆi and
discordance index d̂i .
Before calculating the pure index, we should transform rough interval into definite number.
Song et al. (2017) has proposed this method. We use ∆−1 represents the calculation of changing rough
interval into definite number [14].
The calculation includes the following procedures.
(1) Normalization
zei L =
zei U =
ziL − minziL /∆max
min
(44)
i
L
zU
i − minzi
i
/∆max
min
(45)
U
L
∆max
min = maxzi − minzi
i
(46)
i
L
where ziL and zU
i are the lower limit and the upper limit of the rough number zei respectively; zei and
zei U are the normalized form of ziL and zU
i respectively.
(2) Determine the total normalized definite value by
zei L × 1 − zei L + zei U × zei U
βi =
(47)
1 − zei L + zei U
(3) Compute the final definite value form zei der for zei by
zei der = minziL + β i ∆max
min
(48)
i
Therefore, we can use this method to calculate the concordance index and discordance index.
m
cˆi =
∑
q=1,q6=i
∆−1 cf
iq −
m
d̂i =
∑
q=1,q6=i
m
∑
p=1,p6=i
∆−1 cf
pi
(49)
m
diq −
∑
d pi
(50)
p=1,p6=i
Step 5: Determine the final ranking
According to the cˆi , we can get a priority in concordance. The bigger value of cˆi the higher place
the alternative would get. We use R1i for the ranking in concordance. The same we can get the priority
Sustainability 2018, 10, 2622
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in discordance by d̂i . But on the contrary, the smaller value of d̂i the higher place the alternative would
get. We use R2i for the ranking in discordance. The final ranking is calculated as follows:
Ri =
R1i + R2i
2
(51)
Ri is the final rank of all the alternatives.
4. Case Study
In this section, in order to validate the applicability and effectiveness of the proposed method, we
use an example to illustrate. We assume that there is a manufacturing company. For the purpose of
choosing a good supplier, they set up a panel of 4 experts. The experts come from various departments
including purchasing, quality and production and planning who are involved in the supplier selection
process. And there are 8 suppliers for selection.
4.1. Implementation
4.1.1. Determine the Supplier Evaluation Criteria and Their Weights
Step 1: determine the evaluation criteria of sustainable suppliers
First of all, the experts make a decision of the criteria. In addition to economic criteria,
environmental criteria and social criteria should also be considered for the sustainable supplier
selection. These criteria consist of three parts, we use C1~10 to represent these ten criteria. They are
Economic criteria including quality (C1), response (C2) and cost (C3); Environmental criteria including
environmental management system (C4), carbon emission & resource consumption (C5), design for
the environment (C6), Green image (C7); Social criteria including product liability (C8), employee right
and welfare (C9), social commitment (C10). The detailed introduction is shown in Table 1. We use
A1~8 to represent alternatives, E1~4 to represent experts.
Step 2: determine the weights for the evaluation criteria of sustainable suppliers
After the decision of criteria, experts should evaluate the weight of each criterion. The experts
give their evaluation to the criteria in the Table 2. Firstly, we convert the grades which experts give to
criteria into rough number. Take criterion C1 for example.
Table 2. The grade of each criterion.
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
E1
E2
E3
E4
4
3
6
5
6
6
4
4
6
7
5
6
7
5
4
6
4
3
6
4
4
4
5
5
6
5
3
2
6
5
6
4
7
6
5
5
5
4
7
4
According to the Equations (6)–(13) in Section 3,
d1 = 2
Apr w11 = {4, 4}, Apr w11 = {4, 5, 4, 6}
Apr w12 = {4, 5, 4}, Apr w12 = {5, 6}
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Apr w13 = {4, 4}, Apr w13 = {4, 5, 4, 6}
Apr w14 = {4, 5, 4, 6}, Apr w14 = {6}
√
√
Lim w11 = 2 4 × 4 = 4, Lim w11 = 4 4 × 5 × 4 × 6 = 4.68
√
√
Lim w12 = 3 4 × 5 × 4 = 4.31, Lim w12 = 2 5 × 6 = 5.48
√
√
Lim w13 = 2 4 × 4 = 4, Lim w13 = 4 4 × 5 × 4 × 6 = 4.68
√
Lim w14 = 4 4 × 5 × 4 × 6 = 4.68, Lim w14 = 6
√
√
w1L = 4 4 × 4.31 × 4 × 4.68 = 4.24, w1U = 4 4.68 × 5.48 × 4.68 × 6 = 5.18
The same as the other criteria, following the same procedure, we can get the importance degree of
all the criteria in Table 3.
Table 3. The importance of all the criteria.
Rough Importance
W1
W2
W3
W4
W5
W6
W7
W8
W9
W10
[4.24, 5.18]
[3.57, 4.77]
[5.69, 6.70]
[5.06, 5.42]
[4.68, 5.70]
[5.23, 5.73]
[3.53, 4.37]
[2.63, 3.67]
[6.06, 6.42]
[4.28, 5.60]
4.1.2. Evaluate the Sustainable Suppliers with the Proposed Rough ELECTRE
Step 1: Construct the rough decision matrix
Different expert might hold different view for alternatives and criteria because of their personal
experience and knowledge. And the true information is just contained in the cognitive vagueness.
According to the evaluation towards the alternatives from the experts, we could get the rough number
of each alternative. We take the data for criterion 1 in Table 4 for example.
Table 4. The evaluation for alternative under the criterion 1.
