Multi-Level Grey Risky Multi-Attribute Decision-Making Method
Based on Dynamic Regret Theory
Lili Qian and Guifeng Deng
School of Statistics and Mathematics, Shanghai Lixin University of Accounting and Finance, Shanghai 201209, China
Keywords: Multi-Level Decision-Making, Risky MADM, Dynamic Regret Theory, General Grey Number (GGN),
Kernel, Degree of Greyness.
Abstract: There are some complex decision-making problems with incomplete information and multiple choices in the
uncertain environment, in which the decision-makers are more vulnerable to the influence of their emotional
psychology. Thus a new method based on dynamic regret theory is introduced in this paper for solving the
multi-level risky multi-attribute decision-making problem with general grey numbers (GGNs).
First,
normalize GGNs and calculate the probability of the combined state to construct the combined decision
matrix. Then, the dynamic regret theory is integrated into the grey multi-level decision-making problem by
constructing grey dynamic regret function and grey dynamic perceived utility function. Finally, the regret
equilibrium set is established and the best scheme is selected according to the comprehensive grey perceived
utility value.
The model proposed in this paper can solve the multi-level grey risky decision-making problem
with the consideration of the regret emotion. It not only complements and perfects the multi-level decision-
making method, but also widens the research scope and space of dynamic regret theory.
1 INTRODUCTION
In real life, there are many uncertain decision-making
problems, that is, decision-making involves
fuzziness, randomness, grey and so on. For example,
decision makers may encounter the interval grey
number whose true value is unknown but the range of
the value is known (Liu and Lin, 2006). The interval
grey number is the basic concept in grey system
theory (Deng, 1982) which is an effective tool to
process the systems with some known information
and some unknown information. Decision
information is grey in this circumstance. In addition,
the attribute values in some decision-making
problems may be stochastic variables which will
change with the different state of nature. The
decision-maker can predict various possible natural
states, or quantify this randomness by setting the
probability distribution, but the decision-maker
cannot get the real state in the future. Decision
information is random and risky. This decision-
making problem is risky multi-attribute decision-
making(RMADM). Due to the limitations of human
cognition and the increasing complexity of practical
problems, decision problems are often both grey and
stochastic, which is called grey risky decision-
making. It can objectively describe more uncertain
decision-making problems and has a broad
application background. Luo and Liu (2004) have
constructed a risky decision-making model based on
interval grey number and provided the grey risky
decision-making method.
However, there are still practical decision-making
conditions where interval grey numbers cannot
accurately or fully describe information. With the
continuous in-depth development of research, a kind
of number, called a general grey number (GGN),
which is in union of open or closed grey intervals, is
introduced by Liu, Fang and Yang
(2012). They think
the general grey number can better describe the
uncertainty. Suppose the market share of the products
of a company is estimated in three ways. The interval
grey number [0.67, 0.82] is less accurate than the
general grey number [0.67,0.7] ∪ [0.73,0.75] ∪
[0.77,0.82] in describing the market share. At
present, little research has been done on risky multi-
attribute decision-making with general grey numbers
(GGNs). Qian et al.
(2019) studied the risky multi-
attribute decision-making based on the general grey
number with kernel and greyness and integrated the
regret theory into the problem. Zhou et al. (2017a)
proposed the stochastic multi-criteria decision-
664
Qian, L. and Deng, G.
Multi-Level Grey Risky Multi-Attribute Decision-Making Method Based on Dynamic Regret Theory.
DOI: 10.5220/0012012200003612
In Proceedings of the 3rd International Symposium on Automation, Information and Computing (ISAIC 2022), pages 664-673
ISBN: 978-989-758-622-4; ISSN: 2975-9463
Copyright
c
 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

