Interval Type–2 Fuzzy Decision Making

Abstract

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.

Full text

Interval Type–2 Fuzzy Decision Making1
Thomas Runklera, Simon Couplandb, Robert Johnc
2
aSiemens AG, Corporate Technology, 81730 Munich, Germany, Email:3
Thomas.Runkler@siemens.com4
bCentre for Computational Intelligence, De Montfort University, The Gateway,5
Leicester, LE1 9BH, UK, Email: simonc@dmu.ac.uk6
cLaboratory for Uncertainty in Data and Decision Making (LUCID), University of7
Nottingham, Wollaton Road, Nottingham, NG8 1BB, UK8
Email: Robert.John@nottingham.ac.uk9
Abstract10
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.
Keywords:11
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-
2

0
1u˜
A(x)
u˜
A(x)
Figure 1: Interval type–2 fuzzy set.
membership function u˜
A:X→[0,1], where36
u˜
A(x)≤u˜
A(x) (1)
for all x∈X. 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 x∗is defined as42
x∗= argmax
x∈Xug1(x)∧. . . ∧ugm(x)∧uc1(x)∧. . . ∧ucn(x)(4)
czany, 1987; Gehrke et al., 1996).
3

x∗
0
1
g1g2c1
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
4

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
5

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
x∈Xu˜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
x∈Xu˜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 x∗and 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
x∗
β= argmax
x∈X((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))
6

x∗
x∗
0
1
˜g1˜g2˜c1
Figure 3: Interval type–2 fuzzy decisions.
87
∧. . . ∧((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 x∗and x∗highlighted by circles. The approach leads to equations (9)98
and (10) giving alternative ways of calculating the respective worst and best99
7

0
1˜g˜c
˜g∩˜c
Figure 4: The intersection of an interval type-2 fuzzy goal and constraint
case decisions.100
x∗= argmax
x∈X
(˜
f(x)) (9)
101
x∗= argmax
x∈X
(˜
f(x)) (10)
We can use equations (9) and (10) to calculate the decision for given risk102
value βusing equation 11.103
x∗
β= argmax (1 −β)·µ˜
f(x) + β·µ˜
f(x))(11)
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
8

It is easy to see that x∗
0=x∗and 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 x∗and the best case decision x∗, so110
min x∗, x∗≤x∗
β≤max 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 ˜
g1∩˜
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)≥˜
f(x), ∀x∈X. We can now130
show that (12) holds for convex sets when using the minimum and product131
9

0 0.5 1
0
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∗, x∗≤x≤max x∗, x∗
133
•Region II : x < min x∗, x∗
134
•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∗
βis143
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
10

0
1III III
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
11

14 16 18 20 22 24
0
0.2
0.4
0.6
0.8
1˜
A
˜
B
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
12

6 7 8 9 10 111213 14151617 181920
0
0.2
0.4
0.6
0.8
1
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−(x−7·60
133 )2
+ 0.525 ·e−(x−19·60
290 )2
(13)
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
13

6 7 8 9 101112 13141516 17181920
0
0.2
0.4
0.6
0.8
1
6(14) 7(15) 8(16) 9(17) 10(18)11(19)12(20)
0
0.2
0.4
0.6
0.8
1
morning
evening
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−(x−7·60
3·60 )2
+ 0.75 ·e−(x−19·60
3·60 )2
(14)
198
u(x) = 0.6·e−(x−7·60
1.5·60 )2
+ 0.3·e−(x−19·60
4.5·60 )2
(15)
which represent the lower (dashed) and upper (solid) membership functions199
14

6 7 8 9 10 111213 14151617 181920
0
0.2
0.4
0.6
0.8
1
6(14) 7(15) 8(16) 9(17) 10(18)11(19)12(20)
0
0.2
0.4
0.6
0.8
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
15

8:30(16:30) 8:45(16:45) 9:00(17:00)
0
0.2
0.4
0.6
0.8
1
best case
worst case
type 1
8:46(16:46)
min type 2
8:32(16:32) max type 2
8:59(16:59)
-5.3%
-8.1%
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
16

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
detail.240
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