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Research Article
A Modified TOPSIS Method Based on 𝐷Numbers and Its
Applications in Human Resources Selection
Liguo Fei,1Yong Hu,2Fuyuan Xiao,1Luyuan Chen,1andYongDeng
1,2,3,4
1School of Computer and Information Science, Southwest University, Chongqing 400715, China
2Big Data Decision Institute, Jinan University, Tianhe, Guangzhou 510632, China
3Institute of Integrated Automation, School of Electronic and Information Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi
710049, China
4School of Engineering, Vanderbilt University, Nashville, TN 37235, USA
Correspondence should be addressed to Yong Deng; prof.deng@hotmail.com
Received February ; Accepted April
Academic Editor: Rita Gamberini
Copyright © Liguo Fei et al. is is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Multicriteria decision-making (MCDM) is an important branch of operations research which composes multiple-criteria to make
decision. TOPSIS is an eective method in handling MCDM problem, while there still exist some shortcomings about it. Upon
facing the MCDM problem, various types of uncertainty are inevitable such as incompleteness, fuzziness, and imprecision result
from the powerlessness of human beings subjective judgment. However, the TOPSIS method cannot adequately deal with these
types of uncertainties. In this paper, a -TOPSISmethodisproposedforMCDMproblembasedonaneweectiveandfeasible
representation of uncertain information, called numbers. e -TOPSIS method is an extension of the classical TOPSIS method.
Within the proposed method, numbers theory denotes the decision matrix given by experts considering the interrelation of
multicriteria. An application about human resources selection, which essentially is a multicriteria decision-making problem, is
conducted to demonstrate the eectiveness of the proposed -TOPSIS method.
1. Introduction
Multicriteria decision-making (MCDM) or multiple-criteria
decision analysis is an important branch of operations
research that denitely uses multiple-criteria in decision-
making environments [, ]. In daily life and professional
learning, there exist generally multiple conicting criteria
which need to be considered in making decisions and opti-
mization [, ]. Price and spend are typically one of the main
criteria with regard to a large amount of practical problems.
However, the factor of quality is generally another criterion
which is in conict with the price. For example, the cost,
safety, fuel economy, and comfort should be considered as the
main criteria upon purchasing a car. It is the most benet for
us to select the safest and most comfortable one which has the
bedrock price simultaneously. e best situation is obtaining
the highest returns while reducing the risks to the most extent
with regard to portfolio management. In addition, the stocks
that have the potential of bringing high returns typically also
carry high risks of losing money. In service industry, there
is a couple of conicts between customer satisfaction and
the cost to provide service. Upon making decision, it will be
compelling if multiple-criteria are considered even though
theycamefromandarebasedonsubjectivejudgmentof
human. What is more, it is signicant to reasonably describe
theproblemandpreciselyevaluatetheresultsbasedon
multiple-criteria when the stakes are high. With regard to
the problem of whether to build a chemical plant or not and
where the best site for it is, there exist multiple-criteria that
need to be considered; also, there are multiple parties that will
be aected deeply by the consequences.
Constructing complex problems properly as well as
multiple-criteria taken into account explicitly results in more
reasonable and better decisions. Signicant achievements
in this eld have been made since the beginning of the
modern multicriteria decision-making (MCDM) discipline
Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2016, Article ID 6145196, 14 pages
http://dx.doi.org/10.1155/2016/6145196
Mathematical Problems in Engineering
in the early s. A variety of approaches and methods have
been proposed for MCDM. In [], a novel MCDM method
named FlowSort-GDSS is proposed to sort the failure modes
into priority classes by involving multiple decision-makers,
which has the robust advantages in sorting failures. In the
eld of multiple objective mathematical programming, Evans
and Yu [, ] proposed the vector maximization method
aimed at approximating the nondominated set which is orig-
inally developed for multiple objective linear programming
problems. Torrance [] used elaborate interview techniques
to deal with the problem in MCDM, which exist for eliciting
linear additive utility functions and multiplicative nonlinear
utility functions. And there are many other methods, such
as best worst method [], characteristic objects method [],
fuzzy sets method [–], rough sets [], and analytic hierar-
chy process [–]. In [], the authors aim to systematically
review the applications and methodologies of the MCDM
techniques and approaches, which is a good guidance for
us to fully understand the MCDM. Technique for order
preference by similarity to ideal solution (TOPSIS), which
is proposed in [–], is a ranking method in conception
andapplication.estandardTOPSISmethodologyaimsto
select the alternatives which have the shortest distance from
thepositiveidealsolutionandthelongestdistancefromthe
negative ideal solution at the same time. e positive ideal
solution maximizes the benet attributes and minimizes the
cost attributes, whereas the negative ideal solution maximizes
the cost attributes and minimizes the benet attributes. e
TOPSIS methodology is applied widely in MCDM eld [–
], especially in the fuzzy extension of linguistic variables
[–].