making method with the extended grey number,
which was introduced by Yang
(2007). When dealing
with grey number operation, Zhou et al. (2017a)
adopted the idea and method of interval numbers to a
certain extent, and could not catch the essential
characteristics of grey numbers. As a matter of fact,
the basic concepts such as the kernel, the degree of
greyness, and the field of general grey numbers have
been proposed by Liu et al.
(2012), which can
construct the reduced form of a general grey number
and provide important information of the general grey
number.
In addition, how the psychological behaviour of
the decision-maker affects the actual decision-
making process is also a hot spot of current research.
A large number of practice and research show that in
general, it is difficult for decision-makers to fully
obtain accurate decision-making information. They
are also affected by subjective factors such as their
own emotions, psychological behaviour and
experience intuition, thus making the decision-
making process show the characteristics of irrational
or bounded rationality. That is, decision makers do
not always pursue the maximum utility, but make
satisfactory decisions according to their own
psychological and emotional behaviour and limited
experience cognition. In the grey risky decision-
making problem, the future environment is uncertain,
and the information value is not completely accurate,
which is grey. This dual uncertainty makes the
decision-making more complex. Therefore, decision-
makers are more vulnerable to the influence of
emotional psychology in the stages of information
acquisition, cognitive judgment and overall
evaluation, thus showing a variety of irrational
behaviour characteristics. Researchers hope to find a
better way to explain this personal behaviour
decision-making, which makes the classical expected
utility theory continue to be expanded and improved.
Among them, the most representative research results
are the prospect theory proposed by Kahneman and
Tversky (1979) and the regret theory independently
proposed by Loomes and Sugden (1982) and Bell
(1982). At present, many researchers apply prospect
theory to grey risk decision-making, and rich results
have been achieved. As the emotion that you want to
avoid most in the decision-making process, regret has
also attracted much attention in the field of decision-
making research. Regret theory has gone through a
process of continuous improvement. At first, it was
applied to pair by pair selection, and then it was
extended to a limited number of alternative objects.
Finally, Quiggin (1994) extended it to the case of
general selection sets. So regret theory can be applied
to the case of any infinite alternative actions.
Zeelenberg (1999) believed that anticipated regret
could encourage decision makers to avoid risks and
seek risks at the same time, thus affecting the
decision-making results. Connolly (2002) put
forward the decision judgment theory of regret.
Humphrey(2004) introduced the feedback
information of the abandoned scheme into regret
theory and conducted empirical research. Bleichrodt
(2010) gave the formula for measuring utility
function and regret function. Chorus
(2010) proposed
a random regret model on basis of regret theory and
applied it to the travel traffic choice problem.
Thanks to containing fewer parameters and
simpler calculation steps, regret theory is more and
more applied to risk-based multi-attribute decision-
making(RMADM)problems, and has already become
a hot spot in the field of decision-making research. To
solve the RMADM problem in which the
probabilities of states and the attribute values are both
interval numbers, Zhang et al. (2013)
proposed a
decision analysis method based on regret theory by
calculating the sum of utility and regret value of each
alternative. Zhang et al. (2014) studied group
decision-making method based on regret theory under
multi-dimensional preference information of pair-
wise alternatives; Liang et al. (2015) introduced the
method of stochastic multi-attribute decision making
with 2-tuple aspirations considering regret behaviour.
Although regret theory has already become a hot spot
in the field of decision-making, the research about
grey risky decision-making with regret psychology is
still in its infancy. Qian et al. (2017) established grey
stochastic multi-criteria decision-making model; Guo
et al. (2015)
constructed multi-objective grey target
model based on interval grey number; Zhou et al.
(2017b) presented a solution to the grey random
multi-criteria problem with extended grey numbers
based on regret theory and TOPSIS method; Qian et
al. (2019) introduced the grey extended EDAs
method based on the general grey numbers, and then
combined the regret theory to construct a new grey
risky multi-attribute decision-making model.
It is noteworthy that the actual decision-making is
quite complex. Except for uncertain fuzzy
information or grey information, it also involves
multi-stage and multi-level decision-making, which
is dynamic decision-making. Different from the static
decision-making background, decision-makers will
face multiple choices in the dynamic decision-making
process. Their later decision-making is not only
determined by the expected consideration, but also
affected by the early decision-making. Therefore,
dynamic decision-making is more vulnerable to the
Multi-Level Grey Risky Multi-Attribute Decision-Making Method Based on Dynamic Regret Theory
665

