Management Question

‫المملكة العربية السعودية‬
‫وزارة التعليم‬
‫الجامعة السعودية اإللكترونية‬

Kingdom of Saudi Arabia
Ministry of Education
Saudi Electronic University

College of Administrative and Financial Sciences

Assignment-3
MGT425-Spreadsheet Decision Modeling
Due Date: 30/11/2024 (End of Week-13) @ 23:59
Course Name: Spreadsheet Decision
Modeling
Course Code: MGT425

Student’s Name:

Semester: First

CRN:

Student’s ID Number:

Academic Year: 2024-2025 (1446 H) – 1st Term

For Instructor’s Use only
Instructor’s Name:
Students’ Grade: Marks Obtained/Out of 10 Level of Marks: High/Middle/Low
Instructions – PLEASE READ THEM CAREFULLY
• The Assignment must be submitted on Blackboard (WORD format only)
via allocated folder.
• Assignments submitted through email will not be accepted.
• Students are advised to make their work clear and well presented; marks
may be reduced for poor presentation. This includes filling your information
on the cover page.
• Students must mention question number clearly in their answer.
• Late submission will NOT be accepted.

• Avoid plagiarism, the work should be in your own words, copying from
students or other resources without proper referencing will result in ZERO
marks. No exceptions.
• All answered must be typed using Times New Roman (size 12, doublespaced) font. No pictures containing text will be accepted and will be
considered plagiarism).
• Submissions without this cover page will NOT be accepted.

Course Learning Outcomes-Covered
Aligned (PLOs)
MGT.K.1
(1.1)
MGT.K.3
(1.2)
MGT.S.1
(2.1)
MGT.V.1
(3.1)

Course Learning Outcomes (CLOs)

Question

Find some structured ways of dealing with complex managerial
decision problems.
Explain simple decision models and management science ideas
that provide powerful and (often surprising) qualitative insight
about large spectrum of managerial problems.
Demonstrate the tools for deciding when and which decision
models to use for specific problems.
Build an understanding of the kind of problems that is tackled
using Spreadsheet Modeling and decision analysis.

Question- 2.
Question- 1, 5

Question-4
Question-3

Assignment Instructions:
Assignment Questions: (Marks 10)
PART-A:
Decision Analysis (Critical Thinking)
Question 1: “A decision tree is a specialized model for recognizing the role of uncertainties
in a decision-making situation” In context of this statement explain the Principles for
Building and Analysing the Decision Trees (250-300 Words) (3-Marks)

PART-B:
• Log in to Saudi Digital Library (SDL) via University’s website
• On first page of SDL, choose “English Databases”
• From the list find and click on EBSCO database.
• In the Search Bar of EBSCO find the following article:
Title: A Rough Multi-Criteria Decision-Making Approach for Sustainable Supplier
Selection under Vague Environment: A Case Study.
Author: Huiyun Lu , Shaojun Jiang , Wenyan Song, Xinguo Ming

Read the above case study and answer the following Questions:
Question 2: Explain the decision-making approach discussed in this case study (250-300
words) (2-Marks).
Question 3: Why supplier selection is a typical multi-criteria decision-making process
involving subjectivity and vagueness? (250-300 words) (2-Marks).
Question 4: Discuss the Sustainable supplier selection that is required for manufacturing
companies. (250-300 words) (2-Marks).
Question 5: Write minimum four references in APA style in support of your answers (1Mark).

Answers:
1
2
3
4
5

Restricted – ‫مقيد‬

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
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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

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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.

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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)

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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 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}

Sustainability 2018, 10, 2622

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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 discordance index. Follow
the Equations (37)–(41).
m

∑

cL =

m

∑

p=1,p6=q q=1,q6= p

m

c Lpq

m ( m − 1)

= 22.49, cU =
m

∑

d=

m

∑

m

∑

p=1,p6=q q=1,q6= p

∑

p=1,p6=q q=1,q6= p

cU
pq

= 26.78

m ( m − 1)

d pq

= 0.79

m ( m − 1)

And we can get the concordance Boolean and discordance Boolean matrices in Tables 9 and 10.
Table 9. The concordance Boolean matrix.

A1
A2
A3
A4
A5
A6
A7
A8

A1

A2

A3

A4

A5

A6

A7

A8

1
1
0
0
1
0
0

0
0
0
0
0
1
0

0
1
0
0
1
0
0

1
1
1
1
1
1
1

1
1
1
0
1
1
1

0
1
0
0
0
0
0

1
0
1
0
0
1
1

1
1
1
0
0
1
0
–

Table 10. The discordance Boolean matrix.

