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Interval Type–2 Fuzzy Decision Making



This paper concerns itself with decision making under uncertainty and the consideration of risk. Type-1 fuzzy logic by its (essentially) crisp nature is limited in modelling decision making as there is no uncertainty in the membership function. We are interested in the role that interval type–2 fuzzy sets might play in enhancing decision making. Previous work by Bellman and Zadeh considered decision making to be based on goals and constraint. They deployed type–1 fuzzy sets. This paper extends this notion to interval type–2 fuzzy sets and presents a new approach to using interval type-2 fuzzy sets in a decision making situation taking into account the risk associated with the decision making. The explicit consideration of risk levels increases the solution space of the decision process and thus enables better decisions. We explain the new approach and provide two examples to show how this new approach works.
Interval Type–2 Fuzzy Decision Making1
Thomas Runklera, Simon Couplandb, Robert Johnc
aSiemens AG, Corporate Technology, 81730 Munich, Germany, Email:3
bCentre for Computational Intelligence, De Montfort University, The Gateway,5
Leicester, LE1 9BH, UK, Email:
cLaboratory for Uncertainty in Data and Decision Making (LUCID), University of7
Nottingham, Wollaton Road, Nottingham, NG8 1BB, UK8
This paper concerns itself with decision making under uncertainty and the
consideration of risk. Type-1 fuzzy logic by its (essentially) crisp nature
is limited in modelling decision making as there is no uncertainty in the
membership function. We are interested in the role that interval type–2 fuzzy
sets might play in enhancing decision making. Previous work by Bellman and
Zadeh considered decision making to be based on goals and constraint. They
deployed type–1 fuzzy sets. This paper extends this notion to interval type–2
fuzzy sets and presents a new approach to using interval type-2 fuzzy sets
in a decision making situation taking into account the risk associated with
the decision making. The explicit consideration of risk levels increases the
solution space of the decision process and thus enables better decisions. We
explain the new approach and provide two examples to show how this new
approach works.
fuzzy decision making, interval type–2 fuzzy sets12
Preprint submitted to International Journal of Approximate Reasoning April 18, 2016
1. Introduction13
In this paper we are concerned with decision making under uncertainty.14
In particular, we are interested in the role that interval type–2 fuzzy sets15
might play in enhancing decision making. In part, this has been motivated by16
our recent work on the properties of type-2 defuzzification operators (Runkler17
et al., 2015) where we explored the role of defuzzification of type–2 fuzzy sets18
in decision making. In particular that work explored the semantic meaning19
of interval type–2 fuzzy sets from the perspective of opportunity or risk, in20
respect to defuzzification operators. This led us to explore how risk could21
be modelled using interval type–2 fuzzy sets. Most fuzzy logic based risk22
research relates to applications of risk (e.g. (Mays et al., 1997; Malek et al.,23
2015)). We are interested in the notion of risk from the perspective of how24
different individuals might make decisions with their own notions of risk.25
In the context of this work, by decision making we mean where we have26
a goal(s) that is limited by some constraints. In the case of type–1 fuzzy sets27
the fuzzy decision making process finds an optimal decision when goals and28
constraints are specified by fuzzy sets (Zadeh, 1965). A type–1 fuzzy set is29
defined by a membership function u:X[0,1]. So, they are by their very30
nature crisp and there is no uncertainty around the membership function. In31
this paper we will always consider fuzzy sets over one–dimensional continu-32
ous intervals X= [xmin , xmax]. An interval type–2 fuzzy set (Zadeh, 1975;33
Liang and Mendel, 2000; Mendel et al., 2006) ˜
Ais defined by two member-34
ship functions1, a lower membership function u˜
A:X[0,1] and an upper35
1Interval type–2 fuzzy sets are known to be equivalent to interval–fuzzy sets (Gorzal-
Figure 1: Interval type–2 fuzzy set.
membership function u˜
A:X[0,1], where36
A(x) (1)
for all xX. Fig. 1 shows an example of a triangular interval type–2 fuzzy37
set and its upper (solid) and lower (dashed) membership functions. Fuzzy38
decision making using type–1 fuzzy sets was introduced by Bellman and39
Zadeh (1970). Given a set of goals specified by the membership functions40
{ug1(x), . . . , ugm(x)}(2)
and a set of constraints specified by the membership functions41
{uc1(x), . . . , ucn(x)}(3)
the optimal decision xis defined as42
x= argmax
xXug1(x). . . ugm(x)uc1(x). . . ucn(x)(4)
czany, 1987; Gehrke et al., 1996).