It is obvious that the mentioned approaches play a role
under some specic circumstances, but, in the practical appli-
cations, they show more uncertainties due to the subjective
judgment of experts’ assessment. In order to eectively han-
dle various uncertainties involved in the MCDM problem,
a new representation of uncertain information, called
numbers [], is presented in this paper. It is an extension of
Dempster-Shafer evidence theory. It gives the framework of
nonexclusive hypotheses, applied to many decision-making
problems under uncertain environment [–]. Comparing
with existing methods, numbers theory can eciently
denote uncertain information and more coincide with the
actual conditions.
erefore, in this paper, to well address these issues in
TOPSIS method, an extended version is presented based on
numbers named -TOPSIS, which considers the interre-
lation of multicriteria and handles the fuzzy and uncertain
criteria eectively. e -TOPSIS method can represent
uncertain information more eectively than other group
decision support systems based on classical TOPSIS method,
which cannot adequately handle these types of uncertainties.
An application has been conducted using the -TOPSIS
method in human resources selection, and the result can
be more reasonable because of its consideration about the
interrelation of multiple-criteria.
e remainder of this paper is constituted as follows.
Section introduces the Dempster-Shafer theory and its basic
rules and some necessary related concepts about numbers
theory and its distance function and TOPSIS. e proposed
-TOPSIS method is presented in Section . Section
conducts an application in human resources selection based
on -TOPSIS. Conclusion is given in Section .
2. Preliminaries
2.1. Dempster-Shafer Evidence eory. Dempster-Shafer evi-
dence theory [, ], which is rst developed by Dempster
and later extended by Shafer, is used to manage various types
of uncertain information [–], belonging to the category
of articial intelligence. As a theory widely applied under the
uncertain environment, it needs weaker conditions and has
a wider range of use than the Bayesian probability theory.
When the ignorance is conrmed, Dempster-Shafer theory
couldconvertintoBayesiantheory,soitisoenregardedas
an extension of the Bayesian theory. Dempster-Shafer theory
has the advantage to directly express the “uncertainty” by
assigning the probability to the subsets of the union set
composed of multiple elements, rather than to each of the
single elements. Besides, it has the ability to combine pairs
of bodies of evidence or belief functions to generate a new
evidence or belief function [, ].
e decision-making or optimization in real system is
very complex with incomplete information [–]. With
the superiority in dealing with uncertain information and
the practicability in engineering, a number of applications
of D-S evidence theory have been published in the literature
indicating its widespread for fault diagnosis [, ], pattern
recognition [–], supplier selection [, ], and risk
assessment [, ]. Also, it exerts a great eect on combining
with other theories and methods such as fuzzy numbers [],
decision-making [], and AHP [–]. Moreover, based on
the Dempster-Shafer theory, the generalized evidence theory
has been proposed by Deng to develop the classical evidence
theory [] to handle conicting evidence combination
[]. It should be noted that the combination of dependent
evidence is still an open issue [, ]. For a more detailed
explanation of evidence theory, some basic concepts are
introduced as follows.
Denition 1 (frame of discernment). A frame of discernment
is a set of alternatives perceived as distinct answers to a
question. Suppose is the frame of discernment of research-
ing problem, a nite nonempty set of elements that are
mutually exclusive and exhaustive, indicated by
=1,2,...,𝑖,...,𝑛()
and denote 2𝑈as the power set composed of 2𝑁elements of
, and each element of 2𝑈is regarded as a proposition. Based
on the two conceptions, mass function is dened as below.
Denition 2 (mass function). For a frame of discernment ,
a mass function is a mapping from 2𝑈to [0,1],formally
dened by
:2𝑈→ [0,1]()
Mathematical Problems in Engineering
y
x
O
ZD LD MD HD CD
1
0.5 0.75 1
0.25
(a) Dempster-Shafer evidence theory
y
Ox
ZD LD MD HD CD
1
0.5 0.75 10.25
(b) 𝐷number theory
F : e framework of DSET and DNT.
satisfying ()=0,
𝐴∈2𝑈()=1, ()
where is an empty set and represents the propositions.