A1
A2
A3
A4
A5
A6
A7
A8

A1

A2

A3

A4

A5

A6

A7

A8

0
0
0
0
1
0
0

0
1
0
0
1
1
0

1
0
1
0
1
0
0

0
1
0
0
1
1
1

1
1
1
1
1
1
1

0
0
0
0
0
0
0

0
0
0
0
0
1
0

0
0
0
0
0
1
0
–

According to the Equation (42), we could get the general matrix in Table 11.

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Table 11. The general matrix.

A1
A2
A3
A4
A5
A6
A7
A8

A1

A2

A3

A4

A5

A6

A7

A8

0
0
0
0
1
0
0

0
0
0
0
0
1
0

0
0
0
0
1
0
0

0
1
0
0
1
1
1

1
1
1
0
1
1
1

0
0
0
0
0
0
0

0
0
0
0
0
1
0

0
0
0
0
0
1
0
–

And following
the PEER
general
matrix, we could draw the priority picture like Figure 2.
Sustainability
2018, 10, x FOR
REVIEW

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Figure 2. The relations of alternatives in conventional ELECTRE.

We use
use ‘>’
> A5;
A2A2
> {A4,
A5};A5};
A3 >A3
A5;
{A1,
We
‘>’ indicating
indicatingbetter,
better,then
thenwe
wecould
couldfind
findthat
thatA1
A1
> A5;
> {A4,
> A6
A5;>A6
>
A2,
A3,
A4,
A5,
A7,
A8};
A7
>
{A2,
A4,
A5};
A8
>
{A4,
A5}.
That’s
some
relation
between
all
{A1, A2, A3, A4, A5, A7, A8}; A7 > {A2, A4, A5}; A8 > {A4, A5}. That’s some relation between the
all
alternatives.
the
alternatives.
But we
we cannot
cannot have
have aa rank
rank of
of all
all the
the alternatives
alternatives just
just though
though this
this figure.
figure. Like
Like we
we do
do not
not know
know is
is
But
A4 better
better than
than A5
A5 or
or A5
A5 better
better than
than A4
A4 or
or they
they are
are the
the same.
same. So,
So, we
we bring
bring in
in the
the concept
concept of
of the
the pure
pure
A4
concordance
index
and
discordance
index.
concordance index and discordance index.
Step 4: Calculate the pure concordance index and discordance index
Step 4: Calculate the pure concordance index and discordance index
Before we calculate the pure concordance index and discordance index, we should convert the
Before we calculate the pure concordance index and discordance index, we should convert the
rough concordance matrix into definite number matrix. According to the Equations (44)–(48). We can
rough concordance matrix into definite number matrix. According to the Equations (44)–(48). We can
get the result in Table 12.
get the result in Table 12.
Table 12. The definite number concordance matrix.

A1
A2
A3
A4
A5
A6
A7
A8

A1
35.87
38.39
8.04
7.95
33.91
20.83
18.89

A2
14.50
21.58
14.50
11.78
25.40
27.62
25.12

A3
12.16
29.60
8.05
7.27
29.01
19.06
18.51

A4
40.77
33.42
41.08
27.71
46.10
29.21
35.07

A5
42.17
37.34
43.11
18.63
41.68
31.53
25.65

A6
17.01
26.20
22.63
4.35
9.09
15.08
21.43

A7
28.50
22.47
30.90
17.60
16.55
35.78
29.69

A8
31.35
25.39
32.29
12.58
23.00
29.64
19.55
–

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Table 12. The definite number concordance matrix.

A1
A2
A3
A4
A5
A6
A7
A8

A1

A2

A3

A4

A5

A6

A7

A8

35.87
38.39
8.04
7.95
33.91
20.83
18.89

14.50
21.58
14.50
11.78
25.40
27.62
25.12

12.16
29.60
8.05
7.27
29.01
19.06
18.51

40.77
33.42
41.08
27.71
46.10
29.21
35.07

42.17
37.34
43.11
18.63
41.68
31.53
25.65

17.01
26.20
22.63
4.35
9.09
15.08
21.43

28.50
22.47
30.90
17.60
16.55
35.78
29.69

31.35
25.39
32.29
12.58
23.00
29.64
19.55
–

Then we could calculate the pure concordance index and discordance index of each alternative
and the result is in Table 13.
Table 13. The pure concordance index and discordance index of each supplier.