Figure 2: Type–1 fuzzy decision.
where is a triangular norm such as the minimum or the product operator.43
In the experiments presented in section 4 we will use the minimum operator.44
Fig. 2 shows an example of a type–1 fuzzy decision with two type–1 triangular45
goals g1,g2and one triangular constraint c1. Notice that in fuzzy decision46
making goals and constraints are treated in the same way, so we do not need47
to explicitly distinguish between goals and constraints.48
Successful applications of type-1 fuzzy decision making include environ-49
mental applications such as water resource planning (Afshar et al., 2011) or50
waste management (Kara, 2011), infrastructure planning applications such51
as energy system planning (Kaya and Kahraman, 2010) or location manage-52
ment (Guneri et al., 2009), logistic applications such as supplier selection53
(Bottani and Rizzi, 2008), transportation planning (He et al., 2012), fuzzy54
data fusion (Shell et al., 2010) or optimisation of logistic processes (Sousa55
et al., 2002).56
In this paper we provide a new fuzzy decision making approach using in-57
terval type–2 fuzzy sets within the context of risk. Chen and Wang (Chen and58
Wang, 2013, 2011) deploy interval type-2 fuzzy sets to aid decision making59
through a ranking mechanism and fuzzy multiple attributes decision making.60
Multi-Criteria Group Decision Making and type-2 fuzzy sets are explored by61
Naim and Hagras (Naim and Hagras, 2015) in an extensive comparison of62
different approaches. They are interested in where groups make decisions.63
Lascio et al. (Di Lascio et al., 2007) take a formal mathematical approach to64
type-2 fuzzy decision making. Zhang and Zhang (Zhang and Zhang, 2012)65
extend so called soft sets to type-2 fuzzy sets and provide limited examples of66
type-2 fuzzy soft sets in decision making. An example application is that of67
using type-2 fuzzy sets in multi-criteria decision making for choosing energy68
storage (Ozkan et al., 2015).69
The decision making research using type-2 fuzzy sets does not align the70
decision making with the notion of risk. When making a decision our attitude71
to risk affects our decision making. Our approach then is to consider risk and72
decision making and provide an interval type-2 fuzzy set approach to that.73
The rest of the paper is structured as follows: Section 2 provides an74
overview of interval type-2 fuzzy decision making; Section 3 discusses the75
properties of this type of decision making; Section 4 provides examples of76
the use of the approach and Section 5 provides some closing remarks.77
2. Interval Type–2 Fuzzy Decision Making78
In type–1 fuzzy decision making the membership values of the goals and79
constraints quantify the degrees of utility of the different decision options.80
In interval type–2 fuzzy decision making the utility is subject to uncertainty.81
The upper and lower membership values of each option quantify the lower82
bound (worst case) and upper bound (best case) of the corresponding utility,83
respectively. Hence, it is straightforward to define the worst case interval84
type–2 fuzzy decision as85
x= argmax
xXu˜g1(x). . . u˜gm(x)u˜c1(x). . . u˜cn(x)(5)
and to define the best case interval type–2 fuzzy decision as86
x= argmax
xXu˜g1(x). . . u˜gm(x)u˜c1(x). . . u˜cn(x)(6)
The worst case interval type–2 fuzzy decision maximizes the utility that is
obtained under the worst possible conditions. This decision policy reflects a
cautious or pessimistic decision maker. The best case interval type–2 fuzzy
decision maximizes the utility that is obtained under the best possible condi-
tions. This decision policy reflects a risky or optimistic decision maker. Fig.
3 shows an example of worst case and best case type–2 fuzzy decisions with
two type–2 triangular goals ˜g1, ˜g2and one triangular constraint ˜c1. We do
not want to restrict the interval type–2 fuzzy decision to the worst case and
best case decisions but we want to allow to specify the level of risk β[0,1]
associated with the decision, where risk β= 0 corresponds to the worst case
decision xand risk β= 1 corresponds to the best case decision x. This
leads us to define the interval type–2 fuzzy decision at risk level βas
β= argmax
xX((1 β)·u˜g1(x) + β·u˜g1(x))
. . . ((1 β)·u˜gm(x) + β·u˜gm(x))
((1 β)·u˜c1(x) + β·u˜c1(x))
Figure 3: Interval type–2 fuzzy decisions.
