INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL
ISSN 1841-9836, 13(3), 391-407, June 2018.

Attribute Selection Method based on Objective Data
and Subjective Preferences in MCDM

X. Ma, Y. Feng, Y. Qu, Y. Yu

Xiaofei Ma, Yi Feng
Technology Planning
Dalian Commodity Exchange
Dalian City, China, 116023
maxf@dce.com.cn, fengyi@dce.com.cn

Yi Qu*, Yang Yu
Agricultural Bank of China Data Center
88 Aoni Road, Pudong New Area
Shanghai City, China, 200131
*Corresponding author: 57638907@qq.com
yuyinyang@126.com

Abstract: Decision attributes are important parameters when choosing an alter-
native in a multiple criteria decision-making (MCDM) problem. In order to select
the optimal set of decision attributes, an analysis framework is proposed to illustrate
the attribute selection problem. Then a two-step attribute selection procedure is
presented based on the framework: In the first step, attributes are filtered by using
correlation algorithm. In the second step, a multi-objective optimization model is
constructed to screen attributes from the results of the first step. Finally, a case
study is given to illustrate and verify this method. The advantage of this method is
that both external attribute data and subjective decision preferences are utilized in
a sequential procedure. It enhances the reliability of decision attributes and matches
the actual decision-making scenarios better.
Keywords: attribute selection, multi-criteria decision-making (MCDM), multi-
objective optimization, attribute correlation.

1 Introduction

Multi-criteria decision-making (MCDM) is successfully applied to help decision makers (DMs)
choose optimal alternatives. During the past few decades, various MCDM methods have been
proposed based upon different philosophies such as multi-attribute utility, the analytic hierar-
chy process (AHP), outranking methods and so on. Meanwhile, many decision support systems
(DSSs) have been designed in MCDM to assist DMs in analyzing problems and making decisions
more easily. MCDM deals with a general class of problems that contains multiple attributes,
objectives and criteria [20]; and alternatives are determined based on many qualitative or quan-
titative criteria which are generally complicated and assessed by more relevant attributes. How-
ever, not all of them can be used in decision-making procedure, because they contain plenty of
redundant or "noisy" attributes and it will lead to useless decision alternatives. Therefore, the
effectiveness of an alternative is highly dependent on the set of decision attributes.

The concept of "optimal" can be illustrated by two aspects: (1) Elements of the set are highly
related to the MCDM problem for decision purpose; (2) The set of attributes is parsimonious, and
the selected alternative will be suboptimum if one of these attributes is omitted. The rationale
of attribute selection is similar to feature selection in data mining field. There are a lot of feature
selection methods have been proposed, but only a few of them can be applied in decision-making
process directly which are mentioned in the Literature Review part. Most of these methods can’t

Copyright ©2018 CC BY-NC



392 X. Ma, Y. Feng, Y. Qu, Y. Yu

account for both objective and subjective perspectives, so there is a low reliability of decision
attributes. This study is focused on how to select the optimal set of decision attributes for a
MCDM problem, external attribute data and subjective decision preferences are utilized in a
sequential procedure to enhance the reliability of decision attributes.

The attribute selection method for decision problem should account for both objective and
subjective perspectives, more precisely, this paper merges attribute data (objective) with DMs’
requirement (subjective). Utilize the former for rational, constrained modeling and the latter for
adapting specific problem issues to the decision-making process. In order to obtain the optimal
set of attributes, an analysis framework for attribute selection problem and a two-step screening
procedure is conducted in this paper. The specific procedure is illustrated in Figure 1.

Figure 1: The illustration for the presented method

This new attribute selection method in MCDM contributes to selecting the optimal set of
decision attributes and helping DMs choose optimal alternatives better. Meanwhile, this method
merges objective data and subjective preferences to improve performance for actual decision
scenarios.

2 Literature review

Many feature selection methods have been proposed, and they are usually classified into three
classes: "filter" methods, "wrapper" methods and "embedded" methods [2,7,8,14]. Wu [19] pro-
posed using the fuzzy and grey Delphi methods to identify a set of reliable attributes and,
based on these attributes, transforming big data to a manageable scale to consider their impacts.
Meinshausen et al. [11] demonstrated linear model and Gaussian model of variable selection
consistency in higher dimensional case. Although they have good performance, they cannot be
applied in decision-making process directly (partial principle may be accepted). Only a few
papers mentioned attribute selection in decision-making problems; for instance, Chun [4] con-
sidered the "optimizing" and "satisficing" (a portmanteau of satisfy and suffice) attributes and
deal with the multi-attribute decision problem with sequentially presented decision alternatives;
Dai et al. [6] constructed three attribute selection approaches in context of incomplete deci-
sion systems based on information-theoretical measurement of attribute importance; in order to
screen the critical factors influencing the stability of perilous rock, Meng et al. [12] used fuzzy
compromise TOPSIS method to calculate the importance of attributes in the decision problem.