subjective cognition and psychological behaviour of
decision-makers. For example, the decision maker
will try his best to reduce the regret caused by
previous decisions. The regret caused by decisions in
each stage will not "disappear" in the final evaluation
stage.
Daniel and Rebecca
(2005) introduced regret
theory into multi-stage decision-making, proposed
dynamic regret theory, and put forward specific
behaviour prediction derived from regret equilibrium.
The main idea is that in the dynamic decision-making
process, the decision-maker will use the later choice
to strategically reduce the overall risk he faces, that
is, the decision maker's final regret is determined by
the accumulation of the results of a series of decision-
making actions, and the regret generated in each stage
of decision-making will affect the final benefit. For
multi-stage and multi-level decision-making analysis,
dynamic regret theory can reflect the regret avoidance
psychology of decision-makers in each stage, which
is closer to reality and has broad application
prospects. Cao(2013) applied the dynamic regret
theory to the selection of different types of venture
capital projects in multiple industries. However, few
attempts have been conducted on applying dynamic
regret to risky multi-attribute decision-making,
especially in uncertain grey environment.
As a result, this paper proposes a grey risky multi-
attribute decision-making model based on dynamic
regret theory, in which the information value is taken
in the form of general grey numbers, and the decision
maker's regret avoidance is considered in multi-level
dynamic risky multi-attribute decision-making.
The rest o this paper is organized as follows:
Section 2 reviews basic concepts and methods such
as general grey numbers and dynamic regret theory.
In section3, an approach for risky multi-attribute
decision-making method with general grey numbers
is proposed based on dynamic regret theory. Section
4 illustrates a numerical example to show the
feasibility and the validity. In section5, conclusions
are discussed and drawn.
2 BASIC CONCEPTS
2.1 General Grey Number
Definition 1. ( Liu et al., 2012)
Let
g
±
∈ℜ
be an unknown real number within a
union set of closed or open grey intervals:
1
[, ]
n
i
i
i
gaa
±
=
∈  ,
1, 2, ,in= 
n
is an integer and
0 n<<∞，
,
i
i
aa∈ℜ
and
1
1
ii
ii
a aaa
−
+
≤≤≤
, for any interval
1
[, ] [, ]
n
ii
ii i
i
aa aa
=
⊗∈ ⊂ , then
g
±
is called a
general grey number.
Definition 2. ( Liu et al., 2012)
o
g
g
∧
（
）
is named the simplified form of the
general grey number, if
g
∧
is the “kernel” of a general
grey number
1
[, ]
n
i
i
i
gaa
±
=
∈  and
0
g
is the degree of
greyness of the general grey number. Here,
1
1
=
n
i
i
ga
n
∧∧
=

(1)
is called the “kernel” of the general grey number.
1
()
1
()
()
n
o
ii
i
a
gg
g
μ
μ
∧
±
∧
=
⊗
=
Ω

(2)
is called the degree of greyness.
Ω
is the background
which makes the general number
1
[, ]
n
i
i
i
gaa
±
=
∈ 
come into being and
μ
is the measure of
Ω
.
Proposition1. ( Liu et al., 2012)
For a general grey number
1
[, ]
n
i
i
i
gaa
±
=
∈  , in the
event that all the
i
a
∧
and
()
i
μ
⊗
are known, its
simplified form
o
g
g
∧
（
）
is one-one correspondence
with the general grey number
1
[, ]
n
i
i
i
gaa
±
=
∈  .
2.2 Dynamic Regret Theory
Daniel and Rebecca (2005) put forward dynamic
regret theory on the basis of regret theory, which
introduced regret theory into multi-stage decision-
ISAIC 2022 - International Symposium on Automation, Information and Computing
666