. . . ((1 β)·u˜cn(x) + β·u˜cn(x))(7)
It is worth noting the relationship between equations (5) and (6) and the88
intersection operator. The worst case decision computed in equation (5) may89
also be calculated through the intersection operator when using the same t-90
norm as used by the operator in equations (5), (6) and (7). Equations (5)91
and (6) find the maximum value across the domain Xfrom the minimum92
of all the membership functions at a domain point x. This could equally be93
obtained by finding the highest membership grade across the domain of a94
fuzzy set which is the intersection of all goals and constraints. Let this fuzzy95
set fbe calculated by equation (8) below.96
f= ˜g1. . . ˜gm˜c1. . . ˜cn(8)
Figure 4 depicts the intersection of a single goal and constraint with the97
points xand xhighlighted by circles. The approach leads to equations (9)98
and (10) giving alternative ways of calculating the respective worst and best99
Figure 4: The intersection of an interval type-2 fuzzy goal and constraint
case decisions.100
x= argmax
f(x)) (9)
x= argmax
f(x)) (10)
We can use equations (9) and (10) to calculate the decision for given risk102
value βusing equation 11.103
β= argmax (1 β)·µ˜
f(x) + β·µ˜
where β[0,1]. The next section explores some properties of this approach.104
3. Properties of Interval Type–2 Fuzzy Decision Making105
In this section we investigate in some detail the properties of the interval106
type–2 fuzzy decision at risk level βdefined by (7).107
It is easy to see that x
0=xand x
1=x. It seems reasonable to require108
that for any risk level β[0,1] the decision should be in the interval bounded109
by the worst case decision xand the best case decision x, so110
min x, xx
βmax x, x(12)
for arbitrary t–norms .111
We now consider whether equation (12) holds for all fuzzy sets, placing no112
constraints on the membership functions. For simplicity consider a decision113
with only one goal and no constraint, so for the decision we consider only114
one single type–2 fuzzy set, and we don’t have to worry about the t–norm115
. Fig. 5 shows an example of such a type–2 fuzzy set where the maximum116
of the upper membership function (solid) is at x= 0.5, the maximum of117
the lower membership function (dashed) is at x= 1, but where for the risk118
level β= 0.5 (dotted) we obtain the decision x
0.5= 0, which is outside the119
interval between the worst case and the best case, i.e. x
0.56∈ [x, x]. This120
example proves that (12) does not hold in general.121
We will now consider equation (12) for interval type-2 fuzzy set whose122
membership functions are convex. By convex we mean both the upper and123
lower membership functions are convex. Consider the two convex interval124
type-2 fuzzy sets ˜g1and ˜c1over the domain X. We know that taking the125
minimum or the product of two convex functions will always yield a convex126
function. Therefore ˜g1˜c1and ˜
c1must yield convex functions when127
using the product or minimum t-norm. Let ˜
f= ˜g1˜c1as with equation(8).128
It is obvious that the lower membership function of ˜
fis contained by the129
upper membership function of ˜
fi.e. ˜
f(x), xX. We can now130
show that (12) holds for convex sets when using the minimum and product131
0 0.5 1
Figure 5: Nonconvex example for an interval type–2 fuzzy decision.
t-norms. First divide the domain Xinto three distinct regions.132
Region I: min x, xxmax x, x
Region II : x < min x, x
Region III : max x, x< x135
These regions are depicted in Figure 6. For any value of x
βto be outside136
region I it must be in either region II or III. For x
βto be in region II the137
derivative of either function must negative with respect to x. Since both138
functions are convex this is impossible. For x
βto be in region III the deriva-139
tive of either function must positive with respect to x. Since both functions140
are convex this is impossible. Therefore any value of x
βmust be in region I.141
This completes the proof.142
There is a caveat we must add to this discussion which is that x
only non zero when xis in the support of the intersection of all the goals and144
constraints. If the intersection is an empty set we have no decision agreement.145
Figure 6: Regions in a pair of convex functions.
The next section looks at two examples as to how this decision making146
approach works.147
4. Application Examples148
In this section we illustrate our proposed interval type–2 fuzzy decision149
making approach with two application examples: optimization of the room150
temperature and choosing optimal travel times with low road congestion.151
For the first application example assume you have invited two guests, A152
and B, and wonder to which room temperature you should set the heater.153
You know that A will be completely happy with 17 degrees, and will be com-154
pletely unhappy at less than 16 degrees or more than 19 degrees. And B will155
be completely happy with 20 degrees, and will be completely unhappy for156
less than 18 degrees or more than 22 degrees. This can be modeled using157
14 16 18 20 22 24
Figure 7: Interval type–2 fuzzy decision for the temperature example.
the interval type–2 fuzzy sets shown in Fig. 7, where the upper membership158
functions for ˜
Aand ˜
Bare shown as solid triangles and the lower membership159
functions for ˜
Aand ˜
Bas dashed triangles. Now a cautious decision maker160
will set the temperature to 18.5 degrees (lower circle, at the intersection of161
the lower membership functions, dashed), because then none of the guests162
will be less happy than 25%. And a risky decision maker will set the temper-163
ature to 19 degrees (upper circle, at the intersection of the upper membership164
functions, solid), because in the best case both guests will be 75% happy. In-165
termediate levels of risk between β= 0 and 1 will yield optimal temperatures166
between 18.5 and 19 degrees.167
For the second application example assume that we want to drive to work168
at some time between 6 and 12 o’clock, work for 8 hours, and then drive back.169
From a traffic reporting system we have obtained the traffic density curves for170
the 10 previous work days that are shown in Fig. 8. These curves represent,171
in our view, a sensible view of typical daily traffic density. Note they are non172
6 7 8 9 10 111213 14151617 181920
Figure 8: Traffic example: Observed traffic densities.