In Dempster-Shafer theory, is also named as basic
probability assignment (BPA), and ()is named as assigned
probability, presenting how strong the evidence supports .
is regarded as a focal element when ()>0,andtheunion
of all focal elements are called the core of the mass function.
Considering two pieces of evidence from dierent and
independent information sources, denoted by two BPAs 1
and 2, Dempster’s rule of combination is used to derive a
new BPA fro m t w o BPAs.
Denition 3 (Dempster’s rule of combination). Dempster’s
rule of combination, also known as orthogonal sum, is
expressed by =1⊕2, dened as follows:
()=
1
1−
𝐴1∩𝐴2=𝐴1122, =;
0, = ()
with =
𝐴1∩𝐴2=01122, ()
where is a normalization constant, called conict coe-
cient of two BPAs. Note that the Dempster-Shafer evidence
theory is only applicable to such two BPAs which satisfy the
condition <1.
2.2. Number eory. number theory, proposed by Deng
[], is a generalization of Dempster-Shafer evidence theory.
A wide range of applications have been published based on it,
especially in the uncertain environment and MCDM []. In
the classical Dempster-Shafer theory, there are several strong
hypotheses on the frame of discernment and basic probability
assignment. However, some shortcomings still exist which
limit the representation of some types of information as well
as the restriction of the application in practice. number
theory, considered as an extension and developed method,
makes the following progress.
First, Dempster-Shafer evidence theory deals with the
problem about the strong hypotheses, which means that
elements in the frame of discernment are required to be
mutually exclusive. In general, the frame of discernment
is determined by experts, always involving human being’s
subjective judgments and uncertainty. Hence, the hypothesis
is hard to meet. For example, there are ve anchor points
“zero dependence [ZD],” “low dependence [LD],” “moderate
dependence [MD],” “high dependence [HD],” and “complete
dependence [CD]” corresponding to dependence levels avail-
able to analysts to make judgments. It is inevitable that there
exist some overlaps of human being’s subjective judgments.
number theory is more suitable to the actual situation
based on the framework of nonexclusive hypotheses. e
dierence between Dempster-Shafer theory and number
theory about this is shown in Figure .
Second, the problem solved by number theory without
another hypothesis of Dempster-Shafer theory is related to
basic probability assignment. In Dempster-Shafer theory, the
sum of BPAs must be equal to , which means that the experts
have to make all the judgments and then give the assessment
results. Nevertheless, on the one hand, it would be dicult
to satisfy in some complex environment. On the other hand,
from time to time, it would be unnecessary and redundant to
Mathematical Problems in Engineering
meet the hypothesis, when the framework does not contain
overall situations. From this point of view, number theory
allows the incompleteness of information, having the ability
to adapt to more cases.
ird, compared with Dempster-Shafer theory, num-
ber theory is more suitable to the framework. In Dempster-
Shafer theory, the BPA is calculated through the power
setoftheframeofdiscernment.Itishardtoworkwhen
there are too many elements in it, and it even can not be
acceptedwhenthenumberoftheframeworksistoohigh
to use Dempster’s rule of combination to some degree.
number theory emphasizes the set of problem domains itself.
Uncertain information is represented by numbers so that
the fusion would have less calculation and allows arbitrary
framework.
Even so, numbers theory is still preferable in many
cases, for the advantages of all the three points above, not just
improving one certain aspect. It is dened as follows.
Denition 4 (number). Let a nite nonempty set denote
the problem domain. number function is a mapping
formulated by :→[0,1]()
with ()=0,
𝐵⊆Ω()1 ()
and, compared with the mass function, the structure of the
ex pression s e ems to be si milar. However, in number theory,
the elements of do not require to be mutually exclusive.
In addition, being contrary to the frame of discernment
containing overall events, is suitable to incomplete
information by ∑𝐵⊆Ω ()<1.
Furthermore, for a discrete set ={1,2,...,𝑖,...,𝑛},
where 𝑖∈,andwhen =,𝑖=
𝑗.Aspecialformof
numbers can be expressed by
1=V1
2=V2
...