Attribute Selection Method based on Objective Data
and Subjective Preferences in MCDM 393

Wu [18] used Fuzzy Delphi method to screen out the unnecessary attributes to deal with the
complex interrelationships among the aspects and attributes. Attribute selection procedure can
improve the decision performance, but no papers used this procedure in MCDM problems.

3 Method

3.1 Preliminaries

Trapezoidal fuzzy numbers

Definition 1. Let X be a universe set. A fuzzy ã in a universe of discourse X is characterized
by a membership function µã(x), which associates with each element x in X, a real number in
the interval [0,1]. The function is termed the grade of membership of x in ã.

Definition 2. A tuple à = (a,b,c,d),a ≤ b ≤ c ≤ d, is called a trapezoidal fuzzy number (TFN)
if its membership function is

µã(x) =




(x−a)/(b−a) a ≤ x ≤ b
1 b ≤ x ≤ c
(d−x)/(d− c) c ≤ x ≤ d
0 otherwise

Where a, b, c, d are real numbers.

Definition 3. Given two trapezoidal fuzzy numbers à = (a1,b1,c1,d1), à = (a2,b2,c2,d2), and
a real number λ, the main operations can be expressed as follows:

1○Ã1
⊕
Ã2 = (a1 + a2,b1 + b2,c1 + c2,d1 + d2)

2○Ã1
⊗
Ã2 = (a1a2,b1b2,c1c2,d1d2)

3○λ
⊗
Ã2 = (λa1,λb1,λc1,λd1)

Multi-objective optimization

Let χ be a vector containing n decision variables and in a universe of discourse X. Mathe-
matically, an optimization problem with p objective functions can be expressed as (Mahdi and
Seyed, 2012):

minimize fi(x) for i = 1, 2, ...,p
subject to: gj(x) ≤ 0,j = 1, 2, ...,q,

hj(x) = 0,j = q + 1, 2, ...,m

where x = (x1,x2, ...,xn), xl is the lth decision variable; p and m are respectively the numbers
of objective function and constraint.

A variety of methods can be used to solve this problem. One popular method is to combine
those objectives into a single composite objective so that traditional mathematical programming
methods can be applied. And other approaches are based on the Pareto optimum concept, and
more specifically, let y = (y1,y2, ...,yn) be another vector containing n decision variables in X.

Definition 4. (Domination) x ∈ X dominates y ∈ X, denoted x � y, if ∀j ∈ p, vj(x) ≤ vj(y)
with at least one strict inequality.

Definition 5. (Pareto Optimal) x is a Pareto Optimal (PO) solution, if there is no y ∈ X such
that y � x. The set of all Pareto optimal solutions in X is denoted PO(X).



394 X. Ma, Y. Feng, Y. Qu, Y. Yu

These methods include Multiple Objective Genetic Algorithm (MOGA), Non-dominated
Sorting Genetic Algorithm (NSGA), NSGA-II, Multi-objective Multi-state Genetic Algorithm
(MOMS-GA), Niched-Pareto Genetic Algorithm (NPGA) and Multi-objective Particle Swarm
Optimization (MOPSO) [9,21].

An analysis framework of attribute selection under MCDM

Multi-Criteria Decision Making (MCDM) has been widely applied as a well-known branch of
decision making, and can be further divided into multi-objective decision making (MADM) and
multi-attribute decision making (MODM) [5,13]. The whole hierarchical structure of a MCDM
problem is shown in Figure 2, and it can be unfolded from four hierarchies from top to the
bottom, namely, objective, criteria, attribute and alternative.

(1) Objective Hierarchy: Generally contain multiple decision objectives given by DMs which
reflect different decision requirement.

(2) Criteria Hierarchy: Criteria Hierarchy is to realize a fair comparison of alternatives from
the various aspects of Objective Hierarchy, and it is also developed in a hierarchical fashion,
starting from some general but imprecise criteria description, which is refined into more
precise sub- and sub-sub criteria.

(3) Attribute Hierarchy: Attribute Hierarchy is determined by criteria Hierarchy and Objective
Hierarchy. It is generally assumed that each criterion can be represented by some measur-
able attributes of the consequences arising from implementation of any particular decision
alternative [16], as well as objective. No matter what kind of MCDM methods, decision
attributes are the bottom and direct evaluative parameters to determine alternatives.

(4) Alternative Hierarchy: Consist from candidate alternatives evaluated by the set of decision
attribute.

Figure 2: The hierarchical structure of a MCDM problem

This paper focus on how to obtain the optimal set of decision attributes from Attribute
Hierarchy for evaluating alternatives.