convex and that, as is typical, the uncertainty at the beginning and end of173
the day is larger.174
We start with a type–1 fuzzy approach to model this situation and find175
an optimal decision. Based on the observed traffic densities we estimate176
the average traffic densities using a mixture of two Gaussian membership177
functions as178
u(x) = 0.775 ·e(x7·60
133 )2
+ 0.525 ·e(x19·60
290 )2
Fig. 9 left shows a plot of this membership function which may be associated179
with the linguistic label “traffic”, so for example at 7 o’clock we have 0.775180
traffic. We want to drive to work some time between 6 and 12 o’clock, so for181
the morning traffic we consider the part of the membership function for the182
time between 6:00 and 12:00 (solid curve in Fig. 9 right). We want to drive183
back after 8 hours of work, so for the evening traffic we consider the part184
of the membership function for the time between 14:00 and 20:00, shifted 8185
hours to the left (dashed curve in Fig. 9 right). If we do the morning trip186
6 7 8 9 101112 13141516 17181920
6(14) 7(15) 8(16) 9(17) 10(18)11(19)12(20)
Figure 9: Traffic example: Type–1 fuzzy membership function of the traffic (left) and
type–1 fuzzy decision (right).
at 7:00 and the evening trip at 15:00, for example, then we will have 0.775187
traffic in the morning and about 0.26 traffic in the evening. Our goal is to188
find a travel time, where the traffic in the morning is low and the traffic in the189
evening is low. This yields a fuzzy decision with two goals that correspond to190
the two membership functions shown in in Fig. 9 right. In contrast to the first191
example we are looking for the minimum, not the maximum memberships,192
so we replace the argmax in the decision function by argmin. The optimal193
type–1 fuzzy decision (marked by a circle) is at 8:46 (return 16:46) with a194
traffic of 0.42 for both the morning and the evening trips.195
Next, we consider a type–2 fuzzy approach for this problem. We estimate196
the minimum and maximum bounds of the traffic densities as197
u(x) = 0.95 ·e(x7·60
3·60 )2
+ 0.75 ·e(x19·60
3·60 )2
u(x) = 0.6·e(x7·60
1.5·60 )2
+ 0.3·e(x19·60
4.5·60 )2
which represent the lower (dashed) and upper (solid) membership functions199
6 7 8 9 10 111213 14151617 181920
6(14) 7(15) 8(16) 9(17) 10(18)11(19)12(20)
1morning evening
Figure 10: Traffic example: Interval type–2 fuzzy membership function of the traffic (left)
and type–2 fuzzy decision (right).
of the interval type–2 fuzzy membership function of the traffic, as shown in200
Fig. 10 left. The lower and upper membership functions for both the morning201
and evening trips are shown in Fig. 10 right. The three circles show three202
type–2 fuzzy decisions at different risk levels. A cautious decision maker will203
drive to work at 8:59 and back at 16:59 (upper circle), because the worst case204
traffic is about 0.64. A risky decision maker will drive to work at 8:32 and205
back at 16:32 (lower circle), because the best case traffic is about 0.22. For206
intermediate levels of risk the optimal decision will be to leave between 8:32207
and 8:59 and return 8 hours later. For example, for risk level β= 0.8 we208
obtain the dotted curve which is minimized for leaving at 8:37 and returning209
at 16:37 with a traffic of about 0.3.210
A comparison of the type–1 and type–2 fuzzy decisions is shown in Fig. 11.211
The two almost linear solid curves show the worst case and best case traffic212
for the morning trip times between 8:30 and 9:00, corresponding to evening213
8:30(16:30) 8:45(16:45) 9:00(17:00)
best case
worst case
type 1
min type 2
8:32(16:32) max type 2
Figure 11: Traffic example: Comparison of the worst case and best case traffic for the
type–1 and type–2 fuzzy decisions.
trip times between 16:30 and 17:00. The middle dashed line at 8:46(16:46)214
corresponds to the type–1 fuzzy decision, where the worst case traffic is about215
0.69 and the best case traffic is about 0.23. The left dashed line at 8:32(16:32)216
corresponds to a risky decision maker who picks the minimum type–2 fuzzy217
decision, where the best case traffic is 5.3% lower than the best case for the218
type–1 fuzzy decision. The right dashed line at 8:59(16:59) corresponds to219
a cautious decision maker who picks the maximum type–2 fuzzy decision,220
where the worst case traffic is 8.1% lower than the worst case for the type–1221
fuzzy decision. So if we specify a risk level that we are willing to accept, then222
type–2 fuzzy decision making can take this risk level into account and may223
therefore yield better results than type–1 fuzzy decision making.224
5. Conclusions225
Existing approaches supporting decision making using type-2 fuzzy sets226
ignore the risk associated with these decisions. In this paper we have pre-227
sented a new approach to using interval type–2 fuzzy sets in decision making228
with the notion of risk. The method extends the work of Bellman and Zadeh229
(1970) by replacing the type–1 fuzzy sets with interval type–2 fuzzy sets.230
This brings an extra capability to model more complex decision making, for231
example, allowing trade-offs between different preferences and different atti-232
tudes to risk. The explicit consideration of risk levels increases the solution233
space of the decision process and thus enables better decisions. In a traffic234
application example, the quality of the obtained decision could be improved235
by 5.3–8.1%.236
The paper explores some of the properties of this new approach and with237
two examples shows how it works. We will follow on this work by tackling238
larger, more complex, problems as well as investigating the properties in more239
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... Applying Equation (44) to the real bank data presented in Table 20, we obtained the following countries' risk scores (measured on a scale from 1 (very low risk) to 5 (very high risk)): 1.6840 for France, 1.8316 for the Czech Republic, and 2.5309 for Romania. By using the calculated country risk values from these three countries and real data for bank variables included in the model and presented in Table 21, Equation (89) was applied to calculate the bank performance scores for Societe Generale, Komercni Banka, and BRD Groupe Societe Generale SA. The scores of the banks from the advanced economies, namely Societe Generale (0.1064) from France and Komercni Banka (0.0965) from the Czech Republic, registered higher values than the bank from emerging and developing Europe, namely BRD Groupe Societe Generale SA (0.0507) from Romania. ...