𝑖=V𝑖
...
𝑛=V𝑛
()
or simply denoted as ={(
1,V1),(2,V2),...,(𝑖,V𝑖),...,
(𝑛,V𝑛)},whereV𝑖>0and ∑𝑛
𝑖=1 V𝑖≤1.
Below is the combination rule, a kind of addition opera-
tion to combine two numbers.
Denition 5 (two numbers’ rule of combination). Suppose
1and 2are two numbers, indicated by
1=1
1,V1
1,...,1
𝑖,V1
𝑖,...,1
𝑛,V1
𝑛,
2=2
1,V2
1,...,2
𝑗,V2
𝑗,...,2
𝑚,V2
𝑚, ()
and the combination of 1and 2, which is expressed as =
1⊕2, is dened as follows:
()=V()
with
=1
𝑖+2
𝑗
2,
V=V1
𝑖+V2
𝑗/2
,
=
𝑚
𝑗=1
𝑛
𝑖=1 V1
𝑖+V2
𝑗
2, 𝑛
𝑖=1
V1
𝑖=1, 𝑚
𝑗=1
V2
𝑗=1;
𝑚
𝑗=1
𝑛
𝑖=1 V1
𝑖+V2
𝑗
2+𝑚
𝑗=1 V1
𝑐+V2
𝑗
2, 𝑛
𝑖=1
V1
𝑖<1, 𝑚
𝑗=1
V2
𝑗=1;
𝑚
𝑗=1
𝑛
𝑖=1 V1
𝑖+V2
𝑗
2+𝑛
𝑖=1 V1
𝑖+V2
𝑐
2, 𝑛
𝑖=1
V1
𝑖=1, 𝑚
𝑗=1
V2
𝑗<1;
𝑚
𝑗=1
𝑛
𝑖=1 V1
𝑖+V2
𝑗
2+𝑚
𝑗=1 V1
𝑐+V2
𝑗
2+𝑛
𝑖=1 V1
𝑖+V2
𝑐
2+V1
𝑐+V2
𝑐
2,𝑛
𝑖=1
V1
𝑖<1, 𝑚
𝑗=1
V2
𝑗<1,
()
Mathematical Problems in Engineering
where V1
𝑐=1−∑𝑛
𝑖=1 V1
𝑖and V2
𝑐=1−∑𝑚
𝑗=1 V2
𝑗.Notethat
superscript in the above equations is not exponent when
numbers are more than two.
It must be pointed out that the combination operation
dened in Denition does not preserve the associative
property. It is clear that (1⊕2)⊕3=
1⊕(2⊕
3) =(
1⊕3)⊕2. In order that multiple numbers
can be combined correctly and eciently, a combination
operation for multiple numbersisdevelopedasfol-
lows.
Denition 6 (multiple numbers’ rule of combination).
Let 1,2,...,𝑛be numbers, 𝑗is an order variable
for each 𝑗,indicatedbytuple𝑗,𝜇𝑗, and then the
combination operation of multiple numbers is a mapping
𝐷,suchthat
𝐷1,2,...,𝑛=⋅⋅⋅𝜆1⊕𝜆2⊕⋅⋅⋅⊕𝜆𝑛, ()
where 𝜆𝑖is 𝜇𝑗of the tuple 𝑗,𝜇𝑗having the th lowest
𝑗.
In the meanwhile, an aggregate operation is proposed on
this special numbers, as such.
Denition 7 (numbers’ integration). For ={(1,V1),(2,
V2),...,(𝑖,V𝑖),...,(𝑛,V𝑛)}, the integrating representation of
is dened as
()=𝑛
𝑖=1𝑖V𝑖.()
2.3. Distance Function of Numbers. Anewdistancefunc-
tion to measure the distance between two numbers has
been proposed in [].
In numbers theory, there is no compulsive requirement
that the frame of discernment is a mutually exclusive and
collectively exhaustive set. So a relative matrix is used to
represent the relationship of numbers. e denition of
relation matrix is shown as follows.
2.3.1. Relative Matrix and Intersection Matrix
Denition 8. Let 𝑖and 𝑗denote the number and number
of linguistic constants, 𝑖𝑗 represent the intersection area
between 𝑖and 𝑗,and12 is the union area between 𝑖and
𝑗. e denition of nonexclusive degree 𝑖𝑗 can be shown as
follows:
𝑖𝑗 =𝑖𝑗
𝑖𝑗 .()
U12 L1L2
S12
···
LiLi+1
Sii+1
···
Ln−1 Ln
Sn−1n
F : Example for linguistic constants.