A1
A2
A3
A4
A5
A6
A7
A8

cˆi

d̂i

22.58
69.80
106.31
−169.60
−136.74
125.73
−18.63
0.57

−0.75
−0.15
0.81
1.93
4.16
−4.92
−0.87
−0.21

Step 5: Determine the final ranking
According to the pure concordance index and discordance index, we could get the ranking of
each supplier in concordance and discordance aspects. With the Equation (51), we could get the final
ranking of all the suppliers in Table 14.
Table 14. The final ranking of all the suppliers.

A1
A2
A3
A4
A5
A6
A7
A8

R1i

R2i

Ri

4
3
2
8
7
1
6
5

3
5
6
7
8
1
2
4

2
3
3
7
7
1
3
6

From Table 14, we could see that priority is: A6 > A1 > {A2, A3, A7} > A8 > {A4, A5}.
4.2. Comparisons and Discussion
To further validate the effectiveness and strengths of the approach proposed in this paper, we make
a comparison analysis.
The comparison is conducted between the modified ELECTRE method with rough number (the
rough ELECTRE), fuzzy number (the fuzzy ELECTRE) and crisp number (the conventional ELECTRE).
The results are presented in Figure 3. From the Figure 3, we can see the rank of A2, A3 and A7 are
different with each other in the three methods. In the process of supplier selection, the top three
candidates are critical for the consideration. Different rankings will influence in the final performance
of supply chain.

Sustainability 2018, 10, 2622
Sustainability 2018, 10, x FOR PEER REVIEW
Sustainability 2018, 10, x FOR PEER REVIEW

16 of 20
17 of 21
17 of 21

Figure 3.
3. The
The rank
rank of
of different
different methods.
methods.
Figure
Figure 3. The rank of different methods.

The
The fuzzy
fuzzy methods
methods of
of supplier
supplier selection
selection (e.g.,
(e.g., methods
methods in
in [39,41])
[39,41]) often
often use
use the
the fuzzy
fuzzy number
number
with
fixed
interval
to
deal
with
the
uncertainty
in
supplier
selection,
which
will
cause
information
with
uncertainty
in in
supplier
selection,
which
willwill
cause
information
lost
with fixed
fixedinterval
intervaltotodeal
dealwith
withthe
the
uncertainty
supplier
selection,
which
cause
information
lost
in
process.
Different
the
methods,
the
approach
uses
the
in
decision–making
process.
Different
with with
the
methods,
the proposed
approach
uses the
rough
lost
in decision–making
decision–making
process.
Different
withfuzzy
the fuzzy
fuzzy
methods,
the proposed
proposed
approach
uses
the
rough
number
with
flexible
interval
to
describe
the
uncertainty
and
it
does
not
require
to
subjectively
number
with
flexible
interval
to
describe
the
uncertainty
and
it
does
not
require
to
subjectively
set
rough number with flexible interval to describe the uncertainty and it does not require to subjectively
set
the
fuzzy
membership
function
in
advance.
The
rough
number
can
flexibly
reflect
the
change
of
the
fuzzy
membership
function
in advance.
TheThe
rough
number
can can
flexibly
reflect
the change
of the
set the
fuzzy
membership
function
in advance.
rough
number
flexibly
reflect
the change
of
the
experts’
preference.
For
example,
if
one
expert
provides
the
scores
of
6,
4,
6,
5.
It
then
can
be
experts’
preference.
For example,
if one if
expert
provides
the scores
6, 4, 6,of5. 6,It 4,
then
canItbe
converted
the experts’
preference.
For example,
one expert
provides
theofscores
6, 5.
then
can be
converted
to
[4,
which
have
of
to
fuzzy intervals
ofintervals
[5, 7], [3, of
5],[5,
[5,7],
7] [3,
and
[4,[5,
6],7]
alland
of which
have
interval
of 2. interval
But
the proposed
converted
to fuzzy
fuzzy
intervals
of
[5,
7],
[3, 5],
5],
[5,
7]
and
[4, 6],
6], all
all of
offixed
which
have fixed
fixed
interval
of 2.
2. But
But
the
proposed
approach
transforms
the
original
scores
into
the
flexible
rough
intervals
of
[5.18,
6],
approach
transforms
thetransforms
original scores
into thescores
flexible
rough
intervalsrough
of [5.18,
6], [4, of
5.18],
[5.18,
6]
the proposed
approach
the original
into
the flexible
intervals
[5.18,
6], [4,
[4,
5.18],
[5.18,
6]
[4.47,
5.65],
which
are
4.
change
evaluations
and
which
shown
in Figure
4. Ifin
theFigure
experts
change
their evaluations
3, 4, 6, 4,
5.18],[4.47,
[5.18,5.65],
6] and
and
[4.47,are
5.65],
which
are shown
shown
in
Figure
4. If
If the
the experts
experts
change their
theirinto
evaluations
into
3,
4,
will
into
[5,
[3,
5],
the
ELECTRE
the
will intervals
change
into
[2,change
4], [3, 5],
[5,[2,
7],4],
[3,[3,
5],5],
while
the
ELECTRE
transform
the
intofuzzy
3, 4,
4, 6,
6,intervals
4, the
the fuzzy
fuzzy
intervals
will
change
into
[2,
4],
[3,
5],
[5, 7],
7],
[3,rough
5], while
while
the rough
rough
ELECTRE
transform
the
original
scores
into
[3,
4.12],
[3.63,
4.58],
[4.12,
6]
and
[3.36,
4.58].
Obviously,
the
original
scores
[3, 4.12],
[3.63,
4.58],
and [3.36,
the boundary
of the fuzzy
transform
the into
original
scores
into
[3, [4.12,
4.12], 6][3.63,
4.58], 4.58].
[4.12,Obviously,
6] and [3.36,
4.58]. Obviously,
the
boundary
of
the
fuzzy
interval
has
no
alteration
with
the
change
of
the
experts’
change
in
the
fuzzy
interval
has
no
alteration
with
the
change
of
the
experts’
change
in
the
fuzzy
ELECTRE.
On
the
other
boundary of the fuzzy interval has no alteration with the change of the experts’ change in the fuzzy
ELECTRE.
On
other
ELECTRE
can
identify
the
of
preferences,
hand,
the rough
ELECTRE
can the
identify
the
changes of
expert
preferences,
which
will make
the final
ELECTRE.
On the
the
other hand,
hand,
the rough
rough
ELECTRE
can
identify
the changes
changes
of expert
expert
preferences,
which
will
make
the
final
ranking
more
accurate
and
reasonable.
ranking
more
accurate
andranking
reasonable.
which will
make
the final
more accurate and reasonable.