... The bank performance model's weights are displayed in Table 18 and Equation (89). The most important bank variables for the performance score were EQUITY (0.1512) and ROA (0.1406). ...
... Furthermore, this proposed mathematical method could also be amplified by employing additional decision-making frameworks (ELECTRE, DEMATEL, VIKOR, TOPSIS, or TODIM [88]) with distinct variables for further empirical investigations. Likewise, in future studies, the method of fuzzy-ANP could also be explored based on type-2 fuzzy sets for enhancing decision making [89] and by using the "trapezoidal type-2 intuitionistic fuzzy set [88]. ...
Full-text available
In recent years, bank-related decision analysis has reflected a relevant research area due to key factors that affect the operating environment of banks. This study’s aim is to develop a model based on the linkages between the performance of banks and their operating context, determined by country risk. For this aim, we propose a multi-analytic methodology using fuzzy analytic network process (fuzzy-ANP) with principal component analysis (PCA) that extends existing mathematical methodologies and decision-making approaches. This method was examined in two studies. The first study focused on determining a model for country risk assessment based on the data extracted from 172 countries. Considering the first study’s scores, the second study established a bank performance model under the assumption of country risk, based on data from 496 banks. Our findings show the importance of country risk as a relevant bank performance dimension for decision makers in establishing efficient strategies with a positive impact on long-term performance. The study offers various contributions. From a mathematic methodology perspective, this research advances an original approach that integrates fuzzy-ANP with PCA, providing a consistent and unbiased framework that overcomes human judgement. From a business and economic analysis perspective, this research establishes novelty based on the performance evaluation of banks considering the operating country’s risk.
... The exact numerical value of an entity's membership or providing a specific membership value for any ambiguous entity, however, can be exceedingly difficult in more complex situations. This undertaking is difficult in and of itself since it is uncertain what information is required to create the rules used in fuzzy systems [4]. Because of incorrectly constructed fuzzy rules, membership functions may also experience ambiguity. ...
One of the crucial subjects used in operations research is decision-making processes. Multi-Criteria Decision-Making (MCDM) is one of the topics of this problem that is utilized rather often. In terms of ease and effectiveness, Technique for Order Preference by Similarity to Ideal Solutions (TOPSIS) is one of the most used MCDM techniques. To increase the applicability and simplicity of MCDM methods, Interval Type-2 Fuzzy Numbers (IT2FN), a specific type of Type-2 Fuzzy Sets (T2FS), are widely employed in combination with them. Defuzzification methods are used to find the best solution before the problem's ultimate step of resolution, however. Before the last phase of the process, switching to crisp numbers diminishes the use of the IT2FNs. A brief overview of fuzzy IT2FNs applications is given in this study. The review is based on 25 studies carried out between 2015 and 2023. Supply chains, the environment, energy sources, business, and healthcare are just a few of the application areas that were reviewed and segregated from fuzzy logic articles of various sorts that were the most pertinent and often discussed. The analysis and comparison of fuzzy logic implementations takes into account a wide range of alternatives and criteria, including the use of fuzzy logic mixed with additional techniques, such as the fuzzy Analytic Hierarchical Process (AHP), or improvements for group decision-making, fuzzy sets, hesitant fuzzy sets, intuitionistic fuzzy sets, or improvements for fuzzy logic. The merits and limitations of a variety of FLS types that have been employed for system identification are evaluated. Their effectiveness in various applications is also evaluated.
... They con- Fig. 4 a Type-1 fuzzy function, b Type-2 fuzzy function where edges are blurred (Image source: Mendel and Type 2002) tend that parameter reduction of the interval-valued fuzzy soft sets has not been addressed and therefore proposed four different algorithms to perform parameter reduction to help decision-makers. Finally, the algorithms are compared and summarized on usability, complexity, etc. Runkler et al. (2017) solved a few applications using the interval type-2 model. One of the applications is to help decide when is the best time to drive to work, when the starting time is somewhere between 6 and 12 o'clock, work for 8 h, and then drive back. ...