Next, the relative matrix can be constructed for these
elements based on 𝑖𝑗:
=
1
12 ⋅⋅⋅ 1𝑖 ⋅⋅⋅ 1𝑛
21 1 ⋅⋅⋅ 2𝑖 ⋅⋅⋅ 2𝑛
...... ⋅⋅⋅ ...⋅⋅⋅...
𝑖1 𝑖2 ⋅⋅⋅ 1 ⋅⋅⋅ 𝑖𝑛
...... ⋅⋅⋅ ...⋅⋅⋅...
𝑛1 𝑛2 ⋅⋅⋅ 𝑛𝑖 ⋅⋅⋅ 1
.()
For example, suppose there are linguistic constants
which are shown in Figure . e nonexclusive degree
between two numbers can be obtained by 𝑖𝑗 based on the
intersection area 𝑖𝑗 and the union area 𝑖𝑗 of two linguistic
constants 𝑖and 𝑗.
Denition 9. Aer obtaining the relative matrix between
two subsets which belong to 2Ω, the denition of the inter-
section degree of two subsets can be shown as follows:
1,2= ∑𝑖𝑗
1⋅2,()
where =and 1,2∈2
Ω.denotes the rst element’s
row number of set 1in the relative matrix and is the
rst element’s column number of set 2.|1|expresses the
cardinality of 1and |2|represents the cardinality of 2.In
particular, when =,=1.
2.3.2. Distance between Two Numbers. It is known that
numbers theory is a generalization of the Dempster-Shafer
theory. e body of numbers can be considered as a
discrete random variable whose values are 2Ωby a probability
distribution . erefore, numbercanbeseenasavector
in the vector space. us, the distance function between two
numbers can be dened as follows.
Denition 10. Let 1and 2be two numbers on the same
frame of discernment , containing elements which are
not required to be exclusive to each other. e distance
between 1and 2is
𝐷-number(𝑑1,𝑑2)=1
2→
1−→
2𝑇⋅→
1−→
2, ()
where and are two (2𝑁×2𝑁)-dimensional matrixes.
Mathematical Problems in Engineering
e elements of canberepresentedas
(,)=|∩|
|∪|,,∈2
Ω. ()
e elements of can be denoted as
(,)=∑𝑖𝑗
||⋅||,
=,,∈2Ω,when =,=1, ()
where denotes the rst element’s row number of set 1in the
relative matrix and is the rst element’s column number
of set 2.
2.4. TOPSIS Method. Technique for order preference by sim-
ilarity to ideal solution (TOPSIS), which is proposed in [],
is a ranking method which is applied to MCDM problem.
e standard TOPSIS method is designed to nd alternatives
which have the shortest distance from the positive ideal
solutionandthelongestdistancefromthenegativeideal
solution. e positive ideal solution attempts to seek the
maximization of benet criteria and the minimum of the
cost criteria, whereas the negative ideal solution is just the
opposite.
Denition 11. Construct a decision matrix =(𝑚𝑛),which
includes alternatives and criteria. Normalize the decision
matrix
𝑖𝑗 =𝑖𝑗
∑𝑚
𝑗=1 2
𝑖𝑗 , =1,...,;=1,...,. ()
To obtain the weighted decision matrix using the associated
weightstomultiplythecolumnsofthenormalizeddecision
matrix =V(),
V𝑖𝑗 =𝑗×𝑖𝑗 , =1,...,;=1,...,, ()
where 𝑗is the weight of th criterion.
Determine the positive ideal and negative ideal solutions.
e denitions of the positive ideal solution, represented as
+, and the negative ideal solution, represented as −,are
shown as follows:
+=V+
1,V+
2,...,V+
𝑛
=max
𝑖V𝑖𝑗 |∈𝑏min
𝑖V𝑖𝑗 |∈𝑐
−=V−
1,V−
2,...,V−
𝑛
=min
𝑖V𝑖𝑗 |∈𝑏max
𝑖V𝑖𝑗 |∈𝑐,
()
where 𝑏denotes the set of benet criteria and 𝑐represents
the set of cost criteria.