Attribute Selection Method based on Objective Data
and Subjective Preferences in MCDM 395

Rationale of the proposed method

Let O = {O1,O2, ...,Oγ} be a finite set of objectives, and C = {C1,C2, ...,Cn} be the set of
criteria. In most actual cases, objectives and criteria are always predetermined by MDs according
to different knowledge background and decision experiences, thus we assume that all of objectives
and criteria (or sub-criteria) are given.

We define decision attribute selection procedure as a sequential selection procedure based on
objective and subjective information. It means that a series of screening steps will be applied
in sequence to obtain an optimal set of attributes. A sequential screening can be described as
follows [3]:

Scrh,h−1,...,1(Aoriginal) = Scrh(Scrh−1(...(Scr1(Aoriginal))...)) (1)

where Scrk,k = 1, 2, ...,h are screening steps, and Scrh is the final screening step in this
sequential screening; Aoriginal = {aj|j ∈ N} is the original set of decision attributes.

And two screening steps are conducted in proposed selection procedure, namely, Scr2,1(A) =
Scr2(Scr1(A)).

Scri is a screening step based on real-time attribute data, and the corresponding screening
algorithm is established in the step. The result Scri(A) is to obtain a relatively optimized set of
decision attributes AScr1 by deleting redundant attributes.

Scr2 is a further screening step based on the results of Scr2(A) to acquire the absolutely
optimal set of decision attributes AScr2. Decision goals that represent subjective preference are
utilized in this step, and the selected set should satisfy all decision goals.

The whole procedure of attribute selection is shown by Figure 3

Figure 3: The procedure of the proposed method

where AScr1 = {{A1},{A1}, ...,{Ap}, ...,{An}}, and Ã
b
Scr2

is candidate optimal set of Scr2.

3.2 A new method for attribute selection

Preprocessing of the data

We consider three data types of decision attributes in this paper, namely, real numbers
expressed as α0, intervals expressed as [αL,αU ] and linguistic term set expressed as a linguistic
term set S = {s0,s1, ...,sK}. And real-time attribute data can be listed as attribute value series
in Table 1.

where c ≤ t1 < t2 < ... < tm ≤ d and [c,d] is a time interval.



396 X. Ma, Y. Feng, Y. Qu, Y. Yu

Table 1: Attribute value series

α1 α2 ... αj ...
t1 α1(t1) α2(t1) ... αj(t1) ...
t2 α2(t2) α2(t2) ... αj(t2) ...
...
ti αi(ti) α2(ti) ... αj(ti) ...
...
tm αm(tm) α2(tm) ... αj(tm) ...

Both Scr1 and Scr2 are quantification procedures and therefore, in order to guarantee the
scientificity and preciseness of the presented method, preprocessing of attribute data need to be
carried out firstly (referring to our previous work [10]).

(1) Convert attribute values to TFNs

For a real number α0, it can be denoted as a TFN (α1,α2,α3,α4), where α1 = α2 =
α3 = α4 = α0; for internal value bαL,αUc, the corresponding TFN can also be expressed
as (α1,α2,α3,α4), where α1 = α2 = αL and α3 = α4 = αU. It is a little complex for a
linguistic term set with odd cardinality S = {s0,s1, ...,sK}, and the element sK is converted
to the corresponding TFN as follows:

(α1k,α
2
k,α

3
k,α

4
k) =

max { 2k − 1
2K + 1

, 0

}
,

2k

2K + 1
,

2k + 1

2K + 1
, min

{
2k + 2

2K + 1
, 1

} (2)
where k = 0, 1, ...,K.

(2) Normalize values of attributes

In real-world decision scenarios, some decision attributes are cost ones, which mean the lower
the values of attributes, the better they will be; the others are benefit ones, which mean the
higher the values of attributes, the better they will be. Without loss of generality, we
assume that the attribute values can be expressed as αj(ti) = (α1j (ti),α

2
j (ti),α

3
j (ti),α

4
j (ti)),

and normalized attribute values can be denoted as γj(ti) = (r1j (ti),r
2
j (ti),r

3
j (ti),r

4
j (ti)). The

normalized methods are given as follows:

• For cost attributes:

rvj (ti) =
max
i

(
α4j (ti)

)
−αvj (ti)

max
i

(
α4j (ti)

)
− min

i

(
α1j (ti)

) v = 1, 2, 3, 4. (3)
• For benefit attributes:

rvj (ti) =
αvj (ti) − min

i

(
α1j (ti)

)
max
i

(
α4j (ti)

)
− min

i

(
α1j (ti)

) v = 1, 2, 3, 4. (4)
And the normalized attribute value series can be denoted as
γ1 = {r1(t1),r1(t2), ...,r1(tm)};



Attribute Selection Method based on Objective Data
and Subjective Preferences in MCDM 397

γ2 = {r2(t1),r2(t2), ...,r2(tm)};
...;

γj = {rj(t1),rj(t2), ...,rj(tm)};
....