Figure 4. Different vagueness manipulations for judgements on alternative one of criterion one.
Figure 4. Different vagueness manipulations for judgements on alternative one
one of
of criterion
criterion one.
one.

Moreover,
Moreover, compared
compared with
with the
the traditional
traditional ELECTRE
ELECTRE method
method (e.g.,
(e.g., the
the ELECTRE
ELECTRE method
method used
used
by
by Bırgün
Bırgün and
and Cıhan
Cıhan (2010)
(2010) [40]),
[40]), the
the proposed
proposed method
method provides
provides the
the rank
rank of
of all
all the
the alternatives.
alternatives. In
In
the
the traditional
traditional ELECTRE
ELECTRE method,
method, we
we can
can only
only get
get partial
partial relationships
relationships among
among alternatives.
alternatives. This
This will
will

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Moreover, compared with the traditional ELECTRE method (e.g., the ELECTRE method used by
Bırgün and Cıhan (2010) [40]), the proposed method provides the rank of all the alternatives. In the
traditional ELECTRE method, we can only get partial relationships among alternatives. This will
hinder the managers to directly identify the best supplier. As shown in Figure 2, there is no direct or
indirect relationship between A7 and A3, so we do not know whether A7 performs better than A3 or
not. However, we can get all the relationships in the proposed method based on the calculation of
pure concordance index and discordance index. All the ranks of the suppliers can be provided in the
proposed approach. This is obviously more practical and reasonable than the conventional ELECTRE
method (e.g., the ELECTRE method in [40]). Moreover, different with most of AHP/ANP-based
methods [29,30] and DEA approaches [31,32], the proposed rough ELECTRE method considers
the uncertainty in decision–making process, which makes the final ranking results of suppliers
more accurate.
Theoretically, this study develops a rough multi-criteria decision-making approach for sustainable
supplier selection considering vagueness and subjectivity. 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. The comparisons between the proposed method,
the conventional ELECTRE and the fuzzy ELECTRE reveal that the rough ELECTRE performs better
than the conventional ELECTRE and the fuzzy ELECTRE in dealing with vague and imprecise
information. Besides, this research contributes to modeling the problem of supplier selection based on
the economic, environmental and social aspects. The social aspects are often omitted in the previous
supplier selection methods. Practically, this method provides an effective method to identify the
right suppliers to achieve the success of the sustainable supply chain management. It also provides a
standardized procedure for managers in sustainable supplier selection.
5. Conclusions
To manipulate the vagueness in sustainable supplier selection, a new approach based on the rough
set theory and ELECTRE is developed in this paper. The novel approach integrates both the strength
of rough set theory in handling vagueness and the merit of ELECTRE in modeling multi-criteria
decision-making problem. A case study of sustainable supplier selection for solar air-conditioner
manufacturer is provided to demonstrate the application and potential of the approach. In sum, this
proposed method has the following features:
First, this study considers the social sustainability in the supplier selection, which is often omitted
in the previous literature. This research contributes to modeling the problem of supplier selection
decision within the context of a sustainable supply chain management based on the Triple Bottom Line
(TBL) concept (economic, environmental and social aspects). The sustainability criteria in this study
are generic and can be used for sustainable supplier selection in different industries.
Second, the proposed rough ELECTRE method can flexibly reflect the uncertainty in
decision–making without much priori information. Different with the previous fuzzy methods,
the proposed approach utilizes the lower and upper approximations to describe uncertainty and
it does not require the pre-set fuzzy membership function, which will reduce the decision–making
burdens of managers.
Third, the proposed approach can identify the preference changes of decision makers with flexible
rough intervals. Due to the flexible uncertainty mechanism, the rough number is more sensitive than
fuzzy number to the preference changes of decision makers, which makes the final ranking results
more accurate.
Fourth, different with the conventional ELECTRE revealing partial ranking orders, the proposed
rough ELECTRE method can provide full ranking order of all the alternatives. This is especially useful
for managers to get a comprehensive view of suppliers and make reasonable decision–making in
supplier selection.