Calculate the separation measures between the exist-
ing alternatives and the positive ideal and negative ideal
solutions. e separation measures that are determined by
Euclidean distance, +
𝑖and −
𝑖, of each alternative from the
positive ideal and negative ideal solutions, respectively, are
shown as
+
𝑖=𝑛
𝑗=1 V+
𝑗−V𝑖𝑗2, =1,...,;=1,...,,
−
𝑖=𝑛
𝑗=1 V−
𝑗−V𝑖𝑗2, =1,...,;=1,...,. ()
Obtain the relative closeness to the ideal solution:
𝑖=−
𝑖
−
𝑖++
𝑖, =1,...,. ()
Sort the alternatives based on the relative closeness to the
ideal solution. If alternatives have higher 𝑖,itwillbemore
signicant and should be assigned higher priority.
3. The Modified TOPSIS Method
Based on Numbers
TOPSIS is an eective methodology to handle the problem
in multicriteria decision-making. numberstheoryisanew
representation of uncertain information, which can denote
the more fuzzy conditions. So the combination of TOPSIS
and numbers is a new experiment to make decisions
in an uncertain environment. Next, we will propose the
modied TOPSIS method named -TOPSIS to dea l with
some Gordian knots in MCDM.
3.1. Construct the Decision Matrix
Denition 12. Suppose there is a matrix =(𝑚𝑛),whichis
constructed by alternatives and criteria.
Obtain the weight for each criterion of the matrix, and
assign the weight to corresponding criterion to determine the
weighted matrix =V():
V𝑖𝑗 =𝑗×𝑖𝑗 , =1,...,;=1,...,, ()
where 𝑗is the weight for criterion. Normalize the matrix
to get the decision matrix:
𝑖𝑗 =V𝑖𝑗
∑𝑚
𝑗=1 V2
𝑖𝑗 , =1,...,;=1,...,. ()
3.2. Determine Numbers and Dene Interrelation between
eir Elements. In Section ., the decision matrix has been
constructed; then it will be transformed to numbers as
1({1})=11 1({2})=12 ⋅⋅⋅ 1({})=1𝑛
2({1})=21 2({2})=22 ⋅⋅⋅ 2({})=2𝑛
d...
𝑚({1})=𝑚1 𝑚({2})=𝑚2 ⋅⋅⋅ 𝑚({})=𝑚𝑛
.()
Mathematical Problems in Engineering
e interrelation between criteria is considered in the
-TOPSIS method for more reasonable and more eective
decision-making, which is dened as follows.
Denition 13. Let 𝑖𝑗 denote the inuence relation from
criterion to criterion .Let𝑖𝑗 represent the interrelation
between criterion and criterion ,whichcanalsobeseenas
the intersection of criterion and criterion .en,onegives
the denition of based on shown as follows:
𝑖𝑗 =𝑗𝑖 =1
2×𝑖𝑗 +𝑗𝑖. ()
Denition 14. Let 𝑖𝑗 denote the union set between criterion
and criterion .Let𝑖represent the weight of criterion
from the comprehensive views of four experts. en, one
determines the denition of based on and weights of
criterion and criterion shown as follows:
𝑖𝑗 =𝑗𝑖 =𝑖+𝑗−𝑖𝑗.()
3.3. e Methodology for Proposed -TOPSIS. Firstly, deter-
minethepositiveidealandnegativeidealsolutions.e
positive ideal solution, denoted as +,andthenegative
ideal solution, denoted as −, are dened as follows:
+=+
1,+
2,...,+
𝑛
=max
𝑖𝑖𝑗 |∈𝑏min
𝑖𝑖𝑗 |∈𝑐,
−=−
1,−
2,...,−
𝑛
=min
𝑖𝑖𝑗 |∈𝑏max
𝑖𝑖𝑗 |∈𝑐,
()
where 𝑏is the set of benet criteria and 𝑐is the set of cost
criteria.
Secondly, obtain the separation measures of the existing
alternatives from the positive ideal and negative ideal solu-
tions. e separation measures based on the distance func-
tion of numbers, +
𝑖and −
𝑖,ofeachalternativefrom
the positive ideal and negative ideal solutions, respectively,
are derived from
+
𝑖=𝑛
𝑗=11
2→
+
𝑗−→
𝑖𝑗𝑇⋅→
+
𝑗−→
𝑖𝑗,
=1,...,;=1,...,,
−
𝑖=𝑛
𝑗=11
2→
−
𝑗−→
𝑖𝑗𝑇⋅→
−
𝑗−→
𝑖𝑗,
=1,...,;=1,...,.