The first step for screening Scr1

In Scr1, one "relatively best" representative attribute will be chosen by DMs for each crite-
rion. It means that,

For ∀Cp ∈{C1,C2, ...,Cn}, 1 ≤ p ≤ n,
∃ap1 ∈ Aoriginal,st.Cp ⇔ ap1.
For example, "Economic losses" is a criterion in emergency decision, and "Amount of loss"

is the "relatively best" attribute to evaluate loss degree. Thus the set {ap1|1 ≤ p ≤ n} is a
significant by-product after criteria determination, and actually it is the input of Scr1.

Scr1 is used to screen decision-relevant attributes from Aoriginal by analyzing attribute in-
terrelation. More precisely, we will investigate geometrical relationship between attributes based
on attribute value series; select highly correlative attributes for ap1 from Aoriginal to constitute a
"relatively best" representative set for each criterion Ci, and omit the rest of decision-irrelevant
attributes. The step is entirely implemented on attribute value series without subjective impacts,
and the outcome AScr1 is a smaller and relative optimal set which the elements are determined
by evaluation criteria.

The idea of Scr1 arises from Grey Relational Analysis (GRA) Theory which provides a valid
way to quantify the interrelation of different factors using geometric methods [22]. And the
detailed process of Scr1 is given in the following.

Step 1.1: Normalize attribute value series
It is different from normalization of attribute values, and it guarantees comparability of

different attribute value series. The specific process is

yj = {aj(ti)/Dj, i = 1, 2, ...,m} (5)

Dj =
1

m− 1

m∑
i=2

|aj(ti) −aj(ti−1)| (6)

where yj is normalized attribute value series, and Dj is increment average of yj.
Step 1.2: Calculate increment series

∆yj = {∆yj(ti) = yj(ti) −yj(ti−1), i = 2, 3, ...,m} (7)

Step 1.3: Calculate correlation coefficient
AScr1 is obtained based on the input set {ap1|1 ≤ p ≤ n} and thus, Step 1.3 will be stepwise

conducted for each element ap1. The correlation coefficient of different ti between ap1 and aj can
be defined as follows:

ξ(ti) =

{
sgn(∆yp1(ti) · ∆yj(ti)) ·

min(|∆yp1(ti)|,|∆yj(ti)|)
max(|∆yp1(ti)|,|∆yj(ti)|)

; j 6= p1
0 ∆yp1(ti) · ∆yj(ti) = 0

(8)

where sgn(·) is a sign function; and sgn(∆yp1(ti) · ∆yj(ti)) = 1, if ∆yp1(ti) · ∆yj(ti) >
0; sgn(∆yp1(ti) · ∆yj(ti)) = −1, if ∆yp1(ti) · ∆yj(ti) < 0.

Step 1.4: Calculate correlation degrees
The correlation degree between ap1 and aj can be obtained by



398 X. Ma, Y. Feng, Y. Qu, Y. Yu

τ(ap1,aj) =

∣∣∣∣∣ 1d− c
m∑
i=2

∆ti · ξ(ti)

∣∣∣∣∣ (9)
The correlation degrees reflect the closeness of two attributes; the greater the value of

τ(ap1,aj) is, the closer the two attributes are, and the better the capability of aj is to eval-
uate criterion Cp.

Step 1.5: Set threshold and screen correlation attributes for ap1
For ∀Cp, a threshold of correlation degree τ∗ should be set by DMs, and the attribute aj will

be reserved, if τ(ap1,aj) > τ∗. And the reserved attributes for evaluating Cp can be denoted as
{Ap} = {ap1,ap2, ...,apθp}, where θp is the sum of attributes. In addition, the correlation degree
τ(ap1,apqp) between ap1 and apqp will be abbreviated as zpqp, where 1 ≤ qp ≤ θp.

Step 1.6: Acquire the screening result set AScr1
Repeat Step 1.1 to Step 1.5 for each ap1, where 1 ≤ p ≤ n; and the result set AScr1 can be

obtained, namely,
AScr1 = {{A1},{An}, ...,{An}}.

The second step based on multi-objective optimization Scr2

Implementing decision attribute selection is aimed to assist DMs in evaluating alternatives for
given decision problems and therefore, the selected attributes should be sensitive to the decision
problem as well as familiar to DMs. Sensitivity reflects whether the selected attributes are
accurate enough to represent the decision problem, and familiarity is used to estimate whether
DMs are certain enough to make decision based on the selected attributes. For example, "fire
duration" and "indoor fire load" are more important (sensitive) attributes than "carbon monoxide
concentration" for evaluating fire grades; while it is more reliable for medical staff to estimate
the physical condition of the wounded with "carbon monoxide concentration" than "Radiative
Heat Flux".