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Although the proposed method has some merits in sustainable supplier selection, it also has
several ameliorable aspects which may serve as implications for further study. To make the ranking
results more accurate, it would be favorable for future research to take decision makers’ weighs,
objective criteria weights and subjective criteria weights into consideration. To handle huge number
of suppliers, the proposed rough ELECTRE method will be integrated with DEA method with little
managerial input and output required. Moreover, a computerized tool based on the proposed approach
will be developed to reduce the computation burdens of managers. Besides, more testing work is
necessitated to gain external validity.
Author Contributions: Conceptualization, S.J. and W.S.; Methodology, W.S.; Validation, H.L. and S.J.; Resources,
X.M.; Writing-Original Draft Preparation, H.L.; Writing-Review & Editing, X.M.
Funding: The work described in this paper was supported by the National Natural Science Foundation of China
(Grant No. 71501006 and 71632003), the Technical Research Foundation (JSZL2016601A004), the Open Project of
Henan Key Laboratory of Intelligent Manufacturing of Mechanical Equipment, Zhengzhou University of Light
Industry (No. IM201801), and the Fundamental Research Funds for the Central Universities.
Acknowledgments: The authors would like to thank the editor and the anonymous reviewers for their helpful
comments and suggestions on the drafts of this paper.
Conflicts of Interest: The authors declare no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript:
Di
Ci
Ei
Ai
P
pi
Apr ( pi )
Apr ( pi )
d
RN ( pi )
piL
pU
i
xij
yij
wkj

Lim wkj

Lim wkj

ith decision-maker
ith criterion for supplier selection
ith expert
ith alternative
judgement set
judgement of ith expert
lower approximation set of pi , which contains the elements that smaller than pi in set P
upper approximation set of pi , which contains the elements that bigger than pi in set P
maximum distance of set P
rough interval corresponding to pi
lower approximation of pi
upper approximation of pi
elements of Apr ( pi )
elements of Apr ( pi )
weight of jth criterion with kth expert
lower approximation of wkj
upper approximation of wkj

w jL

lower approximation of the weight of jth criterion

wU
j

upper approximation of the weight of jth criterion

rijk
Rk
reij

kth expert’s judgement on jth criterion in ith alternative
scoring matrix for kth expert
Set of rijk

tij
T
C
D
c
d
F
G
H

rough interval corresponding to reij after normalized
rough scoring matrix
concordance matrix
discordance matrix
concordance index
discordance index
concordance Boolean matrix
discordance Boolean matrix
general Boolean matrix

Sustainability 2018, 10, 2622

cˆi
d̂i
SSCM
ELECTRE
MCDM
EMS
TOPSIS
AHP
ANP
DEA
QFD

19 of 20

pure concordance index
pure discordance index
Sustainable supply chain management
ELimination Et Choix Traduisant la Realité
Multi-criteria decision-making
Environmental management system
Technique for Order Preference by Similarity to an Ideal Solution
Analytical Hierarchy Process
Analytical Network Process
Data Envelopment Analysis
Quality Function Deployment

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