()
Finally, calculate the relative closeness to the ideal solu-
tion:
𝑖=−
𝑖
−
𝑖++
𝑖, =1,...,. ()
D-TOPSIS
Decision matrix
Transform decision
matrix to D numbers
Determine positive
ideal solutions
Determine negative
ideal solutions
Distance function
of D numbers
Calculate the distance between each solution
and positive ideal and negative ideal solutions
Calculate the relative closeness and rank
F : e ow chart of -TOPSIS.
Rank the alternatives according to the relative closeness to the
ideal solution: the alternatives with higher 𝑖are assumed
to be more important and should be given higher priority. e
ow chart of -TOP SIS is show n in Figure .
4. An Application for Human Resources
Selection Based on -TOPSIS
An import and export trading company plans to recruit a
department manager who must satisfy their various require-
ments []. ere are some relevant test items provided by the
human resources department of the company for selecting
the best candidate. e test items include two great aspects:
the objective and the subjective aspects. In addition, the
objective aspect is divided into two sides. e rst one is
knowledge test which includes language test, professional
test, and safety rule test. e other one is skill test which
hastheitemsofprofessionalskillsandcomputerskills.
e subjective aspect is determined by the corresponding
interviews including panel interview and -on- interview.
Now, candidates are qualied for the test, and four experts
rate all the candidates in interviews. e test results for
objective and subjective attributes are shown in Tables and
. What is more, the weights of all the items from four experts
arealsoshowninTable.
e ow chart of the process to select the best candidate
isshowninFigure.Next,wewillillustratethespecicsteps
Mathematical Problems in Engineering
T : e s c o r es of the obj e c t i v e a sp e c ts.
Number Candidates
Objective attributes
Knowledge test Skill test
Language test Professional test Safety rule test Professional skills Computer skills
JamesB.Wang
Carol L. Lee
KenneyC.Wu
RobertM.Liang
SophiaM.Cheng
LilyM.Pai
AbonC.Hsieh
Frank K. Yang
Ted C. Yang
Sue B. Ho
Vincent C. Chen
Rosemary I. Lin
Ruby J. Huang
George K. Wu
Philip C. Tsai
Michael S. Liao
Michelle C. Lin
T : e scores of the subjective aspects from dierent experts for interview.
Number
Subjective attributes
Expert Expert Expert Expert
Panel -on- Panel -on- Panel -on- Panel -on-
Mathematical Problems in Engineering
T : Weight for dierent test items from dierent experts.
Number Attributes Weight
Expert Expert Expert Expert
Language test . . . .
Professional test . . . .
Safety rule test . . . .
Professional skills . . . .
Computer skills . . . .
Panel interview . . . .
-on- interview . . . .
Initialization
Step 1
Step 2
Step 3
Step 4
Step 5
Determine the weight of
each attribute derived
from experts
Construct the decision matrix
Transform the attribute matrix to D numbers
Determine the positive ideal and negative ideal solutions
Calculate the distance between each solution and positive ideal and
negative ideal solutions based on D numbers distance function
Calculate the relative closeness to the ideal solution and rank
Obtain the objective and
each candidate
subjective tests’ score of
F : e ow chart of human resources selection.
about how to select the best one from the candidates for
the company using the new proposed -TOPSIS method.
Step 1. Construct the attribute matrix.
Firstly, we calculate the comprehensive scores of each can-
didate combining the four experts’ advice in the interviews.
And the results are shown in Table .
en, we can obtain the weighted overall results of this
testfromtheobjectiveandsubjectiveaspectsbasedonTables
,,and,whichisshowninTableandcanbeseenasthe
decision matrix.
Step 2. Transformdecisionmatrixtonumbers and obtain
the interrelation between these criteria.
From Step , the decision matrix has be determined.
Now, we need to transform the matrix to numbers. Firstly,
normalize the decision matrix for each item of each candidate
shown in Table . We will represent each test item using ,,
Mathematical Problems in Engineering
T : e comprehensive scores from dierent experts for the
interview.
Number Subjective attributes
Panel interview -on- interview
..
.
.
. .
..
. .
. .
..
. .
. .
.