Multi-objective optimization theory is a feasible way to quantify and synthesize these re-
quirements, and it has solved similar problems in other areas successfully [1]. Thus three main
objectives will be considered in this paper: attribute amount, attribute utility and attribute
familiarity. Attribute amount should be as few as possible without reducing the validity of al-
ternative, and it is the demand of both Scr2 and attribute selection method. Attribute utility is
used to distinguish attribute sensitivity in estimating the same decision problem, and attribute
familiarity explores DMs’ cognition differences for decision attributes. The outcome AScr2 drawn
from Scr2 should satisfy mentioned three goals, and now the optimal set of decision attributes
have been found after Scr1 and Scr2, namely, AScr2 = Scr2(Scr1(Aoriginal)).

A set of zero-one binary vectors {vb} with T = (θ1 +θ2 +...+θn) dimensionality will be defined
to represent different candidate outcome set ÃbScr1, where one represents the attribute belonging
to the set, otherwise it is zero. The optimal set of decision attributes is AScr2 ∈{Ã

b
Scr1
|b ∈ N},

and {ÃbScr1} contains all the subsets of AScr1.
Three objective functions are defined firstly, and we will show that all the objective functions

can be expressed as the functions with independent variable vb.

Defining objective functions (1) Attribute utility function
Attribute utility function measures decision sensitivity of ÃbScr1, and it can be defined as

follows:

U(ÃbScr1 ) = g(MU(Ã
b
Scr1

),W(ÃbScr1 )) (10)



Attribute Selection Method based on Objective Data
and Subjective Preferences in MCDM 399

where

1○ MU(ÃbScr1 ) = v
b ·MU∗ (11)

MU∗ represents attribute utility matrix, and it is diagonal matrix with the diagonal elements
∆a11
∆a11

, ..., ∆a11
∆a1θ1

, ...,
∆ap1
∆apqp

, ..., ∆an1
∆an1

, ..., ∆an1
∆anθn

.
More precisely, MU∗ can be explained by a marginal utility function u(·), namely,

u(apqp) =
∆ap1
∆apqp

1 ≤ p ≤ n, 1 ≤ qp ≤ θp; (12)

where ∆ap1 is increment of ap1, and ∆apqp is the corresponding varying of apqp. And the
function u(·) quantifies the assessment performance on criterion Ci of the attribute apqp according
to calculate the ratio between increments of the two attribute values.

In fact, MU∗ can be predetermined by domain experts, because plenty of field studies have
been conducted to investigate utility relation of attributes.

2○ W(ÃbScr1 ) = W
∗ · (vb)′ (13)

W∗ represents attribute weighting matrix, and it is also a diagonal matrix with diagonal
elements w11, ...,w1θ1, ...,wpqp, ...,wn1, ...,wnθn.

where

wpqp = ωp ·zpqp, 1 ≤ p ≤ n, 1 ≤ qp ≤ θp; (14)

and {ω1,ω2, ...,ωn} are the know weights of C = {C1,C2, ...,Cn}.
g(·) is a dot product function and therefore, U(ÃbScr1 ) can be transformed into the following

function with independent variable vb, namely,

U(vb) = (vb ·MU∗) · (W∗ · (vb)′) (15)

(2) Attribute familiarity function
Attribute familiarity function is established to quantify DMs’ cognition degrees on different

candidate set ÃbScr1, and it is defined as

F(vb) = η ·σ · (vb)′ (16)

In Eq. 16, η = (η1,η2, ...,ηL)1×L is the weight vector of DMs, and L is the number of DMs;
σ is attribute familiarity matrix with the following expression

σ =




σ1(1) σ2(1) ... σT (1)
σ1(2) σ2(2) ... σT (2)
· · ·
· · ·
· · ·

σ1(L) σ2(L) ... σT (L)



L×T

(17)

and σh(l) represents the lth DM’s familiarity degree with attribute ah, where 1 ≤ l ≤ L, and
1 ≤ h ≤ T . Attribute familiarity is usually evaluated qualitatively by DMs, and many methods
can be applied to quantify assessment results and obtain matrix σ, such as AHP and fuzzy set
theory.

(3) Attribute count function



400 X. Ma, Y. Feng, Y. Qu, Y. Yu

Attribute count function calculates the attribute number of a candidate set ÃbScr1, and it can
be defined by means of vector length formula,

Count(ÃbScr1 ) = |v
b|2 (18)

where |vb| represents the length of vector vb.