making, and realized the effective combination of
regret theory and dynamic decision-making.
Different from regret theory, dynamic regret theory
emphasizes that in the process of dynamic decision-
making, the final regret of the decision-maker is
determined by the accumulation of a series of
decision-making actions. The later decision-making
is influenced not only by the expected consideration,
but also by the early decision-making. The decision-
maker will reduce the final regret caused by the
previous decision as much as possible, that is, the
regret caused by the decision-making in each stage
will affect the final income.
Daniel and Rebecca (2005) also described and
analyzed the idea of dynamic regret decision through
a decision problem. There are two parent schemes A
and B, in which A contains two sub schemes
A
a
and
A
b
, and B also contains two sub schemes
B
a
and
B
b
, as shown in Figure 1.
A B
B
a
B
b
A
a
A
b
Figure 1: Dynamic Regret Decision Tree
In the first stage, the decision-maker makes the
choice between the two parent schemes A and B. If
he chooses A, he can only choose
A
a
or
A
b
in the
second stage, and the regrets in both stages should be
considered in the final evaluation. For example, if A
is selected for the first stage and
A
a
is selected for the
second stage, then comparing the utility of
A
a
and
A
b
will produce the regret value of the second stage.
Then fix the selection
A
a
of the second stage, and
compare the maximum utility of
A
a
and the two sub
schemes
B
a
and
B
b
in B to produce the regret value
of the first stage. Finally, the two regret values are
weighted to obtain the comprehensive regret value of
the final choice.
Daniel and Rebecca (2005) combined regret
theory with dynamic decision-making to better solve
the dynamic decision-making problem considering
decision-makers' regret. The basic idea is that the
comprehensive perceived utility value of the
decision-making scheme is made up of the utility
value of the scheme itself and the total regret value.
The total regret value also consists of two parts. One
is the regret value generated by comparing the utility
value with the sub schemes of different parent
schemes at the first stage; the other is the regret value
generated by comparing the utility value with the sub
scheme under the same parent scheme at the second
stage.
11 2 2
() () ( ) ( )Ux vx R x R x θθ =−Δ−Δ
(3)
12
θθ （>0) , （>0)
respectively represent the degree
of regret of the decision-maker for the first stage and
the second stage.
Daniel and Rebecca (2005) believe that the
behavior of the decision maker at the second stage is
actually the result of his own game, constituting a
psychological Nash equilibrium . Thus the set of
regret equilibrium is put forward. Therefore,
considering the decision-making behavior at the first
stage, a parent scheme is selected to maximize the
utility, so as to form a regret equilibrium solution.
3 RMADM APPROACH WITH
GENERAL GREY NUMBERS
BASED ON DYNAMIC REGRET
THEORY
3.1 Problem Description
Consider a two-stage grey risky multi-attribute
decision-making problem. Assume 𝐴=
{
𝐴

,⋯,𝐴

,⋯,𝐴

}
is the set of schemes of the first
stage. In each scheme 𝐴

, there are 𝑞

sub schemes
selected at the second stage. 𝐴

∗
=
𝐴

,⋯,𝐴
,()
,⋯,𝐴


is the set of the sub schemes
of
i
A
. The set of the attributes is 𝐶=
{𝐶

,⋯,𝐶

,⋯,𝐶

}. And
()
1
,,,,
j
m
ωω ω ω = 
is the weighted vector of the attributes, where
1
1, 0, 1, 2, ,
m
jj
j
j
m ωω
=
=≥=


. Suppose the
market faces
l
states, and the set of states is
{
}
(1) (t) ( )
,, ,,
l
SS S S= 
. Assume the
Multi-Level Grey Risky Multi-Attribute Decision-Making Method Based on Dynamic Regret Theory
667

information value of the alternative
,( )ii k
A
relative to
the attribute
j
C
in the state
()t
S
is expressed by the
general grey number
()
,( ),
()
t
ii k j
a ⊗
,
1, ,in= 
，
1, ,
j
m= 
，
() 1, ,
i
ik k q==
，
1, ,tl= 
.Thus the grey decision matrix of all the sub schemes
of
i
A
under the state
()t
S
at the second stage is
obtained as
()
() ()
,( ),
() ()
i
tt
iiikj
qm
Aa
×
⊗= ⊗
. The
problem to be solved now is how to choose the final
best scheme under the consideration of the
psychological behavior of the decision maker's regret
avoidance.
3.2 Combined Processing of Natural
States
For multi-stage and multi-level decision-making
problems, the selection scheme in the first stage
usually involves different countries, regions or
different industries, so the probability of market
situation is also different. Assuming that the market
states faced by each scheme in the first stage are
independent of each other, the combined market state
can be obtained. The market state probabilities of
1
,,,,
in
A
AA
are shown in Table1, and the
probability of
i
A
under the state
()t
S
is
()t
i
p
.
Table 1: The Probability of the State
(1)
S
……
()t
S
……
()l
S
1
A
(1)
1
p
……
()
1
t
p
……
()
1
l
p
…… ……
i
A
(1)
i
p
……
()t
i
p
……
()l
i
p
…… ……
n
A
(1)
n
p
……
()t
n
p
……
()l
n
p
Obviously, there are
n
l
combined states. Assume the
set of combined states is 𝑆
∗
=
{
𝑆