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Audience feedback measurement system is a highly researched field in the scientific and marketing community. Therefore, it is an important task to know what the customer feels about their services or goods. It is also interest to other users who want information and feedback on their services. Cloud Services are essential for up and coming entrepreneurs who want to expand their enterprises without expending too much capital on infrastructure. For this purpose, a cloud ranking or recommendation system is highly necessary to choose a cloud service provider as per their needs. Usually, available rating systems give crisp values as their output, which fails to capture the intrinsic vagueness and uncertainty naturally present while users give their feedback. The main aim of this paper is to minimize uncertainty and vagueness in fuzzy logic systems. Therefore, we propose a general type-2 fuzzy set to interval-valued responses of user ratings on cloud services. The proposed method captures the uncertainty which is present in all ratings that crisp or absolute ratings fail to capture. Specifically, we convert the type-1 fuzzy set to general type-2 fuzzy sets and extract the Z slices which represent the feedback of the users with no loss of data. In particular, we gather data from users over a period of 3 weeks and ascertain the intra- and inter-user agreement on the cloud services.
... This feature inhibits system flexibility, and operational costs are noticeably enhanced. The application of a quadcopter with the aid of cloud computing, image processing [11], and intelligent learning systems (such as deep learning [12,13], deep neural network [14], fuzzy decision making [15][16][17][18], pattern recognition [19], and Bayesian network [20]), can considerably improve the efficiency of existing monitoring systems. A cloud-based urban monitoring system is proposed in this paper based on using a quadcopter and intelligent learning systems. ...
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The application of quadcopter and intelligent learning techniques in urban monitoring systems can improve the flexibility and efficiency features. This paper proposes a cloud-based urban monitoring system that uses deep learning, fuzzy system, image processing, pattern recognition, and Bayesian network. Main objectives of this system are to monitor the climate status, temperature, humidity, and smoke as well as to detect the fire occurrences based on the above intelligent techniques. The quadcopter transmits sensing data of the temperature, humidity, and smoke sensors, geographical coordinates, image frames, and videos to a control station via RF communications. In the control station side, the monitoring capabilities are designed by graphical tools to show urban areas with RGB colors according to the pre-determined data ranges. The evaluation process illustrates simulation results of the deep neural network applied to climate status, effects of the sensors’ data changes on climate status. An illustrative example is used to draw the simulated area by RGB colors. Furthermore, circuit of the quadcopter side is designed using electric devices.
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Este artículo muestra el control del voltaje de salida de un convertidor de potencia tipo boost, los convertidores son sistemas conmutados, por lo cual se eligió un control discontinuo de modos deslizantes, este tipo de control emplea un control equivalente que puede hacer uso del modelo matemático del sistema, y un control discontinuo basado en una función del error de medición. Ante la incertidumbre paramétrica en el circuito y situaciones prácticas pasadas por alto en el modelo del convertidor, como son las caídas de voltaje en el interruptor y el diodo, se emplearon modos deslizantes difusos adaptables, de este modo no es necesario contar con el modelo matemático del convertidor. Una alternativa de justificar la arbitrariedad en el diseño de los conjuntos difusos es emplear conjuntos difusos tipo dos, con los que se abarcarían diferentes formas de conjuntos en la partición difusa, por lo que este tipo de conjuntos es empleado en este esquema de control.
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The humungous amount of fluctuating data is generated from every walk of life, such as cloud users, health care, IoT devices, high-performance computing data a daily basis. Therefore, it is of utmost essential to process those data within a determined span of time. Hence, traditional cluster computing or grid computing is unsuitable for processing those tremendous data, both parallelly and distributed fashion among multi-computing systems. The advent of cloud computing has paved the way to be a viable option for scheduling as well as balancing the loads. Since the nature of data is dynamic, independent and non-preemptive, cloud computing appears to be a prominent solution as it offers a virtualization technology for the dynamic scheduling of cloud requests. The cloud requests should get serviced within a satisfactory time in order to reduce the response time and completion time while effectively improving resource utilization. Since task scheduling is an NP-hard problem, it is thus required to be executed within a polynomial time so as to achieve overall performance. In order to make this possible, some effective mechanisms need to have in place. Due to the complexity of the tasks that can be performed in cloud computing, many metaheuristics, such as BSO, Modified PSO, chaotic JAYA and combined genetic Algorithm with JAYA and quantum-inspired binary chaotic salp swarm algorithm (QBCSSA) as hybrid algorithms are planned to address the various issues associated with the dynamic task scheduling in this paradigm. These algorithms are optimized to provide consistent and robust results. The continuous solutions are transformed into discrete solutions using binary algorithms for representing tasks–VMs assignment in cloud computing. Both task and resource heterogeneities are taken into account to assess the effectiveness of the implemented algorithms. The CloudSim is used as a simulation tool to experiment with the disparate test cases with the considered tasks set and heterogeneous resources. The conflicting quality of service (QoS) scheduling parameters are considered for appraising the efficacy of the proposed algorithms. The real-world benchmark datasets considering both dependent and independent tasks are considered to authenticate the diversifying nature of planned algorithms. This work is apparent as of the simulation outcome where chaotic JAYA (one of the variants of JAYA) and QBCSSA outperform among aforementioned metaheuristic and hybrid algorithms.KeywordsTask schedulingMakespanVM utilizationLoad balancingMetaheuristicModified PSOBSOBinary JAYABinary SAJAYABinary CJAYAGAYAQBCSSACloud computing