Establish multi-objective optimization model for Scr2 After define relevant objective
functions, the multi-objective optimization model of Scr2 is established to obtain the Pareto
Optimal solutions AScr2, as well as the optimal set of decision attribute selection problem.

Maximize U(vb) = (vb ·MU∗) · (W∗ · (vb)′)
F(vb) = η ·σ · (vb)′

(19)

Minimize Count(ÃbScr1 ) = |v
b|2

Subject to : |vb|2 ≥ n

The model Eq. 19 can be solved by aforementioned algorithm, such as MOGA, NSGA,
MOMS-GA and traditional mathematical programming methods, and it depends the specific
model drawn from given decision problems and decision requirements.

4 Empirical results

4.1 Accident scenario and computational settings

In order to validate the presented method, a fire accident scenario will be constructed and
simulated. The fire happens in a factory dormitory, and two workers are asleep; some inflammable
are stacked near the door which arise spontaneous combustion for some reasons. The total layout
of this room is illustrated in Figure 4, containing three beds, two workers marked by fellow
cuboids, fire origin and plenty of fire smoke. Now two experts need to make rescue alternative
based on the collected data; one is fire expert who are engaged in fire research for many years and
have in-depth study of the fire attributes, and the other is an experienced firefighter who know
some medical knowledge. And they are weighted with 0.6 and 0.4 respectively. A fire simulation
software FDS [17] is used to establish fire scenario and generate fire data.

The purpose of simulation is not only to provide a fire scenario, and more importantly, it can
generate relevant fire attributes data for the following experiment. The data can be exported
from FDS containing attribute names, physical dimensions, attribute values and attribute types.
In this paper, we generate twelve fire attributes and 125 data records.

4.2 Select fire attributes by utilizing the presented method

In this fire scenario, three main criteria will be considered, i.e., fire behavior, personal security
and environmental conditions, and corresponding attributes are shown in Table 5

where "Enthalpy" is an interval attribute, "Indoor fire load" and "Circuit aging degree" are
linguistic attributes, and the rest are real number attributes. The bold attributes are "relatively
best" representative attributes for each criterion.

(1) Data preprocessing



Attribute Selection Method based on Objective Data
and Subjective Preferences in MCDM 401

Figure 4: A simulated fire scenario

Table 2: List of criteria and corresponding attributes

Criterion Attributes
Fire behavior Temperature / Enthalpy / Heat Release Rate / Relative

Humidity / Burning Rate / Pressure / Radiative Heat Flux
Personal security Carbon monoxide / Carbon dioxide / oxygen
Environmental conditions Indoor fire load / Circuit aging degree

As aforementioned, the real number attributes and interval attributes can be denoted by
TFNs directly. The linguistic attributes "Flammable of storage items" and "Circuit aging degree"
can be transformed into TFNs in Table 3.

Table 3: Linguistic attributes and corresponding TFNs

Linguistic terms TFNs
Not dangerous (0,0,0.111,0.222)
Medium dangerous (0.111,0.222,0.333,0.444)
Dangerous (0.333,0.444,0.555,0.666)
Very dangerous (0.555,0.666,0.777,0.888)
Highly dangerous (0.777,0.888,0.999,1)

Then we carry out normalization by Eq. 3 and Eq. 4 to obtain corresponding TFNs.
(2) The first step for screening Scr1
Without loss of generality, the criterion "Fire behavior" will be utilized to illustrate the

first screening procedure and the other two criteria "Personal security" and "Environmental
conditions" can be dealt with similarly.



402 X. Ma, Y. Feng, Y. Qu, Y. Yu

The criterion "Fire behavior" is represented by seven attributes, namely, "Temperature",
"Enthalpy", "Heat Release Rate", "Relative Humidity", "Burning Rate", "Pressure" and "Ra-
diative Heat Flux"; and "Temperature" is the "relatively best" representative attribute for the
criterion "Fire" behavior.

Step 1.1 to Step 1.4 are implemented to calculate correlation degrees between ""Tempera-
ture" and the remaining six attributes, and the results are given in Table 4.

Table 4: Correlation degrees between "Temperature" and the other six attributes

Attribute Enthalpy Heat Release
Rate

Relative
Humidity

Burning
Rate

Pressure Radiative
Heat Flux

Correlation degree 0.53745 0.96987 0.33271 0.74787 0.47665 0.88712

Step 1.5: Set threshold of correlation degree τ∗ = 0.6
Thus the attributes will be reserved whose correlation degrees is greater than or equal to 0.6,

i.e., "Burning Rate", "Heat Release Rate" and "Radiative Heat Flux".
Step 1.6: Repeat Step 1.1 to Step 1.5 for each criterion
For criteria "Personal security" and "Environmental conditions", the "relatively best" rep-

resentative attributes are respectively "Carbon monoxide" and "Indoor fire load", and relevant
correlation degrees are shown in Table 5.