,⋯,𝑆

,⋯,𝑆


}
,
and the corresponding probability vector is 𝑃

⃗
=
(
𝑃

,⋯,𝑃

,⋯,𝑃


)
, where 𝑃

is the probability of
occurrence of 𝑆

.
() ()
1
1
=( )
l
n
tt
hii
i
t
Pp θ
=
=
Π

(4)
1, ,
m
hl= 
()
1,
0, .
it
t
i
it
the state of A is S
the state of A is not S
θ

=


According to the combination state, the sub
schemes in the second stage are matrix combined.
Suppose the states of 𝐴

,⋯,𝐴

,⋯,𝐴

are 𝑆
（ 

)
,⋯,𝑆
（ 

)
,⋯,𝑆
（ 

)
respectively, where
1, ,
i
hl= 
.
The decision matrix generated by all the sub
schemes of 𝐴

under the state 𝑆
()
is 𝐴

(ℎ

)
(⊗)=

𝑎
,(),
(ℎ

)
(⊗)



×
, where 𝑖(𝑘)=𝑘=1,⋯,𝑞

，𝑗=
1,⋯,𝑚. Thus the combined decision matrix is
obtained as
𝑌
(

)
(
⊗
)
=
⎝
⎜
⎜
⎛
𝐴

(


)
(⊗)
⋮
𝐴

(


)
(⊗)
⋮
𝐴

(


)
(⊗)
⎠
⎟
⎟
⎞
(


⋯

)
×
=（𝑦

()
(⊗)）
(

⋯

)×
(5)
3.3 Modeling Principle and Method
As different attributes have different dimensions, the
data will be standardized first to facilitate the
calculation.
The grey upper bound effect measurement is
adopted as the conversion formula for the attribute of
benefit type:
()
()
()
()
()
()
max{ }
h
hIj
Ij
h
Ij
I
y
z
y
⊗
⊗=
⊗
(6)
The grey lower bound effect measurement is adopted
as the conversion formula for the attribute of cost
type:
()
()
()
min{ ( )}
()
()
h
Ij
h
I
Ij
h
Ij
y
z
y
⊗
⊗=
⊗
(7)
So the normalized combinatorial decision matrix is
obtained as follows:
1
() ()
()
()=
n
hh
I
jqqm
Zz
++ ×
⊗⊗

（ ())
(8)
The attributes are weighted to obtain a new decision
matrix as follows:
1
() ()
()1
()=
n
hh
Iqq
Xx
++ ×
⊗⊗

（ ())
,
ISAIC 2022 - International Symposium on Automation, Information and Computing
668

()
() ()
1
=
m
hh
IjIj
j
xz ω
=
⊗⊗

()
(9)
So the comprehensive decision matrix in the
combined state can be expressed as below
𝑋
(
⊗
)
=𝑋
(

)
(
⊗
)
,⋯,𝑋
(

)
(
⊗
)
,⋯𝑋
(


)
(
⊗
)

=(𝑥

(⊗))
(

⋯

)×

(10)
Then the regret aversion of the decision maker is
considered by the formula (3).
The power function
() (0 1)vx x
α
α =<<
(11)
is used as the utility function of the attribute
according to Tversky and Kahneman (1992). 𝛼is
named as the risk aversion coefficient. The smaller
the value of 𝛼 is, the greater the risk aversion degree
of the decision-maker is. On basis of the
comprehensive decision matrix, the grey
comprehensive perceived utility value of scheme I
under the state 𝑆

can be obtained:
11 2 2
1
[ ( ) ( ( )) ( ( ))]
n
l
IJIJ
J
UPx R R
α
θθ
=
=⊗−Δ⊗−Δ⊗

(12)
According to Carlos and Elke(2008),
1, 0
()=
0, 0
R
λ
Δ

−Δ≥
Δ

Δ<

,
(13)
1.15≤𝜆≤1.35
is taken as the regret function.
1
1
() max () ()
IJ IJ
I
XX
′
′
∈Ω
Δ⊗= ⊗− ⊗
(14)
2
2
() max () ()
IJ IJ
I
XX
′
′
∈Ω
Δ⊗= ⊗− ⊗
(15)
2
I
Ω=Ω
1
I
Ω=Ω−Ω
Ω
is the set of the first index of all sub scheme sets
under all parent schemes.
I
Ω
is the set of the first
index of the sub scheme sets under the parent scheme
to which the sub scheme with subscript
I
belongs.
According to the calculation results, the optimal
sub scheme
*
i
ij
A
can be found under each parent
scheme
i
A
, thus producing the equilibrium set
***
1
*
1
{,,, }
in
jijnj
DA A A= 
.
Next, the grey comprehensive perceived utility value
of each
*
i
ij
A
is calculated on basis of (12). Here
𝛥

