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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∈Xug1(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∈Xu˜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∈Xu˜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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