Table 5: Relevant correlation degrees for the other two criteria

Criterion Personal security Environmental conditions
Attribute Carbon dioxide oxygen Circuit aging degree
Correlation degree 0.80961 0.93428 0.76865

We reset thresholds of correlation degree for each criteria, viz., τ∗Personal = 0.9 and τ
∗
Environmental =

0.8.
Finally, the result of the first screening AScr1 can be obtained, namely,
AScr1 = {"Temperature", "Heat Release Rate", "Radiative Heat Flux", "Burning Rate",

"Carbon monoxide", "oxygen", "Indoor fire load"}.
(3) The second step for screening Scr2
Before conducting Scr2, MU∗, σ and W∗ should be given firstly. As aforementioned, MU∗

and σ can be determined by relevant research achievements and expert experiences, and therefore,
we collected and integrated evaluation results from ten fire researchers to assign MU∗ and σ as
follows:

MU∗ =




19.5
15.6

7.4
11.2

15.4
18.7

18.2




and σ =
(

9 8 8 9 4 4 5
4 5 4 4 9 9 7

)

where the elements values of MU∗ are in the interval [0, 20], and the greater the value is,
the more sensitive the attribute is for evaluation. The elements values of σ are in the interval [0,
10], and the greater the value is, the more familiar the attribute is to DMs.



Attribute Selection Method based on Objective Data
and Subjective Preferences in MCDM 403

Moreover, W∗ can be acquired with the given criteria weight vector (ω1,ω2,ω3) = (0.4, 0.4, 0.2),
namely,

W∗ =




0.299
0.388

0.215
0.354

0.352
0.374

0.197




After determine relevant parameters MU∗, W∗, σ, and the seven-dimensional independent
variable vb, Scr2 based on the multi-objective optimization model Eq 19 is performed in Matlab
R2013b. We choose the traditional mathematical programming method to solve this model,
because the research [15] has proven that the programming method has better performance than
aforementioned other algorithms to solve such a problem with the zero-one binary vector variable
and the small size of data.

The results are shown in Figure 5 where the curve represents all possible solutions and
the Pareto Optimal solution is the intersection of the curve and the x-axis, namely, vb =
(1, 1, 0, 1, 1, 1, 1) . It means that the attribute "Radiative Heat Flux" represented by zero can be
omitted, and the optimal set of decision attributes is:

AScr2 = {"Temperature", "Heat Release Rate", "Burning Rate", "Carbon monoxide", "oxy-
gen", "Indoor fire load"}.

Figure 5: The results of the second step Scr2

4.3 Validation of the presented method

In order to verify the result is the optimal set for a MCDM problem, multi-attribute utility
theory will be used in the following.

A conclusion underlies the presented method: Removing decision-irrelevant attributes from
the set of decision attributes will improve the effectiveness of decision alternatives. Thus it is
reasonable and feasible to verify our work starting from Scr2, because the purpose of Scr1 is to
omit decision-irrelevant attributes.



404 X. Ma, Y. Feng, Y. Qu, Y. Yu

Multi-attribute utility theory (MAUT) is used to choose decision alternatives by evaluating
the utility values of alternatives which is calculated based on the utility of decision attributes, and
it is an effective way to quantify decision attributes’ impact on decision results. More precisely,
the utility evaluation model of an alternative is given as follows:

u(x1,x2, ...,xn) =

n∑
i=1

kiui(xi) if

n∑
i=1

ki = 1 (20)

where xi is the ith decision attribute, and ki is the corresponding weight; ui(xi) is the at-
tribute utility function of xi, and u(x1,x2, ...,xn) is the utility function of an alternative evaluated
by x1,x2, ...,xn.

Thus the utility values of all possible candidate sets ÃbScr1 are calculated based on Eq. 20,
where attribute utility function ui(xi) can be defined as Eq. 12, and the weights are acquired
by Eq. 14. It should be pointed that normalization of weights need be carried out to guarantee
n∑
i=1

ki = 1. And the results are given in Table 6, where Si represents the candidate set of decision

attributes, and 1 ≤ i ≤ 16. The correspondence between Si and the set of attributes are shown
in Table 7.