Decision-making is a crucial task of our life. In the real world, the decision parameters of most decision-making problems are imprecise and ambiguous in nature, which makes the problems uncertain. To solve uncertain problems, fuzzy sets (FSs) along with its various extensions perform a crucial part in decision-making paradigm. In this thesis, we have presented approaches to tackle with decision-making problems using FS extensions such as the interval type-2 fuzzy set (IT2FS), probabilistic interval-valued intuitionistic hesitant fuzzy set (P-IVIHFS) and interval-valued intuitionistic linear Diophantine fuzzy set (IVILDFS). We have developed an interactive decision-making approach on fuzzy production inventory using granular differentiability. The centroid index, rank index, and correlation coefficient are used in the proposed IT2FS ranking technique. Extended TOPSIS and VIKOR methods are studied to solve multi-attribute group decision-making (MAGDM) problems in the framework of P-IVIHFS. We have introduced IVILDFS as an extension of the linear Diophantine fuzzy set (LDFS). Score functions, accuracy functions and aggregation operators are proposed in the framework of IVILDFS and implemented to tackle decision-making problems. Preservation technology is used in the fuzzy production model to control the deterioration rate of the product, and granular differentiability approach is used for defuzzification. The proposed techniques are illustrated using numerical examples and comparative studies.
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Multi-Criteria Group Decision Making (MCGDM) is a decision tool which is able to find a unique agreement from a group of decision makers (DMs) by evaluating various conflicting criteria. However, most multi-criteria decision making techniques utilizing a group of DMs (MCGDM) do not effectively deal with the large number of possibilities inherent in a domain with a variety of possibilities, different judgments, and ideas on opinions. In recent years, there has been a growing interest in developing MCGDM using type-2 fuzzy systems which provide a framework to handle the encountered uncertainties in decision making models. In addition, fuzzy logic is regarded as an appropriate methodology for decision making systems which are able to simultaneously handle numerical data and linguistic knowledge. In this paper, we will aim to modify the fuzzy logic theories based multi-criteria group decision making models to employ a suite of type-2 fuzzy logic systems in order to provide answers to the problems that are encountered in the real experts’ decision. In the proposed framework, we will present a MCGDM method based on interval type-2 fuzzy logic combined with intuitionistic fuzzy evaluation (from intuitionistic fuzzy sets). This combination handles the linguistic uncertainties by the interval type-2 membership function and simultaneously computes the non-membership degree from the intuitionistic evaluation. However, the interval values with hesitation index cannot fully represent the uncertainty distribution associated with the decision makers. Hence, we will present a final component of our framework employing general type-2 fuzzy logic based approach for MCGDM which is more suited for higher levels of uncertainties. In order to optimally find the type-2 fuzzy sets parameters (including interval type-2 and general type-2), we have employed the Big Bang Big Crunch (BB-BC) optimisation method. In order to validate the efficiency of the proposed systems in handling various DMs’ behaviour and opinion, we will present comparisons which were performed on two different real world decision making problems. As will be shown in the various experiment sections, we found that the proposed type-2 MCGDM based system better agrees with the users’ decision compared to type-1 fuzzy expert system and existing type-1 fuzzy MCDMs including the Fuzzy Logic based TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution). In addition, we will show how the different type-2 fuzzy logic based MCGDM systems compare to each other when increasing the level of uncertainties where the general type-2 MCGDM will outperform the MCGDM based interval type-2 fuzzy logic combined with intuitionistic fuzzy evaluation which will outperform the MCGDM based on interval type-2 fuzzy sets. Hence, this work can be regarded as a step towards producing higher ordered fuzzy logic approach for MCGDM (HFL-MCGDM) which could be applied to complex problems with high uncertainties to produce automated decisions much closer to the group of human experts.
Conference Paper
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Interval type–2 defuzzification maps an interval type–2 fuzzy set to a crisp number. We show that the semantic meaning of the interval type–2 fuzzy set (the associated opportunity or risk) has to be considered in the choice of an appropriate interval type–2 defuzzification method. Motivated by a list of " axioms " for type–1 defuzzification we introduce twelve mathematical properties for interval type–2 defuzzification that serve as a theoretical framework to assess different interval type–2 defuzzification methods. We show that the well–known Karnik–Mendel algorithm violates at least four of these twelve properties.