(
⊗
)
=𝑚𝑎𝑥{𝑋



∗
(
⊗
)
,⋯, 𝑋

(

)


∗
(
⊗
)
,
𝑋

()

∗
(⊗),⋯,𝑋



∗
(⊗)} − 𝑋



∗
(⊗)
(16)
2
() 0Δ⊗=
(17)
4 CASE STUDY
A multinational enterprise wants to invest and
develop new products abroad. After the preliminary
investigation of the team, the sales volume 𝐶

, market
share 𝐶

and development investment cost 𝐶

are
determined as the investigation attributes, and the two
cities 𝐴

、 𝐴

in country 𝐴

and the three cities 𝐴

、 𝐴

、 𝐴

in country 𝐴

are identified as
candidates. Although different countries may face
different future market conditions, they can be
divided into "good", "medium" and "poor", which is
shown in Table.2. Due to the complexity and
uncertainty of decision-making environment
information and the subjective cognition of decision-
makers, the attribute performance values of these five
cities are represented by general grey numbers, as
shown in the Table.3. Now we want to select the best
investment city according to the above conditions and
considering the regret avoidance of decision-makers.
Table 2: The natural state of the two countries
(1)
S
(2)
S
(3)
S
1
A
0.3 0.4 0.3
2
A
0.45 0.35 0.2
Multi-Level Grey Risky Multi-Attribute Decision-Making Method Based on Dynamic Regret Theory
669

Table 3: Attribute value of five cities in two countries under the "good" natural state
1
C
2
C
3
C
11
A
[400,500] [0.3,0.4]

[0.42,0.48] [75,85]

[91,95]
12
A
[350,400]

[404,410] [0.1,0.2]
{0.3}
[60,70]

[72,76]
21
A
[420,440]

[480,500] [0.1,0.2]
{0.3}
[70,80]
{85}
22
A
[420,450] [0.3,0.4] [80,90]

[95,97]
23
A
[330,340]

[350,380] [0.2,0.3]
{0.35}
[70,76]
Table 4: Attribute value of five cities in two countries under the "medium" natural state.
1
C
2
C
3
C
11
A
[380,400]
{420} {0.3}
[0.36,0.42] [75,80]

[85,90]
12
A
[320,380]

[390,400] [0.1,0.2]
{0.3}
[65,75]
{80}
21
A
[400,420]
{430}
[0.4,0.6]
{0.65}
[80,85]
{90}
22
A
[380,400] [0.55,0.65] [75,80]
23
A
[340,360]
{375} {0.2}
[0.3,0.4] [75,80]
{85}
Table 5: Attribute value of five cities in two countries under the "poor" natural state.
1
C
2
C
3
C
11
A
[350,380]

[400,420]
{0.2}
[0.25,0.35] [75,85]
{90}
12
A
[300,350]
{370}
[0.3,0.4]
{75}
[80,85]
21
A
[380,420] [0.3,0.5] [85,90]
22
A
[400,410]
{420}
[0.3,0.5] [80,85]
{87}
23
A
[340,350]

[360,380]
{0.5}
[0.6,0.7] [70,80]
{82}
Step1 The fields of sales volume, market share and
investment cost are determined as
[
300,500
]
，
[
0,1
]
,
[
50,100
]
respectively. The simplified form of
the general grey number (GGN) can be calculated and
obtained by definition 1,2. Thus the decision matrixes
of the two countries in three states "good", "medium"
and "poor" can be obtained as follows:
0.25 0.155 0.271
(1)
1
0.27 0.067 0.272
425 0.4 86.5
()=
391 0.225 69.5
A