.
. .
. .
. .
. .
. .
,,,,andfor convenience. e interrelation between
dierent criteria is shown in Table .
en, the union set and intersection can be obtained
from the experts’ scoring and experience and is shown in
Table based on Denitions and . And, in Figure ,
the interrelation between dierent criteria can be represented
by the network. e dierent size of each node denotes the
weight of dierent criteria from multiple experts, while the
width of the edge reects the interrelation of the dierent
criteria in some ways.
Step 3. Obtain the positive ideal solutions +and negative
ideal solutions −based on ().
We select the positive ideal and negative ideal solutions
from Table . e positive ideal solution is determined by the
highest score of each attribute; similarly, the negative ideal
solution is dened by the lowest score of each attribute. And
theresultsareshowninTable.
Step 4. Calculate the distance between each solution and
positive ideal and negative ideal solutions based on ().
From the above steps, the positive ideal solutions +
and negative ideal solutions −have been obtained. Next,
we will calculate the distance from each alternative scheme to
+and −by (), respectively. e results are shown
in Table .
Step 5. Calculate the relative closeness and rank.
In this step, we calculate the relative closeness to the ideal
solution of each attribute by (). Finally, sort each candidate
a
g
f
e
d
c
b
F : e network chart of interrelation between dierent
criteria.
bytheclosenessvalues.edistancesandrankingresultsare
shown in Table .
e best candidate can be selected easily based on the
ranking results. It is worth noting that the ranking results
will be dierent depending on two factors: (1)the scores in
objective and subjective tests of each criterion and (2)the
interrelation and weights among dierent criteria. And the
major advantages of -TOPSIS are reected in two aspects.
Firstly, it can keep the validity of the traditional TOPSIS
method. In addition, the relationship between multiattributes
is considered for the more reasonable results. e eec-
tiveness of -TOPSIS can be demonstrated by the applica-
tion.
5. Conclusion
In this paper, a new TOPSIS method called -TOPSIS
is proposed to handle MCDM problem using num-
bers to extend the classical TOPSIS method. In the pro-
posed method, the decision matrix determination from
MCDM problem can be transformed to numbers, which
can eectively represent the inevitable uncertainty, such
as incompleteness and imprecision due to the subjective
assessmentofhumanbeings.Andtherelationshipbetween
multiattributes is considered in the process of decision-
making, which is more grounded in reality. An example of
human resources selection is conducted and illustrates the
eectiveness of the proposed -TOPSIS method. In future
research, the theoretical framework of the -TOPSIS needs to
be increasingly perfected. For example, how to scientically
produce the relationship between multicriteria should be
further investigated. Also, the proposed method should be
utilized in other applications to further verify its eective-
ness.
Mathematical Problems in Engineering
T : e weighted overall scores of the test.
Number
Objective attributes Subjective attributes
Knowledge test Skill test
Language test Professional test Safety rule test Professional skills Computer skills Panel interview -on- interview
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
T : Constructing numbers of each candidate.
Number Language test Professional test Safety rule test Professional skills Computer skills Panel interview -on- interview
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
T : e interrelation between dierent criteria.
Relation
. . . . . .
. . . . .
. . . .
. . . .
. . . . .
. . .
. . .
T : e union set and intersection between dierent criteria.
𝑖𝑗 𝑖𝑗
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
Mathematical Problems in Engineering
T : e positive ideal solutions +and negative ideal solutions −.
Ideal solution Language test Professional test Safety rule test Professional skills Computer skills Panel interview -on- interview
+. . . . . . .
−. . . . . . .
T : e relative closeness and ranking results by -TOPSIS
method.
Number +
𝑖−
𝑖𝑖Rank
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
Competing Interests
ere is no conict of interests in this paper.
Authors’ Contributions
Yong Deng designed and performed research. Liguo Fei and
YongHuwrotethepaper.LiguoFeiandYongHuperformed
the computation. Yong Deng, Liguo Fei, Fuyuan Xiao, and
Luyuan Chen analyzed the data. All authors discussed the
resultsandcommentedonthepaper.LiguoFeiandYongHu
contributed equally to this work.
Acknowledgments
e work is partially supported by National Natural Science
Foundation of China (Grant nos. , , and
) and China State Key Laboratory of Virtual Reality
Technology and Systems, Beihang University (Grant no.
BUAA-VR-KF-).
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