C1
A1
A2
A3
A4
A5
A6
A7
A8
E1
E2
E3
E4
6
4
5
4
3
6
7
5
4
3
4
5
5
6
6
4
6
4
6
5
3
4
5
3
5
2
3
5
4
6
7
5
According to the Equations (17)–(26), we use x cab for the cth expert’s evaluation towards alternative
b in criterion a. We can get the rough matrix in Table 5.
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Table 5. The rough matrix.
A1
A2
A3
A4
A5
A6
A7
A8
C1
C2
C3
…
C10
[4.68, 5.70]
[2.63, 3.67]
[3.65, 5.15]
[4.53, 4.93]
[3.23, 4.16]
[5.02, 5.85]
[5.69, 6.70]
[3.66, 4.69]
[5.23, 5.73]
[3.66, 4.69]
[2.22, 3.13]
[5.54, 5.93]
[4.54, 5.38]
[5.69, 6.70]
[4.68, 5.70]
[4.06, 4.41]
[4.24, 5.18]
[5.11, 5.79]
[4.68, 5.70]
[3.23, 4.16]
[4.24, 5.18]
[6.06, 6.42]
[5.23, 5.73]
[3.53, 4.37]
…
…
…
…
…
…
…
…
[3.66, 4.69]
[4.68, 5.70]
[4.67, 6.17]
[4.68, 5.70]
[3.96, 5.29]
[4.68, 5.70]
[5.02, 5.85]
[4.24, 5.18]
Note: not all of the data are provided in Table 5 due to the space limitation.
Then, we normalize the rough matrix. According to the Equations (27)–(30). We can get the result
in Table 6.
Table 6. The normalized weighted decision matrix.
A1
A2
A3
A4
A5
A6
A7
A8
C1
C2
C3
…
C10
[0.57, 0.85]
[0.32, 0.55]
[0.45, 0.77]
[0.55, 0.74]
[0.39, 0.62]
[0.61, 0.87]
[0.69, 1.00]
[0.45, 0.70]
[0.58, 0.86]
[0.41, 0.70]
[0.25, 0.47]
[0.62, 0.89]
[0.51, 0.80]
[0.64, 1.00]
[0.52, 0.85]
[0.45, 0.66]
[0.56, 0.81]
[0.68, 0.90]
[0.62, 0.89]
[0.43, 0.65]
[0.56, 0.81]
[0.80, 1.00]
[0.69, 0.89]
[0.47, 0.68]
…
…
…
…
…
…
…
…
[0.45, 0.76]
[0.58, 0.92]
[0.58, 1.00]
[0.58, 0.92]
[0.49, 0.86]
[0.58, 0.92]
[0.62, 0.95]
[0.53, 0.84]
Step 2: Construct the rough concordance matrix and discordance matrix
In this step, we construct the concordance and discordance matrices according to the normalized
rough decision matrix. For the construct of the concordance matrix, we take alternative1 and alternative
2 for example. At the first, we should find in which criterion A1 performs better than A2, that means
the score in certain criterion, A1 is higher than A2.
According to the Table 6, we could find in criterion 1, 2, 9, A1 performs better than A2. Add up all
these weights of the criteria. We could get the value of c12 = [13.87, 16.37] in the concordance matrix.
And we can get the concordance matrix in Table 7 by repeat these procedures.
Table 7. The concordance matrix.
A1
A2
A3
A4
A5
A6
A7
A8
A1
A2
A3
…
A8
[31.11, 37.19]
[33.63, 39.23]
[7.85, 10.37]
[7.81, 9.97]
[29.01, 35.71]
[19.27, 22.89]
[17.55, 21.08]
[13.87, 16.37]
[19.81, 22.93]
[13.87, 16.37]
[11.35, 14.32]
[22.20, 26.74]
[23.84, 28.67]
[22.09, 26.44]
[11.35, 14.32]
[25.17, 30.63]
[7.81, 9.95]
[7.11, 9.14]
[24.73, 30.11]
[17.04, 21.03]
[16.85, 20.25]
…
…
…
…
…
…
…
…
[27.43, 32.48]
[22.89, 27.12]
[28.13, 33.30]
[12.09, 15.55]
[20.67, 25.25]
[25.47, 31.34]
[17.78, 22.25]
–
For the construct of the discordance matrix. First of all, we find the criterion which A2 is better
than A1. And we could find that they are criterion 3, 4, 5, 6, 7, 8, 10. Then we find the biggest distance
in these criteria. Using it divide the biggest distance between A1 and A2. We can get the value of
d12 = 1. Repeating these procedures and we can get the discordance matrix in Table 8.
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Table 8. The discordance matrix.
A1
A2
A3
A4
A5
A6
A7
A8
A1
A2
A3
A4
A5
A6
A7
A8
0.85
1.00
1.00
1.00
0.46
1.00
0.93
1.00
0.64
1.00
1.00
0.37
0.71
1.00
0.53
1.00
0.73
1.00
0.43
0.88
0.91
0.90
0.49
1.00
1.00
0.00
0.37
0.68
0.20
0.27
0.64
0.65
0.22
0.55
0.32
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.88
1.00
1.00
1.00
1.00
0.32
1.00
1.00
0.97
1.00
1.00
1.00
0.27
0.82
–
Step 3: Determine the general Boolean matrix
Based on concordance and discordance matrix, we construct the concordance Boolean and
discordance Boolean matrices. Calculate the concordance index and discorda