⊗


0.2 0.067 0.188
(1)
2 0.15 0.1 0.23
0.204 0.083 0.12
460 0.225 80
( )= 435 0.35 90.5
350 0.3 73
A


⊗



0.096 0.068 0.2
(2)
1
0.335 0.067 0.187
405 0.345 82.5
()=
372.5 0.225 75
A

⊗


ISAIC 2022 - International Symposium on Automation, Information and Computing
670

0.098 0.174 0.096
(2)
20.10.10.1
0.097 0.127 0.095
420 0.575 86.25
( )= 390 0.6 77.5
362.5 0.275 81.25
A


⊗



0.247 0.12 0.188
(3)
1
0.234 0.1 0.105
387.5 0.25 85
()=
347.5 0.35 78.75
A

⊗


0.2 0.113 0.1
(3)
2 0.049 0.2 0.097
0.152 0.2 0.191
400 0.575 87.5
( )= 412.5 0.4 84.75
357.5 0.4 78.5
A


⊗



Step2 It’s clear there are 9 combined states. The set
of the combined states is
{
}
*
19
,,,,
h
SS S S= 
. And the corresponding probability vector can be
calculated on basis of (4).
()
19
,,,,
h
PP P P=


()
= 0.135, 0.105, 0.06, 0.18, 0.14, 0.08, 0.135 0.105, 0.06，
Step3 The sub schemes in the second stage are matrix
combined under the combination state according to
(5).
The decision matrix under the combined state 𝑆

("good"+"good") is
𝑌
()
(⊗)=
𝐴

()
(⊗)
𝐴

()
(⊗)
≜𝑦

()
(⊗)
×
The decision matrix under the combined state 𝑆

("good"+"medium") is
𝑌
()
(⊗)=
𝐴

()
(⊗)
𝐴

()
(⊗)
≜𝑦

()
(⊗)
×
The decision matrix under the combined state 𝑆

("good"+"poor") is
𝑌
()
(⊗)=
𝐴

()
(⊗)
𝐴

()
(⊗)
≜𝑦

()
(⊗)
×
𝑆

（"medium"+"good"）:
𝑌
()
(⊗)=
𝐴

()
(⊗)
𝐴

()
(⊗)
≜𝑦

()
(⊗)
×
𝑆

（"medium"+"medium"）:
𝑌
()
(⊗)=
𝐴

()
(⊗)
𝐴

()
(⊗)
≜𝑦

()
(⊗)
×
𝑆

（"medium"+"poor"）:
𝑌
()
(⊗)=
𝐴

()
(⊗)
𝐴

()
(⊗)
≜𝑦

()
(⊗)
×
𝑆

（"poor"+"good"）:
𝑌
()
(⊗)=
𝐴

()
(⊗)
𝐴

()
(⊗)
≜𝑦

()
(⊗)
×
𝑆

（"poor"+"medium"）:
𝑌
()
(⊗)=
𝐴

()
(⊗)
𝐴

()
(⊗)
≜𝑦

()
(⊗)
×
𝑆

（"poor"+"poor"）:
𝑌
()
(⊗)=
𝐴

()
(⊗)
𝐴

()
(⊗)
≜𝑦

()
(⊗)
×
Step4 As the beneficial attribute, the sales volume
𝐶

and the market share𝐶

are normalized by (6). As
the cost attribute, development investment cost 𝐶

is
normalized by (7). The normalized decision matrix
under each state is obtained by (8) as follows:
()
() ()
53
() ()
hh
Ij
Zz
×
⊗= ⊗
, here
1, , 9h = 
Step5 The attributes are weighted to obtain a new
decision matrix under each state by (9) as follows:
()
() ()
51
() ()
hh
I
Xx
×
⊗= ⊗
, here
1, , 9h = 
,
() ()
1
() ()
m
hh
IjIj
j
xz ω
=
⊗= ⊗

.
Thus the grey comprehensive decision matrix on
basis of (10) is
𝑋
(
⊗
)
=

𝑋
(

)
(
⊗
)
,⋯,𝑋
(

)
(
⊗
)
,⋯,𝑋
(

)
(
⊗
)