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Energy storage alternatives that help storing excess energy and then using it when the system needs it has become more important in recent years. Determination of the most suitable energy storage alternative can be analyzed by using multi criteria decision making (MCDM) techniques. There are many criteria that affect the best energy storage alternative and the aims are contrasting so, MCDM methodology is a good approach to solve these problems. In this paper, a hybrid MCDM methodology that consists of analytic hierarchy process (AHP) and TOPSIS based on type-2 fuzzy sets is proposed. To obtain more flexible evaluation and more precise results the proposed methodology combines type-2 fuzzy AHP that used to determine the weights of criteria and type-2 fuzzy TOPSIS methodology that analyzes the alternatives with respect to criteria and weights. The proposed methodology has been used to determine the most suitable energy storage alternatives. For this aim, 6 electrical energy storage alternatives are considered with a hierarchical structure of 4 main and 18 sub–criteria. According to results obtained the best electrical energy storage alternative is determined as compressed air energy storage.
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Molodtsov introduced the theory of soft sets, which can be used as a general mathematical tool for dealing with uncertainty. This paper aims to introduce the concept of the type-2 fuzzy soft set by integrating the type-2 fuzzy set theory and the soft set theory. Some operations on the type-2 fuzzy soft sets are given. Furthermore, we investigate the decision making based on type-2 fuzzy soft sets. By means of level soft sets, we propose an adjustable approach to type-2 fuzzy-soft-set based decision making and give some illustrative examples. Moreover, we also introduce the weighted type-2 fuzzy soft set and examine its application to decision making.
Determining the level and importance of concrete deterioration in a given structure is a multivariate decision-making problem with parameters that vary extensively, depending on which specific structural element(s) is/are implicated. This paper presents the development of a tool to measure the extent of the criticality of the concrete deterioration to help quantify if the deterioration warrants intervention for repair. The decision-making logic is based on fuzzy set theory (FST), a well-developed tool used to address uncertainties in decision making. Investigation of the extent of deterioration is conducted using nondestructive testing (NDT) of concrete—specifically, the Schmidt hammer. An assessment is performed using the fuzzy logic analysis. The tool presented herein is generic enough to allow the user to encompass the criteria of the structure at hand and to render a judgment to determine the optimum timing for repair. The objective of this research is the development of a decision-support model that will provide quantifiable support for a decision maker's decisions and the demonstration of its use. This research also highlights an extensive area for further development. It provides a blueprint to achieve the overall goal of assessing deterioration. Through this model, users are able to develop a quantifiable metric to help support their decisions on the appropriate time for intervention and the repair procedure necessary. An analysis of the model illustrates that the system demonstrates utility for practical use.
An increasing number of companies have focused on reducing the amount of waste properly or gaining value from used products. Facilitating the reverse flow of used products from consumers to manufacturers is a difficult and expensive process depending on the product and transportation type and distance. Another alternative is to outsource these activities. Outsourcing management helps companies for better using of time, energy, labor, technology, capital, resources etc. Moreover, working with wrong partners effects manufacturers’ financial and operational situations. In order to get the best services, manufacturers usually invite several outsourcing companies for providing their tenders and then select the best offer. In this stage, using mathematical decision making techniques may help decision makers to get realistic results. In this paper the proposed methodology integrates two multi-criteria decision methods for ranking alternatives. This methodology is applied to a mid-sized firm operating in the field of electrical and electronic equipment. The results indicate that the most important criterion is cost for determining the best alternative. Besides, as it can be seen from the results, the best alternative for the manufacturer is the second alternative. These results propose a guideline for manufacturers for selecting the best alternative. From the results it can easily be seen that this approach shows its potential advantage in selecting suitable alternative due to its sound logic and easily programmable computation procedure.
In this paper, we present a new method for fuzzy multiple attributes decision making based on interval type-2 fuzzy sets. First, we present a new fuzzy ranking method based on the α-cuts of interval type-2 fuzzy sets. Then, based on the proposed fuzzy ranking method of interval type-2 fuzzy sets, we present a new method for fuzzy multiple attributes decision making. The proposed method can overcome the drawbacks of Liu and Su’s method [45] and Wang and Luo’s method [64] due to the fact that it can deal with the ranking of interval type-2 fuzzy sets to distinguish the preference order of the alternatives. The proposed fuzzy multiple attributes decision making method is more flexible and more intelligent than Chen and Lee’s method [21] due to the fact that it not only uses interval type-2 fuzzy sets, but also considers the decision-maker’s attitude towards risks.
Resorcinol and formaldehyde were used as carbon precursors, poly(ethylene oxide)-poly(propylene oxide)-poly(ethylene oxide) triblock copolymer was employed as a soft template, and tetraethylorthosilicate-generated silica was used as hard templates to synthesize spherical mesoporous carbon. The resulting spherical mesoporous carbons were characterized by nitrogen adsorption–desorption isotherms and electron microscopy (SEM and TEM) and used as electrode materials for aqueous electric double-layer capacitors. The average diameters of spherical particles ranged from 2 to 7 μm and the mesopore was ca 2 nm. The highest specific surface area of 1,000 m2/g and mesopore volume of 0.86 cm3/g was obtained. The specific capacitance of 130 F/g was obtained by means of galvanostatic charging/discharging and cycle voltammetry.