Table 6: The utility values of attribute sets

Attribute set S1 S2 S3 S4 S5 S6 S7 S8

Utility value 23.621 24.343 28.496 28.923 29.665 31.745 28.474 29.500
Attribute set S9 S10 S11 S12 S13 S14 S15 S16

Utility value 25.908 28.447 32.631 31.229 30.335 35.774 31.658 33.439

Table 7: The correspondence between Si and the set of attributes

Si The correspondence set Si The correspondence set
S1 (1, 0, 0, 0, 1, 0, 1) S9 (1, 0, 0, 0, 1, 1, 1)
S2 (1, 1, 0, 0, 1, 0, 1) S10 (1, 1, 0, 0, 1, 1, 1)
S3 (1, 0, 1, 0, 1, 0, 1) S11 (1, 0, 1, 0, 1, 1, 1)
S4 (1, 0, 0, 1, 1, 0, 1) S12 (1, 0, 0, 1, 1, 1, 1)
S5 (1, 1, 1, 0, 1, 0, 1) S13 (1, 1, 1, 0, 1, 1, 1)
S6 (1, 1, 0, 1, 1, 0, 1) S14 (1, 1, 0, 1, 1, 1, 1)
S7 (1, 0, 1, 1, 1, 0, 1) S15 (1, 0, 1, 1, 1, 1, 1)
S8 (1, 1, 1, 1, 1, 0, 1) S16 (1, 1, 1, 1, 1, 1, 1)

In Table 6, for ∀i, 1 ≤ i ≤ 16, it has u(S14) > u(Si), where i 6= 14. The set S14 has
the maximum utility value u(S14) = 35.774, and thus the alternative chosen based on S14 (the
optimal set AScr2) is the optimal alternative.

In addition, S1 contains three attributes with minimum utility value u(S1) = 23.621, and S16
contains all attributes with utility value u(S16) = 33.439. It means that too few or too many



Attribute Selection Method based on Objective Data
and Subjective Preferences in MCDM 405

attributes are not available in a MCDM problem; too few attributes are not enough to evaluate
the decision problem, while too many attributes may lead to conflict and interfere decision results.

The set of decision attributes obtained by utilizing the presented method has the maximum
utility value in alternative evaluation, and it demonstrates the effectiveness of the presented
method.

5 Implications

This paper utilizes attribute selection procedure in MCDM problems innovatively. MCDM
problems refer to getting the optimal alternatives which are determined based on many quali-
tative or quantitative criteria, and these criteria are conflicting and assessed by more relevant
attributes. Therefore, the set of decision attributes determines the effectiveness of an alternative.
In order to improve the reliability of MCDM, the method proposed in this study can select the
optimal set of decision attributes which are highly related to the MCDM problem and remove
redundant or "noisy" attributes. Meanwhile, the selection of optimal set can reduce the attribute
dimension and improve the efficiency of decision making while maintaining the great effect.

The method combines external attribute data with subjective decision preferences in the
sequential procedure to improve decision performance. External attribute data are objective
factors, which show the features of the data. These attributes contain its own regularity and
all kinds of information that decision-making needs, so the first step to screen is based on
objective attribute data. Subjective factors show the decision makers’ preferences. Because of
the different profession and background knowledge, decision makers have different recognition of
attributes. Considering the subjective decision preferences contributes to improve the reliability
and accuracy of decision-making. Using the former to build the method and using the constraint
conditions of the latter to adjust the method can make it more in line with the actual situation.

The method combines external attribute data with subjective decision preferences in the
sequential procedure to improve decision performance. External attribute data are objective
factors, which show the features of the data. These attributes contain its own regularity and
all kinds of information that decision-making needs, so the first step to screen is based on
objective attribute data. Subjective factors show the decision makers" preferences. Because of
the different profession and background knowledge, decision makers have different recognition of
attributes. Considering the subjective decision preferences contributes to improve the reliability
and accuracy of decision-making. Using the former to build the method and using the constraint
conditions of the latter to adjust the method can make it more in line with the actual situation.

6 Conclusions

In this paper, a new attribute selection method with the context of MCDM is proposed.
According to the analysis of attribute selection problem, a two-step procedure is established to
reduce original set to the optimal set of decision attributes. GRA theory and multi-objective
optimization method are respectively used to implement these two screening steps. And then a
fire example is shown to illustrate application and validity of screening method. The attribute
selection method presented in this paper is a dynamic and flexible procedure which depends on
decision data resource and decision requirement. It eliminates the drawback that decision at-
tributes are chosen completely subjectively, and it will show better performance for new decision
scenarios.

Attribute selection is also a critical process for many domains, such as classification problem,
clustering problem, risk assessment, credit assessment, etc. And even solution of these issues



406 X. Ma, Y. Feng, Y. Qu, Y. Yu

depended on results of attribute selection. Attribute selection problem is also discussed in data
mining called "Feature Engineering", and still has needs to continue thorough research and the
technique problem to be solved.

Generally it exists plenty of missing data among decision information, and how to select the
optimal decision attributes in this case will be our future research.

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