=𝑥

(⊗)
×
as shown in Table.6.
Table 6: Grey Decision Matrix.
1
S
2
S
3
S
4
S
5
S
6
S
7
S
8
S
9
S
0.135 0.105 0.06 0.18 0.14 0.08 0.135 0.105 0.06
111
()XA
0.911
0.272
0.85
0.272
0.85
0.272
0.913
0.2
0.831
0.2
0.845
0.2
0.809
0.247
0.768
0.247
0.783
0.247
212
()XA 0.809
0.272
0.785
0.272
0.785
0.272
0.809
0.335
0.335
0.767
0.335
0.779 0.88
0.234
0.801
0.234
0.819
0.234
321
()XA 0.829
0.272
0.937
0.272
0.915
0.272
0.867
0.2
0.187
0.948 0.945
0.2
0.867
0.2
0.957
0.174
0.957
0.2
422
()XA 0.871
0.272
0.949
0.272
0.843
0.272
0.92
0.23
0.187
0.962 0.874
0.2
0.92
0.23
0.971
0.1
0.887
0.2
523
()XA 0.815
0.272
0.741
0.272
0.811
0.272
0.861
0.204
0.187
0.76 0.842
0.2
0.861
0.204
0.769
0.127
0.855
0.2
Multi-Level Grey Risky Multi-Attribute Decision-Making Method Based on Dynamic Regret Theory
671

Step6 According to Tversky and Kahneman(1992)
and Chorus(2010), the coefficient of risk aversion of
(13) is set as
=0.88 α
, and
=1.25 λ
, and
12
=0.7 =0.3 θθ ，
.
According to (12)(13)(14)(15), if I=1or 2, the
grey comprehensive perceived utility value of the
scheme can be obtained by the formula
𝑈

=𝑃



[𝑥

(⊗)

−𝜃

𝑅(𝛥

(𝑚𝑎𝑥
 ，  ， 
𝑥

(⊗)
−𝑥

(⊗)) − 𝜃

𝑅(𝑚𝑎𝑥
 ， 
𝑥

(⊗) − 𝑥

(⊗))]
Thus
1 0.272
0.85002U =
，
2 0.272
0.80334U =
.
According to (12)(13)(14)(15), if I=3 or 4 or 5,
the grey comprehensive perceived utility value of the
scheme can be obtained by the formula
𝑈

=𝑃



[𝑥

(⊗)

−𝜃

𝑅(𝛥

(𝑚𝑎𝑥
 ， 
𝑥

(⊗
−𝑥

(⊗)) − 𝜃

𝑅( 𝑚𝑎𝑥
 ，  ， 
𝑥

(⊗) − 𝑥

(⊗))]
Thus
3 0.272
0.91022U =
，
4 0.272
0.924997U =
，
50.272
0.81751U =
.
Step7 The equilibrium set is
*
14
{, }DXX=
according to the above results. Then calculate the
grey comprehensive perceived utility values of the
scheme
14
XX，
on basis of (12)(13)(16)(17) .
𝑈


=𝑃



[𝑥

(⊗)

−𝜃

𝑅(𝑥

(⊗) − 𝑥

(⊗))
−𝜃

𝑅(𝑥

(⊗) − 𝑥

(⊗))]
0.272
0.853=
𝑈


=𝑃



[𝑥

(⊗)

−𝜃

𝑅(𝑥

(⊗) − 𝑥

(⊗))
−𝜃

𝑅(𝑥

(⊗) − 𝑥

(⊗))]
0.272
0.926=
So the best investment city is the city
22
A
of the
country
2
A
.
5 CONCLUSIONS
Aiming at the multi-stage risk multi-attribute
decision-making problem, this paper proposes a
decision-making method based on general grey
number information and dynamic regret theory.
Considering the impact of regret avoidance
psychology on decision-making results, the method
calculates the probability of the combined market
state and constructs the grey regret function as well as
grey comprehensive perceived utility function on
basis of the dynamic regret theory. The set of regret
equilibrium solutions is established according to the
value of grey perceived utility function, and then the
optimal scheme is selected by further comparing the
comprehensive utility value.
The method model proposed in this paper not only
considers the psychological and behavioral state of
regret avoidance of decision makers in the dynamic
decision-making process, but also takes into account
the uncertainty and gray of the dynamic environment.
Compared with the traditional risky multi-attribute
decision-making method, it is closer to reality and has
stronger operability and practicability, which
provides method support and theoretical guidance for
more uncertain decision-making problems in reality.
At the same time, this paper also broadens the
research scope and application space of regret theory
and dynamic regret theory.
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